1.
Computational physics
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Computational physics is the study and implementation of numerical analysis to solve problems in physics for which a quantitative theory already exists. Historically, computational physics was the first application of computers in science. In physics, different theories based on mathematical models provide very precise predictions on how systems behave, unfortunately, it is often the case that solving the mathematical model for a particular system in order to produce a useful prediction is not feasible. This can occur, for instance, when the solution does not have a closed-form expression, in such cases, numerical approximations are required. There is a debate about the status of computation within the scientific method, while computers can be used in experiments for the measurement and recording of data, this clearly does not constitute a computational approach. Physics problems are in very difficult to solve exactly. This is due to several reasons, lack of algebraic and/or analytic solubility, complexity, on the more advanced side, mathematical perturbation theory is also sometimes used. In addition, the computational cost and computational complexity for many-body problems tend to grow quickly, a macroscopic system typically has a size of the order of 1023 constituent particles, so it is somewhat of a problem. Solving quantum mechanical problems is generally of exponential order in the size of the system, because computational physics uses a broad class of problems, it is generally divided amongst the different mathematical problems it numerically solves, or the methods it applies. Furthermore, computational physics encompasses the tuning of the structure to solve the problems. It is possible to find a corresponding computational branch for every field in physics, for example computational mechanics. Computational mechanics consists of fluid dynamics, computational solid mechanics. One subfield at the confluence between CFD and electromagnetic modelling is computational magnetohydrodynamics, the quantum many-body problem leads naturally to the large and rapidly growing field of computational chemistry. Computational solid state physics is an important division of computational physics dealing directly with material science. A field related to computational condensed matter is computational statistical mechanics, computational statistical physics makes heavy use of Monte Carlo-like methods. More broadly, it concerns itself with in the social sciences, network theory, and mathematical models for the propagation of disease. Computational astrophysics is the application of techniques and methods to astrophysical problems. Stickler, E. Schachinger, Basic concepts in computational physics, E. Winsberg, Science in the Age of Computer Simulation
Computational physics
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Computational physics
2.
Numerical analysis
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Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis. Being able to compute the sides of a triangle is important, for instance, in astronomy, carpentry. Numerical analysis continues this tradition of practical mathematical calculations. Much like the Babylonian approximation of the root of 2, modern numerical analysis does not seek exact answers. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors, before the advent of modern computers numerical methods often depended on hand interpolation in large printed tables. Since the mid 20th century, computers calculate the required functions instead and these same interpolation formulas nevertheless continue to be used as part of the software algorithms for solving differential equations. Computing the trajectory of a spacecraft requires the accurate numerical solution of a system of differential equations. Car companies can improve the safety of their vehicles by using computer simulations of car crashes. Such simulations essentially consist of solving differential equations numerically. Hedge funds use tools from all fields of analysis to attempt to calculate the value of stocks. Airlines use sophisticated optimization algorithms to decide ticket prices, airplane and crew assignments, historically, such algorithms were developed within the overlapping field of operations research. Insurance companies use programs for actuarial analysis. The rest of this section outlines several important themes of numerical analysis, the field of numerical analysis predates the invention of modern computers by many centuries. Linear interpolation was already in use more than 2000 years ago, to facilitate computations by hand, large books were produced with formulas and tables of data such as interpolation points and function coefficients. The function values are no very useful when a computer is available. The mechanical calculator was developed as a tool for hand computation. These calculators evolved into electronic computers in the 1940s, and it was found that these computers were also useful for administrative purposes. But the invention of the computer also influenced the field of analysis, since now longer
Numerical analysis
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Babylonian clay tablet
YBC 7289 (c. 1800–1600 BC) with annotations. The approximation of the
square root of 2 is four
sexagesimal figures, which is about six
decimal figures. 1 + 24/60 + 51/60 2 + 10/60 3 = 1.41421296...
Numerical analysis
–
Direct method
Numerical analysis
3.
Scientific visualization
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Scientific visualization is an interdisciplinary branch of science. It is also considered a subset of computer graphics, a branch of computer science, the purpose of scientific visualization is to graphically illustrate scientific data to enable scientists to understand, illustrate, and glean insight from their data. One of the earliest examples of scientific visualisation was Maxwells thermodynamic surface. This prefigured modern scientific techniques that use computer graphics. Scientific visualization using computer graphics gained in popularity as graphics matured, primary applications were scalar fields and vector fields from computer simulations and also measured data. The primary methods for visualizing two-dimensional scalar fields are color mapping and drawing contour lines, 2D vector fields are visualized using glyphs and streamlines or line integral convolution methods. For 3D scalar fields the primary methods are volume rendering and isosurfaces, methods for visualizing vector fields include glyphs such as arrows, streamlines and streaklines, particle tracing, line integral convolution and topological methods. Later, visualization techniques such as hyperstreamlines were developed to visualize 2D, computer animation is the art, technique, and science of creating moving images via the use of computers. It is becoming common to be created by means of 3D computer graphics, though 2D computer graphics are still widely used for stylistic, low bandwidth. Sometimes the target of the animation is the computer itself, but sometimes the target is another medium and it is also referred to as CGI, especially when used in films. Computer simulation is a program, or network of computers. The simultaneous visualization and simulation of a system is called visulation, computer simulations vary from computer programs that run a few minutes, to network-based groups of computers running for hours, to ongoing simulations that run for months. Information visualization focused on the creation of approaches for conveying information in intuitive ways. The key difference between scientific visualization and information visualization is that information visualization is often applied to data that is not generated by scientific inquiry, some examples are graphical representations of data for business, government, news and social media. Interface technology and perception shows how new interfaces and an understanding of underlying perceptual issues create new opportunities for the scientific visualization community. Rendering is the process of generating an image from a model, the model is a description of three-dimensional objects in a strictly defined language or data structure. It would contain geometry, viewpoint, texture, lighting, the image is a digital image or raster graphics image. The term may be by analogy with a rendering of a scene
Scientific visualization
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A scientific visualization of a simulation of a
Rayleigh–Taylor instability caused by two mixing fluids.
Scientific visualization
–
Surface rendering of
Arabidopsis thaliana pollen grains with
confocal microscope.
Scientific visualization
–
Scientific visualization of Fluid Flow:
Surface waves in
water
Scientific visualization
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Chemical imaging of a simultaneous release of SF 6 and NH 3.
4.
Morse/Long-range potential
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Owing to the simplicity of the Morse potential, it is not used in modern spectroscopy. The MLR potential is a version of the Morse potential which has the correct theoretical long-range form of the potential naturally built into it. Cases of particular note include, the c-state of Li2, where the MLR potential was able to bridge a gap of more than 5000 cm−1 in experimental data. Two years later it was found that Dattanis MLR potential was able to predict the energies in the middle of this gap. The accuracy of these predictions was much better than the most sophisticated ab initio techniques at the time and this lithium oscillator strength is related to the radiative lifetime of atomic lithium and is used as a benchmark for atomic clocks and measurements of fundamental constants. It has been said that work by Le Roy et al. was a landmark in diatomic spectral analysis. The a-state of KLi, where a global potential was successfully built despite there only being a small amount of data near the top of the potential. The MLR potential is based on the classic Morse potential which was first introduced in 1929 by Philip M. Morse, a primitive version of the MLR potential was first introduced in 2006 by professor Robert J. Le Roy and colleagues for a study on N2. This primitive form was used on Ca2, KLi and MgH, before the modern version was introduced in 2009 by Le Roy, Dattani. A further extension of the MLR potential referred to as the MLR3 potential was introduced in a 2010 study of Cs2, and it is clear to see that, lim r → ∞ = y n r x, so lim r → ∞ β = β ∞. More sophisticated versions are used for polyatomic molecules, examples of molecules for which the MLR has been used to represent ab initio points are KLi, KBe
Morse/Long-range potential
–
Computational physics
5.
Lennard-Jones potential
–
The Lennard-Jones potential is a mathematically simple model that approximates the interaction between a pair of neutral atoms or molecules. A form of this potential was first proposed in 1924 by John Lennard-Jones. At rm, the function has the value −ε. The distances are related as rm = 21/6σ ≈1. 122σ and these parameters can be fitted to reproduce experimental data or accurate quantum chemistry calculations. Due to its simplicity, the Lennard-Jones potential is used extensively in computer simulations even though more accurate potentials exist. Differentiating the L-J potential with respect to r gives an expression for the net inter-molecular force between 2 molecules and this inter-molecular force may be attractive or repulsive, depending on the value of r. When r is very small, the 2 molecules repel each other, whereas the functional form of the attractive term has a clear physical justification, the repulsive term has no theoretical justification. It is used because it approximates the Pauli repulsion well, and is convenient due to the relative computing efficiency of calculating r12 as the square of r6. The Lennard-Jones potential was improved by the Buckingham potential later proposed by R. A. Buckingham, in which the part is an exponential function. The L-J potential is a good approximation. Due to its simplicity, it is used to describe the properties of gases. It is especially accurate for noble gas atoms, and is an approximation at long and short distances for neutral atoms. The lowest energy arrangement of a number of atoms described by a Lennard-Jones potential is a hexagonal close-packing. On raising temperature, the lowest free energy arrangement becomes cubic close packing, under pressure, the lowest energy structure switches between cubic and hexagonal close packing. Real materials include BCC structures also, other more recent methods, such as the Stockmayer potential, describe the interaction of molecules more accurately. Quantum chemistry methods, Møller–Plesset perturbation theory, coupled cluster method, or full configuration interaction can give accurate results. There are many different ways to formulate the Lennard-Jones potential. This form is a formulation that is used by some simulation software, V L J = A r 12 − B r 6
Lennard-Jones potential
–
Computational physics
Lennard-Jones potential
–
A graph of strength versus distance for the 12-6 Lennard-Jones potential.
6.
Yukawa potential
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The potential is monotone increasing in r and it is negative, implying the force is attractive. In the SI system, the unit of the Yukawa potential is, the Coulomb potential of electromagnetism is an example of a Yukawa potential with e−kmr equal to 1 everywhere. This can be interpreted as saying that the mass m is equal to 0. In interactions between a field and a fermion field, the constant g is equal to the gauge coupling constant between those fields. In the case of the force, the fermions would be a proton. Hideki Yukawa showed in the 1930s that such a potential arises from the exchange of a scalar field such as the field of a massive boson. Since the field mediator is massive the corresponding force has a certain range, if the mass is zero, then the Yukawa potential equals a Coulomb potential, and the range is said to be infinite. In fact, we have, m =0 ⇒ e − m r = e 0 =1, consequently, the equation V Yukawa = − g 2 e − m r r simplifies to the form of the Coulomb potential V Coulomb = − g 21 r. A comparison of the long range potential strength for Yukawa and Coulomb is shown in Figure 2 and it can be seen that the Coulomb potential has effect over a greater distance whereas the Yukawa potential approaches zero rather quickly. However, any Yukawa potential or Coulomb potential are non-zero for any large r, the easiest way to understand that the Yukawa potential is associated with a massive field is by examining its Fourier transform. One has V = − g 23 ∫ e i k ⋅ r 4 π k 2 + m 2 d 3 k where the integral is performed all possible values of the 3-vector momentum k. In this form, the fraction 4 π / is seen to be the propagator or Greens function of the Klein–Gordon equation, the Yukawa potential can be derived as the lowest order amplitude of the interaction of a pair of fermions. The Yukawa interaction couples the fermion field ψ to the meson field ϕ with the coupling term L i n t = g ψ ¯ ϕ ψ. The scattering amplitude for two fermions, one with initial momentum p 1 and the other with momentum p 2, exchanging a meson with momentum k, is given by the Feynman diagram on the right. The Feynman rules for each associate a factor of g with the amplitude. The line in the middle, connecting the two lines, represents the exchange of a meson. The Feynman rule for an exchange is to use the propagator. Thus, we see that the Feynman amplitude for this graph is nothing more than V = − g 24 π k 2 + m 2, from the previous section, this is seen to be the Fourier transform of the Yukawa potential
Yukawa potential
–
Figure 1: A comparison of Yukawa potentials where g=1 and with various values for m.
7.
Finite difference method
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Today, FDMs are the dominant approach to numerical solutions of partial differential equations. First, assuming the function whose derivatives are to be approximated is properly-behaved, by Taylors theorem, we can create a Taylor Series expansion f = f + f ′1. H n + R n, where n. denotes the factorial of n, the error in a methods solution is defined as the difference between the approximation and the exact analytical solution. To use a finite difference method to approximate the solution to a problem and this is usually done by dividing the domain into a uniform grid. Note that this means that finite-difference methods produce sets of numerical approximations to the derivative. An expression of general interest is the truncation error of a method. Typically expressed using Big-O notation, local truncation error refers to the error from an application of a method. That is, it is the quantity f ′ − f i ′ if f ′ refers to the exact value, the remainder term of a Taylor polynomial is convenient for analyzing the local truncation error. Using the Lagrange form of the remainder from the Taylor polynomial for f, N +1, where x 0 < ξ < x 0 + h, the dominant term of the local truncation error can be discovered. For example, again using the formula for the first derivative. 2, and with some algebraic manipulation, this leads to f − f i h = f ′ + f ″2, a final expression of this example and its order is, f − f i h = f ′ + O. This means that, in case, the local truncation error is proportional to the step sizes. The quality and duration of simulated FDM solution depends on the discretization equation selection, the data quality and simulation duration increase significantly with smaller step size. Therefore, a balance between data quality and simulation duration is necessary for practical usage. Large time steps are useful for increasing speed in practice. However, time steps which are too large may create instabilities, the von Neumann method is usually applied to determine the numerical model stability. For example, consider the differential equation u ′ =3 u +2. The last equation is an equation, and solving this equation gives an approximate solution to the differential equation
Finite difference method
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Navier–Stokes differential equations used to simulate airflow around an obstruction.
8.
Finite volume method
–
The finite volume method is a method for representing and evaluating partial differential equations in the form of algebraic equations. Similar to the finite difference method or finite element method, values are calculated at discrete places on a meshed geometry, Finite volume refers to the small volume surrounding each node point on a mesh. In the finite volume method, volume integrals in a differential equation that contain a divergence term are converted to surface integrals. These terms are then evaluated as fluxes at the surfaces of finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. The method is used in computational fluid dynamics packages. Consider a simple 1D advection problem defined by the partial differential equation ∂ ρ ∂ t + ∂ f ∂ x =0, t ≥0. Here, ρ = ρ represents the variable and f = f represents the flux or flow of ρ. Conventionally, positive f represents flow to the right while negative f represents flow to the left, if we assume that equation represents a flowing medium of constant area, we can sub-divide the spatial domain, x, into finite volumes or cells with cell centres indexed as i. Integrating equation in time, we have, ρ = ρ − ∫ t 1 t 2 f x d t and we assume that f is well behaved and that we can reverse the order of integration. Also, recall that flow is normal to the area of the cell. Equation is exact for the averages, i. e. no approximations have been made during its derivation. This method can also be applied to a 2D situation by considering the north and south faces along with the east and west faces around a node. We can also consider the conservation law problem, represented by the following PDE. Here, u represents a vector of states and f represents the corresponding flux tensor, again we can sub-divide the spatial domain into finite volumes or cells. For a particular cell, i, we take the integral over the total volume of the cell, v i. So, finally, we are able to present the general equivalent to. Again, values for the fluxes can be reconstructed by interpolation or extrapolation of the cell averages
Finite volume method
–
Navier–Stokes differential equations used to simulate airflow around an obstruction.
9.
Finite element method
–
The finite element method is a numerical method for solving problems of engineering and mathematical physics. It is also referred to as finite element analysis, typical problem areas of interest include structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. The analytical solution of these problems generally require the solution to boundary value problems for partial differential equations, the finite element method formulation of the problem results in a system of algebraic equations. The method yields approximate values of the unknowns at discrete number of points over the domain, to solve the problem, it subdivides a large problem into smaller, simpler parts that are called finite elements. The simple equations that model these finite elements are assembled into a larger system of equations that models the entire problem. FEM then uses variational methods from the calculus of variations to approximate a solution by minimizing an associated error function, the global system of equations has known solution techniques, and can be calculated from the initial values of the original problem to obtain a numerical answer. To explain the approximation in this process, FEM is commonly introduced as a case of Galerkin method. The process, in language, is to construct an integral of the inner product of the residual. In simple terms, it is a procedure that minimizes the error of approximation by fitting trial functions into the PDE, the residual is the error caused by the trial functions, and the weight functions are polynomial approximation functions that project the residual. These equation sets are the element equations and they are linear if the underlying PDE is linear, and vice versa. In step above, a system of equations is generated from the element equations through a transformation of coordinates from the subdomains local nodes to the domains global nodes. This spatial transformation includes appropriate orientation adjustments as applied in relation to the coordinate system. The process is carried out by FEM software using coordinate data generated from the subdomains. FEM is best understood from its application, known as finite element analysis. FEA as applied in engineering is a tool for performing engineering analysis. It includes the use of mesh generation techniques for dividing a complex problem into small elements, FEA is a good choice for analyzing problems over complicated domains, when the domain changes, when the desired precision varies over the entire domain, or when the solution lacks smoothness. For instance, in a crash simulation it is possible to increase prediction accuracy in important areas like the front of the car. Another example would be in weather prediction, where it is more important to have accurate predictions over developing highly nonlinear phenomena rather than relatively calm areas
Finite element method
–
Visualization of how a car deforms in an asymmetrical crash using finite element analysis. [1]
Finite element method
–
Navier–Stokes differential equations used to simulate airflow around an obstruction.
10.
Monte Carlo method
–
Monte Carlo methods are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. Their essential idea is using randomness to solve problems that might be deterministic in principle and they are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. Monte Carlo methods are used in three distinct problem classes, optimization, numerical integration, and generating draws from a probability distribution. In principle, Monte Carlo methods can be used to any problem having a probabilistic interpretation. By the law of numbers, integrals described by the expected value of some random variable can be approximated by taking the empirical mean of independent samples of the variable. When the probability distribution of the variable is parametrized, mathematicians often use a Markov Chain Monte Carlo sampler, the central idea is to design a judicious Markov chain model with a prescribed stationary probability distribution. That is, in the limit, the samples being generated by the MCMC method will be samples from the desired distribution, by the ergodic theorem, the stationary distribution is approximated by the empirical measures of the random states of the MCMC sampler. In other problems, the objective is generating draws from a sequence of probability distributions satisfying a nonlinear evolution equation, in other instances we are given a flow of probability distributions with an increasing level of sampling complexity. These models can also be seen as the evolution of the law of the states of a nonlinear Markov chain. In contrast with traditional Monte Carlo and Markov chain Monte Carlo methodologies these mean field particle techniques rely on sequential interacting samples, the terminology mean field reflects the fact that each of the samples interacts with the empirical measures of the process. Monte Carlo methods vary, but tend to follow a particular pattern, generate inputs randomly from a probability distribution over the domain. Perform a deterministic computation on the inputs, for example, consider a circle inscribed in a unit square. Given that the circle and the square have a ratio of areas that is π/4, uniformly scatter objects of uniform size over the square. Count the number of objects inside the circle and the number of objects. The ratio of the two counts is an estimate of the ratio of the two areas, which is π/4, multiply the result by 4 to estimate π. In this procedure the domain of inputs is the square that circumscribes our circle and we generate random inputs by scattering grains over the square then perform a computation on each input. Finally, we aggregate the results to obtain our final result, there are two important points to consider here, Firstly, if the grains are not uniformly distributed, then our approximation will be poor. Secondly, there should be a number of inputs
Monte Carlo method
–
Computational physics
11.
Monte Carlo integration
–
In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers. It is a particular Monte Carlo method that computes a definite integral. While other algorithms usually evaluate the integrand at a regular grid and this method is particularly useful for higher-dimensional integrals. There are different methods to perform a Monte Carlo integration, such as sampling, stratified sampling, importance sampling, Sequential Monte Carlo. In numerical integration, methods such as the Trapezoidal rule use a deterministic approach, Monte Carlo integration, on the other hand, employs a non-deterministic approach, each realization provides a different outcome. In Monte Carlo, the outcome is an approximation of the correct value with respective error bars. This is because the law of large numbers ensures that lim N → ∞ Q N = I, given the estimation of I from QN, the error bars of QN can be estimated by the sample variance using the unbiased estimate of the variance. V a r ≡ σ N2 =1 N −1 ∑ i =1 N2. Which leads to V a r = V2 N2 ∑ i =1 N V a r = V2 V a r N = V2 σ N2 N, as long as the sequence is bounded, this variance decreases asymptotically to zero as 1/N. The estimation of the error of QN is thus δ Q N ≈ V a r = V σ N N and this is standard error of the mean multiplied with V. While the naive Monte Carlo works for simple examples, this is not the case in most problems, a large part of the Monte Carlo literature is dedicated in developing strategies to improve the error estimates. In particular, stratified sampling—dividing the region in sub-domains—, and importance sampling—sampling from non-uniform distributions—are two of such techniques, a paradigmatic example of a Monte Carlo integration is the estimation of π. Consider the function H = {1 if x 2 + y 2 ≤10 else, notice that I π = ∫ Ω H d x d y = π. Keep in mind that a random number generator should be used. On each recursion step the integral and the error are estimated using a plain Monte Carlo algorithm, if the error estimate is larger than the required accuracy the integration volume is divided into sub-volumes and the procedure is recursively applied to sub-volumes. The ordinary dividing by two strategy does not work for multi-dimensions as the number of sub-volumes grows far too quickly to keep track, instead one estimates along which dimension a subdivision should bring the most dividends and only subdivides the volume along this dimension. The popular MISER routine implements a similar algorithm, the MISER algorithm is based on recursive stratified sampling. This technique aims to reduce the overall integration error by concentrating integration points in the regions of highest variance, the MISER algorithm proceeds by bisecting the integration region along one coordinate axis to give two sub-regions at each step
Monte Carlo integration
–
An illustration of Monte Carlo integration. In this example, the domain D is the inner circle and the domain E is the square. Because the square's area (4) can be easily calculated, the area of the circle (π*1 2) can be estimated by the ratio (0.8) of the points inside the circle (40) to the total number of points (50), yielding an approximation for the circle's area of 4*0.8 = 3.2 ≈ π*1 2.
12.
Molecular dynamics
–
Molecular dynamics is a computer simulation method for studying the physical movements of atoms and molecules, and is thus a type of N-body simulation. The atoms and molecules are allowed to interact for a period of time. The method was developed within the field of theoretical physics in the late 1950s but is applied today mostly in chemical physics, materials science. Following the earlier successes of Monte Carlo simulations, the method was developed by Fermi, Pasta, in 1957, Alder and Wainwright used an IBM704 computer to simulate perfectly elastic collisions between hard spheres. In 1960, Gibson et al. simulated radiation damage of solid copper by using a Born-Mayer type of repulsive interaction along with a surface force. In 1964, Rahman published landmark simulations of liquid argon that used a Lennard-Jones potential, calculations of system properties, such as the coefficient of self-diffusion, compared well with experimental data. Even before it became possible to simulate molecular dynamics with computers, the idea was to arrange them to replicate the properties of a liquid. I took a number of balls and stuck them together with rods of a selection of different lengths ranging from 2.75 to 4 inches. I tried to do this in the first place as casually as possible, working in my own office, being interrupted every five minutes or so and not remembering what I had done before the interruption. In physics, MD is used to examine the dynamics of atomic-level phenomena that cannot be observed directly, such as thin film growth and it is also used to examine the physical properties of nanotechnological devices that have not been or cannot yet be created. In principle MD can be used for ab initio prediction of protein structure by simulating folding of the chain from random coil. The results of MD simulations can be tested through comparison to experiments that measure molecular dynamics, michael Levitt, who shared the Nobel Prize awarded in part for the application of MD to proteins, wrote in 1999 that CASP participants usually did not use the method due to. A central embarrassment of molecular mechanics, namely that energy minimization or molecular dynamics generally leads to a model that is less like the experimental structure, limits of the method are related to the parameter sets used, and to the underlying molecular mechanics force fields. The neglected contributions include the conformational entropy of the polypeptide chain, Another important factor are intramolecular hydrogen bonds, which are not explicitly included in modern force fields, but described as Coulomb interactions of atomic point charges. This is an approximation because hydrogen bonds have a partially quantum mechanical and chemical nature. Furthermore, electrostatic interactions are calculated using the dielectric constant of vacuum. Using the macroscopic dielectric constant at short distances is questionable. Finally, van der Waals interactions in MD are usually described by Lennard-Jones potentials based on the Fritz London theory that is applicable in vacuum
Molecular dynamics
Molecular dynamics
–
Example of a molecular dynamics simulation in a simple system: deposition of a single
Cu atom on a Cu
(001) surface. Each circle illustrates the position of a single atom; note that the actual atomic interactions used in current simulations are more complex than those of 2-dimensional hard spheres.
13.
Sergei K. Godunov
–
Sergei Konstantinovich Godunov is professor at the Sobolev Institute of Mathematics of the Russian Academy of Sciences in Novosibirsk, Russia. Professor Godunovs most influential work is in the area of applied and it has had a major impact on science and engineering, particularly in the development of methodologies used in Computational Fluid Dynamics and other computational fields. On 1–2 May 1997 a symposium entitled, Godunov-type numerical methods, was held at the University of Michigan to honour Godunov and these methods are widely used to compute continuum processes dominated by wave propagation. On the following day,3 May, Godunov received a degree from the University of Michigan. Godunovs theorem, Linear numerical schemes for solving differential equations, having the property of not generating new extrema. Godunovs scheme is a numerical scheme for solving partial differential equations. 1946-1951 - Department of Mechanics and Mathematics, Moscow State University,1951 - Diploma, Moscow State University. 1954 - Candidate of Physical and Mathematical Sciences,1965 - Doctor of Physical and Mathematical Sciences. 1976 - Corresponding member of the USSR Academy of Sciences,1994 - Member of the Russian Academy of Sciences. 1997 - Honorary professor of the University of Michigan, krylov Prize of the USSR Academy of Sciences. 1993 - M. A. Lavrentiev Prize of the Russian Academy of Sciences, total variation diminishing Upwind scheme Godunov, Sergei K. Ph. D. Dissertation, Difference Methods for Shock Waves, Moscow State University. A Difference Scheme for Numerical Solution of Discontinuous Solution of Hydrodynamic Equations, sbornik,47, 271-306, translated US Joint Publ. Service, JPRS7225 Nov.29,1960, Godunov, Sergei K. and Romenskii, Evgenii I. Elements of Continuum Mechanics and Conservation Laws, Springer, ISBN 0-306-47735-1, Numerical Computation of Internal and External Flows, vol 2, Wiley. Sergei K. Godunov at the Mathematics Genealogy Project Godunovs Personal Web Page Sobolev Institute of Mathematics
Sergei K. Godunov
–
Sergei Godunov
14.
Fluid mechanics
–
Fluid mechanics is a branch of physics concerned with the mechanics of fluids and the forces on them. Fluid mechanics has a range of applications, including for mechanical engineering, civil engineering, chemical engineering, geophysics, astrophysics. Fluid mechanics can be divided into fluid statics, the study of fluids at rest, and fluid dynamics, fluid mechanics, especially fluid dynamics, is an active field of research with many problems that are partly or wholly unsolved. Fluid mechanics can be complex, and can best be solved by numerical methods. A modern discipline, called computational fluid dynamics, is devoted to this approach to solving fluid mechanics problems, Particle image velocimetry, an experimental method for visualizing and analyzing fluid flow, also takes advantage of the highly visual nature of fluid flow. Inviscid flow was further analyzed by mathematicians and viscous flow was explored by a multitude of engineers including Jean Léonard Marie Poiseuille. Fluid statics or hydrostatics is the branch of mechanics that studies fluids at rest. It embraces the study of the conditions under which fluids are at rest in stable equilibrium, and is contrasted with fluid dynamics, hydrostatics is fundamental to hydraulics, the engineering of equipment for storing, transporting and using fluids. It is also relevant to some aspect of geophysics and astrophysics, to meteorology, to medicine, fluid dynamics is a subdiscipline of fluid mechanics that deals with fluid flow—the science of liquids and gases in motion. The solution to a fluid dynamics problem typically involves calculating various properties of the fluid, such as velocity, pressure, density and it has several subdisciplines itself, including aerodynamics and hydrodynamics. Some fluid-dynamical principles are used in engineering and crowd dynamics. Fluid mechanics is a subdiscipline of continuum mechanics, as illustrated in the following table, in a mechanical view, a fluid is a substance that does not support shear stress, that is why a fluid at rest has the shape of its containing vessel. A fluid at rest has no shear stress, the assumptions inherent to a fluid mechanical treatment of a physical system can be expressed in terms of mathematical equations. This can be expressed as an equation in integral form over the control volume, the continuum assumption is an idealization of continuum mechanics under which fluids can be treated as continuous, even though, on a microscopic scale, they are composed of molecules. Fluid properties can vary continuously from one element to another and are average values of the molecular properties. The continuum hypothesis can lead to results in applications like supersonic speed flows. Those problems for which the continuum hypothesis fails, can be solved using statistical mechanics, to determine whether or not the continuum hypothesis applies, the Knudsen number, defined as the ratio of the molecular mean free path to the characteristic length scale, is evaluated. Problems with Knudsen numbers below 0.1 can be evaluated using the continuum hypothesis, the Navier–Stokes equations are differential equations that describe the force balance at a given point within a fluid
Fluid mechanics
–
Balance for some integrated fluid quantity in a
control volume enclosed by a
control surface.
15.
Data structure
–
In computer science, a data structure is a particular way of organizing data in a computer so that it can be used efficiently. Data structures can implement one or more abstract data types, which specify the operations that can be performed on a data structure. In comparison, a structure is a concrete implementation of the specification provided by an ADT. Different kinds of structures are suited to different kinds of applications. For example, relational databases commonly use B-tree indexes for data retrieval, Data structures provide a means to manage large amounts of data efficiently for uses such as large databases and internet indexing services. Usually, efficient data structures are key to designing efficient algorithms, some formal design methods and programming languages emphasize data structures, rather than algorithms, as the key organizing factor in software design. Data structures can be used to organize the storage and retrieval of stored in both main memory and secondary memory. Many data structures use both principles, sometimes combined in non-trivial ways, the implementation of a data structure usually requires writing a set of procedures that create and manipulate instances of that structure. The efficiency of a data structure cannot be analyzed separately from those operations, there are numerous types of data structures, generally built upon simpler primitive data types, An array is a number of elements in a specific order, typically all of the same type. Elements are accessed using an index to specify which element is required. Typical implementations allocate contiguous memory words for the elements of arrays, arrays may be fixed-length or resizable. A linked list is a collection of data elements of any type, called nodes, where each node has itself a value. The principal advantage of a linked list over an array, is that values can always be efficiently inserted and removed without relocating the rest of the list, certain other operations, such as random access to a certain element, are however slower on lists than on arrays. A record is a data structure. A record is a value that contains other values, typically in fixed number and sequence, the elements of records are usually called fields or members. A union is a structure that specifies which of a number of permitted primitive types may be stored in its instances. Contrast with a record, which could be defined to contain a float, enough space is allocated to contain the widest member datatype. A tagged union contains an additional field indicating its current type, a class is a data structure that contains data fields, like a record, as well as various methods which operate on the contents of the record
Data structure
–
A
hash table.
16.
Fluid dynamics
–
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids. It has several subdisciplines, including aerodynamics and hydrodynamics, before the twentieth century, hydrodynamics was synonymous with fluid dynamics. This is still reflected in names of some fluid dynamics topics, like magnetohydrodynamics and hydrodynamic stability, the foundational axioms of fluid dynamics are the conservation laws, specifically, conservation of mass, conservation of linear momentum, and conservation of energy. These are based on mechanics and are modified in quantum mechanics. They are expressed using the Reynolds transport theorem, in addition to the above, fluids are assumed to obey the continuum assumption. Fluids are composed of molecules that collide with one another and solid objects, however, the continuum assumption assumes that fluids are continuous, rather than discrete. The fact that the fluid is made up of molecules is ignored. The unsimplified equations do not have a general solution, so they are primarily of use in Computational Fluid Dynamics. The equations can be simplified in a number of ways, all of which make them easier to solve, some of the simplifications allow some simple fluid dynamics problems to be solved in closed form. Three conservation laws are used to solve fluid dynamics problems, the conservation laws may be applied to a region of the flow called a control volume. A control volume is a volume in space through which fluid is assumed to flow. The integral formulations of the laws are used to describe the change of mass, momentum. Mass continuity, The rate of change of fluid mass inside a control volume must be equal to the net rate of flow into the volume. Mass flow into the system is accounted as positive, and since the vector to the surface is opposite the sense of flow into the system the term is negated. The first term on the right is the net rate at which momentum is convected into the volume, the second term on the right is the force due to pressure on the volumes surfaces. The first two terms on the right are negated since momentum entering the system is accounted as positive, the third term on the right is the net acceleration of the mass within the volume due to any body forces. Surface forces, such as forces, are represented by F surf. The following is the form of the momentum conservation equation
Fluid dynamics
17.
Boundary value problem
–
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to the equation which also satisfies the boundary conditions. Boundary value problems arise in several branches of physics as any physical differential equation will have them, problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems. A large class of important boundary value problems are the Sturm–Liouville problems, the analysis of these problems involves the eigenfunctions of a differential operator. To be useful in applications, a boundary value problem should be well posed and this means that given the input to the problem there exists a unique solution, which depends continuously on the input. Much theoretical work in the field of differential equations is devoted to proving that boundary value problems arising from scientific. Among the earliest boundary value problems to be studied is the Dirichlet problem, of finding the harmonic functions, boundary value problems are similar to initial value problems. Finding the temperature at all points of a bar with one end kept at absolute zero. If the problem is dependent on both space and time, one could specify the value of the problem at a point for all time or at a given time for all space. Concretely, an example of a value is the problem y ″ + y =0 to be solved for the unknown function y with the boundary conditions y =0, y =2. Without the boundary conditions, the solution to this equation is y = A sin + B cos . From the boundary condition y =0 one obtains 0 = A ⋅0 + B ⋅1 which implies that B =0, from the boundary condition y =2 one finds 2 = A ⋅1 and so A =2. One sees that imposing boundary conditions allowed one to determine a unique solution, a boundary condition which specifies the value of the function itself is a Dirichlet boundary condition, or first-type boundary condition. For example, if one end of a rod is held at absolute zero. A boundary condition which specifies the value of the derivative of the function is a Neumann boundary condition. For example, if there is a heater at one end of a rod, then energy would be added at a constant rate. If the boundary has the form of a curve or surface that gives a value to the normal derivative and the variable itself then it is a Cauchy boundary condition. Summary of boundary conditions for the function, y, constants c 0 and c 1 specified by the boundary conditions
Boundary value problem
–
Shows a region where a
differential equation is valid and the associated boundary values
18.
Supercomputer
–
A supercomputer is a computer with a high level of computing performance compared to a general-purpose computer. Performance of a supercomputer is measured in floating-point operations per second instead of instructions per second. As of 2015, there are supercomputers which can perform up to quadrillions of FLOPS and it tops the rankings in the TOP500 supercomputer list. Sunway TaihuLights emergence is also notable for its use of indigenous chips, as of June 2016, China, for the first time, had more computers on the TOP500 list than the United States. However, U. S. built computers held ten of the top 20 positions, in November 2016 the U. S. has five of the top 10, throughout their history, they have been essential in the field of cryptanalysis. The use of multi-core processors combined with centralization is an emerging trend, the history of supercomputing goes back to the 1960s, with the Atlas at the University of Manchester and a series of computers at Control Data Corporation, designed by Seymour Cray. These used innovative designs and parallelism to achieve superior computational peak performance, Cray left CDC in 1972 to form his own company, Cray Research. Four years after leaving CDC, Cray delivered the 80 MHz Cray 1 in 1976, the Cray-2 released in 1985 was an 8 processor liquid cooled computer and Fluorinert was pumped through it as it operated. It performed at 1.9 gigaflops and was the second fastest after M-13 supercomputer in Moscow. Fujitsus Numerical Wind Tunnel supercomputer used 166 vector processors to gain the top spot in 1994 with a speed of 1.7 gigaFLOPS per processor. The Hitachi SR2201 obtained a performance of 600 GFLOPS in 1996 by using 2048 processors connected via a fast three-dimensional crossbar network. The Intel Paragon could have 1000 to 4000 Intel i860 processors in various configurations, the Paragon was a MIMD machine which connected processors via a high speed two dimensional mesh, allowing processes to execute on separate nodes, communicating via the Message Passing Interface. Approaches to supercomputer architecture have taken dramatic turns since the earliest systems were introduced in the 1960s, early supercomputer architectures pioneered by Seymour Cray relied on compact innovative designs and local parallelism to achieve superior computational peak performance. However, in time the demand for increased computational power ushered in the age of massively parallel systems, supercomputers of the 21st century can use over 100,000 processors connected by fast connections. The Connection Machine CM-5 supercomputer is a parallel processing computer capable of many billions of arithmetic operations per second. Throughout the decades, the management of heat density has remained a key issue for most centralized supercomputers, the large amount of heat generated by a system may also have other effects, e. g. reducing the lifetime of other system components. There have been diverse approaches to management, from pumping Fluorinert through the system. Systems with a number of processors generally take one of two paths
Supercomputer
–
IBM 's
Blue Gene/P supercomputer at
Argonne National Laboratory runs over 250,000 processors using normal data center air conditioning, grouped in 72 racks/cabinets connected by a high-speed optical network
Supercomputer
–
A
Cray-1 preserved at the
Deutsches Museum
Supercomputer
–
A
Blue Gene /L cabinet showing the stacked
blades, each holding many processors
Supercomputer
–
An
IBM HS20 blade
19.
Transonic
–
This condition depends not only on the travel speed of the craft, but also on the temperature of the airflow in the vehicles local environment.2, when most of the airflow is supersonic. Between these speeds some of the airflow is supersonic, but a significant fraction is not, most modern jet powered aircraft are engineered to operate at transonic air speeds. Transonic airspeeds see an increase in drag from about Mach 0.8. Attempts to reduce wave drag can be seen on all high-speed aircraft, most notable is the use of swept wings, but another common form is a wasp-waist fuselage as a side effect of the Whitcomb area rule. Severe instability can occur at transonic speeds, shock waves move through the air at the speed of sound. When an object such as an aircraft also moves at the speed of sound, transonic speeds can also occur at the tips of rotor blades of helicopters and aircraft. This puts severe, unequal stresses on the blade and may lead to accidents if it occurs. It is one of the factors of the size of rotors. At transonic speeds supersonic expansion fans form intense low-pressure, low-temperature areas at various points around an aircraft, if the temperature drops below the dew point a visible cloud will form. These clouds remain with the aircraft as it travels and it is not necessary for the aircraft as a whole to reach supersonic speeds for these clouds to form. Typically, the tail of the aircraft will reach supersonic flight while the bow of the aircraft is still in subsonic flight, a bubble of supersonic expansion fans terminating by a wake shockwave surround the tail. As the aircraft continues to accelerate, the supersonic expansion fans will intensify and this is Mach one and the Prandtl–Glauert singularity. In astrophysics, wherever there is evidence of shocks, the close by must be transonic. Interestingly, all black hole accretions are transonic, many such flows also have shocks very close to the black holes. The outflows or jets from young stellar objects or disks around black holes can also be transonic since they start subsonically, supernovae explosions are accompanied by supersonic flows and shock waves. Bow shocks formed in solar winds are a result of transonic winds from a star. It has been thought that a bow shock was present around the heliosphere of our solar system. This was recently found not to be the case according to IBEX data, anti-shock body Subsonic flows Supersonic flows Hypersonic flows Supersonic expansion fans
Transonic
–
Aerodynamic condensation evidences of
supersonic expansion fans around a transonic
F/A-18
Transonic
–
Shock waves may appear as weak optical disturbances above airliners with
supercritical wings
20.
Wind tunnel
–
A wind tunnel is a tool used in aerodynamic research to study the effects of air moving past solid objects. A wind tunnel consists of a passage with the object under test mounted in the middle. Air is made to move past the object by a fan system or other means. The test object, often called a wind tunnel model, is instrumented with sensors to measure aerodynamic forces, pressure distribution. The earliest wind tunnels were invented towards the end of the 19th century, in the days of aeronautic research. In that way an observer could study the flying object in action. The development of wind tunnels accompanied the development of the airplane, large wind tunnels were built during World War II. Wind tunnel testing was considered of importance during the Cold War development of supersonic aircraft. Determining such forces was required before building codes could specify the required strength of such buildings, in these studies, the interaction between the road and the vehicle plays a significant role, and this interaction must be taken into consideration when interpreting the test results. The advances in fluid dynamics modelling on high speed digital computers has reduced the demand for wind tunnel testing. However, CFD results are not completely reliable and wind tunnels are used to verify CFD predictions. Air velocity and pressures are measured in several ways in wind tunnels, air velocity through the test section is determined by Bernoullis principle. Measurement of the pressure, the static pressure, and the temperature rise in the airflow. The direction of airflow around a model can be determined by tufts of yarn attached to the aerodynamic surfaces, the direction of airflow approaching a surface can be visualized by mounting threads in the airflow ahead of and aft of the test model. Smoke or bubbles of liquid can be introduced into the upstream of the test model. Aerodynamic forces on the test model are usually measured with beam balances, connected to the test model with beams, strings, or cables. Pressure distributions can more conveniently be measured by the use of pressure-sensitive paint, the strip is attached to the aerodynamic surface with tape, and it sends signals depicting the pressure distribution along its surface. The aerodynamic properties of an object can not all remain the same for a scaled model, however, by observing certain similarity rules, a very satisfactory correspondence between the aerodynamic properties of a scaled model and a full-size object can be achieved
Wind tunnel
–
NASA wind tunnel with the model of a plane.
Wind tunnel
–
A model
Cessna with helium-filled bubbles showing
pathlines of the
wingtip vortices.
Wind tunnel
–
Replica of the Wright brothers' wind tunnel.
Wind tunnel
–
Eiffel's wind tunnels in the Auteuil laboratory
21.
Space Shuttle
–
The Space Shuttle was a partially reusable low Earth orbital spacecraft system operated by the U. S. National Aeronautics and Space Administration, as part of the Space Shuttle program. Its official program name was Space Transportation System, taken from a 1969 plan for a system of reusable spacecraft of which it was the only item funded for development, the first of four orbital test flights occurred in 1981, leading to operational flights beginning in 1982. Five complete Shuttle systems were built and used on a total of 135 missions from 1981 to 2011, the Shuttle fleets total mission time was 1322 days,19 hours,21 minutes and 23 seconds. Shuttle components included the Orbiter Vehicle, a pair of solid rocket boosters. The Shuttle was launched vertically, like a rocket, with the two SRBs operating in parallel with the OVs three main engines, which were fueled from the ET. The SRBs were jettisoned before the vehicle reached orbit, and the ET was jettisoned just before orbit insertion, at the conclusion of the mission, the orbiter fired its OMS to de-orbit and re-enter the atmosphere. The orbiter then glided as a spaceplane to a landing, usually at the Shuttle Landing Facility of KSC or Rogers Dry Lake in Edwards Air Force Base. After landing at Edwards, the orbiter was back to the KSC on the Shuttle Carrier Aircraft. The first orbiter, Enterprise, was built in 1976, used in Approach, four fully operational orbiters were initially built, Columbia, Challenger, Discovery, and Atlantis. Of these, two were lost in accidents, Challenger in 1986 and Columbia in 2003, with a total of fourteen astronauts killed. A fifth operational orbiter, Endeavour, was built in 1991 to replace Challenger, the Space Shuttle was retired from service upon the conclusion of Atlantiss final flight on July 21,2011. Nixons post-Apollo NASA budgeting withdrew support of all components except the Shuttle. The vehicle consisted of a spaceplane for orbit and re-entry, fueled by liquid hydrogen and liquid oxygen tanks. The first of four orbital test flights occurred in 1981, leading to operational flights beginning in 1982, all launched from the Kennedy Space Center, Florida. The system was retired from service in 2011 after 135 missions, the program ended after Atlantis landed at the Kennedy Space Center on July 21,2011. Major missions included launching numerous satellites and interplanetary probes, conducting space science experiments, the first orbiter vehicle, named Enterprise, was built for the initial Approach and Landing Tests phase and lacked engines, heat shielding, and other equipment necessary for orbital flight. A total of five operational orbiters were built, and of these and it was used for orbital space missions by NASA, the US Department of Defense, the European Space Agency, Japan, and Germany. The United States funded Shuttle development and operations except for the Spacelab modules used on D1, sL-J was partially funded by Japan
Space Shuttle
–
Discovery lifts off at the start of
STS-120.
Space Shuttle
–
STS-129 ready for launch
Space Shuttle
–
President Nixon (right) with
NASA Administrator Fletcher in January 1972, three months before Congress approved funding for the Shuttle program
Space Shuttle
–
STS-1 on the launch pad, December 1980
22.
Hyper-X
–
The X-43 was an unmanned experimental hypersonic aircraft with multiple planned scale variations meant to test various aspects of hypersonic flight. It was part of the X-plane series and specifically of NASAs Hyper-X program and it set several airspeed records for jet-propelled aircraft. The X-43 is the fastest aircraft on record at approximately Mach 9.6, a winged booster rocket with the X-43 placed on top, called a stack, was drop launched from a Boeing B-52 Stratofortress. After the booster rocket brought the stack to the speed and altitude, it was discarded, and the X-43 flew free using its own engine. The first plane in the series, the X-43A, was a single-use vehicle, the X-43 was part of NASAs Hyper-X program, involving the American space agency and contractors such as Boeing, Micro Craft Inc, Orbital Sciences Corporation and General Applied Science Laboratory. Micro Craft Inc. built the X-43A and GASL built its engine, one of the primary goals of NASAs Aeronautics Enterprise, as delineated in the NASA Strategic Plan, specified the development and demonstration of technologies for air-breathing hypersonic flight. Following the cancelation of the National Aerospace Plane program in November 1994, Langley was the lead center and responsible for hypersonic technology development. Dryden was responsible for flight research, phase I was a seven-year, approximately $230 million, program to flight-validate scramjet propulsion, hypersonic aerodynamics and design methods. Subsequent phases were not continued as the X-43 series of aircraft was replaced by the X-51, the X-43A aircraft was a small unpiloted test vehicle measuring just over 3.7 m in length. The vehicle was a body design, where the body of the aircraft provides a significant amount of lift for flight. The aircraft weighed roughly 3,000 pounds, the X-43A was designed to be fully controllable in high-speed flight, even when gliding without propulsion. However, the aircraft was not designed to land and be recovered, test vehicles crashed into the Pacific Ocean when the test was over. Traveling at Mach speeds produces a lot of heat due to the shock waves involved in supersonic drag. At high Mach speeds, heat can become so intense that metal portions of the airframe melt, the X-43A compensated for this by cycling water behind the engine cowl and sidewall leading edges, cooling those surfaces. In tests, the circulation was activated at about Mach 3. The X-43As developers designed the aircrafts airframe to be part of the propulsion system, the engine of the X-43A was primarily fueled with hydrogen. In the successful test, about two pounds of the fuel was used, unlike rockets, scramjet-powered vehicles do not carry oxygen on board for fueling the engine. Removing the need to carry oxygen significantly reduces the vehicles size, in the future, such lighter vehicles could take heavier payloads into space or carry payloads of the same weight much more efficiently
Hyper-X
–
Pegasus booster accelerating NASA's X-43A shortly after ignition during test flight (March 27, 2004)
Hyper-X
–
Artist's concept of X-43A with
scramjet attached to the underside
Hyper-X
–
NASA's B-52B launch aircraft takes off carrying the X-43A hypersonic research vehicle (March 27, 2004)
Hyper-X
–
Full-scale model of the X-43 plane in
Langley's 8-foot (2.4 m), high-temperature
wind tunnel.
23.
Mach number
–
In fluid dynamics, the Mach number is a dimensionless quantity representing the ratio of flow velocity past a boundary to the local speed of sound. M = u c, where, M is the Mach number, u is the flow velocity with respect to the boundaries. By definition, Mach 1 is equal to the speed of sound, Mach 0.65 is 65% of the speed of sound, and Mach 1.35 is 35% faster than the speed of sound. The local speed of sound, and thereby the Mach number, depends on the condition of the surrounding medium, the Mach number is primarily used to determine the approximation with which a flow can be treated as an incompressible flow. The medium can be a gas or a liquid, the boundary can be the boundary of an object immersed in the medium, or of a channel such as a nozzle, diffusers or wind tunnels channeling the medium. As the Mach number is defined as the ratio of two speeds, it is a dimensionless number, if M <0. 2–0.3 and the flow is quasi-steady and isothermal, compressibility effects will be small and simplified incompressible flow equations can be used. The Mach number is named after Austrian physicist and philosopher Ernst Mach, as the Mach number is a dimensionless quantity rather than a unit of measure, with Mach, the number comes after the unit, the second Mach number is Mach 2 instead of 2 Mach. This is somewhat reminiscent of the modern ocean sounding unit mark, which was also unit-first. In the decade preceding faster-than-sound human flight, aeronautical engineers referred to the speed of sound as Machs number, never Mach 1, Mach number is useful because the fluid behaves in a similar manner at a given Mach number, regardless of other variables. As modeled in the International Standard Atmosphere, dry air at sea level, standard temperature of 15 °C. For example, the atmosphere model lapses temperature to −56.5 °C at 11,000 meters altitude. In the following table, the regimes or ranges of Mach values are referred to, generally, NASA defines high hypersonic as any Mach number from 10 to 25, and re-entry speeds as anything greater than Mach 25. Aircraft operating in this include the Space Shuttle and various space planes in development. Flight can be classified in six categories, For comparison. At transonic speeds, the field around the object includes both sub- and supersonic parts. The transonic period begins when first zones of M >1 flow appear around the object, in case of an airfoil, this typically happens above the wing. Supersonic flow can decelerate back to only in a normal shock. As the speed increases, the zone of M >1 flow increases towards both leading and trailing edges
Mach number
–
An
F/A-18 Hornet creating a
vapor cone at
transonic speed just before reaching the speed of sound
24.
Viscous
–
The viscosity of a fluid is a measure of its resistance to gradual deformation by shear stress or tensile stress. For liquids, it corresponds to the concept of thickness, for example. Viscosity is a property of the fluid which opposes the motion between the two surfaces of the fluid in a fluid that are moving at different velocities. For a given velocity pattern, the stress required is proportional to the fluids viscosity, a fluid that has no resistance to shear stress is known as an ideal or inviscid fluid. Zero viscosity is observed only at low temperatures in superfluids. Otherwise, all fluids have positive viscosity, and are said to be viscous or viscid. A fluid with a high viscosity, such as pitch. The word viscosity is derived from the Latin viscum, meaning mistletoe, the dynamic viscosity of a fluid expresses its resistance to shearing flows, where adjacent layers move parallel to each other with different speeds. It can be defined through the situation known as a Couette flow. This fluid has to be homogeneous in the layer and at different shear stresses, if the speed of the top plate is small enough, the fluid particles will move parallel to it, and their speed will vary linearly from zero at the bottom to u at the top. Each layer of fluid will move faster than the one just below it, in particular, the fluid will apply on the top plate a force in the direction opposite to its motion, and an equal but opposite one to the bottom plate. An external force is required in order to keep the top plate moving at constant speed. The magnitude F of this force is found to be proportional to the u and the area A of each plate. The proportionality factor μ in this formula is the viscosity of the fluid, the ratio u/y is called the rate of shear deformation or shear velocity, and is the derivative of the fluid speed in the direction perpendicular to the plates. Isaac Newton expressed the forces by the differential equation τ = μ ∂ u ∂ y, where τ = F/A. This formula assumes that the flow is moving along parallel lines and this equation can be used where the velocity does not vary linearly with y, such as in fluid flowing through a pipe. Use of the Greek letter mu for the dynamic viscosity is common among mechanical and chemical engineers. However, the Greek letter eta is used by chemists, physicists
Viscous
–
Pitch has a viscosity approximately 230 billion (2.3 × 10 11) times that of water.
Viscous
–
A simulation of substances with different viscosities. The substance above has lower viscosity than the substance below
Viscous
–
Example of the viscosity of milk and water. Liquids with higher viscosities make smaller splashes when poured at the same velocity.
Viscous
–
Honey being drizzled.
25.
Euler equations (fluid dynamics)
–
In fluid dynamics, the Euler equations are a set of quasilinear hyperbolic equations governing adiabatic and inviscid flow. They are named after Leonhard Euler, in fact, Euler equations can be obtained by linearization of some more precise continuity equations like Navier–Stokes equations in a local equilibrium state given by a Maxwellian. The Euler equations can be applied to incompressible and to compressible flow – assuming the flow velocity is a solenoidal field, historically, only the incompressible equations have been derived by Euler. However, fluid dynamics literature often refers to the full set – including the energy equation – of the more general compressible equations together as the Euler equations, from the mathematical point of view, Euler equations are notably hyperbolic conservation equations in the case without external field. In fact, like any Cauchy equation, the Euler equations originally formulated in convective form can also be put in the conservation form, the convective form emphasizes changes to the state in a frame of reference moving with the fluid. The Euler equations first appeared in published form in Eulers article Principes généraux du mouvement des fluides and they were among the first partial differential equations to be written down. At the time Euler published his work, the system of equations consisted of the momentum and continuity equations, an additional equation, which was later to be called the adiabatic condition, was supplied by Pierre-Simon Laplace in 1816. G represents body accelerations acting on the continuum, for example gravity, inertial accelerations, electric field acceleration, the first equation is the Euler momentum equation with uniform density. The second equation is the constraint, stating the flow velocity is a solenoidal field. Notably, the continuity equation would be required also in this case as an additional third equation in case of density varying in time or varying in space. The equations above thus represent respectively conservation of mass and momentum, in 3D for example N =3 and the r and u vectors are explicitly and. Flow velocity and pressure are the physical variables. In 3D N =3 and the r and u vectors are explicitly and, although Euler first presented these equations in 1755, many fundamental questions about them remain unanswered. In three space dimensions it is not even known whether solutions of the equations are defined for all time or if they form singularities, in order to make the equations dimensionless, a characteristic length r 0, and a characteristic velocity u 0, need to be defined. These should be such that the dimensionless variables are all of order one. The limit of high Froude numbers is thus notable and can be studied with perturbation theory, the conservation form emphasizes the mathematical properties of Euler equations, and especially the contracted form is often the most convenient one for computational fluid dynamics simulations. Computationally, there are advantages in using the conserved variables. This gives rise to a class of numerical methods called conservative methods
Euler equations (fluid dynamics)
–
The "Streamline curvature theorem" states that the pressure at the upper surface of an airfoil is lower than the pressure far away and that the pressure at the lower surface is higher than the pressure far away; hence the pressure difference between the upper and lower surfaces of an airfoil generates a lift force.
26.
Vorticity
–
Conceptually, vorticity could be determined by marking the part of continuum in a small neighborhood of the point in question, and watching their relative displacements as they move along the flow. The vorticity vector would be twice the angular velocity vector of those particles relative to their center of mass. This quantity must not be confused with the velocity of the particles relative to some other point. More precisely, the vorticity is a pseudovector field ω→, defined as the curl of the flow velocity u→ vector, the definition can be expressed by the vector analysis formula, ω → ≡ ∇ × u →, where ∇ is the del operator. The vorticity of a flow is always perpendicular to the plane of the flow. The vorticity is related to the flows circulation along a path by the Stokes theorem. Namely, for any infinitesimal surface element C with normal direction n→ and area dA, many phenomena, such as the blowing out of a candle by a puff of air, are more readily explained in terms of vorticity rather than the basic concepts of pressure and velocity. This applies, in particular, to the formation and motion of vortex rings, in a mass of continuum that is rotating like a rigid body, the vorticity is twice the angular velocity vector of that rotation. This is the case, for example, of water in a tank that has been spinning for a while around its vertical axis, the vorticity may be nonzero even when all particles are flowing along straight and parallel pathlines, if there is shear. The vorticity will be zero on the axis, and maximum near the walls, conversely, a flow may have zero vorticity even though its particles travel along curved trajectories. An example is the ideal irrotational vortex, where most particles rotate about some straight axis, another way to visualize vorticity is to imagine that, instantaneously, a tiny part of the continuum becomes solid and the rest of the flow disappears. If that tiny new solid particle is rotating, rather than just moving with the flow, mathematically, the vorticity of a three-dimensional flow is a pseudovector field, usually denoted by ω→, defined as the curl or rotational of the velocity field v→ describing the continuum motion. In Cartesian coordinates, ω → = ∇ × v → = × = In words, the evolution of the vorticity field in time is described by the vorticity equation, which can be derived from the Navier–Stokes equations. This is clearly true in the case of 2-D potential flow, Vorticity is a useful tool to understand how the ideal potential flow solutions can be perturbed to model real flows. In general, the presence of viscosity causes a diffusion of vorticity away from the vortex cores into the flow field. This flow is accounted for by the term in the vorticity transport equation. Thus, in cases of very viscous flows, the vorticity will be diffused throughout the flow field, a vortex line or vorticity line is a line which is everywhere tangent to the local vorticity vector. Vortex lines are defined by the relation d x ω x = d y ω y = d z ω z, a vortex tube is the surface in the continuum formed by all vortex-lines passing through a given closed curve in the continuum
Vorticity
–
Continuum mechanics
Vorticity
–
Example flows:
27.
Full potential equation
–
In fluid dynamics, potential flow describes the velocity field as the gradient of a scalar function, the velocity potential. As a result, a flow is characterized by an irrotational velocity field. The irrotationality of a flow is due to the curl of the gradient of a scalar always being equal to zero. In the case of an incompressible flow the velocity potential satisfies Laplaces equation, however, potential flows also have been used to describe compressible flows. The potential flow approach occurs in the modeling of both stationary as well as nonstationary flows, applications of potential flow are for instance, the outer flow field for aerofoils, water waves, electroosmotic flow, and groundwater flow. For flows with strong vorticity effects, the potential flow approximation is not applicable, in fluid dynamics, a potential flow is described by means of a velocity potential φ, being a function of space and time. The flow velocity v is a field equal to the gradient, ∇. Sometimes, also the definition v = −∇φ, with a sign, is used. But here we use the definition above, without the minus sign. From vector calculus it is known, that the curl of a gradient is equal to zero, ∇ × ∇ φ =0, and consequently the vorticity and this implies that a potential flow is an irrotational flow. This has direct consequences for the applicability of potential flow, in flow regions where vorticity is known to be important, such as wakes and boundary layers, potential flow theory is not able to provide reasonable predictions of the flow. Fortunately, there are often large regions of a flow where the assumption of irrotationality is valid, for instance in, flow around aircraft, groundwater flow, acoustics, water waves, and electroosmotic flow. As a result, the velocity potential φ has to satisfy Laplaces equation ∇2 φ =0, in this case the flow can be determined completely from its kinematics, the assumptions of irrotationality and zero divergence of flow. Dynamics only have to be applied afterwards, if one is interested in computing pressures, in two dimensions, potential flow reduces to a very simple system that is analyzed using complex analysis. Potential flow theory can also be used to model irrotational compressible flow, the flow velocity v is again equal to ∇Φ, with Φ the velocity potential. The full potential equation is valid for sub-, trans- and supersonic flow at arbitrary angle of attack and this linear equation is much easier to solve than the full potential equation, it may be recast into Laplaces equation by a simple coordinate stretching in the x-direction. Small-amplitude sound waves can be approximated with the following model, ∂2 φ ∂ t 2 = a ¯2 Δ φ. Note that also the parts of the pressure p and density ρ each individually satisfy the wave equation
Full potential equation
–
Potential-flow
streamlines around a
NACA 0012 airfoil at 11°
angle of attack, with upper and lower
streamtubes identified.
28.
Supersonic
–
Supersonic travel is a rate of travel of an object that exceeds the speed of sound. For objects traveling in dry air of a temperature of 20 °C at sea level, speeds greater than five times the speed of sound are often referred to as hypersonic. Flights during which some parts of the air surrounding an object, such as the ends of rotor blades. This occurs typically somewhere between Mach 0.8 and Mach 1.23, sounds are traveling vibrations in the form of pressure waves in an elastic medium. In gases, sound travels longitudinally at different speeds, mostly depending on the mass and temperature of the gas. Since air temperature and composition varies significantly with altitude, Mach numbers for aircraft may change despite a constant travel speed, in water at room temperature supersonic speed can be considered as any speed greater than 1,440 m/s. In solids, sound waves can be polarized longitudinally or transversely and have higher velocities. Supersonic fracture is crack motion faster than the speed of sound in a brittle material, at the beginning of the 20th century, the term supersonic was used as an adjective to describe sound whose frequency is above the range of normal human hearing. The modern term for this meaning is ultrasonic, the tip of a bullwhip is thought to be the first man-made object to break the sound barrier, resulting in the telltale crack. The wave motion traveling through the bullwhip is what makes it capable of achieving supersonic speeds, most modern fighter aircraft are supersonic aircraft, but there have been supersonic passenger aircraft, namely Concorde and the Tupolev Tu-144. Both these passenger aircraft and some modern fighters are also capable of supercruise, since Concordes final retirement flight on November 26,2003, there are no supersonic passenger aircraft left in service. Some large bombers, such as the Tupolev Tu-160 and Rockwell B-1 Lancer are also supersonic-capable, most modern firearm bullets are supersonic, with rifle projectiles often travelling at speeds approaching and in some cases well exceeding Mach 3. Most spacecraft, most notably the Space Shuttle are supersonic at least during portions of their reentry, during ascent, launch vehicles generally avoid going supersonic below 30 km to reduce air drag. Note that the speed of sound decreases somewhat with altitude, due to lower temperatures found there, at even higher altitudes the temperature starts increasing, with the corresponding increase in the speed of sound. When an inflated balloon is burst, the pieces of latex contracts at a supersonic speed. Supersonic aerodynamics is simpler than subsonic aerodynamics because the airsheets at different points along the plane often cant affect each other, Supersonic jets and rocket vehicles require several times greater thrust to push through the extra aerodynamic drag experienced within the transonic region. Designers use the Supersonic area rule and the Whitcomb area rule to minimize changes in size. However, in applications, a supersonic aircraft will have to operate stably in both subsonic and supersonic profiles, hence aerodynamic design is more complex
Supersonic
–
A
United States Navy F/A-18F Super Hornet in
transonic flight
Supersonic
–
U.S. Navy
F/A-18 approaching the sound barrier. The white cloud forms as a result of the
supersonic expansion fans dropping the air temperature below the dew point.
29.
Hypersonic
–
In aerodynamics, a hypersonic speed is one that is highly supersonic. Since the 1970s, the term has generally assumed to refer to speeds of Mach 5. The hypersonic regime is often defined as speeds where ramjets do not produce net thrust. The peculiarity in hypersonic flows are as follows, Shock layer Aerodynamic heating Entropy layer Real gas effects Low density effects Independence of aerodynamic coefficients with Mach number. As a bodys Mach number increases, the density behind a bow shock generated by the body also increases, consequently, the distance between the bow shock and the body decreases at higher Mach numbers. As Mach numbers increase, the change across the shock also increases. A portion of the kinetic energy associated with flow at high Mach numbers transforms into internal energy in the fluid due to viscous effects. The increase in energy is realized as an increase in temperature. This causes the bottom of the layer to expand, so that the boundary layer over the body grows thicker. Although subsonic and supersonic usually refer to speeds below and above the speed of sound respectively. Generally, NASA defines high hypersonic as any Mach number from 10 to 25, among the aircraft operating in this regime are the Space Shuttle and various developing spaceplanes. In the following table, the regimes or ranges of Mach values are referenced instead of the meanings of subsonic and supersonic. The categorization of airflow relies on a number of similarity parameters, for transonic and compressible flow, the Mach and Reynolds numbers alone allow good categorization of many flow cases. Hypersonic flows, however, require other similarity parameters, first, the analytic equations for the oblique shock angle become nearly independent of Mach number at high Mach numbers. Second, the formation of strong shocks around aerodynamic bodies means that the freestream Reynolds number is useful as an estimate of the behavior of the boundary layer over a body. Finally, the temperature of hypersonic flows mean that real gas effects become important. For this reason, research in hypersonics is often referred to as aerothermodynamics, the introduction of real gas effects means that more variables are required to describe the full state of a gas. This means that for a flow, something between 10 and 100 variables may be required to describe the state of the gas at any given time
Hypersonic
–
NASA X-43 at Mach 7
30.
Linearization
–
In mathematics linearization refers to finding the linear approximation to a function at a given point. In the study of systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or discrete dynamical systems. This method is used in such as engineering, physics, economics. Linearizations of a function are lines—usually lines that can be used for purposes of calculation, in short, linearization approximates the output of a function near x = a. However, what would be an approximation of 4.001 =4 +.001. For any given function y = f, f can be approximated if it is near a known differentiable point, the most basic requisite is that L a = f, where L a is the linearization of f at x = a. The point-slope form of an equation forms an equation of a line, given a point, the general form of this equation is, y − K = M. Using the point, L a becomes y = f + M, because differentiable functions are locally linear, the best slope to substitute in would be the slope of the line tangent to f at x = a. While the concept of local linearity applies the most to points arbitrarily close to x = a, the slope M should be, most accurately, the slope of the tangent line at x = a. Visually, the diagram shows the tangent line of f at x. At f, where h is any positive or negative value. The final equation for the linearization of a function at x = a is, the derivative of f is f ′, and the slope of f at a is f ′. To find 4.001, we can use the fact that 4 =2. The linearization of f = x at x = a is y = a +12 a, substituting in a =4, the linearization at 4 is y =2 + x −44. In this case x =4.001, so 4.001 is approximately 2 +4.001 −44 =2.00025. The true value is close to 2.00024998, so the linearization approximation has an error of less than 1 millionth of a percent. Linearization makes it possible to use tools for studying linear systems to analyze the behavior of a function near a given point. The linearization of a function is the first order term of its Taylor expansion around the point of interest, in stability analysis of autonomous systems, one can use the eigenvalues of the Jacobian matrix evaluated at a hyperbolic equilibrium point to determine the nature of that equilibrium
Linearization
31.
Conformal transformation
–
In mathematics, a conformal map is a function that preserves angles locally. In the most common case, the function has a domain, more formally, let U and V be subsets of C n. A function f, U → V is called conformal at a point u 0 ∈ U if it preserves oriented angles between curves through u 0 with respect to their orientation. Conformal maps preserve both angles and the shapes of small figures, but not necessarily their size or curvature. The conformal property may be described in terms of the Jacobian derivative matrix of a coordinate transformation, if the Jacobian matrix of the transformation is everywhere a scalar times a rotation matrix, then the transformation is conformal. Conformal maps can be defined between domains in higher-dimensional Euclidean spaces, and more generally on a Riemannian or semi-Riemannian manifold, an important family of examples of conformal maps comes from complex analysis. If U is a subset of the complex plane C, then a function f, U → C is conformal if and only if it is holomorphic. If f is antiholomorphic, it preserves angles, but it reverses their orientation. In the literature, there is another definition of conformal maps, since a one-to-one map defined on a non-empty open set cannot be constant, the open mapping theorem forces the inverse function to be holomorphic. Thus, under this definition, a map is conformal if, the two definitions for conformal maps are not equivalent. Being one-to-one and holomorphic implies having a non-zero derivative, however, the exponential function is a holomorphic function with a nonzero derivative, but is not one-to-one since it is periodic. A map of the complex plane onto itself is conformal if. Again, for the conjugate, angles are preserved, but orientation is reversed, an example of the latter is taking the reciprocal of the conjugate, which corresponds to circle inversion with respect to the unit circle. This can also be expressed as taking the reciprocal of the coordinate in circular coordinates. In Riemannian geometry, two Riemannian metrics g and h on smooth manifold M are called equivalent if g = u h for some positive function u on M. The function u is called the conformal factor, a diffeomorphism between two Riemannian manifolds is called a conformal map if the pulled back metric is conformally equivalent to the original one. For example, stereographic projection of a sphere onto the plane augmented with a point at infinity is a conformal map, one can also define a conformal structure on a smooth manifold, as a class of conformally equivalent Riemannian metrics. If a function is harmonic over a domain, and is transformed via a conformal map to another plane domain
Conformal transformation
–
A rectangular grid (top) and its image under a conformal map f (bottom). It is seen that f maps pairs of lines intersecting at 90° to pairs of curves still intersecting at 90°.
32.
Cylinder (geometry)
–
In its simplest form, a cylinder is the surface formed by the points at a fixed distance from a given straight line called the axis of the cylinder. It is one of the most basic curvilinear geometric shapes, commonly the word cylinder is understood to refer to a finite section of a right circular cylinder having a finite height with circular ends perpendicular to the axis as shown in the figure. If the ends are open, it is called an open cylinder, if the ends are closed by flat surfaces it is called a solid cylinder. The formulae for the area and the volume of such a cylinder have been known since deep antiquity. The area of the side is known as the lateral area. An open cylinder does not include either top or bottom elements, the surface area of a closed cylinder is made up the sum of all three components, top, bottom and side. Its surface area is A = 2πr2 + 2πrh = 2πr = πd=L+2B, for a given volume, the closed cylinder with the smallest surface area has h = 2r. Equivalently, for a surface area, the closed cylinder with the largest volume has h = 2r. Cylindric sections are the intersections of cylinders with planes, for a right circular cylinder, there are four possibilities. A plane tangent to the cylinder meets the cylinder in a straight line segment. Moved while parallel to itself, the plane either does not intersect the cylinder or intersects it in two line segments. All other planes intersect the cylinder in an ellipse or, when they are perpendicular to the axis of the cylinder, a cylinder whose cross section is an ellipse, parabola, or hyperbola is called an elliptic cylinder, parabolic cylinder, or hyperbolic cylinder respectively. Elliptic cylinders are also known as cylindroids, but that name is ambiguous, as it can also refer to the Plücker conoid. The volume of a cylinder with height h is V = ∫0 h A d x = ∫0 h π a b d x = π a b ∫0 h d x = π a b h. Even more general than the cylinder is the generalized cylinder. The cylinder is a degenerate quadric because at least one of the coordinates does not appear in the equation, an oblique cylinder has the top and bottom surfaces displaced from one another. There are other unusual types of cylinders. Let the height be h, internal radius r, and external radius R, the volume is given by V = π h
Cylinder (geometry)
–
Tycho Brahe Planetarium building, Copenhagen, its roof being an example of a cylindric section
Cylinder (geometry)
–
A right circular cylinder with radius r and height h.
Cylinder (geometry)
–
In
projective geometry, a cylinder is simply a cone whose
apex is at infinity, which corresponds visually to a cylinder in perspective appearing to be a cone towards the sky.
33.
Airfoil
–
An airfoil or aerofoil is the shape of a wing, blade, or sail. An airfoil-shaped body moved through a fluid produces an aerodynamic force, the component of this force perpendicular to the direction of motion is called lift. The component parallel to the direction of motion is called drag, subsonic flight airfoils have a characteristic shape with a rounded leading edge, followed by a sharp trailing edge, often with a symmetric curvature of upper and lower surfaces. Foils of similar function designed with water as the fluid are called hydrofoils. The lift on an airfoil is primarily the result of its angle of attack, when oriented at a suitable angle, the airfoil deflects the oncoming air, resulting in a force on the airfoil in the direction opposite to the deflection. This force is known as force and can be resolved into two components, lift and drag. Most foil shapes require an angle of attack to generate lift. This turning of the air in the vicinity of the airfoil creates curved streamlines, resulting in pressure on one side. The lift force can be related directly to the average top/bottom velocity difference without computing the pressure by using the concept of circulation, a fixed-wing aircrafts wings, horizontal, and vertical stabilizers are built with airfoil-shaped cross sections, as are helicopter rotor blades. Airfoils are also found in propellers, fans, compressors and turbines, sails are also airfoils, and the underwater surfaces of sailboats, such as the centerboard and keel, are similar in cross-section and operate on the same principles as airfoils. Swimming and flying creatures and even many plants and sessile organisms employ airfoils/hydrofoils, common examples being bird wings, the bodies of fish, an airfoil-shaped wing can create downforce on an automobile or other motor vehicle, improving traction. Any object with an angle of attack in a fluid, such as a flat plate. Airfoils are more efficient lifting shapes, able to more lift. A lift and drag curve obtained in wind tunnel testing is shown on the right, the curve represents an airfoil with a positive camber so some lift is produced at zero angle of attack. With increased angle of attack, lift increases in a linear relation. At about 18 degrees this airfoil stalls, and lift falls off quickly beyond that, the drop in lift can be explained by the action of the upper-surface boundary layer, which separates and greatly thickens over the upper surface at and past the stall angle. The thicker boundary layer also causes an increase in pressure drag, so that the overall drag increases sharply near. Airfoil design is a facet of aerodynamics
Airfoil
–
Lift and Drag curves for a typical airfoil
Airfoil
–
Examples of airfoils in nature and within various vehicles. Though not strictly an airfoil, the dolphin flipper obeys the same principles in a different fluid medium.
Airfoil
–
An airfoil section is displayed at the tip of this
Denney Kitfox aircraft, built in 1991.
Airfoil
–
Airfoil of Kamov Ka-26 helicopters
34.
Lewis Fry Richardson
–
He is also noted for his pioneering work concerning fractals and a method for solving a system of linear equations known as modified Richardson iteration. Lewis Fry Richardson was the youngest of seven born to Catherine Fry. They were a prosperous Quaker family, David Richardson operating a successful tanning, at age 12 he was sent to a Quaker boarding school, Bootham School in York, where he received an education in science, which stimulated an active interest in natural history. In 1898 he went on to Durham College of Science where he took courses in physics, chemistry, botany. He proceeded in 1900 to King’s College, Cambridge, where he was taught physics in the natural sciences tripos by J. J. Thomson, at age 47 he received a doctorate in mathematical psychology from the University of London. Richardsons working life represented his interests, National Physical Laboratory. Manager of the physical and chemical laboratory, Sunbeam Lamp Company, Meteorological Office – as superintendent of Eskdalemuir Observatory. Head of the Physics Department at Westminster Training College, principal, Paisley Technical College, now part of the University of the West of Scotland. Richardson worked from 1916 to 1919 for the Friends Ambulance Unit attached to the 16th French Infantry Division, after the war, he rejoined the Meteorological Office but was compelled to resign on grounds of conscience when it was amalgamated into the Air Ministry in 1920. He subsequently pursued a career on the fringes of the world before retiring in 1940 to research his own ideas. His pacifism had direct consequences on his research interests and he described his ideas thus, “After so much hard reasoning, may one play with a fantasy. Imagine a large hall like a theatre, except that the circles and galleries go right round through the space occupied by the stage. The walls of this chamber are painted to form a map of the globe, the ceiling represents the north polar regions, England is in the gallery, the tropics in the upper circle, Australia on the dress circle and the Antarctic in the pit. A myriad computers are at work upon the weather of the part of the map where each sits, the work of each region is coordinated by an official of higher rank. Numerous little night signs display the instantaneous values so that neighbouring computers can read them, each number is thus displayed in three adjacent zones so as to maintain communication to the North and South on the map. From the floor of the pit a tall pillar rises to half the height of the hall and it carries a large pulpit on its top. In this sits the man in charge of the theatre, he is surrounded by several assistants. One of his duties is to maintain a speed of progress in all parts of the globe
Lewis Fry Richardson
–
Lewis Fry Richardson D.Sc., FRS
35.
ENIAC
–
ENIAC was amongst the earliest electronic general-purpose computers made. It was Turing-complete, digital, and could solve a large class of problems through reprogramming. ENIAC was formally dedicated at the University of Pennsylvania on February 15,1946 and was heralded as a Giant Brain by the press. This combination of speed and programmability allowed for thousands more calculations for problems, ENIACs design and construction was financed by the United States Army, Ordnance Corps, Research and Development Command, led by Major General Gladeon M. Barnes. The total cost was about $487,000, equivalent to $6,740,000 in 2016, ENIAC was designed by John Mauchly and J. Presper Eckert of the University of Pennsylvania, U. S. The team of design engineers assisting the development included Robert F. Shaw, Jeffrey Chuan Chu, Thomas Kite Sharpless, Frank Mural, Arthur Burks, Harry Huskey, in 1946, the researchers resigned from the University of Pennsylvania and formed the Eckert-Mauchly Computer Corporation. ENIAC was a computer, composed of individual panels to perform different functions. Twenty of these modules were accumulators which could not only add and subtract, numbers were passed between these units across several general-purpose buses. In order to achieve its high speed, the panels had to send and receive numbers, compute, save the answer and trigger the next operation, all without any moving parts. Key to its versatility was the ability to branch, it could trigger different operations, depending on the sign of a computed result. By the end of its operation in 1955, ENIAC contained 17,468 vacuum tubes,7200 crystal diodes,1500 relays,70,000 resistors,10,000 capacitors and approximately 5,000,000 hand-soldered joints. It weighed more than 30 short tons, was roughly 2.4 m ×0.9 m ×30 m in size, occupied 167 m2 and this power requirement led to the rumor that whenever the computer was switched on, lights in Philadelphia dimmed. Input was possible from an IBM card reader and an IBM card punch was used for output and these cards could be used to produce printed output offline using an IBM accounting machine, such as the IBM405. While ENIAC had no system to store memory in its inception, in 1953, a 100-word magnetic-core memory built by the Burroughs Corporation was added to ENIAC. ENIAC used ten-position ring counters to store digits, each digit required 36 vacuum tubes,10 of which were the dual triodes making up the flip-flops of the ring counter. ENIAC had 20 ten-digit signed accumulators, which used tens complement representation and it was possible to connect several accumulators to run simultaneously, so the peak speed of operation was potentially much higher, due to parallel operation. The other 9 units in ENIAC were the Initiating Unit, the Cycling Unit, the Master Programmer, the Reader, the Printer, the references by Rojas and Hashagen give more details about the times for operations, which differ somewhat from those stated above. The basic machine cycle was 200 microseconds, or 5,000 cycles per second for operations on the 10-digit numbers, in one of these cycles, ENIAC could write a number to a register, read a number from a register, or add/subtract two numbers
ENIAC
–
ENIAC
ENIAC
–
Glen Beck (background) and
Betty Snyder (foreground) program ENIAC in
BRL building 328. (U.S. Army photo)
ENIAC
–
Cpl. Irwin Goldstein (foreground) sets the switches on one of ENIAC's function tables at the Moore School of Electrical Engineering. (U.S. Army photo) This photo has been artificially darkened, obscuring details such as the women who were present and the IBM equipment in use.
ENIAC
–
A function table from ENIAC on display at Aberdeen Proving Ground museum.
36.
Three-dimensional space
–
Three-dimensional space is a geometric setting in which three values are required to determine the position of an element. This is the meaning of the term dimension. In physics and mathematics, a sequence of n numbers can be understood as a location in n-dimensional space, when n =3, the set of all such locations is called three-dimensional Euclidean space. It is commonly represented by the symbol ℝ3 and this serves as a three-parameter model of the physical universe in which all known matter exists. However, this space is one example of a large variety of spaces in three dimensions called 3-manifolds. Furthermore, in case, these three values can be labeled by any combination of three chosen from the terms width, height, depth, and breadth. In mathematics, analytic geometry describes every point in space by means of three coordinates. Three coordinate axes are given, each perpendicular to the two at the origin, the point at which they cross. They are usually labeled x, y, and z, below are images of the above-mentioned systems. Two distinct points determine a line. Three distinct points are either collinear or determine a unique plane, four distinct points can either be collinear, coplanar or determine the entire space. Two distinct lines can intersect, be parallel or be skew. Two parallel lines, or two intersecting lines, lie in a plane, so skew lines are lines that do not meet. Two distinct planes can either meet in a line or are parallel. Three distinct planes, no pair of which are parallel, can meet in a common line. In the last case, the three lines of intersection of each pair of planes are mutually parallel, a line can lie in a given plane, intersect that plane in a unique point or be parallel to the plane. In the last case, there will be lines in the plane that are parallel to the given line, a hyperplane is a subspace of one dimension less than the dimension of the full space. The hyperplanes of a space are the two-dimensional subspaces, that is
Three-dimensional space
–
Three-dimensional
Cartesian coordinate system with the x -axis pointing towards the observer. (See diagram description for correction.)
37.
Douglas Aircraft
–
The Douglas Aircraft Company was an American aerospace manufacturer based in Southern California. It was founded in 1921 by Donald Wills Douglas, Sr. Douglas Aircraft Company largely operated as a division of McDonnell Douglas after the merger. MD later merged with Boeing in 1997, the Douglas Aircraft Company was founded by Donald Wills Douglas, Sr. on July 22,1921 in Santa Monica, California, following dissolution of the Davis-Douglas Company. An early claim to fame was the first circumnavigation of the world by air in Douglas airplanes in 1924. In 1923, the U. S. Army Air Service was interested in carrying out a mission to circumnavigate the Earth for the first time by aircraft, Donald Douglas proposed a modified Douglas DT to meet the Armys needs. The two-place, open cockpit DT biplane torpedo bomber had previously produced for the U. S. Navy. The DTs were taken from the lines at the companys manufacturing plants in Rock Island, Illinois and Dayton. The modified aircraft known as the Douglas World Cruiser, also was the first major project for Jack Northrop who designed the system for the series. After the prototype was delivered in November 1923, upon the completion of tests on 19 November. Due to the expedition ahead, spare parts, including 15 extra Liberty L-12 engines,14 extra sets of pontoons. These were sent to airports along the route, the last of these aircraft was delivered to the U. S. Army on 11 March 1924. After the success of the World Cruiser, the Army Air Service ordered six similar aircraft as observation aircraft. The success of the DWC established the Douglas Aircraft Company among the aircraft companies of the world. Douglas adopted a logo that showed aircraft circling a globe, replacing the original winged heart logo, the logo evolved into an aircraft, a rocket, and a globe. It was later adopted by the McDonnell Douglas Corporation, and then became the basis of the current logo of the Boeing Company after their 1997 merger, many Douglas aircraft had long service lives. Douglas Aircraft designed and built a variety of aircraft for the U. S. military, including the Navy, Army Air Forces, Marine Corps, Air Force. The company initially built torpedo bombers for the U. S. Navy, within five years, the company was building about 100 aircraft annually. Among the early employees at Douglas were Ed Heinemann, Dutch Kindelberger, and Jack Northrop, the company retained its military market and expanded into amphibian airplanes in the late 1920s, also moving its facilities to Clover Field at Santa Monica, California
Douglas Aircraft
–
Machine tool operator at the Douglas Aircraft plant,
Long Beach, California in World War II. After losing thousands of workers to military service, American manufacturers hired women for production positions, to the point where the typical aircraft plant's workforce was 40% female.
Douglas Aircraft
–
Douglas Aircraft Company
Douglas Aircraft
–
Women at work on bomber, Douglas Aircraft Company, Long Beach, California in October 1942
Douglas Aircraft
–
An ex-USAF
C-47A Skytrain, the military version of the DC-3, on display in England in 2010. This aircraft flew from a base in
Devon, England, during the
Invasion of Normandy.
38.
Boeing
–
The Boeing Company is an American multinational corporation that designs, manufactures, and sells airplanes, rotorcraft, rockets, and satellites worldwide. The company also provides leasing and product support services, Boeing stock is a component of the Dow Jones Industrial Average. The Boeing Companys corporate headquarters are located in Chicago and the company is led by President, Boeing is organized into five primary divisions, Boeing Commercial Airplanes, Boeing Defense, Space & Security, Engineering, Operations & Technology, Boeing Capital, and Boeing Shared Services Group. Boeing bought Heaths shipyard in Seattle on the Duwamish River, which became his first airplane factory. Boeing was incorporated in Seattle by William Boeing, on July 15,1916, Boeing was later incorporated in Delaware, the original Certificate of Incorporation was filed with the Secretary of State of Delaware on July 19,1934. Boeing, who studied at Yale University, worked initially in the timber industry and this knowledge proved invaluable in his subsequent design and assembly of airplanes. The company stayed in Seattle to take advantage of the supply of spruce wood. William Boeing founded his company a few months after the June 15 maiden flight of one of the two B&W seaplanes built with the assistance of George Conrad Westervelt, a U. S. Navy engineer. Boeing and Westervelt decided to build the B&W seaplane after having flown in a Curtiss aircraft, Boeing bought a Glenn Martin Flying Birdcage seaplane and was taught to fly by Glenn Martin himself. Boeing soon crashed the Birdcage and when Martin informed Boeing that replacement parts would not become available for months, Westervelt agreed to build a better airplane and soon produced the B&W Seaplane. This first Boeing airplane was assembled in a hangar located on the northeast shore of Seattles Lake Union. Many of Boeings early planes were seaplanes, on April 6,1917, the U. S. declared War on Germany and later in the year entered World War I. On May 9,1917, the became the Boeing Airplane Company. With the U. S. entering the war, Boeing knew that the U. S. Navy needed seaplanes for training, so Boeing shipped two new Model Cs to Pensacola, Florida, where the planes were flown for the Navy. The Navy liked the Model C and ordered 50 more, the company moved its operations to a larger former shipbuilding facility known as Boeing Plant 1, located on the lower Duwamish River, Washington state. Others, including Boeing, started selling other products, Boeing built dressers, counters, and furniture, along with flat-bottom boats called Sea Sleds. In 1919 the Boeing B-1, flying boat made its first flight and it accommodated one pilot and two passengers and some mail. Over the course of eight years, it made international airmail flights from Seattle to Victoria, on May 24,1920, the Boeing Model 8 made its first flight
Boeing
–
Replica of Boeing's first plane, the
Boeing Model 1, at the
Museum of Flight
Boeing
–
William E. Boeing in 1929
Boeing
–
Boeing 377 Stratocruiser
Boeing
–
The
Boeing 707 in
British Overseas Airways Corporation (BOAC) livery, 1964
39.
Lockheed Corporation
–
The Lockheed Corporation was an American aerospace company. Lockheed was founded in 1912 and later merged with Martin Marietta to form Lockheed Martin in 1995, the Alco Hydro-Aeroplane Company was established in San Francisco in 1912 by the brothers Allan and Malcolm Loughead. Following the Model F-1, the company invested heavily in the design, however, the asking price of $2500 could not compete in a market that was saturated with post World War 1 $350 Curtiss JN-4s and De Haviland trainers. The Loughead Aircraft Manufacturing Company closed its doors in 1921, in 1926, Allan Loughead, Jack Northrop, and Kenneth Jay secured funding to form the Lockheed Aircraft Company in Hollywood. This new company utilized some of the technology originally developed for the Model S-1 to design the Vega Model. In March 1928, the relocated to Burbank, California. From 1926-28 the company produced over 80 aircraft and employed more than 300 workers who by April 1929 were building five aircraft per week, in July 1929, majority shareholder Fred Keeler sold 87% of the Lockheed Aircraft Company to Detroit Aircraft Corporation. In August 1929, Allan Lockheed resigned, the Great Depression ruined the aircraft market, and Detroit Aircraft went bankrupt. A group of headed by brothers Robert and Courtland Gross. The syndicate bought the company for a mere $40,000, ironically, Allan Lockheed himself had planned to bid for his own company, but had raised only $50,000, which he felt was too small a sum for a serious bid. In 1934, Robert E. Gross was named chairman of the new company, the Lockheed Aircraft Corporation and his brother Courtlandt S. Gross was a co-founder and executive, succeeding Robert as Chairman following his death in 1961. The company was named the Lockheed Corporation in 1977, in the 1930s, Lockheed spent $139,400 to develop the Model 10 Electra, a small twin-engined transport. The company sold 40 in the first year of production, amelia Earhart and her navigator, Fred Noonan, flew it in their failed attempt to circumnavigate the world in 1937. Subsequent designs, the Lockheed Model 12 Electra Junior and the Lockheed Model 14 Super Electra expanded their market. The Lockheed Model 14 formed the basis for the Hudson bomber and its primary role was submarine hunting. The Model 14 Super Electra were sold abroad, and more than 100 were license-built in Japan for use by the Imperial Japanese Army, the P-38 was the only American fighter aircraft in production throughout American involvement in the war, from Pearl Harbor to Victory over Japan Day. It filled ground-attack, air-to-air, and even strategic bombing roles in all theaters of the war in which the United States operated, the Lockheed Vega factory was located next to Burbanks Union Airport which it had purchased in 1940. During the war, the area was camouflaged to fool enemy aerial reconnaissance
Lockheed Corporation
–
P-38J Lightning Yippee
Lockheed Corporation
–
P-38 Lightning assembly line at the Lockheed plant,
Burbank, California in World War II. In June 1943, this
assembly line was reconfigured into a mechanized line, which more than doubled the rate of production. The transition to the new system was accomplished in only eight days. During this time production never stopped. It was continued outdoors.
Lockheed Corporation
–
A
Lockheed L-049 Constellation sporting the livery of
Trans World Airlines at the
Pima Air & Space Museum.
Lockheed Corporation
–
The
Lockheed U-2, which first flew in 1955, provided intelligence on
Soviet bloc countries.
40.
Douglas Aircraft Company
–
The Douglas Aircraft Company was an American aerospace manufacturer based in Southern California. It was founded in 1921 by Donald Wills Douglas, Sr. Douglas Aircraft Company largely operated as a division of McDonnell Douglas after the merger. MD later merged with Boeing in 1997, the Douglas Aircraft Company was founded by Donald Wills Douglas, Sr. on July 22,1921 in Santa Monica, California, following dissolution of the Davis-Douglas Company. An early claim to fame was the first circumnavigation of the world by air in Douglas airplanes in 1924. In 1923, the U. S. Army Air Service was interested in carrying out a mission to circumnavigate the Earth for the first time by aircraft, Donald Douglas proposed a modified Douglas DT to meet the Armys needs. The two-place, open cockpit DT biplane torpedo bomber had previously produced for the U. S. Navy. The DTs were taken from the lines at the companys manufacturing plants in Rock Island, Illinois and Dayton. The modified aircraft known as the Douglas World Cruiser, also was the first major project for Jack Northrop who designed the system for the series. After the prototype was delivered in November 1923, upon the completion of tests on 19 November. Due to the expedition ahead, spare parts, including 15 extra Liberty L-12 engines,14 extra sets of pontoons. These were sent to airports along the route, the last of these aircraft was delivered to the U. S. Army on 11 March 1924. After the success of the World Cruiser, the Army Air Service ordered six similar aircraft as observation aircraft. The success of the DWC established the Douglas Aircraft Company among the aircraft companies of the world. Douglas adopted a logo that showed aircraft circling a globe, replacing the original winged heart logo, the logo evolved into an aircraft, a rocket, and a globe. It was later adopted by the McDonnell Douglas Corporation, and then became the basis of the current logo of the Boeing Company after their 1997 merger, many Douglas aircraft had long service lives. Douglas Aircraft designed and built a variety of aircraft for the U. S. military, including the Navy, Army Air Forces, Marine Corps, Air Force. The company initially built torpedo bombers for the U. S. Navy, within five years, the company was building about 100 aircraft annually. Among the early employees at Douglas were Ed Heinemann, Dutch Kindelberger, and Jack Northrop, the company retained its military market and expanded into amphibian airplanes in the late 1920s, also moving its facilities to Clover Field at Santa Monica, California
Douglas Aircraft Company
–
Machine tool operator at the Douglas Aircraft plant,
Long Beach, California in World War II. After losing thousands of workers to military service, American manufacturers hired women for production positions, to the point where the typical aircraft plant's workforce was 40% female.
Douglas Aircraft Company
–
Women at work on bomber, Douglas Aircraft Company, Long Beach, California in October 1942
Douglas Aircraft Company
–
An ex-USAF
C-47A Skytrain, the military version of the DC-3, on display in England in 2010. This aircraft flew from a base in
Devon, England, during the
Invasion of Normandy.
Douglas Aircraft Company
–
Douglas DC-3
41.
McDonnell Aircraft
–
The McDonnell Aircraft Corporation was an American aerospace manufacturer based in St. Louis, Missouri. McDonnell Aircraft later merged with the Douglas Aircraft Company to form McDonnell Douglas in 1967, McDonnell & Associates in Milwaukee, Wisconsin in 1928 to produce a personal aircraft for family use. The economic depression from 1929 ruined his plans and the company collapsed and he went to work for Glenn L. Martin. He left in 1938 to try again with his own firm, McDonnell Aircraft Corporation, based near St. Louis, Missouri, world War II was a major boost to the new company. It grew from 15 employees in 1939 to 5,000 at the end the war and became a significant aircraft parts producer, McDonnell also developed the LBD-1 Gargoyle guided missile. McDonnell Aircraft suffered after the war with an end of government orders and a surplus of aircraft, the advent of the Korean War helped push McDonnell into a major military fighter supply role. In 1943, McDonnell began developing jets when they were invited to bid on a US Navy contest, Dave Lewis joined the company as Chief of Aerodynamics in 1946. He led the development of the legendary F-4 Phantom II in 1954, Lewis became Executive Vice President in 1958, and finally became President and Chief Operating Officer in 1962. Lewis went on to manage Douglas Aircraft Division in 1967 after the McDonnell Douglas merger, in 1969, he returned to St. Louis as President of McDonnell Douglas. The company was now a major employer, but was having problems, with no civilian side of the company, every peacetime downturn in procurement led to lean times at McDonnell. McDonnell Aircraft and Douglas Aircraft began to sound each other out about a merger, inquiries began in 1963, Douglas offered bid invitations from December 1966 and accepted that of McDonnell. The two firms were merged on April 28,1967 as the McDonnell Douglas Corporation. In 1967, with the merger of McDonnell and Douglas Aircraft, Dave Lewis, then president of McDonnell, was named chairman of what was called the Long Beach, Lewis managed the turnaround of the division. McDonnell Douglas would later merge with Boeing in August 1997, boeings defense and space division is based in St. Louis, Missouri, U. S. and is responsible for defense and space products and services. McDonnell Douglass legacy product programs include the F-15 Eagle, AV-8B Harrier II, F/A-18 Hornet, McDonnell, nephew of founder and later President, CEO and Chair of McDonnell Douglas. Francillon, René J. McDonnell Douglas Aircraft since 1920, McDonnell Aircraft history 1939-45 McDonnell Aircraft history 1946-56 McDonnell Aircraft history 1957-67 McDonnell Gemini Space Program 1963-1966 List of all McDonnell model numbers through 1974
McDonnell Aircraft
–
An FH-1 Phantom, in 1948.
McDonnell Aircraft
–
McDonnell F2H Banshee, F3H Demon, and F4H Phantom II.
42.
NASA
–
President Dwight D. Eisenhower established NASA in 1958 with a distinctly civilian orientation encouraging peaceful applications in space science. The National Aeronautics and Space Act was passed on July 29,1958, disestablishing NASAs predecessor, the new agency became operational on October 1,1958. Since that time, most US space exploration efforts have led by NASA, including the Apollo Moon landing missions, the Skylab space station. Currently, NASA is supporting the International Space Station and is overseeing the development of the Orion Multi-Purpose Crew Vehicle, the agency is also responsible for the Launch Services Program which provides oversight of launch operations and countdown management for unmanned NASA launches. NASA shares data with various national and international such as from the Greenhouse Gases Observing Satellite. Since 2011, NASA has been criticized for low cost efficiency, from 1946, the National Advisory Committee for Aeronautics had been experimenting with rocket planes such as the supersonic Bell X-1. In the early 1950s, there was challenge to launch a satellite for the International Geophysical Year. An effort for this was the American Project Vanguard, after the Soviet launch of the worlds first artificial satellite on October 4,1957, the attention of the United States turned toward its own fledgling space efforts. This led to an agreement that a new federal agency based on NACA was needed to conduct all non-military activity in space. The Advanced Research Projects Agency was created in February 1958 to develop technology for military application. On July 29,1958, Eisenhower signed the National Aeronautics and Space Act, a NASA seal was approved by President Eisenhower in 1959. Elements of the Army Ballistic Missile Agency and the United States Naval Research Laboratory were incorporated into NASA, earlier research efforts within the US Air Force and many of ARPAs early space programs were also transferred to NASA. In December 1958, NASA gained control of the Jet Propulsion Laboratory, NASA has conducted many manned and unmanned spaceflight programs throughout its history. Some missions include both manned and unmanned aspects, such as the Galileo probe, which was deployed by astronauts in Earth orbit before being sent unmanned to Jupiter, the experimental rocket-powered aircraft programs started by NACA were extended by NASA as support for manned spaceflight. This was followed by a space capsule program, and in turn by a two-man capsule program. This goal was met in 1969 by the Apollo program, however, reduction of the perceived threat and changing political priorities almost immediately caused the termination of most of these plans. NASA turned its attention to an Apollo-derived temporary space laboratory, to date, NASA has launched a total of 166 manned space missions on rockets, and thirteen X-15 rocket flights above the USAF definition of spaceflight altitude,260,000 feet. The X-15 was an NACA experimental rocket-powered hypersonic research aircraft, developed in conjunction with the US Air Force, the design featured a slender fuselage with fairings along the side containing fuel and early computerized control systems
NASA
–
1963 photo showing Dr. William H. Pickering, (center) JPL Director, President John F. Kennedy, (right). NASA Administrator James Webb in background. They are discussing the
Mariner program, with a model presented.
NASA
–
Seal of NASA
NASA
–
At launch control for the May 28, 1964,
Saturn I SA-6 launch.
Wernher von Braun is at center.
NASA
–
Mercury-Atlas 6 launch on February 20, 1962
43.
Ship
–
Historically, a ship was a sailing vessel with at least three square-rigged masts and a full bowsprit. Ships are generally distinguished from boats, based on size, shape, Ships have been important contributors to human migration and commerce. They have supported the spread of colonization and the trade, but have also served scientific, cultural. After the 16th century, new crops that had come from, Ship transport is responsible for the largest portion of world commerce. As of 2016, there were more than 49,000 merchant ships, of these 28% were oil tankers, 43% were bulk carriers, and 13% were container ships. Military forces operate vessels for naval warfare and to transport and support forces ashore, the top 50 navies had a median fleet of 88 surface vessels each, according to various sources. There is no definition of what distinguishes a ship from a boat. Ships can usually be distinguished from boats based on size and the ability to operate independently for extended periods. A legal definition of ship from Indian case law is a vessel that carries goods by sea, a common notion is that a ship can carry a boat, but not vice versa. American and British 19th Century maritime law distinguished vessels from other craft, ships and boats fall in one legal category, a number of large vessels are usually referred to as boats. Other types of vessel which are traditionally called boats are Great Lakes freighters, riverboats. Though large enough to carry their own boats and heavy cargoes, in most maritime traditions ships have individual names, and modern ships may belong to a ship class often named after its first ship. The first known vessels date back about 10,000 years ago, the first navigators began to use animal skins or woven fabrics as sails. Affixed to the top of a pole set upright in a boat and this allowed men to explore widely, allowing for the settlement of Oceania for example. By around 3000 BC, Ancient Egyptians knew how to assemble wooden planks into a hull and they used woven straps to lash the planks together, and reeds or grass stuffed between the planks helped to seal the seams. Sneferus ancient cedar wood ship Praise of the Two Lands is the first reference recorded to a ship being referred to by name, the ancient Egyptians were perfectly at ease building sailboats. A remarkable example of their skills was the Khufu ship. Aksum was known by the Greeks for having seaports for ships from Greece, a panel found at Mohenjodaro depicted a sailing craft
Ship
–
Italian full-rigged ship Amerigo Vespucci in
New York Harbor, 1976
Ship
–
A
raft is among the simplest boat designs.
Ship
–
Roman
trireme mosaic from Carthage,
Bardo Museum,
Tunis.
Ship
–
A Japanese
atakebune from the 16th century
44.
Helicopter
–
A helicopter is a type of rotorcraft in which lift and thrust are supplied by rotors. This allows the helicopter to take off and land vertically, to hover, and to fly forward, backward and these attributes allow helicopters to be used in congested or isolated areas where fixed-wing aircraft and many forms of VTOL aircraft cannot perform. English language nicknames for helicopter include chopper, copter, helo, heli, Helicopters were developed and built during the first half-century of flight, with the Focke-Wulf Fw 61 being the first operational helicopter in 1936. Some helicopters reached limited production, but it was not until 1942 that a helicopter designed by Igor Sikorsky reached full-scale production, with 131 aircraft built. Though most earlier designs used more than one rotor, it is the single main rotor with anti-torque tail rotor configuration that has become the most common helicopter configuration. Tandem rotor helicopters are also in use due to their greater payload capacity. Coaxial helicopters, tiltrotor aircraft, and compound helicopters are all flying today, quadcopter helicopters pioneered as early as 1907 in France, and other types of multicopter have been developed for specialized applications such as unmanned drones. The earliest references for vertical flight came from China, since around 400 BC, Chinese children have played with bamboo flying toys. This bamboo-copter is spun by rolling a stick attached to a rotor, the spinning creates lift, and the toy flies when released. The 4th-century AD Daoist book Baopuzi by Ge Hong reportedly describes some of the ideas inherent to rotary wing aircraft, designs similar to the Chinese helicopter toy appeared in Renaissance paintings and other works. In the 18th and early 19th centuries Western scientists developed flying machines based on the Chinese toy. It was not until the early 1480s, when Leonardo da Vinci created a design for a machine that could be described as an aerial screw, that any recorded advancement was made towards vertical flight. His notes suggested that he built flying models, but there were no indications for any provision to stop the rotor from making the craft rotate. As scientific knowledge increased and became accepted, people continued to pursue the idea of vertical flight. In July 1754, Russian Mikhail Lomonosov had developed a small coaxial modeled after the Chinese top but powered by a spring device. It was powered by a spring, and was suggested as a method to lift meteorological instruments. Sir George Cayley, influenced by a fascination with the Chinese flying top, developed a model of feathers, similar to that of Launoy and Bienvenu. By the end of the century, he had progressed to using sheets of tin for rotor blades and his writings on his experiments and models would become influential on future aviation pioneers
Helicopter
–
A police department
Bell 206 helicopter
Helicopter
–
A decorated Japanese taketombo bamboo-copter
Helicopter
–
Leonardo's "aerial screw"
Helicopter
–
Prototype created by
M. Lomonosov, 1754
45.
Aircraft
–
An aircraft is a machine that is able to fly by gaining support from the air. It counters the force of gravity by using either static lift or by using the lift of an airfoil. The human activity that surrounds aircraft is called aviation, crewed aircraft are flown by an onboard pilot, but unmanned aerial vehicles may be remotely controlled or self-controlled by onboard computers. Aircraft may be classified by different criteria, such as type, aircraft propulsion, usage. Each of the two World Wars led to technical advances. Consequently, the history of aircraft can be divided into five eras, Pioneers of flight, first World War,1914 to 1918. Aviation between the World Wars,1918 to 1939, Second World War,1939 to 1945. Postwar era, also called the jet age,1945 to the present day, aerostats use buoyancy to float in the air in much the same way that ships float on the water. They are characterized by one or more large gasbags or canopies, filled with a relatively low-density gas such as helium, hydrogen, or hot air, which is less dense than the surrounding air. When the weight of this is added to the weight of the aircraft structure, a balloon was originally any aerostat, while the term airship was used for large, powered aircraft designs – usually fixed-wing. In 1919 Frederick Handley Page was reported as referring to ships of the air, in the 1930s, large intercontinental flying boats were also sometimes referred to as ships of the air or flying-ships. – though none had yet been built, the advent of powered balloons, called dirigible balloons, and later of rigid hulls allowing a great increase in size, began to change the way these words were used. Huge powered aerostats, characterized by an outer framework and separate aerodynamic skin surrounding the gas bags, were produced. There were still no fixed-wing aircraft or non-rigid balloons large enough to be called airships, then several accidents, such as the Hindenburg disaster in 1937, led to the demise of these airships. Nowadays a balloon is an aerostat and an airship is a powered one. A powered, steerable aerostat is called a dirigible, sometimes this term is applied only to non-rigid balloons, and sometimes dirigible balloon is regarded as the definition of an airship. Non-rigid dirigibles are characterized by a moderately aerodynamic gasbag with stabilizing fins at the back and these soon became known as blimps. During the Second World War, this shape was adopted for tethered balloons, in windy weather
Aircraft
–
NASA test aircraft
Aircraft
–
The
Mil Mi-8 is the
most-produced helicopter in history
Aircraft
–
"Voodoo" a modified P 51 Mustang is the 2014 Reno Air Race Champion
Aircraft
–
A hot air
balloon in flight
46.
Wind turbines
–
A wind turbine is a device that converts the winds kinetic energy into electrical power. Wind turbines are manufactured in a range of vertical and horizontal axis types. The smallest turbines are used for such as battery charging for auxiliary power for boats or caravans or to power traffic warning signs. Slightly larger turbines can be used for making contributions to a power supply while selling unused power back to the utility supplier via the electrical grid. Wind turbines were used in Persia about 500–900 A. D, the windwheel of Hero of Alexandria marks one of the first known instances of wind powering a machine in history. However, the first known practical wind turbines were built in Sistan and these Panemone were vertical axle wind turbines, which had long vertical drive shafts with rectangular blades. Made of six to twelve sails covered in reed matting or cloth material, these turbines were used to grind grain or draw up water. Wind turbines first appeared in Europe during the Middle Ages, the first historical records of their use in England date to the 11th or 12th centuries and there are reports of German crusaders taking their windmill-making skills to Syria around 1190. By the 14th century, Dutch wind turbines were in use to areas of the Rhine delta. Advanced wind mills were described by Croatian inventor Fausto Veranzio, in his book Machinae Novae he described vertical axis wind turbines with curved or V-shaped blades. The first electricity-generating wind turbine was a battery charging machine installed in July 1887 by Scottish academic James Blyth to light his home in Marykirk. Some months later American inventor Charles F, although Blyths turbine was considered uneconomical in the United Kingdom electricity generation by wind turbines was more cost effective in countries with widely scattered populations. In Denmark by 1900, there were about 2500 windmills for mechanical loads such as pumps and mills, the largest machines were on 24-meter towers with four-bladed 23-meter diameter rotors. By 1908 there were 72 wind-driven electric generators operating in the United States from 5 kW to 25 kW, around the time of World War I, American windmill makers were producing 100,000 farm windmills each year, mostly for water-pumping. By the 1930s, wind generators for electricity were common on farms, in this period, high-tensile steel was cheap, and the generators were placed atop prefabricated open steel lattice towers. A forerunner of modern wind generators was in service at Yalta. This was a 100 kW generator on a 30-meter tower, connected to the local 6.3 kV distribution system and it was reported to have an annual capacity factor of 32 percent, not much different from current wind machines. In the autumn of 1941, the first megawatt-class wind turbine was synchronized to a utility grid in Vermont, the Smith-Putnam wind turbine only ran for 1,100 hours before suffering a critical failure
Wind turbines
–
Offshore wind farm, using 5 MW turbines
REpower 5M in the
North Sea off the coast of
Belgium.
Wind turbines
–
James Blyth's electricity-generating wind turbine, photographed in 1891
Wind turbines
–
The first automatically operated wind turbine, built in Cleveland in 1887 by Charles F. Brush. It was 60 feet (18 m) tall, weighed 4 tons (3.6 metric tonnes) and powered a 12 kW generator.
Wind turbines
–
Nordex N117/2400 in
Germany, a modern low-wind turbine.
47.
Yacht
–
A yacht /ˈjɒt/ is a recreational boat or ship. In modern use of the term, yachts differ from working ships mainly by their leisure purpose, there are two different classes of yachts, sailing and power boats. With the rise of the steamboat and other types of powerboat, sailing vessels in general came to be perceived as luxury, later the term came to encompass large motor boats for primarily private pleasure purposes as well. Yacht lengths normally range from 10 metres up to dozens of meters, a luxury craft smaller than 12 metres is more commonly called a cabin cruiser or simply a cruiser. A superyacht generally refers to any yacht above 24 m and a megayacht generally refers to any yacht over 50 metres and this size is small in relation to typical cruise liners and oil tankers. A few countries have a special flag worn by recreational boats or ships, although inspired by the national flag, the yacht ensign does not always correspond with the civil or merchant ensign of the state in question. Yacht ensigns differ from merchant ensigns in order to signal that the yacht is not carrying cargo that requires a customs declaration, carrying commercial cargo on a boat with a yacht ensign is deemed to be smuggling in many jurisdictions. Until the 1950s, almost all yachts were made of wood or steel, although wood hulls are still in production, the most common construction material is fibreglass, followed by aluminium, steel, carbon fibre, and ferrocement. The use of wood has changed and is no longer limited to traditional board-based methods, wood is mostly used by hobbyists or wooden boat purists when building an individual boat. Apart from materials like carbon fibre and aramid fibre, spruce veneers laminated with epoxy resins have the best weight-to-strength ratios of all boatbuilding materials. Sailing yachts can range in length from about 6 metres to well over 30 metres. Most privately owned yachts fall in the range of about 7 metres -14 metres, in the United States, sailors tend to refer to smaller yachts as sailboats, while referring to the general sport of sailing as yachting. Within the limited context of racing, a yacht is any sailing vessel taking part in a race. Many modern racing yachts have efficient sail-plans, most notably the Bermuda rig. This capability is the result of a sail-plan and hull design oriented towards this capability, day sailing yachts are usually small, at under 6 metres in length. Sometimes called sailing dinghies, they often have a keel, centreboard. Most day sailing yachts do not have a cabin, as they are designed for hourly or daily use and not for overnight journeys. They may have a cabin, where the front part of the hull has a raised solid roof to provide a place to store equipment or to offer shelter from wind or spray
Yacht
–
Sailing Yacht "Zapata II"
Yacht
–
The "Lazzara" 80' "Alchemist" runs at full speed up the California Coast
Yacht
–
A yacht in
Lorient,
Brittany,
France
Yacht
–
Aerial view of a
yacht club and
marina - Yacht Harbour Residence "Hohe Düne" in
Rostock,
Germany.
48.
University of Stuttgart
–
The University of Stuttgart is a university located in Stuttgart, Germany. It was founded in 1829 and is organized into 10 faculties and it is one of the top nine leading technical universities in Germany with highly ranked programs in civil, mechanical, industrial and electrical engineering. The academic tradition of the University of Stuttgart goes back to its probably most famous student, Gottlieb Daimler. These four universities, in combination with RWTH Aachen are the top five universities of the aforementioned TU9, from 1770 to 1794, the Karlsschule was the first university in Stuttgart. Located in Stuttgart-Hohenheim, it has since 1818 been the University of Hohenheim and is not related to the University of Stuttgart, what is now the University of Stuttgart was founded in 1829, and celebrated its 175th anniversary in 2004. Because of the importance of the technical sciences and instruction in these fields. In 1900 it was awarded the right to grant doctoral degrees in the technical disciplines, the development of the courses of study at the Technical College of Stuttgart led to its renaming in 1967 to the present-day Universität Stuttgart. With this change of name came along a built-up of new fields, such as history of science and technology and the sciences. Since the end of the 1950s, a part of the university has located in the suburb of Stuttgart-Vaihingen. Most technical subjects are located in Vaihingen, while the humanities, the sciences, architecture. As of 2014, University of Stuttgart is ranked 85th in the world in the field of Engineering & Technology according to QS World University Rankings
University of Stuttgart
–
Mensa building at the main campus
University of Stuttgart
–
Campus at Vaihingen
University of Stuttgart
–
International Centrum at the University of Stuttgart
University of Stuttgart
–
Keplerstraße 11 ("K1", right) and 17 ("K2", left) in the city center
49.
MIT
–
The Massachusetts Institute of Technology is a private research university in Cambridge, Massachusetts, often cited as one of the worlds most prestigious universities. Researchers worked on computers, radar, and inertial guidance during World War II, post-war defense research contributed to the rapid expansion of the faculty and campus under James Killian. The current 168-acre campus opened in 1916 and extends over 1 mile along the bank of the Charles River basin. The Institute is traditionally known for its research and education in the sciences and engineering, and more recently in biology, economics, linguistics. Air Force and 6 Fields Medalists have been affiliated with MIT, the school has a strong entrepreneurial culture, and the aggregated revenues of companies founded by MIT alumni would rank as the eleventh-largest economy in the world. In 1859, a proposal was submitted to the Massachusetts General Court to use newly filled lands in Back Bay, Boston for a Conservatory of Art and Science, but the proposal failed. A charter for the incorporation of the Massachusetts Institute of Technology, Rogers, a professor from the University of Virginia, wanted to establish an institution to address rapid scientific and technological advances. The Rogers Plan reflected the German research university model, emphasizing an independent faculty engaged in research, as well as instruction oriented around seminars, two days after the charter was issued, the first battle of the Civil War broke out. After a long delay through the war years, MITs first classes were held in the Mercantile Building in Boston in 1865, in 1863 under the same act, the Commonwealth of Massachusetts founded the Massachusetts Agricultural College, which developed as the University of Massachusetts Amherst. In 1866, the proceeds from sales went toward new buildings in the Back Bay. MIT was informally called Boston Tech, the institute adopted the European polytechnic university model and emphasized laboratory instruction from an early date. Despite chronic financial problems, the institute saw growth in the last two decades of the 19th century under President Francis Amasa Walker. Programs in electrical, chemical, marine, and sanitary engineering were introduced, new buildings were built, the curriculum drifted to a vocational emphasis, with less focus on theoretical science. The fledgling school still suffered from chronic financial shortages which diverted the attention of the MIT leadership, during these Boston Tech years, MIT faculty and alumni rebuffed Harvard University president Charles W. Eliots repeated attempts to merge MIT with Harvard Colleges Lawrence Scientific School. There would be at least six attempts to absorb MIT into Harvard, in its cramped Back Bay location, MIT could not afford to expand its overcrowded facilities, driving a desperate search for a new campus and funding. Eventually the MIT Corporation approved an agreement to merge with Harvard, over the vehement objections of MIT faculty, students. However, a 1917 decision by the Massachusetts Supreme Judicial Court effectively put an end to the merger scheme, the neoclassical New Technology campus was designed by William W. Bosworth and had been funded largely by anonymous donations from a mysterious Mr. Smith, starting in 1912. In January 1920, the donor was revealed to be the industrialist George Eastman of Rochester, New York, who had invented methods of production and processing
MIT
–
Stereographic card showing an MIT mechanical drafting studio, 19th century (photo by
E.L. Allen, left/right inverted)
MIT
–
Massachusetts Institute of Technology
MIT
–
A 1905 map of MIT's Boston campus
MIT
–
Plaque in Building 6 honoring
George Eastman, founder of
Eastman Kodak, who was revealed as the anonymous "Mr. Smith" who helped maintain MIT's independence
50.
Grumman Aircraft
–
The Grumman Aircraft Engineering Corporation, later Grumman Aerospace Corporation, was a leading 20th century U. S. producer of military and civilian aircraft. Founded on December 6,1929, by Leroy Grumman and partners, factory in Baldwin on Long Island, New York. All of the early Grumman employees were former Loening employees, the company was named for Grumman because he was its largest investor. The company filed as a business on December 5,1929, keeping busy by welding aluminum tubing for truck frames, the company eagerly pursued contracts with the US Navy. Grumman designed the first practical floats with a landing gear for the Navy. The first Grumman aircraft was also for the Navy, the Grumman FF-1 and this was followed by a number of other successful designs. Grumman ranked 22nd among United States corporations in the value of wartime production contracts, Grummans first jet aircraft was the F9F Panther, it was followed by the upgraded F9F/F-9 Cougar, and the less well known F-11 Tiger in the 1950s. The companys big postwar successes came in the 1960s with the A-6 Intruder and E-2 Hawkeye and in the 1970s with the Grumman EA-6B Prowler, Grumman products were prominent in the film Top Gun and numerous World War II naval and Marine Corps aviation films. The U. S. Navy still employs the Hawkeye as part of Carrier Air Wings on board aircraft carriers, Grumman was the chief contractor on the Apollo Lunar Module that landed men on the moon. The firm received the contract on November 7,1962, as the Apollo program neared its end, Grumman was one of the main competitors for the contract to design and build the Space Shuttle, but lost to Rockwell International. The company ended up involved in the program nonetheless, as a subcontractor to Rockwell, providing the wings. In 1969 the company changed its name to Grumman Aerospace Corporation, the company built the Grumman Long Life Vehicle, a light transport mail truck designed for and used by the United States Postal Service. The LLV entered service in 1986, Gulfstream business jets continue to be currently manufactured by Gulfstream Aerospace which is a wholly owned subsidiary of General Dynamics. For much of the Cold War period Grumman was the largest corporate employer on Long Island, Grummans products were considered so reliable and ruggedly built that the company was often referred to as the Grumman Iron Works. At its peak in 1986 it employed 23,000 people on Long Island, a portion of the airport property has been used for the Grumman Memorial Park. Northrop Grummans remaining business at the Bethpage campus is the Battle Management and Engagement Systems Division, under the Grumman Olson brand it made the P-600 and P-6800 step vans for UPS. Grumman manufactured fire engines under the name Firecat and aerial tower trucks under the Aerialcat name, the company entered the fire apparatus business in 1976 with its purchase of Howe Fire Apparatus and ended operations in 1992. Grumman canoes were developed in 1944 as World War II was winding down, Company executive William Hoffman used the companys aircraft aluminum to replace the traditional wood design
Grumman Aircraft
–
Grumman Historical Marker
Grumman Aircraft
–
Grumman Corporation
Grumman Aircraft
–
Apollo Spacecraft: Apollo Lunar Module Diagram
Grumman Aircraft
–
F-14 Tomcat at Grumman Memorial Park,
Calverton, New York
51.
New York University
–
New York University is a private nonprofit research university based in New York City. Founded in 1831, NYU is considered one of the worlds most influential research universities, University rankings compiled by Times Higher Education, U. S. News & World Report, and the Academic Ranking of World Universities all rank NYU amongst the top 32 universities in the world. NYU is a part of the creativity, energy and vibrancy that is Manhattan, located with its core in Greenwich Village. Among its faculty and alumni are 37 Nobel Laureates, over 30 Pulitzer Prize winners, over 30 Academy Award winners, alumni include heads of state, royalty, eminent mathematicians, inventors, media figures, Olympic medalists, CEOs of Fortune 500 companies, and astronauts. NYU alumni are among the wealthiest in the world, according to The Princeton Review, NYU is consistently considered by students and parents as a Top Dream College. Albert Gallatin, Secretary of Treasury under Thomas Jefferson and James Madison, declared his intention to establish in this immense, a system of rational and practical education fitting and graciously opened to all. A three-day-long literary and scientific convention held in City Hall in 1830 and these New Yorkers believed the city needed a university designed for young men who would be admitted based upon merit rather than birthright or social class. On April 18,1831, an institution was established, with the support of a group of prominent New York City residents from the merchants, bankers. Albert Gallatin was elected as the institutions first president, the university has been popularly known as New York University since its inception and was officially renamed New York University in 1896. In 1832, NYU held its first classes in rented rooms of four-story Clinton Hall, in 1835, the School of Law, NYUs first professional school, was established. American Chemical Society was founded in 1876 at NYU and it became one of the nations largest universities, with an enrollment of 9,300 in 1917. NYU had its Washington Square campus since its founding, the university purchased a campus at University Heights in the Bronx because of overcrowding on the old campus. NYU also had a desire to follow New York Citys development further uptown, NYUs move to the Bronx occurred in 1894, spearheaded by the efforts of Chancellor Henry Mitchell MacCracken. The University Heights campus was far more spacious than its predecessor was, as a result, most of the universitys operations along with the undergraduate College of Arts and Science and School of Engineering were housed there. NYUs administrative operations were moved to the new campus, but the schools of the university remained at Washington Square. In 1914, Washington Square College was founded as the undergraduate college of NYU. In 1935, NYU opened the Nassau College-Hofstra Memorial of New York University at Hempstead and this extension would later become a fully independent Hofstra University. In 1950, NYU was elected to the Association of American Universities, in the late 1960s and early 1970s, financial crisis gripped the New York City government and the troubles spread to the citys institutions, including NYU
New York University
–
Albert Gallatin
New York University
–
New York University
New York University
–
The University Heights campus, now home to
Bronx Community College
New York University
–
The
Silver Center c. 1900
52.
Cartesian coordinate system
–
Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair. The coordinates can also be defined as the positions of the projections of the point onto the two axis, expressed as signed distances from the origin. One can use the principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes. In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n and these coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes. The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes can be described by Cartesian equations, algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, a familiar example is the concept of the graph of a function. Cartesian coordinates are also tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering. They are the most common system used in computer graphics, computer-aided geometric design. Nicole Oresme, a French cleric and friend of the Dauphin of the 14th Century, used similar to Cartesian coordinates well before the time of Descartes. The adjective Cartesian refers to the French mathematician and philosopher René Descartes who published this idea in 1637 and it was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. Both authors used a single axis in their treatments and have a length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes La Géométrie was translated into Latin in 1649 by Frans van Schooten and these commentators introduced several concepts while trying to clarify the ideas contained in Descartes work. Many other coordinate systems have developed since Descartes, such as the polar coordinates for the plane. The development of the Cartesian coordinate system would play a role in the development of the Calculus by Isaac Newton. The two-coordinate description of the plane was later generalized into the concept of vector spaces. Choosing a Cartesian coordinate system for a one-dimensional space – that is, for a straight line—involves choosing a point O of the line, a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive, and which is negative, we say that the line is oriented from the negative half towards the positive half
Cartesian coordinate system
–
The
right hand rule.
Cartesian coordinate system
–
Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: (2,3) in green, (−3,1) in red, (−1.5,−2.5) in blue, and the origin (0,0) in purple.
Cartesian coordinate system
–
3D Cartesian Coordinate Handedness
53.
Georgia Institute of Technology
–
The Georgia Institute of Technology is a public research university in Atlanta, Georgia, in the United States. It is a part of the University System of Georgia and has campuses in Savannah, Georgia, Metz, France, Athlone, Ireland, Shenzhen, China. The educational institution was founded in 1885 as the Georgia School of Technology as part of Reconstruction plans to build an economy in the post-Civil War Southern United States. Initially, it offered only a degree in mechanical engineering, by 1901, its curriculum had expanded to include electrical, civil, and chemical engineering. In 1948, the changed its name to reflect its evolution from a trade school to a larger and more capable technical institute. Today, Georgia Tech is organized into six colleges and contains about 31 departments/units, with emphasis on science and it is well recognized for its degree programs in engineering, computing, business administration, the sciences, design, and liberal arts. Student athletics, both organized and intramural, are a part of student and alumni life, Georgia Tech fields eight mens and seven womens teams that compete in the NCAA Division I athletics and the Football Bowl Subdivision. Georgia Tech is a member of the Coastal Division in the Atlantic Coast Conference, the idea of a technology school in Georgia was introduced in 1865 during the Reconstruction period. However, because the American South of that era was mainly populated by workers and few technical developments were occurring. In 1882, the Georgia State Legislature authorized a committee, led by Harris and they were impressed by the polytechnic educational models developed at the Massachusetts Institute of Technology and the Worcester County Free Institute of Industrial Science. On October 13,1885, Georgia Governor Henry D. McDaniel signed the bill to create, in 1887, Atlanta pioneer Richard Peters donated to the state 4 acres of the site of a failed garden suburb called Peters Park. The site was bounded on the south by North Avenue, and he then sold five adjoining acres of land to the state for US$10,000. This land was near Atlantas northern city limits at the time of its founding, the surrender of the city took place on the southwestern boundary of the modern Georgia Tech campus in 1864. The Georgia School of Technology opened in the fall of 1888 with two buildings, One building had classrooms to teach students, The second building featured a shop and had a foundry, forge, boiler room, and engine room. It was designed for students to work and produce goods to sell, on October 20,1905, U. S. President Theodore Roosevelt visited Georgia Tech. On the steps of Tech Tower, Roosevelt delivered a speech about the importance of technological education and he then shook hands with every student. Georgia Techs Evening School of Commerce began holding classes in 1912, the evening school admitted its first female student in 1917, although the state legislature did not officially authorize attendance by women until 1920. Annie T. Wise became the first female graduate in 1919 and was Georgia Techs first female faculty member the following year
Georgia Institute of Technology
–
Atlanta during the Civil War (c. 1864)
Georgia Institute of Technology
–
Georgia Institute of Technology
Georgia Institute of Technology
–
An early picture of Georgia Tech
Georgia Institute of Technology
–
Former Georgia Tech President
G. Wayne Clough speaks at a student meeting.
54.
Overflow (software)
–
OVERFLOW - the OVERset grid FLOW solver - is a software package for simulating fluid flow around solid bodies using computational fluid dynamics. It is a compressible 3-D flow solver that solves the time-dependent, Reynolds-averaged, OVERFLOW was developed as part of a collaborative effort between NASAs Johnson Space Center in Houston, Texas and NASA Ames Research Center in Moffett Field, California. The driving force behind this work was the need for evaluating the flow about the Space Shuttle launch vehicle, scientists use OVERFLOW to better understand the aerodynamic forces on a vehicle by evaluating the flowfield surrounding the vehicle. OVERFLOW has also used to simulate the effect of debris on the space shuttle launch vehicle. Computational fluid dynamics Official NASA OVERFLOW CFD Code web site Article on OVERFLOW from NASA Insights
Overflow (software)
–
This image depicts the flowfield around the Space Shuttle Launch Vehicle traveling at Mach 2.46 and at an altitude of 66,000 feet (20,000 m). The surface of the vehicle is colored by the pressure coefficient, and the gray contours represent the density of the surrounding air, as calculated using the OVERFLOW codes.
55.
Geometry
–
Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space
Geometry
–
Visual checking of the
Pythagorean theorem for the (3, 4, 5)
triangle as in the
Chou Pei Suan Ching 500–200 BC.
Geometry
–
An illustration of
Desargues' theorem, an important result in
Euclidean and
projective geometry
Geometry
–
Geometry lessons in the 20th century
Geometry
–
A
European and an
Arab practicing geometry in the 15th century.
56.
Computer-aided design
–
Computer-aided design is the use of computer systems to aid in the creation, modification, analysis, or optimization of a design. CAD software is used to increase the productivity of the designer, improve the quality of design, improve communications through documentation, CAD output is often in the form of electronic files for print, machining, or other manufacturing operations. The term CADD is also used and its use in designing electronic systems is known as electronic design automation, or EDA. In mechanical design it is known as mechanical design automation or computer-aided drafting, however, it involves more than just shapes. CAD may be used to design curves and figures in space, or curves, surfaces. CAD is also used to produce computer animation for special effects in movies, advertising and technical manuals. The modern ubiquity and power of computers means that even perfume bottles, because of its enormous economic importance, CAD has been a major driving force for research in computational geometry, computer graphics, and discrete differential geometry. The design of models for object shapes, in particular, is occasionally called computer-aided geometric design. Eventually CAD provided the designer with the ability to perform engineering calculations, during this transition, calculations were still performed either by hand or by those individuals who could run computer programs. CAD was a change in the engineering industry, where draftsmen, designers. It did not eliminate departments, as much as it merged departments and empowered draftsman, CAD is just another example of the pervasive effect computers were beginning to have on industry. Current computer-aided design software packages range from 2D vector-based drafting systems to 3D solid, modern CAD packages can also frequently allow rotations in three dimensions, allowing viewing of a designed object from any desired angle, even from the inside looking out. Some CAD software is capable of mathematical modeling, in which case it may be marketed as CAD. CAD technology is used in the design of tools and machinery and in the drafting and design of all types of buildings and it can also be used to design objects. Furthermore, many CAD applications now offer advanced rendering and animation capabilities so engineers can better visualize their product designs, 4D BIM is a type of virtual construction engineering simulation incorporating time or schedule related information for project management. CAD has become an important technology within the scope of computer-aided technologies, with benefits such as lower product development costs. CAD enables designers to layout and develop work on screen, print it out, computer-aided design is one of the many tools used by engineers and designers and is used in many ways depending on the profession of the user and the type of software in question. Document management and revision control using Product Data Management, potential blockage of view corridors and shadow studies are also frequently analyzed through the use of CAD
Computer-aided design
–
Example: 3D CAD model
Computer-aided design
–
Example: 2D CAD drawing
Computer-aided design
–
CAD rendering of
Sialk ziggurat based on archeological evidence
57.
Volume
–
Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance or shape occupies or contains. Volume is often quantified numerically using the SI derived unit, the cubic metre, three dimensional mathematical shapes are also assigned volumes. Volumes of some simple shapes, such as regular, straight-edged, volumes of a complicated shape can be calculated by integral calculus if a formula exists for the shapes boundary. Where a variance in shape and volume occurs, such as those that exist between different human beings, these can be calculated using techniques such as the Body Volume Index. One-dimensional figures and two-dimensional shapes are assigned zero volume in the three-dimensional space, the volume of a solid can be determined by fluid displacement. Displacement of liquid can also be used to determine the volume of a gas, the combined volume of two substances is usually greater than the volume of one of the substances. However, sometimes one substance dissolves in the other and the volume is not additive. In differential geometry, volume is expressed by means of the volume form, in thermodynamics, volume is a fundamental parameter, and is a conjugate variable to pressure. Any unit of length gives a unit of volume, the volume of a cube whose sides have the given length. For example, a cubic centimetre is the volume of a cube whose sides are one centimetre in length, in the International System of Units, the standard unit of volume is the cubic metre. The metric system also includes the litre as a unit of volume, thus 1 litre =3 =1000 cubic centimetres =0.001 cubic metres, so 1 cubic metre =1000 litres. Small amounts of liquid are often measured in millilitres, where 1 millilitre =0.001 litres =1 cubic centimetre. Capacity is defined by the Oxford English Dictionary as the applied to the content of a vessel, and to liquids, grain, or the like. Capacity is not identical in meaning to volume, though closely related, Units of capacity are the SI litre and its derived units, and Imperial units such as gill, pint, gallon, and others. Units of volume are the cubes of units of length, in SI the units of volume and capacity are closely related, one litre is exactly 1 cubic decimetre, the capacity of a cube with a 10 cm side. In other systems the conversion is not trivial, the capacity of a fuel tank is rarely stated in cubic feet, for example. The density of an object is defined as the ratio of the mass to the volume, the inverse of density is specific volume which is defined as volume divided by mass. Specific volume is an important in thermodynamics where the volume of a working fluid is often an important parameter of a system being studied
Volume
–
A
measuring cup can be used to measure volumes of
liquids. This cup measures volume in units of
cups,
fluid ounces, and
millilitres.
58.
Enthalpy
–
Enthalpy /ˈɛnθəlpi/ is a measurement of energy in a thermodynamic system. It is the thermodynamic quantity equivalent to the heat content of a system. It is equal to the energy of the system plus the product of pressure. Enthalpy is defined as a function that depends only on the prevailing equilibrium state identified by the systems internal energy, pressure. The unit of measurement for enthalpy in the International System of Units is the joule, but other historical, conventional units are still in use, such as the British thermal unit and the calorie. At constant pressure, the enthalpy change equals the energy transferred from the environment through heating or work other than expansion work, the total enthalpy, H, of a system cannot be measured directly. The same situation exists in classical mechanics, only a change or difference in energy carries physical meaning. Enthalpy itself is a potential, so in order to measure the enthalpy of a system, we must refer to a defined reference point, therefore what we measure is the change in enthalpy. The ΔH is a change in endothermic reactions, and negative in heat-releasing exothermic processes. For processes under constant pressure, ΔH is equal to the change in the energy of the system. This means that the change in enthalpy under such conditions is the heat absorbed by the material through a reaction or by external heat transfer. Enthalpies for chemical substances at constant pressure assume standard state, most commonly 1 bar pressure, standard state does not, strictly speaking, specify a temperature, but expressions for enthalpy generally reference the standard heat of formation at 25 °C. Enthalpy of ideal gases and incompressible solids and liquids does not depend on pressure, unlike entropy, real materials at common temperatures and pressures usually closely approximate this behavior, which greatly simplifies enthalpy calculation and use in practical designs and analyses. The word enthalpy stems from the Ancient Greek verb enthalpein, which means to warm in and it combines the Classical Greek prefix ἐν- en-, meaning to put into, and the verb θάλπειν thalpein, meaning to heat. The word enthalpy is often attributed to Benoît Paul Émile Clapeyron. This misconception was popularized by the 1927 publication of The Mollier Steam Tables, however, neither the concept, the word, nor the symbol for enthalpy existed until well after Clapeyrons death. The earliest writings to contain the concept of enthalpy did not appear until 1875, however, Gibbs did not use the word enthalpy in his writings. The actual word first appears in the literature in a 1909 publication by J. P. Dalton
Enthalpy
Enthalpy
–
Fig.1 During
steady, continuous operation, an energy balance applied to an open system equates shaft work performed by the system to heat added plus net enthalpy added
Enthalpy
–
Fig.3 Two open systems in the steady state. Fluid enters the system (dotted rectangle) at point 1 and leaves it at point 2. The mass flow is. a: schematic diagram of the throttling process. b: schematic diagram of a compressor. A power P is applied and a heat flow is released to the surroundings at ambient temperature T a.
59.
Random-access memory
–
Random-access memory is a form of computer data storage which stores frequently used program instructions to increase the general speed of a system. A random-access memory device allows data items to be read or written in almost the same amount of time irrespective of the location of data inside the memory. RAM contains multiplexing and demultiplexing circuitry, to connect the lines to the addressed storage for reading or writing the entry. Usually more than one bit of storage is accessed by the same address, in todays technology, random-access memory takes the form of integrated circuits. RAM is normally associated with types of memory, where stored information is lost if power is removed. Other types of non-volatile memories exist that allow access for read operations. These include most types of ROM and a type of memory called NOR-Flash. Integrated-circuit RAM chips came into the market in the early 1970s, with the first commercially available DRAM chip, early computers used relays, mechanical counters or delay lines for main memory functions. Ultrasonic delay lines could only reproduce data in the order it was written, drum memory could be expanded at relatively low cost but efficient retrieval of memory items required knowledge of the physical layout of the drum to optimize speed. Latches built out of vacuum tube triodes, and later, out of transistors, were used for smaller and faster memories such as registers. Such registers were relatively large and too costly to use for large amounts of data, the first practical form of random-access memory was the Williams tube starting in 1947. It stored data as electrically charged spots on the face of a cathode ray tube, since the electron beam of the CRT could read and write the spots on the tube in any order, memory was random access. The capacity of the Williams tube was a few hundred to around a thousand bits, but it was smaller, faster. In fact, rather than the Williams tube memory being designed for the SSEM, magnetic-core memory was invented in 1947 and developed up until the mid-1970s. It became a form of random-access memory, relying on an array of magnetized rings. By changing the sense of each rings magnetization, data could be stored with one bit stored per ring, since every ring had a combination of address wires to select and read or write it, access to any memory location in any sequence was possible. Magnetic core memory was the form of memory system until displaced by solid-state memory in integrated circuits. Data was stored in the capacitance of each transistor, and had to be periodically refreshed every few milliseconds before the charge could leak away
Random-access memory
–
Example of
writable volatile random-access memory: Synchronous
Dynamic RAM modules, primarily used as main memory in
personal computers,
workstations, and
servers.
Random-access memory
–
These IBM
tabulating machines from the 1930s used
mechanical counters to store information
Random-access memory
–
A portion of a
core memory with a modern flash RAM
SD card on top
Random-access memory
–
1 Megabit chip – one of the last models developed by
VEB Carl Zeiss Jena in 1989
60.
Reynolds number
–
The Reynolds number is an important dimensionless quantity in fluid mechanics used to help predict flow patterns in different fluid flow situations. It has wide applications, ranging from liquid flow in a pipe to the passage of air over an aircraft wing. The concept was introduced by George Gabriel Stokes in 1851, but the Reynolds number was named by Arnold Sommerfeld in 1908 after Osborne Reynolds, who popularized its use in 1883. A similar effect is created by the introduction of a stream of higher velocity fluid and this relative movement generates fluid friction, which is a factor in developing turbulent flow. Counteracting this effect is the viscosity of the fluid, which as it increases, progressively inhibits turbulence, the Reynolds number quantifies the relative importance of these two types of forces for given flow conditions, and is a guide to when turbulent flow will occur in a particular situation. Such scaling is not linear and the application of Reynolds numbers to both situations allows scaling factors to be developed, the Reynolds number can be defined for several different situations where a fluid is in relative motion to a surface. These definitions generally include the properties of density and viscosity, plus a velocity. This dimension is a matter of convention – for example radius and diameter are equally valid to describe spheres or circles, for aircraft or ships, the length or width can be used. For flow in a pipe or a sphere moving in a fluid the internal diameter is used today. Other shapes such as pipes or non-spherical objects have an equivalent diameter defined. For fluids of variable density such as gases or fluids of variable viscosity such as non-Newtonian fluids. The velocity may also be a matter of convention in some circumstances, in practice, matching the Reynolds number is not on its own sufficient to guarantee similitude. Fluid flow is chaotic, and very small changes to shape. Nevertheless, Reynolds numbers are an important guide and are widely used. Osborne Reynolds famously studied the conditions in which the flow of fluid in pipes transitioned from laminar flow to turbulent flow, when the velocity was low, the dyed layer remained distinct through the entire length of the large tube. When the velocity was increased, the broke up at a given point. The point at which this happened was the point from laminar to turbulent flow. From these experiments came the dimensionless Reynolds number for dynamic similarity—the ratio of forces to viscous forces
Reynolds number
–
Sir George Stokes, introduced Reynolds numbers
Reynolds number
–
Osborne Reynolds popularised the concept
Reynolds number
–
The
Moody diagram, which describes the
Darcy–Weisbach friction factor f as a function of the Reynolds number and relative pipe roughness.
61.
Discretization
–
In mathematics, discretization concerns the process of transferring continuous functions, models, and equations into discrete counterparts. This process is carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers. Dichotomization is the case of discretization in which the number of discrete classes is 2. Discretization is also related to mathematics, and is an important component of granular computing. In this context, discretization may also refer to modification of variable or category granularity, whenever continuous data is discretized, there is always some amount of discretization error. The goal is to reduce the amount to a level considered negligible for the modeling purposes at hand, discretization is not always distinguished from quantization in any clearly defined way. The two terms share a semantic field, the same is true of discretization error and quantization error. Mathematical methods relating to discretization include the Euler–Maruyama method and the zero-order hold, discretization is also concerned with the transformation of continuous differential equations into discrete difference equations, suitable for numerical computing. It can, however, be computed by first constructing a matrix, the discretized process noise is then evaluated by multiplying the transpose of the lower-right partition of G with the upper-right partition of G, Q d = T. Now we want to discretise the above expression and we assume that u is constant during each timestep. Exact discretization may sometimes be due to the heavy matrix exponential and integral operations involved. It is much easier to calculate an approximate model, based on that for small timesteps e A T ≈ I + A T. The approximate solution then becomes, x ≈ x + T B u Other possible approximations are e A T ≈ −1 and e A T ≈ −1, each of them have different stability properties. The last one is known as the transform, or Tustin transform. In statistics and machine learning, discretization refers to the process of converting continuous features or variables to discretized or nominal features and this can be useful when creating probability mass functions. Discrete space Time-scale calculus Discrete event simulation Stochastic simulation Finite volume method for unsteady flow Discrete time, introduction to random signals and applied Kalman filtering. Philadelphia, PA, USA, Saunders College Publishing, computing integrals involving the matrix exponential. Digital control and estimation, a unified approach
Discretization
–
A solution to a discretized partial differential equation, obtained with the
finite element method.
62.
Flux limiters
–
Flux limiters are used in high resolution schemes – numerical schemes used to solve problems in science and engineering, particularly fluid dynamics, described by partial differential equations. Use of flux limiters, together with a high resolution scheme. In general, the term flux limiter is used when the acts on system fluxes. They are used in high resolution schemes for solving problems described by PDEs, for smoothly changing waves, the flux limiters do not operate and the spatial derivatives can be represented by higher order approximations without introducing spurious oscillations. Consider the 1D semi-discrete scheme below, d u i d t +1 Δ x i =0, R i = u i − u i −1 u i +1 − u i. The limiter function is constrained to be greater than or equal to zero, therefore, when the limiter is equal to zero, the flux is represented by a low resolution scheme. Similarly, when the limiter is equal to 1, it is represented by a high resolution scheme, the various limiters have differing switching characteristics and are selected according to the particular problem and solution scheme. No particular limiter has been found to work well for all problems, HQUICK ϕ h q =2, lim r → ∞ ϕ h q =4. Koren – third-order accurate for sufficiently smooth data ϕ k n = max, minmod – symmetric ϕ m m = max, lim r → ∞ ϕ m m =1. Monotonized central – symmetric ϕ m c = max, lim r → ∞ ϕ m c =2. Osher ϕ o s = max, lim r → ∞ ϕ o s = β. ospre – symmetric ϕ o p =1.5, smart ϕ s m = max, lim r → ∞ ϕ s m =4. Superbee – symmetric ϕ s b = max, lim r → ∞ ϕ s b =2, Sweby – symmetric ϕ s w = max, lim r → ∞ ϕ s w = β. UMIST ϕ u m = max, lim r → ∞ ϕ u m =2, Van Albada 1 – symmetric ϕ v a 1 = r 2 + r r 2 +1, lim r → ∞ ϕ v a 1 =1. Van Albada 2 – alternative form used on high spatial order schemes ϕ v a 2 =2 r r 2 +1, lim r → ∞ ϕ v a 2 =0. Van Leer – symmetric ϕ v l = r + | r |1 + | r |, all the above limiters indicated as being symmetric, exhibit the following symmetry property, ϕ r = ϕ. This is a property as it ensures that the limiting actions for forward and backward gradients operate in the same way. Unless indicated to the contrary, the above functions are second order TVD. This means that they are designed such that they pass through a region of the solution, known as the TVD region
Flux limiters
–
Admissible limiter region for second-order TVD schemes.
63.
Total variation diminishing
–
In numerical methods, total variation diminishing is a property of certain discretization schemes used to solve hyperbolic partial differential equations. The most notable application of method is in computational fluid dynamics. The concept of TVD was introduced by Ami Harten, a numerical method is said to be total variation diminishing if, T V ≤ T V. A numerical scheme is said to be monotonicity preserving if the properties are maintained, If u n is monotonically increasing in space. Harten 1983 proved the following properties for a scheme, A monotone scheme is TVD. In Computational Fluid Dynamics, TVD scheme is employed to capture sharper shock predictions without any misleading oscillations when variation of field variable “Ø” is discontinuous, to capture the variation fine grids are needed and the computation becomes heavy and therefore uneconomic. The use of coarse grids with central difference scheme, upwind scheme, hybrid difference scheme, TVD scheme enables sharper shock predictions on coarse grids saving computation time and as the scheme preserves monotonicity there are no spurious oscillations in the solution. Note that f + is the function when the flow is in positive direction i. e. from left to right. So, f r + is a function of ϕ P − ϕ L ϕ R − ϕ L. Likewise when the flow is in direction, P is negative. Monotone schemes are attractive for solving engineering and scientific problems because they do not produce non-physical solutions, Godunovs theorem proves that linear schemes which preserve monotonicity are, at most, only first order accurate. Higher order linear schemes, although more accurate for smooth solutions, are not TVD, to overcome these drawbacks, various high-resolution, non-linear techniques have been developed, often using flux/slope limiters. Flux limiters Godunovs theorem High-resolution scheme MUSCL scheme Sergei K. Godunov Total variation Hirsch, Numerical Computation of Internal and External Flows, Vol 2, Wiley. Computational Gas Dynamics, Cambridge University Press, toro, E. F. Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer-Verlag. Anderson, D. A. and Pletcher, R. H. Computational Fluid Mechanics and Heat Transfer, principles of Computational Fluid Dynamics, Springer-Verlag. Anil W. Date Introduction to Computational Fluid Dynamics, Cambridge University Press
Total variation diminishing
–
A picture showing the control volume with velocities at the faces,nodes and the distance between them, where 'P' is the node at the center.
64.
Large eddy simulation
–
Large eddy simulation is a mathematical model for turbulence used in computational fluid dynamics. It was initially proposed in 1963 by Joseph Smagorinsky to simulate atmospheric air currents, LES is currently applied in a wide variety of engineering applications, including combustion, acoustics, and simulations of the atmospheric boundary layer. The simulation of turbulent flows by numerically solving the Navier–Stokes equations requires resolving a wide range of time and length scales. Such a low-pass filtering, which can be viewed as a time- and spatial-averaging, an LES filter can be applied to a spatial and temporal field ϕ and perform a spatial filtering operation, a temporal filtering operation, or both. The filtered field, denoted with a bar, is defined as and this can also be written as, ϕ ¯ = G ⋆ ϕ. The filter kernel G has an associated cutoff length scale Δ, scales smaller than these are eliminated from ϕ ¯. Using the above definition, any field ϕ may be split up into a filtered and sub-filtered portion. It is important to note that the large eddy simulation filtering operation does not satisfy the properties of a Reynolds operator, the governing equations of LES are obtained by filtering the partial differential equations governing the flow field ρ u. There are differences between the incompressible and compressible LES governing equations, which lead to the definition of a new filtering operation, the nonlinear filtered advection term u i u j ¯ is the chief cause of difficulty in LES modeling. It requires knowledge of the velocity field, which is unknown. The analysis that follows illustrates the difficulty caused by the nonlinearity, namely, leonard decomposed this stress tensor as τ i j r = L i j + C i j + R i j and provided physical interpretations for each term. Modeling the unclosed term τ i j r is the task of SFS models and this is made challenging by the fact that the sub-filter scale stress tensor τ i j r must account for interactions among all scales, including filtered scales with unfiltered scales. The filtered diffusive flux J ϕ ¯ is unclosed, unless a form is assumed for it. Q i j is defined analogously to τ i j r, q i j = ϕ ¯ u ¯ j − ϕ u j ¯ and this sub-filter tensor also requires a sub-filter model. This equation models the changes in time of the variables u i ¯. Since the unfiltered variables u i are not known, it is impossible to directly calculate ∂ u i u j ∂ x j ¯, however, the quantity ∂ u i ¯ u j ¯ ∂ x j is known. Let τ i j = u i u j ¯ − u i ¯ u j ¯, for the governing equations of compressible flow, each equation, starting with the conservation of mass, is filtered. This gives, ∂ ρ ¯ ∂ t + ∂ u i ρ ¯ ∂ x i =0 which results in an additional sub-filter term, however, it is desirable to avoid having to model the sub-filter scales of the mass conservation equation
Large eddy simulation
–
Large eddy simulation of a turbulent gas velocity field.
65.
Wavelet
–
A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases, and then decreases back to zero. It can typically be visualized as a brief oscillation like one recorded by a seismograph or heart monitor, generally, wavelets are intentionally crafted to have specific properties that make them useful for signal processing. Wavelets can be combined, using a reverse, shift, multiply and integrate technique called convolution, for example, a wavelet could be created to have a frequency of Middle C and a short duration of roughly a 32nd note. If this wavelet were to be convolved with a created from the recording of a song. Mathematically, the wavelet will correlate with the if the unknown signal contains information of similar frequency. This concept of correlation is at the core of many applications of wavelet theory. As a mathematical tool, wavelets can be used to extract information from different kinds of data, including – but certainly not limited to – audio signals. Sets of wavelets are generally needed to analyze data fully, a set of complementary wavelets will decompose data without gaps or overlap so that the decomposition process is mathematically reversible. Thus, sets of complementary wavelets are useful in wavelet based compression/decompression algorithms where it is desirable to recover the information with minimal loss. This is accomplished through coherent states, the word wavelet has been used for decades in digital signal processing and exploration geophysics. The equivalent French word ondelette meaning small wave was used by Morlet, Wavelet theory is applicable to several subjects. All wavelet transforms may be considered forms of representation for continuous-time signals. Almost all practically useful discrete wavelet transforms use discrete-time filterbanks and these filter banks are called the wavelet and scaling coefficients in wavelets nomenclature. These filterbanks may contain either finite impulse response or infinite impulse response filters, the product of the uncertainties of time and frequency response scale has a lower bound. Thus, in the scaleogram of a wavelet transform of this signal, such an event marks an entire region in the time-scale plane. Also, discrete wavelet bases may be considered in the context of other forms of the uncertainty principle, Wavelet transforms are broadly divided into three classes, continuous, discrete and multiresolution-based. In continuous wavelet transforms, a signal of finite energy is projected on a continuous family of frequency bands. For instance the signal may be represented on every frequency band of the form for all frequencies f >0
Wavelet
–
Seismic wavelet
Wavelet
–
Meyer
66.
Vortex stretching
–
Vortex stretching is associated with a particular term in the vorticity equation. For example, vorticity transport in an inviscid flow is governed by D ω → D t = v →. The source term on the hand side is the vortex stretching term. It amplifies the vorticity ω → when the velocity is diverging in the parallel to ω →. A simple example of stretching in a viscous flow is provided by the Burgers vortex. Vortex stretching is at the core of the description of the energy cascade from the large scales to the small scales in turbulence. In general, in turbulence fluid elements are more lengthened than squeezed, in the end, this results in more vortex stretching than vortex squeezing. For incompressible flow—due to volume conservation of fluid elements—the lengthening implies thinning of the elements in the directions perpendicular to the stretching direction. This reduces the length scale of the associated vorticity. Finally, at the scales of the order of the Kolmogorov microscales. Vorticity and turbulence, Springer, ISBN 0-387-94197-5 Tennekes, H. Lumley, a First Course in Turbulence, Cambridge, MA, MIT Press, ISBN 0-262-20019-8
Vortex stretching
–
Studies of vortices in turbulent fluid motion by
Leonardo da Vinci.
67.
Probability density function
–
In a more precise sense, the PDF is used to specify the probability of the random variable falling within a particular range of values, as opposed to taking on any one value. The probability density function is everywhere, and its integral over the entire space is equal to one. The terms probability distribution function and probability function have also sometimes used to denote the probability density function. However, this use is not standard among probabilists and statisticians, further confusion of terminology exists because density function has also been used for what is here called the probability mass function. In general though, the PMF is used in the context of random variables. Suppose a species of bacteria typically lives 4 to 6 hours, what is the probability that a bacterium lives exactly 5 hours. A lot of bacteria live for approximately 5 hours, but there is no chance that any given bacterium dies at exactly 5.0000000000, instead we might ask, What is the probability that the bacterium dies between 5 hours and 5.01 hours. Lets say the answer is 0.02, next, What is the probability that the bacterium dies between 5 hours and 5.001 hours. The answer is probably around 0.002, since this is 1/10th of the previous interval, the probability that the bacterium dies between 5 hours and 5.0001 hours is probably about 0.0002, and so on. In these three examples, the ratio / is approximately constant, and equal to 2 per hour, for example, there is 0.02 probability of dying in the 0. 01-hour interval between 5 and 5.01 hours, and =2 hour−1. This quantity 2 hour−1 is called the probability density for dying at around 5 hours, therefore, in response to the question What is the probability that the bacterium dies at 5 hours. A literally correct but unhelpful answer is 0, but an answer can be written as dt. This is the probability that the bacterium dies within a window of time around 5 hours. For example, the probability that it lives longer than 5 hours, there is a probability density function f with f =2 hour−1. The integral of f over any window of time is the probability that the dies in that window. A probability density function is most commonly associated with absolutely continuous univariate distributions, a random variable X has density fX, where fX is a non-negative Lebesgue-integrable function, if, Pr = ∫ a b f X d x. That is, f is any function with the property that. In the continuous univariate case above, the measure is the Lebesgue measure
Probability density function
–
Boxplot and probability density function of a
normal distribution N (0, σ 2).
68.
N-body problem
–
In physics, the n-body problem is the problem of predicting the individual motions of a group of celestial objects interacting with each other gravitationally. Solving this problem has been motivated by the desire to understand the motions of the Sun, Moon, planets, in the 20th century, understanding the dynamics of globular cluster star systems became an important n-body problem. The n-body problem in general relativity is more difficult to solve. Having done so, he and others soon discovered over the course of a few years, Newton realized it was because gravitational interactive forces amongst all the planets was affecting all their orbits. Thus came the awareness and rise of the problem in the early 17th century. Ironically, this conformity led to the wrong approach, after Newtons time the n-body problem historically was not stated correctly because it did not include a reference to those gravitational interactive forces. Newton does not say it directly but implies in his Principia the n-body problem is unsolvable because of gravitational interactive forces. Newton said in his Principia, paragraph 21, And hence it is that the force is found in both bodies. The Sun attracts Jupiter and the planets, Jupiter attracts its satellites. Two bodies can be drawn to other by the contraction of rope between them. Newton concluded via his third law of motion according to this Law all bodies must attract each other. This last statement, which implies the existence of gravitational forces, is key. The problem of finding the solution of the n-body problem was considered very important. Indeed, in the late 19th century King Oscar II of Sweden, advised by Gösta Mittag-Leffler, in case the problem could not be solved, any other important contribution to classical mechanics would then be considered to be prizeworthy. The prize was awarded to Poincaré, even though he did not solve the original problem, the version finally printed contained many important ideas which led to the development of chaos theory. The problem as stated originally was finally solved by Karl Fritiof Sundman for n =3. The n-body problem considers n point masses mi, i =1,2, …, n in a reference frame in three dimensional space ℝ3 moving under the influence of mutual gravitational attraction. Each mass mi has a position vector qi, Newtons second law says that mass times acceleration mi d2qi/dt2 is equal to the sum of the forces on the mass
N-body problem
–
The Real Motion v.s. Kepler's Apparent Motion
N-body problem
–
Restricted 3-Body Problem
69.
Volume of fluid method
–
In computational fluid dynamics, the volume of fluid method is a free-surface modelling technique, i. e. a numerical technique for tracking and locating the free surface. It belongs to the class of Eulerian methods which are characterized by a mesh that is stationary or is moving in a certain prescribed manner to accommodate the evolving shape of the interface. As such, VOF is an advection scheme—a numerical recipe that allows the programmer to track the shape and position of the interface, the Navier–Stokes equations describing the motion of the flow have to be solved separately. The same applies for all other advection algorithms, the volume of fluid method is based on earlier Marker-and-cell methods. First accounts of what is now known as VOF have been given by Noh & Woodward in 1976, since VOF method surpassed MAC by lowering computer storage requirements, it quickly became popular. Early applications include Torrey et al. from Los Alamos, who created VOF codes for NASA, first implementations of VOF suffered from imperfect interface description, which was later remedied by introducing a Piecewise-Linear Interface Calculation scheme. Using VOF with PLIC is a standard, used in number of computer codes, such as FLOW-3D, Gerris, ANSYS Fluent. The method is based on the idea of a so-called fraction function C and it is a scalar function, defined as the integral of a fluids characteristic function in the control volume, namely the volume of a computational grid cell. The volume fraction of fluid is tracked through every cell in the computational grid. When a cell is empty with no traced fluid inside, the value of C is zero, when the cell is full, C =1, C is a discontinuous function, its value jumps from 0 to 1 when the argument moves into interior of traced phase. The normal direction of the interface is found where the value of C changes most rapidly. With this method, the free-surface is not defined sharply, instead it is distributed over the height of a cell, thus, in order to attain accurate results, local grid refinements have to be done. The refinement criterion is simple, cells with 0 < C <1 have to be refined, a method for this, known as the marker and micro-cell method, has been developed by Raad and his colleagues in 1997. For each cell, properties such as density ρ are calculated by a volume average of all fluids in the cell ρ = ∑ m =1 n ρ m C m. These properties are used to solve a single momentum equation through the domain. The VOF method is computationally friendly, as it only one additional equation. The method is characterized by its capability of dealing with highly non-linear problems in which the free-surface experiences sharp topological changes. By using the VOF method, one also evades the use of complicated mesh deformation algorithms used by surface-tracking methods, the major difficulty associated with the method is the smearing of the free-surface
Volume of fluid method
–
An illustration of fluid simulation using VOF method.
70.
Two-phase flow
–
In fluid mechanics, two-phase flow is a flow of gas and liquid usually in a pipe. Two-phase flow is an example of multiphase flow. Two-phase flow can occur in various forms, the widely-accepted method to categorize two-phase flows is to consider the velocity of each phase as if there is not other phases available. The parameter is a concept called Superficial velocity. Historically, probably the most commonly studied cases of two-phase flow are in power systems. Coal and gas-fired power stations used very large boilers to produce steam for use in turbines, in such cases, pressurised water is passed through heated pipes and it changes to steam as it moves through the pipe. The design of boilers requires an understanding of two-phase flow heat-transfer and pressure drop behaviour. Even more critically, nuclear reactors use water to heat from the reactor core using two-phase flow. A great deal of study has been performed on the nature of flow in such cases, so that engineers can design against possible failures in pipework, loss of pressure. Another case where two-phase flow can occur is in pump cavitation, here a pump is operating close to the vapor pressure of the fluid being pumped. If pressure drops further, which can happen locally near the vanes for the pump, for example, then a change can occur. Similar effects can occur on marine propellors, wherever it occurs. When the vapor bubble collapses, it can produce very large pressure spikes, the above two-phase flow cases are for a single fluid occurring by itself as two different phases, such as steam and water. The term two-phase flow is applied to mixtures of different fluids having different phases, such as air and water, or oil. Sometimes even three-phase flow is considered, such as in oil, other interesting areas where two-phase flow is studied includes in climate systems such as clouds, and in groundwater flow, in which the movement of water and air through the soil is studied. Other examples of two-phase flow include bubbles, rain, waves on the sea, foam, fountains, mousse, cryogenics, several features make two-phase flow an interesting and challenging branch of fluid mechanics, Surface tension makes all dynamical problems nonlinear. In the case of air and water at temperature and pressure. Similar differences are typical of water liquid/water vapor densities, the sound speed changes dramatically for materials undergoing phase change, and can be orders of magnitude different
Two-phase flow
–
Different modes of two-phase flows.
71.
Level set method
–
Level set methods are a conceptual framework for using level sets as a tool for numerical analysis of surfaces and shapes. The advantage of the level set model is one can perform numerical computations involving curves and surfaces on a fixed Cartesian grid without having to parameterize these objects. Also, the level set method makes it easy to follow shapes that change topology. All these make the level set method a great tool for modeling time-varying objects, like inflation of an airbag, the figure on the right illustrates several important ideas about the level set method. In the upper-left corner we see a shape, that is, below it, the red surface is the graph of a level set function φ determining this shape, and the flat blue region represents the xy-plane. The boundary of the shape is then the level set of φ. In the top row we see the shape changing its topology by splitting in two and it would be quite hard to describe this transformation numerically by parameterizing the boundary of the shape and following its evolution. One would need an algorithm able to detect the moment the shape splits in two, and then construct parameterizations for the two newly obtained curves, on the other hand, if we look at the bottom row, we see that the level set function merely translated downward. Thus, in two dimensions, the level set method amounts to representing a closed curve Γ using an auxiliary function φ, Γ is represented as the zero level set of φ by Γ =, and the level set method manipulates Γ implicitly, through the function φ. This function φ is assumed to take positive values inside the region delimited by the curve Γ and this is a partial differential equation, in particular a Hamilton–Jacobi equation, and can be solved numerically, for example by using finite differences on a Cartesian grid. The numerical solution of the level set equation, however, requires sophisticated techniques, simple finite difference methods fail quickly. Instead, the shape of the set may get severely distorted. Further sophisticated methods to deal with this difficulty have been developed, consider a unit circle in R2, shrinking in on itself at a constant rate, i. e. each point on the boundary of the circle moves along its inwards pointing normal at some fixed speed. The circle will shrink, and eventually collapse down to a point, if an initial distance field is constructed on the initial circle, the normalised gradient of this field will be the circle normal. If the field has a constant value subtracted from it in time, the level of the new fields will also be circular. This is due to this being effectively the temporal integration of the Eikonal equation with a fixed front velocity, the level set method was developed in the 1980s by the American mathematicians Stanley Osher and James Sethian. It has become popular in many disciplines, such as processing, computer graphics, computational geometry, optimization. A number of level set data structures have developed to facilitate the use of the level set method in computer applications
Level set method
–
An illustration of the level set method
Level set method
72.
Ordinary differential equations
–
In mathematics, an ordinary differential equation is a differential equation containing one or more functions of one independent variable and its derivatives. The term ordinary is used in contrast with the partial differential equation which may be with respect to more than one independent variable. ODEs that are linear equations have exact closed-form solutions that can be added and multiplied by coefficients. Graphical and numerical methods, applied by hand or by computer, may approximate solutions of ODEs and perhaps yield useful information, often sufficing in the absence of exact, Ordinary differential equations arise in many contexts of mathematics and science. Mathematical descriptions of change use differentials and derivatives, often, quantities are defined as the rate of change of other quantities, or gradients of quantities, which is how they enter differential equations. Specific mathematical fields include geometry and analytical mechanics, scientific fields include much of physics and astronomy, meteorology, chemistry, biology, ecology and population modelling, economics. Many mathematicians have studied differential equations and contributed to the field, including Newton, Leibniz, the Bernoulli family, Riccati, Clairaut, dAlembert, in general, F is a function of the position x of the particle at time t. The unknown function x appears on both sides of the equation, and is indicated in the notation F. In what follows, let y be a dependent variable and x an independent variable, the notation for differentiation varies depending upon the author and upon which notation is most useful for the task at hand. Given F, a function of x, y, and derivatives of y, then an equation of the form F = y is called an explicit ordinary differential equation of order n. The function r is called the term, leading to two further important classifications, Homogeneous If r =0, and consequently one automatic solution is the trivial solution. The solution of a homogeneous equation is a complementary function. The additional solution to the function is the particular integral. The general solution to an equation can be written as y = yc + yp. Non-linear A differential equation that cannot be written in the form of a linear combination, a number of coupled differential equations form a system of equations. In column vector form, = These are not necessarily linear, the implicit analogue is, F =0 where 0 = is the zero vector. In the same sources, implicit ODE systems with a singular Jacobian are termed differential algebraic equations and this distinction is not merely one of terminology, DAEs have fundamentally different characteristics and are generally more involved to solve than ODE systems. Given a differential equation F =0 a function u, I ⊂ R → R is called the solution or integral curve for F, if u is n-times differentiable on I, and F =0 x ∈ I
Ordinary differential equations
–
Navier–Stokes differential equations used to simulate airflow around an obstruction.
73.
Preconditioner
–
In mathematics, preconditioning is the application of a transformation, called the preconditioner, that conditions a given problem into a form that is more suitable for numerical solving methods. Preconditioning is typically related to reducing a condition number of the problem, the preconditioned problem is then usually solved by an iterative method. In linear algebra and numerical analysis, a preconditioner P of a matrix A is a such that P −1 A has a smaller condition number than A. It is also common to call T = P −1 the preconditioner, rather than P, preconditioned iterative solvers typically outperform direct solvers, e. g. Gaussian elimination, for large, especially for sparse, matrices. Iterative solvers can be used as methods, i. e. become the only choice if the coefficient matrix A is not stored explicitly. Alternatively, one may solve the left preconditioned system, P −1 =0, both of systems give the same solution as the original system so long as the preconditioner matrix P is nonsingular. The left preconditioning is more common, the goal of this preconditioned system is to reduce the condition number of the left or right preconditioned system matrix P −1 A or A P −1, respectively. The preconditioned matrix P −1 A or A P −1 is almost never explicitly formed, only the action of applying the preconditioner solve operation P −1 to a given vector need to be computed in iterative methods. Typically there is a trade-off in the choice of P, since the operator P −1 must be applied at each step of the iterative linear solver, it should have a small cost of applying the P −1 operation. The cheapest preconditioner would therefore be P = I since then P −1 = I, clearly, this results in the original linear system and the preconditioner does nothing. One therefore chooses P as somewhere between two extremes, in an attempt to achieve a minimal number of linear iterations while keeping the operator P −1 as simple as possible. Some examples of typical preconditioning approaches are detailed below, preconditioned iterative methods for A x − b =0 are, in most cases, mathematically equivalent to standard iterative methods applied to the preconditioned system P −1 =0. For example, the standard Richardson iteration for solving A x − b =0 is x n +1 = x n − γ n, n ≥0. Applied to the preconditioned system P −1 =0, it turns into a preconditioned method x n +1 = x n − γ n P −1, n ≥0. Examples of popular preconditioned iterative methods for linear systems include the preconditioned conjugate gradient method, the biconjugate gradient method, and generalized minimal residual method. Iterative methods, which use scalar products to compute the iterative parameters, for a symmetric positive definite matrix A the preconditioner P is typically chosen to be symmetric positive definite as well. The preconditioned operator P −1 A is then also symmetric positive definite, in this case, the desired effect in applying a preconditioner is to make the quadratic form of the preconditioned operator P −1 A with respect to the P -based scalar product to be nearly spherical. Denoting T = P −1, we highlight that preconditioning is practically implemented as multiplying some vector r by T, i. e. computing the product T r
Preconditioner
–
Illustration of gradient descent
74.
Aorta
–
The aorta is the main artery in the human body, originating from the left ventricle of the heart and extending down to the abdomen, where it splits into two smaller arteries. The aorta distributes oxygenated blood to all parts of the body through the systemic circulation, in anatomical sources, the aorta is usually divided into sections. One way of classifying a part of the aorta is by anatomical compartment, the aorta then continues downward as the abdominal aorta diaphragm to the aortic bifurcation. Another system divides the aorta with respect to its course and the direction of blood flow, in this system, the aorta starts as the ascending aorta then travels superiorly from the heart and then makes a hairpin turn known as the aortic arch. Following the aortic arch, the aorta then travels inferiorly as the descending aorta, the descending aorta has two parts. The aorta begins to descend in the cavity, and consequently is known as the thoracic aorta. After the aorta passes through the diaphragm, it is known as the abdominal aorta, the aorta ends by dividing into two major blood vessels, the common iliac arteries and a smaller midline vessel, the median sacral artery. The ascending aorta begins at the opening of the valve in the left ventricle of the heart. It runs through a common pericardial sheath with the pulmonary trunk and these two blood vessels twist around each other, causing the aorta to start out posterior to the pulmonary trunk, but end by twisting to its right and anterior side. The transition from ascending aorta to aortic arch is at the reflection on the aorta. At the root of the aorta, the lumen has three small pockets between the cusps of the aortic valve and the wall of the aorta, which are called the aortic sinuses or the sinuses of Valsalva. The left aortic sinus contains the origin of the coronary artery. Together, these two arteries supply the heart, the posterior aortic sinus does not give rise to a coronary artery. For this reason the left, right and posterior aortic sinuses are also called left-coronary, right-coronary and non-coronary sinuses, in addition to these blood vessels, the aortic arch crosses the left main bronchus. Between the aortic arch and the trunk is a network of autonomic nerve fibers. The left vagus nerve, which passes anterior to the arch, gives off a major branch, the recurrent laryngeal nerve. It then runs back to the neck, the aortic arch has three major branches, from proximal to distal, they are the brachiocephalic trunk, the left common carotid artery, and the left subclavian artery. The brachiocephalic trunk supplies the right side of the head and neck as well as the arm and chest wall
Aorta
–
A pig's aorta cut open showing also some leaving arteries.
Aorta
–
Course of the aorta in the thorax (anterior view), starting posterior to the
main pulmonary artery, then anterior to the right
pulmonary arteries, the
trachea and the
esophagus, then turning posteriorly to course dorsally to these structures.
Aorta
–
Major Aorta anatomy displaying Ascending Aorta, Brachiocephalic trunk, Left Common Carotid Artery, Left Subclavian Artery, Aortic Isthmus, Aortic Arch and Descending Thoracic Aorta
75.
Blade element theory
–
Blade element theory is a mathematical process originally designed by William Froude, David W. Taylor and Stefan Drzewiecki to determine the behavior of propellers. It involves breaking a blade down into small parts then determining the forces on each of these small blade elements. These forces are then integrated along the blade and over one rotor revolution in order to obtain the forces. One of the key lies in modelling the induced velocity on the rotor disk. Because of this the blade element theory is combined with the momentum theory to provide additional relationships necessary to describe the induced velocity on the rotor disk. At the most basic level of approximation a uniform induced velocity on the disk is assumed and this approach is sometimes called the Froude-Finsterwalder equation. The most simple forward flight inflow models are first harmonic models, TU Berlin Review paper on forward flight inflow models by Robert Chen, NASA
Blade element theory
–
Velocities and forces for a blade element
76.
Central differencing scheme
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In applied mathematics, the central differencing scheme is a finite difference method. The finite difference method optimizes the approximation for the operator in the central node of the considered patch. The central differencing scheme is one of the used to solve the integrated convection-diffusion equation. The right hand side of the equation which basically highlights the diffusion terms can be represented using central difference approximation. This equation represents flux balance in a control volume, the left hand side gives the net convective flux and the right hand side contains the net diffusive flux and the generation or destruction of the property within the control volume. In the absence of source term equation one becomes d d x = d d x → Equation 2, continuity equation, d d x =0 → Equation 3. When pe is zero φ is spread in all equally and as Pe increases φ at a point largely depends on upstream value. But central differencing scheme does not possess Transportiveness at higher pe since Φ at a point is average of neighbouring nodes for all Pe, the Taylor series truncation error of the central differencing scheme is second order. Central differencing scheme will be only if Pe <2. Owing to this limitation central differencing is not a suitable discretisation practice for general purpose flow calculations, central difference type schemes are currently being applied on a regular basis in the solution of the Euler equations and Navier–Stokes equations. The results using central differencing approximation have demonstrated improvements in accuracy in smooth regions. The central difference schemes have a parameter in conjunction with the fourth-difference dissipation. This dissipation is needed to approach a steady state and this scheme is more accurate than the first order upwind scheme if Peclet number is less than 2. The central differencing scheme is somewhat more dissipative and this scheme leads to oscillations in the solution or divergence if the local Peclet number is larger than 2. Anderson, ISBN 0-07-001685-2 Computational Fluid Dynamics volume 1 – Klaus A. Hoffmann, Steve T
Central differencing scheme
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Figure 1.Comparison of different schemes
77.
Finite element analysis
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The finite element method is a numerical method for solving problems of engineering and mathematical physics. It is also referred to as finite element analysis, typical problem areas of interest include structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. The analytical solution of these problems generally require the solution to boundary value problems for partial differential equations, the finite element method formulation of the problem results in a system of algebraic equations. The method yields approximate values of the unknowns at discrete number of points over the domain, to solve the problem, it subdivides a large problem into smaller, simpler parts that are called finite elements. The simple equations that model these finite elements are assembled into a larger system of equations that models the entire problem. FEM then uses variational methods from the calculus of variations to approximate a solution by minimizing an associated error function, the global system of equations has known solution techniques, and can be calculated from the initial values of the original problem to obtain a numerical answer. To explain the approximation in this process, FEM is commonly introduced as a case of Galerkin method. The process, in language, is to construct an integral of the inner product of the residual. In simple terms, it is a procedure that minimizes the error of approximation by fitting trial functions into the PDE, the residual is the error caused by the trial functions, and the weight functions are polynomial approximation functions that project the residual. These equation sets are the element equations and they are linear if the underlying PDE is linear, and vice versa. In step above, a system of equations is generated from the element equations through a transformation of coordinates from the subdomains local nodes to the domains global nodes. This spatial transformation includes appropriate orientation adjustments as applied in relation to the coordinate system. The process is carried out by FEM software using coordinate data generated from the subdomains. FEM is best understood from its application, known as finite element analysis. FEA as applied in engineering is a tool for performing engineering analysis. It includes the use of mesh generation techniques for dividing a complex problem into small elements, FEA is a good choice for analyzing problems over complicated domains, when the domain changes, when the desired precision varies over the entire domain, or when the solution lacks smoothness. For instance, in a crash simulation it is possible to increase prediction accuracy in important areas like the front of the car. Another example would be in weather prediction, where it is more important to have accurate predictions over developing highly nonlinear phenomena rather than relatively calm areas
Finite element analysis
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Visualization of how a car deforms in an asymmetrical crash using finite element analysis. [1]
Finite element analysis
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Navier–Stokes differential equations used to simulate airflow around an obstruction.
78.
Shape optimization
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Shape optimization is part of the field of optimal control theory. The typical problem is to find the shape which is optimal in that it minimizes a certain cost functional while satisfying given constraints, in many cases, the functional being solved depends on the solution of a given partial differential equation defined on the variable domain. Topology optimization is, in addition, concerned with the number of connected components/boundaries belonging to the domain, such methods are needed since typically shape optimization methods work in a subset of allowable shapes which have fixed topological properties, such as having a fixed number of holes in them. Topological optimization techniques can then help work around the limitations of pure shape optimization, mathematically, shape optimization can be posed as the problem of finding a bounded set Ω, minimizing a functional F, possibly subject to a constraint of the form G =0. Sometimes additional constraints need to be imposed to that end to ensure well-posedness of the problem, Shape optimization is an infinite-dimensional optimization problem. Furthermore, the space of allowable shapes over which the optimization is performed does not admit a vector space structure, Shape optimization problems are usually solved numerically, by using iterative methods. That is, one starts with a guess for a shape. To solve a shape optimization problem, one needs to find a way to represent a shape in the computer memory, one approach is to follow the boundary of the shape. For that, one can sample the shape boundary in a dense and uniform manner. Then, one can evolve the shape by gradually moving the boundary points and this is called the Lagrangian approach. Another approach is to consider a function defined on a box around the shape, which is positive inside of the shape, zero on the boundary of the shape. One can then evolve this function instead of the shape itself, one can consider a rectangular grid on the box and sample the function at the grid points. As the shape evolves, the points do not change. This approach, of using a grid, is called the Eulerian approach. The idea of using a function to represent the shape is at the basis of the level set method, a third approach is to think of the shape evolution as of a flow problem. Mathematically, if Ω0 is the shape, and Ω t is the shape at time t, one considers the diffeomorphisms f t, Ω0 → Ω t. The idea is again that shapes are difficult entities to be dealt with directly, then the Gâteaux or shape derivative of F at Ω0 with respect to the shape is the limit of d F = lim s →0 F − F s if this limit exists. This gives an idea of gradient descent, where the boundary ∂ Ω is evolved in the direction of negative shape gradient in order to reduce the value of the cost functional
Shape optimization
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Example: Shape optimization as applied to building geometry. Example provided courtesy of Formsolver.com
Shape optimization
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Example: Optimization shape families resulting from differing goal parameters. Example provided courtesy of Formsolver.com
79.
Dortmund University of Technology
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TU Dortmund University is a university in Dortmund, North Rhine-Westphalia, Germany with over 30,000 students, and over 3,000 staff. It is situated in the Ruhr area, the fourth largest urban area in Europe, the university is highly ranked in terms of its research performance in the areas of physics, electrical engineering, chemistry and economics. The University of Dortmund was founded in 1968, during the decline of the coal and its establishment was seen as an important move in the economic change from heavy industry to technology. The universitys main areas of research are the sciences, engineering, pedagogy/teacher training in a wide spectrum of subjects, special education. The University of Dortmund was originally designed to be a technical university, in 2006, The University of Dortmund hosted the 11th Federation of International Robot-soccer Association RoboWorld Cup. The universitys robot soccer team, the Dortmund Droids, became world champion in the RoboWorld Cup 2002. Following the Zeitgeist of the late 1960s in Germany, the university was built auf der grünen Wiese about 2 miles outside of downtown Dortmund, one of the most prominent buildings in the university is the Mathetower, which houses the faculty of Mathematics. The first point of registration for. de-domains was at the Dortmund University Department of Computer Science, former president of Germany, Johannes Rau was awarded an honorary degree from the university in 2004. Carl Djerassi was awarded an honorary degree for his science-in-fiction in 2009. ESDP-Network ConRuhr Official homepage of the TU Dortmund University
Dortmund University of Technology
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Dortmund University's Mathetower
Dortmund University of Technology
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Official logo of the TU Dortmund University
Dortmund University of Technology
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Student hostels
Dortmund University of Technology
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Campus Food Court
80.
National Diet Library
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The National Diet Library is the only national library in Japan. It was established in 1948 for the purpose of assisting members of the National Diet of Japan in researching matters of public policy, the library is similar in purpose and scope to the United States Library of Congress. The National Diet Library consists of two facilities in Tokyo and Kyoto, and several other branch libraries throughout Japan. The Diets power in prewar Japan was limited, and its need for information was correspondingly small, the original Diet libraries never developed either the collections or the services which might have made them vital adjuncts of genuinely responsible legislative activity. Until Japans defeat, moreover, the executive had controlled all political documents, depriving the people and the Diet of access to vital information. The U. S. occupation forces under General Douglas MacArthur deemed reform of the Diet library system to be an important part of the democratization of Japan after its defeat in World War II. In 1946, each house of the Diet formed its own National Diet Library Standing Committee, hani Gorō, a Marxist historian who had been imprisoned during the war for thought crimes and had been elected to the House of Councillors after the war, spearheaded the reform efforts. Hani envisioned the new body as both a citadel of popular sovereignty, and the means of realizing a peaceful revolution, the National Diet Library opened in June 1948 in the present-day State Guest-House with an initial collection of 100,000 volumes. The first Librarian of the Diet Library was the politician Tokujirō Kanamori, the philosopher Masakazu Nakai served as the first Vice Librarian. In 1949, the NDL merged with the National Library and became the national library in Japan. At this time the collection gained a million volumes previously housed in the former National Library in Ueno. In 1961, the NDL opened at its present location in Nagatachō, in 1986, the NDLs Annex was completed to accommodate a combined total of 12 million books and periodicals. The Kansai-kan, which opened in October 2002 in the Kansai Science City, has a collection of 6 million items, in May 2002, the NDL opened a new branch, the International Library of Childrens Literature, in the former building of the Imperial Library in Ueno. This branch contains some 400,000 items of literature from around the world. Though the NDLs original mandate was to be a library for the National Diet. In the fiscal year ending March 2004, for example, the library reported more than 250,000 reference inquiries, in contrast, as Japans national library, the NDL collects copies of all publications published in Japan. The NDL has an extensive collection of some 30 million pages of documents relating to the Occupation of Japan after World War II. This collection include the documents prepared by General Headquarters and the Supreme Commander of the Allied Powers, the Far Eastern Commission, the NDL maintains a collection of some 530,000 books and booklets and 2 million microform titles relating to the sciences
National Diet Library
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Tokyo Main Library of the National Diet Library
National Diet Library
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Kansai-kan of the National Diet Library
National Diet Library
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The National Diet Library
National Diet Library
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Main building in Tokyo