1.
Computational physics
–
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
–
Computational physics
2.
Numerical analysis
–
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
–
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.
Computer simulation
–
Computer simulations reproduce the behavior of a system using a model. Simulation of a system is represented as the running of the systems model and it can be used to explore and gain new insights into new technology and to estimate the performance of systems too complex for analytical solutions. The scale of events being simulated by computer simulations has far exceeded anything possible using traditional paper-and-pencil mathematical modeling, other examples include a 1-billion-atom model of material deformation, a 2. Because of the computational cost of simulation, computer experiments are used to perform such as uncertainty quantification. A computer model is the algorithms and equations used to capture the behavior of the system being modeled, by contrast, computer simulation is the actual running of the program that contains these equations or algorithms. Simulation, therefore, is the process of running a model, thus one would not build a simulation, instead, one would build a model, and then either run the model or equivalently run a simulation. It was a simulation of 12 hard spheres using a Monte Carlo algorithm, Computer simulation is often used as an adjunct to, or substitute for, modeling systems for which simple closed form analytic solutions are not possible. The external data requirements of simulations and models vary widely, for some, the input might be just a few numbers, while others might require terabytes of information. Because of this variety, and because diverse simulation systems have common elements. Systems that accept data from external sources must be careful in knowing what they are receiving. While it is easy for computers to read in values from text or binary files, often they are expressed as error bars, a minimum and maximum deviation from the value range within which the true value lie. Even small errors in the data can accumulate into substantial error later in the simulation. While all computer analysis is subject to the GIGO restriction, this is true of digital simulation. Indeed, observation of this inherent, cumulative error in digital systems was the main catalyst for the development of chaos theory, another way of categorizing models is to look at the underlying data structures. For time-stepped simulations, there are two classes, Simulations which store their data in regular grids and require only next-neighbor access are called stencil codes. Many CFD applications belong to this category, if the underlying graph is not a regular grid, the model may belong to the meshfree method class. Equations define the relationships between elements of the system and attempt to find a state in which the system is in equilibrium. Such models are used in simulating physical systems, as a simpler modeling case before dynamic simulation is attempted
Computer simulation
–
Computer simulation of the process of
osmosis
Computer simulation
–
A 48-hour computer simulation of
Typhoon Mawar using the
Weather Research and Forecasting model
4.
Scientific visualization
–
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
–
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
–
Chemical imaging of a simultaneous release of SF 6 and NH 3.
5.
Morse/Long-range potential
–
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
6.
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.
7.
Yukawa potential
–
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.
8.
Morse potential
–
The Morse potential, named after physicist Philip M. Morse, is a convenient interatomic interaction model for the potential energy of a diatomic molecule. It is an approximation for the vibrational structure of the molecule than the QHO because it explicitly includes the effects of bond breaking. It also accounts for the anharmonicity of real bonds and the transition probability for overtone. The Morse potential can also be used to other interactions such as the interaction between an atom and a surface. Due to its simplicity, it is not used in modern spectroscopy, however, its mathematical form inspired the MLR potential, which is the most popular potential energy function used for fitting spectroscopic data. The dissociation energy of the bond can be calculated by subtracting the zero point energy E from the depth of the well. Since the zero of energy is arbitrary, the equation for the Morse potential can be rewritten any number of ways by adding or subtracting a constant value. This form approaches zero at r and equals − D e at its minimum. It clearly shows that the Morse potential is the combination of a short-range repulsion term, like the quantum harmonic oscillator, the energies and eigenstates of the Morse potential can be found using operator methods. One approach involves applying the method to the Hamiltonian. Whereas the energy spacing between levels in the quantum harmonic oscillator is constant at h ν0, the energy between adjacent levels decreases with increasing v in the Morse oscillator. Mathematically, the spacing of Morse levels is E − E = h ν0 −2 /2 D e and this trend matches the anharmonicity found in real molecules. However, this equation fails above some value of v m where E − E is calculated to be zero or negative, specifically, v m =2 D e − h ν0 h ν0. This failure is due to the number of bound levels in the Morse potential. For energies above v m, all the energy levels are allowed. Below v m, E is an approximation for the true vibrational structure in non-rotating diatomic molecules. An important extension of the Morse potential that made the Morse form very useful for spectroscopy is the MLR potential. The MLR potential is used as a standard for representing spectroscopic and/or virial data of diatomic molecules by a potential energy curve
Morse potential
–
Computational physics
9.
Finite difference method
–
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
–
Navier–Stokes differential equations used to simulate airflow around an obstruction.
10.
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.
11.
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.
12.
Riemann solver
–
A Riemann solver is a numerical method used to solve a Riemann problem. They are heavily used in fluid dynamics and computational magnetohydrodynamics. Godunov is credited with introducing the first exact Riemann solver for the Euler equations, modern solvers are able to simulate relativistic effects and magnetic fields. For the hydrodynamic case latest research showed the possibility to avoid the iterations to calculate the exact solution for the Euler equations. As iterative solutions are too costly, especially in Magnetohydrodynamics, some approximations have to be made, the most popular solvers are, Roe used the linearisation of the Jacobian, which he then solves exactly. The HLLE solver is a solution to the Riemann problem, which is only based on the integral form of the conservation laws. The stability and robustness of the HLLE solver is closely related to the signal velocities, the description of the HLLE scheme in the book mentioned below is incomplete and partially wrong. The reader is referred to the original paper, actually, the HLLE scheme is based on a new stability theory for discontinuities in fluids, which was never published. The HLLC solver was introduced by Toro and it restores the missing Rarefaction wave by some estimates, like linearisations, these can be simple but also more advanced exists like using the Roe average velocity for the middle wave speed. They are quite robust and efficient but somewhat more diffusive and these solvers were introduced by Nishikawa and Kitamura, in order to overcome the carbuncle problems of the Roe solver and the excessive diffusion of the HLLE solver at the same time. In particular, the one derived from the Roe and HLLE solvers, called Rotated-RHLL solver, is extremely robust, godunovs scheme Computational fluid dynamics Computational magnetohydrodynamics Toro, Eleuterio F. Riemann Solvers and Numerical Methods for Fluid Dynamics, Berlin, Springer Verlag, ISBN 3-540-65966-8
Riemann solver
–
Computational physics
13.
Smoothed-particle hydrodynamics
–
Smoothed-particle hydrodynamics is a computational method used for simulating the dynamics of continuum media, such as solid mechanics and fluid flows. It was developed by Gingold and Monaghan and Lucy initially for astrophysical problems and it has been used in many fields of research, including astrophysics, ballistics, volcanology, and oceanography. It is a mesh-free Lagrangian method, and the resolution of the method can easily be adjusted with respect to such as the density. The smoothed-particle hydrodynamics method works by dividing the fluid into a set of discrete elements and these particles have a spatial distance, over which their properties are smoothed by a kernel function. This means that the quantity of any particle can be obtained by summing the relevant properties of all the particles which lie within the range of the kernel. For example, using Monaghans popular cubic spline kernel the temperature at position r depends on the temperatures of all the particles within a radial distance 2 h of r. The contributions of each particle to a property are weighted according to their distance from the particle of interest, mathematically, this is governed by the kernel function. Kernel functions commonly used include the Gaussian function and the cubic spline, the latter function is exactly zero for particles further away than two smoothing lengths. This has the advantage of saving computational effort by not including the relatively minor contributions from distant particles, similarly, the spatial derivative of a quantity can be obtained easily by virtue of the linearity of the derivative. ∇ A = ∑ j m j A j ρ j ∇ W, although the size of the smoothing length can be fixed in both space and time, this does not take advantage of the full power of SPH. By assigning each particle its own smoothing length and allowing it to vary with time, for example, in a very dense region where many particles are close together the smoothing length can be made relatively short, yielding high spatial resolution. Conversely, in low-density regions where particles are far apart and the resolution is low. Combined with an equation of state and an integrator, SPH can simulate hydrodynamic flows efficiently, however, the traditional artificial viscosity formulation used in SPH tends to smear out shocks and contact discontinuities to a much greater extent than state-of-the-art grid-based schemes. The Lagrangian-based adaptivity of SPH is analogous to the adaptivity present in grid-based adaptive mesh refinement codes, in some ways it is actually simpler because SPH particles lack any explicit topology relating them, unlike the elements in FEM. Adaptivity in SPH can be introduced in two ways, either by changing the particle smoothing lengths or by splitting SPH particles into daughter particles with smaller smoothing lengths, the first method is common in astrophysical simulations where the particles naturally evolve into states with large density differences. However, in hydrodynamics simulations where the density is constant this is not a suitable method for adaptivity. For this reason particle splitting can be employed, with conditions for splitting ranging from distance to a free surface through to material shear. Often in astrophysics, one wishes to model self-gravity in addition to pure hydrodynamics, the particle-based nature of SPH makes it ideal to combine with a particle-based gravity solver, for instance tree gravity code, particle mesh, or particle-particle particle-mesh
Smoothed-particle hydrodynamics
–
Fig. SPH simulation of ocean waves using FLUIDS v.1 (Hoetzlein)
14.
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
15.
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.
16.
N-body simulation
–
In physics and astronomy, an N-body simulation is a simulation of a dynamical system of particles, usually under the influence of physical forces, such as gravity. In physical cosmology, N-body simulations are used to study processes of non-linear structure formation such as galaxy filaments, direct N-body simulations are used to study the dynamical evolution of star clusters. The particles treated by the simulation may or may not correspond to objects which are particulate in nature. For example, an N-body simulation of a star cluster might have a particle per star and this quantity need not have any physical significance, but must be chosen as a compromise between accuracy and manageable computer requirements. These calculations are used in situations where interactions between objects, such as stars or planets, are important to the evolution of the system. The first direct N-body simulations were carried out by Sebastian von Hoerner at the Astronomisches Rechen-Institut in Heidelberg, regularization is a mathematical trick to remove the singularity in the Newtonian law of gravitation for two particles which approach each other arbitrarily close. Sverre Aarseths codes are used to study the dynamics of star clusters, planetary systems, many simulations are large enough that the effects of general relativity in establishing a Friedmann-Lemaitre-Robertson-Walker cosmology are significant. This is incorporated in the simulation as a measure of distance in a comoving coordinate system. The boundary conditions of these simulations are usually periodic, so that one edge of the simulation volume matches up with the opposite edge. N-body simulations are simple in principle, because they merely involve integrating the 6N ordinary differential equations defining the particle motions in Newtonian gravity, therefore, a number of refinements are commonly used. There are two basic approximation schemes to decrease the time for such simulations. These can reduce the complexity to O or better, at the loss of accuracy. This can dramatically reduce the number of particle pair interactions that must be computed, for simulations where particles are not evenly distributed, the well-separated pair decomposition methods of Callahan and Kosaraju yield optimal O time per iteration with fixed dimension. The gravitational field can now be found by multiplying by k →, since this method is limited by the mesh size, in practice a smaller mesh or some other technique is used to compute the small-scale forces. Sometimes an adaptive mesh is used, in which the cells are much smaller in the denser regions of the simulation. Several different gravitational perturbation algorithms are used to get accurate estimates of the path of objects in the solar system. People often decide to put a satellite in a frozen orbit and it is possible to find a frozen orbit without calculating the actual path of the satellite. Some characteristics of the paths of a system of particles can be calculated directly
N-body simulation
17.
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.
18.
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
19.
Stanislaw Ulam
–
Stanisław Marcin Ulam was a Polish-American mathematician. In pure and applied mathematics, he proved some theorems and proposed several conjectures, born into a wealthy Polish Jewish family, Ulam studied mathematics at the Lwów Polytechnic Institute, where he earned his PhD in 1933 under the supervision of Kazimierz Kuratowski. In 1935, John von Neumann, whom Ulam had met in Warsaw, invited him to come to the Institute for Advanced Study in Princeton, New Jersey, for a few months. From 1936 to 1939, he spent summers in Poland and academic years at Harvard University in Cambridge, Massachusetts, on 20 August 1939, he sailed for America for the last time with his 17-year-old brother Adam Ulam. He became an assistant professor at the University of Wisconsin–Madison in 1940, in October 1943, he received an invitation from Hans Bethe to join the Manhattan Project at the secret Los Alamos Laboratory in New Mexico. There, he worked on the calculations to predict the behavior of the explosive lenses that were needed by an implosion-type weapon. He was assigned to Edward Tellers group, where he worked on Tellers Super bomb for Teller, after the war he left to become an associate professor at the University of Southern California, but returned to Los Alamos in 1946 to work on thermonuclear weapons. With the aid of a cadre of female computers, including his wife Françoise Aron Ulam, in January 1951, Ulam and Teller came up with the Teller–Ulam design, which is the basis for all thermonuclear weapons. With Fermi and John Pasta, Ulam studied the Fermi–Pasta–Ulam problem, Ulam was born in Lemberg, Galicia, on 13 April 1909. At this time, Galicia was in the Kingdom of Galicia and Lodomeria of the Austro-Hungarian Empire, in 1918, it became part of the newly restored Poland, the Second Polish Republic, and the city took its Polish name again, Lwów. The Ulams were a wealthy Polish Jewish family of bankers, industrialists, Ulams immediate family was well-to-do but hardly rich. His father, Józef Ulam, was born in Lwów and was a lawyer and his uncle, Michał Ulam, was an architect, building contractor, and lumber industrialist. From 1916 until 1918, Józefs family lived temporarily in Vienna, after they returned, Lwów became the epicenter of the Polish–Ukrainian War, during which the city experienced a Ukrainian siege. In 1919, Ulam entered Lwów Gymnasium Nr, VII, from which he graduated in 1927. He then studied mathematics at the Lwów Polytechnic Institute, under the supervision of Kazimierz Kuratowski, he received his Master of Arts degree in 1932, and became a Doctor of Science in 1933. At the age of 20, in 1929, he published his first paper Concerning Function of Sets in the journal Fundamenta Mathematicae. From 1931 until 1935, he traveled to and studied in Wilno, Vienna, Zurich, Paris, and Cambridge, England, along with Stanisław Mazur, Mark Kac, Włodzimierz Stożek, Kuratowski, and others, Ulam was a member of the Lwów School of Mathematics. Its founders were Hugo Steinhaus and Stefan Banach, who were professors at the University of Lwów, mathematicians of this school met for long hours at the Scottish Café, where the problems they discussed were collected in the Scottish Book, a thick notebook provided by Banachs wife
Stanislaw Ulam
–
Stanisław Ulam
Stanislaw Ulam
–
The
Scottish Café 's building now houses the Universal Bank in
Lviv, the present name of Lwów.
Stanislaw Ulam
–
Stan Ulam Holding the
FERMIAC
Stanislaw Ulam
–
Ivy Mike, the first full test of the Teller–Ulam design (a
staged fusion bomb), with a
yield of 10.4 megatons on 1 November 1952
20.
John von Neumann
–
John von Neumann was a Hungarian-American mathematician, physicist, inventor, computer scientist, and polymath. He made major contributions to a number of fields, including mathematics, physics, economics, computing, and statistics. He published over 150 papers in his life, about 60 in pure mathematics,20 in physics, and 60 in applied mathematics and his last work, an unfinished manuscript written while in the hospital, was later published in book form as The Computer and the Brain. His analysis of the structure of self-replication preceded the discovery of the structure of DNA, also, my work on various forms of operator theory, Berlin 1930 and Princeton 1935–1939, on the ergodic theorem, Princeton, 1931–1932. During World War II he worked on the Manhattan Project, developing the mathematical models behind the lenses used in the implosion-type nuclear weapon. After the war, he served on the General Advisory Committee of the United States Atomic Energy Commission, along with theoretical physicist Edward Teller, mathematician Stanislaw Ulam, and others, he worked out key steps in the nuclear physics involved in thermonuclear reactions and the hydrogen bomb. Von Neumann was born Neumann János Lajos to a wealthy, acculturated, Von Neumanns place of birth was Budapest in the Kingdom of Hungary which was then part of the Austro-Hungarian Empire. He was the eldest of three children and he had two younger brothers, Michael, born in 1907, and Nicholas, who was born in 1911. His father, Neumann Miksa was a banker, who held a doctorate in law and he had moved to Budapest from Pécs at the end of the 1880s. Miksas father and grandfather were both born in Ond, Zemplén County, northern Hungary, johns mother was Kann Margit, her parents were Jakab Kann and Katalin Meisels. Three generations of the Kann family lived in apartments above the Kann-Heller offices in Budapest. In 1913, his father was elevated to the nobility for his service to the Austro-Hungarian Empire by Emperor Franz Joseph, the Neumann family thus acquired the hereditary appellation Margittai, meaning of Marghita. The family had no connection with the town, the appellation was chosen in reference to Margaret, Neumann János became Margittai Neumann János, which he later changed to the German Johann von Neumann. Von Neumann was a child prodigy, as a 6 year old, he could multiply and divide two 8-digit numbers in his head, and could converse in Ancient Greek. When he once caught his mother staring aimlessly, the 6 year old von Neumann asked her, formal schooling did not start in Hungary until the age of ten. Instead, governesses taught von Neumann, his brothers and his cousins, Max believed that knowledge of languages other than Hungarian was essential, so the children were tutored in English, French, German and Italian. A copy was contained in a private library Max purchased, One of the rooms in the apartment was converted into a library and reading room, with bookshelves from ceiling to floor. Von Neumann entered the Lutheran Fasori Evangelikus Gimnázium in 1911 and this was one of the best schools in Budapest, part of a brilliant education system designed for the elite
John von Neumann
–
Excerpt from the university calendars for 1928 and 1928–1929 of the
Friedrich-Wilhelms-Universität Berlin announcing Neumann's lectures on axiomatic set theory and logics, problems in quantum mechanics and special mathematical functions. Notable colleagues were
Georg Feigl,
Issai Schur,
Erhard Schmidt,
Leó Szilárd,
Heinz Hopf, Adolf Hammerstein and
Ludwig Bieberbach.
John von Neumann
–
John von Neumann in the 1940s
John von Neumann
–
Julian Bigelow, Herman Goldstine, J. Robert Oppenheimer and John von Neumann at the Princeton Institute for Advanced Study.
John von Neumann
–
Von Neumann's gravestone
21.
Boris Galerkin
–
Boris Grigoryevich Galerkin, born in Polotsk, Vitebsk Governorate, Russian Empire, was a Jewish Soviet mathematician and an engineer. Galerkin was born on March 41871 in Polotsk, Vitebsk Governorate, Russian Empire, now part of Belarus, to Girsh-Shleym Galerkin and Perla Basia Galerkina. His parents owned a house in the town, but the homecraft they made did not bring money, so at the age of 12. He had finished school in Polotsk, but still needed the exams from a year which granted him the right to continue education at a higher level. He passed those in Minsk in 1893, as an external student, the same year he was enrolled at the St. Petersburg Technological Institute, at the mechanics department. Due to the lack of funds Boris Grigoryevich had to combine studying at the institute with working as a draftsman, in some point of his life, he married Revekka Treivas, a second niece. They did not have any children, like many other students and technologists, he was involved in political activities, and joined the social-democratic group. In 1899, the year of graduating from the institute, he became a member of the Russian Social-Democratic Party and this provides a plausible explanation for his frequent job changes. From the end of 1903 he was an engineer on the construction of the China Far East Railway, half a year later he became the head at the North mechanical. He participated in organizing the Union of Engineers in St. Petersburg and, in 1906, Boris Grigoryevich became a member of the Social-Democratic Partys St. Petersburg Committee and did not work anywhere else. In prison, known as Kresty, Boris Grigoryevich lost interest to revolutionary activities and devoted himself to science, prison conditions of that time were giving such opportunities. And what is more, in his work-book it is written that Boris Grigoryevich worked as an engineer at designing and constructing the power plant from 1907. This fact was not explained, and Boris Grigoryevich did not like to others about his revolutionary youth. Later, in Soviet questionnaires he would not give clear answers on the persistent questions about membership in different parties. Of course, he was familiar with the fate of old Party members, Galerkins life could become the price if this fact became known to the public. Same year his first scientific work was published by the institutes Transactions, the article was titled A theory of longitudinal curving and an experience of longitudinal curving theory application to many-storied frames, frames with rigid junctions and frame systems. The length of the title was indicative of the length of the work itself,130 pages and it was written in the Kresty prison. In the summer of 1909 Boris Grigoryevich had a trip abroad to see constructions, during the next four years, i. e. before World War I, he and many other institute staff visited Europe to stimulate their scientific interests
Boris Galerkin
–
Boris Galerkin
22.
Edward Norton Lorenz
–
Edward Norton Lorenz was an American mathematician, meteorologist, and a pioneer of chaos theory, along with Mary Cartwright. He introduced the strange attractor notion and coined the term butterfly effect, Lorenz was born in West Hartford, Connecticut. He studied mathematics at both Dartmouth College in New Hampshire and Harvard University in Cambridge, Massachusetts, from 1942 until 1946, he served as a meteorologist for the United States Army Air Corps. After his return from World War II, he decided to study meteorology, Lorenz earned two degrees in the area from the Massachusetts Institute of Technology where he later was a professor for many years. He was a Professor Emeritus at MIT from 1987 until his death, during the 1950s, Lorenz became skeptical of the appropriateness of the linear statistical models in meteorology, as most atmospheric phenomena involved in weather forecasting are non-linear. His work on the topic culminated in the publication of his 1963 paper Deterministic Nonperiodic Flow in Journal of the Atmospheric Sciences, and with it and his description of the butterfly effect followed in 1969. He was awarded the Kyoto Prize for basic sciences, in the field of earth and planetary sciences, in 1991, the Buys Ballot Award in 2004, in his later years, Lorenz lived in Cambridge, Massachusetts. He was an outdoorsman, who enjoyed hiking, climbing. He kept up with these pursuits until very late in his life, according to his daughter, Cheryl Lorenz, Lorenz had finished a paper a week ago with a colleague. On April 16,2008, Lorenz died at his home in Cambridge at the age of 90,1969 Carl-Gustaf Rossby Research Medal, American Meteorological Society. 1973 Symons Gold Medal, Royal Meteorological Society,1975 Fellow, National Academy of Sciences. 1981 Member, Norwegian Academy of Science and Letters,1983 Crafoord Prize, Royal Swedish Academy of Sciences. 1984 Honorary Member, Royal Meteorological Society,2004 Lomonosov Gold Medal Lorenz built a mathematical model of the way air moves around in the atmosphere. As Lorenz studied weather patterns he began to realize that the weather patterns did not always behave as predicted, minute variations in the initial values of variables in his twelve-variable computer weather model would result in grossly divergent weather patterns. This sensitive dependence on conditions, which came to be known as the butterfly effect. Lorenz published several books and articles, a selection,1955 Available potential energy and the maintenance of the general circulation. 1967 The nature and theory of the circulation of atmosphere. No.218 Three approaches to atmospheric predictability, bulletin of the American Meteorological Society
Edward Norton Lorenz
–
Edward Norton Lorenz
23.
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.
24.
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.
25.
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
26.
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
27.
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
28.
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
29.
Turbulence
–
Turbulence or turbulent flow is a flow regime in fluid dynamics characterized by chaotic changes in pressure and flow velocity. It is in contrast to a flow regime, which occurs when a fluid flows in parallel layers. Turbulence is caused by kinetic energy in parts of a fluid flow. For this reason turbulence is easier to create in low viscosity fluids, in general terms, in turbulent flow, unsteady vortices appear of many sizes which interact with each other, consequently drag due to friction effects increases. This would increase the energy needed to pump fluid through a pipe, however this effect can also be exploited by such as aerodynamic spoilers on aircraft, which deliberately spoil the laminar flow to increase drag and reduce lift. The onset of turbulence can be predicted by a constant called the Reynolds number. However, turbulence has long resisted detailed physical analysis, and the interactions within turbulence creates a complex situation. Richard Feynman has described turbulence as the most important unsolved problem of classical physics, smoke rising from a cigarette is mostly turbulent flow. However, for the first few centimeters the flow is laminar, the smoke plume becomes turbulent as its Reynolds number increases, due to its flow velocity and characteristic length increasing. If the golf ball were smooth, the boundary layer flow over the front of the sphere would be laminar at typical conditions. However, the layer would separate early, as the pressure gradient switched from favorable to unfavorable. To prevent this happening, the surface is dimpled to perturb the boundary layer. This results in higher skin friction, but moves the point of boundary layer separation further along, resulting in form drag. The flow conditions in industrial equipment and machines. The external flow over all kind of such as cars, airplanes, ships. The motions of matter in stellar atmospheres, a jet exhausting from a nozzle into a quiescent fluid. As the flow emerges into this external fluid, shear layers originating at the lips of the nozzle are created and these layers separate the fast moving jet from the external fluid, and at a certain critical Reynolds number they become unstable and break down to turbulence. Biologically generated turbulence resulting from swimming animals affects ocean mixing, snow fences work by inducing turbulence in the wind, forcing it to drop much of its snow load near the fence
Turbulence
–
Flow visualization of a turbulent jet, made by
laser-induced fluorescence. The jet exhibits a wide range of length scales, an important characteristic of turbulent flows.
Turbulence
–
Laminar and turbulent water flow over the hull of a submarine
Turbulence
–
Turbulence in the
tip vortex from an
airplane wing
30.
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
31.
Flight test
–
The flight test phase can range from the test of a single new system for an existing vehicle to the complete development and certification of a new aircraft, launch vehicle, or reusable spacecraft. Therefore, the duration of a flight test program can vary from a few weeks to many years. There are typically two categories of flight test programs – commercial and military, commercial flight testing is conducted to certify that the aircraft meets all applicable safety and performance requirements of the government certifying agency. Since commercial aircraft development is funded by the aircraft manufacturer and/or private investors. These civil agencies are concerned with the safety and that the pilot’s flight manual accurately reports the aircraft’s performance. The market will determine the suitability to operators. Normally, the certification agency does not get involved in flight testing until the manufacturer has found. Military programs differ from commercial in that the government contracts with the manufacturer to design. These performance requirements are documented to the manufacturer in the aircraft specification, in this case, the government is the customer and has a direct stake in the aircraft’s ability to perform the mission. Since the government is funding the program, it is involved in the aircraft design. Often military test pilots and engineers are integrated as part of the flight test team. The final phase of the aircraft flight test is the Operational Test. OT is conducted by a government-only test team with the dictate to certify that the aircraft is suitable, Flight testing of military aircraft is often conducted at military flight test facilities. The US Navy tests aircraft at Naval Air Station Patuxent River, the U. S. Air Force Test Pilot School and the U. S. Naval Test Pilot School are the programs designed to teach military test personnel. In the UK, most military flight testing is conducted by three organizations, the RAF, BAE Systems and QinetiQ, all launch vehicles, as well as a few reusable spacecraft, must necessarily be designed to deal with aerodynamic flight loads while moving through the atmosphere. Many launch vehicles are flight tested, with more extensive data collection. Reusable spacecraft or reusable booster test programs are more involved. For both commercial and military aircraft, as well as vehicles, flight test preparation begins well before the test vehicle is ready to fly
Flight test
–
Static pressure probe on the nose of a
Sukhoi Superjet 100 prototype
Flight test
Flight test
–
Flight test pressure probes and water tanks in
747-8I prototype
Flight test
–
Static pressure probe rig aboard
Boeing 747-8I prototype; A long tube, rolled up inside the barrel, is connected to a probe which can be deployed far behind the tail of the aircraft
32.
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
33.
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.
34.
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
35.
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.
36.
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.
37.
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:
38.
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.
39.
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.
40.
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
41.
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
42.
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°.
43.
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.
44.
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
45.
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
46.
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.
47.
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.)
48.
Los Alamos National Lab
–
Los Alamos National Laboratory is one of two laboratories in the United States in which classified work towards the design of nuclear weapons has been undertaken. LANL is a United States Department of Energy national laboratory, managed and operated by Los Alamos National Security, located in Los Alamos, the laboratory is one of the largest science and technology institutions in the world. It conducts multidisciplinary research in such as national security, space exploration, renewable energy, medicine, nanotechnology. General Leslie Groves wanted a central laboratory at a location for safety. It should be at least 200 miles from international boundaries and west of the Mississippi, major John Dudley suggested Oak City, Utah or Jemez Springs, New Mexico but both were rejected. Manhattan Project scientific director J. Robert Oppenheimer had spent much time in his youth in the New Mexico area, Dudley had rejected the school as not meeting Groves’ criteria, but as soon as Groves saw it he said in effect This is the place. Oppenheimer became the laboratorys first director, during the Manhattan Project, Los Alamos hosted thousands of employees, including many Nobel Prize-winning scientists. The location was a total secret and its only mailing address was a post office box, number 1663, in Santa Fe, New Mexico. Eventually two other post office boxes were used,180 and 1539, also in Santa Fe, though its contract with the University of California was initially intended to be temporary, the relationship was maintained long after the war. The work of the laboratory culminated in the creation of several devices, one of which was used in the first nuclear test near Alamogordo, New Mexico, codenamed Trinity. The other two were weapons, Little Boy and Fat Man, which were used in the attacks on Hiroshima, the Laboratory received the Army-Navy ‘E’ Award for Excellence in production on October 16,1945. Many of the original Los Alamos luminaries chose to leave the laboratory, in the years since the 1940s, Los Alamos was responsible for the development of the hydrogen bomb, and many other variants of nuclear weapons. In 1952, Lawrence Livermore National Laboratory was founded to act as Los Alamos competitor, Los Alamos and Livermore served as the primary classified laboratories in the U. S. national laboratory system, designing all the countrys nuclear arsenal. Additional work included basic research, particle accelerator development, health physics. Many nuclear tests were undertaken in the Marshall Islands and at the Nevada Test Site, during the late-1950s, a number of scientists including Dr. J. Robert Bob Beyster left Los Alamos to work for General Atomics in San Diego. Three major nuclear-related accidents have occurred at LANL, criticality accidents occurred in August 1945 and May 1946, and a third accident occurred during an annual physical inventory in December 1958. Several buildings associated with the Manhattan Project at Los Alamos were declared a National Historic Landmark in 1965, Los Alamos nuclear work is currently thought to relate primarily to computer simulations and stockpile stewardship. The development of the Dual-Axis Radiographic Hydrodynamic Test Facility will allow complex simulations of nuclear tests to take place without full explosive yields, the lab has made intense efforts for humanitarian causes through its scientific research in medicine
Los Alamos National Lab
–
Aerial view
Los Alamos National Lab
–
Los Alamos National Laboratory
Los Alamos National Lab
–
The first stages of the explosion of the
Trinity nuclear test.
Los Alamos National Lab
–
Sites
49.
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.
50.
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