In fluid dynamics, vortex shedding is an oscillating flow that takes place when a fluid such as air or water flows past a bluff body at certain velocities, depending on the size and shape of the body. In this flow, vortices are created at the back of the body and detach periodically from either side of the body. See Von Kármán vortex street; the fluid flow past the object creates alternating low-pressure vortices on the downstream side of the object. The object will tend to move toward the low-pressure zone. If the bluff structure is not mounted rigidly and the frequency of vortex shedding matches the resonance frequency of the structure the structure can begin to resonate, vibrating with harmonic oscillations driven by the energy of the flow; this vibration is the cause for overhead power line wires "singing in the wind", for the fluttering of automobile whip radio antennas at some speeds. Tall chimneys constructed of thin-walled steel tubes can be sufficiently flexible that, in air flow with a speed in the critical range, vortex shedding can drive the chimney into violent oscillations that can damage or destroy the chimney.
Vortex shedding was one of the causes proposed for the failure of the original Tacoma Narrows Bridge in 1940, but was rejected because the frequency of the vortex shedding did not match that of the bridge. The bridge failed by aeroelastic flutter. A thrill ride, "VertiGo" at Cedar Point in Sandusky, Ohio suffered vortex shedding during the winter of 2001, causing one of the three towers to collapse; the ride was closed for the winter at the time. In northeastern Iran, the Hashemi-Nejad natural gas refinery's flare stacks suffered vortex shedding seven times from 1975 to 2003; some simulation and analyses were done, which revealed that the main cause was the interaction of the pilot flame and flare stack. The problem was solved by removing the pilot; the frequency at which vortex shedding takes place for an infinite cylinder is related to the Strouhal number by the following equation: S t = f D V Where S t is the dimensionless Strouhal number, f is the vortex shedding frequency, D is the diameter of the cylinder, V is the flow velocity.
The Strouhal number depends on the Reynolds number R e, but a value of 0.22 is used. Fairings can be fitted to a structure to streamline the flow past the structure, such as on an aircraft wing. Tall metal smokestacks or other tubular structures such as antenna masts or tethered cables can be fitted with an external corkscrew fin to deliberately introduce turbulence, so the load is less variable and resonant load frequencies have negligible amplitudes; the effectiveness of helical strakes for reducing vortex induced vibration was discovered in 1957 by Christopher Scruton and D. E. J. Walshe at the National Physics Laboratory in Great Britain, they are therefore described as Scruton strakes. For maximum effectiveness each fin or strake should have a height of about 10 percent of the cylinder diameter and a pitch for each fin of 5 times the cylinder diameter. A tuned mass damper can be used to mitigate vortex shedding in chimneys. A Stockbridge damper is used to mitigate aeolian vibrations caused by vortex shedding on overhead power lines.
Aeroelastic flutter - vibration-induced vortices - by way of contrast Vortex Vortex-induced vibration Von Kármán vortex street Flow visualisation of the vortex shedding mechanism on circular cylinder using hydrogen bubbles illuminated by a laser sheet in a water channel. Courtesy of G. R. S. Assi
The Weber number is a dimensionless number in fluid mechanics, useful in analysing fluid flows where there is an interface between two different fluids for multiphase flows with curved surfaces. It is named after Moritz Weber, it can be thought of as a measure of the relative importance of the fluid's inertia compared to its surface tension. The quantity is useful in analyzing the formation of droplets and bubbles; the Weber number may be written as: W e = ρ v 2 l σ where ρ is the density of the fluid. V is its velocity. L is its characteristic length the droplet diameter. Σ is the surface tension. The modified Weber number, W e ∗ = W e 12 equals the ratio of the kinetic energy on impact to the surface energy, W e ∗ = E k i n E s u r f,where E k i n = π ρ l 3 v 2 12 and E s u r f = π l 2 σ. One application of the Weber number is the study of heat pipes; when the momentum flux in the vapor core of the heat pipe is high, there is a possibility that the shear stress exerted on the liquid in the wick can be large enough to entrain droplets into the vapor flow.
The Weber number is the dimensionless parameter that determines the onset of this phenomenon called the entrainment limit. In this case the Weber number is defined as the ratio of the momentum in the vapor layer divided by the surface tension force restraining the liquid, where the characteristic length is the surface pore size. Weast, R. Lide, D. Astle, M. Beyer, W.. CRC Handbook of Chemistry and Physics. 70th ed. Boca Raton, Florida: CRC Press, Inc.. F-373,376
In fluid dynamics, Görtler vortices are secondary flows that appear in a boundary layer flow along a concave wall. If the boundary layer is thin compared to the radius of curvature of the wall, the pressure remains constant across the boundary layer. On the other hand, if the boundary layer thickness is comparable to the radius of curvature, the centrifugal action creates a pressure variation across the boundary layer; this leads to the centrifugal instability of the boundary layer and consequent formation of Görtler vortices. The onset of Görtler vortices can be predicted using the dimensionless number called Görtler number, it is the ratio of centrifugal effects to the viscous effects in the boundary layer and is defined as G = U e θ ν 1 / 2 where U e = external velocity θ = momentum thickness ν = kinematic viscosity R = radius of curvature of the wallGörtler instability occurs when G exceeds about 0.3. A similar phenomenon arising from the same centrifugal action is sometimes observed in rotational flows which do not follow a curved wall, such as the rib vortices seen in the wakes of cylinders and generated behind moving structures.
Görtler, H.. "Dreidimensionales zur Stabilitätstheorie laminarer Grenzschichten". Journal of Applied Mathematics and Mechanics. 35: 362–363. Bibcode:1955ZaMM...35..360.. Doi:10.1002/zamm.19550350906. Saric, W. S.. "Görtler vortices". Annu. Rev. Fluid Mech. 26: 379–409. Bibcode:1994AnRFM..26..379S. Doi:10.1146/annurev.fl.26.010194.002115
In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities and units of measure and tracking these dimensions as calculations or comparisons are performed. The conversion of units from one dimensional unit to another is somewhat complex. Dimensional analysis, or more the factor-label method known as the unit-factor method, is a used technique for such conversions using the rules of algebra; the concept of physical dimension was introduced by Joseph Fourier in 1822. Physical quantities that are of the same kind have the same dimension and can be directly compared to each other if they are expressed in differing units of measure. If physical quantities have different dimensions, they cannot be expressed in terms of similar units and cannot be compared in quantity. For example, asking whether a kilogram is larger than an hour is meaningless. Any physically meaningful equation will have the same dimensions on its left and right sides, a property known as dimensional homogeneity.
Checking for dimensional homogeneity is a common application of dimensional analysis, serving as a plausibility check on derived equations and computations. It serves as a guide and constraint in deriving equations that may describe a physical system in the absence of a more rigorous derivation. Many parameters and measurements in the physical sciences and engineering are expressed as a concrete number – a numerical quantity and a corresponding dimensional unit. A quantity is expressed in terms of several other quantities. Compound relations with "per" are expressed with division, e.g. 60 mi/1 h. Other relations can involve powers, or combinations thereof. A set of base units for a system of measurement is a conventionally chosen set of units, none of which can be expressed as a combination of the others, in terms of which all the remaining units of the system can be expressed. For example, units for length and time are chosen as base units. Units for volume, can be factored into the base units of length, thus they are considered derived or compound units.
Sometimes the names of units obscure the fact. For example, a newton is a unit of force; the newton is defined as 1 N = 1 kg⋅m⋅s−2. Percentages are dimensionless quantities, since they are ratios of two quantities with the same dimensions. In other words, the % sign can be read as "hundredths", since 1% = 1/100. Taking a derivative with respect to a quantity adds the dimension of the variable one is differentiating with respect to, in the denominator. Thus: position has the dimension L. In economics, one distinguishes between stocks and flows: a stock has units of "units", while a flow is a derivative of a stock, has units of "units/time". In some contexts, dimensional quantities are expressed as dimensionless quantities or percentages by omitting some dimensions. For example, debt-to-GDP ratios are expressed as percentages: total debt outstanding divided by annual GDP – but one may argue that in comparing a stock to a flow, annual GDP should have dimensions of currency/time, thus Debt-to-GDP should have units of years, which indicates that Debt-to-GDP is the number of years needed for a constant GDP to pay the debt, if all GDP is spent on the debt and the debt is otherwise unchanged.
In dimensional analysis, a ratio which converts one unit of measure into another without changing the quantity is called a conversion factor. For example, kPa and bar are both units of pressure, 100 kPa = 1 bar; the rules of algebra allow both sides of an equation to be divided by the same expression, so this is equivalent to 100 kPa / 1 bar = 1. Since any quantity can be multiplied by 1 without changing it, the expression "100 kPa / 1 bar" can be used to convert from bars to kPa by multiplying it with the quantity to be converted, including units. For example, 5 bar × 100 kPa / 1 bar = 500 kPa because 5 × 100 / 1 = 500, bar/bar cancels out, so 5 bar = 500 kPa; the most basic rule of dimensional analysis is that of dimensional homogeneity. 1. Only commensurable quantities may be compared, added, or subtracted. However, the dimensions form an abelian group under multiplication, so: 2. One may take ratios of incommensurable quantities, multiply or divide them. For example, it makes no sense to ask whether 1 hour is more, the same, or less than 1 kilometer, as these have different dimensions, nor to add 1 hour to 1 kilometer.
However, it makes perfect sense to ask whether 1 mile is more, the same, or less than 1 kilometer being the same dimension of physical quantity though the units are different. On the other hand, if an object travels 100 km in 2 hours, one may divide these and conclude that the object's average speed was 50 km/h; the rule implies that in a physically mea
In continuum mechanics, the Froude number is a dimensionless number defined as the ratio of the flow inertia to the external field. Named after William Froude, the Froude number is based on the speed–length ratio which he defined as: F r = u 0 g 0 l 0 where u0 is a characteristic flow velocity, g0 is in general a characteristic external field, l0 is a characteristic length; the Froude number has some analogy with the Mach number. In theoretical fluid dynamics the Froude number is not considered since the equations are considered in the high Froude limit of negligible external field, leading to homogeneous equations that preserve the mathematical aspects. For example, homogeneous Euler equations are conservation equations. However, in naval architecture the Froude number is a significant figure used to determine the resistance of a submerged object moving through water. Dynamics of vessels that have the same Froude number are compared as they produce a similar wake if their size or geometry are otherwise different.
In open channel flows, Bélanger introduced first the ratio of the flow velocity to the square root of the gravity acceleration times the flow depth. When the ratio was less than unity, the flow behaved like a fluvial motion, like a torrential flow motion when the ratio was greater than unity. Quantifying resistance of floating objects is credited to William Froude, who used a series of scale models to measure the resistance each model offered when towed at a given speed. Froude's observations led him to derive the Wave-Line Theory which first described the resistance of a shape as being a function of the waves caused by varying pressures around the hull as it moves through the water; the naval constructor Ferdinand Reech had put forward the concept in 1852 for testing ships and propellers. Speed–length ratio was defined by Froude in his Law of Comparison in 1868 in dimensional terms as: speed–length ratio = u LWL where: u = flow speed LWL = length of waterlineThe term was converted into non-dimensional terms and was given Froude's name in recognition of the work he did.
In France, it is sometimes called Reech–Froude number after Ferdinand Reech. To show how the Froude number is linked to general continuum mechanics and not only to hydrodynamics we start from the nondimensionalisation of Cauchy momentum equation. In order to make the equations dimensionless, a characteristic length r0, a characteristic velocity u0, need to be defined; these should be chosen such. The following dimensionless variables are thus obtained: ρ ∗ ≡ ρ ρ 0, u ∗ ≡ u u 0, r ∗ ≡ r r 0, t ∗ ≡ u 0 r 0 t, ∇ ∗ ≡ r 0 ∇, g ∗ ≡ g g 0, σ ∗ ≡ σ p 0, Substitution of these inverse relations in the Euler momentum equations, definition of the Froude number: F r = u 0 g 0 r 0, the Euler number: E u = p 0 ρ 0 u 0 2, the equations are expressed: Cauchy-type equations in the high Froude limit Fr → ∞ are named free equations. On the other hand, in the low Euler limit Eu → 0 general Cauchy momentum equation becomes an inhomogeneous Burgers equation: This is an inhomogeneous pure advection equation, as much as the Stokes equation is a pure diffusion equation.
Euler momentum equation is a Cauchy momentum equation with the Pascal law being the stress constitutive relation: σ = p I in nondimensional Lagrangian form is: D u D t + E u ∇ p ρ = 1 F r 2 g ^ Free Euler equations are conservative
The Damköhler numbers are dimensionless numbers used in chemical engineering to relate the chemical reaction timescale to the transport phenomena rate occurring in a system. It is named after German chemist Gerhard Damköhler; the Karlovitz number is related to the Damköhler number by Da = 1/Ka. In its most used form, the Damköhler number relates the reaction timescale to the convection time scale, volumetric flow rate, through the reactor for continuous or semibatch chemical processes: D a = reaction rate convective mass transport rate In reacting systems that include interphase mass transport, the second Damköhler number is defined as the ratio of the chemical reaction rate to the mass transfer rate D a I I = reaction rate diffusive mass transfer rate It is defined as the ratio of the characteristic fluidic and chemical time scales: D a = flow time scale chemical time scale Since the reaction timescale is determined by the reaction rate, the exact formula for the Damköhler number varies according to the rate law equation.
For a general chemical reaction A → B of nth order, the Damköhler number for a convective flow system is defined as: D a = k C 0 n − 1 τ where: k = kinetics reaction rate constant C0 = initial concentration n = reaction order τ = mean residence time or space timeOn the other hand, the second Damköhler number is defined as: D a I I = k C 0 n − 1 k g a where kg is the global mass transport coefficient a is the interfacial areaThe value of Da provides a quick estimate of the degree of conversion that can be achieved. As a rule of thumb, when Da is less than 0.1 a conversion of less than 10% is achieved, when Da is greater than 10 a conversion of more than 90% is expected. The limit D a → ∞ is called the Burke–Schumann limit. From the general mole balance on some species A, where for a CSTR steady state and perfect mixing are assumed, in − out + generation = accumulation F A 0 − F A + r A V = 0 F A − F A 0 = r A V Assuming a constant volumetric flow rate v 0, the case for a liquid reactor or a gas phase reaction with no net generation of moles, v 0 = r A V = r A V v 0 = r A τ where the space time is defined to be the ratio of the reactor volume to volumetric flow rate.
It is the time required for a slug of fluid to pass through the reactor. For a decomposition reaction, the rate of reaction is proportional to some power of the concentration of A. In addition, for a single reaction a conversion may be defined in terms of the limiting reactant, for the simple decomposition, species A = − k C A n τ = − k C A 0 n τ n X = k C A 0 n − 1 τ n 0 = n X − 1 D a
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 local flow velocity with respect to the boundaries, c is the speed of sound in the medium. By definition, at Mach 1, the local flow velocity u is equal to the speed of sound. At Mach 0.65, u is 65% of the speed of sound, and, at Mach 1.35, u is 35% faster than the speed of sound. The local speed of sound, thereby the Mach number, depends on the condition of the surrounding medium, in particular the temperature; the Mach number is used to determine the approximation with which a flow can be treated as an incompressible flow. The medium can be a liquid; the boundary can be traveling in the medium, or it can be stationary while the medium flows along it, or they can both be moving, with different velocities: what matters is their relative velocity with respect to each other. 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, is a designation proposed by aeronautical engineer Jakob Ackeret; as the Mach number is a dimensionless quantity rather than a unit of measure, with Mach, the number comes after the unit. This is somewhat reminiscent of the early modern ocean sounding unit mark, unit-first, may have influenced the use of the term Mach. In the decade preceding faster-than-sound human flight, aeronautical engineers referred to the speed of sound as Mach's 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 mean sea level, standard temperature of 15 °C, the speed of sound is 340.3 meters per second.
The speed of sound is not a constant. For example, the standard atmosphere model lapses temperature to −56.5 °C at 11,000 meters altitude, with a corresponding speed of sound of 295.0 meters per second, 86.7% of the sea level value. While the terms subsonic and supersonic, in the purest sense, refer to speeds below and above the local speed of sound aerodynamicists use the same terms to talk about particular ranges of Mach values; this occurs because of the presence of a transonic regime around M = 1 where approximations of the Navier-Stokes equations used for subsonic design no longer apply. Meanwhile, the supersonic regime is used to talk about the set of Mach numbers for which linearised theory may be used, where for example the flow is not chemically reacting, where heat-transfer between air and vehicle may be reasonably neglected in calculations. In the following table, the regimes or ranges of Mach values are referred to, not the pure meanings of the words subsonic and supersonic. NASA defines high hypersonic as any Mach number from 10 to 25, re-entry speeds as anything greater than Mach 25.
Aircraft operating in this regime include the Space Shuttle and various space planes in development. Flight can be classified in six categories: For comparison: the required speed for low Earth orbit is 7.5 km/s = Mach 25.4 in air at high altitudes. At transonic speeds, the flow field around the object includes both sub- and supersonic parts; the transonic period begins. In case of an airfoil, this happens above the wing. Supersonic flow can decelerate back to subsonic only in a normal shock; as the speed increases, the zone of M > 1 flow increases towards both trailing edges. As M = 1 is reached and passed, the normal shock reaches the trailing edge and becomes a weak oblique shock: the flow decelerates over the shock, but remains supersonic. A normal shock is created ahead of the object, the only subsonic zone in the flow field is a small area around the object's leading edge. Fig. 1. Mach number in transonic airflow around an airfoil; when an aircraft exceeds Mach 1, a large pressure difference is created just in front of the aircraft.
This abrupt pressure difference, called a shock wave, spreads backward and outward from the aircraft in a cone shape. It is this shock wave. A person inside the aircraft will not hear this; the higher the speed, the more narrow the cone. At supersonic speed, the shock wave starts to take its cone shape and flow