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UNIT I BASIC CONCEPTS AND ISENTROPIC FLOWS

In physics, fluid dynamicsis a sub-discipline of fluid mechanicsthat deals with fluid

flowthe natural science of fluids (liquids and gases) in motion. It has several

subdisciplines itself, including aerodynamics(the study of air and other gases in motion)

and hydodynamics(the study of liquids in motion). Fluid dynamics has a wide range of

applications, including calculating forcesandmomentson aircraft, determining the mass

flow rate of petroleum through pipelines, predicting weather patterns, understanding

nebulae in interstellarspace and reportedly modeling fission weapon detonation. ome of

its principles are even used in traffic engineering, where traffic is treated as a continuous

fluid.

Fluid dynamics offers a systematic structure that underlies these practical disciplines, that

embraces empirical and semi-empirical laws derived from flow measurementand used to

solve practical problems. !he solution to a fluid dynamics problem typically involves

calculating various properties of the fluid, such as velocity, pressure, density, and

temperature,as functions of space and time.

"istorically, hydrodynamicsmeant something different than it does today. #efore the

twentieth century, hydrodynamics was synonymous with fluid dynamics. !his is still

reflected in names of some fluid dynamics topics, li$e magnetohydrodynamics and

hydrodynamic stabilityboth also applicable in, as well as being applied to, gases.a

!he foundational a%ioms of fluid dynamics are the conservation laws, specifically,

conservation of mass,conservation of linear momentum (also $nown as&ewton's econd

aw of otion), and conservation of energy (also $nown as First aw of

!hermodynamics). !hese are based on classical mechanicsand are modified in quantum

mechanics and general relativity. !hey are e%pressed using the *eynolds !ransport!heorem.

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 ob+ects. "owever, the

continuum assumption considers fluids to be continuous, rather than discrete.

http://en.wikipedia.org/wiki/Physicshttp://en.wikipedia.org/wiki/Fluid_mechanicshttp://en.wikipedia.org/wiki/Natural_sciencehttp://en.wikipedia.org/wiki/Fluidhttp://en.wikipedia.org/wiki/Liquidhttp://en.wikipedia.org/wiki/Gashttp://en.wikipedia.org/wiki/Aerodynamicshttp://en.wikipedia.org/wiki/Forcehttp://en.wikipedia.org/wiki/Moment_(physics)http://en.wikipedia.org/wiki/Aircrafthttp://en.wikipedia.org/wiki/Mass_flow_ratehttp://en.wikipedia.org/wiki/Mass_flow_ratehttp://en.wikipedia.org/wiki/Petroleumhttp://en.wikipedia.org/wiki/Weatherhttp://en.wikipedia.org/wiki/Nebulahttp://en.wikipedia.org/wiki/Interstellarhttp://en.wikipedia.org/wiki/Traffic_engineering_(transportation)http://en.wikipedia.org/wiki/Flow_measurementhttp://en.wikipedia.org/wiki/Velocityhttp://en.wikipedia.org/wiki/Pressurehttp://en.wikipedia.org/wiki/Densityhttp://en.wikipedia.org/wiki/Temperaturehttp://en.wikipedia.org/wiki/Magnetohydrodynamicshttp://en.wikipedia.org/wiki/Hydrodynamic_stabilityhttp://en.wikipedia.org/wiki/Conservation_lawhttp://en.wikipedia.org/wiki/Conservation_of_masshttp://en.wikipedia.org/wiki/Conservation_of_momentumhttp://en.wikipedia.org/wiki/Newton's_laws_of_motionhttp://en.wikipedia.org/wiki/Newton's_laws_of_motionhttp://en.wikipedia.org/wiki/Conservation_of_energyhttp://en.wikipedia.org/wiki/First_Law_of_Thermodynamicshttp://en.wikipedia.org/wiki/First_Law_of_Thermodynamicshttp://en.wikipedia.org/wiki/Classical_mechanicshttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/General_relativityhttp://en.wikipedia.org/wiki/Reynolds_transport_theoremhttp://en.wikipedia.org/wiki/Reynolds_transport_theoremhttp://en.wikipedia.org/wiki/Physicshttp://en.wikipedia.org/wiki/Fluid_mechanicshttp://en.wikipedia.org/wiki/Natural_sciencehttp://en.wikipedia.org/wiki/Fluidhttp://en.wikipedia.org/wiki/Liquidhttp://en.wikipedia.org/wiki/Gashttp://en.wikipedia.org/wiki/Aerodynamicshttp://en.wikipedia.org/wiki/Forcehttp://en.wikipedia.org/wiki/Moment_(physics)http://en.wikipedia.org/wiki/Aircrafthttp://en.wikipedia.org/wiki/Mass_flow_ratehttp://en.wikipedia.org/wiki/Mass_flow_ratehttp://en.wikipedia.org/wiki/Petroleumhttp://en.wikipedia.org/wiki/Weatherhttp://en.wikipedia.org/wiki/Nebulahttp://en.wikipedia.org/wiki/Interstellarhttp://en.wikipedia.org/wiki/Traffic_engineering_(transportation)http://en.wikipedia.org/wiki/Flow_measurementhttp://en.wikipedia.org/wiki/Velocityhttp://en.wikipedia.org/wiki/Pressurehttp://en.wikipedia.org/wiki/Densityhttp://en.wikipedia.org/wiki/Temperaturehttp://en.wikipedia.org/wiki/Magnetohydrodynamicshttp://en.wikipedia.org/wiki/Hydrodynamic_stabilityhttp://en.wikipedia.org/wiki/Conservation_lawhttp://en.wikipedia.org/wiki/Conservation_of_masshttp://en.wikipedia.org/wiki/Conservation_of_momentumhttp://en.wikipedia.org/wiki/Newton's_laws_of_motionhttp://en.wikipedia.org/wiki/Newton's_laws_of_motionhttp://en.wikipedia.org/wiki/Conservation_of_energyhttp://en.wikipedia.org/wiki/First_Law_of_Thermodynamicshttp://en.wikipedia.org/wiki/First_Law_of_Thermodynamicshttp://en.wikipedia.org/wiki/Classical_mechanicshttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/General_relativityhttp://en.wikipedia.org/wiki/Reynolds_transport_theoremhttp://en.wikipedia.org/wiki/Reynolds_transport_theorem

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onsequently, properties such as density, pressure, temperature, and velocity are ta$en to

be well-defined at infinitesimally small points, and are assumed to vary continuously

from one point to another. !he fact that the fluid is made up of discrete molecules is

ignored.

For fluids which are sufficiently dense to be a continuum, do not contain ionied species,

and have velocities small in relation to the speed of light, the momentum equations for

&ewtonian fluidsare the&avier-to$es equations, which is a non-linear set ofdifferential

equations that describes the flow of a fluid whose stress depends linearly on velocity

gradients and pressure. !he unsimplified equations do not have a general closed-form

solution, so they are primarily of use in omputational Fluid ynamics. !he equations

can be simplified in a number of ways, all of which ma$e them easier to solve. ome ofthem allow appropriate fluid dynamics problems to be solved in closed form.

In addition to the mass, momentum, and energy conservation equations, a

thermodynamical equation of state giving the pressure as a function of otherthermodynamic variables for the fluid is required to completely specify the problem. /n

e%ample of this would be theperfect gas equation of state0

wherepispressure, 1is density,Ruis the gas constant,Mis the molar massand T istemperature.

Com!"ssi#l" $s incom!"ssi#l" flow

/ll fluids are compressibleto some e%tent, that is changes in pressure or temperature will

result in changes in density. "owever, in many situations the changes in pressure and

temperature are sufficiently small that the changes in density are negligible. In this case

the flow can be modeled as an incompressible flow. 2therwise the more general

compressible flow equations must be used.

athematically, incompressibility is e%pressed by saying that the density 1 of a fluid

parcel does not change as it moves in the flow field, i.e.,

http://en.wikipedia.org/wiki/Newtonian_fluidhttp://en.wikipedia.org/wiki/Navier-Stokes_equationshttp://en.wikipedia.org/wiki/Non-linearhttp://en.wikipedia.org/wiki/Differential_equationshttp://en.wikipedia.org/wiki/Differential_equationshttp://en.wikipedia.org/wiki/Solution_in_closed_formhttp://en.wikipedia.org/wiki/Solution_in_closed_formhttp://en.wikipedia.org/wiki/Computational_Fluid_Dynamicshttp://en.wikipedia.org/wiki/Thermodynamicshttp://en.wikipedia.org/wiki/Ideal_gas_lawhttp://en.wikipedia.org/wiki/Pressurehttp://en.wikipedia.org/wiki/Pressurehttp://en.wikipedia.org/wiki/Densityhttp://en.wikipedia.org/wiki/Gas_constanthttp://en.wikipedia.org/wiki/Molar_masshttp://en.wikipedia.org/wiki/Molar_masshttp://en.wikipedia.org/wiki/Temperaturehttp://en.wikipedia.org/wiki/Temperaturehttp://en.wikipedia.org/wiki/Compressibilityhttp://en.wikipedia.org/wiki/Incompressible_flowhttp://en.wikipedia.org/wiki/Compressible_flowhttp://en.wikipedia.org/wiki/Newtonian_fluidhttp://en.wikipedia.org/wiki/Navier-Stokes_equationshttp://en.wikipedia.org/wiki/Non-linearhttp://en.wikipedia.org/wiki/Differential_equationshttp://en.wikipedia.org/wiki/Differential_equationshttp://en.wikipedia.org/wiki/Solution_in_closed_formhttp://en.wikipedia.org/wiki/Solution_in_closed_formhttp://en.wikipedia.org/wiki/Computational_Fluid_Dynamicshttp://en.wikipedia.org/wiki/Thermodynamicshttp://en.wikipedia.org/wiki/Ideal_gas_lawhttp://en.wikipedia.org/wiki/Pressurehttp://en.wikipedia.org/wiki/Densityhttp://en.wikipedia.org/wiki/Gas_constanthttp://en.wikipedia.org/wiki/Molar_masshttp://en.wikipedia.org/wiki/Temperaturehttp://en.wikipedia.org/wiki/Compressibilityhttp://en.wikipedia.org/wiki/Incompressible_flowhttp://en.wikipedia.org/wiki/Compressible_flow

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where 3 t is the substantial derivative, which is the sum of local and convective

derivatives. !his additional constraint simplifies the governing equations, especially in

the case when the fluid has a uniform density.

For flow of gases, to determine whether to use compressible or incompressible fluid

dynamics, the ach number of the flow is to be evaluated. /s a rough guide,

compressible effects can be ignored at ach numbers below appro%imately 4.5. For

liquids, whether the incompressible assumption is valid depends on the fluid properties

(specifically the critical pressure and temperature of the fluid) and the flow conditions

(how close to the critical pressure the actual flow pressure becomes). /cousticproblems

always require allowing compressibility, since sound waves are compression waves

involving changes in pressure and density of the medium through which they propagate.

$iscous $s in$iscid flow

6iscousproblems are those in which fluid friction has significant effects on the fluid

motion.

!he*eynolds number, which is a ratio between inertial and viscous forces, can be used

to evaluate whether viscous or inviscid equations are appropriate to the problem.

to$es flowis flow at very low *eynolds numbers,Re778, such that inertial forces can

be neglected compared to viscous forces.

2n the contrary, high *eynolds numbers indicate that the inertial forces are more

significant than the viscous (friction) forces. !herefore, we may assume the flow to be an

inviscid flow, an appro%imation in which we neglect viscositycompletely, compared to

inertial terms.

!his idea can wor$ fairly well when the *eynolds number is high. "owever, certain

problems such as those involving solid boundaries, may require that the viscosity be

included. 6iscosity often cannot be neglected near solid boundaries because the no-slip

conditioncan generate a thin region of large strain rate ($nown as #oundary layer) which

http://en.wikipedia.org/wiki/Substantial_derivativehttp://en.wikipedia.org/wiki/Convective_derivativehttp://en.wikipedia.org/wiki/Convective_derivativehttp://en.wikipedia.org/wiki/Mach_numberhttp://en.wikipedia.org/wiki/Mach_numberhttp://en.wikipedia.org/wiki/Acousticshttp://en.wikipedia.org/wiki/Sound_waveshttp://en.wikipedia.org/wiki/Viscosityhttp://en.wikipedia.org/wiki/Reynolds_numberhttp://en.wikipedia.org/wiki/Reynolds_numberhttp://en.wikipedia.org/wiki/Stokes_flowhttp://en.wikipedia.org/wiki/Inviscid_flowhttp://en.wikipedia.org/wiki/Viscosityhttp://en.wikipedia.org/wiki/No-slip_conditionhttp://en.wikipedia.org/wiki/No-slip_conditionhttp://en.wikipedia.org/wiki/Boundary_layerhttp://en.wikipedia.org/wiki/Substantial_derivativehttp://en.wikipedia.org/wiki/Convective_derivativehttp://en.wikipedia.org/wiki/Convective_derivativehttp://en.wikipedia.org/wiki/Mach_numberhttp://en.wikipedia.org/wiki/Acousticshttp://en.wikipedia.org/wiki/Sound_waveshttp://en.wikipedia.org/wiki/Viscosityhttp://en.wikipedia.org/wiki/Reynolds_numberhttp://en.wikipedia.org/wiki/Stokes_flowhttp://en.wikipedia.org/wiki/Inviscid_flowhttp://en.wikipedia.org/wiki/Viscosityhttp://en.wikipedia.org/wiki/No-slip_conditionhttp://en.wikipedia.org/wiki/No-slip_conditionhttp://en.wikipedia.org/wiki/Boundary_layer

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enhances the effect of even a small amount of viscosity,and thus generating vorticity.

!herefore, to calculate net forces on bodies (such as wings) we should use viscous flow

equations. /s illustrated by d'/lembert's parado%, a body in an inviscid fluid will

e%perience no drag force. !he standard equations of inviscid flow are the 9uler equations.

/nother often used model, especially in computational fluid dynamics, is to use the 9uler

equations away from the body and the boundary layerequations, which incorporates

viscosity, in a region close to the body.

!he 9uler equations can be integrated along a streamline to get #ernoulli's equation.

:hen the flow is everywhere irrotationaland inviscid, #ernoulli's equation can be used

throughout the flow field. uch flows are calledpotential flows.

S%"ady $s uns%"ady flow

:hen all the time derivatives of a flow field vanish, the flow is considered to be a s%"ady

flow. teady-state flow refers to the condition where the fluid properties at a point in the

system do not change over time. 2therwise, flow is called unsteady. :hether a particular

flow is steady or unsteady, can depend on the chosen frame of reference. For instance,

laminar flow over a sphere is steady in the frame of reference that is stationary with

respect to the sphere. In a frame of reference that is stationary with respect to abac$ground flow, the flow is unsteady.

!urbulent flows are unsteady by definition. / turbulent flow can, however, be

statistically stationary.

!he random field U(x,t) is statistically stationary if all statistics are invariant under a shift

in time.

!his roughly means that all statistical properties are constant in time. 2ften, the mean

field is the ob+ect of interest, and this is constant too in a statistically stationary flow.

teady flows are often more tractable than otherwise similar unsteady flows. !he

governing equations of a steady problem have one dimension fewer (time) than the

http://en.wikipedia.org/wiki/Viscosityhttp://en.wikipedia.org/wiki/Vorticityhttp://en.wikipedia.org/wiki/D'Alembert's_paradoxhttp://en.wikipedia.org/wiki/Euler_equations_(fluid_dynamics)http://en.wikipedia.org/wiki/Boundary_layerhttp://en.wikipedia.org/wiki/Bernoulli's_equationhttp://en.wikipedia.org/wiki/Lamellar_fieldhttp://en.wikipedia.org/wiki/Potential_flowhttp://en.wikipedia.org/wiki/Frame_of_referencehttp://en.wikipedia.org/wiki/Spherehttp://en.wikipedia.org/wiki/Turbulencehttp://en.wikipedia.org/wiki/Stationary_processhttp://en.wikipedia.org/wiki/Viscosityhttp://en.wikipedia.org/wiki/Vorticityhttp://en.wikipedia.org/wiki/D'Alembert's_paradoxhttp://en.wikipedia.org/wiki/Euler_equations_(fluid_dynamics)http://en.wikipedia.org/wiki/Boundary_layerhttp://en.wikipedia.org/wiki/Bernoulli's_equationhttp://en.wikipedia.org/wiki/Lamellar_fieldhttp://en.wikipedia.org/wiki/Potential_flowhttp://en.wikipedia.org/wiki/Frame_of_referencehttp://en.wikipedia.org/wiki/Spherehttp://en.wikipedia.org/wiki/Turbulencehttp://en.wikipedia.org/wiki/Stationary_process

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governing equations of the same problem without ta$ing advantage of the steadiness of

the flow field.

Lamina $s %u#ul"n% flow

!urbulenceis flow characteried by recirculation,eddies,and apparent randomness.Flow

in which turbulence is not e%hibited is called laminar. It should be noted, however, that

the presence of eddies or recirculation alone does not necessarily indicate turbulent flow

these phenomena may be present in laminar flow as well. athematically, turbulent

flow is often represented via a *eynolds decomposition, in which the flow is bro$en

down into the sum of an averagecomponent and a perturbation component.

It is believed that turbulent flows can be described well through the use of the &avier;

to$es equations. irect numerical simulation (&), based on the &avier;to$es

equations, ma$es it possible to simulate turbulent flows at moderate *eynolds numbers.

*estrictions depend on the power of the computer used and the efficiency of the solution

algorithm. !he results of & have been found to agree well with e%perimental data for

some flows.

ost flows of interest have *eynolds numbers much too high for & to be a viable

option, given the state of computational power for the ne%t few decades. /ny flight

vehicle large enough to carry a human ( @ 5 m), moving faster than AB $m3h (B4 m3s) is

well beyond the limit of & simulation (*e C = million). !ransport aircraft wings (such

as on an/irbus /544or #oeing A=A) have *eynolds numbers of =4 million (based on the

wing chord). In order to solve these real-life flow problems, turbulence models will be a

necessity for the foreseeable future. *eynolds-averaged &avier;to$es equations

(*/&) combined with turbulence modeling provides a model of the effects of the

turbulent flow. uch a modeling mainly provides the additional momentum transfer by

the*eynolds stresses,although the turbulence also enhances the heatand mass transfer.

/nother promising methodology is large eddy simulation(9), especially in the guise

of detached eddy simulation (9)which is a combination of */& turbulence

modeling and large eddy simulation.

http://en.wikipedia.org/wiki/Turbulencehttp://en.wikipedia.org/wiki/Eddy_(fluid_dynamics)http://en.wikipedia.org/wiki/Randomhttp://en.wikipedia.org/wiki/Laminar_flowhttp://en.wikipedia.org/wiki/Reynolds_decompositionhttp://en.wikipedia.org/wiki/Averagehttp://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equationshttp://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equationshttp://en.wikipedia.org/wiki/Direct_numerical_simulationhttp://en.wikipedia.org/wiki/Fluid_dynamics#cite_note-3%23cite_note-3http://en.wikipedia.org/wiki/Fluid_dynamics#cite_note-4%23cite_note-4http://en.wikipedia.org/wiki/Airbus_A300http://en.wikipedia.org/wiki/Airbus_A300http://en.wikipedia.org/wiki/Boeing_747http://en.wikipedia.org/wiki/Reynolds-averaged_Navier%E2%80%93Stokes_equationshttp://en.wikipedia.org/wiki/Turbulence_modelinghttp://en.wikipedia.org/wiki/Reynolds_stresseshttp://en.wikipedia.org/wiki/Reynolds_stresseshttp://en.wikipedia.org/wiki/Reynolds_stresseshttp://en.wikipedia.org/wiki/Heat_transferhttp://en.wikipedia.org/wiki/Mass_transferhttp://en.wikipedia.org/wiki/Large_eddy_simulationhttp://en.wikipedia.org/wiki/Detached_eddy_simulationhttp://en.wikipedia.org/wiki/Turbulencehttp://en.wikipedia.org/wiki/Eddy_(fluid_dynamics)http://en.wikipedia.org/wiki/Randomhttp://en.wikipedia.org/wiki/Laminar_flowhttp://en.wikipedia.org/wiki/Reynolds_decompositionhttp://en.wikipedia.org/wiki/Averagehttp://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equationshttp://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equationshttp://en.wikipedia.org/wiki/Direct_numerical_simulationhttp://en.wikipedia.org/wiki/Fluid_dynamics#cite_note-3%23cite_note-3http://en.wikipedia.org/wiki/Fluid_dynamics#cite_note-4%23cite_note-4http://en.wikipedia.org/wiki/Airbus_A300http://en.wikipedia.org/wiki/Boeing_747http://en.wikipedia.org/wiki/Reynolds-averaged_Navier%E2%80%93Stokes_equationshttp://en.wikipedia.org/wiki/Turbulence_modelinghttp://en.wikipedia.org/wiki/Reynolds_stresseshttp://en.wikipedia.org/wiki/Heat_transferhttp://en.wikipedia.org/wiki/Mass_transferhttp://en.wikipedia.org/wiki/Large_eddy_simulationhttp://en.wikipedia.org/wiki/Detached_eddy_simulation

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!he Boussin"s& a!!o'ima%ionneglects variations in density e%cept to calculate

buoyancy forces. It is often used in free convection problems where density

changes are small.

Lu#ica%ion %h"oy and ("l")Shaw flow e%ploits the large aspect ratioof the

domain to show that certain terms in the equations are small and so can be

neglected.

Sl"nd")#ody %h"oyis a methodology used in to$es flowproblems to estimate

the force on, or flow field around, a long slender ob+ect in a viscous fluid.

!he shallow)wa%" "&ua%ions can be used to describe a layer of relatively

inviscid fluid with a free surface, in which surface gradientsare small.

!he Boussin"s& "&ua%ionsare applicable to surface waveson thic$er layers of

fluid and with steeper surface slopes.

Dacy*s lawis used for flow inporous media, and wor$s with variables averaged

over several pore-widths.

In rotating systems, the &uasi)+"os%o!hic a!!o'ima%ion assumes an almost

perfect balance betweenpressure gradientsand the oriolis force. It is useful in

the study of atmospheric dynamics.

Terminology in incompressible fluid dynamics

!he concepts of total pressure and dynamic pressurearise from #ernoulli's equationand

are significant in the study of all fluid flows. (!hese two pressures are not pressures in the

usual sensethey cannot be measured using an aneroid, #ourdon tube or mercury

column.) !o avoid potential ambiguity when referring to pressure in fluid dynamics,

many authors use the term static pressure to distinguish it from total pressure and

dynamic pressure. tatic pressureis identical topressureand can be identified for every

point in a fluid flow field.

In Aerodynamics, .D. lancy writes0 To distinguish it from the total and dynamic

pressures, the actual pressure of the fluid, which is associated not with its motion ut

with its state, is often referred to as the static pressure, ut where the term pressure

alone is used it refers to this static pressure!

http://en.wikipedia.org/wiki/Boussinesq_approximation_(buoyancy)http://en.wikipedia.org/wiki/Boussinesq_approximation_(buoyancy)http://en.wikipedia.org/wiki/Buoyancyhttp://en.wikipedia.org/wiki/Convectionhttp://en.wikipedia.org/wiki/Lubrication_theoryhttp://en.wikipedia.org/wiki/Hele-Shaw_flowhttp://en.wikipedia.org/wiki/Hele-Shaw_flowhttp://en.wikipedia.org/wiki/Aspect_ratiohttp://en.wikipedia.org/wiki/Slender-body_theoryhttp://en.wikipedia.org/wiki/Stokes_flowhttp://en.wikipedia.org/wiki/Shallow-water_equationshttp://en.wikipedia.org/wiki/Free_surfacehttp://en.wikipedia.org/wiki/Slopehttp://en.wikipedia.org/wiki/Boussinesq_equations_(water_waves)http://en.wikipedia.org/wiki/Surface_waveshttp://en.wikipedia.org/wiki/Slopehttp://en.wikipedia.org/wiki/Darcy's_lawhttp://en.wikipedia.org/wiki/Porous_mediumhttp://en.wikipedia.org/wiki/Balanced_flow#Geostrophic_flowhttp://en.wikipedia.org/wiki/Pressure_gradienthttp://en.wikipedia.org/wiki/Coriolis_forcehttp://en.wikipedia.org/wiki/Atmospheric_dynamicshttp://en.wikipedia.org/wiki/Atmospheric_dynamicshttp://en.wikipedia.org/wiki/Dynamic_pressurehttp://en.wikipedia.org/wiki/Bernoulli's_equationhttp://en.wikipedia.org/wiki/Pressurehttp://en.wikipedia.org/wiki/Static_pressurehttp://en.wikipedia.org/wiki/Static_pressurehttp://en.wikipedia.org/wiki/Pressurehttp://en.wikipedia.org/wiki/Fluid_dynamics#cite_note-5%23cite_note-5http://en.wikipedia.org/wiki/Fluid_dynamics#cite_note-5%23cite_note-5http://en.wikipedia.org/wiki/Boussinesq_approximation_(buoyancy)http://en.wikipedia.org/wiki/Buoyancyhttp://en.wikipedia.org/wiki/Convectionhttp://en.wikipedia.org/wiki/Lubrication_theoryhttp://en.wikipedia.org/wiki/Hele-Shaw_flowhttp://en.wikipedia.org/wiki/Aspect_ratiohttp://en.wikipedia.org/wiki/Slender-body_theoryhttp://en.wikipedia.org/wiki/Stokes_flowhttp://en.wikipedia.org/wiki/Shallow-water_equationshttp://en.wikipedia.org/wiki/Free_surfacehttp://en.wikipedia.org/wiki/Slopehttp://en.wikipedia.org/wiki/Boussinesq_equations_(water_waves)http://en.wikipedia.org/wiki/Surface_waveshttp://en.wikipedia.org/wiki/Slopehttp://en.wikipedia.org/wiki/Darcy's_lawhttp://en.wikipedia.org/wiki/Porous_mediumhttp://en.wikipedia.org/wiki/Balanced_flow#Geostrophic_flowhttp://en.wikipedia.org/wiki/Pressure_gradienthttp://en.wikipedia.org/wiki/Coriolis_forcehttp://en.wikipedia.org/wiki/Atmospheric_dynamicshttp://en.wikipedia.org/wiki/Dynamic_pressurehttp://en.wikipedia.org/wiki/Bernoulli's_equationhttp://en.wikipedia.org/wiki/Pressurehttp://en.wikipedia.org/wiki/Static_pressurehttp://en.wikipedia.org/wiki/Static_pressurehttp://en.wikipedia.org/wiki/Pressurehttp://en.wikipedia.org/wiki/Fluid_dynamics#cite_note-5%23cite_note-5

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/ point in a fluid flow where the flow has come to rest (i.e. speed is equal to ero

ad+acent to some solid body immersed in the fluid flow) is of special significance. It is of

such importance that it is given a special namea stagnation point. !he static pressure at

the stagnation point is of special significance and is given its own namestagnation

pressure. In incompressible flows, the stagnation pressure at a stagnation point is equal to

the total pressure throughout the flow field.

Terminology in compressible fluid dynamics

In a compressible fluid, such as air, the temperature and density are essential when

determining the state of the fluid. In addition to the concept of total pressure (also $nown

as stagnation pressure), the concepts of total (or stagnation) temperature and total (or

stagnation) density are also essential in any study of compressible fluid flows. !o avoid

potential ambiguity when referring to temperature and density, many authors use the

terms static temperature and static density. tatic temperature is identical to temperature

and static density is identical to density and both can be identified for every point in a

fluid flow field.

!he temperature and density at a stagnation pointare called stagnation temperature and

stagnation density.

/ similar approach is also ta$en with the thermodynamic properties of compressible

fluids. any authors use the terms total (or stagnation) enthalpyand total (or stagnation)

entropy. !he terms static enthalpy and static entropy appear to be less common, but

where they are used they mean nothing more than enthalpy and entropy respectively, and

the prefi% GstaticG is being used to avoid ambiguity with their 'total' or 'stagnation'

counterparts. #ecause the 'total' flow conditions are defined by isentropically bringing the

fluid to rest, the total (or stagnation) entropy is by definition always equal to the GstaticG

entropy.

!he ach number is commonly used both with ob+ects traveling at high speed in a fluid,

and with high-speed fluid flows inside channels such as noles, diffusers or wind

tunnels. /s it is defined as a ratio of two speeds, it is a dimensionless number. /t

http://en.wikipedia.org/wiki/Stagnation_pointhttp://en.wikipedia.org/wiki/Stagnation_pressurehttp://en.wikipedia.org/wiki/Stagnation_pressurehttp://en.wikipedia.org/wiki/Stagnation_pressurehttp://en.wikipedia.org/wiki/Stagnation_pointhttp://en.wikipedia.org/wiki/Enthalpyhttp://en.wikipedia.org/wiki/Enthalpyhttp://en.wikipedia.org/wiki/Entropyhttp://en.wikipedia.org/wiki/Nozzlehttp://en.wikipedia.org/wiki/Diffuserhttp://en.wikipedia.org/wiki/Wind_tunnelhttp://en.wikipedia.org/wiki/Wind_tunnelhttp://en.wikipedia.org/wiki/Dimensionless_numberhttp://en.wikipedia.org/wiki/Stagnation_pointhttp://en.wikipedia.org/wiki/Stagnation_pressurehttp://en.wikipedia.org/wiki/Stagnation_pressurehttp://en.wikipedia.org/wiki/Stagnation_pressurehttp://en.wikipedia.org/wiki/Stagnation_pointhttp://en.wikipedia.org/wiki/Enthalpyhttp://en.wikipedia.org/wiki/Entropyhttp://en.wikipedia.org/wiki/Nozzlehttp://en.wikipedia.org/wiki/Diffuserhttp://en.wikipedia.org/wiki/Wind_tunnelhttp://en.wikipedia.org/wiki/Wind_tunnelhttp://en.wikipedia.org/wiki/Dimensionless_number

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tandard ea evel conditions (corresponding to a temperatureof 8? degrees elsius),

the speed of sound is 5=4.5 m3s(8BB? $m3h, or AE8.B mph, or EE8.? $nots, or 888Eft3s)

in the 9arth's atmosphere. !he speed represented by ach 8 is not a constant for

e%ample, it is mostly dependent on temperature and atmospheric composition and largely

independent of pressure. In the stratosphere, where the temperatures are constant, it does

not vary with altitude even though the air pressure changes significantly with altitude.

ince the speed of sound increases as the temperature increases, the actual speed of an

ob+ect traveling at ach 8 will depend on the fluid temperature around it. ach number

is useful because the fluid behaves in a similar way at the same ach number. o, an

aircraft traveling at ach 8 at B4H or EHF will e%perience shoc$ waves in much the

same manner as when it is traveling at ach 8 at 88,444 m (5E,444 ft) at -?4H or -?F,even though it is traveling at only EJ of its speed at higher temperature li$e B4H or

EHF.

High-speed flow around objects

Flight can be roughly classified in si% categories0

R"+im" Su#sonic Tansonic Sonic Su!"sonic (y!"sonic(i+h)

hy!"sonic

,ach 74.A? 4.A?;8.B 8.4 8.B;?.4 ?.4;84.4 @84.4

For comparison0 the required speed for low 9arth orbitis appro%imately A.? $m3s C ach

B?.= in air at high altitudes. !he speed of light in a vacuum corresponds to a ach

number of appro%imately 8,444 (relative to air at sea level).

/t transonic speeds, the flow field around the ob+ect includes both sub- and supersonic

parts. !he transonic period begins when first ones of @8 flow appear around the

ob+ect. In case of an airfoil (such as an aircraft's wing), this typically happens above thewing. upersonic flow can decelerate bac$ to subsonic only in a normal shoc$ this

typically happens before the trailing edge. (Fig.8a)

/s the speed increases, the one ofM@8 flow increases towards both leading and trailing

edges. /sMC8 is reached and passed, the normal shoc$ reaches the trailing edge and

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becomes a wea$ oblique shoc$0 the flow decelerates over the shoc$, but remains

supersonic. / normal shoc$ is created ahead of the ob+ect, and the only subsonic one in

the flow field is a small area around the ob+ect's leading edge. (Fig.8b)

(a) (b)

Fi+- .-Mach numer in transonic airflow around an airfoil" M#$ %a& and M'$ %&!

:hen an aircraft e%ceeds ach 8 (i.e. the sound barrier) a large pressure difference is

created +ust in front of the aircraft. !his abrupt pressure difference, called a shoc$ wave,

spreads bac$ward and outward from the aircraft in a cone shape (a so-called ach cone).

It is this shoc$ wave that causes the sonic boom heard as a fast moving aircraft travels

overhead. / person inside the aircraft will not hear this. !he higher the speed, the more

narrow the cone at +ust over MC8 it is hardly a cone at all, but closer to a slightly

concave plane.

/t fully supersonic speed, the shoc$ wave starts to ta$e its cone shape and flow is either

completely supersonic, or (in case of a blunt ob+ect), only a very small subsonic flow area

remains between the ob+ect's nose and the shoc$ wave it creates ahead of itself. (In the

case of a sharp ob+ect, there is no air between the nose and the shoc$ wave0 the shoc$

wave starts from the nose.)

/s the ach number increases, so does the strength of the shoc$ waveand the ach

cone becomes increasingly narrow. /s the fluid flow crosses the shoc$ wave, its speed is

reduced and temperature, pressure, and density increase. !he stronger the shoc$, the

greater the changes. /t high enough ach numbers the temperature increases so much

over the shoc$ that ioniation and dissociation of gas molecules behind the shoc$ wave

begin. uch flows are called hypersonic.

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It is clear that any ob+ect traveling at hypersonic speeds will li$ewise be e%posed to the

same e%treme temperatures as the gas behind the nose shoc$ wave, and hence choice of

heat-resistant materials becomes important.

High-speed flow in a channel

/s a flow in a channel crosses MC8 becomes supersonic, one significant change ta$es

place. !he conservation of mass flow rateleads one to e%pect that contracting the flow

channel would increase the flow speed (i.e. ma$ing the channel narrower results in faster

air flow) and at subsonic speeds this holds true. "owever, once the flow becomes

supersonic, the relationship of flow area and speed is reversed0 e%panding the channel

actually increases the speed.

!he obvious result is that in order to accelerate a flow to supersonic, one needs a

convergent-divergent nole, where the converging section accelerates the flow toMC8,

sonic speeds, and the diverging section continues the acceleration. uch noles are

called de aval nolesand in e%treme cases they are able to reach incredible, hypersonic

speeds (ach 85 at B4H).

/n aircraft achmeteror electronic flight information system (9FI) can display ach

number derived from stagnation pressure (pitot tube) and static pressure.

Ci%ical ,ach num#"

In aerodynamics, the ci%ical ,ach num#" /,c0of an aircraft is the lowest ach

numberat which the airflow over a small region of the wing reaches the speed of sound.

For all aircraft in flight, the airflow around the aircraft is not e%actly the same as the

airspeed of the aircraft due to the airflow speeding up and slowing down to travel around

the aircraft structure. /t the ritical ach number, local airflow in some areas near the

airframe reaches the speed of sound, even though the aircraft itself has an airspeed lower

than ach 8.4. !his creates a wea$ shoc$ wave. /t speeds faster than the ritical ach

number0

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drag coefficientincreases suddenly, causingdramatically increased drag

in aircraft not designed for transonicor supersonicspeeds, changes to the airflow

over the flight control surfaces lead to deterioration in control of the aircraft.

In aircraft not designed to fly at the ritical ach number, shoc$ waves in the flow over

the wing and tailplane were sufficient to stall the wing, ma$e control surfaces ineffective

or lead to loss of control such as ach tuc$. !he phenomena associated with problems at

the ritical ach number became $nown as compressibility. ompressibility led to a

number of accidents involving high-speed military and e%perimental aircraft in the 8K54s

and 8K=4s.

/lthough un$nown at the time, compressibility was the cause of the phenomenon $nown

as the sound barrier. ubsonic aircraftsuch as the upermarine pitfire, #F 84K, L-?8

ustang, Mloster eteor, e BEB, L-4have relatively thic$, unswept wings and are

incapable of reaching ach 8.4. In 8K=A, huc$ Neagerflew the#ell O-8 to ach 8.4

and beyond, and the sound barrier was finally bro$en.

9arly transonic military aircraft such as the "aw$er "unter and F-E abre were

designed to fly satisfactorily faster than their ritical ach number. !hey did not possess

sufficient engine thrust to reach ach 8.4 in level flight but could be dived to ach 8.4and beyond, and remain controllable. odern passenger-carrying +et aircraft such as

/irbus and #oeing aircraft have a%imum 2perating ach numbers slower than ach

8.4.

upersonicaircraft, such as oncorde,the 9nglish 9lectric ightning, oc$heed F-84=,

assault irage III, and iM B8are designed to e%ceed ach 8.4 in level flight. !hey

have very thin wings. !heir ritical ach numbers are higher than those of subsonic and

transonic aircraft but less than ach 8.4.

!he actual ritical ach number varies from wing to wing. In general a thic$er wing will

have a lower ritical ach number, because a thic$er wing accelerates the airflow to a

faster speed than a thinner one. For instance, the fairly thic$ wing on the L-5 ightning

led to a ritical ach number of about .EK, a speed it could reach with some ease in

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dives, which led to a number of crashes. !he much thinner wing on the upermarine

pitfirecaused this aircraft to have a ritical ach number of about 4.K

Eff"c%s of ,ach num#" and com!"ssi#ili%y

:e study the effects of ach number and compressibility on strain-rate and vorticity

dynamics in decaying isotropic turbulence employing direct numerical simulations. ince

local ach number and dilatation are two direct indicators of compressibility of a fluid

element, we use these quantities as conditioning parameters to e%amine the various

aspects of turbulence dynamics. everal interesting observations along with the

underlying physics pertaining to the inertial (vorte% stretching and self-straining) and

pressure (pressure "essian and baroclinic) terms in the budget of strain-rate and vorticity

dynamics will be presented in the tal$. !he contrasting nature of these physical effects in

e%panding vs. contracting and supersonic vs. subsonic fluid elements will be highlighted.

P&I!-II F2: !"*2PM" P!

Rayl"i+h Flow1

*ayleigh flow refers todia#a%icflow through a constant area duct where the

effect of heat addition or re+ection is considered. Com!"ssi#ili%yeffects often come

into consideration, although the *ayleigh flow model certainly also applies to

incom!"ssi#l" flow. For this model, the duct area remains constant and no mass is

added within the duct. !herefore, unli$e Fanno flow, thes%a+na%ion %"m!"a%u"is

a variable. !he heat addition causes a decrease in s%a+na%ion !"ssu", which is

$nown as the *ayleigh effect and is critical in the design of combustion systems.

"eat addition will cause both su!"sonicand su#sonic,ach num#"sto approach

ach 8, resulting in cho2"d flow.onversely, heat re+ection decreases a subsonic

ach number and increases a supersonic ach number along the duct. It can be

shown that for calorically perfect flows the ma%imum "n%o!y occurs at ,C 8.

*ayleigh flow is named after 3ohn S%u%%4 5d Baon Rayl"i+h.

Fanno Flow1

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Fanno flow refers to adiabaticthrough a constant area duct where the effect

of friction is considered. ompressibilityflow effects often come into consideration,

although the Fanno flow model certainly also applies to incompressible flow. For this

model, the duct area remains constant, the flow is assumed to be steady and one-

dimensional, and no mass is added within the duct. !he Fanno flow model is considered

an irreversible process due to viscous effects. !he viscous friction causes the flow

properties to change along the duct. !he frictional effect is modeled as a shear stress at

the wall acting on the fluid with uniform properties over any cross section of the duct.

For a flow with an upstream ach numbergreater than 8.4 in a sufficiently long

enough duct, deceleration occurs and the flow can become cho$ed.2n the other hand, for

a flow with an upstream ach number less than 8.4, acceleration occurs and the flow can

become cho$ed in a sufficiently long duct. It can be shown that for flow of calorically per

!he Fanno flow model begins with a differential equationthat relates the change in ach

number with respect to the length of the duct, dM(dx. 2ther terms in the differential

equation are the heat capacity ratio, ), the Fanning friction factor, f, and the hydraulic

diameter,*h0

6aia%ion of Fluid Po!"%i"s1

E&ua%ions of fluid dynamics

!he foundational a%ioms of fluid dynamics are the conservation laws, specifically,

conservation of mass,conservation of linear momentum(also $nown as&ewton's econd

aw of otion), and conservation of energy (also $nown as First aw of

!hermodynamics). !hese are based on classical mechanicsand are modified in quantum

mechanics and general relativity. !hey are e%pressed using the *eynolds !ransport

!heorem.

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 ob+ects. "owever, the

continuum assumption considers fluids to be continuous, rather than discrete.

onsequently, properties such as density, pressure, temperature, and velocity are ta$en to

be well-defined at infinitesimally small points, and are assumed to vary continuously

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from one point to another. !he fact that the fluid is made up of discrete molecules is

ignored.

For fluids which are sufficiently dense to be a continuum, do not contain ionied species,

and have velocities small in relation to the speed of light, the momentum equations for

&ewtonian fluidsare the&avier-to$es equations, which is a non-linearset ofdifferential

equations that describes the flow of a fluid whose stress depends linearly on velocity

gradients and pressure. !he unsimplified equations do not have a general closed-form

solution, so they are primarily of use in omputational Fluid ynamics. !he equations

can be simplified in a number of ways, all of which ma$e them easier to solve. ome of

them allow appropriate fluid dynamics problems to be solved in closed form.

In addition to the mass, momentum, and energy conservation equations, a

thermodynamical equation of state giving the pressure as a function of other

thermodynamic variables for the fluid is required to completely specify the problem. /n

e%ample of this would be theperfect gas equation of state0

wherep is pressure, 1 is density, Ru is the gas constant, M is the molar massand T is

temperature.

Com!"ssi#l" $s incom!"ssi#l" flow

/ll fluids are compressibleto some e%tent, that is changes in pressure or temperature will

result in changes in density. "owever, in many situations the changes in pressure and

temperature are sufficiently small that the changes in density are negligible. In this case

the flow can be modeled as an incompressible flow. 2therwise the more general

compressible flowequations must be used.

athematically, incompressibility is e%pressed by saying that the density 1 of a fluid

parcel does not change as it moves in the flow field, i.e.,

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included. 6iscosity often cannot be neglected near solid boundaries because the no-slip

conditioncan generate a thin region of large strain rate ($nown as #oundary layer) which

enhances the effect of even a small amount of viscosity,and thus generating vorticity.

!herefore, to calculate net forces on bodies (such as wings) we should use viscous flow

equations. /s illustrated by d'/lembert's parado%, a body in an inviscid fluid will

e%perience no drag force. !he standard equations of inviscid flow are the 9uler equations.

/nother often used model, especially in computational fluid dynamics, is to use the 9uler

equations away from the body and the boundary layerequations, which incorporates

viscosity, in a region close to the body.

!he 9uler equations can be integrated along a streamline to get #ernoulli's equation.

:hen the flow is everywhere irrotationaland inviscid, #ernoulli's equation can be usedthroughout the flow field. uch flows are calledpotential flows.

S%"ady $s uns%"ady flow

"ydrodynamics simulation of the *ayleigh;!aylor instability

:hen all the time derivatives of a flow field vanish, the flow is considered to be a s%"ady

flow. teady-state flow refers to the condition where the fluid properties at a point in the

system do not change over time. 2therwise, flow is called unsteady. :hether a particular

http://en.wikipedia.org/wiki/No-slip_conditionhttp://en.wikipedia.org/wiki/No-slip_conditionhttp://en.wikipedia.org/wiki/Boundary_layerhttp://en.wikipedia.org/wiki/Viscosityhttp://en.wikipedia.org/wiki/Viscosityhttp://en.wikipedia.org/wiki/Vorticityhttp://en.wikipedia.org/wiki/D'Alembert's_paradoxhttp://en.wikipedia.org/wiki/Euler_equations_(fluid_dynamics)http://en.wikipedia.org/wiki/Boundary_layerhttp://en.wikipedia.org/wiki/Bernoulli's_equationhttp://en.wikipedia.org/wiki/Lamellar_fieldhttp://en.wikipedia.org/wiki/Potential_flowhttp://en.wikipedia.org/wiki/Rayleigh%E2%80%93Taylor_instabilityhttp://en.wikipedia.org/wiki/File:HD-Rayleigh-Taylor.gifhttp://en.wikipedia.org/wiki/No-slip_conditionhttp://en.wikipedia.org/wiki/No-slip_conditionhttp://en.wikipedia.org/wiki/Boundary_layerhttp://en.wikipedia.org/wiki/Viscosityhttp://en.wikipedia.org/wiki/Vorticityhttp://en.wikipedia.org/wiki/D'Alembert's_paradoxhttp://en.wikipedia.org/wiki/Euler_equations_(fluid_dynamics)http://en.wikipedia.org/wiki/Boundary_layerhttp://en.wikipedia.org/wiki/Bernoulli's_equationhttp://en.wikipedia.org/wiki/Lamellar_fieldhttp://en.wikipedia.org/wiki/Potential_flowhttp://en.wikipedia.org/wiki/Rayleigh%E2%80%93Taylor_instability

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flow is steady or unsteady, can depend on the chosen frame of reference. For instance,

laminar flow over a sphere is steady in the frame of reference that is stationary with

respect to the sphere. In a frame of reference that is stationary with respect to a

bac$ground flow, the flow is unsteady.

!urbulent flows are unsteady by definition. / turbulent flow can, however, be

statistically stationary. /ccording to Lope

!he random field U(x,t) is statistically stationary if all statistics are invariant under a shift

in time.

!his roughly means that all statistical properties are constant in time. 2ften, the mean

field is the ob+ect of interest, and this is constant too in a statistically stationary flow.

teady flows are often more tractable than otherwise similar unsteady flows. !he

governing equations of a steady problem have one dimension fewer (time) than the

governing equations of the same problem without ta$ing advantage of the steadiness of

the flow field.

Lamina $s %u#ul"n% flow

!urbulenceis flow characteried by recirculation,eddies,and apparent randomness.Flow

in which turbulence is not e%hibited is called laminar. It should be noted, however, that

the presence of eddies or recirculation alone does not necessarily indicate turbulent flow

these phenomena may be present in laminar flow as well. athematically, turbulent

flow is often represented via a *eynolds decomposition, in which the flow is bro$en

down into the sum of an averagecomponent and a perturbation component.

It is believed that turbulent flows can be described well through the use of the &avier;

to$es equations. irect numerical simulation (&), based on the &avier;to$es

equations, ma$es it possible to simulate turbulent flows at moderate *eynolds numbers.

*estrictions depend on the power of the computer used and the efficiency of the solution

http://en.wikipedia.org/wiki/Frame_of_referencehttp://en.wikipedia.org/wiki/Spherehttp://en.wikipedia.org/wiki/Turbulencehttp://en.wikipedia.org/wiki/Stationary_processhttp://en.wikipedia.org/wiki/Turbulencehttp://en.wikipedia.org/wiki/Eddy_(fluid_dynamics)http://en.wikipedia.org/wiki/Eddy_(fluid_dynamics)http://en.wikipedia.org/wiki/Eddy_(fluid_dynamics)http://en.wikipedia.org/wiki/Randomhttp://en.wikipedia.org/wiki/Randomhttp://en.wikipedia.org/wiki/Laminar_flowhttp://en.wikipedia.org/wiki/Reynolds_decompositionhttp://en.wikipedia.org/wiki/Reynolds_decompositionhttp://en.wikipedia.org/wiki/Averagehttp://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equationshttp://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equationshttp://en.wikipedia.org/wiki/Direct_numerical_simulationhttp://en.wikipedia.org/wiki/Frame_of_referencehttp://en.wikipedia.org/wiki/Spherehttp://en.wikipedia.org/wiki/Turbulencehttp://en.wikipedia.org/wiki/Stationary_processhttp://en.wikipedia.org/wiki/Turbulencehttp://en.wikipedia.org/wiki/Eddy_(fluid_dynamics)http://en.wikipedia.org/wiki/Randomhttp://en.wikipedia.org/wiki/Laminar_flowhttp://en.wikipedia.org/wiki/Reynolds_decompositionhttp://en.wikipedia.org/wiki/Averagehttp://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equationshttp://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equationshttp://en.wikipedia.org/wiki/Direct_numerical_simulation

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algorithm. !he results of & have been found to agree well with e%perimental data for

some flows.

ost flows of interest have *eynolds numbers much too high for & to be a viable

option, given the state of computational power for the ne%t few decades. /ny flight

vehicle large enough to carry a human ( @ 5 m), moving faster than AB $m3h (B4 m3s) is

well beyond the limit of & simulation (*e C = million). !ransport aircraft wings (such

as on an/irbus /544or #oeing A=A) have *eynolds numbers of =4 million (based on the

wing chord). In order to solve these real-life flow problems, turbulence models will be a

necessity for the foreseeable future. *eynolds-averaged &avier;to$es equations

(*/&) combined with turbulence modeling provides a model of the effects of the

turbulent flow. uch a modeling mainly provides the additional momentum transfer bythe*eynolds stresses,although the turbulence also enhances the heatand mass transfer.

/nother promising methodology is large eddy simulation(9), especially in the guise

of detached eddy simulation (9)which is a combination of */& turbulence

modeling and large eddy simulation.

N"w%onian $s non)N"w%onian fluids

ir Isaac &ewton showed how stress and the rate of strain are very close to linearlyrelated for many familiar fluids, such as water and air. !hese &ewtonian fluids are

modeled by a coefficient calledviscosity,which depends on the specific fluid.

"owever, some of the other materials, such as emulsions and slurries and some visco-

elastic materials (e.g. blood, some polymers), have more complicated non-Newtonian

stress-strain behaviours. !hese materials includesticky liquidssuch as late%, honey, and

lubricants which are studied in the sub-discipline of rheology.

Su#sonic $s %ansonic4 su!"sonic and hy!"sonic flows

http://en.wikipedia.org/wiki/Airbus_A300http://en.wikipedia.org/wiki/Airbus_A300http://en.wikipedia.org/wiki/Boeing_747http://en.wikipedia.org/wiki/Reynolds-averaged_Navier%E2%80%93Stokes_equationshttp://en.wikipedia.org/wiki/Turbulence_modelinghttp://en.wikipedia.org/wiki/Reynolds_stresseshttp://en.wikipedia.org/wiki/Reynolds_stresseshttp://en.wikipedia.org/wiki/Reynolds_stresseshttp://en.wikipedia.org/wiki/Heat_transferhttp://en.wikipedia.org/wiki/Mass_transferhttp://en.wikipedia.org/wiki/Large_eddy_simulationhttp://en.wikipedia.org/wiki/Detached_eddy_simulationhttp://en.wikipedia.org/wiki/Isaac_Newtonhttp://en.wikipedia.org/wiki/Stress_(physics)http://en.wikipedia.org/wiki/Strain_(materials_science)http://en.wikipedia.org/wiki/Strain_(materials_science)http://en.wikipedia.org/wiki/Waterhttp://en.wikipedia.org/wiki/Earth's_atmospherehttp://en.wikipedia.org/wiki/Newtonian_fluidhttp://en.wikipedia.org/wiki/Viscosityhttp://en.wikipedia.org/wiki/Viscosityhttp://en.wikipedia.org/wiki/Viscosityhttp://en.wikipedia.org/wiki/Bloodhttp://en.wikipedia.org/wiki/Polymerhttp://en.wikipedia.org/wiki/Non-Newtonian_fluidhttp://en.wikipedia.org/wiki/Latexhttp://en.wikipedia.org/wiki/Honeyhttp://en.wikipedia.org/wiki/Rheologyhttp://en.wikipedia.org/wiki/Airbus_A300http://en.wikipedia.org/wiki/Boeing_747http://en.wikipedia.org/wiki/Reynolds-averaged_Navier%E2%80%93Stokes_equationshttp://en.wikipedia.org/wiki/Turbulence_modelinghttp://en.wikipedia.org/wiki/Reynolds_stresseshttp://en.wikipedia.org/wiki/Heat_transferhttp://en.wikipedia.org/wiki/Mass_transferhttp://en.wikipedia.org/wiki/Large_eddy_simulationhttp://en.wikipedia.org/wiki/Detached_eddy_simulationhttp://en.wikipedia.org/wiki/Isaac_Newtonhttp://en.wikipedia.org/wiki/Stress_(physics)http://en.wikipedia.org/wiki/Strain_(materials_science)http://en.wikipedia.org/wiki/Waterhttp://en.wikipedia.org/wiki/Earth's_atmospherehttp://en.wikipedia.org/wiki/Newtonian_fluidhttp://en.wikipedia.org/wiki/Viscosityhttp://en.wikipedia.org/wiki/Bloodhttp://en.wikipedia.org/wiki/Polymerhttp://en.wikipedia.org/wiki/Non-Newtonian_fluidhttp://en.wikipedia.org/wiki/Latexhttp://en.wikipedia.org/wiki/Honeyhttp://en.wikipedia.org/wiki/Rheology

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:hile many terrestrial flows (e.g. flow of water through a pipe) occur at low mach

numbers, many flows of practical interest (e.g. in aerodynamics) occur at high fractions

of the ach &umber C8 or in e%cess of it (supersonic flows). &ew phenomena occur at

these ach number regimes (e.g. shoc$ waves for supersonic flow, transonic instability

in a regime of flows with nearly equal to 8, non-equilibrium chemical behavior due to

ioniation in hypersonic flows) and it is necessary to treat each of these flow regimes

separately.

,a+n"%ohydodynamics

agnetohydrodynamics is the multi-disciplinary study of the flow of electrically

conducting fluids in electromagnetic fields. 9%amples of such fluids include plasmas,

liquid metals, and salt water. !he fluid flow equations are solved simultaneously with

a%well's equationsof electromagnetism.

Us" of Ta#l"s and Cha%s1

Fanno Flow1

Fanno flow refers to adiabatic flow through a constant area duct where the effect ---!he

equation above can be used to plot the Fanno line

Rayl"i+h flow

*ayleigh flow refers to adiabatic flow through a constant area duct where the effect ---

!herefore, unli$e Fanno flow , the stagnation

UNIT III NOR,AL AND OBLI7UE S(OC8S

/ shoc2 wa$" (also called shoc2 fon%or simply Gshoc2G) is a type of propagating

disturbance. i$e an ordinary wave, it carries energy and can propagate through a

medium (solid, liquid, gas orplasma) or in some cases in the absence of a materialmedium, through afieldsuch as the electromagnetic field. hoc$ waves are characteried

by an abrupt, nearly discontinuous change in the characteristics of the medium. /cross

a shoc$ there is always an e%tremely rapid rise inpressure, temperature anddensity of the

flow. In supersonic flows, e%pansion is achieved through an e%pansion fan. / shoc$ wave

travels through most media at a higher speed than an ordinary wave.

http://en.wikipedia.org/wiki/Magnetohydrodynamicshttp://en.wikipedia.org/wiki/Electrical_conductionhttp://en.wikipedia.org/wiki/Electrical_conductionhttp://en.wikipedia.org/wiki/Electrical_conductionhttp://en.wikipedia.org/wiki/Electromagnetismhttp://en.wikipedia.org/wiki/Electromagnetismhttp://en.wikipedia.org/wiki/Electromagnetismhttp://en.wikipedia.org/wiki/Plasma_(physics)http://en.wikipedia.org/wiki/Saline_waterhttp://en.wikipedia.org/wiki/Maxwell's_equationshttp://en.wikipedia.org/wiki/Rayleigh_flowhttp://en.wikipedia.org/wiki/Wavehttp://en.wikipedia.org/wiki/Solidhttp://en.wikipedia.org/wiki/Liquidhttp://en.wikipedia.org/wiki/Gashttp://en.wikipedia.org/wiki/Plasma_(physics)http://en.wikipedia.org/wiki/Field_(physics)http://en.wikipedia.org/wiki/Electromagnetic_fieldhttp://en.wikipedia.org/wiki/Shock_wave#cite_note-0%23cite_note-0http://en.wikipedia.org/wiki/Pressurehttp://en.wikipedia.org/wiki/Temperaturehttp://en.wikipedia.org/wiki/Densityhttp://en.wikipedia.org/wiki/Expansion_fanhttp://en.wikipedia.org/wiki/Magnetohydrodynamicshttp://en.wikipedia.org/wiki/Electrical_conductionhttp://en.wikipedia.org/wiki/Electrical_conductionhttp://en.wikipedia.org/wiki/Electromagnetismhttp://en.wikipedia.org/wiki/Plasma_(physics)http://en.wikipedia.org/wiki/Saline_waterhttp://en.wikipedia.org/wiki/Maxwell's_equationshttp://en.wikipedia.org/wiki/Rayleigh_flowhttp://en.wikipedia.org/wiki/Wavehttp://en.wikipedia.org/wiki/Solidhttp://en.wikipedia.org/wiki/Liquidhttp://en.wikipedia.org/wiki/Gashttp://en.wikipedia.org/wiki/Plasma_(physics)http://en.wikipedia.org/wiki/Field_(physics)http://en.wikipedia.org/wiki/Electromagnetic_fieldhttp://en.wikipedia.org/wiki/Shock_wave#cite_note-0%23cite_note-0http://en.wikipedia.org/wiki/Pressurehttp://en.wikipedia.org/wiki/Temperaturehttp://en.wikipedia.org/wiki/Densityhttp://en.wikipedia.org/wiki/Expansion_fan

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Pnli$esolitons(another $ind of nonlinear wave), the energy of a shoc$ wave dissipates

relatively quic$ly with distance. /lso, the accompanying e%pansion wave approaches and

eventually merges with the shoc$ wave, partially cancelling it out. !hus the sonic boom

associated with the passage of a supersonic aircraft is the sound wave resulting from the

degradation and merging of the shoc$ wave and the e%pansion wave produced by the

aircraft.

:hen a shoc$ wave passes through matter, the total energy is preserved but the energy

which can be e%tracted as wor$ decreases and entropy increases. !his, for e%ample,

creates additional dragforce on aircraft with shoc$s.

hoc$ waves can be0

&ormal0 at K4H (perpendicular) to the shoc$ medium's flow direction.

2blique0 at an angle to the direction of flow.

#ow0 2ccurs upstream of the front (bow) of a blunt ob+ect when the upstream

velocity e%ceeds ach 8.

ome other terms

hoc$ Front0 an alternative name for the shoc$ wave itself

ontact Front0 in a shoc$ wave caused by a driver gas (for e%ample the GimpactG

of a high e%plosive on the surrounding air), the boundary between the driver

(e%plosive products) and the driven (air) gases. !he ontact Front trails the hoc$

Front.

In supersonic flows

Lressure-time diagram at an e%ternal observation point for the case of a supersonic ob+ect

propagating past the observer. !he leading edge of the ob+ect causes a shoc$ (left, in red)

and the trailing edge of the ob+ect causes an e%pansion (right, in blue).

:hen an ob+ect (or disturbance) moves faster than the information about it can be

propagated into the surrounding fluid, fluid near the disturbance cannot react or Gget out

http://en.wikipedia.org/wiki/Solitonhttp://en.wikipedia.org/wiki/Sonic_boomhttp://en.wikipedia.org/wiki/Wave_draghttp://en.wikipedia.org/wiki/Surface_normalhttp://en.wikipedia.org/wiki/Oblique_shockhttp://en.wikipedia.org/wiki/Bow_shock_(aerodynamics)http://en.wikipedia.org/wiki/Bow_(ship)http://en.wikipedia.org/wiki/Solitonhttp://en.wikipedia.org/wiki/Sonic_boomhttp://en.wikipedia.org/wiki/Wave_draghttp://en.wikipedia.org/wiki/Surface_normalhttp://en.wikipedia.org/wiki/Oblique_shockhttp://en.wikipedia.org/wiki/Bow_shock_(aerodynamics)http://en.wikipedia.org/wiki/Bow_(ship)

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of the wayG before the disturbance arrives. In a shoc$ wave the properties of the fluid

(density, pressure, temperature, velocity, ach number) change almost instantaneously.

easurements of the thic$ness of shoc$ waves have resulted in values appro%imately one

order of magnitudegreater than the mean free pathof the gas investigated.

hoc$ waves form when the speed of a gas changes by more than the speed of sound.

/t the region where this occurs sound waves traveling against the flow reach a point

where they cannot travel any further upstream and the pressure progressively builds in

that region, and a high pressure shoc$ wave rapidly forms.

hoc$ waves are not conventional sound waves a shoc$ wave ta$es the form of a very

sharp change in the gas properties on the order of a few mean free paths(roughly micro-

meters at atmospheric conditions) in thic$ness. hoc$ waves in air are heard as a loud

Gcrac$G or GsnapG noise. 2ver longer distances a shoc$ wave can change from a nonlinear

wave into a linear wave, degenerating into a conventional sound wave as it heats the air

and loses energy. !he sound wave is heard as the familiar GthudG or GthumpG of a sonic

boom, commonly created by the supersonic flight of aircraft.

!he shoc$ wave is one of several different ways in which a gas in a supersonic flow can

be compressed. ome other methods are isentropic compressions, including Lrandtl-eyer compressions. !he method of compression of a gas results in different

temperatures and densities for a given pressure ratio, which can be analytically calculated

for a non-reacting gas. / shoc$ wave compression results in a loss of total pressure,

meaning that it is a less efficient method of compressing gases for some purposes, for

instance in the inta$e of a scram+et. !he appearance of pressure-drag on supersonic

aircraft is mostly due to the effect of shoc$ compression on the flow.

Due to nonlinear steepening

hoc$ waves can form due to steepening of ordinary waves. !he best-$nown e%ample of

this phenomenon is ocean wavesthat formbrea$erson the shore. In shallow water, the

speed of surface waves is dependent on the depth of the water. /n incoming ocean wave

has a slightly higher wave speed near the crest of each wave than near the troughs

http://en.wikipedia.org/wiki/Densityhttp://en.wikipedia.org/wiki/Pressurehttp://en.wikipedia.org/wiki/Temperaturehttp://en.wikipedia.org/wiki/Velocityhttp://en.wikipedia.org/wiki/Mach_numberhttp://en.wikipedia.org/wiki/Order_of_magnitudehttp://en.wikipedia.org/wiki/Mean_free_pathhttp://en.wikipedia.org/wiki/Speed_of_soundhttp://en.wikipedia.org/wiki/Shock_wave#cite_note-1%23cite_note-1http://en.wikipedia.org/wiki/Mean_free_pathhttp://en.wikipedia.org/wiki/Sonic_boomhttp://en.wikipedia.org/wiki/Sonic_boomhttp://en.wikipedia.org/wiki/Supersonichttp://en.wikipedia.org/wiki/Isentropic_process#Isentropic_flowhttp://en.wikipedia.org/wiki/Ludwig_Prandtlhttp://en.wikipedia.org/wiki/Scramjethttp://en.wikipedia.org/wiki/Ocean_wavehttp://en.wikipedia.org/wiki/Breakerhttp://en.wikipedia.org/wiki/Shorehttp://en.wikipedia.org/wiki/Densityhttp://en.wikipedia.org/wiki/Pressurehttp://en.wikipedia.org/wiki/Temperaturehttp://en.wikipedia.org/wiki/Velocityhttp://en.wikipedia.org/wiki/Mach_numberhttp://en.wikipedia.org/wiki/Order_of_magnitudehttp://en.wikipedia.org/wiki/Mean_free_pathhttp://en.wikipedia.org/wiki/Speed_of_soundhttp://en.wikipedia.org/wiki/Shock_wave#cite_note-1%23cite_note-1http://en.wikipedia.org/wiki/Mean_free_pathhttp://en.wikipedia.org/wiki/Sonic_boomhttp://en.wikipedia.org/wiki/Sonic_boomhttp://en.wikipedia.org/wiki/Supersonichttp://en.wikipedia.org/wiki/Isentropic_process#Isentropic_flowhttp://en.wikipedia.org/wiki/Ludwig_Prandtlhttp://en.wikipedia.org/wiki/Scramjethttp://en.wikipedia.org/wiki/Ocean_wavehttp://en.wikipedia.org/wiki/Breakerhttp://en.wikipedia.org/wiki/Shore

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between waves, because the wave height is not infinitesimal compared to the depth of the

water. !he crests overta$e the troughs until the leading edge of the wave forms a vertical

face and spills over to form a turbulent shoc$ (a brea$er) that dissipates the wave's energy

as sound and heat.

imilar phenomena affect strong sound wavesin gas or plasma, due to the dependence of

the sound speedon temperatureandpressure. trong waves heat the medium near each

pressure front, due to adiabatic compression of the air itself, so that high pressure fronts

outrun the corresponding pressure troughs. :hile shoc$ formation by this process does

not normally happen to sound waves in 9arth's atmosphere, it is thought to be one

mechanism by which the solar chromosphere and corona are heated, via waves that

propagate up from the solar interior.

Analogies

/ shoc$ wave may be described as the furthest point upstream of a moving ob+ect which

G$nowsG about the approach of the ob+ect. In this description, the shoc$ wave position is

defined as the boundary between the one having no information about the shoc$-driving

event, and the one aware of the shoc$-driving event, analogous with the light cone

described in the theory of special relativity.

!o get a shoc$ wave something has to be travelling faster than the local speed of sound.

In that case some parts of the air around the aircraft are travelling at e%actly the speed of

sound with the aircraft, so that the sound waves leaving the aircraft pile up on each other,

similar to a tailbac$ on a road, and a shoc$ wave forms, the pressure increases, and then

spreads out sideways. #ecause of this amplification effect, a shoc$ wave is very intense,

more li$e an e%plosion when heard (not coincidentally, since e%plosions create shoc$

waves).

/nalogous phenomena are $nown outside fluid mechanics. For e%ample, particles

accelerated beyond the speed of lightin a refractive medium(where the speed of light is

less than that in a vacuum, such as water) create visible shoc$ effects, a phenomenon

$nown as heren$ov radiation.

http://en.wikipedia.org/wiki/Sound_wavehttp://en.wikipedia.org/wiki/Sound_speedhttp://en.wikipedia.org/wiki/Temperaturehttp://en.wikipedia.org/wiki/Temperaturehttp://en.wikipedia.org/wiki/Pressurehttp://en.wikipedia.org/wiki/Sunhttp://en.wikipedia.org/wiki/Sunhttp://en.wikipedia.org/wiki/Chromospherehttp://en.wikipedia.org/wiki/Coronahttp://en.wikipedia.org/wiki/Light_conehttp://en.wikipedia.org/wiki/Special_relativityhttp://en.wikipedia.org/wiki/Speed_of_lighthttp://en.wikipedia.org/wiki/Refractionhttp://en.wikipedia.org/wiki/Refractionhttp://en.wikipedia.org/wiki/Vacuumhttp://en.wikipedia.org/wiki/Waterhttp://en.wikipedia.org/wiki/Cherenkov_radiationhttp://en.wikipedia.org/wiki/Sound_wavehttp://en.wikipedia.org/wiki/Sound_speedhttp://en.wikipedia.org/wiki/Temperaturehttp://en.wikipedia.org/wiki/Pressurehttp://en.wikipedia.org/wiki/Sunhttp://en.wikipedia.org/wiki/Chromospherehttp://en.wikipedia.org/wiki/Coronahttp://en.wikipedia.org/wiki/Light_conehttp://en.wikipedia.org/wiki/Special_relativityhttp://en.wikipedia.org/wiki/Speed_of_lighthttp://en.wikipedia.org/wiki/Refractionhttp://en.wikipedia.org/wiki/Vacuumhttp://en.wikipedia.org/wiki/Waterhttp://en.wikipedia.org/wiki/Cherenkov_radiation

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Examples

#elow are a number of e%amples of shoc$ waves, broadly grouped with similar shoc$

phenomena0

hoc$ wave propagating into a stationary medium, ahead of the fireball of an e%plosion.

!he shoc$ is made visible by the shadow effect(!rinity e%plosion.)

Moving shock

Psually consists of a shoc$wave propagating into a stationary medium

In this case, the gas ahead of the shoc$ is stationary (in the laboratory frame), and

the gas behind the shoc$ is supersonic in the laboratory frame. !he shoc$

propagates with a wave front which is normal (at right angles) to the direction of

flow. !he speed of the shoc$ is a function of the original pressure ratio between

the two bodies of gas.

http://en.wikipedia.org/wiki/Shadowgraphhttp://en.wikipedia.org/wiki/Shadowgraphhttp://en.wikipedia.org/wiki/File:Trinity_explosion_film_strip.jpghttp://en.wikipedia.org/wiki/Shadowgraph

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oving shoc$sare usually generated by the interaction of two bodies of gas at

different pressure, with a shoc$ wave propagating into the lower pressure gas, and

an e%pansion wave propagating into the higher pressure gas.

9%amples0 #alloon bursting, hoc$ tube, shoc$ wave from e%plosion

etonation wave

ain article0 etonation

/ detonation wave is essentially a shoc$ supported by a trailing e%othermic

reaction. It involves a wave traveling through a highly combustible or chemically

unstable medium, such as an o%ygen-methane mi%ture or a high e%plosive. !he

chemical reaction of the medium occurs following the shoc$ wave, and the

chemical energy of the reaction drives the wave forward.

/ detonation wave follows slightly different rules from an ordinary shoc$ since it

is driven by the chemical reaction occurring behind the shoc$ wave front. In the

simplest theory for detonations, an unsupported, self-propagating detonation wave

proceeds at the hapman-Douguetvelocity. / detonation will also cause a shoc$

of type 8, above to propagate into the surrounding air due to the overpressure

induced by the e%plosion.

:hen a shoc$wave is created by high e%plosives such as !&! (which has a

detonation velocityof E,K44 m3s), it will always travel at high, supersonicvelocity

from its point of origin.

hadowgraphof the detached shoc$ on a bullet in supersonic flight, published by 9rnst

ach in 8A.

http://en.wikipedia.org/wiki/Moving_shockhttp://en.wikipedia.org/wiki/Shock_tubehttp://en.wikipedia.org/wiki/Blast_wavehttp://en.wikipedia.org/wiki/Detonationhttp://en.wikipedia.org/wiki/Detonationhttp://en.wikipedia.org/wiki/Exothermic_reactionhttp://en.wikipedia.org/wiki/Exothermic_reactionhttp://en.wikipedia.org/wiki/Chapman-Jouguet_conditionhttp://en.wikipedia.org/wiki/Chapman-Jouguet_conditionhttp://en.wikipedia.org/wiki/High_explosivehttp://en.wikipedia.org/wiki/Trinitrotoluenehttp://en.wikipedia.org/wiki/Detonation_velocityhttp://en.wikipedia.org/wiki/Detonation_velocityhttp://en.wikipedia.org/wiki/Supersonichttp://en.wikipedia.org/wiki/Supersonichttp://en.wikipedia.org/wiki/Velocityhttp://en.wikipedia.org/wiki/Shadowgraphhttp://en.wikipedia.org/wiki/File:Supersonic_Bullet.jpghttp://en.wikipedia.org/wiki/Moving_shockhttp://en.wikipedia.org/wiki/Shock_tubehttp://en.wikipedia.org/wiki/Blast_wavehttp://en.wikipedia.org/wiki/Detonationhttp://en.wikipedia.org/wiki/Detonationhttp://en.wikipedia.org/wiki/Exothermic_reactionhttp://en.wikipedia.org/wiki/Exothermic_reactionhttp://en.wikipedia.org/wiki/Chapman-Jouguet_conditionhttp://en.wikipedia.org/wiki/High_explosivehttp://en.wikipedia.org/wiki/Trinitrotoluenehttp://en.wikipedia.org/wiki/Detonation_velocityhttp://en.wikipedia.org/wiki/Supersonichttp://en.wikipedia.org/wiki/Velocityhttp://en.wikipedia.org/wiki/Shadowgraph

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etached shock

!hese shoc$s are curved, and form a small distance in front of the body. irectly

in front of the body, they stand at K4 degrees to the oncoming flow, and then

curve around the body. etached shoc$s allow the same type of analytic

calculations as for the attached shoc$, for the flow near the shoc$. !hey are a

topic of continuing interest, because the rules governing the shoc$'s distance

ahead of the blunt body are complicated, and are a function of the body's shape.

/dditionally, the shoc$ standoff distance varies drastically with the temperature

for a non-ideal gas, causing large differences in the heat transfer to the thermal

protection system of the vehicle. ee the e%tended discussion on this topic at

/tmospheric reentry. !hese follow the Gstrong-shoc$G solutions of the analytic

equations, meaning that for some oblique shoc$s very close to the deflection

angle limit, the downstream ach number is subsonic. ee also bow shoc$ or

oblique shoc$

uch a shoc$ occurs when the ma%imum deflection angle is e%ceeded. / detached

shoc$ is commonly seen on blunt bodies, but may also be seen on sharp bodies at

low ach numbers.

9%amples0 pace return vehicles (/pollo, pace shuttle), bullets, the boundary

(#ow shoc$) of a magnetosphere. !he name Gbow shoc$G comes from the

e%ample of abow wave, the detached shoc$ formed at the bow (front) of a ship or

boat moving through water, whose slow surface wave speed is easily e%ceeded

(see ocean surface wave).

!ttached shock

!hese shoc$s appear as GattachedG to the tip of a sharp body moving at supersonic

speeds.

9%amples0 upersonic wedges and cones with small ape% angles !he attached shoc$ wave is a classic structure in aerodynamics because, for a

perfect gas and inviscid flow field, an analytic solution is available, such that the

pressure ratio, temperature ratio, angle of the wedge and the downstream ach

number can all be calculated $nowing the upstream ach number and the shoc$

angle. maller shoc$ angles are associated with higher upstream ach numbers,

http://en.wikipedia.org/wiki/Atmospheric_reentryhttp://en.wikipedia.org/wiki/Bow_shock_(aerodynamics)http://en.wikipedia.org/wiki/Oblique_shockhttp://en.wikipedia.org/wiki/Bow_shockhttp://en.wikipedia.org/wiki/Magnetospherehttp://en.wikipedia.org/wiki/Bow_wavehttp://en.wikipedia.org/wiki/Ocean_surface_wavehttp://en.wikipedia.org/wiki/Atmospheric_reentryhttp://en.wikipedia.org/wiki/Bow_shock_(aerodynamics)http://en.wikipedia.org/wiki/Oblique_shockhttp://en.wikipedia.org/wiki/Bow_shockhttp://en.wikipedia.org/wiki/Magnetospherehttp://en.wikipedia.org/wiki/Bow_wavehttp://en.wikipedia.org/wiki/Ocean_surface_wave

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and the special case where the shoc$ wave is at K4 degrees to the oncoming flow

(&ormal shoc$), is associated with a ach number of one. !hese follow the

Gwea$-shoc$G solutions of the analytic equations.

"ecompression shock !hese shoc$s appear when the flow over a transonic body is decelerated to

subsonic speeds.

9%amples0 !ransonic wings, turbines

:here the flow over the suction side of a transonic wing is accelerated to a

supersonic speed, the resulting re-compression can be by either Lrandtl-eyer

compression or by the formation of a normal shoc$. !his shoc$ is of particular

interest to ma$ers of transonic devices because it can cause separation of the

boundary layer at the point where it touches the transonic profile. !his can then

lead to full separation and stall on the profile, higher drag, or shoc$-buffet, a

condition where the separation and the shoc$ interact in a resonance condition,

causing resonating loads on the underlying structure.

Shock in a pipe flow

!his shoc$ appears when supersonic flow in a pipe is decelerated.

9%amples0 upersonic ram+et, scram+et,needle valve

In this case the gas ahead of the shoc$ is supersonic (in the laboratory frame), and

the gas behind the shoc$ system is either supersonic (olique shocks) or subsonic

(a normal shock) (/lthough for some oblique shoc$s very close to the deflection

angle limit, the downstream ach number is subsonic.) !he shoc$ is the result of

the deceleration of the gas by a converging duct, or by the growth of the boundary

layer on the wall of a parallel duct.

http://en.wikipedia.org/wiki/Ramjethttp://en.wikipedia.org/wiki/Scramjethttp://en.wikipedia.org/wiki/Ramjethttp://en.wikipedia.org/wiki/Scramjet

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Shock waves in rapid granular flows

hoc$ waves can also occur in rapid flows of dense granular materials down inclined

channels or slopes. trong shoc$s in rapid dense granular flows can be studied

theoretically and analyed to compare with e%perimental data. onsider a configurationin which the rapidly moving material down the chute impinges on an obstruction wall

erected perpendicular at the end of a long and steep channel. Impact leads to a sudden

change in the flow regime from a fast moving supercritical thin layer to a stagnant thic$

heap. !his flow configuration is particularly interesting because it is analogous to some

hydraulic and aerodynamic situations associated with flow regime changes