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| '''Compressible flow''' is the area of [[fluid mechanics]] that deals with [[fluid]]s in which the fluid [[density]] varies significantly in response to a change in [[pressure]]. [[Compressibility]] effects are typically considered significant if the [[Mach number]] (the ratio of the flow velocity to the local [[speed of sound]]) of the flow exceeds 0.3, or if the fluid undergoes very large pressure changes. The most distinct differences between the compressible and [[incompressible flow|incompressible]] flow models are that the compressible flow model allows for the existence of [[shock wave]]s and [[choked flow]].
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| ==Definition==
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| Compressible flow describes the behaviour of fluids that experience significant variations in density. For flows in which the density does not vary significantly, the analysis of the behaviour of such flows may be simplified greatly by assuming a constant density. This is an idealization, which leads to the theory of [[incompressible flow]]. However, in the many cases dealing with [[gas]]es (especially at higher velocities) and those cases dealing with [[liquid]]s with large pressure changes, the significant variations in density can occur, and the flow should be analysed as a compressible flow if accurate results are to be obtained.<ref name="white">White, Frank M. (2003) ''Fluid Mechanics'', 5th ed. [[McGraw-Hill]]. ISBN 0-07-119911-X. pp. 599-688.</ref>
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| Allowing for a change in density brings an additional variable into the analysis. This is in contrast to incompressible flows, which can usually be solved by considering only the equations from [[conservation of mass]] and [[conservation of momentum]]. Usually, the principle of [[conservation of energy]] is included. However, this introduces another variable ([[temperature]]), and so a fourth equation (such as the ideal gas equation) is required to relate the temperature to the other thermodynamic properties in order to fully describe the flow.
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| When defining what is meant by a compressible flow, it is useful to compare the density to a reference value, such as the stagnation density, <math>\rho_{0}</math>, which is the density of the fluid if it were to be slowed down [[Isentropic process|isentropically]] to stationary. As a general rule of thumb, if the change in density relative to the stagnation density is greater than 5%, then the fluid should be analysed as a compressible flow. For an [[ideal gas]] with a [[heat capacity ratio|ratio of specific heats]] of 1.4, this occurs at a Mach number greater than approximately 0.3. Below this value, however, whether or not a specific case should be treated as compressible or incompressible depends largely on the level of accuracy that is required.<ref name="anderson">{{cite book| author = John David Anderson| title = Fundamentals of Aerodynamics| year = 2007| isbn = 978-0-07-125408-3| pages = 483-755 }}</ref>
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| == Compressible Flow Phenomena ==
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| Two of the most distinctive phenomena which occur in compressible flow are the possibility of [[choked flow]](see [[#Internal Flows|Internal Flows]]) and the presence of [[acoustic wave]]s, which may also be referred to as either compression or expansion waves, depending on whether they lead to an increase or decrease in pressure.<ref name="white"/>
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| === Shock Waves ===
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| [[Shock wave]]s are one of the most common examples of compressible flow phenomena. A shock is characterised by a discontinuous change in the thermodynamic properties. In one dimensional flows, shock waves can form when a series of compression waves coalesce, or when a membrane separating two regions of differing pressure is suddenly removed. This is the technique often used to produce shock waves in shock tubes (see [[#Shock Tubes|Shock Tubes]]).
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| In two and three dimensional supersonic flows, [[Oblique shock|oblique shock waves]] occur as a result of a change in direction of the flow. A classic example of these shock waves are those shock waves that form off the nose of a supersonic aircraft.
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| == Aerodynamics ==
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| [[Aerodynamics]] is a subfield of [[fluid dynamics]] and [[gas dynamics]], and is primarily concerned with obtaining the forces that air exerts on an object. For Mach numbers greater than about 0.3, density changes are significant, and the flow should be considered compressible for an accurate representation of reality.
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| === Subsonic Aerodynamics ===
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| Due to the complexities of compressible flow theory, it is often easier to calculate the incompressible flow characteristics first, and then employ a correction factor to obtain the actual flow properties. Several correction factors exist with varying degrees of complexity and accuracy.
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| ====Prandtl–Glauert transformation====
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| The [[Prandtl-Glauert transformation]] is found by linearizing the potential equations associated with compressible, inviscid flow. The Prandtl–Glauert transformation or Prandtl–Glauert rule (also Prandtl–Glauert–Ackeret rule) is an approximation function which allows comparison of aerodynamical processes occurring at different Mach numbers. It was discovered that the linearized pressures in such a flow were equal to those found from incompressible flow theory multiplied by a correction factor. This correction factor is given below.:<ref>Erich Truckenbrodt: Fluidmechanik Band 2, 4. Auflage, Springer Verlag, 1996, p. 178-179
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| </ref> | |
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| :<math>c_{p} = \frac {c_{p0}} {\sqrt {1-{M}^2}}.</math>
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| where
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| * c<sub>p</sub> is the compressible pressure coefficient
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| * c<sub>p0</sub> is the [[incompressible]] pressure coefficient
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| * ''M'' is the Mach number.
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| This correction factor is correct only for two-dimensional flow.
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| For general three-dimensional flows, it is necessary to apply the full [[Prandtl-Glauert transformation]] to the geometry, and then apply Göthert's Rule <ref>Göthert, B.H. Plane and Three-Dimensional Flow at High Subsonic Speeds (Extension of the Prandtl Rule). NACA TM 1105, 1946.</ref> to get the physical pressure coefficient and forces.
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| This 2D Prandtl-Glauert Rule, or the general 3D Göthert's Rule, work well until transonic flow starts to appear, typically for Mach numbers below 0.7 for 2D airfoils.
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| ====Karman-Tsien correction factor====
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| The Karman-Tsien transformation is a nonlinear correction factor to find the pressure coefficient of a compressible, inviscid flow. It is an empirically derived correction factor that tends to slightly overestimate the magnitude of the fluid's pressure. In order to employ this correction factor, the incompressible, inviscid fluid pressure must be known from previous investigation.<ref>Shapiro (1953), ''The Dynamics and Thermodynamics of Compressible Fluid Flow, Volume 1'', p.344</ref>
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| :<math>C_P=\frac {C_{P0}} {\sqrt{1-M^2}+\frac{C_{P0}}{2}(M^2/(1+\sqrt{1-M^2}))}</math> | |
| where
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| * c<sub>p</sub> is the compressible pressure coefficient
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| * c<sub>p0</sub> is the [[incompressible]] pressure coefficient
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| * ''M'' is the Mach number.
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| Like the Prandtl-Glauert Rule, this is only valid for 2D flows, and only until transonic flow starts to appear.
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| === Supersonic Aerodynamics ===
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| As with subsonic aerodynamics, a compressibility correction factor can be derived by linearising the governing equations. The supersonic correction factor is similar to the Prandtl-Glauert transformation, but the terms under the square root sign are reversed.
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| :<math>c_{p} = \frac {c_{p0}} {\sqrt {{M}^2-1}}.</math>
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| where
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| * c<sub>p</sub> is the compressible pressure coefficient
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| * c<sub>p0</sub> is the [[incompressible]] pressure coefficient
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| * ''M'' is the Mach number.
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| Again this is correct only for 2D flows. For validity it also requires that there is no transonic flow, which requires that the body be sufficiently slender and the freestream Mach number be sufficiently high, typically greater than 1.3.
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| === Transonic Aerodynamics ===
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| [[Transonic speed|Transonic flow]] typically occurs in flows with Mach numbers between 0.8 and 1.2. Under these conditions, some of the flow is supersonic and some is subsonic. At these velocities, the correction factors derived using linearized theory breaks down due to a singularity that occurs at a Mach number of 1. In addition, severe instabilities caused by the formation of local shock waves and the existence of both subsonic and supersonic flow (which behave completely differently) makes the solution of the governing equations rather difficult. However, the analysis of compressible flows in the transonic regime has led to some developments which help reduce the increases in drag caused by compressibility effects, including the use of [[swept wing]]s and the [[Area rule|Whitcomb area rule]].
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| == Internal Flows ==
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| If the flow of a fluid is confined by a surface, it is referred to as an [[internal flow]]. This includes the flow of fluids through pipes and ducts, and often arise in industrial and manufacturing processes, and is vital in the analysis of propulsion systems.
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| One example is in [[die casting]] or [[injection molding]] processes. This involves injecting a liquid material (such as a thermosetting plastic for injection molding or molten metal for die casting) at very high pressures into a cavity. The air that is already in the cavity is displaced very rapidly, and compressibility needs to be considered in the design of the die if problems with air entrapment are to be avoided.
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| === Effect of area changes ===
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| Compressible flows play a big role in determining the behaviour of [[nozzle]]s. Subsonic and supersonic flow react differently to changes in cross sectional area. While subsonic flow flowing through a converging duct (narrowing down from a wide diameter to a smaller diameter in the direction of the flow) will experience an increase in velocity, a supersonic flow through an identical duct will experience a decrease in velocity. In general, flow through a converging nozzle will always tend towards Mach 1. If the area convergence is great enough that the speed of sound is reached, a phenomenon known as "choking" occurs. In this case, the flow is choked, and either the flow rate of the fluid entering the pipe is limited, or shock waves form in the nozzle such that the Mach number at the point of minimum area (called the throat) remains unity. Similarly, subsonic flow through a diverging nozzle will always be slowed, and supersonic flow will accelerate. The Mach number of the flow can be directly related to the area by the relation<ref name="anderson"/>
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| :<math>\frac{A}{A^*} = \frac{1}{M}\left[\frac{1+\left(\frac{\gamma-1}{2}\right)M^{2}}{\left(\frac{\gamma+1}{2}\right)}\right]^{\frac{\gamma+1}{2(\gamma-1)}}</math>
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| where <math>A</math> and <math>M</math> are the area and Mach number at a point in the nozzle, <math>\gamma</math> is the ratio of specific heats, and <math>A^*</math> is the area that would cause the flow velocity to reach a Mach number of 1 (i.e. the area at the throat, provided that the nozzle is choked).
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| Thus, for a subsonic flow to be accelerated to supersonic velocities, the nozzle needs to have a converging section in which the flow is subsonic, a throat, at which the flow velocity is the local speed of sound, and a diverging section with supersonic flow. Such an arrangement is called a [[de Laval nozzle]], and is commonly used in propulsion systems such as [[Rocket engine|rocket]] and supersonic [[jet engine]]s.
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| Note that Mach 1 can be a very high speed for a hot gas, since the speed of sound varies as the square root of absolute temperature. Thus the speed reached at a nozzle throat can be far higher than the speed of sound under standard atmospheric conditions. This fact is used extensively in rocketry where hypersonic flows are required, and where propellant mixtures are deliberately chosen to further increase the sonic speed.
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| === Effect of friction ===
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| [[Friction]] has a similar effect as an area change on compressible flow. In a pipe of constant cross sectional area in which the walls exert a frictional force on the flow, the flow velocity will tend toward the speed of sound. In other words, subsonic flow through a pipe with friction will accelerate, and supersonic flow will decelerate. If the pipe length is long enough that the flow velocity would pass through unity, then the flow chokes such that the flow exiting the pipe is at Mach 1. As with the nozzle, this is achieved either through the flow rate at the inlet being limited, or the formation of shock waves in the pipe (for supersonic flows). For the [[Adiabatic process|adiabatic]] flow of an ideal gas model, the effects of friction may be calculated using the [[Fanno flow]] model. For a constant friction factor, the model is given by<ref name="zucker">Zucker, R.D., Biblarz, O., ''Fundamentals of Gas Dynamics'', John Wiley & Sons, 2002.</ref>
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| :<math>\ 4\frac{fL^*}{D_h} = \left(\frac{1 - M^2}{\gamma M^2}\right) + \left(\frac{\gamma + 1}{2\gamma}\right)\ln\left[\frac{M^2}{\left(\frac{2}{\gamma + 1}\right)\left(1 + \frac{\gamma - 1}{2}M^2\right)}\right]</math>
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| where <math>f</math> is the [[Fanning friction factor]], <math>L^*</math> is the required pipe length passed the point being considered that would result in the flow choking, and <math>D_h</math> is the [[hydraulic diameter]] of the pipe.<ref>Hodge, B.K., and Koenig, K., ''Compressible Fluid Dynamics with Personal Computer Applications'', Prentice Hall, 1995.</ref>
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| === Effect of heat transfer ===
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| Adding heat to a fluid flowing at subsonic velocities in a pipe will cause the flow to accelerate, and adding heat to supersonic flow in a pipe will cause the flow to decelerate. As with the cases of friction and area change discussed above, adding more heat than that required to reach a Mach number of 1 will result in the flow choking.
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| For an ideal gas in a constant area pipe, the effect of heat addition to the pipe may be calculated using the [[Rayleigh flow]] model, which describes how the Mach number varies with changes in the [[stagnation temperature]]. The stagnation temperature at a point is the temperature that the fluid would reach if it were to be slowed isentropically to stationary. As heat is added to the system, the stagnation temperature increases. The Rayleigh flow model is given by
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| :<math>\ \frac{T_0}{T_0^*} = \frac{2\left(\gamma + 1\right)M^2}{\left(1 + \gamma M^2\right)^2}\left(1 + \frac{\gamma - 1}{2}M^2\right) </math>
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| where <math>T_0</math> and <math>T_0^*</math> represent the stagnation temperatures at the point under consideration, and at the point at which the Mach number is 1 respectively.<ref name="zucker"/>
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| == Shock Tubes ==
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| In addition to measurements of rates of chemical kinetics, [[shock tube]]s have been used to measure dissociation energies and molecular relaxation rates, investigate shock wave behaviour, and they have been used in aerodynamic tests. The fluid flow in the driven gas (the gas behind the shock wave) can be used much as a wind tunnel, allowing higher temperatures and pressures replicating the conditions in the turbine sections of jet engines. However, test times are limited to a few milliseconds, either by the arrival of the contact surface or the reflected shock wave.
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| They have been further developed into shock tunnels, with an added nozzle and dump tank. The resultant high temperature hypersonic flow can be used to simulate atmospheric re-entry of spacecraft or hypersonic craft, again with limited testing times.
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| == See also ==
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| * [[Gas dynamics]]
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| * [[Transonic flow]]
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| * [[Choked flow]]
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| * [[Hypersonic flow]]
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| * [[Fanno flow]]
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| * [[Rayleigh flow]]
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| * [[Isothermal flow]]
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| * [[Mach number]]
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| * [[Aerodynamics]]
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| * [[Nozzle]]
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| ==References==
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| {{Reflist}}
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| * {{cite book
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| | last = Shapiro | first = Ascher H.
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| | title = The Dynamics and Thermodynamics of Compressible Fluid Flow, Volume 1
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| | publisher = [[Ronald Press]]
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| | isbn = 978-0-471-06691-0 | year = 1953 }}
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| * {{cite book| last = Anderson| first = John D.| title = Modern Compressible Flow| year = 2004| publisher = [[McGraw-Hill]]| isbn = 0-07-124136-1 }}
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| * {{cite book| coauthors = Roshko A.| last = Liepmann| first = H. W.| title = Elements of Gasdynamics| year = 2002| publisher = [[Dover Publications]]| isbn = 0-486-41963-0 }}
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| * {{cite book| last = von Mises| first = Richard| title = Mathematical Theory of Compressible Fluid Flow| year = 2004| publisher = [[Dover Publications]]| isbn = 0-486-43941-0 }}
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| * {{cite book| last = Meyer| first = Richard E.| title = Introduction to Mathematical Fluid Dynamics| year = 2007| publisher = [[Dover Publications]]| isbn = 0-486-45887-3 }}
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| * {{cite book| last = Saad| first = Michael A.| title = Compressible Fluid Flow| year = 1985| publisher = [[Prentice Hall]]| isbn = 0-13-163486-0 }}
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| * {{cite book| last = Schreier| first = S.| title = Compressible Flow| year = 1982| publisher = [[Wiley-Interscience]]| isbn = 0-471-05691-X }}
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| * {{cite book| last = Lakshminarayana| first = B.| title = Fluid Dynamics and Heat Transfer of Turbomachinery| year = 1995| publisher = [[Wiley-Interscience]]| isbn = 978-0-471-85546-0 }}
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| ==External links==
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| * [http://www.grc.nasa.gov/WWW/K-12/airplane/bgc.html NASA Beginner's Guide to Compressible Aerodynamics]
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| * [https://engineering.purdue.edu/~wassgren/applet/java/comp_calculator/Index.html Purdue University Compressible Flow Calculators]
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| * [http://www.potto.org/gasDynamics/node40.php]
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| {{DEFAULTSORT:Compressible Flow}}
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| [[Category:Fluid mechanics]]
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| [[Category:Aerodynamics]]
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