Understanding Compressibility of a Flow Comprendiendo La Comprensibilidad De Un Flujo

Understanding Compressibility of a flow Comprendiendo la comprensibilidad de un flujo

Recibido: marzo de 2016 | Revisado: abril de 2016 | Aceptado: mayo de 2016

Luis A. Arriola1 Abstract The coverage of the physical phenomena experienced in compressible flow is somehow difficult to understand. However, here we provide a brief explanation of the assumptions used in the analysis of compressible flows. A strong foundation for more advanced and focused study and understanding of what causes compressible flows to differ from incompressible flows and how they can be analyzed is vital to resolve realistic problems in aerodynamics. Compressibility issues arise when fluids are moving at velocities higher than the speed of sound which is the equivalent of Mach number 0.3. It is vital to understand the behavior of the fluid beyond this number because compressibility factors affect the fluid and it would be necessary to account for such factors in order to get results closer to the real solution. Keywords: compressible, flow, incompressible, aerodynamics, Mach number, speed, sound

Resumen La cobertura de los fenómenos físicos experi- mentados en el flujo compresible es algo difícil de entender. Sin embargo, aquí damos una ex- plicación breve acerca de las hipótesis empleadas en el análisis de flujo compresible. Una base só- lida para un estudio más avanzado y enfocado, y la comprensión de las causas de flujo compresible para diferenciar de los flujos incompresibles y la forma en que se pueden analizar es vital para resolver problemas reales en aerodinámica. Los asuntos de comprensibilidad aparecen cuando los fluidos se están moviendo con una velocidad mayor que la velocidad del sonido la cual es equivalente al valor del número Mach 0.3. Es importante entender el comportamiento del fluido por encima de este valor de número Mach porque los factores de comprensibilidad afectan al fluido y se necesitaría tomar en cuenta tales factores para conseguir resultados cercanos a la solución real. 1 Facultad de Ingeniería y Arquitectura Escuela Profesional de Ciencias Aeronáuticas Universidad de San Martín de Porres, Palabras clave: compresible, flujo, incompre- Lima - Perú sible, aerodinámica, número Mach, velocidad, [email protected] sonido

Introduction However, the simple definition of compressible flow as one in which the density is High speed flows are frequently encoun- variable requires more elaboration. Consider tered in engineering applications. Flow a small element of fluid of volume “v” as features such as shock waves in nozzles or shown in Figure 1. The pressure exerted on supersonic wind tunnels, expansion waves, the sides of the element by the neighboring and oblique or bow shock waves in front of fluid is p. rapidly moving objects are all examples of in- triguing phenomena that occur due to high speeds and compressibility of fluids. Com- pressible fluid flow theory is intended to co- ver compressible behavior of fluids. Here, we Figure 1. Small element of fluid of volume v provide a brief introduction to compressible flows, including fundamentals of isentropic compressibility and isothermal compressibility. This phenomenon takes place in certain regions of speed of the Assumeflow such the aspressure high is now increased by an infinitesimal amount dp. The volume of the element subsonic, transonic, sonic, supersonic and Figure 1. Small element of fluid of volume v hypersonic velocities. will be correspondingly compressed by the amount dv. Since the volume is reduced, dv is a Assume the pressure is now increased by However, normal shocks, oblique and ex- negative quantity. The compressibilityan infinitesimal of amountthe fluid, dp. Ʈ, isThe defined volume by of Anderson, Jr. (2003) as pansion shocks may appear in these pheno- Figure 1. Small element of fluid of volume v the element will be correspondingly com- mena contributing to the complexity of the pressed by the amount dv. Since the volu- solution. Properties of the flow may change me is reduced, dv is a negative quantity. The dramatically right after the shocks affecting compressibility of the fluid, Ʈ, is defined by the whole solution of the fluid flow in this Anderson, Jr. (2003) as area and the surroundingAssume the pressure is now increased by an infinitesimal amount dp. The volume of the element Compressible flow is routinely definedDiscussion aswill variable be correspondingly density flow; compressedthis is in bycontrast the amount to dv. Since the volume is reduced, dv is a τ= - (2) (2) incompressible flow, where theCompressible density is assumedflownegative is toroutinely be quantity. constant defined The throughout. compressibility Obviously, of the in fluid, real1 𝑑𝑑𝑑𝑑 Ʈ,𝑣𝑣𝑣𝑣 is defined by Anderson, Jr. (2003) as as variable density flow; this is in contrast 𝑣𝑣𝑣𝑣 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 life every flow of every fluidto incompressible is compressible flow, to somewhere greaterthe density or lesser extent;Physically, hence, the a compressibilitytruly is the frac- is assumed to be constant throughout. Ob- tional change in volume of the fluid element constant density (incompressible)viously, inflow real is life a everymyth. flow However, of every as fluidpreviously is permentioned, unit change for inalmost pressure. However, equa- transferred into or out of the gas through the boundaries of the system. Therefore, if the temperature compressible to some greater or lesser extent; tion 2 is not sufficiently precise. We know all liquid flows as well as forhence, the flows a truly of constant some gases density under (incompres certain conditions,- from the experience density changes that when a gas is compres- of the fluid element is heldsed, constant its temperature (due to some tends heat to transferincrease, mechanism), de- then the isothermal sible) flow is a myth. However,Physically, as previously the compressibility is the fractional change in volume of the fluid element per unit are so small that the assumption of constant density can be made with reasonablepending accuracy. on the Inamount such of heat transferred mentioned, for almost allcompressibility liquid flows as is welldefined by Anderson,τ= -Jr. (2003) as (2) as for the flows of some gases under certain into or out of the gas through the bounda- cases, Bernoulli’s equation 1, defined by Anderson,change Jr. (1989) in pressure. can be appliedHowever, with equation confidence.2 is1 not𝑑𝑑𝑑𝑑 𝑣𝑣𝑣𝑣 sufficiently precise. We know from experience that conditions, the density changes are so small ries of the system. Therefore, if the tempe- 𝑣𝑣𝑣𝑣 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 that the assumption of constantwhen a densitygas is compressed,can rature its oftemperature the fluid elementtends to is increase, held constant depending on the amount of heat be made with reasonable accuracy. In such (due to some heat transfer mechanism), then cases, Bernoulli’s equation 1, defined by An- the isothermal compressibility is defined by 5 derson, Jr. (1989) can be applied with con- Anderson, Jr. (2003) as fidence. Physically, the compressibility is the fractional= change( ) in volume (3)of the fluid element per unit(3 ) + ρ = (1) (1) 1 1 𝜕𝜕𝜕𝜕𝑣𝑣𝑣𝑣 2change in pressure. However, equation 2𝑇𝑇𝑇𝑇 is not sufficiently𝑇𝑇𝑇𝑇 precise. We know from experience that 𝑝𝑝𝑝𝑝 2 𝑉𝑉𝑉𝑉 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝜏𝜏𝜏𝜏 − 𝑣𝑣𝑣𝑣 𝜕𝜕𝜕𝜕𝑑𝑑𝑑𝑑 when a gas is compressed, its temperature tends to increase, depending on the amount of heat 58 | Campus | V. XXI | No. 21 | enero-junio | 2016 | On the other hand, if no heat is added to or taken away from the fluid element (if the compression5

is adiabatic), and if no other dissipative transport mechanisms such as viscosity and diffusion are However, the simple definition of compressible flow as one in which the density is variable important (if the compression is reversible), then the compression of the fluid element takes place requires more elaboration. Consider a small element of fluid of volume “v” as shown in figure 1. isentropically as described by Shapiro (1953), and the isentropic compressibility is defined by The pressure exerted on the sides of the element by the neighboring fluid is p. Anderson, Jr. (2003) as

= ( ) (4) 1 𝜕𝜕𝜕𝜕𝑣𝑣𝑣𝑣 𝜏𝜏𝜏𝜏𝑠𝑠𝑠𝑠 − 𝑣𝑣𝑣𝑣 𝜕𝜕𝜕𝜕𝑑𝑑𝑑𝑑 𝑠𝑠𝑠𝑠

where the subscript s denotes that the partial derivative4 is taken at constant entropy.

Compressibility is a property of the fluid. Liquids have very low values of compressibility (ƮT for

-10 2 -5 2 water is 5x10 m / N at 1 atm) whereas gases have high compressibility (ƮT for air is 10 m / N

6 transferred into or out of the gas through the boundaries of the system. Therefore, if the temperature of the fluid element is held constant (due to some heat transfer mechanism), then the isothermal compressibility is defined by Anderson, Jr. (2003) as

= ( ) (3) 1 𝜕𝜕𝜕𝜕𝑣𝑣𝑣𝑣 𝜏𝜏𝜏𝜏𝑇𝑇𝑇𝑇 − 𝑣𝑣𝑣𝑣 𝜕𝜕𝜕𝜕𝑑𝑑𝑑𝑑 𝑇𝑇𝑇𝑇

Understanding Compressibility of a flow On the other hand, if no heat is added to or taken away from the fluid element (if the compression is adiabatic), and if no other Ondissipative the other transport hand, if mechanismsno heat is added such to as viscosityges in theand pressure. diffusion In are particular, we shall see or taken away from the fluid element (if the that high-speed flows generally involve lar- important (if the compressioncompression is reversible), is adiabatic), then the andcompression if no other of the fluidge pressure element gradients. takes place For a given change in dissipative transport mechanisms such as pressure, dp, due to the flow, equation 7 de- isentropically as describedviscosity by Shapiro and diffusion (1953), andare importanttherelatively isentropic large(if the compressibilitypre ssuremonstrates gradients is thatcan defined createthe resulting byhigh velocities change withoutin den- much change in density. compression is reversible), then the compres- sity will be small for liquids (which have low sion of the fluid element takes place isentro- values of Ʈ), and large for gases (which have Anderson, Jr. (2003) as Hence, such flows are usually assumed to be incompressible, where ρ is constant. On the other pically as described by Shapiro (1953), and high values of Ʈ). Therefore, for the flow of the isentropic compressibility is defined by liquids, relatively large pressure gradients can Anderson, Jr. (2003) as hand, for the flow of gasescreate with high their velocities attendant without large values much of change Ʈ, moderate to strong pressure in density. Hence, such flows are usually gradients lead to substantialassumed changes to inbe the incompressible, density via equation where7. At ρ theis same time, such pressure at 1 atm). If the fluid element is assumed to have unit mass, v is (4)the specific volume (volume per = ( ) constant. On the other(4) hand, for the flow 1 𝜕𝜕𝜕𝜕𝑣𝑣𝑣𝑣gradients create large velocity changes in the gas. Such flows are defined as compressible flows, unit mass), and the density is of gases with their attendant large values of where the subscript𝜏𝜏𝜏𝜏𝑠𝑠𝑠𝑠 −s denotes𝑣𝑣𝑣𝑣 𝜕𝜕𝜕𝜕𝑑𝑑𝑑𝑑 that𝑠𝑠𝑠𝑠 the partial Ʈ, moderate to strong pressure gradients derivative is taken at constantwhere entropy. ρ is a variable. lead to substantial changes in the density via equation 7. At the same time, such pressure at 1 atm). If the fluid elementCompressibility is assumed to is have a property unit mass, of the v fluid.is the specificgradients volume create (volume large velocityper changes in the where the subscript s denotes that the partial derivative is taken at constant entropy. Liquids have very low values ofIf thecompressibi velocity of- gasesgas. is less Such than flows 0.3 than are the defined speed of as sound compressible (M<0.3), the associated pressure at 1 atm). If the fluid elementƮ is assumed to have -10unit 2 mass, v is the specific volume (volume per unit mass), and the densitylity is ( T for water is 5x10= m / N at 1 atm) flows, whereρ is a variable.(5) whereas gases have high compressibilitychanges are small,(Ʈ and even though Ʈ is large for gases, dp in equation 7 may still be small enough Compressibilityunit mass), and isthe a densityproperty is of the fluid. Liquids 1have very low valuesT of compressibility (ƮT for for air is 10 -5 m2/ N at 1 atm). If the fluid If the velocity of gases is less than 0.3 than 𝑣𝑣𝑣𝑣 -10 2 𝜌𝜌𝜌𝜌 -5 2 waterIn terms is 5x10 of density, m / Nequation at 1element atm)2 becomeswhereas is assumed gases to have have high unitto dictate compressib mass, a vsmall is ility dρ .(Ʈthe ForT forspeed this air reason, ofis 10sound them (M<0.3),/low N -speed the flow associated of gases can be assumed to be the specific volume (volume per unit mass), pressure changes are small, and even though and the density is Ʈ is large for gases, dp in equation 7 may still = incompressible as shown in figure 2. This is velocities(5) less than 250 mi/h (112 m/s). On the other be small enough to dictate a small dρ. For = ( 1) (6) = (5) this reason, the low-speed(5) flow of gases can 1 𝑑𝑑𝑑𝑑𝜌𝜌𝜌𝜌 hand, for flow velocities higher than 0.3 of the 6speed of sound (M>0.3), the associated pressure 𝜌𝜌𝜌𝜌 1𝑣𝑣𝑣𝑣 be assumed to be incompressible as shown in In terms of density, equation 2 becomesƮ 𝜌𝜌𝜌𝜌 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 In terms of density,𝜌𝜌𝜌𝜌 equation𝑣𝑣𝑣𝑣 changes, 2 becomes dp are relativelyFigure large, 2. and This having is velocities a large value less than of Ʈ for250 gases, mi/h large changes in density, In terms of density, equation 2 becomes (112 m/s). On the other hand, for flow velocities higher than 0.3 of the speed of sound = ( ) dρ are produced via equation 7 (6) Therefore, whenever the fluid experiences a change in pressure,(6) dp, the corresponding(M>0.3), the associated change in pressure changes, dp 1 𝑑𝑑𝑑𝑑𝜌𝜌𝜌𝜌 = ( ) are relatively large, and(6 having) a large value Ʈ ρ density will be dρ, where from equationƮ 6 𝜌𝜌𝜌𝜌1 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝜌𝜌𝜌𝜌 of for gases, large changes in density, d Therefore, whenever the fluid experiences are produced via equation 7 a change in pressure,Ʈ 𝜌𝜌𝜌𝜌 dp,𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 the corresponding change in density will be dρ, where from Therefore, whenever the equationfluid experiences 6 a change in pressure, dp, the corresponding change in densityTherefore, will whenever be dρ, where the fluidfrom experiencesequation 6 =a change in pressure,(7) dp, the corresponding change( 7in) density will be dρ, where from equation 6 To this point,𝑑𝑑𝑑𝑑𝜌𝜌𝜌𝜌 we have𝜌𝜌𝜌𝜌Ʈ considered𝑑𝑑𝑑𝑑𝑝𝑝𝑝𝑝 just the fluid itself, with compressibility being a property of the fluid. Now assume that the fluid To this point, we have consideredis in motion. just According the fluid= itself, to Zucrow with compressibilityand Ho- being a property of (the7) ffman (1976), such flows are initiated and = (7) fluid. Now assume that themaintained fluid is inby𝑑𝑑𝑑𝑑 motion.forces𝜌𝜌𝜌𝜌 𝜌𝜌𝜌𝜌on AccordingƮ the𝑑𝑑𝑑𝑑𝑝𝑝𝑝𝑝 fluid, to usuallyZucrow andFigure Hoffman 2. Compressibility(1976), such Range of 0.3 M created by, or at least accompanied by, chan- Figure 2. Compressibility Range of 0.3 M flows are initiated and maintained by𝑑𝑑𝑑𝑑𝜌𝜌𝜌𝜌 forces𝜌𝜌𝜌𝜌 Ʈon𝑑𝑑𝑑𝑑 𝑝𝑝𝑝𝑝the fluid, usually created by, or at least To this point, we have considered just the fluid itself, with compressibility being a property of the accompanied by, changes| Campus in the | pressure.V. XXI | No. In 21particular, | enero-junio we shall | 2016 see | that high-speed flows 59 To this point, we have considered just the fluid itself, with compressibility being a property of the generallyfluid. Now involve assume large that pressure the fluid gradients. is in motion. For a According given chan toge Zucrowin pressure, and dp,Hoffman due to the(1976), flow, s uchEq. fluid. Now assume that the fluid is in motion. According to Zucrow and Hoffman (1976), such 7flows demonstrates are initiated that theand resulting maintained change by in forcesdensity on will the be smallfluid, forusually liquids created (which haveby, or low at values least flows are initiated and maintained by forces on the fluid, usually created by, or at least 8 ofaccompanied Ʈ), and large by, forchanges gases in (which the pressure. have high In values particular, of Ʈ). we Therefore, shall see for that the high flow-speed of liquids, flows generallyaccompanied involve by, largechanges pressure in the gradients. pressure. For In a givenparticular, chan gewe in shallpressure, see dp,that due high to -thespeed flow, flows Eq. 7 7generally demonstrates involve that large the resultingpressure gradients.change in densityFor a given will chanbe smallge in for pressure, liquids (whichdp, due haveto the low flow, values Eq. of7 demonstrates Ʈ), and large that for the gases resulting (which change have in highdensity values will be of small Ʈ). Therefore, for liquids for (which the flowhave low of liquids, values of Ʈ), and large for gases (which have high values of Ʈ). Therefore, for the flow of liquids, 7

Luis A. Arriola

Conclusion

In summary, a compressible flow is con- sity, dρ/ρ, which is too large to be ignored. sidered as one where the change in pressure, For most practical problems, if the density dp, over a characteristic length of the flow, changes by 5% or more, the flow is conside- multiplied by the compressibility via equa- red to be compressible. tion 7, results in a fractional change in den-

References

Anderson, Jr., J. D. (1989). Introduction to Shapiro, A. (1953). The dynamics and ther- Flight (3rd ed.). New York: McGraw modynamics of compressible fluid flow. Hill. New York: The Ronald Press Com- pany. Anderson, Jr., J. D. (2003). Modern Com- pressible Flow with Historical Perspecti- Zucrow, M. & Hoffman, J. (1976). Gas ve. New York: McGraw Hill. Dynamics. New York: Wiley

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