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Compressible Flow Notes

By Dr. Dianne J. DeTurris

Table of Contents

Nomenclature .......................................................................................................................................................... 22.0 Review of Thermodynamics and Compressible Flow .................................................................................. 3

2.1 Thermodynamics Review ............................................................................................................................. 32.2 Stagnation Properties .................................................................................................................................. 42.3 Gas Dynamics - Isentropic Flow With Area Change ............................................................................... 6

2.3.1 Critical Mass Flow Rate .......................................................................................................................... 72.4 Normal Shocks ........................................................................................................................................... 102.5 Oblique Shocks and Expansion Waves .................................................................................................... 12

2.5.1 Oblique Shock Waves ........................................................................................................................... 122.5.2 Expansion Waves .................................................................................................................................. 15

2.6 Effect of Back Pressure on Flow in Converging-Diverging Nozzles ..................................................... 17References ............................................................................................................................................................. 18

December 17, 1903

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Nomenclature

units Compressible flow: a speed of sound m/sec h enthalpy J/kg M Mach number T static temperature K P static pressure Pa P0 stagnation pressure Pa !m mass flow rate kg/sec

Cp specific heat at constant pressure J/kgK Cv specific heat at constant volume J/kgK R gas constant m2/sec2K ρ density kg/m3 V velocity m/sec u internal energy J/kg z altitude m g gravity m/sec2 s entropy J/kgK ε oblique shock wave angle degree ν specific volume m3/kg

Air at STP Pa=14.7 psi=101325 N/m2

R= 287 m2/sec2K, 1716 ft2/sec2R

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2.0 Review of Thermodynamics and Compressible Flow

This chapter is intended as a review of the thermodynamic principles that are used in deriving compressible flow equations. The gas dynamics concepts are used as a foundation for the cycle analysis of jet engines. At the very least, this review will serve as an introduction to the nomenclature that will be used in the remainder of these course notes. The compressible flow equations presented in this section are tabulated in many textbooks, such as Zucrow & Hoffman, Anderson, Saad, and Shapiro. They can also be found online. The tabulated equations include isentropic flow, normal shocks, oblique shocks, and expansion waves.

2.1 Thermodynamics Review

An ideal gas has the thermodynamic properties of being both thermally and calorically perfect. For a thermally perfect gas, the equation of state, P RTρ= , holds where R is gas constant defined as

massmolecularconstgasuniversal

MR

R u

−

−−==

For a calorically perfect gas, a relationship between the gas constant and the specific heats of the gas can be written as RCC vp =− , where the specific heat at constant pressure, Cp, and the specific heat at constant volume, Cv, are both constant. With the definition of the ratio of

specific heats as v

p

CC

=γ , the relationship between the specific heats can also be written a

1−

=γγRCp

The First Law of Thermodynamics relates the energy of a system to heat and work.

ΔE =Q −W = ΔU +ΔKE +ΔPE where ∆𝐸 is the total energy change of system, Q is the heat transferred to the system and W is the work done by the system. An isentropic process is defined as adiabatic and reversible. The heat transfer is zero for an adiabatic process (∆𝑄=0), and a reversible process is defined as one in which there is no friction (∆𝑈). The first law written in intensive properties is

gzVue ++=2

2

specific total energy = internal energy + kinetic energy + potential energy The Second Law of Thermodynamics and entropy for a closed system with an internally

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reversible process can be written as Tds du Pdv= + To define property changes for an isentropic process, it is necessary to define enthalpy from the second law in terms of internal energy, h u Pdv≡ + and for an ideal gas, this becomes dTCdh p= which integrates to Δh =CpΔT

For entropy change in a reversible process, the Second Law can also be written as

vdvR

TdTCds v += or p

dT dPds C RT P

= − where integration of the second equation gives

2 22 1

1 1

ln lnpT Ps s C RT P

− = −

Recall that Cp is constant for a calorically perfect gas. If s1=s2, the change of state is isentropic and

1

2 2

1 1

T PT P

γγ−

⎡ ⎤= ⎢ ⎥⎣ ⎦

This shows that the relationship between pressure and temperature ratio’s in a system for an isentropic process change is only a function of the ratio of specific heats of the gas. 2.2 Stagnation Properties

Stagnation conditions are defined at a fluid-solid boundary where the fluid element has been isentropically brought to rest. To illustrate this, consider the rocket with a rounded nosecone flying through the air as shown in Figure 2.1. The incoming flow that touches the foremost tip of the nose necessarily has a velocity of zero. Because the flow exactly at the center neither gets diverted up toward the top or down toward the bottom, it must slow down and stop at the tip. This stopping process is isentropic and the point at the tip of the nose is called a stagnation point. The flow that gets stopped at any other point on the body goes through a boundary layer where it is acted upon by frictional (irreversible) forces. This defines a static condition, which is not isentropic.

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Rocket nosecone

flow stagnation

point

Figure 2.1 Stagnation Point at Tip of Nose of Rocket

For incompressible flow, the Bernoulli equation gives the relationship between static and stagnation pressure as

20 2

1 VPP ρ+=

total = static + dynamic The stagnation (total) pressure, P0, is defined at the stagnation point where the fluid velocity is zero. Figure 2.2 depicts a sketch of a pitot tube, the opening at the front indicates where the stagnation pressure can be read by virtue of the flow being stopped isentropically at that location. The static pressure, P, is measured from the wall where the flow passes through the boundary layer and is not isentropically brought to rest. This measurement technique is used to calculate the velocity of a moving vehicle in both incompressible and compressible flow. stagnation pressure

P0 pitot tube

boundary layer static pressure, P

Figure 2.2 Pitot Tube Measures Stagnation Pressure

For compressible flow, the above Bernoulli equation does not hold. The new relationship between static and stagnation conditions is derived from the one dimensional energy equation. The stagnation (total) temperature, To, is defined in the same way as P0 at the same point in the flow where the flow has been isentropically brought to rest and the velocity is equal to zero. The stagnation enthalpy, ho, is a constant that is equal to the sum of the static enthalpy and the kinetic energy of the system. In terms of enthalpy, the energy equation can be written as

2 2

0 0 2 2p pV Vh C T h C T const= = + = + =

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Dividing both sides by CpT and substituting in1−

=γγRCp , the above equation yields an

equation in which the stagnation to static temperature ratio can be written in terms of Mach number and ratio of specific heats.

222

0

211

21

21 M

RTV

CR

TCV

TT

pp

−+=+=+=γ

γγ

Note that the Mach number in the above equation is defined as the ratio of directed kinetic energy to random kinetic energy in a flow. This translates to velocity of the flow over the speed of sound, 𝑀 = !

!. The square of the acoustic speed of sound, a, is defined as the ratio of

the isentropic bulk modulus, Ks, and the density. For a perfect gas, this equation can also be written as

RTPKa s γ

ργ

ρ===2

Note that the definition of stagnation speed of sound becomes 00 RTa γ= . Using the isentropic thermodynamic relationship above at the end of the thermodynamics review section, the corresponding pressure ratio can be written as

12100

211

−−

⎥⎦

⎤⎢⎣

⎡ −+=⎥

⎦

⎤⎢⎣

⎡=

γγ

γγ

γ MTT

PP

In the same way, stagnation density can be defined by the equation of state as

0

00 RT

P=ρ

and the ratio of static to stagnation density can also be written in terms of Mach number and the ratio of specific heats

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20

211

−

⎥⎦

⎤⎢⎣

⎡ −+=

γγρρ

M

2.3 Gas Dynamics - Isentropic Flow With Area Change The relationship between area change and flow properties is different for subsonic and supersonic flow. The relationship can be shown mathematically from the governing equations for fluids. For steady, 1D, isentropic flow with no body forces, the logarithmic differentiation form of the continuity and momentum equations becomes

0=++VdV

AdAd

ρρ

and

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2 0dP dVVVρ

+ =

If the definition of speed of sound in differential form 2 dPadρ

= is added,

then the momentum equation can be rewritten as

0222

=+=+VdVMd

VdVVda

ρρ

ρρ

Combining this with continuity yields

02 =++−VdV

AdA

VdVM

which can be written as

0)1( 2 =−=VdVM

AdA

This relationship shows that if the Mach number is subsonic, as area decreases, velocity increases. However if the Mach number is supersonic, then as area decreases, the velocity also decreases. This means that subsonic and supersonic flow have opposite reactions to area changes. It is easy to demonstrate this principle using a converging-diverging nozzle. 2.3.1 Critical Mass Flow Rate A converging-diverging, or DeLaval nozzle, has a converging section followed by a diverging section. The location where the area is smallest is known as the throat, shown in Figure 2.3. The continuity equation requires that mass flow rate is constant through a duct with simple area change, therefore, the nozzle can change the flow velocity without changing the mass flow rate. If the limiting condition at the throat is met, the flow can either accelerate from subsonic to supersonic, or accelerate to sonic and then slow back down to subsonic. At its limit, the mass flow rate is equal to the critical mass flow rate, !m* , and the flow achieves exactly Mach 1 at the throat. For an isentropic nozzle, the stagnation properties like pressure and temperature remain constant throughout the nozzle.

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M=1 flow

ßthroat à

*mm !! < for M<1 *mm !! = for M=1

Figure 2.3 Critical Mass Flow Rate

Now consider a converging-diverging nozzle attached to a tank of compressed air as sketched in Figure 2.4. The air outside of the tank has ambient conditions of Pa and Ta. The velocity of the flow in the tank is zero, therefore, the pressure and temperature inside the tank represent the stagnation conditions of the flow at that point. If a valve is opened and the flow travels through the nozzle, it accelerates until it reaches the throat and then accelerates or decelerates from there depending on whether or not the mass flow rate is at its maximum. If we consider the flow through the nozzle to be isentropic, then the stagnation properties remain the same as in the tank throughout. The static temperature and pressure at the exit, Pe and Te, depend on the exit Mach number of the flow.

PP ≠0 and TT ≠0 unless V=0

Figure 2.4 Static and Stagnation Conditions in a Tank and CD Nozzle

The flow from the tank to the nozzle exit starts out at zero velocity and then can either remain subsonic or accelerate to supersonic through the nozzle. The condition that occurs at the throat of the nozzle when the flow is choked is referred to as the critical condition. The area at the

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throat when M=1 is known as A*. The critical flow area can also be written in terms of M and gamma by starting with the continuity equation

!m = !m* = ρAV = ρ*A*V *

Since by definition the mass flow rate is constant, the area ratio becomes

VV

AA

ρρ **

* =

Combining the equation of state, the energy equation and isentropic relations, the ratio can be written in terms of Mach number as

)1(21

2* )

211)(

12(1 −

+

⎥⎦

⎤⎢⎣

⎡ −+

+=

γγ

γγ

MMA

A

This equation is tabulated in compressible flow tables for isentropic flow. The same process is used to obtain the critical speed of sound. Starting with energy equation

22

22

2

21

10VTCVTCh pp +=+=

use 1−

=γγRCp and RTa γ=2

to get 2121

22

22

21

21 VaVa

+−

=+− γγ

⎥⎦

⎤⎢⎣

⎡ −+=⎥⎦

⎤⎢⎣

⎡ −+ 2

222

21

21 2

11211 MaMa γγ then set M2=1

⎥⎦

⎤⎢⎣

⎡ +=⎥⎦

⎤⎢⎣

⎡ −+

21*

211 222 γγ aMa

The critical temperature, T*, is easier to obtain, needing only the isentropic relation

20

211 M

TT −

+=γ

10

set M=1, then T=T* and

012* TT+

=γ

The corresponding critical pressure is defined as

0

1

12* PP

−

⎟⎟⎠

⎞⎜⎜⎝

⎛

+=

γγ

γ

The previously discussed throat parameter, the critical mass flow rate, *m! , is the maximum mass flow rate possible in a converging diverging duct. The equation is derived from continuity, the equation of state and the isentropic relations for pressure and temperature.

!m = ρAV =PRTAV =

PA γ

RTVγRT

= PMA γRT

using M=1, P RTρ= and 1

00−

⎥⎦

⎤⎢⎣

⎡=

γγ

TT

PP and 20

211 M

TT −

+=γ

you can write

!m* = A*γP0a0

2γ +1⎡

⎣⎢

⎤

⎦⎥

γ+12(γ−1)

This equation enables finding the critical mass flow rate just by knowing the stagnation conditions and the geometry at the throat. This relation is very useful in sizing nozzles for supersonic flow. 2.4 Normal Shocks A normal shock is modeled as a pure discontinuity with isentropic flow on either side. The actual shock is on the order of 0.00001 inch thick. The shock wavefront is perpendicular to the flow direction and is considered an instantaneous compression of gas. At the point where the shock occurs, the flow is slowed from supersonic to subsonic and the kinetic energy in the flow turns to heat, which increases entropy. A normal shock is not a reversible process. Flow across a normal shock is not isentropic and the flow properties change. The velocity decreases, the static pressure increases, the static temperature increases and the static density

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increases. The Mach number decreases. The stagnation temperature remains the same, but the stagnation pressure drops across the shock. Shocks can be mathematically modeled assuming the following:

1) Frictionless streamtube 2) 1D, constant area 3) Adiabatic 4) No external work 5) No body forces 6) Shock perpendicular to streamlines 7) Irreversible

The governing equations for flow across a discontinuity are as follows, with 1 representing before the shock and 2 after.

Continuity !m = ρ1V1 = ρ2V2 for a constant area

Momentum 2 21 1 1 2 2 2P V P Vρ ρ+ = + , which is not the same as for isentropic flow

Energy: 2 21 2

1 2 02 2V Vh h h+ = + = , which holds for an adiabatic flow

The condition that the flow is adiabatic means that the stagnation enthalpy does not change across a normal shock. Since Δh0 =CpΔT0 , then also T01=T02. Note that all other flow properties will change across a shock. The fact that the stagnation temperatures are constant allows us to find the change in Mach number across the shock by starting with the isentropic equations for temperature.

21

2

212

112112

MTT M

γ

γ

−+

=−

+

Now adding the equation of state, the continuity equation and the definition of Mach number allows the solving for the ratio T2/T1 as

2 22 22 2 1 2 2 2 2 2

1 2 1 1 1 1 1 11 1

M RTT P R P V P P MT R P P V P P MM RT

γρρ γ

⎡ ⎤ ⎡ ⎤= = = = ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦

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To get the static pressure ratio P2/P1 in terms of Mach number, go back to the momentum equation and the definition of the speed of sound, which gives

2

2 12

1 2

11

P MP M

γγ

+=

+

Then combining

2 2 2212 1 2

221 2 12

11 121 112

MT M MT M MM

γγ

γ γ

−+ ⎡ ⎤ ⎡ ⎤+

= = ⎢ ⎥ ⎢ ⎥− +⎣ ⎦ ⎣ ⎦+

The solution of this equation for M2 in terms of M1 can eventually be written as

21

22

21

21

2 11

MM

M

γγ

γ

+−

=−

−

Similar equations can be written for the static pressure, static temperature and stagnation pressure changes across the normal shock. The equations for these property changes are tabulated in compressible flow tables such that it is not necessary to calculate the values directly from the equations. 2.5 Oblique Shocks and Expansion Waves Oblique shocks occur when a shock hits a compression corner, expansion waves occur when a shock hits an expansion corner, a simple schematic is shown for each in Figure 2.5. First oblique shocks will be discussed, then expansion fans. oblique shock M2 M1 expansion fan M1 M2

Figure 2.5 Oblique Shock and Expansion Fan 2.5.1 Oblique Shock Waves An oblique shock wave is similar to a normal shock in that it is an instantaneous discontinuity, however it forms an angle of less than 90 degrees with the upstream flow.

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Flow direction changes across an oblique shock, unlike across a normal shock. The Mach number on the far side of a weak oblique shock wave is supersonic. The flow across an oblique shock can be analytically modeled by taking the components of the velocity vectors before and after the shock and using the normal components to relate to the shock characteristics as discussed in Section 2.4. The flow is assumed to be steady, two-dimensional planar, adiabatic, with no external work and negligible body forces. Figure 2.6 shows the flow angles are defined such that the flow deflection angle (δ ) designates the angle that the flow turns. The oblique shock wave angle (ε ) is the difference between the incoming flow and the shock wave. VN2 VT2 V1 V2 VT1 VN1 ε δ

Figure 2.6 - Oblique Shock Wave angle and Flow Deflection Angle From the geometries shown, it can be seen that

1 1 sinNV V ε= and 1 1 cosTV V ε= The governing equations are written in terms of the normal and tangential components of the flow. Continuity: 1 1 2 2N NV Vρ ρ= Momentum: 2 2

1 1 1 2 2 2N NP V P Vρ ρ+ = + normal

1 1 1 2 2 2N T N TV V V Vρ ρ= tangential

Energy: 2 21 2

1 22 2N

P pV VC T C T+ = +

From combining the continuity and tangential momentum we can say that 1 2T TV V= because property changes occur only normal to shocks. We also know that by definition 1 2N NV V> since the governing equations for the normal component are the same as for normal shocks. Also, the normal component must be supersonic before the shock and subsonic after the shock which requires that the flow deflect TOWARD the shock.

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The following procedure can be used to solve for the Mach number after the shock given the incoming Mach number and the flow deflection angle. From these two inputs, the oblique shock wave angle is obtained either graphically or from the tables of compressible flow which can be found in numerous textbooks or online, for example see the compressible aerodynamics calculator at http://www.dept.aoe.vt.edu/~devenpor/aoe3114/calc.html The normal shock method starts by first finding the normal component of the incoming Mach number using the relation 1 1 sinNM M ε= . Then, look up the Mach number on the far side of the shock from normal shock tables 1 2N NM M⇒ . Then, the Mach number after the shock can be found by

22 sin( )

NMMε δ

=−

This procedure can also be used to find the static pressure and temperature changes across the oblique shock.

2 2

1 1

P TandP T

⇒

If the flow deflection angle of an oblique shock becomes too large, the shock detaches from the corner and forms a detached shock. In the case of the cone shown in Figure 2.7, this shape shock structure is called a bow shock.

Figure 2.7 Bow Shock in Front of Blunt cone

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2.5.2 Expansion Waves Recall that oblique shocks compress at a turn in the flow, but expansion waves expand around a corner. Figure 2.8 shows Mach number increasing around an expansion corner.

M1 expansion fan M2>M1 M2

Figure 2.8 Mach Increase Across an Expansion Fan An expansion of a continuous succession of Mach waves is called a Prandtl-Meyer expansion wave. Prandtl-Meyer flow is valid for a gradual expansion of supersonic flow along a curved surface or for expansion around a corner. The flow through an expansion fan has consistent properties because the flow is considered a continuous expansion region. Mach number increases, but pressure, temperature and density decrease through an expansion wave. The expansion takes place through a continuous succession of Mach waves, and for each Mach wave the entropy change is zero, therefore, the expansion is isentropic. The Prandtl-Meyer Angle, ν , is the angle through which initially sonic flow M1 must be deflected to a supersonic M2. The forward and rearward Mach lines for M1=1 are shown in Figure 2.9. For M1=1, ν 1 = 0, and ν 2=δ . forward Mach line If M1=1, α 1=90, P=P* rearward Mach line δ =ν 2 M2>1

Figure 2.9 Prandtl-Meyer Angle Definition The Prandtl-Meyer angle is defined for a starting Mach number of 1, but in most cases the supersonic flow starts at a Mach number greater than one. For all general cases,

12 ννδ −=

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The governing differential equation for Prandtl-Meyer flow is

VdVM 12 −−=∂δ

The effects of flow deflection angle on Mach number is best understood with a definition of

Mach angle which is 1sinM

α = , and is illustrated below in Figure 2.10.

M 1 α 2 1M − Figure 2.10 Definition of Mach Angle The definition comes from a geometric study of the supersonic flow. Consider a beeper traveling right to left at supersonic speed V, as shown in Figure 2.11. It emits a beep at time t. At time t +Δ t, the beeper has moved outside of the radius that the sound is traveling, which is aΔ t. The Mach cone is the locus of beeper sound wave fronts, the Mach angle is α .

Mach Cone aΔ t

α

t+Δ t VΔ t t

direction of travel

Figure 2.11 The Mach Cone

1sin a tV t M

αΔ

= =Δ

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However, using the definition of Mach angle, a relationship can be made between the Mach number and the Prandtl-Meyer angle. The relationship [Z&H] can be written as

( )1 2 1 21 1tan 1 tan 11 1

M Mγ γν

γ γ− −+ −

= − − + −− +

Since the Prandtl-Meyer angle is defined for both M1 and M2, it is easy to calculate the necessary flow deflection for a given Mach number change using 12 ννδ −= . One final note to remember about expansion waves is that they are isentropic so stagnation properties are constant across the expansion. 2.6 Effect of Back Pressure on Flow in Converging-Diverging Nozzles The compilation of the compressible flow phenomena presented above is illustrated by a discussion of the effects of back pressure on supersonic flow in a converging-diverging nozzle. If the flow from a compressed gas tank opens into a converging-diverging duct, the flow will start out as subsonic and remain subsonic throughout the entire duct until the mass flow rate of the gas becomes a maximum. These ideas are illustrated in Figure 2.12 taken from Shapiro. Once the critical mass flow rate has been achieved, the flow will reach Mach 1 at the throat. Then in the diverging section of the nozzle, the flow can either accelerate to supersonic or decelerate back to subsonic. The pressure lines labeled (1) and (2) in the figure represent these two conditions. If the flow is supersonic, then the geometry of the duct and the back pressure behind the exit of the nozzle determine whether or not shocks will form in the diverging section, shown as (3) on the diagram. If the exit pressure is greater than the ambient pressure, the flow forms an expansion wave outside the nozzle and the flow is called underexpanded, shown as (7). If the exit pressure is less than the ambient pressure, an oblique shock wave forms at the exit and the flow is considered overexpanded. If the exit pressure is significantly less than the ambient pressure, the oblique shock will begin to move into the nozzle duct itself until eventually the shock gets back to the throat and the flow is forced to become subsonic everywhere.

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The right hand side of the diagram summarizes the flow in four regions. Region I is subsonic, Region II is supersonic with normal shocks. Region III is supersonic with oblique shocks, and Region IV has expansion waves. Figure 2.12 Effect of Back Pressure with Distance Along a Converging Diverging Nozzle [Shapiro]

References 1) Hill, P., and Peterson, C., “Mechanics and Thermodynamics of Propulsion”, Second Edition, Addison-

Wesley, 1992. 2) Zucrow, M.J., and Hoffman, J.D., “Gas Dynamics Volume 1”, Wiley, 1976. 3) Anderson, J.D., “Modern Compressible Flow with Historical Perspective”, McGraw-Hill, 1982. 4) Shapiro, A.H., “The Dynamics and Thermodynamics of Compressible Fluid Flow”, Volume 1, Wiley, 1953. 5) Wark, K., “Thermodynamics”, 3rd edition, McGraw-Hill, 1977. 6) Moran, M., Shapiro, H., Boettner, D., and Bailey, M., “Fundamentals of Engineering Thermodynamics (7th

Edition), Wiley, 2011.