Fluid MechanicsChapter 9 – Introduction to compressible Wow

last edited April 2, 2018

9.1 Motivation 1939.2 Compressibility and its consequences 193

9.2.1 Problem description 1939.2.2 The speed of pressure 1949.2.3 The measure of compressibility 196

9.3 Thermodynamics of isentropic Wow for a perfect gas 1969.3.1 Principle 1969.3.2 Pressure, temperature, density and the speed of sound 198

9.4 Speed and cross-sectional area 2009.5 Isentropic Wow in converging and diverging nozzles 2029.6 The perpendicular shock wave 2049.7 Compressible Wow beyond frictionless pipe Wow 2069.8 Exercises 207

These lecture notes are based on textbooks by White [13], Çengel & al.[16], and Munson & al.[18].

9.1 Motivation

Video: pre-lecture brieVng forthis chapter

by o.c. (CC-by)https://youtu.be/Lk8BMRC_Cyw

In this chapter we investigate the fundamentals of compressible Wows: thosein which density varies signiVcantly. This study should allow us to answertwo questions:

• What happens in ideal (reversible) compressible Wows?

• What is a shock wave, and how does it inWuence a Wow?

9.2 Compressibility and its consequences

9.2.1 Problem description

Ever since we have begun describing Wuid Wow in an extensive manner inchapter 4, we have restricted ourselves to incompressible Wows — those inwhich the Wuid density ρ is uniform and constant. Now, ρ becomes oneadditional unknown property Veld.

In some cases, solving for an additional unknown is not particularly challeng-ing. For example, if we wish to calculate water temperature in a heat transfercase, or species concentration in a reacting Wow, it is possible to Vrst solvefor the Wow, and then solve for those particular property Velds within thesolution; thus the additional unknowns are solved separately with additionalequations.

The mathematical treatment of density, however, is much worse, because ρappears in the momentum and mass conservation equations. Density directlyaUects the velocity Velds, and so we have solve for ρ, p and ~V simultaneously.

From a physical point of view, compressibility brings two new eUects:193

• Density changes in Wuids translate into large changes in Wuid mechani-cal energy; these can be come very large in comparison to other formsof Wuid energy, in particular kinetic energy. When ρ varies, Wuids“perform work upon themselves”, accumulating and expending thisenergy, which translates into kinetic energy changes, and irreversiblelosses as heat transfer.

• When density is allowed to vary, pressure travels at Vnite speed withinthe Wuid — one could say that the Wuid becomes appreciably “squishy”.The delay resulting from this Vnite pressure propagation speed be-comes very signiVcant if the Wuid velocity is high, and it may evenprevent the propagation of pressure waves in the upstream directionaltogether.

In this chapter, we wish to quantify both of those eUects. By now, aftereight chapters already where ρ was just an input constant, the reader shouldreadily agree we have no hope of coming up with a general solution tocompressible Wow. Nevertheless, it is possible to reproduce and quantifythe most important aspects of compressibility eUects with relatively simple,one-dimensional air Wow cases — and this will suXce for us in this lastexploratory chapter.

9.2.2 The speed of pressure

We start by considering the speed of pressure. In compressible Wuids, pressurechanges propagate with a speed that increases with the magnitude of thepressure change. When this magnitude tends to zero (and the pressure wavereduces to that of a weak sound wave, with fully-reversible Wuid compressionand expansion) then the speed tends to a given speed which we name speedof sound and noted c . At ambient conditions, this speed is approximately1 000 km/h in air and roughly 5 000 km/h in water.

Let us, as a thought experiment, create a small pressure change inside a staticWuid, and then travel along with the pressure wave. As we follow a wavewhich travels from right to left, we perceive Wuid movement from left to right(Vg. 9.1). We construct a small control volume encompassing and movingwith the wave — at the speed of sound c .

Control volume Wuid Wow analysis does not frighten us anymore. From massconservation (eq. 3/5 p.64) we write:

ρcA = (ρ + dρ) (c − dV )A (9/1)

Figure 9.1 – An inVnitesimal pressure-diUerence wave is represented traveling fromright to left inside a stationary Wuid, in a one-dimensional tube. From the point ofview of a control volume attached to (and traveling with) the wave, the Wow is fromleft to right.

Figure CC-0 o.c.

194

This equation 9/1 relates the density change across the wave to the velocitychange between inlet and outlet. Re-arranging, and focusing only on the casewhere the pressure wave is extremely weak (as are sound waves in practice),we see the product dρ dV vanish and obtain:

ρc = ρc − ρ dV + c dρ − dρ dV

ρ dV = c dρ (9/2)

In equation 9/2, which relates the speed of sound c to the density ρ, we wouldnow like to eliminate the dV term. For this, we turn to the control volumemomentum equation (eq: 3/8 p.65):

Fnet = −minVin + moutVout

pA − (p + dp)A = −ρcAc + (ρ − dρ) (c − dV )A(c − dV )

− dpA = ρcA [−c + (c − dV )]dpc= ρ dV (9/3)

We can now combine eqs. (9/2) and (9/3) to obtain:

c =

√dpdρ

(9/4)

This can be generalized for any pressure wave, and we state that

c =

öp

∂ρ

�����s=cst.(9/5)

where the s = cst. subscript indicates that entropy s is constant.

So now, we see that in a Wuid the speed of sound —the speed of pressurewaves when they travel in a reversible (constant-entropy) manner— is thesquare root of the derivative of pressure with respect to density. This allowsus to compare it to another term, the bulk modulus of elasticity K , whichrepresents the diUerential amount of pressure that one has to apply on abody to see its density increase by a certain percentage:

K ≡dpdρρ

(9/6)

where K is expressed in Pa.

Inserting eq. 9/6 into eq. 9/5, we obtain a relationship which we had intro-duced already in chapter 6 (as eq. 6/20 p.138):

c =

√Ks=cst.

ρ(9/7)

With this equation 9/7 we see that in any Wuid, the square of the speed ofsound increases inversely proportionally to the density and proportionally tothe modulus of elasticity (the “hardness”) of the Wuid.

195

9.2.3 The measure of compressibility

The most useful measure of compressibility in Wuid mechanics is the Machnumber, which we already deVned (with eq. 0/10 p.15) as:

[Ma] ≡V

c(9/8)

As we saw in chapter 6 (§6.4.3 p.138), [Ma]2 is proportional to the ratio ofnet force to elastic force on Wuid particles: at high Mach numbers the forcesrelated to the compression and expansion of the Wuid dominate its dynamics.

Not all compressible Wows feature high Mach numbers. In large-scale atmo-spheric weather for example, air Wows mush slower than the speed of soundyet large density changes can occur because of the vertical (static) pressuregradient. The same is true in low-speed centrifugal machines because ofcentrifugal pressure gradients. In this chapter however, we choose to focuson the simplest of compressible Wows: one-directional gas Wows with Machnumbers approaching or exceeding 1.

9.3 Thermodynamics of isentropic Wowfor a perfect gas

9.3.1 Principle

When a gas is compressed or expanded, its pressure and temperature varytogether with its density. Before we consider the dynamics of such phenom-ena, we need a robust model to relate those properties. This is the realm ofthermodynamics.

In short, from a macroscopic point of view, a gas behaves as a mechanicalspring (Vg. 9.2): one needs to provide it with energy as work in order tocompress it, and one recovers energy as work when it expands. Threeimportant points must be made regarding these compressions and expansions:

1. When gases are heated during the movement, the work transfers are in-creased; conversely they are decreased when the gas is cooled (Vg. 9.3).Cases where no heat transfer occurs are called adiabatic;

Figure 9.2 – When they are compressed or expanded, the behavior of Wuids can bemodeled as if they were mechanical springs: they gain energy during compressions,and lose energy during expansions. These spring can be thought of as being “fragile”:when the movements are sudden, compressions involve more mechanical energy,and expansions involve less.

Figure CC-by-sa Olivier Cleynen

196

Figure 9.3 – Pressure-volume diagram of a perfect gas during a compression. Whenthe Wuid is heated up, more work is required; when it is cooled, less work is required.Opposite trends are observed during expansions.

Figure CC-0 Olivier Cleynen

Figure 9.4 – When no heat transfer occurs, the a compression or expansion is saidto be adiabatic. The temperature of Wuids can still vary signiVcantly because of thecompression or expansion. When adiabatic evolutions are inVnitely smooth, theyare termed isentropic.

Figure CC-0 Olivier Cleynen

197

2. The temperature of gases tends to increase when they are compressed,and tends to decrease when they expand, even when no heat transferoccurs (Vg. 9.4);

3. A gas behaves as a “fragile” spring: the faster a compression occurs,and the more work is required to perform it. Conversely, the fasteran expansion occurs, and the less work is recovered from it. Caseswhere evolutions occur inVnitely slowly are used as a reference forcomparison: such cases are termed reversible. Cases where the evolu-tion is both adiabatic and reversible are named isentropic (iso-entropic,because they occur at constant entropy).

9.3.2 Pressure, temperature, density and the speed of sound

The quantitative relations between pressure, temperature and density ofperfect gas during isentropic evolutions are classical results in the study ofthermodynamics. In this section, we wish to re-write them with particularfocus placed on the speed of sound and the Mach number. The goal is toarrive to eqs. 9/20 to 9/22 p.199, which we need in the upcoming sections todescribe supersonic Wow of gases.

Let us then focus on the isentropic evolutions of a perfect gas. It is knownfrom the study of thermodynamics that in isentropic Wow from a point 1 toa point 2, the properties of a perfect gas are related one to another by therelations: (

T1

T2

)=

(v2

v1

)γ−1

(9/9)(T1

T2

)=

(p1

p2

) γ−1γ

(9/10)(p1

p2

)=

(v2

v1

)γ(9/11)

for a perfect gas, during an isentropic (constant-entropy) evolution.

It follows from eq. 9/11 that in an isentropic evolution, a perfect gas behavessuch that pρ−γ = k = cst., which enables us to re-write equation 9/5 for aperfect gas as:

c =

öp

∂ρ

�����s=cst.=

√kγ ργ−1 =

√pρ−γγ ργ−1 =

√γpρ−1

The speed of sound in a perfect gas can therefore be expressed as:

c =√γRT (9/12)

[Ma] =V√γRT

(9/13)

for any evolution in a perfect gas.

Thus, in a perfect gas, the speed of sound depends only on the local tempera-ture.

We now introduce the concept of stagnation or total properties, a measure ofthe total amount of speciVc energy possessed by a Wuid particle at a given198

instant. They are the properties that the Wuid would have if it was brought torest in an isentropic manner:

h0 ≡ h +12V 2 (9/14)

T0 ≡ T +1cp

12V 2 (9/15)

h0 = cpT0 (9/16)

Using equation 9/13, we can re-write eq. (9/15) as:

T0

T= 1 +

1cpT

12V 2 = 1 +

1γRγ−1T

12V 2

= 1 +(γ − 1

2

) V 2

c2 = 1 +(γ − 1

2

)[Ma]2

Eqs. (9/9) to (9/11) can thus be re-written to express the transition from Wowcondition to stagnation condition:(T0

T

)= 1 +

(γ − 12

)[Ma]2 (9/17)(

p0

p

)=

[1 +

(γ − 12

)[Ma]2

] γγ−1

(9/18)(ρ0

ρ

)=

[1 +

(γ − 12

)[Ma]2

] 1γ−1

(9/19)

These three equations (9/17) to (9/19) express, for any point inside the Wowof a perfect gas, the local Mach number [Ma] as a function of the ratio ofthe local (current) property (T , p or ρ) to the local total property (T0, p0, ρ0).These ratios are always inferior or equal to one, by deVnition.

Now, if now we deVne critical conditions as those that would occur if theWuid was exactly at [Ma] = 1, noting them with an asterisk, we can re-writeeqs. (9/17) to (9/19) comparing stagnation properties to critical properties:(

T ∗

T0

)=

2γ + 1

(9/20)(p∗

p0

)=

[2

γ + 1

] γγ−1

(9/21)(ρ∗

ρ0

)=

[2

γ + 1

] 1γ−1

(9/22)

Here, we compare the local total property (e.g. the total temperature T0) tothe value the property needs to have for the Wow to be at sonic speed (e.g.the critical temperature T ∗). This ratio does not depend at all on the WuidWow: it is merely a function of the (invariant) characteristics of the gas.

This is it! It is important to understand that nothing in this section describesa particular Wow movement. We have merely derived a convenient toolexpressing Wow properties as a function of their stagnation properties andthe local Mach number, which we shall put to good use further down.

199

9.4 Speed and cross-sectional area

Let us now go back to Wuid dynamics. The last time we considered frictionlesspipe Wow, in chapter 5, we situation was rather simple. From the one-dimensional mass conservation equation, eq. 5/1:

ρ1V1A1 = ρ2V2A2 (9/23)

it was possible to relate cross-sectional area A and velocity V easily. Nowthat ρ is also allowed to vary, the situation is decidedly more interesting.

We wish to let static, pressurized air accelerate as fast as possible: letting Vincrease at a cost of a decease in ρ according to the rules of an isentropic,steady expansion. We know however, that according to mass conservation,ρVA has to remain constant all along the acceleration, so that only one givennozzle geometry (A = f (p,T )) will permit the desired acceleration. How canwe Vnd this geometry?

We start from the conservation of mass inside our isentropic accelerationnozzle, and diUerentiating the terms:

ρAV = cst.

ln ρ + lnA + lnV = cst.dρρ+

dAA+

dVV= 0

for any steady Wow.

Let us set this equation 9/24 aside for a moment. During our isentropicexpansion, the total energy of the Wuid remains constant, thus the totalspeciVc enthalpy h0 is constant,

h +V 2

2= cst.

and here also we diUerentiate, obtaining:

dh +V dV = 0 (9/24)

In last relation, since the process is reversible and without external energyinput, we can write the Vrst term as dh = dp/ρ, yielding:

1ρ

dp +V dV = 0

1ρ= −

1dp

V dV (9/25)

for isentropic gas Wow.

This expression of 1/ρ is the relation we were after. Now, coming back toequation 9/24 and inserting eq. (9/25), we get:

−dρdp

V dV +dAA+

dVV= 0

dVV

[1 −

dρdp

V 2]+

dAA= 0

200

In this last relation, since we are in an isentropic Wow, we can insert anexpression for the speed of sound, c2 =

∂p∂ρ

���s derived from equation 9/5, ob-taining:

dVV

[1 −

1c2V

2]+

dAA= 0

dVV

[1 − [Ma]2

]+

dAA= 0

This leads us to the following equation relating area change and speed changein compressible gas Wow:

dVV=

dAA

1

[Ma]2− 1

(9/26)

for isentropic, one-dimensional gas Wow.

This equation 9/26 is one of the most stunning equations of Wuid dynamics. Itrelates the change in Wow cross-section area, dA, to the change in speed dV ,for lossless Wow. Let us examine how these two terms relate one to another,as displayed in Vgure 9.5:

• When [Ma] < 1, the term 1/([Ma]2− 1) is negative, and dA and dV are

always of opposite sign: an increase in area amounts to a decrease inspeed, and a decrease in area amounts to an increase in speed.

• When [Ma] > 1, dA and dV have the same sign, which —perhapsdisturbingly— means that an increase in area amounts to an increase inspeed, and a decrease in area amounts to a decrease in speed.

Thus, if we want to let a static, compressed Wuid accelerate as fast as possible,we need a converging nozzle up to [Ma] = 1, and a diverging nozzle onwards.

201

Figure 9.5 – Changes in nozzle cross-section area have opposite eUects when theMach number [Ma] is lower or greater than 1.Note that because temperature changes as the gas compresses and expands, the localMach number varies as the Wow changes speed. It is thus possible for the Wow to gofrom one regime to another during an expansion or compression. These eUects arestudied further down and described in Vgure 9.7 p.204.

Figure CC-0 o.c.

9.5 Isentropic Wow in converging and divergingnozzles

The change of between the subsonic and supersonic Wow regimes, by deVni-tion, occurs at Mach one. When the Wow is isentropic, equation 9/26 abovetells us that the [Ma] = 1 case can only occur at a minimum area.

Of course, there is no guarantee that a pipe Wow will be able to becomesupersonic (Vg. 9.6). For this, very large pressure ratios between inlet andoutlet are required. Nevertheless, whichever the pressure ratio, accordingto eq. 9/26, a converging nozzle will only let an expanding Wuid accelerateto Mach one (critical conditions). Changing the nozzle outlet area itself willnot permit further acceleration unless an increase in nozzle area is generatedbeyond the critical point.

This is a surprising behavior. In a converging-only nozzle, increasing theinlet pressure increases the outlet velocity only up until [Ma] = 1 is reachedat the outlet. From there on, further increases in inlet pressure have no eUecton the outlet pressure, which remains Vxed at poutlet = p

∗. The Wow thenreaches a maximum mass Wow rate mmax. This condition is called chokedWow.

Once the Wow is choked, adding a diverging section further down the nozzlemay enable further acceleration, if the pressure ratio allows for it. Neverthe-less, the pressure pT ≡ pthroat at the throat remains constant at pT = p

∗: fromthis point onwards, downstream conditions (in particular the pressure) arenot propagated upstream anymore: the gas travels faster than the pressurewaves inside it. With pT still equal to the critical pressure p∗, the mass Wowthrough the nozzle cannot exceed mmax.

202

Figure 9.6 – Non-dimensionalized properties of a gas undergoing an isentropic (i.e.shock-less) expansion through a converging-diverging nozzle.

Figure CC-0 o.c.

We can see, therefore, that it follows from equation 9/26 that there existsa maximum mass Wow for any duct, which depends only on the stagnationproperties of the Wuid and the throat cross-sectional area.

The mass Wow, at any point in the duct, is quantiVed as:

m = ρcA[Ma]

=p

RT

√γRTA[Ma]

m = A[Ma]p0

√γ

RT0

1 +

(γ − 1)[Ma]2

2

−γ−12(γ−1)

(9/27)

and m will now reach a maximum when the Wow is choked, i.e. reaches[Ma] = 1 at the throat, where its properties will be p∗ = pT, T ∗ = TT, and thearea A∗ = AT:

mmax =

[2

γ + 1

] γ+12(γ−1)

A∗p0

√γ

RT0(9/28)

A general description of the properties of a perfect gas Wowing through aconverging-diverging duct is given in Vgs. 9.6 and 9.7.

203

Figure 9.7 – The pressure of a gas attempting an isentropic expansion through aconverging-diverging duct, represented as a function of distance for diUerent valuesof outlet pressure pb. The green curves show the reversible evolutions.As long as the back pressure pb is high enough, the Wow is subsonic everywhere,and it remains reversible everywhere. When pT reaches p∗, the Wow downstream ofthe throat becomes supersonic. In that case, if the back pressure is low enough, afully-reversible expansion will occur. If not, then a shock wave will occur and theWow will return to subsonic regime downstream of the shock.

Figure CC-0 o.c.

9.6 The perpendicular shock wave

The movement of an object in a Wuid at supersonic speeds (or a supersonicWuid Wow over a stationary object, which produces the same eUects) provokespressure Veld changes that cannot propagate upstream. Thus, if a directionchange or a pressure increase is required to Wow past an obstacle, Wuidparticles will be subjected to very sudden changes. These localized changesare called shock waves.

The design of bodies moving at supersonic speeds usually focuses on ensuringpressure and direction changes that are very gradual, and at the limit (e.g. faraway from a supersonic aircraft) these are very small.The limiting case at the other side of the spectrum is that of the normal shockwave, the most loss-inducing (and simple!) type of shock wave. This willtypically occur in front of a blunt supersonic body.

A perpendicular shock wave (Vg. 9.8) is studied with the following hypothesis:

• It is a constant energy process;

• It it a one-dimensional process (i.e. the streamlines are all parallel toone another and perpendicular to the shock wave).204

With these two hypothesis, we can compare the Wow conditions upstreamand downstream of a shock wave occurring in a tube using the continuityequation, which we wrote above as eq. 9/23:

ρ1A1V1 = ρ2A2V2 (9/29)

The energy equation in this case shrinks down to:

T01 = T02 (9/30)

Since, from equation 9/17 p.199, we have:

T01

T1= 1 +

(γ − 12

)[Ma]2

1

T02

T2= 1 +

(γ − 12

)[Ma]2

2

and given that T01 = T02, dividing one by the other yields:

T2

T1=

1 + γ−12 [Ma]2

1

1 + γ−12 [Ma]2

2

(9/31)

In a similar fashion, we can obtain:

p2

p1=

1 + γ [Ma]21

1 + γ [Ma]22

(9/32)

Let us set aside these two equations 9/31 and 9/32 and compare conditionsupstream and downstream starting with mass conservation, and re-writing itas a function of the Mach number:

ρ1V1 = ρ2V2

ρ1

ρ2=V2

V1=

[Ma]2a2

[Ma]1a1

=[Ma]2

√γRT2

[Ma]1√γRT1

[Ma]1

[Ma]2=p2

p1

(T1

T2

) 12

(9/33)

So far, we have not obtained anything spectacular, but now, inserting bothequations 9/31 and 9/32 into this last equation 9/33, we obtain the staggering

Figure 9.8 – A normal shock wave occurring in a constant-section duct. All propertieschange except for the total temperature T0, which, since the transformation is atconstant energy, remain constant.

Figure CC-0 o.c.

205

result:

[Ma]1

[Ma]2=

1 + γ [Ma]21

1 + γ [Ma]22

*,

1 + γ−12 [Ma]2

2

1 + γ−12 [Ma]2

1

+-

12

(9/34)

This equation is not interesting because of its mathematical properties, butbecause of its contents: the ratio [Ma]1/[Ma]2 depends only on [Ma]1 and[Ma]2! In other words, if the shock wave is perpendicular, there is onlyone possible post-shock Mach number for any given upstream Machnumber.

After some algebra, [Ma]2 can be expressed as

[Ma]22 =

[Ma]21 +

2γ−1

2[Ma]21γγ−1 − 1

(9/35)

Values for [Ma]2 as a function of [Ma]1 according to equation 9/35 for par-ticular values of γ can be tabulated or recorded graphically, allowing us topredict properties of the Wow across a shock wave. We can therefore quantifythe pressure and temperature increases, and velocity decrease, that occurthough a perpendicular shock wave.

9.7 Compressible Wow beyond frictionless pipeWow

In this chapter, we have restricted ourselves to one-directional steady Wow,i.e. Wows inside a pipe; within these Wows, we have neglected all losses butthose which occur in perpendicular shock waves. This is enough to describeand understand the most important phenomena in compressible Wow.

Of course, in reality most compressible Wows are much more complex thanthat. Like other Wows, compressible Wows have boundary layers, feature lam-inarity and turbulence, heat transfer, unsteadiness, and more. Shock wavesoccur not just perpendicularly to Wow direction, but also in oblique patterns,reWecting against walls and interacting with boundary layers. Temperaturevariations in compressible Wow are often high enough to provoke phasechanges, which complicates the calculation of thermodynamic properties aswell as Wow dynamics and may result in spectacular visual patterns.

Video: watch a 140-thousand-ton rocket lift away humans ontheir Vrst ever trip to the moon.Compressible Wow of expand-ing exhaust gases is what al-lows the maximum amount ofthrust to be obtained from agiven amount of combustion en-ergy in rocket engines.

by NASA (public domain)https://youtu.be/eZDaM3MHNqg

All those phenomena are beyond the scope of this study, but the reader canhopefully climb up to more advanced topics such as high-speed aerodynamics,or rocket engine and turbomachine design, where they are prominentlyfeatured and can surely make for some exciting learning opportunities.

206