Ch11 Comp Flow

Fluid Mechanics – II : Chapter 11 11

Chapter 11

Analysis ofCompressible Flow

.

Fluid Mechanics - II


Introduction • So far we mainly studied incompressible flow analysis. Any study of fluid mechanics

would, however, be incomplete without a brief introduction to compressible flow behavior

• Fluid compressibility is vital in numerous engineering applications of fluid mechanics. For example, the measurement of high-speed flow velocities requires compressible flow theory. Many aircraft fly fast enough to involve a compressible flow field.

• The variation of fluid density for compressible flows requires attention to density and other fluid property relationships. The fluid equation of state, often unimportant for incompressible flows, is vital in the analysis of compressible flows. Also, temperature variations for compressible flows are usually significant and thus the energy equation is important

• The changes in density for Compressible flows, makes the use of Bernoulli's Eqn (and subsequently building upon it) is not applicable

• For simplicity, in this introductory study of compressibility effects we mainly consider the steady, one-dimensional, constant (including zero) viscosity, compressible flow of an ideal gas

• In this chapter, one-dimensional flow refers to flow involving uniform distributions of fluid properties over any flow cross-section area


Ideal Gas Relations• Before we can proceed to study compressible flow equations, we need to become

more familiar with the fluid we will work with, the ideal gas. • Recall Thermodynamics-I Course, (we must Specifically recall how to evaluate ideal

gas property changes). The summary of eqns used as basics are produced below:• Eqn of State of Ideal Gas ; P = RT where R=/MGAS• Specific Heats :

• Specific Internal Energy & Enthalpy (u , h)and

• Specific Heat Ratio (k) some books uses symbol

.

• Change in Entropy .

PP

vv T

hCandTuC

dTCuuT

Tv

2

1

12dTChh

T

TP

2

1

12

RCC vP

v

P

CCk

11

kRCand

kRkC vP

1

2

1

2

2

1

1

212 lnlnlnln

PPR

TTCR

TTCsss Pv


Ideal Gas Relations• Isentropic Flows: S = 0

• Definition of Ideal Gas and Perfect Gas !!!! • Go through Ex 11.1 and 11.2, to check your existing knowledge and

revise if needed

kkk

TT

PP

1

21

1

2

1

2

tconsPPv kk tan

1

2

1

2

2

1

1

2 lnlnlnlnPPR

TTCR

TTC Pv

012 sss


Mach Number and Speed of Sound• We have studied the definition of Mach Number in Chap. 1 & 7 as a dimensionless

property• It’s an important property for study of compressible flows as all flow properties can be

expressed as a function of Mach Number. We know that M = V/c, where c is the speed of sound in the fluid under study

• The expression of “c” can be derived by applying Continuity & Momentum Eqns to a weak wave traveling in a fluid as

– For Isentropic Flows :

– For Ideal Gas :

– For any Fluid : : where Ev is Fluid’s bulk modulus of elasticity

• See values of “c” given in Tables B1 to B4 for water and air for various temperatures• Example 11.3

Pc

s

Pc

vEc

kRTc


Categories of Compressible Flow• In Section 3.8.1, we learned that the effects of compressibility become more

significant as the Mach number increases. For example, the error associated with using V2/2 in calculating the stagnation pressure of an ideal gas increases at larger Mach numbers

• From Fig. 3.24

• We can conclude that incompressible flows can only occur at low Mach numbers• Flow is generally considered to be incompressible when M < 0.3


Categories of Compressible Flow / Contd.

Variation Fluid Properties with Mach Number



• Experience has also demonstrated that compressibility can have a large influence on other important flow variables

• For example, in Fig. 11.2 the variation of drag coefficient with Reynolds number and Mach number is shown for air flow over a sphere

• .


Categories of Compressible Flow / Contd.• Compressibility effects can be of considerable importance• To further illustrate some curious features of compressible flow, a simplified example is

considered. Imagine the emission of weak pressure pulses from a point source• These pressure waves are spherical and expand radially outward from the point source at

the speed of sound

• .



• The discussion done so far suggests that Fluid Flows are categorized as:– Incompressible flow: M < 0.3 Unrestricted, nearly symmetrical and

instantaneous pressure communication– Compressible subsonic flow: 0.3 < M <1.0 Unrestricted but noticeably

asymmetrical pressure communication– Compressible supersonic flow: M > 1.0 Formation of Mach wave;

pressure communication restricted to zone of action• In addition to the above-mentioned categories of flows, two other

regimes are commonly referred to are:– Transonic flows : 0.8 < M < 1.2 : Modern aircraft are mainly powered by gas

turbine engines that involve transonic flows– Hypersonic flows: M > 5 : When a space shuttle reenters the earth’s

atmosphere, the flow is hypersonic. Future aircraft may be expected to operate from subsonic to hypersonic flow conditions

• Example 11.4


Isentropic Flow of Ideal Gas• We now are ready to study further details about the steady, one-dimensional, isentropic flow of an

ideal gas with constant specific heat values CP and Cv. • Isentropic Flow means :

– Adiabatic (No Heat Interactions)– No shaft work (Work =0)– Frictionless Flow (Viscosity =0 or inviscid flow) or Reversible Flow

• Also for simplicity we consider Ideal Gas flow which is one-dimensional and steady (d/dt =0 Specific Heat are constant, and velocity & fluid property changes in the stream wise direction only ; i.e. flow properties at any section are uniformly distributed )

• For Isentropic flows (adiabatic and reversible flows), the changes in Entropy are always ZERO ( s = 0 )

• Stagnation (Total) Properties remains constant for the whole domain of flow, i.e. h0 = constant, where

• For perfect gas; h0 = Cp To = constant– T0 =constant (T02 =T01)– P0 =constant (P02 =P01) etc.

• On a T-S diagram (or h-S diagram) the Static and Stagnation properties are reflected by a straight line

constant2

2

0 uhh


Isentropic Flow of Ideal Gas /Contd.

Incompressible Flows:•Stagnation Pressure; P0 = P + ½ V2 + Z

(3.15)•Velocity for Z=constant; V= [ 2(P0 – P)/ ]1/2

Compressible Flows:•The above relations are not valid a DENSITY is not constant and we express above relations as function of Mach Number ( such expression we touched in Chapter 3; see eqn 3,25 )

The graphical presentation of relations between Total & Static Properties are shown on a T-S diagram valid for both; Compressible & Incompressible Flows


Effect of Variations in Flow Area• Continuity Eqn ; “ uA= constant “ in differential form can be written as [11.43]• We would like to see how the flow velocity changes with cross sectional area only, i.e.

eliminating and d terms from above Eqn• Using isentropic relations and definition of Velocity of Sound, the term d/ can be

expressed as f(u,M) as given below

• Substituting this Eqn into 11.43 we get :

[11.48]• This Eqn called the area-velocity relation and provides us with a lot of very useful

information about flow speeds with changing Area • Similarly we can derive density change eqn as

[11.49]

0A

dAudud

uduM

udu

aud 2

2

2

AdA

Mudu

112

AdA

MMd

12

2


Effect of Variations in Flow Area /Contd.For subsonic flow (M < 1):• The term M2 – 1 is negative. So Eqn 10.25 becomes

dA/A - du/u • Therefore, for subsonic flow, to increase the velocity we must have a convergent duct, and to

decrease the velocity we must have a divergent duct as illustrated in Fig 10.7 belowFor supersonic flow (M > 1):

The term M2 – 1 is positive. So Eqn 10.25 becomes dA/A du/u

• Therefore, for supersonic flow, to increase the velocity we must have a divergent duct, and to decrease the velocity we must have a convergent duct as illustrated in Fig 10.7 below

• By re-arranging Eqn 11-.48, we get [11.50]• For M =1, Eq. 11.50 requires du/dA =0, which implies that the area associated with M=1 is

either a minimum or a maximum amount

uduM

AdA 12

12 MuA

dudA

11.48


Effect of Variations in Flow Area /Contd.


Convergent – Divergent (CD) Duct• For the steady flow of an ideal gas, it can be argued that the sonic condition ( M = 1 ) can be attained

in a converging-diverging duct at the minimum area location• This minimum area location is often called the throat of the CD duct• Furthermore, to achieve supersonic flow from a subsonic state in a duct, a converging-diverging area

variation is necessary• We will now develop equations to determine how important flow properties vary in a isentropic steady

flow in ducts• It can be done in different several approaches. The Text Book uses Isentropic Relations developed

earlier combined with eqn. of motion, enthalpy of ideal gas and eqn. of state• We will use another simpler approach using following (already studied) eqns:

– Enthalpy of Ideal Gas ; dh = cpdT

– Total Enthalpy ;

– Specific Heat for Ideal Gas ;

– Mach no ; M=V/a V2 = M2 a2 where

• Dividing both sides by CP we get ; [11.56]

1

Rc p

tconsVhh tan2

2

0

RTa 2

22

22

0RTMhVhh

TCMh P212

1

pcR

hMh2

12

20 2

11 Mhh

20 2

11 MTT


Convergent – Divergent Duct /Contd.

• Using Isentropic relations of an Ideal Gas, and Eqn. 11.56, we get the equations for all prime properties of the gas as indicated below : [11.59]

[11.60]

• Recall that, the difference in pressure across a duct is the flow deriving potential. If inlet of duct is connected to a reservoir, then reservoir pressure and temperature are Total Pressure & Total Temperature (Stagnation Properties) for the system, which remains constant for Isentropic flow

• The more is the pressure difference higher would be mach number

• See T-S diagram• Also we know that

• Thus

or

1000

TT

pp

20 2

11 MTT

1

20 2

11

MPP

11

20 2

11

M

20 2

11 MTT

222

2110 2

112

11 MTMTT

2

2

21

1

2

211

211

M

M

TT

M=0

M+ve

M+ve


• .


Chocked Flow ( M =1 ) or * Condition• As we discussed, the more is the pressure difference (say P0 - Pe ), higher would

be mach number• If we keep increasing Total pressure (or reduce the outlet pressure; back pressure),

then Velocity (or Mach Number) in a convergent duct will keep increasing reach till M reaches Unity ( M = 1 )

• This condition is called chocked duct (or chocking condition; * ), further increase in pressure difference, the flow will remain at chocking conditions (no increase in flow rate or Mach no in the duct)

• The isentropic eqns [11.56,11.59 & 11.60] for chocked conditions ( * conditions ) thus become), ,

for = 1.4 ,

• Examples 11.5 to 11.7.

11

11

*0

21

211

2

12

11*0

TT

11

*0

21

211

PP

5774.12

1 11

*0

2.1

21

*0

TT

528.0893.12

1

0

*1

*0

PPor

PP


Chocked State


Compressible Nozzle Chocking• Consider the duct of Fig 10.9. The area, Mach number, and velocity at

the throat are denoted by A*, M*, u*. At any other section of the duct they are A, M, u

• Apply the equation of continuity between sections A and A* :[10.26]

• Since u* = a* , this equation becomes [10.27]

where is the stagnation density and is constant in isentropic flow

uAAu

ua

ua

uu

AA

0

0


Compressible Nozzle Flow /Contd.

• We also have the following relations from chapter 8:[10.28]

[10.29]

[10.30]

• Squaring Eqn 10.27 gives

• Substituting for each of the ratios on the right hand side of this equation from Eqns (10.28) to (10.30) and simplifying, gives

[10.32]

• This is called the Area-Mach Number Relation ; M = f (A/A*). In other words, the Mach number at any location in the duct is a function of the ratio of the local duct area to the sonic throat area, (remember A > A* always)

11

0 12

11

20

211

M

22

1

22

12

2

1 MM

Mau

22

0

2

0

2

ua

AA

11

22

2

211

121

MMA

A


Compressible Nozzle Flow /Contd.

• The results for A/A* as a function of M, obtained from (10.32) are available in tabular form (App. A)

• For subsonic values of M, the area ratio A/A* decreases as M increases; i.e., the duct converges

• For supersonic values of M, the area ratio A/A* increases as M increases; i.e., the duct diverges

• The table also shows the double-valued nature of Mach number as a function of A/A*; for example, for A/A* = 2, we have either M = 0.31 or M = 2.2

• .

04/27/23 Fluid Mechanics – II : Chapter 11 24

04/27/23 Fluid Mechanics – II : Chapter 11 25

Fluid Mechanics – II : Chapter 11 26Chapter 10 26

The Convergent-Divergent Nozzle• We will now study the physics of the flow through a

convergent-divergent nozzle • CD Nozzle shown in Fig 10.10a have large area ratio

at the inlet Ai /A*, and that the flow at the inlet is fed from a large gas reservoir. The flow is sonic at the throat; hence M = 1 and At = A*. ( Recall * properties ratios; Eqns 8.44-46) ; P*/P0 =0.5283

• The reservoir pressure and temperature are p0 and T0 respectively and are constant

• The area distribution A = A(x) is specified; so the area ratio A /A* is known at every station along the nozzle

• Subscript “ t ” denotes conditions at throat, and “e ”, conditions at the exit

• Assume that we have an isentropic expansion of the gas through the nozzle to a supersonic Mach number Me = Me,6 at the exit (the reason for the subscript 6 will become clear later). Let’s say the corresponding exit pressure is pe,6


Isentropic Nozzle Flow• Area-Mach Number Relation reflect TWO possible solution in divergent passage

after the throat, (if there are no SHOCKS waves or flow remains Isentropic)

• Examples 11.8 to 11.10


Non-Isentropic Flows of Ideal Gas• Constant Area Duct Flow

– Adiabatic Constant Area Duct Flow with Friction ( Fanno Flow )

– Frictionless Constant Area Duct Flow with Heat Transfer ( Rayleigh Flow)

• Flow having Normal Shocks

• Flow having Oblique Shocks (2-D Flow)


Assignments / Self Study

• Examples; 11.1 to 11.10• Do at least 15 (from 11.1 to 11.47)

problems from Chapter 11 of text book

Ch11 Comp Flow

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