high speed aerodynamics notes

8/22/2019 high speed aerodynamics notes

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NPTELIITKharagpur:Prof.K.P.Sinhamahapatra,Dept.ofAerospaceEngineering

1

Module1: Brief Review

of

Thermodynamics

Lecture1:

Compressible Aerodynamics


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2

Thermodynamics

Experimental results are the basis of any physical theory. The experimental basis of

thermodynamics is formalized in the principal laws. The law of conservation of energy is one of

these principal laws. It introduces the concept of internal energy of a system. The other principal

laws of thermodynamics introduce and define the properties and concepts of temperature and

entropy. Classical thermodynamics is concerned, at any rate as the bulk of the subject stands,

with equilibrium state of uniform matter, that is, with states in which all local mechanical,

physical and thermal quantities are virtually independent of both position and time.

Thermodynamical results may be applied directly to fluids at rest when their properties are

uniform. A very little is known of the thermodynamics of non-equilibrium states. However,

observation shows that results for equilibrium states are approximately valid for the non-

equilibrium non-uniform states common in practical fluid dynamics; large through the departures

from equilibrium in a moving fluid may appear to be, they are apparently small in their effect on

thermodynamical relationships.

Fluid mechanics of perfect fluids (without viscosity and heat conductivity) is an extension of

equilibrium thermodynamics to moving fluids. In addition to internal energy, kinetic energy of the

fluid needs to be considered. The ratio of the kinetic energy per unit mass to the internal energy

per unit mass is a characteristic dimensionless quantity of the flow problem and in the simplest

cases is directly proportional to the square of Mach number.

In fluid mechanics of low speed flow, thermodynamic considerations are not needed: the heat

content of the fluid is then so large compared to the kinetic energy of the flow that the

temperature remains nearly constant even if the whole kinetic energy is transformed into heat.

The opposite can be true in high-speed blow problems.

Thermodynamic system a quantity of matter separated from the surroundings or the

environment by an enclosure

Enclosure = a closed surface with its properties defined everywhere may or may not transmit

heat, work or mass.


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3The concepts of thermodynamics are helpful in fluid mechanics for the additional reason that in

both subjects the objective is a set of results which apply to matter as generally as possible,

without regard for the different molecular properties and mechanism at work. Additional results

may be obtained by taking into account any known molecular properties of a fluid (with the aid

of kinetic theory in case of certain gases).

It is a fact of experience that the state of a given mass of fluid in equilibrium (spatial and

temporal uniformity) under the simplest possible conditions is specified uniquely by two

parameters, which for convenience may be chosen as the specific volume

1= and the

pressure p. All other quantities describing the state of the fluid are function of these two

parameters of state. One of the most important of these quantities is the temperature. A mass

of fluid in equilibrium has the same temperature as a test mass of fluid also in equilibrium if thetwo masses remain in equilibrium when placed in thermal contact (Zeroth law). The relation

between the temperature T and the two parameters of state may be written as

( ), , 0f p T =

This exhibits formally the arbitrariness of the choice of the two parameters of state. The

equation is called the equation of state. Generally written as ( )Tpp ,= and is called thermal

equation of state. Another important quantity describing the state of the fluid is the internal

energy per unit mass e. The change in the internal energy of the system (mass of fluid) at rest

consequent on a change of state is defined by the first law of thermodynamics, as being such as

to satisfy the conservation of energy when account is taken of both heat given to the fluid and

work done on the fluid. Thus if the state of a given uniform mass of fluid is changed by a gain of

heat of amount Q per unit mass and by the performance of work on the fluid of amount Wper

unit mass, then

WQe +=

),( Tvee = is the caloric equation of state

The internal energy e is a function of the state parameters, and the change which may be eitherinfinitesimal or finite depends only on the initial and final states, but Q and W are measures of

external effects and may separately depend on the particular way in which the transition

between the two states is made. If the mass of fluid is thermally isolated from its surrounding so

that no exchange of heat can occur, 0=Q , and the change of state of the fluid is adiabatic.


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4The most important way of performing work on the system is compression. Analytic expression

can be obtained if the change is reversible. This implies that the change is carried out so slowly

that fluid passes through a succession of equilibrium states, the direction of the change being

without effect. At each stage, the pressure is uniform p, so the work done on unit mass of fluid

for small decrease in volume is p . Thus for a reversible transition from one state to another

neighbouring state,

e q p =

The particular path by which the initial and final equilibrium states are joined is relevant here,

because p is not in general a function of alone.

Another practical quantity of some importance is the specific heat of the fluid, which is the

amount of heat given to unit mass of the fluid per unit rise in temperature in a small reversible

change. The specific heat may be written as

TQ

c

=

This is not uniquely determined until the conditions under which the reversible changes occur

are specified: An equilibrium state is a point on a ( ),p plane (indicator diagram) and a small

reversible change ( ) ,p starting from a point A may proceed in any direction.

If the only work done on the fluid is that done by compression, the heat Q which must be

supplied to unit mass is determined as

pE

pp

EQ

p

+

+

=

nm

A

A A

p A

isothermal

adiabatic


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5and the change in temperatures is

p

Tp

p

TT

+

=

The specific heat thus depends on the ratio

p

, and hence on the choice of direction of the

change from A. Two particular well defined choices are changes parallel to the axes of the

indicator diagram giving the principal specific heats

;ppp

pT

pT

E

T

QC

+

=

=

=

=

T

E

T

QC

T various sinusoidally as the point representing the final state moves round a circle of small

radius centered on A, being zero on the isotherm through A and maximum in a direction mG

normal to the isotherm. Likewise, Q varies sinusoidally, being zero on the adiabate through A

and a maximum in a direction nG

normal to it. The components of the unit vectors being

pmm , and pnn ,

( )( )max

max

Tm

QnCp

= ,( )

( )max

max

Tm

QnC

p

p

=

Since pmm and pnn are the gradients of the isothermal and adiabatic lines, the ratio of

the principal specific heats is

==

pp

p

m

m

n

n

C

C

Tadiab

pp

=

or

=

aTpp

Extensive variables: value depends on the mass of the system, LikeM, E, V, S.

Intensive variables: variables that do not depend on the total mass of the system, like p, T.

Both sets refer to state variables only. For every extensive variable an intensive variable (per

unit mass, or specific) can be introduced.

The weighted ratio p p in a small reversible change is the bulk modulus of

elasticity of fluid. For fluid dynamical purposes, its reciprocalp

or1

p

is more useful.

This is called the coefficient of compressibility. Like specific heat, the bulk modulus or the

coefficient of compressibility takes a different value for each direction of change. Adiabatic and


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6isothermal changes correspond to two particular directions with special significance and the first

law requires the ratio of the two corresponding bulk module to equal the ratio of the principal

specific heats.

____________________________________________________________________________

___________________________________________________________________________


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1

Module1: Brief Review

of

Thermodynamics

Lecture2:

Compressible Aerodynamics (Contd.)


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2Thermal equation of state for all gases for low densities approaches

p = PRT, T in K or R, C + 273.16 or F + 459.69

This defines a family of perfect gases, one for each value ofR. Any gas at low enough density

approaches a perfect gas with a particular value ofR. the caloric equation of state for a perfect

gas is e = constant T TC

=

Every real gas can be liquefied. The highest temperature at which this is possible is called

critical temperature Tc, the corresponding pressure and density are called critical pressure pc

and critical density c . Critical variables are characteristics of a gas and depend on

intermolecular forces. At the critical point2

20.

p p

v v

= =

An equation of state for a real gas must

involve at least two parameters besides R, say pc and Tc as in Van-der-Walls equation.

=

RTpRTp

1

1

cc ppRTp 27,8

27 2 ==

The internal energy of a Van-der-Walls gas is

( ) ( )o oe e T e T

= =

It is clearly possible to draw lines defining the direction of a small reversible change involving no

gain or less of heat through each point of the indicator diagram, and to regard the family of

these adiabatic lines of equal value of some new function of state. The properties of this

function are the subject of the second law. The second law implies the existence of another

extensive property of the fluid in equilibrium (even for systems with more than 2 independent

parameters of state) termed the entropy such that in a reversible transition from an equilibrium

state to another neighbouring equilibrium state, the increase in entropy is proportional to the

heat given to the fluid and that the constant of proportionality itself is a function of state,depends only on the temperature and can be chosen as the reciprocal of the temperature. With

entropy per unit mass of a fluid s, we have

Tds =dq = the infinitesimal amount of hat given reversibly.

This is the means by which the thermodynamic or absolute scale of temperature is defined.


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3

An adiabatic reversible transition takes place at constant entropy, and the process is called

isentropic. It is a consequence of the second law that in an adiabatic irreversible change the

entropy can not diminish.

Hence, for a small reversible change in which work is done on fluid only by compression

pesT +=

Since the equation contains only functions of state, the relation must be valid for any

infinitesimal transition in which work is done by compression, whether reversible or not. If the

transition is irreversible QsT , and pw

Another function of state which like internal energy and entropy proves to be convenient for use

in fluid mechanics particularly when effects of compressibility of the fluid are important is the

enthalpy or heat function. The enthalpy of unit mass of fluid is

peh +=

psTdppdeh +=++=

The relation involves only state functions. For a reversible small change at constant pressure

qh =

Helmholtz free energy

F =E Ts per unit mass

TspTssTEF ==

Thus, the gain in free energy per unit mass in a small isothermal change, whether reversible or

not, is equal to p . When this small isothermal change is reversible the gain in free energy is

equal to the work done on the system.

Another form of free energy is Gibbs free energy defined as

G =E + pV Ts =H Ts

G V p s T =

Using and s as the 2 independent parameters

Ts

Ep

EpEsT

s

=

=

+=

,


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4

s

E

2

may be obtained in two different ways.

s

T

s

E

s

p

s

E

=

=

22

,

Hence

s

T

s

p

=

Similarly,

spp

T

s

=

;p TT

s p s

T p T

= =

may be obtained by forming the double derivative, in two different ways, of the functions

,

h F

E p E Ts and E p Ts H Ts G + + = =

The four relations given above are known as Maxwells thermodynamic relations.

Alternatively, from the first relation

s ss

T T p

p v

=

s p s

p p Ts s

= =

spp

T

s

=

Coefficient of thermal expansion of fluid is

1

pT

=

It plays an important role in considerations of the action of gravity on a fluid of non-uniform

temperature.

Specific heats using entropy

T

sT

T

QC

==


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5

,p

pT

sTC

=

=

T

sTC

Using ( )

T

sT

T

ssTss

+

== ,,

pTp T

s

T

s

T

s

+

=

Since

=

T

ps

T

p

p

pC C T

T T

=

RHS can be calculated from the equation of state.

In terms of easily measurable quantities,

p

T p p

pC C T

T T

=

2

T p T p

p pT T

T T

= =

Relationship between increments ofS and E consequent on small changes in two parameters

(useful for flow of fluid with non-uniform temperature)

( )pTSS ,=

pp

sT

T

ss

Tp

+

=

pT

p

p Tp

s

T

C

T

s

=

=

,

pT

TCp

TT

TCs p

p

p

=

=

or pEpTTCsT p +==

Except s and E , others are directly observable

p

T

T

p

CC

C

pT

TC

pTp

pp

=


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6

pp

TT

T

p

=

1

from which it will be possible to see whether one term is dominant

( )( )psHHsEE

,

,

=

= Canonical equation of state

TS

p

Couple of relations for perfect gas

( )Tss .=

ds

dTT

sds

T

+

=

dT

p

T

dTC

+ [from definition ofC and Maxwell relation]

dR

T

dTC

+

Integrating between state (1) & (2)

dR

T

dTCss +=

2

1

2

112

For a calorically perfect gas RCCp ,, are constant

1

2

1

2

12lnln

RT

TCss +=

Similarly, ( )Tpss ,=

dpp

sdT

T

sds

Tp

+

=

dpTT

dTCp

p

= [definition & Maxwell relation]

Integrating

2 22 1

1 1

ln lnp

T ps s C R

T p =

Conjugate variables


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7Isentropic relations

/

2 2 2 2

1 1 1 1

ln ln

pC R

p

T p p T C R

T p p T

= =

,1

=

RCp Hence

1

1

2

1

2

=

T

T

p

p

Similarly,

2 2

1 1

ln lnT

R CT

=

or,

RC

T

T

=

1

2

1

2 1

1

=

R

C

11

1

2

1

2

=

T

Tor

1

1

1

2

1

2

=

T

T

Hence,

1

1

2

1

2

1

2

=

=

T

T

p

p

Total or stagnation conditions

If a fluid element passing through a point where the local pressure, temperature, density,velocity, etc are p, T, , V, is brought to rest adiabatically the flow parameters will change and

the corresponding values are called total values. The temperature of the fluid element after it

brought to rest (imagine) is called total temperature To. For a calorically perfect gas, the

corresponding total enthalpy is ho =Cp To.

Considering the energy equation for inviscid flow and assuming adiabatic flow with negligible

body forces,

( )

21

2

De V pV p V V p

Dt

+ = = +

G G G

From continuity equationDt

DV

1=

G

pVDt

DpVVe

Dt

D=

+

GGG

2

1


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8

Now( )Dt

pD

Dt

Dp

Dt

Dp

=

t

ppVDt

DpVVpeDt

D

==

++

GGG

2

1

or 02

2

=

=

+

t

pVh

Dt

D for steady flow.

The time rate of change of 22

1Vh + following a moving fluid element is zero.

2

2

1Vh + = constant along a path or streamline (steady, adiabatic, inviscid flow)

The total enthalpy ho is enthalpy at a point if the fluid element were brought to rest adiabatically

ohV

h =+2

2

The combination21

2h V+ in the equation can be replaced by ho.

The energy equation for steady, adiabatic, inviscid flow is then 0=Dt

Dho

The total enthalpy is constant along a streamline. If all the stream lines of the flow originate froma common uniform free stream, then ho is constant for each line. Hence, for steady, inviscid,

adiabatic flow, the energy equation becomes

ho = constant everywhere

or, Alternatively To = constant for calorically perfect gas.

For a general non-adiabatic flow, the forms of the energy equation are not valid but the

definition of total quantities hold locally at each point of the flow. At point 1, the local static

enthalpy and velocity are h1 and V1 and the total enthalpy2

111

2

1Vhho += ; and at point 2,

2

2

22

21 hVho += . But 21 oo hh . Only if the flow between point 1 and 2 is adiabatic 21 oo hh =

The total (stagnation) pressure and density are the pressure and density if the fluid is brought to

rest adiabatically and reversibly, i.e., isentropically (temperature oT ). The definition of op and


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9

o involve isentropic assumption. However, the concept of total pressure and density can be

applied throughout any general non-isentropic flow. If the flow is non-isentropic between points

1 and 2,1 2

,o o

p p 21 oo . But if the flow is isentropic between points 1 and 2 then

,21 oo pp = and 21 oo = . If the general flow field is isentropic throughout then both op and

o are constant throughout.

Sonic temperature (T*) In a subsonic flow if a fluid element is speeded up to sonic velocity,

adiabatically, the temperature it would have at such sonic condition is T*. In a supersonic flow,

the fluid element is slowed down to sonic velocity adiabatically.

----xxx---


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NPTEL IIT Kharagpur: Prof. K.P. Sinhamahapatra, Dept. of Aerospace Engineering

1

Module2:

One-Dimensional Gas Dynamics

Lecture3:Governing Equations


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2

One-Dimensional Gas Dynamics

The definition applies to flow in a channel or tube which may be described by specifying the

variation of the cross-sectional area (A) along its axis (x), ( )xAA = , and in which the flow

properties are uniform over each cross-section ( ) ,p p x = ( )x = , etc. The flow quantities

may be time dependant, i.e., ( ) ( )txpptxuu ,,, == . If there are sections over which the flow

conditions are not uniform it is still possible to apply the results between sections where they are

uniform, i.e., one-dimensional. At non-uniform stations, the results applied to suitable mean

values. Furthermore, the one-dimensional results are applicable to the individual stream tubes of

a general 3-D motion; x being along the stream tube.

For an incompressible flow, complete information about a one-dimensional flow is obtained from

the kinematic relation: u is inversely proportional to A. The pressure is then obtained from the

Bernoullis equation. For a compressible flow the relation between velocity and area also depends

on density variation since the governing equations are interdependent.

1-D continu ity equation

If the flow is unsteady then the mass contained between sections 1 and 2, x distance apart,

changes at the rate ( )xAt

where xA is the mass contained. The rate of change must be

equal to the flow through 1 minus flow through 2, i.e., the net inflow

( ) ( )A x uA xt x

=

or ( ) ( ) 0=

+

uA

xA

t

x

u


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3

If the flow in the tube is steady, the continuity equation is

( ) 0d

uAdx

=

This implies that the mass of fluid that passes a given section must pass all the other sections

downstream. At any two sections where conditions are uniform 1 1 1 2 2 2u A u A = .

This equation is general, since it holds even if the conditions between the sections are not

uniform. If the flow is uniform at every section, the equation can be written as

== muA Constant

Eulers equation or Momentum equation

1u u p

ut x x

+ =

for steady flow, the first term is zero, and the derivatives become total derivatives.

0=+

dpduu

or =+

dpu

2

2

1constant

It is often convenient to express Eulers equation in an alternative form that describes the changes

in momentum of the fluid within a fixed control space.

Multiplying Eulers equation by A and the continuity equation by u

x

pA

x

uuA

t

uA

=

+

( ) ( ) 0=

+

uA

xuA

tu

Adding the two yields the one-dimensional momentum equation

( ) ( ) ( ) xA

ppAxx

p

AAuxuAt

+

=

=

+

2

Integrating this between any two sections gives

( ) ( ) ( ) +=+ 2

122111

2

112

2

22

2

1pdAApApAuAudxuA

t

The first integral is the momentum of the fluid enclosed between 1 and 2 and the last integral may

be evaluated by defining a mean pressure mp .


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4

( ) ( ) ( ) ( )1222111

2

112

2

22

2

1AApApApAuAudxuA

tm +=+

The integral form of the momentum equations is more general, since it is valid even when there

are dissipative processes within the control space, provided that the reference sections are

equilibrium states. During the integration of the differential momentum equation the forces on

adjacent internal faces cancel as they are equal and opposite and only the forces and the fluxes

at the boundaries of the control space are left out. If there is a non-equilibrium region inside this

space, it does not affect the integrated result.

For steady flow in a duct of constant area, the momentum equation becomes

21

2

11

2

22 ppuu =

Energy equation

For a fluid flow problem the basic thermodynamic quantity is the enthalpy, rather than internal

energy due to the presence of flow work. In adiabatic flow through a resistance the total enthalpy

per unit mass upstream and downstream of the resistance is the same.

Lets select a definite portion of the flowing fluid, between sections 1 and 2 for the system.

During a small time interval in which the fluid is displaced to a

region bounded by sections 1 and 2 , a quantity of heat, q, is

added. According to the first law,

q + work done = increase in energy

Assume that the volume displaced at 1 is the specific volume 1 corresponding to a unit mass,

then for steady conditions the displacement at 2 is also for unit mass with specific volume 2 . The

work done on the system during this displacement is 2211 pp . The local energy of the system

Unsteadychange in space

Transport or flux of momentuminto the space through end

sections

Force in the x direction due topressures on the end sections

and on the walls


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5

is2

2

1ue + per unit mass. Comparing the energy of the system after the displacement with that

before, the net increase in energy is

+

+

2

11

2

22

2

1

2

1ueue

Hence, the steady flow energy equation is

+

+=+

2

11

2

2222112

1

2

1ueueppq

or22

212 12

1

2

1uuhhq +=

And, the adiabatic flow energy equation becomes

2

11

2

22 2

1

2

1uhuh +=+

These equations relate conditions at two equilibrium states. They are valid even if there are

viscous stresses, heat transfer, or other non-equilibrium conditions between the two sections

provided sections1 and 2 are equilibrium states.

If equilibrium exists all along, the equilibrium equation is valid everywhere and may be written as

21

2h u+ = constant

or 0=+ ududh

For a thermally perfect gas this becomes 0=+ ududTCp

And for a thermally and calorically perfect gas21

2p

C T u+ = constant

At a place where u = 0 and the fluid is in equilibrium

2

0

1

2h u h+ = = constant

h0 is called reservoir or stagnation enthalpy, the enthalpy of the fluid in a large reservoir where

velocity is practically zero.

If there is no heat addition to the flow between two reservoirs, then the enthalpy of both the

reservoirs is same, 0h . Since 0 0ph C T= for a perfect gas, the stagnation temperatures in the two

reservoirs are also same.


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6

0 0 0 0,h h T T

= =

From second law of thermodynamics 0 0s s

Since for a perfect gas0

0 0/

0 0 0

ln ln 0p

p Ts s R C

Tp

= +

1o

o

p

p >

Downstream total pressure must be less than upstream total pressure

This is true for any gas, since it follows from the definition of entropy,1

Tds dh dp

= ,

An increase in entropy, at constant stagnation enthalpy, will be associated with a decrease of

stagnation pressure.

The increase of entropy, and the corresponding decrease of stagnation pressure, represents an

irreversible process. Entropy is being produced in the flow between the reservoirs. The flow is not

in equilibrium throughout. Only if the flow is in equilibrium throughout, entropy will not be produced

and the flow will be isentropic. Only in such isentropic flow 0 0 0 0,s s p p = =

The reservoir conditions or stagnation conditions are also called total conditions. The terms are

used to define conditions at any point in the flow. The total conditions at any point in the flow arethe conditions that would be attained if the flow there were brought to rest isentropically. For

stagnation conditions to exist it is not enough that the velocity be zero, it is also necessary that

equilibrium conditions exist.

Since the imaginary local stagnation process is isentropic, the total entropy at any point is by

definition equal to local static entropy sso = . Since oo TT =

, then the local entropy for a perfect

gas is related to total pressure by

ln oo

ps s R

p

=

A flow which is in equilibrium and adiabatic is isentropic. For adiabatic, non-conducting flow the

energy equation

0=+ ududh applies all along the flow


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7

Similarly, in the absence of friction forces, Eulers equation

0=+ dpudu is applicable everywhere

0= dpdh

01

=

=

dpdh

Tds ors = constant along the flow

Thus an adiabatic, non-conducting, inviscid flow is isentropic. In this case, either the momentum

or the energy equation can be replaced by the equation

s = constant

For a perfect gas, using 0dp

dh

= , this condition may be written as

1

o o o

p Tp T

= =

The conditions of equilibrium cannot be strictly attained in a real, non-uniform flow, since a fluid

particle must adjust itself continuously to the new conditions that it encounters. Thus, entropy

production is never strictly zero.


23/243


Module2:One-dimensional gas dynamics

Lecture4:Governing Equations(Contd.)


24/243


Speed of sound & Mach number

Speed of sound is the speed at which small disturbances or waves are propagated through a

compressible fluid or an elastic medium in general. Its relation to the compressibility of the fluid is

given by

2

s

pa

= , isentropic compressibility 1s

sp

=

s

s

K==

1, sK is isentropic bulk modulus

The disturbances (the temperature and velocity gradients) produced in a fluid by a sound wave

are so small that each fluid particle undergoes a nearly isentropic process. In a perfect gas

=p .const

RTp

a

== 2

In a flowing fluid, the speed of sound is a significant measure of the effects of compressibility

when it is compared to the speed of the flow. This introduces the dimensionless parameter called

Mach numbera

uM =

M will vary from point to point in a flow because of change in u and a . In an adiabatic flow an

increase in u always corresponds to an increase of M. A flow is called subsonic ifM < 1 and it is

called supersonic ifM > 1.

Area-veloc ity Relations

For a steady adiabatic flow in a stream tube of varying area the continuity equation is

0=++A

dA

u

dud

For incompressible flow 0=d and this gives the simple result that increase or decrease of

velocity is proportional to decrease or increase of area. The change in density modifies this simple

relation. Using Eulers equation for steady flow

d

d

dpdpduu ==


25/243


Since, adiabatic, inviscid flow is isentropics

d

dp

d

dp

=

daduu

2=

or

u

duM

d 2=

At very low Mach numbers the density changes are so small compared to the velocity

changes, that they may be neglected in flow computation and it may be considered that =

constant. Hence, equivalent definitions of incompressible flow are =a or 0=M .

The continuity equation now becomes

02 =++A

dA

u

du

u

duM

or 21 M

AdA

u

du

=

(1) AtM = 0, a decrease in area gives a proportional increase in velocity

(2) For 0


26/243


Some Impor tant Relations

In adiabatic flow the energy equation for a thermally and calorically perfect gas is

opp TCTCu =+2

2

1

Now, ( ) TCTCRTa pp 1112 =

==

or1

2

=

aTCp

112

222

=

+

oaau

Multiplying by2

1

a

T

T

a

aM oo ==+

2

2

2 12

1

2

2

11 M

T

To +=

Hence, the isentropic relations become

( )12

2

11

+=

M

ppo

( )112

2

11

+=

Mo

In the above equations, the values of oT and oa are constant throughout the flow and can be taken

as the actual reservoir value. The values of op and o are the local reservoir values. They are

constant throughout only if the flow is isentropic.

Instead of the reservoir, any other point in the flow can be used for evaluating the constant in the

energy equation. The throat, where M = 1, is a very useful point. The flow variables at the throat

are called sonic and are denoted by superscript *. The flow speed and sound speed are u and

a . At sonic conditionM = 1 and = au . The energy equation then gives


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11

1

2

1

1212

2

22222

=

+=

+=

+

oaaauau

ooT

T

a

a

=+

=1

22

2

Thus for a given fluid the sonic and the reservoir temperatures are in a fixed ratio, so that T* is

constant throughout in an adiabatic flow.

For air 913.0,833.0 ==

oo a

a

T

T

Using the isentropic relations withM = 1,

528.01

2 1=

+=

op

p

643.01

2 1=

+=

o

It is not necessary that a throat actually exist in the flow for sonic values to be used as reference.

The speed ratio

=Ma

u is a convenient quantity in many situations.

Using2

22

1

1

2

1

12

+=

+ a

au

Or,( ) 22

1

1

1

2

1

1

1

2

1

+=

+

MM

( )( ) ( ) 22

2

1

1

1

12

+=

+

MM

M

or( )

( ) 2

2

2

22

2

11

2

1

12

1

M

M

M

MM

+

+

=+

+=

12

1

2+

+=

M

Alternatively


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( )11

2

2

1

2

12

2

22

+

=

+

=

MM

MM

M*< 1 for M < 1, andM* > 1 for M > 1

Using RTp = and eliminating T from the energy equation for adiabatic steady flow,

2

2 1 1

o

o

pu p

+ =

For isentropic conditions,

o

opp =

11

o o

o o o o

p p pp

p

= =

Hence, the energy equation becomes

o

o

oo

o p

pppu

112

12

=

+

This is the steady state Bernoullis equation for an adiabatic compressible flow.

In a compressible flow, the dynamic pressure2

2

1u (used for normalizing pressure and forces) is

not simply the difference between stagnation and static pressure. It depends on Mach number as

well as static pressure.

2222

21

21

21

21 2 MpMpaMu

===

Hence,2

2 2

21

1 1

2 2

p

p p p p pC

M pU p M

= = =

For isentropic flow, this becomes

( )( )

2 1

2 2

2 121

2 1p

MC

M M

+

= +

M and U are reference quantities,2 2

2 2

2 2,

U uM M

a a

= =

M or a can be eliminated from the above definition using the energy equation in the form


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2 22 2

2 1 2 1

U au a

+ = +

12

2

2 2

2 11 1 1

2p

uC M

M U

= +


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1

Module2:

One-dimensional gas dynamics

Lecture5:Governing Equations(Contd.)


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2

Flow through a Constant Area Duct and Normal Shock

The inviscid incompressible flow through a uniform duct has only one possible solution, which is

of uniform flow. However, there are two possible solutions when the flow is compressible. If there

is no change in entropy anywhere, the only possible solution is the uniform flow. An alternative

solution which contains a jump in the parameters is also possible when there is a change in

entropy or a non-equilibrium region between the two stations.

Considering two sections (1) and (2) where equilibrium exists but which may contain non-

equilibrium region between them, all the conservation laws apply, and hence

2211 uu =

2

222

2

111upup +=+

2

22

2

112

1

2

1uhuh +=+

There is no restriction on the size or details of the dissipative region as long as the reference

sections are outside it. The non-equilibrium region may be idealized by a vanishingly thin region,

across which the flow parameters may jump. The control sections (1) & (2) may be brought

arbitrarily close to it. Such a discontinuity is called shock wave. A real fluid cannot have an actual

discontinuity and this is just an idealization of the very high gradients that actually occur in a

shock wave. These severe gradients produce viscous stress and heat transfer (non-equilibrium

conditions) inside the shock.

Non-equilibrium region (shock)Non-equilibrium region

(1) (2) (1) (2)


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3

The three conservation statements given above are the general equations for a steady, inviscid

adiabatic flow and hence, for a normal shock. For a thermally and calorically perfect gas, the

equations can be solved explicitly in terms of the Mach number M1. The results always apply

locally to the conditions on either side of a shock, provided it is normal to the streamline.

Both sides of the momentum equation divided by the appropriate side of the continuity equation

22

2

222

11

2

111

u

up

u

up

+=

+

or1

2

1

2

2

2

11

1

22

221

u

a

u

a

u

p

u

puu

==

Using the energy equation for a perfect gas, in terms of sound speed

2 2 2 2

21 1 2 21 1

2 1 2 1 2 1

u a u aa

+

+ = + =

2

21

= auu

This is known as Prandtl or Meyer relation.

The equation can be stated, in terms of the speed ratio, as

=

1

21

MM

1=>

=

=

= + + = <


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4

( ) ( ) ( )

1

1 0 1 1 0 1 1 4 1 1

1

0

1 1 1,

2 2 2

0

x a t

u x t a s x a t a s x a t a s a t x a t

x a t

>

= + = <


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5

wave passing through these same points differ in wave speed by the amount dc . Since a finite

wave may be thought of as a succession of infinitesimal pressure pulses, each element of the

wave may be analyzed as an acoustic wave. As long as the velocity and temperature gradients

are moderate, the viscous and heat conduction effects are negligible. Hence, each part of the

wave travels at the local speed of sound with respect to the fluid in which it is propagating. The

propagation velocity of a part of the wave with respect to fixed coordinates is then,

c u a= +

The propagation velocity of an adjacent part of the wave is

c dc u du a da+ = + + +

dc du da = +

dc du da

dp dp dp = +

Now for a right ward wave1 1

,du dc da

dp a dp dp a = = +


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6

Since the entire fluid was originally at rest with uniform pressure and temperature and each

particle of fluid undergoes isentropic changes, the increments in pressure and density between

adjacent fluid-particles obey

2 2dp da d dp d d dp

a a

d dp dp d dp d d

= = =

11

2

s

d dp

d ddc

dpdp ad

= +

Replacing density by specific volume

=

1

d

d

d

d

d

d

d

d

d

d 22

1===

( )

2

22

2

s

s

d pddc

dpdp ad

=

For a thermodynamically stable fluid,

( )sdp

dmust be negative. Hence, the isentrope must have

a negative slope on the p diagram. Consequently, the sign ofdcdp

depends only on the sign

of2

2

s

d pd

, i.e., on whether the isentrope on the p diagram is concave upwards or

concave downwards. Hence, higher-pressure parts of the wave overtake the lower-pressure parts

whendp

dcis positive. Consequently, a compression wave steepens as it progresses and an

expansion wave flattens as it progresses. Opposite happens ifdp

dc is negative.


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7

(i) Compression waves steepen and expansion waves flatten when 02

2

>

sd

pd

, or the

isentrope is concave upward. This is the usual case for all real fluids.

(ii) Compression waves flatten and expansion waves steepen if

2

2 0,s

d p

d

. Hence, compression waves

steepen and expansion waves flatten in a perfect gas.

From the point of view of an observer moving with the local particle velocity the acoustic theory

applies locally. Relative to an observer moving with the local fluid velocity, the wave at that point

propagates with the local acoustic speed ( )1

2dpa

d= whereas relative to the fixed frame of

reference in the undisturbed fluid, it propagates with the speed c a u= + . Considering both left

and right moving waves, the local wave speed at any point is given byuac +=

The wave speed is no longer constant since a is a variable and u may no longer be neglected.

To evaluate these in terms of the density, the acoustic theory is applied locally. Using the

isentropic relation for a perfect gas 1

1

pp

= to eliminate p from

pa =2 ,

2

1

11

=

aa

The particle velocity is evaluated in terms of the density by applying locally

=

dadu

This becomes integrable ifa is replaced by above


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8

( )12

1

1

1

1

21

1

2

1

aaad

au

=

==

or uaa2

2

1

=

1

1

2c a u

+ = +

or

++=

11

11

2

1

1

1

ac

where 1a is the speed of sound in the undisturbed fluid and 1 is the density of undisturbed fluid.


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1

Module2:

One-dimensional gas dynamics

Lecture5:Governing Equations (Contd.)


53/243


2

Propagation of Fini te waves

Consider the propagation of a simple wave of finite amplitude. Assume that the initial density

distribution is as shown in the figure ( )00 == tt

For the right ward propagating wave

++=

11

11

2

1

1

1

ac

Hence, the wave speed is higher than 1a in regions of condensation ( )1 > and lower than 1a

in regions rarefaction. Thus the wave distorts as it propagates, the regions of higher condensation

tending to overtake those in regions of lower condensation. In regions of higher condensation, the

characteristic lines are inclined more, since the slope is inversely proportional to the wave speed.

In terms of the compression and expansion regions the net effect is to steepen compression

regions and to flatten expansion regions in which the characteristic lines converge and diverge

respectively. In a compression region, the characteristics lines would eventually cross leading to

the situation 3t t= . But this would be physically impossible, for it implies three values of density at

a given point. Actually, well before this happens, the velocity and temperature gradients in the

compression regions become so large that friction and heat transfer effects become important.

These have a diffusive action which counteracts the steepening tendency. The two opposingeffects achieve a balance and the compression portion of the wave become stationary, in the

sense that it propagates without further distortion. It is then a shock wave.

t

t3

t2

t1

x

t0 =0 ( )0,~ xs


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3

In compression regions, the isentropic relations are valid until friction and heat transfer become

important. When a stationary balance between the diffusive and steepening (non-linear) terms

has been reached, the conditions across the wave front are given by the shock wave relations.

The intermediate unsteady, non-isentropic states can be treated only with the full unsteady

equations including viscous and heat transfer terms.

An expansion wave always remains isentropic as it tends to flatten and so reduce the velocity and

temperature gradients further. It never achieves stationary condition, corresponding to the fact

that there are no expansion shocks.

Centered Expansion wave

Consider a duct containing fluid enclosed by a piston. If the piston is withdrawn an expansion

wave is produced. If the piston starts impulsively, with speedp

u , the distribution of particle

velocity in the first instant is a step. However, the expansion wave begins to flatten as soon as the

wave starts propagating. At some later time1t the particle velocity has a linear distribution and

the pressure has a corresponding distribution.

u3 =- |up|

p3

p4

u =0

a4

(4)

up

x

x =-|up| t

Piston path

Expansion path

t

t1(3)

(4)

x =a4 t

tuatcxp

+==

2

143

(3)


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4

The front of any isentropic wave propagates at the speed of sound of the undisturbed fluid. Thus,

the front of the wave propagates with speed 4a into the undisturbed fluid that is in the opposite

direction of the piston and fluid motion. The wave speed in the portions of the wave behind the

front is given by

uac2

14

++=

Wave speed c decreases continuously through the wave since 0u < . The fan of straight lines are

lines of constant c and thus of constant u and . These lines are the characteristics. With

increasing time the fan becomes wider and the wave becomes flatter and the gradients of

velocity, density, temperature become smaller. Thus the wave remains isentropic. The terminating

characteristic is given by puactx

2

143

+==

and slopes to right or left depending on

whether 41

2pa u

>M ), the wave angle decreases with

decrease in the wedge angle. When decreases to zero, decreases to the limiting value , given

by

01sin 22

1 =M or

=

M

1sin 1

The jump in the flow quantities is then zero and, hence the strength of the wave is zero. The flow is

continuous without any disturbance. There is nothing unique about the point where this wave

originates; it might be any point in the flow. The angle is simply a characteristic angle associated

M2

M1

M1

M1

M2

M2

M1


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3

with 1M . It is called the Mach angle. The lines of inclination which may be drawn at any point in

the flow field are called Mach lines or Mach waves.

If the flow is nonuniform varies with M and the Mach lines are curved. At any point P in a 2-D flow

field, there are always two lines which intersect the streamline at the angle . In 3-D flow, the Mach

lines or characteristics define a conical surface with vertex at P. A 2-D supersonic flow is always

associated with two families of Mach lines denoted by the labels (+) and (). Those in the (+) set run

to the right of the streamlines and those in the () set run to the left. They are also called

characteristics from the mathematical theory of hyperbolic PDEs. These are analogous to the two

families of characteristics that trace the propagation of 1-D waves in the x-t plane. Like the

characteristics in the x-t plane, Mach lines have a distinguished direction, the direction of flow or the

direction of increasing time. This is related to the fact that there is no upstream influence in

supersonic flow.

First-order approximation for weak oblique shocks

For small deflection angles , the oblique shock equations reduce to very simple expressions. The

approximate relation that can be used to derive others is

+ tan

211sin 21

221 MM

For small , the value of is close to either2

or , depending on whether 12 M .

For 12


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4

22 2 1

12

1

1sin 1

2 1

MM

M

+

, as

1

1tantan

2

1

=

M

The pressure is then approximated to

12

1

2

1

1

12

=

M

M

p

p

p

pp

The changes in other flow quantities are also proportional to the deflection angle '' . The change of

entropy is proportional to the third power of the shock strength and hence to third power of deflection

angle3

s

The difference between the wave angle and the Mach angle , to first order accuracy, can be

found as follows,

Let , = +


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5

Hence, for a finite deflection angle, the direction of the wave differs from the Mach direction by an

amount , which is of the same order as .

The change in flow speed can be obtained as

( )( )

2

2 22 2 2 2

2 2

22 2 2 2 2

1 1 1

1 tan 1 cos

tan 1 cos1

u

w u

w u u

+ ++ = = = =

+ + +

Now,

22 2 1

2 21 1

1 2cos 1 sin 1

1

M

M M

= =

Similarly, ( ) 2cos can be obtained form 2cos by replacing by .

The final result, after dropping all terms of order2 and higher

2

21 1

11

w

w M

or

12

11

=

Mw

w


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1

Module3:Waves in Supersonic Flow

Lecture11:Waves in Supersonic Flow (Contd.)


71/243


2

Supersonic compression by turning

A shock wave passing through a fluid increases the pressure and density of the fluid. Hence, shock

waves can be used to compress a flow. A simple method for compressing a supersonic flow is to turn

it through an oblique shock by deflecting the wall through an angle . The turn may be subdivided

into several segments which make smaller corners of angle so that compression occurs through

successive weaker oblique shocks. These shocks divide the field near the wall into segments of

uniform flow. Away from the wall the shocks tend to intersect each other since they are convergent. In

the near wall region each segment of the flow is independent of the next one and may be constructed

step by step proceeding downstream. This property of limited upstream influence exists as long as

the deflection does not become so great that the flow becomes subsonic.

For each wave in the multiple shock p and ( )3s .

The overall pressure and entropy changes are

1 ~kp p n

( ) ( )( ) ( )3 2 2

1 ~ ~ks s n n

Thus, when the compression is achieved through a large number of weak shocks, the entropy

increase can be reduced significantly compared to a single shock giving the same net deflection. It

decreases as2

1

n. By continuing the process of subdivision, the segments can be made vanishingly

small ( 0 ), and in the limit, the smooth turn or isentropic compression is obtained.

When the shocks become vanishingly weak, they are almost straight Mach lines. Each segment of

uniform flow becomes vanishingly narrow and finally coincides with a Mach line. Thus, the flow

inclination and Mach number are constant on each Mach line. Thus, in the limit of smooth flow, the

velocities and flow inclination are continuous, but their derivatives may still be discontinuous. The

approximate expression for the change of speed across a very weak shock

12

1

=

Mw

w

becomes the differential equation


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3

)(1

2

1

M

M

d

w

wd

=

=

Due to the convergence of the Mach lines, the change form 1M to 2M on the streamline b occurs in a

shorter distance than on the streamline a. Hence, the gradients of velocity and temperature on b are

higher than those on a. An intersection of Mach lines would imply an infinitely high gradient for there

would be two values of M at one point. However, this cannot occur since in the region where Mach

lines converge and the gradients become very high the conditions are no longer isentropic. Before the

Mach lines cross a shock wave is developed. Far from the corner, there would be a simple oblique

shock for 1M and . The convergence of Mach lines in a compression is a typical nonlinear effect:

decreasing Mach number and increasing flow inclination both tend to make successive Mach lines

steeper.

If a wall is placed along one of the streamlines, say b, where the gradients are still small enough for

the flow to be isentropic; then an isentropic compression in a curved channel is obtained. Since this

flow is isentropic, it may be reversed without violating the second law of thermodynamics.

a

b

M2

M2

M1

M1


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4

Supersonic Expansion by Turning

Flow round a concave turn, that is turns in which the wall is deflected in to the flow, undergoes

compression through shock wave/Mach lines. Expansion takes place in a flow over a convex corner.

In this case a turn through a single oblique wave is not possible.

Since 1 2 = , 2u must be greater than 1u decrease in entropy. Hence, expansion shocks are not

possible.

The non-linear mechanism that steepens a compression produces the opposite effect in expansion.Instead of being convergent, the Mach lines are divergent.

2 w1

u1

w2

1


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5

Consequently, there is a tendency to decrease gradients. Thus an expansion is isentropic throughout.

The expansion at a corner occurs though a centered wave defined by a fan of straight Mach lines.

The flow up to the corner is uniform at Mach number 1M and thus the leading Mach wave must be

straight at the Mach angle 1 . The terminating Mach lines stands at the angle 2 (corresponding

to 2M ) to downstream wall. This centered wave is more often called a Prandtl-Meyer expansion fan.

Using the differential relation between and M in an isentropic compression or expansion by turning

2

1

1

2M1 M2

M2

M1


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6

w

dwMd 12 =

( )Mw

dwMK ==+ 1

2

Now aMw = and

220

2

11

2

aM

a

= +

+

=+=2

2

11

1

MM

dM

a

da

M

dM

w

dw

Hence M

dM

M

MM

+

=

2

2

2

11

1)(

( )1 2 1 21 1

tan 1 tan 11 1

M M

+ = +

This function is known as the Prandtl-Meyer function. The constant of integration is chosen arbitrarily

so that 0= corresponds to 1=M . The corresponding values of the flow properties are obtained

from isentropic relations.

A supersonic Mach number M is always associated with a definite value of the function . As M

varies from 1 to , increases monotonically form 0 to max , where

max

1

1 2.27685rad 130.454 for 1.42 1

+

= = =

D

In a compression turn decreases, whereas in an expansion turn it increases, in each case by an

amount equal to the flow deflection. Knowing the initial value ( )1 1M = , the value of for a given


76/243


7

value of then gives the corresponding value ofM . Usually, the value of 1 0,is set to since only

the deflection matters.

For compression

11 =

,,M

1

11,M

,,M

11 +=

111 ,, M

For expansion


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1


Lecture12:Waves in Supersonic Flow (Contd.)


78/243


2

Simple and Non-simple regions

The isentropic compression and expansion waves are distinguished by the straight Mach lines with

constant conditions on each one and by the simple relation between flow deflection and Prandtl-

Meyer function. A wave belongs to one of two families (+ or ), depending on whether the wall that

produces it is to the left or right of flow respectively. In the region where two simple waves of opposite

family interact with each other, the flow is non-simple. The relation between and is not the simple

one given by = 1 . These regions may be treated by the method of characteristics.

Reflection & Intersection of oblique shocks

An oblique shock incident on a wall is reflected. The incident shock deflects the flow through an

angle toward the wall. Hence, a reflected shock of the opposite family is required to turn it back

again an amount of to satisfy the zero normal velocity on the wall.

Non simple region

Simple wave

Simple wave ()


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3

The deflections produced by the two shocks are equal in magnitude but the pressure ratios are not,

since 12 MM < . The strength of the reflection is defined by the overall pressure ratio which equals the

product of the individual shock strengths.

3 3 2

1 2 1

p p p

p p p=

Usually, the reflection is not specular, i.e., the inclination of the reflected shock is not the same as

the inclination of the incident shock. The shock angles are different since both Mach number and

flow inclination ahead of the second shock are smaller than those ahead of the first shock. The two

effects are opposite and the result depends on the particular values of 1M and . An explicit relation

cannot be found but the values can be found easily.

The wall streamline, in the reflection case, may also be identified with the central streamline of thesymmetric flow in the intersection of two shocks of equal strength but of opposite families. The shocks

pass through each other but are slightly bent in the process. The flow downstream of the shock

system is parallel to the initial flow.

p2p1

On wall

On streamline

p3p2

p1

M3M2M1


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4

(Intersection of equal strength shocks)

If the intersecting shocks are of unequal strengths, the flow experiences different changes in

traversing the shock wave system. The streamline through the intersection point divides the flow intotwo portions. The two portions have the same pressure and the same flow direction. The direction is

not necessarily that of the free stream. These two requirements determine the final direction and the

final pressure 3p . All other parameters are then determined, but they do not have the same values on

the two sides of the dividing streamline. A slip stream or shear layer develops since the magnitudes

of the velocity on either side of it are different. It is also called a contact surface, because the

temperature and density on either side are different. These differences are related to the net entropy

changes experienced by the fluid on the two sides of the intersection.

M3

Flow symmetric about the

central streamline

M3M2M1

M2


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5

Dividing streamlineor slip stream

, M2M3

p3

p3

M3

M2

M1


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1


Lecture13:Waves in Supersonic Flow(Contd.)


83/243


2

Intersection of shocks of the same family

Two shocks of the same family produced by, for example, successive corners in the same wall,

cannot pass through each other. They coalesce to form a single stronger branch.

The flow on either side of the intersection point, o, experiences different entropy changes and a

slipstream is produced. An additional wave oe , of the opposite family, is needed to equalize the

pressures on the two sides of the slipstream. This may be either a compression or an expansion

wave depending on the particular configuration and Mach number. However, it is very much weaker

than the primary waves. If the second shock bo is much weaker than the first one ao , thenoe is

usually a compression. In this case the second shock is partly transmitted along oc , thus

augmenting the first one and partly reflected along oe .

In the interaction of an expansion wave with a shock wave of the same family, the main effect is an

attenuation of the shock, but there is also a partial reflection of the expansion along Mach lines of the

opposite family. These reflected waves are always very mach weaker than the primary ones and may

be neglected in all but the strongest interactions. Instead of the single slipstream, there is a whole

region of vorticity, that is, an entropy field downstream of the interaction.

Slip stream

Weak reflection

d

e

ba

c

o


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3

Detached shocks

For a given supersonic stream if the wall deflection is max , > the flow cannot negotiate the turn

through an attached oblique shock. The observed flow configurations are, as example,

M1

M1

Attenuation of shock by expansion wave

Weak reflections

M=1

bcM=M1=max


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4

The flow is compressed through a curved shock, detached from the body and stands at some

distance ahead of it. The shape of the shock and the detachment distance depend on the geometry

and flow Mach number. On the central streamline, the shock is normal and the flow behind the shock

is subsonic. On the nearby streamlines, the shock is nearly normal and the flow is compressed tosubsonic conditions. Further out, the shock becomes weaker and less steep, approaching

asymptotically to the Mach angle. Thus conditions along the detached shock wave contain the whole

range of the oblique shock solution for the given Mach number. In such configurations, shock

inclination corresponding to strong solution is found. When the flow behind the shock is subsonic, the

shock is no longer independent of the downstream conditions. A change in geometry or pressure in

the subsonic portion affects the entire flow up to the shock and the shock needs to adjust itself to the

new conditions. In the case of a blunt-nosed body, the shock wave is detached at all Mach numbers.

A wedge of half-angle max > is a blunt-nosed body so far as the oncoming flow is concerned.

The sequence of events in the flow over a wedge with after-body with decreasing Mach number is as

follows: (1) when 1M is sufficiently high the shock wave is attached to the nose, the straight portion is

M>1

M


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5

independent of the shoulder and after-body. The shock angle increases as 1M decreases. (2) At a

certain reduced Mach number, the flow after the shock becomes subsonic. The shoulder now affects

the whole shock, which may be curved, even though still attached. These conditions correspond to

the region between the lines 2 1M = and max = in the M relationship. (3) At the Mach

number corresponding to max , the shock wave starts to detach. This is called detachment Mach

number. (4) With further decrease of 1M , the detached shock moves upstream of the nose.

A similar sequence of events occurs in flow over a cone with cylindrical after-body. The detachment

Mach numbers are lower than for wedges.


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Lecture14:Waves in Supersonic Flow

(Contd.)


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2

Mach Reflection:

The appearance of subsonic regions in the flow complicates the problem. The complications are also

encountered in shock reflections, when they are too strong to give the simple or regular reflections. If

2M after the incident shock is lower than the detachment Mach number for , then no solution with

simple oblique wave is possible. A three-shock Mach reflection appears that satisfies the downstream

conditions.

A normal, or, nearly normal, shock that appears near the wall forms with the incident and reflected

shocks a triple intersection point at O. Due to the difference in entropy on streamlines above and

below the triple point, the streamline that extends downstream from the triple point is a slipstream.

The nearly normal shock is termed shock stem.

The subsonic region behind the shock stem makes a local description of the configuration impossible.

The triple point solution that occurs in a particular problem and the location of the triple point are

determined by the downstream conditions which influence the subsonic part of the flow.

Shock-Expansion Theory

Slip stream

0M1M21


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Oblique shock wave and simple isentropic wave relations can be used to analyze many 2-D

supersonic flow problems, particularly for geometries with straight segments.

(1) Diamond-section airfoil: Consider a diamond section or double-wedge section airfoil with

semi-vertex angle . Assume the semi-vertex angle to be sufficiently smaller than

max associated with the free stream Mach number 1M . An attached oblique shock appears at

the nose that compresses the

flow to pressure 2p .On the straight portion, downstream of the shock the flow remains uniform at 2M .

The centered expansion at the shoulder expands the flow to pressure 3p and the trailing edge shock

recompresses it to nearly the free stream pressure ( 14 pp ). Hence, an overpressure acts on the

forward face and an under-pressure acts on the rearward face. Since the pressure on the two straight

portions is unequal, a drag force acts on the airfoil. This drag force is given by

( ) ( )2 3 2 3cosD p p t p p t= per unit span

t is the section thickness at the shoulder. Pressure values 2p and 3p can be obtained using the

shock and expansion relations. This drag exists only in supersonic flow and is called supersonic

wave drag.

p3

p4

p2

p1

3

4

2

M1, p1

M1>12

t

2


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(2) Flat plate at incidence: Consider a flat plate of chord c set at an angle of attack . Due to

no upstream influence, the streamlines ahead of the leading edge are straight and the upper surface

flow is independent of lower surface. The flow on the upper surface turns at the nose through a

centered expansion by the angle whereas on the lower side the flow is turned through a

compression angleby an oblique shock. The reverse happens at the trailing edge.

From the uniform pressures on the two sides, the lift and drag forces are

( )3 2 cosL p p c =

( )3 2sinD p p c =

The shock on the lower surface at the nose is weaker than the shock at the trailing edge on the upper

surface (shock at higher Mach number). Hence, the increase in entropy for flow on the two sides is

not same and consequently the streamline from the trailing edge is a slipstream inclined at a small

angle relative to the free stream.

(3) Curved airfo il section

An attached shock forms at the nose. Subsequently, continuous expansion occurs along the surface.

The flow leaves at the trailing edge through an oblique shock. For the shocks to be attached, it is

required that nose and tail be wedge shaped with half angle less than max . Since the flow over the

curved wall varies continuously, no simple expression for lift and drag forces is obtained in this case.

Slip stream

3

2

M1>1

M1, p1

p1

p2

p3


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If a larger portion of the flow field is considered, then the shocks and expansion waves will interact.

The expansion fans attenuate the oblique shocks, making them weak and curved. At large distances

they approach asymptotically the free-stream Mach lines. Due to the interaction the waves will reflect.

The reflected wave system will alter the flow field. In shock-expansion theory, the reflected waves are

neglected. For a diamond airfoil and a lifting flat plate, the reflected waves do not intercept the airfoil

at all. Hence, the shock-expansion results are not affected.


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Lecture15:Waves in Supersonic Flow

(Contd.)


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2

Thin Airfoil Theory

The shock-expansion theory provides numerical solutions as long as the shocks are attached. The

method is quite simple and general for computing lift and drag. The results, in general, cannot be

expressed in a concise analytic form. However, for a thin airfoil at small angle of attack all the flow

deflections are small and the shock-expansion theory can be approximated by the approximate

relations for weak shocks and expansions. The basic approximate expression for computing pressure

change is

12

2

M

M

p

p

When the first-order weak wave approximation is valid, p andMwill not be much different

from 1p and 1M . Hence,

12

1

2

1

M

M

p

p

If all pressure changes are referenced to free-stream pressure 1p and all deflections to the free-stream

direction, then

12

1

2

1

1

1

M

M

p

ppwhere is inclination relative to free

stream

1

22

2

1 21

1

1

2

12

11

1

=

=

Mp

pp

Mu

ppCp

Thus for a flat plate at a small angle of attack , the pressure coefficients on the upper and lower

surfaces are


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3

2

1

2

1p

CM

=

The lift and drag coefficients are

( ) ( )2

21

1 1

cos 4cos

1 12

l u

l u

L p p

p p cC C C

Mu c

= =

( )( )

2

22

11 1

sin 4sin

1 12

l u

l u

D p p

p p cC C C

Mu c

= =

2

12

11

4

D

L

CM

C =

The aerodynamic centre is at mid-chord.

For the diamond section aerofoil with nose angle 2 the pressure coefficients, on the front and rear

faces, at zero-incidence are

2

1

2

1pC

M

=

2 32

1

4

1p p

M

=

2

1 1

1

2u

( ) ( )2

2

2 3 2 3 1 12

1

4 1

21D p p t p p c u c

M

= = =

Hence,

22

2 2

1 1

4 4

1 1D

tC

cM M

= =


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For a general airfoil that has thickness, camber and angle of attack, the pressure coefficient on the

upper and lower surfaces are

2 2

1 1

2 2,

1 1

U L

U Lp p

dy dyC C

dx dxM M

= =

The profile can be resolved into a symmetric thickness distribution ( )xt and a camber line of zero

thickness ( )xc at an incidence

U c tdy d d

dx dx dx

= +

c tL d ddydx dx dx

=

Hence,

( )

2

1 12

1 12

1

14

1 2

2 1L U

c cc

p po o

ud

L u C C dx dxdxM

= = +

2

1 1

1

2 L U

cUL

p po

dydyD u C C dx

dx dx

= +

2221 1

2

1

12

2

1

cUL

o

udydy

dx

dx dxM

= +


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22 21 1

2

1

14

2

1

cc t

o

ud d

dxdx dxM

= + +

The lift and drag coefficients are given as

2 2

2

201

4 1

1

c

c t

D

d dC dx

c dx dxM

= + +

The lift coefficient depends only on angle of attack but the drag coefficient also depends on camber

and thickness. The drag splits into three parts: a drag due to lift, a drag due to thickness and a drag

due to camber.

2 2

1 1

4 1 4

1 1

cc

Lo

dC dx

c dxM M

= + =


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Module4:Fiber in Ducts

Lecture 16:Flow in ducts, (Nozzles and diffusers)

and wind tunnels


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2

The flow can be assumed to be one-dimensional, that is, conditions across each section are uniform.

The conditions at any two sections in a steady flow are related by the equation

222111 AuAu =

Using the sonic condition as reference

= AuuA

When the flow is purely subsonic,

A is a fictitious area that does not occur in the flow. But, if sonic

and supersonic conditions are attained in the flow, then == tAA area of the actual throat

Since = au ,

u

a

u

a

A

A o

o

==

We have

1

12,

1o

= +

2

1

2

1

12

+

+=

M

u

a

1

1

2

2

11

+=

Mo

The isentropic area-Mach number relation becomes

( )( )1

1

2

2

2

2

11

1

21 +

+

+=

M

MA

A

Area-pressure relation

1

1

2

1

2

1

1

2

11

1

2

2

1

1

+

+

==

oo p

pp

p

u

u

A

A


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Mass flow rate per unit area

o

o

TT

T

RRT

puu

RT

pu

A

m 1

===

2112o

pM

RT

= +

Defining a mass flow parameter as

WpT

Am om 1=

, where W is the molecular weight

2112

mM M

= +

, = universal gas constant

In terms of stagnation quantities and Mach number

( )121

2

2

11

+

+

=

M

M

T

p

RA

m

o

o

Hence, for a given Mach number, the flow rate is proportional to the stagnation pressure and

inversely proportional to the square root of stagnation temperature.o

o

pTm

is used as a non-

dimensional mass flow parameter for turbomachinery performances.

In can be seen that the mass flow rate attains a maxima when 1M = . Hence,


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o

o

T

p

RA

m

A

m 11

max 1

2 +

+==

Hence, for a given gas, the maximum flow per unit area depends on o

o

pT

. For fixed ando op T and

passage, the maximum flow that can pass is relatively large for gases of high molecular weight and

small for gases of low molecular weight.

The fact that the curve of mass flow rate per unit area has a maximum is connected with the

interesting and important effect called choking.

1

isentropic relations

chart orM ,1

o

,

1

oTT

1

A

A.

A is constant . Hence, ( ) ( )1

1

2

2 = A

AA

A

AA

2A

A orchart

relationsisentropic

,2M ,

2

o

2

oTT

Since op and oT are constant, 2p & 2T can be obtained as

,

1

2

1

2

=

o

o

p

p

pp

pp

1

2

1

2

=

o

o

T

T

TT

T

T

Now, for a given area ratio1

2

AA

, 2M can be computed for given 1M . The plotted results look like


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(1) For a given initial Mach number 1M and a given area ratio2

1

AA

, there are either two solutions for

final 2M or none at all. When there are two solutions, one is subsonic and the other is supersonic.

Which one of the two occurs depends, the part, on whether a throat exists between sections (1) and

(2), since in order to change the regime the flow must pass a throat at 1M = .

For example if 1M is subsonic and the passage is converging, then 2M must be subsonic. But if the

passage is converging-diverging and has a throat between (1) and (2), the flow at section (2) may be

either subsonic

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