COMPRESSIBLE FLOW

1. Explain compressibility, compressible flow and incompressible flow.

Compressibility is the fractional change in volume of the fluid element per

unit change in pressure. It is given by,

dpdv

v1

−=τ

Since the volume is reduced, dv is negative quantity. Compressibility is a

property of the fluid. Liquids have very low values of compressibility (order

of 10-10) whereas gases have high compressibility. (order of 10-5).

Per unit mass,

dpdρ

ρτ 1=

or dpd ⋅⋅= τρρ

Now consider the fluid in motion. Such flows are initiated and maintained by

forces on the fluid, usually created by or at least accompanied by changes in

the pressure.

High speed flows involve large pressure gradients. For a given change in dp,

due to the flow, the resulting change in density will be small for liquids and

large for gases. Therefore, for the flow of liquids, relatively large pressure

gradients can create high velocities without change in density. Hence, such

flows are usually assumed to be incompressible, where ρ is constant.

For flow of gases with large value of compressibility, moderate to strong

pressure gradients lead to substantial changes in the density. At the same time,

such pressure gradients create large velocity changes in the gas. Such flows

are called compressible flows, where ρ is variable.

For gas velocities less than about 0.3 of the speed of sound, the associated

pressure changes are small and even through compressibility is large for gases,

dp may still be small enough to dictate a small ρd . So the low speed flow

gases can be assumed to be incompressible.

If the density changes by 5 percent of more, the flow is considered to be

compressible

2. Classify the flow based on Mach number.

Subsonic flow: M < 1 at every point, and hence the flow velocity is

everywhere less than the speed of sound. This flow is characterized by smooth

streamlines and continuously varying properties. Here, the flow is forewarned

by the presence of body.

For airfoils in common use, if M<0.8, the flowfield is generally completely

subsonic.

Transonic flow: 0.8 < M < 1.2. If M remains subsonic, but is sufficiently near

1, the flow expansion over the top surface of the airfoil may result in locally

supersonic regions. M < 1 but is high enough to produce a pocket of locally

supersonic flow. This pocket terminates with a shock wave across which there

is a discontinuous and sometimes rather severe change in flow properties. If M

is slightly increased above unity, this shock pattern will move to the trailing

edge of the airfoil, and a second shock called bow shock appears upstream of

the leading edge.

Supersonic flow: 1 < M < 5. M > 1 everywhere in the flowfield. The flow is

not forewarned by the presence of the body. The flow is supersonic both

upstream and downstream of the oplique shock.

Hypersonic flow: M > 5. The temperature, pressure and density of the flow

increase almost explosively across the shock. Oblique shock wave moves

closer to the surface and the flowfield between the shock and the body

becomes very hot- indeed hot enough to dissociate or even ionize the gas.

3. What is one dimensional flow and quasi-one dimensional flow?

One dimensional flow: It is one in which the flow field properties vary only

with one coordinate direction.

A = constant, p = p(x), u = u(x), T = T(x)

Example: Normal shock

Quasi one dimensional flow: The area varies gradually along x, and it is

convenient and sufficiently accurate to neglect the y and z flow variations and

to assume that the flow properties are functions of x only.

A = A(x), p = p(x), u = u(x), T = T(x)

Example: Nozzle.

4. Write the steady, one dimensional flow equations.

Continuity: 2211 uu ρρ =

Momentum equation: 2222

2111 upup ρρ +=+

Energy equation: 22

22

2

21

1vhqvh +=++

5. Define sound wave and speed of sound.

Sound wave: Weak wave is called sound wave. (if the changes through the

wave are strong, it is identified as a shock wave).

Speed of sound: Rate of propagation of small pressure disturbances through

the medium of interest.

s

pa ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=ρ

is the general relation.

For a perfect gas, RTa γ=

6. Define static temperature, stagnation temperature, static pressure and

total pressure.

Static temperature: It is the characteristic temperature of the flow. It has no

contribution from velocity.

Stagnation temperature: Temperature measured when the flow is brought to

rest adiabatically.

Static pressure: Pressure which has no contribution from velocity.

Stagnation pressure: Pressure measured when the flow is brought to rest

isentropically.

7. What is Crocco Number?

maxVVCR = where Vmax = 02 TC p

8. Define normal shock. Write the normal shock relations.

Shock is formed due to coalescing of weak pressure disturbances.

Normal shock: Shock waves that are perpendicular to freestream. Shock is

very thin region (10-5 cm).

Normal shock relations:

Continuity: 2211 uu ρρ =

Momentum equation: 2222

2111 upup ρρ +=+

Energy equation: 22

22

2

21

1vhvh +=+

9. State the property variations across the normal shock wave.

Normal shock is assumed to be a discontinuity across which the flow

properties suddenly change. The flow is supersonic ahead of the shock and

subsonic behind the shock.

Static pressure, static temperature and density increases across the shock

wave.

Velocity and Mach number decreases across the shock.

Since flow across the shock is adiabatic, stagnation temperature is constant

and stagnation pressure decreases across the shock.

10. Define Mach wave and Mach angle.

Infinitely weak normal shock is defined as Mach wave. (when M1 = 1 and M2

= 1).

Also infinitely weak oblique shock is called Mach wave.

The wavefronts form a disturbance envelope given by a straight line which is

tangent to the family of circles. This line of disturbances is defined as Mach

wave.

Mach wave can be either compression wave or an expansion fan.

The angle which the Mach wave makes with respect to the direction of motion

is called Mach angle μ .

Mach angle ⎟⎠⎞

⎜⎝⎛= −

M1sin 1μ

11. What happens to properties behind the normal shock when M1 tends to

infinity?

378.02

1lim 2 =−

=γ

γM

11lim

1

2

−+

=γγ

ρρ

= 6

∞=1

2limpp

∞=1

2limTT

12. Write the Prandtl equation for shocks.

1*2

*1 =MM

13. What really causes the entropy across a shock wave?

The changes across the shock wave occur over a very short distance, on the

order of 10-5 cm. Hence the velocity and temperature gradients inside the

shock structure itself are very large. In regions of large gradients, the viscous

effect of viscosity and thermal conduction become important. In turn, these are

dissipative, irreversible phenomena that generate entropy. Therefore, the net

entropy increase is provided by nature in the form of friction and thermal

conduction inside the shock wave structure itself.

Shock is always possible from supersonic flow to subsonic flow and not from

subsonic flow to supersonic flow because entropy change is less than zero

which is violation of second law of thermodynamics.

14. Explain Hugoniot equation.

Because the static pressure always increases across a shock wave, the wave

itself can also be visualized as a thermodynamic device which compresses the

gas. Indeed, the changes across a normal shock wave can be expressed in

terms of purely thermodynamic variable without explicit reference to a

velocity or Mach numbers.

)(2 21

2112 vvppee −

+=−

This is called Hugoniot equation. The change in internal energy equals the

mean pressure across the shock times the change in specific volume.

A plot of p2 = f(p1,v1, v2) on a pv graph is called the Hugoniot curve.

For a given decrease in specific volume, a shock wave creates a higher

pressure increase than an isentropic compression. However, the shock wave

costs more because of the increase and consequent total pressure loss.

i.e.shock compression is less efficient than the isentropic compression.

For a calorically perfect gas,

2

1

2

1

1

2

11

111

vv

vv

pp

−⎟⎟⎠

⎞⎜⎜⎝

⎛−+

−⎟⎟⎠

⎞⎜⎜⎝

⎛−+

=

γγγγ

v1 and v2 are specific volumes.

15. Write the equation for shock strength.

Shock strength = 1

12

ppp −

Entropy change across the shock is proportional to cube of shock strength.

16. What are property changes across the moving shock?

Ratio of static properties does not change.(i.e. same as that of the normal

shock)

Total temperature and total pressure across and behind the moving shock are

different.

Mach number behind the shock need not be subsonic.

17. Write the governing equations for one dimensional flow with heat

addition.

Continuity: 2211 uu ρρ =

Momentum equation: 2222

2111 upup ρρ +=+

Energy equation: 22

22

2

21

1vhqvh +=++

18. Write the properties changes due to heat addition.

For supersonic flow in region 1, i.e. M1 > 1, when heat is added

a. Mach number decreases. M2 < M1

b. Pressure increases p2 > p1

c. Temperature increases T2 > T1

d. Total temperature increases T02 > T01

e. Total pressure decreases P02 < P01

f. Velocity decreases u2 < u1

For subsonic flow in region 1, i.e. M1 < 1, when heat is added

a. Mach number decreases. M2 > M1

b. Pressure decreases p2 < p1

c. Temperature increases for M < 2/1−γ and decreases for M > 2/1−γ

d. Total temperature increases T02 > T01

e. Total pressure decreases P02 < P01

f. Velocity increases u2 > u1

For heat extraction, all of the above trends are opposite.

19. Explain Rayleigh curve.

Lower branch of the Rayleigh curve corresponds to supersonic flow and the

upper branch corresponds to subsonic flow. If the flow is supersonic, heat

addition will decrease M till the flow will become sonic. For this condition,

the flow is said to be chocked. If heat is added further, then a normal shock

will form inside the nozzle and conditions in initial region will suddenly

become subsonic.

If the flow is subsonic, heat addition will increase M till the flow will become

sonic. For this condition, the flow is said to be chocked. If heat is added

further, then a series of pressure waves will propagate upstream and nature

will adjust the conditions in the initial region to a lower subsonic M, to the left

of initial region.

20. Write the properties changes due to friction.

For supersonic flow in region 1, i.e. M1 > 1, due to friction

a. Mach number decreases. M2 < M1

b. Pressure increases p2 > p1

c. Temperature increases T2 > T1

d. Total temperature remains constant T02 = T01

e. Total pressure decreases P02 < P01

f. Velocity decreases u2 < u1

For subsonic flow in region 1, i.e. M1 < 1, due to friction

a. Mach number decreases. M2 > M1

b. Pressure decreases p2 < p1

c. Temperature decreases T2 < T1

d. Total temperature remains constant T02 = T01

e. Total pressure decreases P02 < P01

f. Velocity increases u2 > u1

21. Write the governing equations for one dimensional flow with heat

addition.

Continuity: 2211 uu ρρ =

Momentum equation: ∫++=+L

wdxD

upup0

2222

2111

4 τρρ

where shear stress 2

21 vfw ρτ =

Energy equation: 22

22

2

21

1vhqvh +=++

22. Explain Fanno curve.

Friction always drives the Mach number toward 1, decelerating a supersonic

flow and accelerating a subsonic flow.

Lower branch of the Fanno curve corresponds to supersonic flow and the

upper branch corresponds to subsonic flow. If the flow is supersonic, friction

will decrease M till the flow will become sonic for a certain duct length L. For

this condition, the flow is said to be chocked. If duct length is increased

further, then a normal shock will form inside the nozzle and conditions in

initial region will suddenly become subsonic.

If the flow is subsonic, friction will increase M till the flow will become sonic.

For this condition, the flow is said to be chocked. If duct length is increased

further, then a series of pressure waves will propagate upstream and nature

will adjust the conditions in the initial region to a lower subsonic M, to the left

of initial region.

23. Define oblique shock and expansion wave.

Oblique shock: It usually occurs when the flow is turned into itself. The

change in flow direction takes place across the shock wave which is oblique to

the freestream direction. Across the shock wave, the Mach number decreases

and the pressure, temperature and density increase.

Expansion wave: It occurs when the supersonic flow is turned away from

itself. All flow properties through an expansion wave change smoothly and

continuously, with the exception of the wall streamline which change

discontinuously at the corner. [the change of properties take place slowly and

isentropically through the innumerable rays of fan]

Across the expansion wave, the Mach number increases and the pressure,

temperature and density decrease.

24. Define shock wave angle and deflection angle.

Shock wave angle: Angle which the original streamline makes with the plane

of the shock.

Deflection angle: It is the angle by which the original streamline is deflected.

25. Write the oblique shock relations.

Oblique shock relations:

Continuity: 2211 uu ρρ =

Momentum equation: 2222

2111 upup ρρ +=+

Energy equation: 22

22

2

21

1vhvh +=+

The tangential component of the flow velocity is preserved across an oblique

shock wave. In all the above equations, the velocities are normal to the wave.

Therefore, the changes across an oblique shock wave are governed by the

normal component of freestream velocity.

βsin11 MM n =

and )sin(

22 θβ −= nMM

26. What happen if maximum deflection angle is greater than the deflection

angle?

If maximum deflection angle is greater than the deflection angle, then no

solution exists for a straight oblique shock wave. Instead, the shock will be

curved and detached.

27. Explain weak and strong shocks.

Changes across the shock are more severe as wave angle increases, the large

value of wave angle is called the strong shock solution and the small value of

wave angle is called the weak shock solution. In nature, the weak shock

solution is favored and usually occurs. Whether weak shock or strong shock

solution occurs is determined by the backpressure.

In the strong shock solution, M2 is subsonic. In the weak shock solution, M2 is

supersonic.

28. Write the possible range of oblique shock wave angle. What happen if

upstream Mach number is increased keeping flow deflection angle is

constant and vice versa?

Range: ⎟⎠⎞

⎜⎝⎛−

M1sin 1 to 900.

As M1 increases (holding deflection angle constant), the shock wave becomes

stronger and wave angle decreases.

As deflection angle increases (holding M1 constant), the shock wave becomes

stronger and wave angle increases.

29. State the differences between supersonic flow over wedges and cones.

The flow over wedge is 2 dimensional whereas the flow over the cone is three

dimensional. The flow streamlines behind the shock are straight and parallel to

the wedge surface. The flowfield between the shock and the cone surface is no

longer uniform. The streamlines are curved, and the pressure at the cone

surface ps is not same as p2 immediately behind the shock.

Addition of third dimension provides extra space to move through, hence

relieving some of the obstructions set up by the presence of the body. This is

called three dimensional relieving effect which is characteristic of all three-

dimensional flows. This relieving effect results in a weaker shock wave than

for the wedge of the same angle.

30. What is slip line?

Slip line is a line across which the entropy changes discontinuously.

31. What is sonic boom?

A noise caused by a shock wave that emanates from an aircraft or other object

traveling at or above sonic velocity.

32. What is shock tube?

A shock tube is a device that allows the experimental study of many of the

phenomena associated with the occurrence of shock waves. It consists of a

long tube divided into two sections by a diaphragm. One section of the tube

called the driver section contains a relatively high pressure gas. The second

section called the expansion section contains a low pressure gas that may be

different from that of the high pressure gas. Upon rupture of the diaphragm, a

shock wave propagates into the stagnant gas in the expansion section.

33. Write the Prandtl – Meyer function.

1tan)1(11tan

11)( 2121 −−−

+−

−+

= −− MMMγγ

γγν

This is the direction measured from M = 1 though which the flow has turned

by an isentropic process to reach the given M.

34. Define nozzle and diffuser.

Nozzle: Any device that accelerates the flow. Velocity increases at the

expense of pressure drop.

Diffuser: Any device that decelerates the flow. Pressure increases at the

expense of kinetic energy.

35. Write the governing equations for quasi one dimensional steady flow.

Continuity: 222111 uAuA ρρ =

Momentum equation: 222222

211111

2

1

uApApdAuApAA

A

ρρ +=++ ∫

Where integral term represents the pressure force.

Energy equation: 22

22

2

21

1vhvh +=+

36. Write the area-velocity relation and explain.

[ ]12 −= MVdV

AdA

a. For M tends to zero, which in the limit corresponds to incompressible

flow.

b. For 0 < M < 1, an increase in velocity is associated with a decrease in

area and vice versa. The velocity increases in a converging duct and

decreases in a diverging duct.

c. For M > 1, an increase in velocity is associated with an increase in area

and vive versa. For supersonic flow, the velocity increases in a

diverging duct and decreases in a converging duct.

d. For M = 1 yields dA/A = 0, which mathematically corresponds to a

minimum or maximum in the area distribution. The minimum in area is

the only physically realistic solution.

37. What is chocked mass flow rate?

Maximum mass flow rate for the given stagnation condition is called chocked

mass flow rate.

)1(21

0

0* 1

2 −+

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=γγ

γγ

TP

RAm

38. What is the effect of back pressure on nozzle exit plane pressure?

If the exit Mach number is subsonic, Pe must be equal to Pb.

In sonic and supersonic flow, Pe need not be equal to Pb. There is enough fluid

dynamic mechanism like oblique shock and expansion fans to achieve the

pressure equalization.

39. Draw the operating characteristics of a convergent nozzle and explain.

a. When Pb = P0, there is no flow.

b. Reduce Pb slightly from the previous. Flow takes place, accelerates in

the nozzle. Assuming exit subsonic flow , Pe = Pb.

c. Reduce Pb further such that the nozzle is chocked. Pe = P*. Chocked

mass flow rate is achieved.

d. Further reducing the pressure Pb upto nozzle exit, no change. After that

expansion fans are formed.

40. Draw the operating characteristics of a convergent – divergent nozzle and

explan.

To generate supersonic flow, only possible way is to use Convergent –

Divergent nozzle.

a. When Pb = P0, there is no flow. Curve ‘a’

b. Pb is reduced slightly. Curve ‘b’ corresponds to the subsonic flow

through out with Pe = Pb. Me is subsonic.

c. Curve ‘c’ corresponds to again a subsonic flow. But higher Me than

that at ‘b’ above.

d. Curve ‘d’ is that corresponds to the value of Pb which chock the throat

first time. Pressure ratio corresponds to this is first critical pressure

ratio. This can be found from the isentropic tables corresponding to the

given Ae/A* and choosing the subsonic Mach number.

e. Choose a pressure value such that the flow goes through isentropically

all through the nozzle with supersonic exit flow of Pe = Pb. Me

corresponds to this is the supersonic solution in the Ae/A* Vs M graph.

Curve ‘I’ has the pressure ratio corresponding to this is third critical

pressure ratio. (perfectly expanded condition of the CD nozzle)

f. As Pb is lowered more, say to condition ‘f’, the shock wave takes up

positions successively farther downstream occurring at greater

upstream Mach numbers and undergoing greater pressure rises.

g. One particular ratio may be imposed that will cause the normal shock

to stand in the nozzle exit plane. Curve ‘g’. This particular pressure

ratio is called the second critical pressure ratio.

h. For pressure ratios between first and second critical ratios, the exit

plane outflow is subsonic and exit pressure equals back pressure. At

the second critical pressure ratio, the exit plane pressure is double

valued.

i. With the outflow supersonic, the exit plane pressure need not equal the

back pressure and adjustment to the higher back pressure may take

place externally.

j. Overexpanded Nozzle: When the exit pressure is less than back

pressure, oblique shock will get formed at the exit plane to attain

pressure equalization with Pb.

k. Underexpanded Nozzle: When the exit pressure is more than that of

back pressure, expansion fans will be formed at the exit plane to

equalize the pressure.

41. What happen if a pipe is added to the nozzle on either side?

Addition of pipe on either side decreases the mass flow rate since stagnation

pressure is less.

42. What is wind tunnel?

A chamber where steady flow of air or smoke is blown over an object, such as

an airfoil, to calculate its aerodynamic forces, such as lift and drag.

43. Draw the Fanno curve and Rayleigh curve in one graph and mark the

normal shock on it.

44. Draw T-S or h-S diagram for flow with Heat addition, flow with friction,

isentropic process and normal shock

Questions asked by Dr.V. Babu on 13.08.2005 to R. Sivakumar.

1. How will you decide the whether the flow is compressible or

incompressible.

Ans: Refer Question no. 1 in Compressible flow q& a.

2. There is a flow through the duct with a max. velocity of 20 m/s.

Combustion is takes place inside the duct. Say whether it is

compressible or incompressible. What is the effect of temperature

rise due to heat addition?

The flow is incompressible since the pressure gradient (or pressure

difference) is very small. Eventhough the temperature rises due to heat

addition, the density is not changing much. The temperature rise

increases the speed of sound and the flow velocity is less. It decreases

the Mach number further, so the flow is incompressible.

3. Draw the T- s diagram for the following supersonic intake.

4. Draw the T –s diagram for the following two diagrams. How to

calculate the maximum mass flow rate for this? Which case has

more mass flow rate?

Case I is having the maximum mass flow rate because the stagnation

pressure loss is less when compared to case II.

5. What is Mach wave? Is it a compression wave or an expansion

wave.

Ans: Refer Question no. 10 in Compressible flow q& a.

6. Draw the supersonic flow along a compression corner. Mark the

wave pattern for that corner. How it looks like? What happened if

they coalescence?

The mach angle increases for decreasing the Mach number. So the

mach wave becomes wider and wider. When they coalescence, strong

oblique shock will result. Across the mach wave, the property changes

takes place smoothly.

7. Draw the mach wave for an expansion corner. What happened here

to the Mach wave?

The mach angle increases for decreasing the Mach number. So the mach

wave becomes wider and wider. Here the mach wave comes closer and

closer. They won’t coalescence.

8. Draw the T-s diagram for the following two diagrams. How to

calculate the mass flow rate for these cases. How to get the same

mass flow rate for this first case without making any modification in

the setup.

The maximum mass flow rate formula is to be used only for choked and

the isentropic condition. So for the first case, this formula is not

applicable. We have to use our basic continuity equation to calculate the

mass flow rate for the second case.

Choked condition is not possible in the first case because of stagnation

pressure loss. Therefore to get the same mass flow rate of the first case,

is to change the inlet condition. That is increasing the stagnation pressure

is the only way. Choked condition is never achieved in the first case.

9. Draw the fanno curve and Rayleigh curve. What happened if you

increase the length more than L* in case of Fanno flow and adding

more heat after the choked condition is achieved in case of Rayleigh

flow? Mark it in the T- s diagram.

If the length is increased more than L*, then shock will form inside the

duct. Mass flow remains constant in both Fanno and Rayleigh flows.

Increasing the length more and more will result in the shock moving

towards left and finally it occur outside the duct.

If the heat added further and further, shock form outside the duct.

Mass flow rate remains constant in both the cases. So if we change the

mass flow rate, the values are to be represented in the another Fanno or

Rayleigh curve and not on the same curve.