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COMPRESSIBLE FLOW

SOME FUNDAMENTAL ASPECTS

OF COMPRESSIBLE FLOW

Mach number

aVM ==

sound of speed velocitygas number,mach

RTPa γργ

==

1<M : subsonic

1=M : transonic

1>M : supersonic

1>>M : hypersonic

Part three : Mach Number 27

COMPRESSIBLE FLOW

Isentropic flow in a streamtube

In order to illustrate the importance of the Mach number

in determining the conditions under which

compressibility must be taken in account, isentropic

flow, i.e., frictionless adiabatic flow, through a

streamtube will be first considered.

From previous chapter, we know that ;

VdV

PV

PdP 2ρ

−= and γρ

2aP=

the above equation can be written as :

VdVM

VdV

aV

PdP 2

2

2

γγ −=−= (1)

This equation shows that the magnitude of the fractional

pressure change, induced by a given fractional velocity

change, depends on the square of Mach number.


COMPRESSIBLE FLOW

Next, consider the energy equation. Since adiabatic

flow is being considered ;

VdVM

cR

VdV

TcV

TdT

pp

22 γ

−=−=

Since; γ11−=−= vp ccR and 1−= γγ

pcR

Above equation can be written as ;

VdVM

TdT 2)1( −−= γ (2)

Lastly, consider the equation of state;

TdTd

PdP

+=ρρ

combining above equation with eq.(1) and eq.(2)


COMPRESSIBLE FLOW

VdVM

VdVMd 22 )1( −+−= γγ

ρρ

This equation indicates that:

2MV

dV

d−=ρ

ρ

(negative sign means, density decrease when velocity

increased)

at M=0.1 , %1−=V

dV

dρ

ρ

at M=0.33 , %11−=V

dV

dρ

ρ

At low mach number, density changes will be

insignificant.


COMPRESSIBLE FLOW

Normally at M<0.3, the fluid is assumed

incompressible.

It should also be noted that above equation can we

written as ;

2)1( MVdVTdT

−−= γ

Similarly, the temperature difference is neglected at

lower value of Mach number.


COMPRESSIBLE FLOW

Mach waves

Disturbances tend to propagated ahead of the body in

motion to “warn” the gas of the approach of the body.

This is due to pressure at the surface is higher than

surrounding gas and pressure waves spread out from the

body.

The pressure waves spread out at the of sound

Effect of the velocity of the body relative to the speed

of sound (pressure wave velocity) on the flow field.


COMPRESSIBLE FLOW

Consider for subsonic flow M<1, figure (1).

Speed of the body u and speed of sound a, where u<a.

Body position at a, b, c and d at time interval t. Waves

generated at time 0, t, 2t and 3t. Since u<a, a body

moves slower than the waves and therefore a body will

never overtake it.


COMPRESSIBLE FLOW

If u>a, then M>1, the flow is supersonic, a body moves

faster than the waves and will overtake it, (figure (2)).

The waves lie within a cone which has its vertex at the

body at the time considered. On gas within this cone

“aware” of the presence of the body. Vertex angle α is

called Mach angle, where ;

Mua 1sin ==α

The cone is therefore termed a conical Mach wave.


COMPRESSIBLE FLOW


COMPRESSIBLE FLOW

ONE-DIMENSIONAL

ISENTROPIC FLOW

INTRODUCTION

An adiabatic flow (a flow in which there is no heat

exchange) in which viscous losses are negligible, i.e., it

is an adiabatic frictionless flow.

Although no real flow is entirely isentropic, there are

many flows of great practical importance in which the

major portion of the flow can be assumed to be

isentropic.

Part four : One-Dimensional Isentropic Flow 37

COMPRESSIBLE FLOW

For example, in internal duct flows there are many

important cases where the effects of viscosity and heat

transfer are restricted to thin layers adjacent to the walls,

i.e., are only important in the wall boundary layers, and

the rest of the flow can be assumed to be isentropic.

Even when non-isentropic effects become important, it

is often possible to calculate the flow by assuming it to

be isentropic and to then apply an empirical correction

factor to the solution so obtained to account for the

non-isentropic effect, for example, in the design nozzle.


COMPRESSIBLE FLOW

GOVERNING EQUATION

By definition, the entropy remains constant in an

isentropic flow.

cP=γρ (c:constant) (4.1)

From equation (4.1) γ

ρρ

⎟⎟⎠

⎞⎜⎜⎝

⎛=

1

2

1

2

PP


COMPRESSIBLE FLOW

Hence, since the general equation of state gives ;

22

2

11

1

TP

TP

ρρ= or

1

2

1

2

1

2

ρρ

PP

TT

=

It follows that in isentropic flow ;

γγγ

ρρ

1

1

2

1

1

2

1

2

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

PP

TT

Recalling that RTa γ= , that ;

γγγ

ρρ

ρρ 2

1

1

22

1

1

221

1

2

1

2

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=TT

aa

eq.(4.5)

The steady flow adiabatic energy equation is next

applied between the point 1 and point 2. This gives ;

22

22

2

21

1VTcVTc pp +=+


COMPRESSIBLE FLOW

It can be written as ;

)2(1)2(1 1

212 TcVT p+

=2

221 TcVT p+

rom ;

F

222V

=2

122

McR

RTV

Tc pp

−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡ γγγ

o, it follows that ; S

222

1

212

12 )1(1 MT −+=

γ

1 )1(1 MT −+ γ eq.(4.6)

his equation applies in adiabatic flow. If friction T

effects are also negligible, i.e., if the flow is isentropic,

eq.(4.6) cam be used in conjunction with the isentropic

state relations given in eq.(4.5) to obtain ;


COMPRESSIBLE FLOW

1

222

1

212

1

1

2

)1(1)1(1 −

⎥⎦

⎤⎢⎣

⎡−+−+

=γγ

γγ

MM

PP

and

11

222

1

212

1

1

2

)1(1)1(1 −

⎥⎦

⎤⎢⎣

⎡−+−+

=γ

γγ

ρρ

MM

astly, it is called that the continuity equation gives ; L

222111 AVAV ρρ =

hich can be rearranged to give ; w

⎟⎟⎞

⎜⎛

⎠⎜⎝⎟⎟⎠

⎜⎝ 112 VA ρ

⎞⎜⎛

= 221 VA ρ


COMPRESSIBLE FLOW

STAGNATION CONDITIONS

Stagnation conditions are those that would exist if the

flow at any point in fluid stream was isentropically

brought to rest.

If the entire flow is essentially isentropic and if the

velocity is essentially zero at some point in the flow,

then the stagnation conditions will be those existing at

the zero velocity point.


COMPRESSIBLE FLOW

However, even when the flow is non-isentropic, the

concept of the stagnation conditions is still useful, the

stagnation conditions at a point the being the conditions

that would exist if the local flow were brought to rest

isentropically.

If the equations derived in the previous section are

applied between a point in the flow where the pressure,

density, temperature and Mach number are P, ρ, T, M

respectively, then if the stagnation conditions are

denoted by the subscript 0, the stagnation pressure,

density and temperature will, since the Mach number is

zero at the point where the stagnation conditions exist,

be given by ;


COMPRESSIBLE FLOW

120

211

−

⎥⎦⎤

⎢⎣⎡ −+=

γγ

γ MPP

11

20

211

−

⎥⎦⎤

⎢⎣⎡ −+=

γγρρ M

⎥⎦⎤

⎢⎣⎡ −+= 20

211 M

TT γ

( for the particular case of 4.1=γ )


COMPRESSIBLE FLOW

CRITICAL CONDITIONS

The critical conditions are those that would exist if the

flow was isentropically accelerated or decelerated until

the Mach number was unity, (M = 1)

These critical conditions are usually denoted by an

asterisk.

By setting M2=1, we found ;

⎥⎦

⎤⎢⎣

⎡+−

++

= 2*

11

12 M

TT

γγ

γ

21

2*

11

12

⎥⎦

⎤⎢⎣

⎡+−

++

= Maa

γγ

γ

12

*

11

12 −

⎥⎦

⎤⎢⎣

⎡+−

++

=γγ

γγ

γM

PP


COMPRESSIBLE FLOW

11

2*

11

12 −

⎥⎦

⎤⎢⎣

⎡+−

++

=γ

γγ

γρρ M

By setting M2=0, we found ;

12

0

*

+=γT

T

12

0

*

+=

γaa

1

0

*

12 −

⎥⎦

⎤⎢⎣

⎡+

=γγ

γPP

11

0

*

12 −

⎥⎦

⎤⎢⎣

⎡+

=γ

γρρ

For the case of air flow ;

833.00

*

=TT

, 528.00

*

=PP

, 634.00

*

=ρρ


COMPRESSIBLE FLOW

MAXIMUM DISCHARGE VELOCITY

Also known as “maximum escape velocity”, is the

velocity that would be generated if a gas was

adiabatically expanded until its temperature has

dropped to absolute zero.

Using the adiabatic energy equation gives the maximum

discharge velocity as :

0

22

22

ˆTcTcVVpP =+=

This can be rearranged to give ;

02 2)2(ˆ TcTcVV pP =+=

12)

12(ˆ

20

22

−=

−+=

γγaaVV


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