08/03/2022
Gas Dynamics
COMPRESSIBLE FLOW
B. Huyssen
8 March 2022
OBJECTIVES




Appreciate the consequences of compressibility in gas
flows
Understand why a nozzle must have a diverging
section to accelerate a gas to supersonic speeds
Predict the occurrence of shocks and calculate
property changes across a shock wave
Understand the effects of friction and heat transfer on
compressible flows
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OUTLINE
Stagnation properties
Chapter 7
Speed of Sound and Mach Number
Chapter 8
One-Dimensional Isentropic Flow
Chapter 8
COMPRESSIBLE FLOW
 Incompressible flow assumes constant density
 Compressible flow assumes that changes in pressure
and temperature results in changes in density
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COMPRESSIBLE FLOW
The amount by which a substance can be compresses is called compressibility τ.
Consider a section normal to the streamtubes. If υ is the specific volume(volume
per unit mass), for an increase of pressure dp we will have a decrease of specific
volume dv(negative quantity)
 

1 d
 dp
1

d  
d
2
d  dp
The streamlines will diverge in order to get the mass flow past the midsection
The density is not consider constant anymore when M>0.3, Bernoulli equation needs
to be corrected to consider compressibility effect. For subsonic speed, v<a, the changes
in pressure that are generated by the airfoil in movement are smaller relative to the free
stream static pressure and compressibility effect are neglected. As the speed increases
even the flow is still subsonic, the density decreases as the pressure decreases and
velocity increases. The streamlines will diverge in order to get the mass flow past the
midsection of the airfoil. The perturbations(u,v) caused by the airfoil extends vertically
to a greater distance.
STAGNATION PROPERTIES
Compressible flow combines fluid dynamics and thermodynamics. It
involves a significant change in density.
Recall definition of specific enthalpy defined by
unit of mass for calorically perfect gas
h  e  p
h  c pT
dh  de  pd  vdp
e  c T
specific heat capacity at pressure and constant volume for air at standard conditions
which is the sum of specific internal energy e and
flow energy pυ. (if potential and kinetic energy are zero
Enthalpy is the total energy of the a fluid.)
Assume the flow adiabatic and body forces are zero.
For high-speed flows, since the kinetic energy is high,
the enthalpy and kinetic energy are combined into
stagnation enthalpy h0, that represents the total
energy of a flowing fluid stream per unit of mass.
h
V2
 h0
2
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STAGNATION PROPERTIES
The energy eq for steady one dimensional flow is
Therefore, stagnation enthalpy remains constant along a streamline during
steady-flow process, and if all streamlines originate from a common uniform
freestream, than h0 is the same for each streamline!
STAGNATION PROPERTIES
If a fluid were brought to a complete stop (V2 = 0)
Therefore, h0 represents the enthalpy of a fluid when it is
brought to rest adiabatically.
During a stagnation process, kinetic energy is converted to
enthalpy which results in increase in the fluid temperature
and pressure.
Properties at this point are called stagnation properties
(which are identified by subscript 0)
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STAGNATION PROPERTIES
STAGNATION
PROPERTIES
ISENTROPIC PROCESS
For an isentropic process we can define the total density and total pressure in function
of static density and pressure. From the Second law of thermodynamics if the process
is reversible ds  q  ds
𝜕𝑤
𝑝𝑣
irrev
T
From the First Thermodynamic law
𝑑𝑒
For an ideal gas pv=RT, dh = cpdT,
T2
p
 R ln 2
T1
p1
Relates p, ρ, T
Tds  dq  de  pd
𝜕𝑤
Tds  dh dp
From definition of dh=de+pdv+vdp
s  c p ln
𝜕𝑞
ds  c p
p2   2

p1  1
dh vdp

T
T
dT
dp
R
T
T
cp
p2  T2  R
 
p1  T1 
Must be isentropic flow

ds 

  T2   1
  
 T 
  1

p2  T2   1
 
p1  T1 

cp
cv
Ratio of specific heat capacity at
pressure and constant volume for
air at standard conditions
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TOTAL CONDITIONS
Define total temperature T0= value of
temperature of the fluid element after it
has been brought to rest adiabatically.
From the energy equation for steady,
adiabatic, inviscid flow the total
enthalpy is constant along a streamline:
h
V2
 const  h0
2
h  c pT
Along a streamline the total temperature
is the sum of static temperature plus the
dynamic temperature all per unit of
mass.
1
c pT1  V12  c pT0
2
For a calorically perfect gas the total
temperature stays constant along a
streamline.
TOTAL CONDITIONS
T0 represents the temperature an ideal gas obtained when it is brought to rest
adiabatically.
1
c pT1  V12  c pT0
2
T1 
1 2
V1  T0
2c p
V2/2Cp corresponds to the temperature rise, and is called the dynamic
temperature
For ideal gas with constant specific heats, if the process is isentropic the
stagnation pressure and density can be expressed as

p0  T0   1
 
p1  T1 
1
 0  T0   1
 
1  T1 
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TABULATION OF ISENTROPIC FLOW PROPERTIES
TOTAL CONDITIONS
When using stagnation enthalpies, there is no need to explicitly use
kinetic energy in the energy balance.
h0  c pT0
Where h01 and h02 are stagnation enthalpies at states 1 and 2, q is the
heat transfer and w the work at states 1 and 2 per unit of mass.
If the fluid is an ideal gas with constant specific heats
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TOTAL CONDITIONS
From definition of total temperature we can express the temperature in function of
Mach
1
c pT1  V12  c pT0
2
T0
V12
 1
2c pT1
T1
T0
 1 2
M1
 1
T1
2
Only Mach dictates the ratio of total to static temperature
a 2  RT
a2
T
R
Requires adiabatic flow, but does
not have to be isentropic
If the flow goes through an isentropic compression to zero
velocity
R is specific gas constant
cv
p0    1 2 
M1 
 1 
p1 
2

Known M, T, p and ρ in a specific
point we can get the total condition.
0    1 2 
M1 
 1 
2
1 


cp

1
 1
 1
TOTAL CONDITIONS
T0
 1 2
M1
 1
T1
2
If the flow is adiabatic
h
V2
 const  h0
2
1
c pT  V 2  c pT0
2
From the stagnation properties if the flow is Isentropic (a process that is both adiabatic
and reversible)


p2   2   T2   1
  

p1  1   T1 
p0    1 2 
M1 
 1 
p1 
2

0    1 2 
M1 
 1 
2
1 


1
 1
 1
Flow properties depends only from
Mach number!
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EXAMPLE
GIVEN
the isentropic flow over an airfoil. The free stream
conditions correspond to a standard altitude of 3000 m and M∞=0.82. At a
given point on the airfoil M=1.
FIND
p and T at this point.
SPEED OF SOUND
FUNCTION ONLY OF T
Important parameter in compressible flow is the speed of sound.
Speed at which infinitesimally small pressure wave travels. Consider a
sound wave propagating through a gas with a velocity a.
In b you hop on the wave and ride with it. The gas upstream is coming to you
with velocity a, behind you is receding away from you a relative velocity a+da.
The flow is adiabatic, no heat source. Flow trough sound wave is isentropic
too. Process is adiabatic and reversible.
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SPEED OF SOUND
Consider a duct with a moving piston at
constant increment of velocity dV
The piston creates an infinitesimal
disturbance, a sonic wave that propagates to
the right at speed a.
Fluid to left of wave front, moving at dV,
experiences incremental change in
properties
Fluid to right of wave front maintains
original properties
a
SPEED OF SOUND
Construct Control Volume that encloses wave front and moves with it. If you
are riding the wave it seems that the fluid on the right is moving towards the
wave with a speed of a. The fluid on the left with speed of a-dV while the
observer is stationary.
1)Mass balance
m  AV
Aa    d Aa  dV 
a
a
Aa  Aa  dV  ad  ddV 
cancel
ad  dV  0
1)
dV 
Neglect
H.O.T.
ad

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SPEED OF SOUND
a  dV 
a2
h
 h  dh 
2
2
2) Energy balance ein = eout
2
h
a2
a 2  2adV  dV 2
 h  dh 
2
2
a
Neglect
H.O.T.
cancel
Cance l
a
dh  adV  0
dh  adV
2)
SPEED OF SOUND
3) Conservation of Energy: first law of thermodynamics
ds 
Second law of thermodynamics
If the process is reversible
ds 
q
T
q  de  pdv
q
 dsirre
T
Tds  de  pdv
From the definition of enthalpy dh  de  pdv  vdp
Tds  dh  vdp  dh 
dP

Since the sonic wave is adiabatic and nearly isentropic, using the thermodynamic
relations from first and second law:
Tds  dh 
dP

dh 
dP

3)
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SPEED OF SOUND
1) dV 
2)
ad

Combing this with mass and energy conservation gives
dP
dh  adV

a
ad

dp
a2 
d
3) dh  dP

For flow through the sound wave we have an isentropic process
and assuming the gas is calorically perfect

cp
cv
 P 
a 2   
   S
 P 
a 2    
  
 1.4 Ratio of specific heat capacity at pressure and constant volume for air
at standard conditions
For an ideal gas
 P 
  RT  
  RT
a 2      
   S
  
R=cp-cv is specific gas constant
a  RT
SPEED OF SOUND
a  RT
Since
R is specific gas constant
Ratio of specific heat capacity γ
is only a function of T
Speed of sound is only a
function of temperature
It varies with the fluid too!
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DYNAMIC PRESSURE FOR COMPRESSIBLE
FLOWS
Dynamic pressure is defined as q = ½ρV2
For high speed flows, where Mach number is used frequently, it is
convenient to express q in terms of pressure p and Mach number, M, rather
than ρ and V
Derive an equation for q = q(p,M)
1
V 2
2
1
1 p

V 2 
q  V 2 
2
2 p
2

V2 
q  p 2  pM 2
2 a
2
q
a2 

p V 2
 p 
q 
p


2
p M 2
SONIC CONDITIONS
From the stagnation properties
p0    1 2 
M1 
 1 
2
p1 

0    1 2 
M1 
 1 
2
1 


1
 1
 1
Flow properties depends only from
Mach number!
If at one point of the flow M =1 flow is in sonic condition at that point. We can get the
value of T*,p*,ρ* in function of total values:
T0   1

T*
2
+
p0    1 


p*  2 
0    1 


*  2 

 1
γ=1.4
1
 1
T
 0.833
T0
p*
 0.528
p0
*
 0.634
0
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SONIC CONDITIONS
Note the ratio is inverted in this table compared to the book appendix!
  1.3
  1.33
  1.667
  1.4
SONIC CONDITIONS
Energy eq for adiabatic flow
For one dimensional flow
1
1
cPT1  u12  cPT2  u22
2
2
cp 
a  RT
2
R
 1
2
1
1
a1
a
 V12  2  V22
 1 2
 1 2
If point 2 is stagnation point, a0 is stagnation ( or total) speed of sound
2
1
a2
 u 2  0  const
 1 2
 1
a
2
2
a1
a
a2
1
1
 u12  2  u 22  0  const
 1 2
 1 2
 1
a and u are at given point in the flow and a0 is the stagnation speed of sound ASSOCIATED with
the same point. Valid for two points along the streamline.
If point 2 is at sonic condition u=a*
2
*2
a
1
1
 u2 
 a *2
 1 2
 1 2
a
  1 *2 a02
a 
2  1
 1
1
 1
a2
 u22 
a *2
2  1
 1 2
2
a0 and a* are constant along a given streamline in a steady adiabatic
inviscid flow. If all streamlines emanate from the same uniform
freestream conditions than they are constants trough the entire field.
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SONIC CONDITIONS
a1
1
 1
 u12 
a *2
2  1
 1 2
2
Characteristic Mach number in function
of actual M for one dimensional flow
2
Characteristic Mach number
u
M*  
a
 1
1
a1 1 1 2 1
a *2 2
 u1

  1 u12 2 u12 2  1
u1
Ratio of local velocity to the speed
of sound at sonic condition not
the actual local value M.
M *2 
  1M 2
2    1M 2
Characteristic Mach number is the Mach number attained by the fluid when the
throat section is hypothetically brought to sonic conditions.
M  1
M* is the local velocity nondimensionalized with the
respect to the sonic velocity (ex. at the throat)
M is the local velocity nondimensionalized with the
respect to the local sonic velocity.
M 1
M  1
IF
M  1
M 
 1
 1
M 1
M 1
M 
EXAMPLE
GIVEN
a point in an airflow where the local M=3.5, P=0.3atm,
T=180K.
FIND
the local values of p0,T0,T*, a*, M* at this point.
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QUASI ONE-DIMENSIONAL ISENTROPIC FLOW
Variation of Fluid Velocity with Flow Area
If the area varies moderately we can assume the variables of the flow field vary only
with x. For flow through nozzles, diffusers, and turbine blade passages, flow quantities
vary primarily in the flow direction. It can be approximated as 1D isentropic flow.
Need to introduce energy equation and isentropic relations. Let’s consider a moving
fluid element along a streamline. From the energy equation for steady,
adiabatic, inviscid flow the total enthalpy is constant:
Differentiate
h
dh  udu  0
u2
 const  h0
2
From second law of thermodynamics if there are no shock waves the flow is isentropic:
Tds  dh dP

1
dh  dP 

Specific volume
m  Au  const
d


dA du

0
A u
1

 udu 
dP
1

dP
Consider the mass balance for a steady flow process.
Differentiate and divide by mass flow rate (AV)
Combining with the other equations
dA
1
d
dP 

A u 2

QUASI ONE-DIMENSIONAL ISENTROPIC FLOW
Variation
Fluid
Velocity
Variation
ofof
Fluid
Velocity
withwith
FlowFlow
Area Area
dA
1
d
 2 dP 
A u

 P 
a 2   
   S
u2
Using thermodynamic relations and rearranging
dA dP  u 2 
1  

A u 2  a 2 
Area-velocity relation

dA dP
 2 1 M 2
A u

This is an important relationship that describes the M with variation of the area
For M < 1, (1 - M2) is positive  dA and dP have the same sign.
Pressure of fluid must increase as the flow area of the duct increases,
and must decrease as the flow area decreases
For M > 1, (1 - M2) is negative  dA and dP have opposite signs.
Pressure must increase as the flow area decreases, and must decrease as
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QUASI ONE-DIMENSIONAL ISENTROPIC FLOW
Variation of Fluid Velocity with Flow Area

dA dP
 2 1 M 2
A u

A relationship between dA and du can be derived by substituting
u = -dP/du

dA
du

1 M 2
A
u

Since A and u are positive
For subsonic flow (M < 1)
 udu 
1

dP
This eq governs the shape of a nozzle or
a diffuser in subsonic or supersonic
isentropic flow
dA/du < 0 increase of velocity, du is associated with a
decrease in area, dA
For supersonic flow (M > 1) dA/du > 0 increase of velocity, du is associated with a
increase in area, dA
For sonic flow (M = 1)
dA/du = 0
even dA=0 the du exists, that is the
minimum area
QUASI
ONE-DIMENSIONAL
ISENTROPIC
FLOW
ONE-DIMENSIONAL
ISENTROPIC
FLOW
Comparison of flow properties in subsonic and supersonic nozzles and diffusers
Max velocity is at sonic condition
If we want to
accelerate the flow
converging
If we want to
decelerate the flow
diverging
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