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1MAE 5420 - Compressible Fluid Flow
Section 3: Lecture 1
Introduction to One-Dimensional
Compressible Flow
Anderson: Chapter 3 pp. 71-86
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2MAE 5420 - Compressible Fluid Flow
Review, General Integral Form of
Equations of Compressible Flow
• Continuity (conservation of mass)
• Newton’s Second law-- Time rate of change of momentumEquals
integral of external forces
! "V!>
• ds
!>#$
%&
C .V
''' =((t
"dvc.v.
'''#
$)%
&*
! fb">
dvC .V .
### " p( )C .S .
## dS">
+ | f">
$ dS">%
&'(
C .S .
## |dS
SC .S .
">
=
!V">
• ds">%
&'(
C .S .
## V">
+))t
!V">%
&'( dv
C .V .
###
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3MAE 5420 - Compressible Fluid Flow
Review (concluded)
• Conservation of Energy--
!!t
" e +V2
2
#
$%
&
'( dv
#
$%%
&
'((
C .V .
))) + "V*>
• d S*>
e +V2
2
#
$%
&
'(
C .S .
)) =
(" f*>
dv) •V*>
C .V .
))) * (pd S*>
) •V*>
C .S .
)) + " q•#
$&' dv
#$%
&'(
C .V .
)))
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4MAE 5420 - Compressible Fluid Flow
One Dimensional Flow Approximations
• Many Useful and practical Flow Situations can be
Approximated by one-dimensional flow analyses
• “Air-goes-in, Air-goes-out, or both”
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5MAE 5420 - Compressible Fluid Flow
Control Volume for 1-D Flow
• Between C.V. entry (1) and exit (2), there could be
1) Normal Shock wave (supersonic engine duct),
2) Heat could be added or subtracted, (heat exchanger), or
3) There could we work performed (turbine element)
Flow Characterized by motion only along longitudinal axis
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6MAE 5420 - Compressible Fluid Flow
Continuity for Steady 1-D Flow
• The general equation for Continuity greatly simplifies for
steady 1-D flow
! "V!>
• ds
!>#$
%&
C .S .
'' =((t
"dvc.v.
'''#
$)%
&*
!V">
• ds
">#$
%&
C .S .
'' = 0
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7MAE 5420 - Compressible Fluid Flow
Continuity for Steady 1-D Flow (cont’d)
• Evaluating surface integral across the C.S.
!V">
• ds
">#$
%&
C .S .
'' = 0
V
dS
dS
dS
dS
C.S.
!!
!V">
• ds
">#$
%&'' = 0
!V">
• ds
">#$
%&'' = 0
!V">
• ds
">#$
%&'' = !VAe
!V">
• ds
">#$
%&'' = "!VAi
Stephen Whitmore
Stephen Whitmore
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Stephen Whitmore
Stephen Whitmore
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8MAE 5420 - Compressible Fluid Flow
Continuity for Steady 1-D Flow (concluded)
• For true 1-D flow Ae=Ai
V
dS
dS
dS
dS
C.S.
!!
!V">
• ds
">#$
%&'' = 0
!V">
• ds
">#$
%&'' = 0
!V">
• ds
">#$
%&'' = !VAe
!V">
• ds
">#$
%&'' = "!VAi
! "iVi= "
eVe
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9MAE 5420 - Compressible Fluid Flow
Momentum for Steady 1-D Flow
• General equation for Momentum greatly simplifies if we assume
inviscid
Flow with no body forces
! fb">
dvC .V .
### " p( )C .S .
## dS">
+ | f">
$ dS">%
&'(
C .S .
## |dS
SC .S .
">
=
!V">
• ds">%
&'(
C .S .
## V">
+))t
!V">%
&'( dv
C .V .
###
!V">
• ds">#
$%&
C .S .
'' V">
= " p( )C .S .
'' dS">
Stephen Whitmore
Stephen Whitmore
Stephen Whitmore
Stephen Whitmore
Stephen Whitmore
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10MAE 5420 - Compressible Fluid Flow
Momentum for Steady 1-D Flow (cont’d)
• General equation for Momentum greatly simplifies if we assume
inviscid
Flow with no body forces
• Evaluating momentum surface integral across the C.S.
V
dS
dS
dS
dS
C.S.
!!
!V">
• ds
">#$
%&'' V
">
= 0
!V">
• ds
">#$
%&'' V
">
= 0
!V">
• ds
">#$
%&'' V
">
= "!iVi
2Ai
!V">
• ds
">#$
%&'' V
">
= !eVe
2Ae
Stephen Whitmore
Stephen Whitmore
Stephen Whitmore
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11MAE 5420 - Compressible Fluid Flow
Momentum for Steady 1-D Flow (concluded)
• Evaluating pressure surface integral across the C.S.Because of
flow “symmetry” upper and lower flow surfaces
pressure forces “cancel out”
V
dS
dS
dS
dS
C.S.
!!
! p( )"" dS!>
= ! peAe
! p( )"" dS!>
= piAi
• Summing up terms and once again enforcing, Ai = Ae
pi+ !
iVi
2= p
e+ !
eVe
2
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Stephen Whitmore
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Stephen Whitmore
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12MAE 5420 - Compressible Fluid Flow
Energy Equation for Steady 1-D Flow
• Assume no body forces, and steady flow
!!t
" e +V2
2
#
$%
&
'( dv
#
$%%
&
'((
C .V .
))) + "V*>
• d S*>
e +V2
2
#
$%
&
'(
C .S .
)) =
(" f*>
dv) •V*>
C .V .
))) * (pd S*>
) •V*>
C .S .
)) + " q•#
$&' dv
#$%
&'(
C .V .
)))
• Let ! q•"
#$% dv
"#&
$%'
C .V .
((( ) Q•
Stephen Whitmore
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13MAE 5420 - Compressible Fluid Flow
Energy Equation for Steady 1-D Flow (cont’d)
• Energy equation reduces to
Q•
! (pd S!>
) •V!>
C .S .
"" = # e +V2
2
$
%&
'
() V
!>
• d S!>
C .S .
""
V
dS
dS
dS
dS
C.S.
!!
• Evaluating
Surface Integrals
(pd S!>
) •V!>
C .S .
"" = 0
(pd S!>
) •V!>
C .S .
"" = 0
! (pd S!>
) •V!>
C .S .
"" = piViAi
! (pd S!>
) •V!>
C .S .
"" = ! peVeAe
Stephen Whitmore
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14MAE 5420 - Compressible Fluid Flow
Energy Equation for Steady 1-D Flow (cont’d)
• Continuing with Surface Integrals
V
dS
dS
dS
dS
C.S.
!!
! e +V2
2
"
#$
%
&' V
(>
• d S
(>
C .S .
)) * 0
! e +V2
2
"
#$
%
&' V
(>
• d S
(>
C .S .
)) * 0
! e +V2
2
"
#$
%
&' V
(>
• d S
(>
C .S .
)) * !i ei +Vi
2
2
"
#$
%
&'Vi • Ai
! e +V2
2
"
#$
%
&' V
(>
• d S
(>
C .S .
)) * !e ee +Ve
2
2
"
#$
%
&'Ve • Ae
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15MAE 5420 - Compressible Fluid Flow
Energy Equation for Steady 1-D Flow (cont’d)
• Collecting Terms and enforcing Ai=Ae=Ac
• Dividing thru by
Q•
Ac+ piVi + !i ei +
Vi2
2
"
#$
%
&'Vi = peVe + !e ee +
Ve2
2
"
#$
%
&'Ve
Q•
!iViAc+pi
!i+ ei +
Vi2
2=pe
!e+ ee +
Ve2
2=
Q•
m•+pi
!i+ ei +
Vi2
2=pe
!e+ ee +
Ve2
2
!iVi
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16MAE 5420 - Compressible Fluid Flow
Energy Equation for Steady 1-D Flow (concluded)
• Recalling from the basic thermodynamics lecture(definition of
Enthalpy)
• The energy equation for 1-D steady, inviscid flow becomes
h = e + pv = e +p
!
q•
+ hi +Vi2
2= he +
Ve2
2
Q•
m•! q
•
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17MAE 5420 - Compressible Fluid Flow
1-D, Steady, Flow: Collected Equations
• Continuity
• Momentum
• Energy
! "iVi= "
eVe
! pi+ "
iVi
2= p
e+ "
eVe
2
! q•
+ hi +Vi2
2= he +
Ve2
2
Stephen Whitmore
Stephen Whitmore
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18MAE 5420 - Compressible Fluid Flow
Mach Number: Revisited
• From the definition of cv
cv= c
p! R
g"
cv= R
g
cp
Rg
!1#
$%
&
'( = Rg
cp
cp! c
v
!1#
$%
&
'( =
Rg
cp! c
p+ c
v
cp! c
v
#
$%
&
'( = Rg
cv
cp! c
v
#
$%
&
'( =
Rg
) !1#$%
&'(
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19MAE 5420 - Compressible Fluid Flow
Mach Number: Revisited (cont’d)
• Recall from the fundamental definition
• But if we take the ratio or kinetic to internal energy
Of a fluid element
M =V
c=
V
! RgT
V2
2
e=
V2
2
cvT
• calorically perfect gas
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20MAE 5420 - Compressible Fluid Flow
Mach Number: Revisited (cont’d)
• Then
i.e. Mach number is a measure of the ratio of the fluid
Kinetic energy to the fluid internal energy (direct motion
To random thermal motion of gas molecules)
V2
2
e=
V2
2
cvT
=
V2
2
Rg
! "1#$%
&'(T
=
!!
V2
2
Rg
! "1#$%
&'(T
=! ! "1( )2
V2
! RgT
=! ! "1( )2
M2
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21MAE 5420 - Compressible Fluid Flow
Mach Number: Revisited (cont’d)
• Look at the Steady Flow Energy equation With no heat
addition
d h +V2
2
!"#
$%&= 0' dh +VdV = 0
hi+Vi
2
2= h
e+Ve
2
2! h +
V2
2= const
• Look at Differential form
Stephen Whitmore
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22MAE 5420 - Compressible Fluid Flow
Mach Number: Revisited (cont’d)
• For a reversible process
• With no heat addition, ds=0
• Subbing into previous
Tds = dh ! vdp
dh = vdp =dp
!
dp = !"VdV (“euler’s equation”)
Stephen Whitmore
Stephen Whitmore
Stephen Whitmore
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23MAE 5420 - Compressible Fluid Flow
Mach Number: Revisited (cont’d)
• Now look at Continuity equation
!iVi= !
eVe" !AV = const
d !AV( )!AV
= 0 =d!
!+dA
A+dV
V
• Substituting in Euler’s Equation dp = !"VdV
d!
!+dA
A+
"dp
!V
V= 0
Stephen Whitmore
Stephen Whitmore
Stephen Whitmore
Stephen Whitmore
Stephen Whitmore
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24MAE 5420 - Compressible Fluid Flow
Mach Number: Revisited (cont’d)
• Solving for dA/A
• But since we are considering a process with ds=0
and …
• Substituting in Euler’s Equation
dA
A=dp
!1
V2"d!!
=dp
!1
V2"d!dp
#$%
&'(
d!
dp=1
c2
dA
A=dp
!1
V2"1
c2
#$%
&'(=1" M 2( )!V 2
dp
dA
A==
1! M 2( )"V 2
!"VdV( ) = M 2 !1( )dV
V
#$%
&'(
dp = !"VdV
Stephen Whitmore
Stephen Whitmore
Stephen Whitmore
Stephen Whitmore
Stephen Whitmore
Stephen Whitmore
Stephen Whitmore
Stephen Whitmore
Stephen Whitmore
Stephen Whitmore
Stephen Whitmore
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Stephen Whitmore
Stephen Whitmore
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25MAE 5420 - Compressible Fluid Flow
Mach Number: Revisited (cont’d)
• Rearranging gives the TWO relationships
Whose ramifications are fundamental to this class
dV
dA
!"#
$%&=
1
M2 '1( )
V
A
dp
dA=
'1M
2 '1( )(V 2
A
Stephen Whitmore
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26MAE 5420 - Compressible Fluid Flow
Mach Number: Revisited (cont’d)
dV
dA
!"#
$%&=
1
M2 '1( )
V
A
dp
dA=
'1M
2 '1( )(V 2
A
M < 1!dV
dA
"#$
%&'< 0!
dp
dA
"#$
%&'> 0
M > 1!dV
dA
"#$
%&'> 0!
dp
dA
"#$
%&'< 0
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27MAE 5420 - Compressible Fluid Flow
Fundamental Properties of Supersonic
and Supersonic Flow
Stephen Whitmore
Stephen WhitmoreSub
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3MAE 5540 - Propulsion Systems
… Hence the shape of the rocket Nozzle
Stephen Whitmore
Stephen Whitmore
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MAE 6350 - Propulsion Systems II!
Why does a rocket !nozzle look like this?"
Stephen WhitmoredA/A > 0
Stephen Whitmore
Stephen Whitmore
Stephen Whitmore
Stephen Whitmore
Stephen Whitmoreexpanding/accelerating flow
Stephen Whitmore
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4MAE 5540 - Propulsion Systems
What is a NOZZLE
• FUNCTION of rocket nozzle is to convert thermal energy
in propellants into kinetic energy as efficiently as
possible
• Nozzle is substantial part of the total engine mass.
• Many of the historical data suggest that 50% of solid rocket
failures
stemmed from nozzle problems.
The design of the nozzle must trade off:
1. Nozzle size (needed to get better performance) against nozzle
weight
penalty.
2. Complexity of the shape for shock-free performance vs. cost
of
fabrication
Stephen Whitmore
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5MAE 5540 - Propulsion Systems
Temperature/Entropy
Diagram for a Typical Nozzle
Tds = !q + dsirrev
cp =dh
dT
!"#
$%&p
q•
+ h1+V1
2
2= h
2+V2
2
2
Isentropic
Nozzle
q•
cp
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28MAE 5420 - Compressible Fluid Flow
De Laval Nozzle (revisited)
AeAI At
pIVIAIr
peVeAer
ptVtAtr
• But De Laval Discovered that when the Nozzle throat
Area was adjusted downward until so that the pressure ratio
pt / pI < 0.5484 -> then the exit Pressure dropped
dramatically
And the exit velocity rose significantly … Which is counter
to
What Bernoulli’s law predicts … he had inadvertently
Generated supersonic flow! …
High Pressure Inlet
Stephen Whitmore
