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1MAE 5420 - Compressible Fluid Flow
Section 6 Lecture 2:Prandtl-Meyer ExpansionWaves
• Anderson, Chapter 4 pp.167-190
2MAE 5420 - Compressible Fluid Flow
What Happens when
•M1 = 3.0, p1=1atm, =1.4, T1=288°K, =0.00001°
• Explicit Solver for
! = M1
2 "1( )2
" 3 1+# "12
M1
2$%&
'()1+
# +1
2M
1
2$%&
'()tan
2 *( ) =8.0
! =
M1
2 "1( )3
" 9 1+# "12
M1
2$%&
'()1+
# "12
M1
2+# +1
4M
1
4$%&
'()tan
2 *( )
+ 3=1.0
!!
!
3MAE 5420 - Compressible Fluid Flow
What Happens when (cont’d)
tan !( ) =
M1
2 "1( ) + 2# cos4$% + cos
"1 &( )3
'()
*+,
3 1+- "12
M1
2./0
123tan 4( )
=19.47° µ =180
!sin
"1 1
M1
#
$%
&
'( = 19.47
o
• “mach line”
•M1 = 3.0, p1=1atm, =1.4, T1=288°K, =0.00001°!!
!
4MAE 5420 - Compressible Fluid Flow
What Happens when (cont’d)
• Compute Normal Component of Free stream mach Number
Mn1= M
1sin! = 1.0000
• p2
p1
= 1+2!
! +1( )Mn
1
2"1( ) = 1.0 (NO COMPRESSION!)
•M1 = 3.0, p1=1atm, =1.4, T1=288°K, =0.00001°!!
5MAE 5420 - Compressible Fluid Flow
Expansion Waves• So if
>0 .. Compression around corner
=0 … no compression across shock
!
"
M1
M2
!
!
6MAE 5420 - Compressible Fluid Flow
Expansion Waves (concluded)
• Then it follows that <0 .. We get an expansion wave (Prandtl-Meyer)
• Flow accelerates around corner
• Continuous flow region …sometimes called “expansion fan”
• Each mach wave is infinitesimally weak
isentropic flow region
• Flow stream lines are curved and smooth through fan
!
7MAE 5420 - Compressible Fluid Flow
Prandtl-Meyer Expansion Fan: MathematicalAnalysis
• Consider flow expansion around an infinitesimal corner
Infinitesimal Expansion Fan Flow Geometry
V
V+dV dθ
µ
Mach Wave
dθµdθ V
• From Law of SinesV
sin!2" µ " d#$
%&'()
=V + dV
sin!2+ µ
$%&
'()
8MAE 5420 - Compressible Fluid Flow
Prandtl-Meyer Expansion Fan: MathematicalAnalysis (cont’d)
• Using the trigonometric identities
• And …
sin!2+ µ
"#$
%&'= sin
!2
"#$
%&'cos µ( ) + sin µ( )cos
!2
"#$
%&'= cos µ( )
sin!2( µ
"#$
%&'= sin
!2
"#$
%&'cos µ( ) ( sin µ( )cos
!2
"#$
%&'= cos µ( )
sin!2" µ " d#$
%&'()= sin
!2" µ
$%&
'()cos d#( ) " cos
!2" µ
$%&
'()sin d#( ) =
cos µ( )cos d#( ) " cos!2
$%&
'()cos µ( ) + sin µ( )sin
!2
$%&
'()
*
+,
-
./sin d#( ) =
cos µ( )cos d#( ) " sin µ( )sin d#( )
9MAE 5420 - Compressible Fluid Flow
Prandtl-Meyer Expansion Fan: MathematicalAnalysis (cont’d)
• Substitution gives
V
cos µ( )cos d!( ) " sin µ( )sin d!( )=V + dV
cos µ( )#
1+dV
V=
cos µ( )cos µ( )cos d!( ) " sin µ( )sin d!( )
• Since dθ is considered to be infinitesimal
cos d!( ) = 1
sin d!( ) = d!
10MAE 5420 - Compressible Fluid Flow
Prandtl-Meyer Expansion Fan: MathematicalAnalysis (cont’d)
• and the equation reduces to
1+dV
V=
cos µ( )cos µ( ) ! sin µ( ) d"( )
=1
1! tan µ( ) d"( )
• Exploiting the form of the power series (expanded about x=0)
1
1! x= 1! x( )
|x=0!
1
1! x( )2
|x=0
(!1)"
#$$
%
&''x ! 0( ) + ....O x
2( )(
11MAE 5420 - Compressible Fluid Flow
Prandtl-Meyer Expansion Fan: MathematicalAnalysis (cont’d)
• Then
• Since dV is infinitesimal … truncate after first order term
1
1!dV
V
= 1+1
1!dV
V
"#$
%&'2
|dV
V=0
dV
V! 0"
#$%&'+O
dV
V
2"
#$%
&'
1
1!dV
V
" 1+dV
V#
1
1!dV
V
"1
1! tan µ( ) d$( )
12MAE 5420 - Compressible Fluid Flow
Prandtl-Meyer Expansion Fan: MathematicalAnalysis (cont’d)
• Solve for d in terms of dV/V
1
1!dV
V
" 1+dV
V#
1
1!dV
V
"1
1! tan µ( ) d$( )
1! tan µ( ) d$( ) = 1!dV
V# d$ =
1
tan µ( )
dV
V
• Since disturbance is infinitesimal (mach wave)
sin µ( ) =1
M
!
13MAE 5420 - Compressible Fluid Flow
Prandtl-Meyer Expansion Fan: MathematicalAnalysis (cont’d)
• Performing some algebraic and trigonometric voodoo
sin µ( ) =1
M! sin
2 µ( ) =1
M2=sin
2 µ( ) + cos2 µ( )M
2!
M2 =
sin2 µ( ) + cos2 µ( )sin
2 µ( )= 1+
1
tan2 µ( )
!1
tan2 µ( )
= M 2"1
1
tan µ( )= M
2"1
• and …. d! =
1
tan µ( )
dV
V= M
2"1
dV
V• Valid forReal and ideal gas
14MAE 5420 - Compressible Fluid Flow
Prandtl-Meyer Expansion Fan: MathematicalAnalysis (cont’d)
• For a finite deflection the O.D.E is integrated over the complete expansion fan
! = M2 "1
dV
VM1
M2
#
• Write in terms of mach by …
V = M ! c" dV = dM ! c + M ! dc"
dV
V=dM ! c + M ! dc
M ! c=dM
M+dc
c
15MAE 5420 - Compressible Fluid Flow
Prandtl-Meyer Expansion Fan: MathematicalAnalysis (cont’d)
• Substituting in
! = M2 "1
dV
VM1
M2
# = M2 "1
dM
M+dc
c
$%&
'()
M1
M2
#
• For a calorically perfect adiabatic gas flow
And T0 is constantc0= ! R
gT0"
c0
c
#$%
&'(=
T0
T
#$%
&'(= 1+
! )12
M2*
+,-./
16MAE 5420 - Compressible Fluid Flow
Prandtl-Meyer Expansion Fan: MathematicalAnalysis (cont’d)
• Solving for c, differentiating, and normalizing by c
dc
c=
!1
2
"#$
%&'
1
1+( !12
M2"
#$%&'
(( !1)M dM( )
1+( !12
M2"
#$%&'
= !
(( !1)2
M dM( )
1+( !12
M2"
#$%&'
17MAE 5420 - Compressible Fluid Flow
Prandtl-Meyer Expansion Fan: MathematicalAnalysis (cont’d)
• Returning to the integral for
! = M2 "1
dV
VM1
M2
# = M2 "1
dM
M+ "
($ "1)2
M dM( )
1+$ "12
M2%
&'()*
%
&
'''
(
)
***M1
M2
#
!
18MAE 5420 - Compressible Fluid Flow
Prandtl-Meyer Expansion Fan: MathematicalAnalysis (cont’d)
• Simplification gives
! = M2 "1
dM
M1+ "
(# "1)2
M2
1+# "12
M2$
%&'()
$
%
&&&
'
(
)))M1
M2
* =
M2 "1
dM
M
1+# "12
M2$
%&'()"(# "1)2
M2
1+# "12
M2$
%&'()
$
%
&&&
'
(
)))=
M1
M2
*M
2 "1dM
M
1+# "12
M2$
%&'()
$
%
&&&
'
(
)))
M1
M2
*
19MAE 5420 - Compressible Fluid Flow
Prandtl-Meyer Expansion Fan: MathematicalAnalysis (cont’d)
• Evaluate integral by performing substitution
M2 !1
dM
M
1+" !12
M2#
$%&'(
) * Ln[M ] + u*dM
M= du,M
2= e
2u,-.
/01
M2 !1
dM
M
1+" !12
M2#
$%&'(
) =e2u !1
1+" !12
e2u#
$%&'(
) du
20MAE 5420 - Compressible Fluid Flow
Prandtl-Meyer Expansion Fan: MathematicalAnalysis (cont’d)
• Standard Integral Table Form
e2u !1
1+" !12
e2u#
$%&'(
) du*emx !1
1+ bemx( )) du
emx !1
1+ bemx( )" du = !
2
mtan
!1emx !1 !
2 b +1( )m
tan!1 b
b +1emx !1( )
b +1 bm
• From tables (CRC math handbook)
21MAE 5420 - Compressible Fluid Flow
Prandtl-Meyer Expansion Fan: MathematicalAnalysis (cont’d)
• Substituting m = 2,b =! "12,e
mx= M
2#$%
&'(
M2 !1
dM
M
1+" !12
M2#
$%&'(
) = !2
2tan
!1M
2 !1 +2
2
" !12
+1
" !12
tan!1
" !12
" !12
+1
M2 !1( )
*
+,,
-,,
.
/,,
0,,
=
" +1
" !1tan
!1 " !1" +1
M2 !1( )
*+,
-,
./,
0,! tan!1
M2 !1
22MAE 5420 - Compressible Fluid Flow
Prandtl-Meyer Expansion Fan: MathematicalAnalysis (cont’d)
• Collected Equations
• Or more simply
! =
" +1
" #1tan
#1 " #1" +1
M2
2 #1( )$%&
'&
()&
*&# tan#1
M2
2 #1+
,--
.
/00#
" +1
" #1tan
#1 " #1" +1
M1
2 #1( )$%&
'&
()&
*&# tan#1
M1
2 #1+
,--
.
/00
! ="(M2) #"(M
1)$"(M ) = % +1
% #1tan
#1 % #1% +1
M2 #1( )
&'(
)(
*+(
,(# tan#1
M2 #1
“Prandtl-Meyer Function”
Implicit function … more Newton!
23MAE 5420 - Compressible Fluid Flow
Prandtl-Meyer Expansion Fan: MathematicalAnalysis (cont’d)
• And we already know the derivative
d
dM!(M )[ ] =
1
M
M2 "1
1+# "12
M2$
%&'()
24MAE 5420 - Compressible Fluid Flow
Newton Solver Algorithm
! ="(M2) #"(M
1)$"(M
2) = ! +"(M
1)
! +"(M1) ="(M
2) ="(M
2 j( ) ) +#"#M
$%&
'()
j( )M
2* M
2 j( )( ) +O(M 2) + ....
• Expand in Taylor’s series
• Truncate after first order terms and solve for M2(j)
M2 j+1( ) =
! +"(M1) #"(M
2 j( ) )$% &'
("(M
)*+
,-.
j( )
+ M2 j( )
• Note: use radians!
25MAE 5420 - Compressible Fluid Flow
M2 versus M1,
M1= 5
M1= 3
M1= 1
!
26MAE 5420 - Compressible Fluid Flow
Pressure and Temperature ChangeAcross Expansion Fan
• Because each mach wave is infinitesimal, expansion is isentropic
- P02 = P01- T02 = T01
p2
p1
=P0
1
p1
!p2
P02
=
1+" #12
M1
2
1+" #12
M2
2
$
%
&&&
'
(
)))
"" #1
T2
T1
=T 0
1
T1
!T2
T 02
=
1+" #12
M1
2
1+" #12
M2
2
$
%
&&&
'
(
)))
27MAE 5420 - Compressible Fluid Flow
Numerical Example
• M1=1.5, p1=81.4 kPa, T1=255.6°K, =1.4, =20°
• Compute M2, p2, P02, T2, T02
M1
M220°
!!
28MAE 5420 - Compressible Fluid Flow
Numerical Example (cont’d)
• M1=1.5
1
1.5! "# $
asin180
%& = 41.81°
!(M1) =
" +1
" #1tan
#1 " #1" +1
M1
2 #1( )$%&
'&
()&
*&# tan#1
M1
2 #1
1.4 1+
1.4 1!" #$ %
0.51.4 1!1.4 1+
1.521!( )" #
$ %0.5
" #& '$ %
atan 1.521!( )0.5
( )atan!
=0.207785 radians
Corner entrance mach angle
Prandtl-Meyer Function
µ1= sin
!1 1
M1
"
#$
%
&' =
29MAE 5420 - Compressible Fluid Flow
Numerical Example (cont’d)
• Compute (M2)
!(M2) = " +!(M
1) =
20!
1800.207785+ =0.5569
• Use Iterative Solver to Compute M2
M2 j+1( ) =
! +"(M1) #"(M
2 j( ) )$% &'
("(M
)*+
,-.
j( )
+ M2 j( ) M2=2.2067
!
30MAE 5420 - Compressible Fluid Flow
Numerical Example (cont’d)
• Pressure• Because each mach wave is infinitesimal, expansion is isentropic
- P02 = P01- T02 = T01
p2= p
1
1+! "12
M1
2
1+! "12
M2
2
#
$
%%%
&
'
(((
!! "1
=
11.4 1!2
1.52
+
11.4 1!2
2.20672
+" #$ %$ %$ %$ %& ' 1.4
1.4 1!" #& '
81.4 =27.655 kPa
31MAE 5420 - Compressible Fluid Flow
Numerical Example (cont’d)
• Temperature• Because each mach wave is infinitesimal, expansion is isentropic
- P02 = P01- T02 = T01
=187.76°K
T2= T
1
1+! "12
M1
2
1+! "12
M2
2
#
$
%%%
&
'
(((
=
11.4 1!2
1.52
+
11.4 1!2
2.20672
+" #$ %$ %$ %$ %& '
255.6
32MAE 5420 - Compressible Fluid Flow
Numerical Example (cont’d)
• Exit Mach Angle
µ2= sin
!1 1
M2
"
#$
%
&' !( =
1
2.2067! "# $
asin180
%& 20' = 6.947°
33MAE 5420 - Compressible Fluid Flow
Numerical Example (concluded)
M1
M220°
M1= 1.5
p1= 81.4kPa
T1= 255.56
oK
µ1= 41.81
o
!
"
##
$
##
%
&
##
'
##
M2= 2.2067
p2= 27.655kPa
T2= 187.76
oK
µ2= 6.947
o
!
"
##
$
##
%
&
##
'
##
34MAE 5420 - Compressible Fluid Flow
Maximum Turning Angle
p2! 0" M
2!#
$(#) = % +1
% &1tan
&1 % &1% +1
#2 &1( )'()
*)
+,)
-)& tan&1 #2 &1 =
% +1
% &1&1
.
/01
2342
5max
=% +1
% &1&1
.
/01
2342&
% +1
% &1tan
&1 % &1% +1
M1
2 &1( )'()
*)
+,)
-)& tan&1
M1
2 &1
p2
p1
=P0
1
p1
!p2
P02
=
1+" #12
M1
2
1+" #12
M2
2
$
%
&&&
'
(
)))
"" #1
35MAE 5420 - Compressible Fluid Flow
Maximum Turning Angle (cont’d)• Plotting as a max function of Mach number
• {T2, p2} = 0
!
36MAE 5420 - Compressible Fluid Flow
Maximum Turning Angle (concluded)
Separated Flow Region
• In Reality Viscous Effects Dominate and Flow Separates around steep corner before pressure expands to vacuum
37MAE 5420 - Compressible Fluid Flow
Section 6: Home Work• M1=4, p1=0.01 atm, T1=217°K, =1.25, 1=15°, 2=15°
• Compute After each corner
- Entry and exit Mach wave angles or shock angles- Mach number, - static & total pressure- temperature
M1
M2M315°
15°
! ! !