analytical models for -...
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Agenda
The University of TennesseeSpace Institute
• Part of the UT system
• Graduate institute
• Space related research
www.utsi.edu
The Advanced Theoretical Research Center
Research Topics
Propulsion,Internal Flow
ModelsInviscid
Viscous
Unsteady flow,
Instability
Particle-Mean Flow Interaction
Asymptotic Methods
Compressibility Effects
Comparison, Initialization,
validation
Flowrate, Dimensionless
numbers, Aspect ratio… etc…
Analytical Models
Control parameters
Physical ProcessesBenchmark
Stability, Vorticity, Lift, Drag,
Steepening, Compressibility
Propulsion Devices
Solid
• Fuel & oxidizer are premixed
• Simple and compact• Requires little servicing• Cannot be throttled
Liquid
• Fuel & oxidizer are stored in separate tanks
• Requires a feed mechanism
• Can be throttled• More control over
combustion process
The Space Shuttle
• The space shuttle uses solid & liquid propulsion
Hybrid
• Combination of solids & liquids
• Can be throttled
• Compact• Low
efficiency
The Taylor-Culick Problem
0
1
z
r
A B
212sin( )z rψ π=
212
212
1 sin( )
cos( )
r
z
u rr
u z r
π
π π
= − =
1. Culick, F. E. C. 1966 Rotational axisymmetric mean flowand damping of acoustic waves in a solid propellantrocket. AIAA Jl 4, 1462-1464.
2. Taylor, G. I., “Fluid Flow in Regions Bounded by PorousSurfaces,” Proceedings of the Royal Society, London,Series A, Vol. 234, No. 1199, 1956, pp. 456-475.
Arbitrary Injection
2 212
0
const; uniform
cos( / ); Berman (half cosine)( ) ( ,0)
[1 ( / ) ]; laminar ( 2) and turbulent
(1 / ) ; turbulent ( 1/ 7)
c
cz m
cm
c
U
U r au r u r
U r a m
U r a m
π
== =
− = − =
Assumptions
• Steady• Inviscid• Rotational• Axisymmetric
1. Majdalani and Saad, “The Taylor-Culick Profile with Arbitrary Headwall Injection,” Physics of Fluids, Vol. 19, No. 6, 2007.
2. Saad and Majdalani, “The Taylor Profile in Porous Channels with Arbitrary Headwall Injection,” AIAA Paper 2007-4120, June 2007.
3. Maicke and Majdalani, “The Compressible Taylor Flow in Slab Rocket Motors,” AIAA Paper 2006-4957, July 2005/Now in Journal of Fluid Mechanics
4. Majdalani, “The Compressible Taylor-Culick Flow,” AIAA Paper 2005-3542, July 2005/Now in Proc. Royal Soc.
5. Majdalani and Vyas, “Inviscid Models of the Classic Hybrid Rocket,” AIAA Paper 2004-3474, July 2004. Now in AIAA Progress Series.
6. Zhou and Majdalani, “Improved Mean Flow Solution for Slab Rocket Motors with Regressing Walls,”Journal of Propulsion and Power, Vol. 18, No. 3, 2002, pp. 703-711.
7. Zhou and Majdalani, “Improved Mean Flow Solution for Slab Rocket Motors with Regressing Walls,” AIAA Paper 2000-3191, July 2000.
8. Majdalani and Zhou, “Moderate-to-Large Injection and Suction Driven Channel Flows with Expanding or Contracting Walls,” Journal of Applied Mathematics and Mechanics, Vol. 83, No. 3, 2003, pp. 181-196.
9. Majdalani, Zhou and Dawson, “Two-Dimensional Viscous Flow between Slowly Expanding or Contracting Walls with Weak Permeability,” Journal of Biomechanics, 2002.
10. Sams, Majdalani and Saad, “Higher Flowfield Approximations for Solid Rocket Motors with Tapered Bores,” AIAA Paper 2004-4051, July 2004. Now in Journal of Propulsion and Power.
11. Saad, Sams and Majdalani, “Analytical and CFD Approximations for Tapered Slab Rocket Motors,” AIAAPaper 2004-4060, July 2004. Now in Physics of Fluids.
Related Rocket Core Flow Models
Normalization
2 2; ; ; ;w w
z r pz r a pa a U a U
ψψρ
= = ∇ = ∇ = =
0; ; ; r zr z c
w w w w
Uu u au u uU U U U
= = = =ΩΩ
Principal Equations
0 vorticity transport equation∇× × =uΩ
vorticity equation= ∇×Ω u
Boundary Conditions
0
( ,0) 0 (no flow across centerline)(0, ) ( ) (headwall injection profile)( ,1) 1 (constant sidewall mass addition)( ,1) 0 (no slip)
r
z
r
z
u zu r u ru zu z
= = = − =
Key Relations
20 ( )rF C rψ ψ∇× × = → Ω = =uΩ
2 2
2 21 0rr rr z
ψ ψ ψ∂ ∂ ∂= ∇× → − + + Ω =
∂∂ ∂Ω u
1ru
r zψ∂
= −∂
1zu
r rψ∂
=∂
2 22 2
2 21 0C rr rr z
ψ ψ ψ ψ∂ ∂ ∂− + + =
∂∂ ∂
Vorticity-Stream Function Eqn.2 2
2 22 2
1 0C rr rz r
ψ ψ ψ ψ∂ ∂ ∂+ − + =
∂∂ ∂
0
1
0
1 ( ,0)1 ( , ) (c) ( )(a) lim 0
1 ( , )(1, ) (d) 1(b) 0r
r
rr z u rr rr z
r zzr zr
ψψ
ψψ=
→
∂∂ == ∂ ∂ ∂∂ == ∂∂
2 21 12 2( , ) ( ) cos( ) sin( )r z z A Cr B Crψ α β = + +
Eigenfunction Expansion
212
0( , ) ( )sin[ (2 1) ]n n
nr z z n rψ α β π
∞
=
= + +∑
( )1 210 20
4 ( ) cos[ ] d(2 1)n u r n r r r
nβ π
π= +
+ ∫
( )0
1 1nn
nα
∞
=
− =∑
Classic Solution
( ) 0
0
11 1
0 0n
nn n n
αα
α
∞
=
=− = ⇔ = ∀ >
∑
Majdalani and Saad, “The Taylor-Culick Profile with Arbitrary Headwall Injection,”Physics of Fluids, Vol. 19, No. 6, 2007.
1 210 20
4 ( )cos[( ) ] d
(2 1)n
u r n r r r
n
πβ
π
+=
+∫
( ) 212
0( , ) sin[( ) ]n n
nr z z n rψ α β π
∞
=
= + +∑
0 10 0n n
αα
= = ∀ >
Majdalani and Saad, “The Taylor-Culick Profile with Arbitrary Headwall Injection,”Physics of Fluids, Vol. 19, No. 6, 2007.
1 210 20
4 ( )cos[( ) ] d
(2 1)n
u r n r r r
n
πβ
π
+=
+∫
( ) 212
0( , ) sin[( ) ]n n
nr z z n rψ α β π
∞
=
= + +∑
0 10 0n n
αα
= = ∀ >
Energy-Based Solutions
• The choice of the sidewall injection sequence is arbitrary
• Optimize kinetic energy to derive possible forms
( )0
1 1nn
nα
∞
=
− =∑
Cumulative Kinetic Energy
2 21, , ,2
0 0( , ) ( )L L n r n z n
n nE r z E u u
∞ ∞
= =
= = +∑ ∑
( )
1,
,
sin
(2 1)cosr n nr
z n n n
u
u z n
α η
π α β η
= −
= + +
212 (2 1)n rη π≡ +
( )22 2 2 2 2 212
0sin (2 1) cosL n n n
nE r z nα η π α β η
∞−
=
= + + + ∑
2 21, , ,2( , ) ( )L n r n z nE r z u u= +
Total Kinetic Energy
( )
2 1
0 0 0
1 22 2 2 2 2 20 0
0
d d d
sin (2 1) cos d d
LV L
Ln n n
n
E E r r z
r z n r r z
πθ
π α η π α β η∞
−
=
=
= + + +
∫ ∫ ∫
∑∫ ∫
( )3 3 2 1 2 2 2 2112
0V n n n n n n n
nE L a L b L c L dπ α α α π
∞− − − −
=
= + + +∑
2 2(2 1) ; 3 ; 33Cin[(2 1) ]
n n n n n n n
n
a n b a c ad n
β βπ
= + = =
= + ( ) 10
Cin( ) 1 cos dx
x t t t−= −∫
Optimization
0( 1) 1n
V nn
g E λ α∞
=
= + − −
∑
( , ) 0 0,1,2...,ng nα λ∇ = = ∞
0( 1) 1 0n
nn
g αλ
∞
=
∂= − − =
∂ ∑3 3 3 2 2 2 1
0
2 2 1
0
2 ( 1) ( )
12 ( )
ii i i
i
i ii
L L b a L d
a L d
π π πλ
π
∞− − −
=∞
− − −
=
+ − += −
+
∑
∑
( ) 3 3 1 2 2112 2 2 ( 1) 0; 0,1,2...,n
n n n n nn
g L a L b L d nπ α α π λα
− − −∂= + + + − = = ∞
∂
3 212
3 3 2 2
6( 1)( )
nn
nn n
b LL a L d
λ πα
π π − −
− += −
+
1 22 2
3
( 1)( )
nn
nn n
S b Sa L d S
απ − −
− −=
+
1 2 2 11
0
1 2 2 12
0
3 2
2 ( 1) ( )
( )
2
ii i i
i
i ii
S L b a L d
S L a L d
S LS
π
π
∞− − − −
=
∞− − − −
=
= + − +
= +
=
∑
∑
Kinetic Energy Density
0 5 10 152
3
4
5
6
7
2 / 3π∞ =E
uc = 1 uniform Berman Poiseuille inert headwall
L
E
3VE
L=E( )3 3 2 1 2 2 2 21
120
V n n n n n n nn
E L a L b L c L dπ α α α π∞
− − − −
=
= + + +∑
Simplification
• Find a way to get rid of L-1
1 2 2 1 1 2 2 1
0 0
2 2 2 2 1
0
( 1) 2 ( 1) ( ) ( )
2( ) ( )
n ii i i n i i
i in
n n i ii
L b a L d b L a L d
a L d a L d
π πα
π π
∞ ∞− − − − − − − −
= =∞
− − − − −
=
− + − + − +
=+ +
∑ ∑
∑
Critical Length
• Define a critical aspect ratio
0.075cr ∞ ∞− ≤E E EcrL
0 5 10 152
3
4
5
6
7
2 / 3π∞ =E
uc = 1
uniform Berman Poiseuille inert headwall
L
E
Critical Length
• For solids (equal burning rates)
20.9620.9921
uniformBermanPoiseuille
crL=
Large L Approximation
• For lengths greater than the critical length
2 28( 1)(2 1)
n
n nα
π−
=+
1 2 2 1 1 2 2 1
0 0
2 2 2 2 1
0
( 1) 2 ( 1) ( ) ( )
2( ) ( )
n ii i i n i i
i in
n n i ii
L b a L d b L a L d
a L d a L d
π πα
π π
∞ ∞− − − − − − − −
= =∞
− − − − −
=
− + − + − +
=+ +
∑ ∑
∑
Streamlines – Least KE
0.0
0.2
0.4
0.6
0.8
1.0
y
0.0
0.2
0.4
0.6
0.8
1.0
y
Generalization
Type I Solutions with KE<KETC
( )( )
22 2 2
18 ( 1)(2 1) 2 1
nn
nA
n nα
π−−
=+ +
( ) ( )( )
1; 2
2 1
nq
n q
Aq q
nα − −
= ≥+
( )( )0
1( 1) 1
2 1
nqnq
n
A
n
∞
=
−− =
+∑
( )0
1 1( )(1 2 )2 1
q qq
n
Aqn ζ∞ −
−
=
= =−+∑
( ) ( )
( ) ( )
( )( )
0
1 1; 2
( )(1 2 ) 2 12 1 2 1
n n
n qqq q
k
q qq nn k
αζ
−∞ −−
=
− −= = ≥
− ++ +∑
( )1; 0
lim0; elsewherenq
nqα −
→∞
==
1( ) q
kq kζ ∞ −
== ∑
2 4 6 8 10 122.0
2.2
2.4
2.6
2.8
a)
uc= 1L = 10
E
2 4 6 8 10 122.0
2.2
2.4
2.6
2.8
2
3VE
L
q b)
L = 20 uniform Berman Poiseuille inert headwall
Type II Solutions with KE>KETC
( )( )
; 22 1
qn q
Bq q
nα + = ≥
+( )0
( 1) 12 1
qnq
n
B
n
∞
=
− =+
∑
( ) ( )31
4 4
0
1 4( , ) ( , )1 2 1
q
qn q
n
Bq qn ζ ζ∞
−
=
= =−− +∑
( ) ( )
( ) ( )
( )31
4 4
0
2 1 4 2 1( , ) ( , )1 2 1
q qq
nk q
k
n nq
q qkα
ζ ζ
− −+
∞−
=
+ += =
−− +∑
( )1; 0
lim0; elsewherenq
nqα +
→∞
==
( )
0( , ) q
kq kζ α α
∞−
=
= +∑
2 4 6 8 10 122.4
2.8
3.2
3.6
4.0
4.4 uc= 1
L = 20 uniform Berman Poiseuille inert headwall
qb)
2 4 6 8 10 122.4
2.8
3.2
3.6
4.0
4.4 uc= 1
L = 10
a)
Energy Bands
1 2 4 6 8 10 20 40
4
8
121620
(e)
uc
L = 10Poiseuille
1 2 4 6 8 10 20 402
4
6
81012
(f) uc
Type I Taylor-Culick
+ + + Type II
L = 20Poiseuille
Streamlines – Most KE(Type II, q = 2)
0.0
0.2
0.4
0.6
0.8
1.0
y
0.0
0.2
0.4
0.6
0.8
1.0
y
Headwall injection Stream function Axial velocity
0 ( )u r ( , )r zψ − ( , )zu r z−
0 2 20
8 ( 1) sin(2 1)
n
nz
nη
π
∞
=
−+∑
0
8 ( 1) cos(2 1)
n
nz
nη
π
∞
=
−+∑
cu ( )2 20
4 ( 1)2 sin(2 1)
n
cn
z un
ηπ
∞
=
−+
+∑ ( )0
4 ( 1)2 cos(2 1)
n
cn
z un
ηπ
∞
=
−+
+∑
( )212coscu rπ ( )21
2 2 20
8 ( 1)sin sin(2 1)
nc
n
ur z
nπ η
π π
∞
=
−+
+∑ ( )212
0
8 ( 1)cos cos(2 1)
n
cn
u r zn
π ηπ
∞
=
−+
+∑
( )21cu r− ( ) ( )2 2
0
8 1 ( 1) sin2 12 1
n c
n
uz
nnη
ππ
∞
=
− +
+ + ∑
( ) ( )0
8 1 ( 1) cos2 1 2 1
n c
n
uz
n nη
π π
∞
=
− +
+ + ∑
Headwall injection Stream function Axial velocity
0 ( )u r ( , )r zψ + ( , )zu r z+
0 20
sin(2 1)n
zn
η∞
= +∑C
0
cos(2 1)n
zn
ηπ∞
= +∑C
cu ( )2 20
4 sin1(2 1)
nc
n
z un
ηπ
∞
=
+ − + ∑ C
( )20
4 cos1(2 1)
nc
n
z un
ηππ
∞
=
+ − + ∑ C
( )212coscu rπ ( )21
2 20
sinsin(2 1)
c
n
u zrn
ηππ
∞
=
++∑C
( )212
0
coscos(2 1)c
n
zu rn
ηπ π∞
=
++∑C
( )21cu r− ( ) ( )3 2
0
8 sin2 1 2 1
c
n
uzn n
ηπ
∞
=
+
+ + ∑ C
( ) ( )3
0
8 cos2 12 1
c
n
uznn
ηππ
∞
=
+
++ ∑ C
Featured Solutions
0 1
-1
0
r
Type II
Type Iq = 2
43
(b)
23 4
Taylor-Culick
ur
0 10
5
Taylor-C
ulick
44
3
3
least kinetic energy stateType I
Type II
zuz
most kinetic energy state
(c)
q = 2
2
r
0
30
60
90
3 2q = 2 3
Taylor-Culick
θ
θ °
Type II
(a)
Type I
ru
Velocities
Velocities
Type I Type II
Asymptotic Limits
( ) ( ) ( )2
2 22 2
0 0
4 4 (2 2)2 1 2 1(2 1) [ ( )]
qq q
qk n
qq k nq
ζζ
−∞ ∞− −− ∞ ∞
∞ ∞ ∞= =
− −= + + = −
∑ ∑E E E
( ) ( ) ( ) ( )2
2 2231
0 0 4 4
4 (4 4) (2 2)1 2 1 2 1[ ( , ) ( , )]
q qk q q
k n
qq k nq q
ζζ ζ
−∞ ∞− −+ ∞ ∞
∞ ∞ ∞= =
− −= − + + = −
∑ ∑E E E
3 /12 2.5838π∞∞ ≡ E
Energy Bracketing
2 3 4 5 6
4
Taylor-Culick
18.9 %
47 %
∞E
Type I Type II
q
Final Remarks1. The approximate solutions are quasi-viscous2. Two families of solutions
1. Type I with increasing energy levels2. Type II with decreasing energy levels3. The original Taylor-Culick solution is a special case
3. Max energy is 47 % higher than Taylor-Culick’swhile the minimum energy is 19% lower : 66% energy band
4. Our Lcr is 21: uniform/Berman/Poiseuille5. The Energy solutions provide an avenue for
constructing a two way coupling framework for stability analysis
Thank You