ANALYTICAL MODELS

FOR

PROPULSION DEVICES

http://tsaad.utsi.edu - tsaad@utsi.edu

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