The Application of Multipole Expansions to Unsteady Combustion Problems*

T. Lieuwen and B.T. Zinn

Schools of Aerospace and Mechanical Engineering

Georgia Institute of Technology

Atlanta, GA

*Research Supported by AGTSR and AFOSR; Dr. Dan Fant and Dr. Mitat Birkan, Contract Monitors

Background

• Behavior of unsteady combustion systems controlled by complex interactions that occur between the combustion process and acoustic waves– Combustion instabilities

– Pulse Combustors

– Combustion Noise

• Predicting or controlling the behavior of these systems requires capabilities for understanding and modeling these interactions

Approximate Models of Unsteady Combustion Systems

• Approximate techniques to analyze acoustics of combustion chambers are well developed– e.g., Galerkin based techniques

– Unified approaches allow consistent treatment of nonlinearities, mean flow effects, etc.

• Approximate techniques to analyze combustion process are not well developed– primarily ad-hoc approaches

Modeling Approaches

• Wave Equation with distributed heat source

LCombustion Region Interfaces

'q)1(i'pk'p 22

q1’….. qn’

Modeling Approaches

• “Concentrated Combustion” Approximation

Plane Acoustic Disturbances

p’2(t)-p’1(t)=0

v’2(t)-v’1(t)=Q’(t)

V

dV)t,x('q)t('Q

1 2

Modeling Approaches

• “Concentrated Combustion” Approximation– Peracchio, A.A., Proscia, W.M., ASME paper # 98-GT-269

(1998).

– Lieuwen, T., Zinn B.T., AIAA Paper # 98-0641 (1998).

– Dowling, A.P., J. Fluid Mech, 346:271-290 (1997).

– Fleifil, M. et al., Comb. and Flame, 106:487:510 (1996)

– Culick, F., Burnley, V., Swenson, G., J. Prop. Power, (1995)

– Many others

• Combustion Process treated as a single lumped parameter

How accurate is the Concentrated Combustion Approximation?

0

50

100

150

200

250

300

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

L/

Err

or

(%

)

L

“Exact” Rate of Energy addition

Approximate Rate of Energy addition

“Exact” Rate of Energy addition

Error=

Concentrated Combustion Approximation

• Source of error - in general, information needed to describe combustion process - acoustic coupling is not contained in one lumped quantity, Q’

Can Approach be Generalized to Reduce the Error?

Distributed Heat Release

(i.e., infinite parameter model)

Single Lumped Parameter Model

Multiple Lumped Parameter Model

Increasing Model Complexity

Multipole Expansions

• Classical Acoustics - Sound Radiation often described in terms of “fundamental sources”

– “Monopole”

– “Dipole”

Multipole Expansions

• Farfield radiation from an arbitrary compact body can be decomposed into the radiation from these fundamental sources

Can decompose sound field of this body:

p’(farfield) = monopole component + (L/)* dipole component + higher order poles

L

Multipole Expansions

• Application to Ducted Problems:– Develop expression of form:

p’2-p’1=(L/)Q’1+(L/ )2Q’2 +...

v’2-v’1=Q’+(L/ )Q’1+(L/ ) 2Q’2 + ...

Multipole Expansions

• Result to O(L/):

p’2-p’1=(L/)Q’1

v’2-v’1=Q’

• Can generalize to arbitrary order in L/

V

1 dV)L

x('q'Q

Example

)(1

' ikxn

ikxn

tin eBeAe

cv

)(' ikxn

ikxn

tin eBeAep

• Equations describing plane waves:

• Boundary Conditions:

– x=0: p’=poe-it

– x=Lcomb: p’ =0

• Heat Release distribution: q’=sin(x/L)e-it

Plane Acoustic Disturbances

LCombustion Region Interfaces

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 0.2 0.4 0.6 0.8 1 1.2

x/Lcomb

|p'|

ExactValueO(1)

O(KL)

O(KL^2)

Combustion Region

Result - L/=0.02

O(L/)

O(L/)2

Result - L/=0.08

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1 1.2

x/Lcomb

|p'|

Exact Value

O(1)

O(KL)

O(KL^2)

Combustion Region

O(L/)

O(L/)2

Predicted Combustion Driving

0

50

100

150

200

250

300

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

L/

Err

or

(%

)

O(1)

O(kL)

O(kL2)O(L/)

O(L/)2

O(1)

“Exact” Rate of Energy addition

Approximate Rate of Energy addition

“Exact” Rate of Energy addition

Error=

Summary and Conclusions

• Developed method that improves current capabilities for modeling combustion - acoustic interactions

• Method generalizes “Concentrated Combustion” approximation– Describe combustion process as a series of lumped

elements

Supporting Slides

Multipole Expansions

• where:

dS)x('vi4

S

)L

(OR

eD)

L(

R

eS)x('p

S

sn

2

ikRikR

Monopole term Dipole term Higher order poles

Multipole Expansions

Lumped Heat Source Model (i.e., concentrated

combustion)

Example of two parameter Heat Source Model

Q’(t)

Qa’(t)

Qb’(t)

Illustration of Method

• Energy Equation:

• Integrate Energy Equation over combustion region volume:

• Obtain: v’2-v’1=Q’+O(L/ )

VS

iidVp

p

iQdS ''

nv'

'q)1('vpt

'p

Illustration of Method

• Evaluating Volumetric Pressure term: – Take moment of momentum equation, integrate over combustor

volume, and integrate by parts

• Obtain more accurate energy balance:

• v’2-v’1- (L/)(p’2-p’1) =Q’+O(L/ )2

S

ii

V

ii

V

dSpdVxiN

dVp )''(1

' nxv

Illustration of Method

• In same way, improved momentum balance:

• p’2-p’1- (L/)(v’2-v’1) = (L/)Q’1+O(L/ )2

• where

V

1 dV)L

x('q'Q