the application of multipole expansions to unsteady combustion problems*
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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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
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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
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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
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Modeling Approaches
• Wave Equation with distributed heat source
LCombustion Region Interfaces
'q)1(i'pk'p 22
q1’….. qn’
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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
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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
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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=
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Concentrated Combustion Approximation
• Source of error - in general, information needed to describe combustion process - acoustic coupling is not contained in one lumped quantity, Q’
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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
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Multipole Expansions
• Classical Acoustics - Sound Radiation often described in terms of “fundamental sources”
– “Monopole”
– “Dipole”
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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
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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 + ...
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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
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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
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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
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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
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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=
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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
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Supporting Slides
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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
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Multipole Expansions
Lumped Heat Source Model (i.e., concentrated
combustion)
Example of two parameter Heat Source Model
Q’(t)
Qa’(t)
Qb’(t)
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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
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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
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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