mahmood silieti eduardo divo alain kassab
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IPE 2003 Tuscaloosa, Alabama 1
An Inverse BEM/GA Approach to Determining Heat Transfer
Coefficient Distributions Within Film Cooling Holes/Slots
Mahmood SilietiEduardo DivoAlain Kassab
Mechanical, Materials, and Aerospace Engineering Department
University of Central Florida, Orlando, FL, USA
IPE 2003 Tuscaloosa, Alabama 2
Overview:
• Motivation
• Procedure
• Problem Setup
• Conjugate Heat Transfer Solution
• Direct BEM Conduction Solution/Verification
• Inverse Problem and Objective Function
• Optimization Technique: Genetic Algorithms
• Numerical Results
• Conclusions and Extensions
IPE 2003 Tuscaloosa, Alabama 3
Find end Wall Film Cooling Effectiveness
Motivation:
rcr
rAW
TT
TT
,
crAWref hTTT
and heat transfer coefficients (HTC)
Can measure film effectiveness usingoptical thermography:
which also provides
To define endwall HTC.
IPE 2003 Tuscaloosa, Alabama 4
Closed loop Transonic Test Rig at UCF funded by SWPC
s
kgm
Ma
air 5.7
8.0
IPE 2003 Tuscaloosa, Alabama 5
Objective of this feasibility study is to find a means of determining HTC correlation in film hole to be used in later 3D inverse problem analysis for endwall HTC
To find endwall HTC: will solve 3D inverse conduction problem using endwall temperature measurements, however, HTC in film hole is unknown?
,....)Pr,(Re, ninclinatioc fh
Each type of film cooling hole is subject of a single hole calibrationExperiment that will yield to be used in correlation)Re(Re P
There are 10 types of film cooling holes in this experiment, several of these are shaped and all are inclined.
IPE 2003 Tuscaloosa, Alabama 6
Procedure:• Conjugate Heat Transfer (CHT) simulation of 2-D film cooling
slot of end wall
• T measured using temperature sensitive paint (TPS)
• q measured using an optical thermographic technique under development at UCF
qT , T
h (or q) = ?
• Results from CHT simulation used to model
experimentally measured surface heat flux and temperature.
• Inverse Problem:Input: T and q at exposed endwall surfacesOutput: h (or q) at slot surface using the
boundary element method (BEM) and a genetic algorithm (GA)
Measured T & q
Measured T & q
IPE 2003 Tuscaloosa, Alabama 7
Setup for CHT Simulation: 2-D Film Cooling Slot
Cooling Slot
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Mesh has been created using Gambit (FLUENT grid generator)
Fluid Grid Nodes=41,112Solid Grid Nodes=2,104
IPE 2003 Tuscaloosa, Alabama 9
Main Air Flow: Turbulent Boundary Layer profile (1/7)th. Temperature= 350 K
Coolant Air Flow:
Uniform Pressure =105800 Pa
Temperature= 300 K
Fluid is Air: compressible, other properties are function of temperature
Solid is Steel: properties are linear function of temperature
CHT Simulation conditions chosen to match experiment to be carried out in wind-tunnel
IPE 2003 Tuscaloosa, Alabama 10
CHT Solver:
Commercial Code “Fluent” Finite Volume
Full Navier-Stokes Equation for compressible turbulent flow
“RNG “k
CHT Results:
Results are converged at least for all residuals
( mass, momentum, energy, & )
510
k
IPE 2003 Tuscaloosa, Alabama 11
IPE 2003 Tuscaloosa, Alabama 12
IPE 2003 Tuscaloosa, Alabama 13
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Direct BEM Conduction Solution:
measured T
Numerical consistency check of BEM (in-house) and CHT (commercial) code.
BEM surface mesh and CHT surface mesh are different radial basis function (RBF) interpolation used to pass information from one grid to the other.
Input CHT wall temperatures at solid surfaces to BEM and check BEM computed heat fluxes.
IPE 2003 Tuscaloosa, Alabama 17
++++++++++++++++++++++++++
++++++++++++++++++++++++++++++++++
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
BEM Methodology
Surface Mesh only: (we use quadratic discontinuous elements)
Governing Equation: (Boundary Integral Equation for Laplace Eqn.)
Where: G(x,) = (-1/ 2k) ln r(x,) in 2D
H(x,) = -kG(x,) / n
q(x) = -kT(x) / n
C() = 1 if
C() = 1/2 if
IPE 2003 Tuscaloosa, Alabama 18
T: 297 299 301 303 305 307 309 311 313 315 317 319 321 323 325 327 329 331 333 335 337 339 341 343
Discretized BIE is collocated at the boundary points, leading to
Introducing Boundary Conditions:
BEM Methodology
Contour plot of direct BEM temperature distribution:
IPE 2003 Tuscaloosa, Alabama 19
Direct BEM Results: BEM fluxes consistent with FLUENT fluxes
element20 40 60
-15000
-10000
-5000
0
5000
10000
15000
Qbem
Qcfd
element20 40 60 80 100 120 140
-2000
-1500
-1000
-500
0
500
1000
1500
2000
Qbem
Qcfd
21
30
1
55
9170
61
135
1
IPE 2003 Tuscaloosa, Alabama 20
Direct BEM Results: Heat Fluxes @ Cooling Slot
element21 22 23 24 25 26 27 28 29 30
9000
10000
11000
12000
13000
14000
15000
Qbem
Qcfd
element61 62 63 64 65 66 67 68 69 70
0
500
1000
1500
Qbem
Qcfd
21
30 61
70
IPE 2003 Tuscaloosa, Alabama 21
Inverse Problem:
Cauchy conditions (T and q) imposed at the surfaces exposed to hot and cold gases.
Both temperature and flux are unknown on the surfaces of the cooling slot.
h (or q) = ?
qT ,T
IPE 2003 Tuscaloosa, Alabama 22
Identification of heat fluxes in the cooling slot to match over-specified boundary data at the exposed surfaces.
jjj rrrrf
),(
Inverse Problem:
Parametric representation of heat flux in cooling slot using radial basis functions (RBF)
Objective function is to minimize
AN
jjjjABEM rrfqq
1
),()(
m
CFDi
BEM
N
ii
mA qq
NqS
1
2)ˆ(1
)(
Anchor point
BEM node
IPE 2003 Tuscaloosa, Alabama 23
Optimization Technique: Genetic Algorithms
• Non-gradient-based global search technique based on Darwinian evolution and operated by rules of natural selection:
“Survival of the fittest”
• Represent the design variables by a string of binary bits.
• Generate a population of individuals genetically characterized by one chromosome or binary string.
• Evaluate the fitness of each individual to identify its likelihood of propagating its genetic material.
• Select and reproduce pairs of individuals to generate new generation subject to a probability of mutation.
/S15)q4,q3,q2,q1,Z(q
1q
2q
3q
4q
5q
genes
Chromosome
10110110
01011011
10100111
10001101
01101110
IPE 2003 Tuscaloosa, Alabama 24
Optimization Technique: Genetic Algorithms
Advantages: - Very robust
- Almost guaranteed global optimal
- Inherent regularization
Disadvantage: - Very slow
Solution: - Parallelize process in a Computer Cluster by assigning different individuals to different nodes in the cluster. (Very efficient parallelization as very little communication is necessary)
IPE 2003 Tuscaloosa, Alabama 25
Parameters: - population size = 50
- probability of jump mutation = 4%
- probability of creep mutation = 20%
- number of bits per parameter = 8 (255 steps)
- number of children = 1
- ellitistic generation = 1
- parameter bound = searches for q between qmin and qmax
block#1 (-15,000 to 15,000)
block#2 (-2,000 to 2,000)
Optimization Technique: Parallel Genetic Algorithms
IPE 2003 Tuscaloosa, Alabama 26
• Inverse BEM Results: Evolution of objective function for Heat fluxes
(T, q) =?
qT ,
qT ,
(T, q) =?
qT ,
qT ,
T
Generation
Ob
ject
ive
Fu
nct
ion
(S)
20 40 60 80 100100
150
200
250
300
Generation
Ob
ject
ive
Fu
nct
ion
(S)
20 40 60 80 10020
30
40
50
60
70
80
90
100
IPE 2003 Tuscaloosa, Alabama 27
Inverse BEM Results: Temperature Distribution @ Cooling Slot
21
30 61
70
element21 22 23 24 25 26 27 28 29 30
315
316
317
318
Tbem
Tcfd
element61 62 63 64 65 66 67 68 69 70
297
297.25
297.5
297.75
298
Tbem
Tcfd
IPE 2003 Tuscaloosa, Alabama 28
Inverse BEM Results: Heat Fluxes
21
30
1
55
9170
61
135
141
Element20 40 60
-15000
-10000
-5000
0
5000
10000
15000
QGA
QCFD
Element20 40 60 80 100 120 140
-2000
-1500
-1000
-500
0
500
1000
1500
2000
QGA
QCFD
IPE 2003 Tuscaloosa, Alabama 29
Inverse BEM Results: Heat Fluxes @ Cooling Slots
21
30 61
70
+ +
+
+
+
Element21 22 23 24 25 26 27 28 29 30
8000
9000
10000
11000
12000
13000
14000
15000
QGA
QCFD
QAP+
+
+
+
+
+
Element61 62 63 64 65 66 67 68 69 70
0
500
1000
1500
QGA
QCFD
QAP+
IPE 2003 Tuscaloosa, Alabama 30
Conclusions and Extensions:
• Methodology shows promise in predicting the temperature and the heat fluxes within the slot.
• Add more anchor points to capture the changes in heat fluxes. Add a regularization term to reduce unwanted oscillations associated with more anchor points.
• Need to study the effect of input error in temperature and heat flux on resolution.
• Apply the methodology to multiple slots.
• Apply the methodology to 3-d single and multiple film-cooling holes.
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