mahmood silieti eduardo divo alain kassab

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


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


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.


Closed loop Transonic Test Rig at UCF funded by SWPC

s

kgm

Ma

air 5.7

8.0


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.


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


Setup for CHT Simulation: 2-D Film Cooling Slot

Cooling Slot


Mesh has been created using Gambit (FLUENT grid generator)

Fluid Grid Nodes=41,112Solid Grid Nodes=2,104


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


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


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.


++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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


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:


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


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


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


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


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


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)


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


• 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


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


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


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+


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.

mahmood silieti eduardo divo alain kassab

Documents

film hole

d film cooling slotipe

type of film cooling

endwall htc

bem surface mesh

cht surface mesh

d inverse conduction

types of film cooling