IPE 2003 Tuscaloosa, Alabama1 An Inverse BEM/GA Approach to Determining Heat Transfer Coefficient Distributions Within Film Cooling Holes/Slots Mahmood Silieti Eduardo Divo Alain Kassab Mechanical, Materials, and Aerospace Engineering Department University of Central Florida, Orlando, FL, USA

IPE 2003 Tuscaloosa, Alabama2 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, Alabama3 Find end Wall Film Cooling Effectiveness Motivation: and heat transfer coefficients (HTC) Can measure film effectiveness using optical thermography: which also provides To define endwall HTC.

IPE 2003 Tuscaloosa, Alabama4 Closed loop Transonic Test Rig at UCF funded by SWPC

IPE 2003 Tuscaloosa, Alabama5 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? Each type of film cooling hole is subject of a single hole calibration Experiment that will yield to be used in correlation There are 10 types of film cooling holes in this experiment, several of these are shaped and all are inclined.

IPE 2003 Tuscaloosa, Alabama6 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 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 surfaces Output: h (or q) at slot surface using the boundary element method (BEM) and a genetic algorithm (GA) Measured T & q

IPE 2003 Tuscaloosa, Alabama7 Setup for CHT Simulation: 2-D Film Cooling Slot Cooling Slot

IPE 2003 Tuscaloosa, Alabama8 Mesh has been created using Gambit (FLUENT grid generator) Fluid Grid Nodes=41,112 Solid Grid Nodes=2,104

IPE 2003 Tuscaloosa, Alabama9 Main Air Flow: Turbulent Boundary Layer profile (1/7)th. Temperature= 350 K Coolant Air Flow: Uniform Pressure = 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, Alabama10 CHT Solver:  Commercial Code “Fluent” Finite Volume  Full Navier-Stokes Equation for compressible turbulent flow “RNG “ CHT Results:  Results are converged at least for all residuals ( mass, momentum, energy, & )

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IPE 2003 Tuscaloosa, Alabama16 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, Alabama17 BEM Methodology  Surface Mesh only: (we use quadratic discontinuous elements)  Governing Equation: (Boundary Integral Equation for Laplace Eqn.) Where:G(x,  ) = (-1/ 2  k) ln r(x,  ) in 2D H(x,  ) = -k  G(x,  ) /  n q(x) = -k  T(x) /  n C(  ) = 1 if    C(  ) = 1/2 if   

IPE 2003 Tuscaloosa, Alabama18  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, Alabama19 Direct BEM Results: BEM fluxes consistent with FLUENT fluxes

IPE 2003 Tuscaloosa, Alabama20 Direct BEM Results: Heat Cooling Slot

IPE 2003 Tuscaloosa, Alabama21 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) = ?

IPE 2003 Tuscaloosa, Alabama22  Identification of heat fluxes in the cooling slot to match over-specified boundary data at the exposed surfaces. Inverse Problem:  Parametric representation of heat flux in cooling slot using radial basis functions (RBF)  Objective function is to minimize Anchor point BEM node

IPE 2003 Tuscaloosa, Alabama23  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. genes Chromosome

IPE 2003 Tuscaloosa, Alabama24 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, Alabama25 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 q min and q max block#1 (-15,000 to 15,000) block#2 (-2,000 to 2,000) Optimization Technique: Parallel Genetic Algorithms

IPE 2003 Tuscaloosa, Alabama26 Inverse BEM Results: Evolution of objective function for Heat fluxes (T, q) =?

IPE 2003 Tuscaloosa, Alabama27 Inverse BEM Results: Temperature Cooling Slot

IPE 2003 Tuscaloosa, Alabama28 Inverse BEM Results: Heat Fluxes

IPE 2003 Tuscaloosa, Alabama29 Inverse BEM Results: Heat Cooling Slots Element Q GA Q CFD Q AP +

IPE 2003 Tuscaloosa, Alabama30 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.