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Phd Thesis, Universit´Elibre De Bruxelles, Chauss´Eede Waterloo, 72, 1640 Rhode- St-Gen`Ese,Belgium, 2004

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Phd Thesis, Universit´Elibre De Bruxelles, Chauss´Eede Waterloo, 72, 1640 Rhode- St-Gen`Ese,Belgium, 2004 Universit´eLibre de Bruxelles von Karman Institute for Fluid Dynamics Aeronautics and Aerospace Department PhD. Thesis Algorithmic Developments for a Multiphysics Framework Thomas Wuilbaut Promoter: Prof. Herman Deconinck Contact information: Thomas Wuilbaut von Karman Institute for Fluid Dynamics 72 Chauss´eede Waterloo 1640 Rhode-St-Gen`ese BELGIUM email: [email protected] webpage: http://www.vki.ac.be/~wuilbaut To Olivia and her wonderful mother... Summary v Summary In this doctoral work, we adress various problems arising when dealing with multi-physical simulations using a segregated (non-monolithic) approach. We concentrate on a few specific problems and focus on the solution of aeroelastic flutter for linear elastic structures in compressible flows, conju- gate heat transfer for re-entry vehicles including thermo-chemical reactions and finally, industrial electro-chemical plating processes which often include stiff source terms. These problems are often solved using specifically devel- oped solvers, but these cannot easily be reused for different purposes. We have therefore considered the development of a flexible and reusable software platform for the simulation of multi-physics problems. We have based this development on the COOLFluiD framework developed at the von Karman Institute in collaboration with a group of partner institutions. For the solution of fluid flow problems involving compressible flows, we have used the Finite Volume method and we have focused on the applica- tion of the method to moving and deforming computational domains using the Arbitrary Lagrangian Eulerian formulation. Validation on a series of testcases (including turbulence) is shown. In parallel, novel time integration methods have been derived from two popular time discretization methods. they allow to reduce the computational effort needed for unsteady flow com- putations. Good numerical properties have been obtained for both methods. For the computations on deforming domains, a series of mesh deformation techniques are described and compared. In particular, the effect of the stiff- ness definition is analyzed for the Solid material analogy technique. Using the techniques developed, large movements can be obtained while preserving a good mesh quality. In order to account for very large movements for which mesh deformation techniques lead to badly behaved meshes, remeshing is also considered. We also focus on the numerical discretization of a class of physical models that are often associated with fluid flows in coupled problems. For the elliptic problems considered here (elasticity, heat conduction and electro- chemical potential problems), the implementation of a Finite Element solver is presented. Standard techniques are described and applied for a variety of vi Summary problems, both steady and unsteady. Finally, we discuss the coupling of the fluid flow solver with the finite ele- ment solver for a series of applications. We concentrate only on loosely and strongly coupled algorithms and the issues associated with their use and implementation. The treatment of non-conformal meshes at the interface between two coupled computational domains is discussed and the problem of the conservation of global quantities is analyzed. The software develop- ment of a flexible multi-physics framework is also detailed. Then, several coupling algorithms are described and assessed for testcases in aeroelasticity and conjugate heat transfer showing the integration of the fluid and solid solvers within a multi-physics framework. A novel strongly coupled algo- rithm, based on a Jacobian-Free Newton-Krylov method is also presented and applied to stiff coupled electrochemical potential problems. Contents vii Contents Summary v 1 Introduction 1 I Fluid Modeling and Simulation 5 2 Finite Volume Method on Moving Geometries 7 2.1 Numerical Methods for Compressible Flows . .7 2.1.1 Finite Difference Method . .8 2.1.2 Finite Element Method . .8 2.1.3 Finite Volume Method . .9 2.1.4 High Order Methods . .9 2.2 Finite Volume Method . .9 2.2.1 Navier-Stokes Equations . 11 2.2.2 Implicit Jacobian Computation . 12 2.2.3 Higher-Order Reconstruction . 13 2.2.4 Limiters . 13 2.3 Arbitrary Lagrangian Eulerian Formulation . 14 2.3.1 FVM on a time-dependent domain . 14 2.3.2 ALE Formulation of Numerical Fluxes . 15 2.3.3 Boundary Conditions . 18 2.3.4 Results . 19 2.4 Turbulence Models . 22 2.4.1 Spalart-Almaras . 24 2.4.2 k-omega . 24 2.4.3 Results . 30 3 Time Discretization 39 3.1 Discretization Methods and Geometric Conservation Law . 40 3.1.1 Geometric Conservation Law . 40 3.1.2 GCL-respecting Time Integration schemes . 40 3.2 Improved Time Discretization methods . 43 viii Contents 3.2.1 Linearized Crank-Nicholson . 43 3.2.2 Results . 45 3.3 Non-Oscillatory Implicit Time Discretization Scheme . 49 3.3.1 Introduction . 49 3.3.2 Proposed Scheme . 51 3.3.3 Results . 54 3.3.4 Conclusions . 65 II A Class of Coupled Physical Models 67 4 Physical Modeling 71 4.1 Heat Conduction . 71 4.1.1 Unsteady Heat Conduction . 71 4.2 Electro-Chemistry . 71 4.2.1 Electro-Chemical Deposition Processes . 71 4.2.2 Potential Model . 73 4.3 Elasticity . 74 4.3.1 Linear Elasticity . 75 4.3.2 Geometrically Non-Linear Elasticity . 75 5 Numerical Discretization: Finite Element Method 77 5.1 Formulation - Steady Problems . 77 5.2 Boundary Conditions . 79 5.2.1 Dirichlet Boundary Conditions . 79 5.2.2 Neumann Boundary Conditions . 79 5.2.3 Robin Boundary Conditions . 79 5.3 Matrix conditioning . 80 5.3.1 High Order Elements . 80 5.4 Unsteady Formulation . 81 5.4.1 Modal Analysis . 81 5.4.2 Time Integration . 82 5.5 Results . 83 5.5.1 Heat Diffusion/Electro-Chemistry . 83 5.5.2 Linear Elasticity . 85 III Mesh Deformation 91 Contents ix 6 Mesh Deformation Techniques 93 6.1 Spring Analogy . 94 6.1.1 Concepts . 94 6.1.2 Variants . 99 6.2 Elastic Body Analogy . 100 6.2.1 Stiffness Definition . 101 6.2.2 Large Displacements . 105 6.2.3 Computational Cost . 106 6.2.4 Wall Distance Computation . 106 7 Remeshing Algorithms 109 7.1 Introduction . 109 7.2 Interpolation Techniques . 109 7.2.1 Closest Neighbour . 110 7.2.2 Shepard's method . 111 7.2.3 Consistent Interpolation . 111 IV Multi-Physics Coupling 115 8 Non-Conformal Interfaces 119 8.1 Mesh Matching Techniques . 120 8.1.1 Interpolation Based Mesh Matching Techniques . 120 8.1.2 Results . 127 8.2 Conservation at the interface . 128 8.2.1 Fluid-Structure Interaction . 129 8.2.2 Heat Transfer . 131 9 Software Issues 133 9.1 COOLFluiD Framework . 133 9.2 Multi-Method, Multi-Physics - Changes to the framework . 134 9.3 Coupler Layer . 136 9.3.1 SubSystemCoupler module . 137 9.3.2 Maestro module . 138 9.3.3 Boundary Conditions . 139 9.4 Parallelization . 139 10 Coupling Algorithms 143 10.1 Introduction . 143 10.2 Fluid-Structure Interaction . 143 x Contents 10.2.1 Static aeroelasticity . 144 10.2.2 Dynamic aeroelasticity . 144 10.2.3 Coupling Algorithms . 145 10.2.4 2D Supersonic Panel Flutter . 147 10.2.5 3D Transonic Aeroelastic Wing Flutter . 152 10.3 Conjugate Heat Transfer . 158 10.3.1 Coupling Algorithms . 158 10.3.2 Heated Flat Plate . 161 10.3.3 Expert Reentry Vehicle in CNEQ . 165 10.4 Electro-Chemistry . 179 10.4.1 Standard Coupling Algorithms . 179 10.4.2 Jacobian-Free Newton Krylov Algorithm . 181 10.4.3 Conclusion . 197 V Conclusions & Appendices 199 11 Conclusions 201 A Kheops Material Properties 209 Bibliography 213 List of Tables 231 List of Figures 233 Acknowledgements 239 List of Symbols xi List of Symbols Symbols R(U) residual vector 1 unit vector th 1k k component of the unit vector ~x spatial coordinates ~n normal vector t[s] time ρ[kg · m−3] density T [K] Temperature λ[W=m · K] thermal conductivity ! relaxation parameter δ Kronecker's delta r ≡ (@=@x; @=@y:::) Hamilton's nabla operator Fluid U vector of conserved variables Fi(U) convective flux along the direction i Gi(U) diffusive flux along the direction i S(U) source term Ω computational domain Ωe control volume (Cell-centered Finite Volume) xii List of Symbols Γ computational domain boundary surface −1 ~vF [m · s ] fluid velocity −1 ~vG[m · s ] mesh velocity −1 ~vr[m · s ] relative velocity ~vF − ~vG u; v; w[m · s−1] fluid velocity along x, y and z directions ν normal of the averaged mesh configuration κ mesh velocity projected along the normal direction p[P a] static pressure E[J · kg−1] specific total energy H[J · kg−1] specific total enthalpy M Mach number a local speed of sound R Riemann Invariant R J:kg−1:mol−1 Ideal gas constant @F A = @U Jacobian matrix Λ Vector of Eigenvalues L Vector of the left Eigenvectors R Vector of the right Eigenvectors ! angular frequency Re Reynolds number τij viscous stress tensor µ[P a · s] dynamic viscosity ν[m2=s] kinematic viscosity 2 νt[m =s] turbulent eddy viscosity List of Symbols xiii k[m2=s2] turbulent kinetic energy ![s−1] specific dissipation rate of turbulent kinetic energy "[m2=s3] dissipation rate of turbulent kinetic energy β Coefficient of Menter's Blended turbulence models β∗ Coefficient of Menter's Blended turbulence models γ Coefficient of Menter's Blended turbulence models σ Coefficient of Menter's Blended turbulence models y0[m] Distance to the first cell Vf Flutter Speed Index Solid u; v; w[m] displacements in the x, y and z directions U vector of displacements U_ vector of velocities U¨ vector of accelerations K stiffness matrix M mass matrix f external force vector ! test function Nj shape function at node j E [P a] Young Modulus G [P a] Shear Modulus ν Poisson ratio µ Lam´eparameter λ Lam´eparameter xiv List of Symbols σij Stress tensor "ij Strain tensor Cij Elasticity
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