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1 A Method for Time Accurate Turbulent Compressible Fluid Flow Simulation with Moving Boundary Components Employing Local Remeshing O. Hassan, K. Morgan and N. P. Weatherill School of Engineering, University of Wales Swansea, United Kingdom Workshop on Mesh Refinement of Unsteady Flows 7 December 2005 - Oxford University

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2 Problem of Interest and Adopted Approach Solution Algorithm Unstructured Mesh Generation Techniques Mesh Adaptation Techniques Mesh Adaptation for Unsteady Flow Problem with Moving Boundary Parallel Implementation Conclusion Outline

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3 DASA - F16 Airbus - A380 Dassault - Falcon Airbus - A340 Industrial End-User Configurations

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4 The governing equations are the Navier-Stokes equations The application requires the ability to model complex geometries The simulation of turbulence affords a real challenge For many applications, the Euler equations are appropriate Computational requirements for realistic geometries can be expected to be large Adopted Approach Unstructured grid technology provides the required flexibility for these (and other) applications

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5 Governing Equations The Favre Averaged Navier Stokes Equations Where and Turbulent is modelled by adding the one equation model of Spalart and Allmaras

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6 Solution Algorithm Edge Based Data Structure: Typical interior node I The ALE term for an interior node I is: Where: should lead to a numerical ALE flux that satisfies GCL Resulting Equation

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7 Solution Algorithm The turbulent viscosity equation is discretised in a similar fashion Stabilisation achieved by replacing the actual flux function by JST flux function Discontinuity capturing achieved by the addition of a switched artificial diffusion For steady state Runge-Kutta relaxation and local timestepping is utilised Convergence acceleration is achieved by using the Full Approximation Storage (FAS) Multigrid scheme Coarse grids are achieved by agglomeration Volume weighted operator is used for restriction Injection is used for prolongation Parallel implementation allows agglomeration across partitions

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8 Steady Turbulent Flow Pressure distribution Cl vs Cd Cl vs AIAA test case: Drag Prediction Workshop 2001 M = 0.75 Re = 3 x 10 6 1.6 million points 35 viscous layers5 grids levels

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9 Time Discretisation For unsteady problems, the second order approximation is adopted An implicit formulation is employed and this removes the stability constraints associated with explicit schemes At each time step, the equation is solved by explicit relaxation with multi- grid acceleration This approach can be thought of as converging the set of steady state equations with the addition of the time source for every physical timestep No significant memory penalties compared to explicit procedures

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10 The surface is defined as a set of: Surface Components: bi-cubic patches, NURBS Curve components: cubic splines, NURBS Mesh control: Background Mesh Point, Line, circular and planar sources Curvature Controlled Mesh Generation : the gap between the element and the surface k 1, k 2 : the two principle curvatures

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11 Surface mesh generation: Advancing Front Volume mesh generation: Delaunay Triangulation with automatic point insertion ( Requires 100Mb/10 6 elements) Boundary layer generation: Hybrid meshes by the Advancing Layer method Mesh Generation

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12 Mesh Minimum Volume Minimum Dihedral Angle Generated3.26 x 10 -9 0.098 20-102.32 x 10 -7 7.32 70-102.36 x 10 -7 12.27 Improved Volume Mesh Quality 3D Edge Swap 3D Edge Collapse 3D Nodal Smoothing 3D Local Re-generation

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13 Super surfaces eliminate small and distorted patches generated by the CAD systems Merge neighboring surfaces based on continuity of the normal Improved Surface Mesh Quality Patch Independent Remeshing Starting from any triangulation, re-triangulate using edge splitting, edge collapse and edge swapping.

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14 Mesh Adaptation Error Analysis Error indicator based on posterior error analysis are also possible, but not very practical for unsteady flow. A Error indicator based upon interpolation theory and accounts for directionality is employed. Assuming exact nodal values, estimate the local error for each elements as: Equidistribution of the error results in a mesh spacing for the new mesh: In 2D/3D: Apply the 1D criterion separately to each principal direction of the Hessian Matrix

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15 Mesh Adaptation Mesh Enrichment Advantages: Simple and quick to implement Trivial interpolation Disadvantages: Multiple refinement results in large meshes De-refinement require excessive storage Incorporating stretching results is distorted elements Not suitable for unsteady flow with moving components

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16 Mesh Adaptation Mesh Enrichment Special care is required in 3D to ensure compatibility of adjacent elements Special care is also needed to ensure the validity of the mesh after projecting the added points onto the surface Geometry Initial Grid Adapted Grid

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17 Mesh Movement Replace the sides of the mesh by spring Spring stiffness depends on the flow properties Where Move the nodes until nodal equilibrium Solve by iteration Mesh Adaptation

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18 Mesh Adaptation Mesh Movement Advantages: Simple and quick to implement Can handle moving components Disadvantages: Expensive interpolation Initial mesh may lack the required resolution to resolve all the flow features Hard to control the quality of the moved mesh Coupling of mesh movement and mesh enrichment can over come most of the restrictions.

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19 Mesh Adaptation Adaptive Remeshing Use the current mesh as a background mesh At each node compute the mesh parameters using the equidistribution criterion Use the geometry definition to regenerate the mesh

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20 Adaptive Remeshing Using a Background Mesh Advantages: Simple and quick to implement Can handle moving components High quality meshes Disadvantages: Expensive to regenerate the complete mesh Mesh Adaptation

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21 Adaptation Mesh Enrichemnet Mesh Movement Remeshing No coarsening beyond the initial mesh Multiple refinement can generate generate distorted elements Memory intensive in 3D Reliable moving methods are expensive Not easy to guarantee valid mesh May not have enough initial points Time consuming in 3D Essential for unsteady flow with large moving boundaries Efficient for unsteady flow with small moving boundaries Useful for steady state For unsteady flow with moving boundary components it is essential to develop a scheme which utilise the advantages of the various methods

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22 Adaptation for Unsteady Flow Generate initial mesh Calculate initial quality measure, q o and spacing, Loop over timesteps Update coordinates of moving nodes Apply deforming mesh algorithm Compute new quality measure, q n Form holes from marked elements Remesh hole Interpolate solution Compute q o, o F Mark element to be deleted T Compute new spacing required based on the equidistribution criterion, n

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23 Interaction between a strong shock and an object of complex shape Density contours 1228 elements 650 points 3780 elements 1946 points 8554 elements 4348 points 11726 elements 5956 points

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24 Confined Blast Wave From Rupturing Cylindrical Pressure Vessel Experimental Apparatus Comparison of pressure history at various transducers 1499 < Number of elements < 21022

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25 Surface Adaptation The Geometry definition is utilised for the regeneration of the surface portion of the hole

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26 Surface Adaptation

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27 Unsteady Simulation B60 Configuration M = 0.801 0 = 2.78 0 m = 1 0 Z m = 1m 745198 Elements 135760 Points

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28 Unsteady Inviscid Flow Store Separation Simulation init = zero degrees M= 0.5 Degree Container motion computed 2.7 million tetrahedra 15 time steps 50 multigrid cycle per time step Geometry for a complete F16 Configuration 8h Wall clock time Solver: 16 R14000 CPUs Preprocessing and adaption : 1 CPU

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29 Store Separation Simulation Surface Pressure Distribution

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30 Adaptation for Unsteady Turbulent Flow Generate initial mesh Calculate minimum dihedral angle (Di o ) Store the layer number for all nodes in the boundary layer Loop over physical time-steps Update coordinates of moving nodes and the boundary layer nodes Apply deforming mesh algorithmCalculate new dihedral angles (Di n ) Di n <0 Form holes from marked elements Interpolate solution & Compute Di o F Mark elements to be deleted T Check elements intersection Check if any boundary layer node can grow further Remesh hole

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31 Unsteady Turbulent Flow NACA64A010 Aerofoil: Prescribed sinusoidal oscillation Amplitude 1.01 degrees init = zero degrees St = 15.567 3D stacked hybrid mesh 173720 nodes 300 multigrid cycle/time step 32time steps per cycle 16 R14000 processors One movement cycle 5 h clock time Estimated speed up 175

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32 Lift Polar k = 90 k = 0. k = 270 Unsteady Turbulent Flow

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33 Shuttle Booster Separation Simulation init = zero degrees M= 0.85 Degree Re = 3 * 10 6 Prescribed Shuttle movement Initial mesh: 2.9 million elements Final mesh: 3.3 million elements 20 time steps 300 multigrid cycle per time step 36h Wall clock time Solver: 16 R14000 CPUs Preprocessing and adaption : 1 CPU Unsteady Turbulent Flow

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34 Meshes on the symmetry plane Shuttle Booster Separation Simulation Unsteady Turbulent Flow Cut through the volume mesh

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35 Shuttle Booster Separation Simulation Unsteady Turbulent Flow Meshes of the symmetry plane after remeshing Cut through the volume mesh after remeshing

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36 Shuttle Booster Separation Simulation Unsteady Turbulent Flow

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37 Elements are select to be remeshed in each domain separately Selection Based on Deviation from Prescribed Spacing Selection Based on Element Quality Selection Based on Intersection Tests To determine intersection of elements due to moving geometries, one ovelapping ghost layer of elements is used. If intersection with the ghost element has occurred, the search will also take place in to the domain which own the ghost cell. Parallel Implementation

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38 Parallel Implementation 1 3 1 2 1 1 2 2 Domain 4 Domain 3 Domain 2 Domain 1 Interfaces Constraint Repartitioning is employed to ensure that each region to be remeshed will be contained completely on one process

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39 Unsteady Inviscid Flow Store Separation Simulation init = 0.46 degrees M= 0.96 Container motion computed 2.1-2.3 million Nodes 12.1-13.4 million Elements 40 Physical timesteps with sub-cycling 10.4 h on 24 Processors Geometry for a complete F18 Configuration

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40 Unsteady Inviscid Flow Store Separation Simulation

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41 Unsteady Inviscid Flow CFD Solution 40% Motion Application 57% Mesh Deformation 10.3% Volume mesh Analysis 3.7% Volume remeshing 37% Re-partitioning 6% I/O 3% Store Separation Simulation

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42 Conclusions A hybrid unstructured finite volume method for aerodynamic flows has been presented Turbulent flows are treated via the one equation Spalart and Allmaras model Computational performance is enhanced by the use of multigrid acceleration and parallelisation Transient moving boundary flows are treated by an ALE approach Mesh movement and adaptive remeshing have been employed to handle the deformation due to the moving components Adaptive remeshing was extended to meshes with stretched elements in the boundary layers Parallel implementation of the adapted remeshing has been completed A number of challenging problems have been simulated and the agreement with available experimental observations is good

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