C3.8 CRM wing/body Case

C3.8 CRM wing/body Case 1. Code description XFlow is a high-order discontinuous Galerkin (DG) finite element solver written in ANSI C, intended to be run on Linux-type platforms. Relevant supported equation sets include compressible Euler, Navier- Stokes, and RANS with the Spalart-Allmaras model. High-order is achieved compactly within elements using various high-order bases on triangles, tetrahedra, quadrilaterals, and hexahedra. Parallel runs are supported using domain partitioning and MPI communication. Visual post-processing is performed with an in-house plotter and with TecPlot. Output-based hp-adaptivity is available using discrete adjoints. 2. Case summary Convergence to steady state on the initial mesh (Figure 2(a)) at p = 1 was achieved using the constrained pseudo-transient continuation method with a greedy line-search method for updating the state [2]. One step of Reynolds number continuation was used, see Figure 1(a). The target lift is sought via Newton s method where the lift sensitivity with respect to the angle of attack is obtained using the lift adjoint. For more details on the procedure, see reference [1]. For the workshop, we intend to perform hp-adaptation on a combination of lift and drag. Figure 1 shows the residual and lift convergence history for the initial p = 1 solution. 10 12 10 10 Re = 5 10 5 Re = 5 10 6 Re = 5 10 6, C L = 0.5 1 0.9 0.8 Re = 5 10 5 Re = 5 10 6 Re = 5 10 6, C L = 0.5 10 8 0.7 0.6 R 10 6 Lift 0.5 10 4 0.4 0.3 10 2 10 0 0 500 1000 1500 2000 Iteration (a) Residual convergence for initial, p = 1 solution 0.2 0.1 0 0 500 1000 1500 2000 Iteration (b) Lift coefficient convergence for initial, p = 1 solution Figure 1: CRM wing-body, M = 0.85, C L = 0.5, Re = 5 10 6 : residual and lift histories for the initial solution. Runs were performed on NASA a Pleaides supercomputer using 800 Harpertown cores. TauBench unit on this hardware corresponds to 7.7s. One 3. Meshes The initial high-order curved mesh was generated by first creating a multiblock linear mesh using ICEM CFD (with the geometry provided on the workshop website), and then agglomerating 3 3 blocks of linear cells into q = 3 high-order elements. This mesh is the same as provided in the HO workshop s website. 2013 High-Order CFD Workshop 1 University of Michigan XFlow Group

4. Results The figures below show the initial, p = 1 solution for this case. (a) Initial density contours (p = 1). (b) Initial ν contours (p = 1). Figure 2: CRM wing-body, M = 0.8, C L = 0.5, Re = 5 10 6 : Initial density and ν contours. References [1] Marco Ceze and Krzysztof J. Fidkowski. Drag prediction using adaptive discontinuous finite elements. In 51st AIAA Aerospace Sciences Meeting and Exhibit, 2013. [2] Marco Antonio de Barros Ceze. A Robust hp-adaptation Method for Discontinuous Galerkin Discretizations Applied to Aerodynamic Flows. PhD thesis, The University of Michigan, 2013. 2013 High-Order CFD Workshop 2 University of Michigan XFlow Group

Case 3.8: CRM wing/body Case University of Michigan - XFlow Department of Aerospace Engineering University of Michigan UofM 2 nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C3.8 1/17

XFlow Code Discontinuous Galerkin spatial discretization. Physicality-constrained pseudo-transient continuation (CPTC) nonlinear solver with line-search for improving convergence. Exact Jacobian with element-line-jacobi preconditioner and GMRES linear solver. Roe solver for inviscid flux and BR2 for viscous discretization. MPI parallelization. Node-edge weighted mesh partitioning. Support for curved meshes. Oliver/Allmaras modification to original SA turbulence model. Shock-capturing via element-wise constant artificial viscosity. Output-based anisotropic hp-adaptation. UofM 2 nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C3.8 2/17

Initial mesh - ver.0 Cubic (q = 3) mesh generated by agglomerating 3 linear cells in each direction. Drag-driven anisotropic h-adaptation at fixed lift with p = 1. y + 100 based on a flat-plate correlation for C f. Linear mesh (1218375 elements). Agglomerated cubic mesh (45125 elements). UofM 2 nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C3.8 3/17

C L driver Adjoint-based parameter correction. General framework for computing sensitivities using adjoints. Mesh Initial Conditions J target, ε tol, α guess Solve R(α, U) = 0 True J J target ε tol Finished False Update α α + (J J target)δα Ψ T δr Compute δr = R(α + δα, U) Solve T R Ψ = J U U UofM 2 nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C3.8 4/17

Results with ver.0 mesh Could not achieve C L = 0.5 on the initial mesh. Gray shaded region: range of DPW5 data computed on fine mesh ( 50M cells). Last adapted mesh has 2M DOFs. Error estimate has room for improvement. UofM 2 nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C3.8 5/17

Results with ver.0 mesh Mach contours at 37% of the span. Separation appeared on coarse mesh due to lack of spatial resolution. Initial mesh (α = 2.8 ) UofM 1st drag-adapted mesh (α = 2.675 ) 2nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C3.8 6/17

Results with ver.0 mesh Mach contours at 37% of the span. Smaller differences in the Mach contours after the first adaptive step. 1st drag-adapted mesh (α = 2.675 ) UofM 5th drag-adapted mesh (α = 2.1598 ) 2nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C3.8 7/17

Results with ver.0 mesh C p comparison with NASA s experimental data. 13% of the reference span 50% of the reference span 0.5 0.5 -Cp 0 -Cp 0-0.5 Initial mesh 1st 2nd 3rd 4th 5th Exp. data -0.5 Initial mesh 1st 2nd 3rd 4th 5th Cp 0 0.2 0.4 0.6 0.8 1 X/Chord 0 0.2 0.4 0.6 0.8 1 X/Chord UofM 2 nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C3.8 8/17

Results with ver.0 mesh High-order mesh and geometric irregularities. Each linear block follows: N node = (q N i + 1) (q N j + 1) (q N k + 1). Geometric irregularities of order g q cause wavy surface and possibly larger drag results. UofM 2 nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C3.8 9/17

Drag error estimation under fixed lift In a fixed-lift run we solve: { R(α, U) = 0 L(α, U) = 0, where L(α, U) = Lift(α, U) Lift target. We seek variations of the drag function that satisfy the constraints above, so: δl = δd + Ψ T R } {{ δr + Ψ } C δl = 0, where Ψ } {{ } C = D L. (a) (b) a is the drag error estimate for fixed α, b is the influence of the lift error in the drag error due to the lift constraint. UofM 2 nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C3.8 10/17

Drag adaptation under fixed lift Analogous to unconstrained adaptation: η κ H = η Drag Ψ c η Lift κ H + κ H. Drag-based, isotropic h adaptation. Comparison between unconstrained and constrained adaptation. NACA 0012- M = 0.5, C L = 0.0182, Re = 5000 Lift convergence Drag convergence 0.057 0.0568 0.0566 Ref. value Drag only Drag+Lift 0.0564 Drag coefficient 0.0562 0.056 0.0558 0.0556 0.0554 0.0552 0.055 2000 3000 4000 5000 6000 7000 8000 9000 ndof UofM 2 nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C3.8 11/17

NACA 0012- M = 0.5, C L = 0.0182, Re = 5000 Comparison between unconstrained and constrained adaptation. Drag-based, isotropic h adaptation. 3 2.5 Angle of attack Ref. value Drag only Drag+Lift 10 1 Drag error Drag only Drag+Lift 10 2 Angle of attack 2 1.5 Drag coefficient error 10 3 10 4 1 0.5 2000 3000 4000 5000 6000 7000 8000 9000 ndof 10 5 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 ndof UofM 2 nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C3.8 12/17

NACA 0012- M = 0.5, C L = 0.0182, Re = 5000 Unconstrained Constrained UofM 2 nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C3.8 13/17

Mesh update Provided in the HOW website. p = 1 solution: C L = 0.5000032, C D = 0.05462245, α = 3.9264 Workunits = 8.5104 10 6 Density contours Pressure contours - zoom UofM 2 nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C3.8 14/17

Mesh update Provided in the HOW website. p = 1 solution: C L = 0.5000032, C D = 0.05462245, α = 3.9264 Workunits = 8.5104 10 6 Density contours Pressure contours - zoom UofM 2 nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C3.8 15/17

Mesh update R 10 12 10 10 10 8 10 6 10 4 10 2 Residual and Lift convergence. Residual Re = 5 10 5 Re = 5 10 6 Re = 5 10 6, C L = 0.5 10 0 0 500 1000 1500 2000 Iteration Lift 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 Lift (constraint) Re = 5 10 5 Re = 5 10 6 Re = 5 10 6, C L = 0.5 0 0 500 1000 1500 2000 Iteration UofM 2 nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C3.8 16/17

Next steps Only recently that we realized the we need the lift error estimate. Perform hp-adaptation for combined lift and drag. We hope to get results with similar quality as the ones obtained for C3.2. Lack of robustness of lift error estimate is worrisome. UofM 2 nd International Workshop on High-Order CFD Methods, May 27-28, Cologne, Germany C3.8 17/17