Verification of Moving Mesh Discretizations

Transcription

1 Verification of Moving Mesh Discretizations Krzysztof J. Fidkowski High Order CFD Workshop Kissimmee, Florida January 6, 2018

2 How can we verify moving mesh results? Goal: Demonstrate accuracy of flow solutions on moving domains and meshes Considerations: Inviscid versus viscous term discretization Solution field norm versus scalar outputs Integration quadrature requirements Mapping singularities Geometric conservation Boundary condition consistency Mesh Motion Verification Introduction 2/19

3 The arbitrary Lagrangian-Eulerian (ALE) method ALE Idea: solve transformed PDE on a static reference domain Reference domain: X, u X, H X ux t NdA + X H X X(u X, q X) = 0 Mapping X, t x( X, t) G = x X g = det(g) u X = gu q X = gg T q v G = x t H X = gg 1 H u X G 1 v G nda = gg T NdA NdA = g 1 G T nda Physical domain: x, u, H u t nda x + H(u, q) = 0 Key definitions X = reference-domain coordinates x = physical-domain coordinates g = determinant of Jacobian matrix v G = grid velocity, x/ t u = physical state u X = reference state H = physical flux vector H X = reference flux vector Mesh Motion Verification Introduction 3/19

4 The transformed equations Integrate the evolution PDE over a time-varying volume v(t), apply the divergence theorem, transform terms to reference space, and apply the divergence theorem again, u X t + X H X (u X, X u X ) = 0, X where u X = gu H X = gg 1 H ux G 1 v G X is the gradient with respect to the reference coordinates The transformed flux, H X, separates into inviscid and viscous contributions, H X = F X + G X, FX = gg 1 F + ux G 1 v G, GX = gg 1 G Mesh Motion Verification Introduction 4/19

5 ALE implementation To minimize code intrusion, express the reference-space fluxes in terms of the physical fluxes FX = gg 1 F ux G 1 v G = gg 1 ( F u vg ) G X = gg 1 G = gg 1 K q = } G 1 {{ KG T } q X K X Linearizations must be performed w.r.t reference states Viscous stabilization does not change (same as physical)! Boundary conditions (e.g. no slip) must incorporate the boundary velocity, v G. Mesh Motion Verification Introduction 5/19

6 Outputs General form of boundary flux integral outputs: J = o T Ĥ n da, o R s is a weight function that defines the output For 2D Navier-Stokes: o = [0; cos α; sin α; 0] Drag o = [0; y; x; 0] Moment about origin o = [0; v Gx ; v Gy ; 0] Power Ω For example, if v G = v 0 + ω r, where r is a position vector relative to some origin of rotation, J = ( v 0 + ω r) f surf da = v 0 f surf da + ω ( r f surf ) da Ω Ω } {{ } Ω } {{ } v 0 F net ω T net Mesh Motion Verification Introduction 6/19

7 Inviscid field norm check: Euler vortex Analytical vortex solution to the Euler equations Sinusoidal interior mesh deformation for testing x(t) = X + 2 sin(2πx/20) sin(2πy/15) sin(2πt) y(t) = Y sin(2πx/20) sin(2πy/15) sin(4πt) Pressure at final time Deformed mesh Mesh Motion Verification Introduction 7/19

8 Euler vortex results Look at L 2 error norm at end of the simulation 10 0 DIRK4 with small t 2 L 2 error norm h = sqrt(1/nelem) DG, p=1 DG, p=2 DG, HDG, p=1 HDG, p=2 HDG, Mesh Motion Verification Introduction 8/19

9 Inviscid boundary output: bump M=0.2, structured mesh, sinusoidal mesh motion Mach contours mesh at t = 0.0 mesh at t = 0.5 mesh at t = 1.5 Mesh Motion Verification Introduction 9/19

10 Inviscid boundary output: bump Monitor instantaneous drag output on the bump Compare moving mesh results to a static mesh Coarse (p = 3, N t = 40) and fine (p = 4, N t = 60) solutions Verification: error decreases with increasing resolution drag e-05,N_t=40, motion,n_t=40,n_t=60, motion,n_t=60 6e-05 4e time Mesh Motion Verification Introduction 10/19

11 Viscous boundary output: flat plate M=0.2, Re = 1000, adapted mesh, sinusoidal mesh motion Mach contours mesh at t = 0.0 mesh at t = 0.5 mesh at t = 1.5 Mesh Motion Verification Introduction 11/19

12 Viscous boundary output: flat plate Monitor instantaneous drag output on the flat plate Compare moving mesh results to a static mesh Coarse (p = 2, N t = 20) and fine (p = 3, N t = 40) solutions Verification: error decreases with increasing resolution drag p=2,n_t=20, motion p=2,n_t=20,n_t=40, motion,n_t= time Mesh Motion Verification Introduction 12/19

13 NACA 0012 airfoil in pitch/plunge motion Smooth plunge, h(t), and pitch, θ(t), motion, 0 t 1 h(t) = 1 cos πt, θ(t) = π cos 2πt. 2 NACA 0012, Re c = 5000, M = 0.2, steady-state IC t = 0 t = 0.5 t = 1.0 Mesh Motion Verification Introduction 13/19

14 Outputs Impulse = Work = F y dt = 1 0 airfoil ŷ f surf ds dt ] [ v 0 F net + ω T net dt = Work done on airfoil appears as total energy deficit in flow 1 0 airfoil v G f surf ds dt Power/Work Instantaneous power Time integrated power Delta total energy in flow Time Mesh Motion Verification Introduction 14/19

15 Outputs Impulse = Work = F y dt = 1 0 airfoil ŷ f surf ds dt ] [ v 0 F net + ω T net dt = Work done on airfoil appears as total energy deficit in flow 1 0 airfoil v G f surf ds dt Power/Work Instantaneous power Time integrated power Delta total energy in flow Time Mesh Motion Verification Introduction 14/19

16 Worskhop runs meshes: quasi-uniform refinement Mesh 0: 664 elements (farfield at 100c) Mesh 2: 7389 elements Mesh 1: 2137 elements Mesh 3: elements Mesh Motion Verification Introduction 15/19

17 Worskhop runs: details Considered spatial orders p = 1, 2, 3, 4, 5 ESDIRK5 (5th order) time stepping Number of time steps depends on mesh and order: p = 1 p = 2 p = 3 p = 4 p = 5 Mesh Mesh Mesh (these values were determined empirically) Error computed relative to a truth solution from a p = 5, N t = 800, ESDIRK5 run on Mesh 3 Mesh Motion Verification Introduction 16/19

18 Worskhop result: case 1, pure heaving h(t) = t2 (3 t), θ(t) = LiftInt error p= p= /(dof) 1/2 LiftInt error p= p= work units Truth lift integral = Mesh Motion Verification Introduction 17/19

19 Worskhop result: case 1, pure heaving h(t) = t2 (3 t), θ(t) = 0 4 PowerInt error 10 3 p=1 p= /(dof) 1/2 PowerInt error 10 3 p=1 p= work units Truth power integral = Mesh Motion Verification Introduction 17/19

20 Worskhop result: case 2, flow aligning h(t) = t2 (3 t), θ(t) = π 4 3 t2 (t 2 4t + 4) LiftInt error p=1 p=2 LiftInt error p=1 p= /(dof) 1/ work units Truth lift integral = Mesh Motion Verification Introduction 18/19

21 Worskhop result: case 2, flow aligning h(t) = t2 (3 t), θ(t) = π 4 3 t2 (t 2 4t + 4) PowerInt error p=1 p= /(dof) 1/2 PowerInt error 10 3 p=1 p= work units Truth power integral = Mesh Motion Verification Introduction 18/19

22 Worskhop result: case 3, energy extracting h(t) = t2 16 ( 8t3 + 51t 2 111t + 84), θ(t) = 4π 9 t2 (t 2 4t + 4) LiftInt error 10 3 p=1 p= /(dof) 1/2 LiftInt error 10 3 p=1 p= work units Truth lift integral = Mesh Motion Verification Introduction 19/19

23 Worskhop result: case 3, energy extracting h(t) = t2 16 ( 8t3 + 51t 2 111t + 84), θ(t) = 4π 9 t2 (t 2 4t + 4) PowerInt error 10 3 p=1 p= /(dof) 1/2 PowerInt error 10 3 p=1 p= work units Truth power integral = Mesh Motion Verification Introduction 19/19