An Embedded Boundary Method with Adaptive Mesh Refinements

Transcription

1 An Embedded Boundary Method with Adaptive Mesh Refinements Marcos Vanella and Elias Balaras 8 th World Congress on Computational Mechanics, WCCM8 5 th European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2008 June 30 July 5, Venice, Italy.

2 Outline Motivation Adaptive Mesh Refinement Method Embedded Boundary strategy Accuracy Study Numerical Results Summary

3 Motivation Flows with Moving boundaries are problems of great practical importance in engineering and biology disciplines. Commonly adopted boundary conforming formulations (ALE) are difficult in problems involving large boundary motions, deformations. Immersed boundary approaches are tied to Cartesian grids that do not allow flexibility in grid refinement. Combine advantages of Cartesian grids with adaptively increasing resolution in particular zones of the flow domain. 30c

4 Motivation Develop a scheme that allows for the grid high resolution to follow zones of the flow where high gradients take place. Save computational effort in outer regions. Two flexible profiles swimming upstream. Vorticity Contours.

5 AMR Method Adaptive Mesh Refinement Topology: Divide the domain in sub-blocks. Each subgrid block has a structured Cartesian topology, and is part of a tree data structure that covers the entire computational domain. Local refinement of a sub-grid block is performed by bisection in each coordinate direction. Number of cells in each sub-block remains constant. Level 0 Level 1 Level 2 Level3

6 AMR Method Spatial-Temporal discretization: Explicit Fractional step method. A single-block Cartesian grid solver is employed in each sub-grid block: standard staggered grid in each subblock second-order central finitedifferences We use the Paramesh toolkit (developed by P. MacNeice and K. Olson)* for the implementation of the AMR process. A multigrid solver is used for the Poisson equation. standard staggered grid coarse-fine interface * MacNeice, P., Olson, K. M., Mobarry, C., defainchtein, R., and Packer, C. Paramesh: a parallel adaptive mesh renement community toolkit. Comput. Phys. Commun. 126 (2000),

7 AMR Method Guard Cell filling: Step 1: coarse guard-cell filling Step 2: coarse guard-cell filling coarse-fine interface Guard-cells must be filled at block edges in order to complete the differencing stencil At least quadratic interpolation is required to maintain 2nd order accuracy of numerical method. Normal velocities to interface jumps are fixed on the coarse boundaries to match the fine mass fluxes.

8 Embedded Boundary Method Direct Forcing at Lagrangian Points The discretized momentum equation using the fractional step method is: u u p k i k k 1 k 1 i i ( k 1) ( k 2) k = γkhui + ρkhui αk + fi t xi By an interpolation step, intermediate velocities are obtained on each surface marker from surrounding Eulerian points U k = I( u ) The body force term is evaluated at the Lagrangian marker point F U U RHS t k k UΨ Ui k k 1 k i =, i = i + i i Uhlmann M., (2005). An immersed boundary method with direct forcing for the simulation of particulate flows. J. Comput. Phys. 209 (2):

9 Embedded Boundary Method Direct Forcing at Lagrangian Points U k k = I( u ) i i is interpolated to the marker point using a 5 or 7 point stencil and moving least squares interpolation (MLS) k F i U ne k = l k i φeu ie e= 1 is extrapolated to the Eulerian points of the stencil using the same operator as the interpolation, properly scaled. A h nl l l k l k i = lφ e il l = 1 ; l = V El f c F c The Momentum transfer is preserved by the extrapolation process. Lancaster P. and Salkauskas K. (1981). Surfaces generated by moving squares methods. Math. Comput. 37,

10 Accuracy Study Taylor Green Vortex Compare numerical solution to analytical solution of 2D Navier-Stokes equations Domain: [π/2, 5π/2]x [π/2, 5π/2] Homogeneous Dirichlet/Neumann velocity boundary conditions and Neumann pressure boundary condition u = e 2t cos x sin y v = e 2t sin x cos y p = e 4 t 4 ( cos2x + cos2y) u p

11 Accuracy Study Taylor Green Vortex Domain with 2 refinement levels linear interpolation Domain with 2 refinement levels quadratic interpolation Uniform domain Linear interpolation does not maintain 2nd order accuracy of numerical scheme

12 Numerical Results Flow around a fixed sphere at Re = 300: Domain size 40 D x 20 D x 20 D Boundary Conditions: Uc=1 and Convective in x, periodic in y, z. Level 0: 64 x 32 x 32 grid cells. Blocks 16 3 cells. 4 Levels to advance 130 time units. 5 Levels in the following 6 shedding periods ( =0.019 of an average of 5 points in the B.L. thickness). 6.4 million points. The resulting St = 0.132, while in Johnson and Patel (1999) St = 0.137

13 Numerical Results AMR Johnson and Patel (1999)* Roos and Willmarth (1971)** CDmean Amplitude CD CLmean Amplitude CL * Johnson T. A. and Patel V. C. Flow past a sphere up to a Reynolds number of 300. J. Fluid Mech. 378 (1999), ** Roos, F. W., and Willmarth, W. W. Some experimental results on sphere and disk drag. AIAA J. 9 (1971),

14 Numerical Results Mean and rms streamwise vorticity in the axial direction behind the sphere. Last 4 periods of computation were used for these statistics. Isosurface of Q colored by values of streamwise vorticity for t = Vorticity ranges from -1 (blue) to +1 (red) using 40 contours.

15 Development Hovering Musca Domestica at Re = 300: (a) Re = Utmax*Lr/ν = 300. Boundary Conditions: Dirichlet in z, periodic in x, y. Level 0: 32 x 32 x 32 grid cells. Blocks 16 3 cells. Coarse calculation maximum 3 levels of refinement ( x =0.019). Integration for 4 flapping cycles. Harmonic prescriptions for beating angle and geometric angle of attack. IB Treatment of the wings as membranes. (c) (b)

16 Development Hovering Musca Domestica at Re = 300: p = -0.3, and 0.3 pressure contours (blue to orange). Slice at 0.85 span. Q = 0.5 isocontour.

17 Summary Combine the computational efficiency of IB Cartesian solvers with the resolution capabilities of the AMR scheme. Concentrate the computational resources in the zones where large gradients are found. Follow in time the evolution of the solids/fluid system reducing the total number of grid points. The AMR method exhibits similar accuracy using refined grids as a single block solver applied to a mesh of the same level of highest resolution. Evaluation of different 3D problems, including insect flight, and the response of the method on fluid-structure interaction problems.

18 Thank You

19 Accuracy Studies Lid Driven Cavity Flow with Immersed Cylinder Assess the error on the solution as the grid resolution is increased, for the problem with immersed boundary. Ulid Reynolds number = Unitary tangential velocity on top lid with no penetration and no slip on other boundaries. Cylinder in center of Cavity D = 0.4LR Grids: - 36 x x x x LR x 540 LR

20 Accuracy Studies Lid Driven Cavity Flow with Immersed Cylinder Second order space accuracy is found in L2 and Linf norms of the error on velocities respect to the finer grid result.