An Embedded Boundary Method with Adaptive Mesh Refinements

Marcos Vanella and Elias Balaras

8th World Congress on Computational Mechanics, WCCM85th European Congress on Computational Methods in Applied Sciences

and Engineering, ECCOMAS 2008

June 30 – July 5, 2008. Venice, Italy.

Outline

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

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

• 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.

Motivation

Divide the domain in sub-blocks. Each sub-grid 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 2

Level 0

Level 1

Level3

Adaptive Mesh Refinement Topology:

AMR Method

• Explicit Fractional step method.

• A single-block Cartesian grid solver is employed in each sub-grid block:

• standard staggered grid in each sub-block

• second-order central finite-differences

• 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

Spatial-Temporal discretization:

AMR Method

* 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), 330-354.

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.

coarse-fine interface

Step 1: coarse guard-cell filling

Step 2: coarse guard-cell filling

Guard Cell filling:

AMR Method

Direct Forcing at Lagrangian Points

Embedded Boundary Method

The body force term is evaluated at the Lagrangian marker point †

1,k

k k k kii i i i

U UF U U RHSt

−Ψ −= = +∆

† Uhlmann M., (2005). An immersed boundary method with direct forcing for the simulation of particulate flows. J. Comput. Phys. 209 (2): 448-476.

The discretized momentum equation using the fractional step method is:

( ) ( )1 1

1 2k k k

k k ki ik i k i k i

i

u u pH u H u ft x

γ ρ α− −

− −− ∂= + − +

∆ ∂

( )k ki iU I u=

By an interpolation step, intermediate velocities are obtained on each surface marker from surrounding Eulerian points

Direct Forcing at Lagrangian Points

Embedded Boundary Method

is interpolated to the marker point using a 5 or 7 point stenciland moving least squares interpolation (MLS) †

( )k ki iU I u=

† Lancaster P. and Salkauskas K. (1981). Surfaces generated by moving squares methods. Math. Comput. 37, 141-158.

1

nek l ki e ie

eU uφ

=

=∑is extrapolated to the Eulerian points of

the stencil using the same operator as the interpolation, properly scaled.

The Momentum transfer is preserved by the extrapolation process.

kiF

1

;l lnl

k l ki l e il l E l

l

A hf c F cV

φ=

= =∆∑

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 yv = e−2t sin x cos y

p = − e−4 t

4cos2x + cos2y( )

u p

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

Accuracy Study

Flow around a fixed sphere at Re = 300:

Numerical Results

• 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 163 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

* Johnson T. A. and Patel V. C. Flow past a sphere up to a Reynolds number of 300. J. Fluid Mech. 378 (1999), 19-70.

** Roos, F. W., and Willmarth, W. W. Some experimental results on sphere and disk drag. AIAA J. 9 (1971), 285-291.

-0.0690.0594CLmean

-0.00160.0019Amplitude

CL

-0.00350.0039Amplitude CD

0.6290.6560.634CDmean

Roos and Willmarth (1971)**

Johnson and Patel (1999)*AMR

Numerical Results

Isosurface of Q colored by values of streamwise vorticity for t = 180.96. Vorticity ranges from -1 (blue) to +1 (red) using 40 contours.

Mean and rms streamwise vorticity in the axial direction behind the sphere. Last 4 periods of computation were used for these statistics.

Numerical Results

DevelopmentHovering Musca Domestica at Re = 300:

• Re = Utmax*Lr/ν = 300.• Boundary Conditions: Dirichlet in z, periodic in x, y.• Level 0: 32 x 32 x 32 grid cells. Blocks 163 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.

(a)

(b)

(c)

DevelopmentHovering Musca Domestica at Re = 300:

Q = 0.5 isocontour.p = -0.3, -0.15 and 0.3 pressure contours (blue to orange). Slice

at 0.85 span.

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 theresponse of the method on fluid-structure interaction problems.

Thank You

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.

• Reynolds number = 1000.

• 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 36

- 60 x 60

- 108 x 108

- 180 x 180

- 540 x 540 LR

Ulid

0.4LR

Accuracy Studies

Lid Driven Cavity Flow with Immersed Cylinder

• Second order space accuracyis found in L2 and Linf norms of the error on velocities respect to the finer grid result.

Accuracy Studies