LOCAL GRID REFINEMENT FOR AN IMMERSED BOUNDARY RANS SOLVER

For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics 1801 Alexander Bell Drive, Suite 500, Reston, VA 20191 AIAA 2004-0586 LOCAL GRID REF

For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics

1801 Alexander Bell Drive, Suite 500, Reston, VA 20191

AIAA 2004-0586

LOCAL GRID REFINEMENT FOR AN

IMMERSED BOUNDARY RANS SOLVER

G Iaccarino, G Kalitzin, P Moin

Center for Turbulence Research Stanford University

Stanford, CA 94305

B Khalighi

Research and Development Center General Motors Corporation

Warren, MI 48090-9055

5-8 January 2004 / Reno, NV

42nd AIAA Aerospace Sciences Meeting and Exhibit

LOCAL GRID REFINEMENT FOR

AN IMMERSED BOUNDARY RANS SOLVER

Gianluca Iaccarino, Georgi Kalitzin, Parviz Moin

Center for Turbulence Research Stanford University Stanford, CA 934305 Bahram Khalighi‡

Research and Development Center General Motors Corporation Warren, MI 48090-9050

ABSTRACT

A RANS solver based on the Immersed Boundary

technique is extended to handle locally refined

grids in order to increase the resolution close to

the boundaries for high Reynolds number

simulations A novel data management

architecture is introduced to take advantage of the

quasi-structured nature of the grids and to obtain

fully implicit, fast and robust solutions when

several levels of refinement are introduced The

mesh refinement is fully anisotropic, and can

handle n-to-one cell connectivity It is generated in

a fully automatic way by coarsening a fine

structured grid A conjugate gradient-based

algorithm is used to solve the Navier-Stokes

equations and validation cases include a two- and

a three-dimensional problem

INTRODUCTION

Recently, a Reynolds-Averaged Navier-Stokes

(RANS) solver based on the Immersed Boundary

approach, IBRANS, has been developed [1]; it

uses Cartesian (non-uniform) grids and forcing

terms in the governing equations to account for

complex non-grid-conforming boundaries It is well

known that very large meshes are required to

achieve appropriate resolution of the boundary

layers at high Reynolds numbers This limits the

applicability of pure Cartesian solvers and requires

the adoption of a more flexible meshing strategy

Local Grid Refinement (LGR) techniques allow to

split selected cells in smaller elements thus

increasing the local resolutions; the resulting

algorithms are typically complicated and

memory/CPU intensive In many respects the

computational mesh is considered as an unstructured grid with hanging nodes at the interface between refined and not refined regions [2,3]

The OCTREE approach2 generates a relation between the original element (father) and the newly generated cells obtained by splitting it (kids) This relation can be recursively applied to build connectivity trees To collect the information about the neighbors of a cell, the tree is followed

up to the top where native connectivity information

is stored This scheme is very simple when the cells are split in a consistent way (isotropic refinement) and in [2] it has been used for an inviscid flow solver in a very efficient fashion

The fully unstructured approach3, on the other hand, handles the elements with hanging nodes

as polyhedra with N faces, shared with N neighbors In this case the connectivity information

is stored in the usual way with the only complexity that a very large number of neighbors can be present for each cell (in three-dimensions with one (two) levels of isotropic refinement a cell can have

24 (96) neighbors)

A novel grid refinement technique for Cartesian grids has been introduced in [4]; it uses an underlying structured grid to build the connectivity information and a simple interpolation formula to treat the hanging nodes

In the present work, this LGR scheme for structured grids is extended to deal with a finite volume formulation in a fully conservative fashion The implementation of LGR in the IBRANS code substantially extends the resolution capability of the solver The main features of the present implementation are the low memory storage requirements (typically 20% of a fully unstructured solver) and its efficiency

The present paper describes the basic algorithm

and the details of the data structure used In

addition, it discusses the generation of locally

refined grids in combination with the immersed

boundary algorithm The validation of the present

scheme is reported for a two-dimensional problem

by comparing solutions obtained using LGR and a

structured grid Calculations for a three-

dimensional problem are presented and compared

to experimental data

IBRANS SIMULATION SYSTEM

The IBRANS simulation system1 consists of three

components: the pre-processor that handles the

geometry modeling and the grid generation, the

flow solver and the post-processor that produces

flow maps on the immersed surface

The pre-processor performs three major functions:

the grid generation, the interface and interior cell

determination and the evaluation of the weighting

coefficients for the immersed boundary

interpolation

Geometries are imported in Stereo-Lithography

Format (STL) The only requirement on the STL

triangulation is to be “water-tight” This allows a

unique set of cells to be determined as interior

cells Several STL files can be handled so that the

geometry can be split in components as

appropriate

After the geometry is read, it is placed on a

structured Cartesian grid This grid covers the

entire computational domain, which includes the

region inside the body The generation of this grid

is extremely simple and automatic The mesh

stretching is based on the location of the

immersed boundary and cells are clustered near

the object surface

Figure 1 Immersed Boundary tagging

Given the geometry and the (underlying) grid the tagging procedure takes place; the cells are classified in exterior (fluid) interior (solid) and partially interior (interface) This is shown in Figure

1 The tagging procedure is carried out using a ray-tracing technique5

The RANS equations are discretized with a second-order, cell centered, fully implicit finite volume scheme The implicitly under-relaxed equations are reduced to a linear system in the form:

a(i,j)n!"

(i,j)n+1

+aWn!"

Wn+1+aEn!"

En+1+aNn!"

Nn+1+aSn!"

Sn+1=S(i,j)n (1) where the indices W,E,N,S refer to the neighboring cells that have a common face with cell (i,j), e.g E for cell (i+1,j) ! is either one of the velocity components, the pressure or a turbulence variable

Turbulence is modeled with the two-equation KG model1 This is a modified version of the Wilcox

k-# model7 where # is substituted with a variable g that is defined as: g=1/($*#%&'( The variable g is zero at the wall This simplifies the enforcement of the IB conditions11

A SIMPLE procedure6 is used to obtain an intermediate velocity field that is corrected to divergence-free conditions using the solution of a Poisson equation for the pressure The transport equations are solved only in the fluid cells; the solid cells are not considered and the interface values are obtained through interpolation enforcing boundary conditions at the geometry walls Note, that the pressure equation is solved in all cells, avoiding the need to specify pressure wall boundary conditions The treament of the interface cells has been described elsewhere8-11 and will not

be repeated here

LOCAL GRID REFINEMENT ALGORITHM

Local Grid Refinement (LGR) allows an efficient clustering of cells in the vicinity of the immersed boundary The present implementation is an

extension of the classical adaptive mesh

refinement (AMR) technique for non-isotropic refinement It can also be interpreted as a generalization of the procedure used for building

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coarse grids for geometric multigrid on structured

meshes

The basic idea was introduced in Durbin &

Iaccarino4 for a finite difference discretization The

LGR grid is considered as a coarsened version of

a fine, structured grid; on this underlying grid the

cells are defined as usual by a couple of vertices

with indices (i,j) and (i+1,j+1) (the following

discussion of the algorithm is for two-dimensions

although the extension to three-dimensions is

straightforward)

On the LGR grid each element is bounded by the

grid lines passing through the vertices (i,j) and

(i+)i,j+)j) The effective element size in this case

is not constant ()i*)j and both indices depend on

(i,j)) Therefore, the cells are not organized in a

structured way with one-to-one neighbors in each

Cartesian direction This requires a modification of

the algorithm to deal with hanging nodes In

Durbin & Iaccarino4 this was simply based on a

second-order interpolation In the current finite

volume implementation, flux conservation is

enforced and the algorithm resembles the

unstructured face-based algorithm described in

Ferziger & Peric6 and, more closely, the AMR

discretization used in Ham et al.3

In contrast to equation 1 the RANS equations for

the LGR algorithm reduce to a linear system of the

form:

a(i,j)n!"

(i,j)n+1++aWn!"

Wn+1++aEn!"

En+1++aNn!"

Nn+1 ++aSn!"

Sn+1=S(i,j)n (2) where the sum is over the neighboring cells, e.g

+aWn!"

Wn+1=aW1n!"

W1n+1+aW2n!"

W2n+1 for the cell distribution in Figure 2

Each face-flux is computed using the two adjacent

cells and for each cell the fluxes (in general more

than two) are collected to build the corresponding

diffusive and convective operators The implicit

discretization yields a sparse matrix with elements

not organized in five diagonals as for its structured

counterpart This complexity in the matrix structure

prevents the use of the SIP procedure as it was

employed in [1] A Krilov-type algorithm with a

simple Jacobi pre-conditioner was implemented

Standard conjugate gradient is used for the

pressure equation and the BiCGStab12 for the

momentum and turbulent scalars

An algebraic multigrid technique available in the

Livermore Hypre library13

(High Performance Preconditioners) has also been used; in two

dimensions this yield a substantial advantage with respect to the conjugate gradient solver In three dimensions the advantage was only marginal Additional work is required to fully evaluate the efficiency of the algebraic multigrid for this class of problem Therefore, only results obtained using the conjugate gradient are presented in the following

Figure 2 Data management for hanging nodes: (a)

LGR grid showing a cell P and its neighbors (b) cell identification array, ID, on the fine underlying grid showing one-to-one connectivity (c) mapping of the boundary of a physical cell based on the structured lines

of the underlying grid

The major advantage of the present LGR

approach with respect to classical

OCTREE-based2 and fully-unstructured3 algorithms lies in

the economy and flexibility of storing and retrieving

connectivity information due to the presence of the

underlying grid In particular, only N cells are

effectively defined on a NixNj underlying grid and

they are defined by the two couples (i,j) and

(i+)i,j+)j), Fig 2c The total storage cost is 4N

integers In addition, an array of integers, ID(i,j), is

defined on the fine grid to store the

correspondence between the underlying cell and

the actual LGR element, Fig 2b In other words,

all the underlying cells included in the range [i:i+)i

-1] and [j:j+)j-1] are tagged using the LGR cell

number The total storage required is, therefore,

NixNj The connectivity information for each cell

are retrieved consistently to a structured

framework by indirectly querying the array ID(i,j)

The neighbors of an LGR cell are ID(i-1,k) and

ID(i+1,k) for k ranging in [j: j+)j-1] in the positive and

negative i-direction respectively It is evident that

the approach handles multiple hanging nodes for

each cell and eventually, allows the reconstruction

of additional connectivity information without any

increase in storage; for example it is

straightforward to identify all the vertex-based

neighbors

As an example of the effectiveness of the present

LGR algorithm, the solution of the RANS

equations for a three-dimensional problem

(discussed later in more details) is considered

Four levels of grid refinements are used and a

total of about one million cells are considered The

OCTREE and fully implicit approaches are

compared to the present LGR in Fig 3 in terms of

memory requirement and CPU necessary to build

the systems (note that the solution is based for all

the approaches on the same conjugate gradient

algorithm) It must also be mentioned that the

present numerical scheme requires the use of

gradients for all variables at the cell centers to

build a second order discretization of the fluxes3

The comparison in Fig 3 shows that in terms of

memory the present algorithm is still substantially

more expensive (three-fold increase) than a

corresponding structured grid scheme with the

same number of cells On the other hand it only

requires a portion of the memory used by the

OCTREE and fully implicit algorithm In terms of

CPU the advantage is also considerable (note that

the solution time for the linear systems is not

included in Fig 3)

Figure 3 Comparison of the present LGR technique

with the OCTREE and the fully unstructured algorithm for a two-dimensional problem.

REFINEMENT CRITERIA

The generation of LGR grids is carried out by creating the underlying (fine) grid as discussed in details in [1] and coarsening it in the regions away from the immersed boundary The advantage of this approach is that all the cell tagging (ray tracing) can be performed on a structured grid taking full advantage of the alignment of the cell centers and the grid nodes The coarsening and the generation of the connectivity information is the last step of the grid generation process

Another important aspect of the application of the LGR is the selection of the refinement/coarsening criteria In the present implementation LGR is used to increase the resolution in the surroundings

of the immersed boundary and, therefore, the only criteria used is the geometrical distance between each cell and the boundary itself A Heavyside tag function (generated using the ray tracing technique as mentioned before) is used to mark the cells inside and outside the immersed body; The cells are tagged as “fluid” or “solid” if the cell center is outside or inside the immersed boundary, respectively (interface elements are not important

at this stage and, therefore, a cell is tagged by considering only the position of its cell center) An integer value +/-1 is assigned to each cell The gradient of this function is non-zero only at the immersed boundary and it is dependent on the local grid size This gradient is used to select the cells to be refined8 The grid is refined until a user specified resolution is achieved at the boundary

A smoothing function can be applied on the +/-1 tagging function to obtain a smeared interface that

4

will allow a smoother transition between coarse

and refined regions

TEST CASES

Two test problems are considered: a

two-dimensional airfoil and a three-two-dimensional

pick-up truck

The first test case is the flow around an airfoil in

close vicinity to the ground The domain size is

4CxC with C the chord of the airfoil The airfoil is

located at 1C from the inlet and 0.1C from the

ground The airfoil geometry is reported in Fig 4;

three V-shaped grooves are present on the lower

surface to increase the geometry complexity and

therefore the grid complexity

Figure 4 LGR grid for a NACA airfoil with V-

grooves

The locally refined grid is reported in Fig 4

Calculations have also been carried out on a

structured grid which is the underlying fine grid

used as a starting point for the coarsening

procedure that eventually yield the LGR grid In

other words, the LGR grid represents a subset of

the structured grid; only about 10% of the cells are

effectively used in the LGR computation

Calculations have been carried out for both grids

to evaluate the accuracy and speed of the LGR

model The results are presented in terms of

pressure distributions on the airfoil in Fig 5, for a

Reynolds number based on the chord of 300,000

The pressure distributions are in remarkable

agreement as the turbulent kinetic energy,

reported in Fig 6 In Fig 7 a comparison of the

convergence histories is presented to demonstrate

the considerable speed of the LGR solver as

opposed to the algorithm applied to the underlying

structured grid It must be noted that due to the

increased computational cost of the LGR (cfn Fig

3) the effective savings in a calculation are in the

order of 40-50%

Figure 5 Pressure distribution on the airfoil surface

Figure 6 Turbulent kinetic energy distribution for the

structured (top) and the LGR (bottom) grid

Figure 7 Convergence history for the airfoil

simulations

The second problem considered is flow around the pick-up truck geometry, presented in [1] The computational domain considered corresponds to the wind tunnel test chamber used in the experiments14 The size of the domain is 12Lx1.05Lx1.25L where L is the length of the model The Reynolds numbers, based on L, is 288,000 and inflow conditions are specified as

constant velocity with a low level of turbulent

intensity (~1%)

The baseline computational grid consists of

354x164x70 cells and this corresponds to the

results presented in [1]; it is used herein as a

reference A new grid with local grid refinement

has been used; it consists of about 3 million cells

(the underlying grid corresponds to

564x256x156=~22million cells) The grids in the

symmetry plane are shown in Fig 8.

Figure 8 Computational LGR (top) and structured

(bottom) grid in the symmetry plane for the GM pick-up

model

PIV measurements are available for this problem

from [14] in a plane behind the cabin, as illustrated

in Fig 9 The comparison between the

experimental data and the IBRANS simulations

carried out the structured and the LGR grid are

reported in Fig 10

Figure 9 Location of the PIV measurement sheet for

the velocity profiles in Figure 11

The agreement is satisfactory, showing that the structured grid captures all the qualitative details

of the complex three-dimensional separated flow1

In particular, both calculations show the strong downwash at the tail of the pick-up illustrated by the experiments

(a)

(b)

(c)

Figure 10 Streamlines in the symmetry plane for the

pick-up; (a) experiments, (b) LGR grid (c) structured grid

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A more quantitative comparison is presented in

Fig 11 where the velocity profiles in the wake

regions are compared The agreement is again

satisfactory but it shows that the LGR grid

captures the position of the shear layer (velocity

gradient) more accurately than the structured grid

This is a direct result of the increased resolution in the immediate vicinity of the immersed boundary that allows for a better description of the shear layer detaching from the top of the cabin

Figure 11 Comparisons of velocity component in x-direction obtained using the LGR grid (solid lines) and the

structured grid (dashed lines) PIV measurements are represented by symbols

Figure 12 Comparisons of surface pressure distributions on the symmetry plane

It is worth noting that the worst agreement is

observed at the tail gate where the simulations

overestimate the velocity (profile x/L=0.98 and

x/L=1.04) This is probably due to unsteady effects

or to inaccuracy of the turbulence model

employed Unsteady simulations are currently

ongoing to evaluate precisely this effect

Finally in Fig 12 the pressure distribution on the

symmetry plane of the pick-up is presented and

compared to the measurements The same

comparison, reported in [1] provided an

encouraging overall agreement but a substantial

limitation in the accuracy of representing the

pressure peaks These are located at the base

and the top of the windshield where the boundary

layer are extremely thin The structured grid

captures substantially smoothed peaks with

15-20% error The LGR grid on the other hand

provides a better resolution in the vicinity of the

immersed boundary and therefore improves

dramatically the agreement with the experiments

CONCLUSIONS

A local grid refinement technique based on a novel

quasi-structured approach has been developed

and applied to the simulation of a two- and a

three-dimensional problem

The technique is based on the use of an

underlying, structured grid to build the connectivity

information and the actual computational grid with

hanging nodes A fully conservative second-order

discretization is employed in the context of a

RANS solver The immersed boundary technique

is used to represent curved boundaries on

non-aligned Cartesian meshes

The LGR represents an useful extension of the

solver providing increased resolution in the close

vicinity of the immersed boundary without the

classical structured grids penalty of grid lines

propagating to the boundaries

The validation cases demonstrate that the current

algorithm is in satisfactory agreement with the

structured solver applied on the underlying grid

The savings in terms of CPU are about 40-50%

and the memory penalty is substantially smaller

than classical AMR approaches

REFERENCES

1

Kalitzin, G., & Iaccarino, G., “Towards an Immersed Boundary RANS Flow Solver”, AIAA Paper

2003-0770

2

Berger, M, & Aftosmis, M, “Aspects (and aspect ratios) of Cartesian Mesh Methods”, 16th Int Conf Of Numerical Methods in Fluid Dynamics, 1998

3Ham, F E., Lien, F S., & Strong, A B., “A Cartesian Grid Method with Transient Anisotropic Adaptation”,

J Comp Phys., V 179, pp 469-494, 2002

4

Durbin P.A., & Iaccarino, G “Adaptive Grid Refinement for Structured Grids”, J Comp Physics, Vol.128, pp.110-121, 2002

5O’Rourke, “Computational Geometry in C”, John

Wiley, 1998

6

Ferziger, J.H., & Peric, M “Computational Methods

for Fluid Dynamics”, Springer 2002

7

Wilcox, D.C “Turbulence Modeling for CFD”, DCW

Industries, 1993

8Iaccarino, G & Verzicco, R., “Immersed Boundary Technique for Turbulent Flow Simulations” to appear

in Applied Mech Review, 2003

9

Majumdar, S., Iaccarino, G & Durbin, P A., “RANS Solver with Adaptive Structured Boundary Non-Conforming Grids”, CTR Annual Briefs, 2001

10 Kalitzin, G., & Iaccarino, G., “Turbulence Modeling in an Immersed Boundary RANS Method”, CTR Annual Briefs, 2002

11

Kalitzin, G., & Iaccarino, G., “Toward Immersed Boundary Simulations of High Reynolds Number Flows”, CTR Annual Briefs, 2003

12 Van der Vorst, “BICGSTAB, A Fast and Smoothly Converging Variant of BICG for the Solution of Non-Symmetric Linear Systems”, SIAM J Numer Anal, V 5, pp 530-558, 1992

13 “HYPRE, High Performance Preconditioners,, Vsn 1.8.1b”, Center for Applied Scientific Computing, Lawrence Livermore National Laboratory,

2003

14

Bernal, L & Khalighi, B, Personal Communication

8