LOCAL GRID REFINEMENT FOR AN IMMERSED BOUNDARY RANS SOLVER

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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

42
nd
Aerospace Sciences Meeting & Exhibit
5-8 January 2004 / Reno, NV

42nd AIAA Aerospace Sciences Meeting and Exhibit
5 - 8 January 2004, Reno, Nevada
AIAA 2004-586
Copyright © 2004 by . Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
AIAA-2004-0586

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 approach
2
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 approach
3
, 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.

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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 system
1
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 technique
5
.
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
+a
W
n
!

"
W
n+1
+a
E
n
!
"
E
n+1
+a
N
n
!
"
N
n+1
+a
S
n
!
"
S
n+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
model
1
. This is a modified version of the Wilcox k-
# model
7
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 conditions
11
.
A SIMPLE procedure
6
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 elsewhere
8-11
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 &
Iaccarino
4
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 & Iaccarino
4
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 & Peric
6
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
++a
W
n
!
"
W
n+1
++a
E
n
!
"
E
n+1
++a
N
n
!

"
N
n+1

++a
S
n
!
"
S
n+1
=S
(i,j)
n
(2)
where the sum is over the neighboring cells, e.g.
+a
W
n
!
"
W
n+1
=a
W1
n
!
"
W1
n+1

+a
W2
n
!
"
W2
n+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 BiCGStab
12
for the
momentum and turbulent scalars.
An algebraic multigrid technique available in the
Livermore Hypre library
13
(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.



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The major advantage of the present LGR
approach with respect to classical OCTREE-
based
2
and fully-unstructured
3
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 N
i
xN
j
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,
N
i
xN
j
. 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 fluxes
3
.
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 refined
8
. 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

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will allow a smoother transition between coarse
and refined regions.


TEST CASES
Two test problems are considered: a two-
dimensional airfoil and a three-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
experiments
14
. 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

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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 flow
1
.
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

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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”, 16
th
Int. Conf. Of

Numerical Methods in Fluid Dynamics, 1998.
3
Ham, 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.
5
O’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
8
Iaccarino, 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



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