1
UNSTRUCTURED MESH SOLVER FOR THE CONVECTION DIFFUSION EQUATION
Zachary A. Sterba
Department of Mechanical Engineering
Purdue University
West Lafayette, Indiana
Email: [email protected]
Hongwei Li
Department of Mechanical Engineering
Purdue University
West Lafayette, Indiana
Email: [email protected]
ABSTRACT
The steadystate convectiondiffusion equation is
solved on a nonorthogonal unstructured mesh using the
upwind difference method and the finite volume scheme. A
Fortran90 code loads a twodimensional mesh of arbitrary
shape containing cells of any polygonal shape, or a mixture of
several polygonal shapes. Data structures used are chosen with
much
attention
given
to
memory
and
computational
efficiency. This mesh is then solved for a test case of uniform
flow, where Dirichlet boundary conditions are specified, and
the high Peclet number assumption is specified at the output. It
is then demonstrated that the solution corresponds to a
uniform mesh solution of the same problem. The solver is then
tested on a large mesh with irregular boundaries, and a
mixture of cell geometries to demonstrate its ability to handle
highly complicated input. The convergence of this large mesh
was unacceptably slow, demonstrating that any future work
will demand a faster and more sophisticated solution scheme.
NOMENCLATURE
a
p
,a
nb
matrix coefficient at a cell and its neighbor
A
f
face area vector
b
forcing term
b
gradient vector resulting from least squares fit
D
f
diffusion term
Dir
n
keeps track of whether the normal vector of face n
points into or out of the cell in question
e
ξ
unit vector in the cell centroid to cell centroid
direction
f
a cell’s face; all faces of a cell
F
f
convection term
G
gradient coefficient matrix
J
flux vector
h
synonymous with
Δ
ξ
i, j
unit vectors in the
x
 and
y
directions
nb
a cell’s neighbor; all neighbors of a cell
L
face “area” (i.e. length) vector in twodimensions
M
geometry matrix for least squares fit
S
source term
T
p
,
T
nb
function value (i.e. temperature) at a cell and its
neighbor
T
function value vector for use in least squares fit
V
velocity vector
V
b
boundary velocity vector
V
end n
the index of vertex at end n of a face where n is 1
or 2 — is useful in linking cells and faces
X, Y
centroid location
Δ
ξ
distance between cells
Δ
V
the volume of one cell
Г
diffusion coefficient
η
vertextovertex direction
ξ
cell centroid to cell centroid direction
ρ
density
ℑ
secondary gradient term
MOTIVATION
Finite volume models generally use one of two mesh
types; these types are characterized by cell connectivity.
Structured meshes have a regular, predictable connectivity
where all grid points have the same number of neighbors
(Figs. 1a, 1b). Unstructured meshes, however, allow a grid
point to have any number of neighbors (fig. 1c) (Owen).
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 Fall '10
 NA
 Mechanical Engineering, Partial Differential Equations, Vector Calculus, Partial differential equation, Vector field, Gradient, Boundary conditions

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