Next: Discretization of boundary conditions
Up: Numerical Method
Previous: Boundary conditions
 
Contents
Computational grid and spatial discretization
In [
1] only uniform grids were used.
Therefore we present the difference stencils used for the staggered grid.
A 2D example is shown in figure
![[*]](file:/usr/share/latex2html/icons/crossref.png)
.
Figure:
2D staggered mesh
|
For convenience we denote the coordinates of the mesh lines by
![$ x_0,...,x_M$](img73.png)
,
![$ y_0,...,y_N$](img74.png)
and
![$ z_0,...,z_K$](img75.png)
. These values completely define the computational grid. How to tell
navcalc these
values is explained in chapter
![[*]](file:/usr/share/latex2html/icons/crossref.png)
.
Here and in the following we denote by
cells the rectangular subdomains
![$ [x_i,x_{i+1}]\times[y_j,y_{j+1}]\times[z_k,z_{k+1}]$](img76.png)
.
The computational domain
![$ \Omega$](img60.png)
is a union of cells.
Velocity components and pressure values are defined on the nodes:
,
where
![$ i,j,k \in \mathbb{Z}$](img86.png)
.
The following stencils are used for the spatial discretization. We use the notation
![$\displaystyle \Delta x_i=x_i-x_{i-1}$](img87.png)
and
The values
![$ \Delta y_j$](img89.png)
,
![$ \Delta y_{j+1/2}$](img90.png)
,
![$ \Delta z_k$](img91.png)
,
![$ \Delta z_{k+1/2}$](img92.png)
are defined analogously.
To preserve the second order accuracy of the stencils a smooth distribution of
the grid spaces
![$ \Delta x_i$](img93.png)
,.. is required. To obtain such smooth grids use the
GridGen utility.
Diffusive terms:
The other diffusive terms are discretized in a similar fashion.
Stencils similar to the one for
![$ \frac{\partial^2 u}{\partial y^2}$](img98.png)
are used for
![$ \frac{\partial^2 \Theta}{\partial x^2}$](img99.png)
,
![$ \frac{\partial^2 \Theta}{\partial y^2}$](img100.png)
and
![$ \frac{\partial^2 \Theta}{\partial z^2}$](img101.png)
, where
![$ \Theta$](img102.png)
is either
![$ T$](img39.png)
or
![$ C$](img40.png)
.
Convective terms:
Five different discretizations of the convective terms are possible:
- Donor-Cell (hybrid-scheme) (1st/2nd order)
- Quadratic upwind interpolation for convective kinematics (QUICK) (2nd-Order)
- Hybrid-Linear Parabolic Arppoximation (HLPA) (2nd-Order)
- Sharp and Monotonic Algorithm for Realistic Transport (SMART) (2nd-Order)
- Variable-Order Non-Oscillatory Scheme (VONOS) (2nd/3rd-Order) (default)
To select one of these schemes, you have to set the appropriate
variable in the scene description file. How to do this is explained in
section
![[*]](file:/usr/share/latex2html/icons/crossref.png)
. By default the VONOS-scheme is used. In the following
the Donor-Cell scheme is briefly described. More details about the
other schemes can be found in [
3].
Second order convective terms:
Stencils similar to the one for
![$ \frac{\partial vu}{\partial y}$](img107.png)
are used for
the discretization of the convective terms in (
![[*]](file:/usr/share/latex2html/icons/crossref.png)
) and (
![[*]](file:/usr/share/latex2html/icons/crossref.png)
), e.g. we have
The unknown values, e.g.
![$ v_{i+1/2,j,k}$](img109.png)
are computed by linear interpolation.
First order upwind:
Stencils similar to the one for
![$ \frac{\partial vu}{\partial y}$](img107.png)
are used for
the upwind discretization of the convective terms in (
![[*]](file:/usr/share/latex2html/icons/crossref.png)
) and (
![[*]](file:/usr/share/latex2html/icons/crossref.png)
).
First and second order terms can be blended using a parameter
![$ \alpha$](img44.png)
by, e.g.
The blending parameter
![$ \alpha$](img44.png)
is user definable (see chapter
![[*]](file:/usr/share/latex2html/icons/crossref.png)
) and may be chosen
different for the equations of momentum and energy or transport of a scalar.
Laplacian for pressure:
We employ a conservative discretization which is simply the nested application of the centered difference for
the pressure gradient and the centered difference for the natural discretization of the divergence, e.g.
Poisson solvers:
For the solution of the linear equation arising from discretization of the
pressure poisson equation, the following numerical methods are implemented:
- Successive Overrelaxation (SOR)
- Symmetric SOR (forward/backward)
- Red-Black scheme
- 8-Color SOR
- 8-Color Symmetric SOR (fw/bw)
- BiCGStab
To select a method, select the corresponding option in the scene
description file(see section
on how to do this).
By default, the Poisson-equation is solved using the BiCGStab-method.
Next: Discretization of boundary conditions
Up: Numerical Method
Previous: Boundary conditions
 
Contents
Martin Engel
2004-03-15