Term Project
Computational Fluid DynamicsProf. S.SaidiSharif University of
Technology International Campus Kish Island
Students name: Aliakbar kabir Students ID.: 89250861
Winter 2012
1.
Definitions
First step of solving the problem is to describe the definitions
of the problem requirement, to have better understanding of the
problem. 1.1. Staggered Grid
The solution procedure for the transport of a general property
developed in without presenting pressure will of course be enlisted
to solve the momentum equations. Matters are, however, not
completely straightforward since there are problems associated with
the pressure source terms of the momentum equations that need
special treatment. The finite volume method starts, as always, with
the discretization of the flow domain and of the relevant transport
equations. First we need to decide where to store the velocities.
It seems logical to define these at the same locations as the
scalar variables such as pressure, temperature etc. However, if the
velocities and pressures are both defined at the nodes of an
ordinary control volume a highly non-uniform pressure field can act
like a uniform field in the discretized momentum equations. This
can be demonstrated with the simple two-dimensional situation shown
in Figure 1.1, where a uniform grid is used for simplicity.
Fig. 1.1 A 'checker-board' pressure fiel
Let us assume that we have somehow obtained a highly irregular
'checkerboard' pressure field with values as shown in Figure
1.1.1|Page
If the pressures at 'e' and 'w' are obtained by linear
interpolation the pressure gradient term dp/dx in the u-momentum
equation is given by
1.1.1
Similarly, the pressure gradient dp/dy for the v-momentum
equation is evaluated as:
1.1.2
The pressure at the central node (P) does not appear in (1.1.1)
and (1.1.2). Substituting the appropriate values from the
'checker-board' pressure field in Figure 1.1 into formulae
(1.1.1-1.1.2) we find that all the discretized gradients are zero
at all the nodal points even though the pressure field exhibits
spatial oscillations in both directions. As a result, this pressure
field would give the same (zero) momentum source in the discretized
equations as a uniform pressure field. This behavior is obviously
nonphysical. It is clear that, if the velocities are defined at the
scalar grid nodes, the influence of pressure is not properly
represented in the discretized momentum equations. A remedy for
this problem is to use a staggered grid for the velocity components
(Harlow and Welch, 1965). The idea is to evaluate scalar variables,
such as pressure, density, temperature etc., at ordinary nodal
points but to calculate velocity components on staggered grids
centered on the cell faces. The arrangement for a two-dimensional
flow calculation is shown in Figure 1.2.
2|Page
The scalar variables, including pressure, are stored at the
nodes marked (). The velocities are defined at the (scalar) cell
faces in between the nodes and are indicated by arrows. Horizontal
( ) arrows indicate the locations for uvelocities and vertical ( )
ones denote those for v-velocities. In addition to the E,W, N,S
notation Figure 1.2 also introduces a new system of notation based
on a numbering of grid lines and cell faces. For the moment we
continue to use the original E, W, N, S notation the uvelocities
are stored at scalar cell faces e and w and the v-velocities at
faces 'n' and 's'.
Fig. 1.2
We observe that the control volumes for u and v are different
from the scalar control volumes and different from each other. The
scalar control volumes are sometimes referred to as the pressure
control volumes because, as we shall see later, the discretized
continuity equation is turned into a pressure correction equation,
which is evaluated on scalar control volumes.3|Page
In the staggered grid arrangement, the pressure nodes coincide
with the cell faces of the u-control volume. The pressure gradient
term dp/dx is given by
1.1.3
Where , is the width of the u-control volume. Similarly dp/dy
for the v-control volume shown is given by
1.1.4
where
, is width of the v-control volume.
If we consider the 'checker-board' pressure field again,
substitution of the appropriate nodal pressure values into
equations (1.1.3) and (1.1.4) now yields very significant non-zero
pressure gradient terms. The staggering of the velocity avoids the
unrealistic behavior of the discretized momentum equation for
spatially oscillating pressures like the 'checker-board' field. A
further advantage of the staggered grid arrangement is that it
generates velocities at exactly the locations where they are
required for the scalar transport convection diffusion -
computations. Hence, no interpolation is needed to calculate
velocities at the scalar cell faces. 1.2. SIMPLE Family
Algorithm
The acronym SIMPLE stands for Semi-Implicit Method for
PressureLinked Equations. The algorithm was originally put forward
by Patankar and Spalding (1972) and is essentially a
guess-and-correct procedure for the calculation of pressure on the
staggered grid arrangement introduced above. The method is
illustrated by considering the two-dimensional laminar steady flow
equations in Cartesian co-ordinates. The SIMPLE algorithm gives a
method of calculating pressure and velocities. The method is
iterative and when other scalars are coupled to the momentum
equations, the calculation needs to be done sequentially.4|Page
1.3.
SIMPLER
The SIMPLER (SIMPLE Revised) algorithm of Patankar (1980) is an
improved version of SIMPLE. In this algorithm the discretized
continuity equation:
1.3.1
is used to derive a discretized equation for pressure, instead
of a pressure correction equation as in SIMPLE. Thus the
intermediate pressure field is obtained directly without the use of
a correction. Velocities are, however, still obtained through the
velocity corrections of SIMPLE.
1.3.2
1.3.3
1.3.4 1.3.5
1.3.6
The discretized momentum equations are re-arranged as:
5|Page
1.3.7
1.3.8
In the SIMPLER algorithm pseudo-velocities and are now defined
as follows:
1.3.9
1.3.10
Equations (1.3.7) and (1.3.8) can now be written as:1.3.11
1.3.12
The definition for d, introduced in (1.3.6), is applied in
(1.3.11-1.3.12). Substituting for and , from these equations into
the discretized continuity equation (1.3.1), using similar forms
for and results in:
1.3.13
Equation (1.3.13) may be re-arranged to give a discretized
pressure equation:6|Page
1.3.14
Note that the coefficients of equation (1.3.14) are the same as
those in the discretized pressure correction equation, with the
difference that the source term b is evaluated using the
pseudo-velocities. Subsequently, the discretized momentum
equations:
are solved using the pressure field obtained above. This yields
the velocity components u* and v*. The velocity correction
equations (1.3.2-1.3.5) are used in the SIMPLER algorithm to obtain
corrected velocities. Therefore, the p'-equation
must also be solved to obtain the pressure corrections needed
for the velocity corrections. The full sequence of operations is
described in Figure 1.3.1.
7|Page
Fig 1.3.1
8|Page
2.
Assessment of the hybrid differencing scheme
The hybrid difference scheme exploits the favorable properties
of the upwind and central differencing schemes. It switches to the
upwind differencing when the central differencing produces
inaccurate results at high Pe numbers. The scheme is fully
conservative and since the coefficients are always positive it is
unconditionally bounded. It satisfies the transportiveness
requirement by using an upwind formulation for large values of
Peclet number. The scheme produces physically realistic solutions
and is highly stable when compared with the higher order schemes.
Hybrid differencing has been widely used in various computational
fluid dynamics (CFD) procedures and has proved to be very useful
for predicting practical flows. The disadvantage is that the
accuracy in terms of Taylor series truncation error is only
firstorder.
2.1. diffusion
Hybrid differencing scheme for multi-dimensional convection-
The hybrid differencing scheme can easily be extended to two-
and three dimensional problems by repeated application of the
derivation in each new coordinate direction. The discretized
equation that covers all cases is given by:
with central coefficient
and the coefficients of this equation for the hybrid
differencing scheme are as follows:
9|Page
In the above expressions the values of F and D are calculated
with the following formulae:
10 | P a g e
3. Results Part B: Cavity Problem
o Grid of cavity problem
11 | P a g e
o Residual-Iteration Plot
12 | P a g e
o Velocity and Pressure Contour
Pressure contour
U-component
13 | P a g e
V-component
Velocity contour with u-v Vectors
14 | P a g e
U-component with Stream lines Part C: Channel Problem o Pressure
Grid
15 | P a g e
o Residual-Iteration Plots
16 | P a g e
o Velocity and Pressure Contour
Pressure Contour
17 | P a g e
U-Component
V-Component
18 | P a g e
o Velocity Profiles along the Symmetry Line at Different
Levels
19 | P a g e
o Pressure Drop
Pressure Drop along X-axis
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4.
Code 4.1. Main code
// Kabir.cpp : Defines the entry point for the console
application. // #include #include #include #include #include
#include "stdafx.h" "Mesh.h" "discrete_func.h" "assemblage.h"
"Solver.h" "Results.h" mesh u_mesh,v_mesh,p_mesh,mesh; FILE *pp;
int Nx,Ny,i,j=0,k,n_repeat,n_p,n_v,n_run,project; double *p_prine;
double *u_prine; double *v_prine; double
L,H,Rx,Ry,U0,u,v,rho,vis,eror_u,eror_v,eror_p,alfa_p,alfa_u,max_u_pr=1,max_v_pr=1,b_prin1=1,ma
x_b_pr,r_b,u_prin1=1,v_prin1=1; int _tmain(int argc, _TCHAR*
argv[]) { pp=fopen("E:\Residuals.dat","wb"); if (pp==NULL) {
printf("Cannot open file.\n"); exit(1); } project=2; if
(project==1) { L=1; H=1; Nx=30; Ny=30; n_run=0; Rx=1; Ry=1; rho=23;
vis=1; U0=1; n_p=5; n_v=20; n_repeat=50000; alfa_p=.3; alfa_u=.6; }
else if(project==2) { L=5; H=1; Nx=60; Ny=15; n_run=0; Rx=1; Ry=1;
rho=1; vis=.01; U0=1.8; n_p=30; n_v=6; n_repeat=12000; alfa_p=.5;
alfa_u=1; } p_mesh=p_mesh.mesh_generate(L,H,Rx,Ry,Nx,Ny);
21 | P a g e
v_mesh=v_mesh.v_mesh_generate(p_mesh,Nx,Ny);
u_mesh=v_mesh.u_mesh_generate(p_mesh,Nx,Ny); p_prine=new
double[p_mesh.n_cv]; v_prine=new double[p_mesh.n_cv]; u_prine=new
double[p_mesh.n_cv];
fprintf(pp,"%s\t%s\t%s\t%s\t%s\t\n","variables=","\"repeat\"","\"continuity\"","\"uvelocity\"","\"v-velocity\"");
//initial guess// for (i=0;i
