silo.c

# we solve a case of 2D silo with orifice at the bottom by using the poisson equation \displaystyle \nabla^2 c = 0 with a flux at the top of the silo This system can be solved with the poisson solver.

#include "grid/multigrid.h"
#include "poisson.h"
#include "utils.h"

Concentration source, and some obvious variables

scalar c[],s[];
double dx,dy,W;

Boundary conditions: a flux at the top of silo an orifice with size W at the bottom of the silo

c[top]    =  neumann(1);
c[bottom] =  fabs(x)<= W/2. ? dirichlet(0): neumann(0);
c[right]  =  neumann(0);
c[left]   =  neumann(0);

## Parameters The size of the domain L0. 0<x<L0 0<y<L0

int main() {
    L0 = 1.;
    X0 = -L0/2.;
    Y0 = -L0/2.;
    N =  64;
    dx = L0/N;
    dy = L0/N;
    W = 0.25;
    init_grid (N);

##Solve Laplace Initialisation of the solution of laplace equation with poisson \displaystyle \nabla^2 c = s with a source term s=0

    foreach()
    c[] = s[] = 0.;
    boundary ({c, s});
    poisson (c,s);

Save the results

numerical solution

    FILE *  fpc = fopen("Fcsilo.txt", "w");
    for (double x =-L0/2 ; x <= L0/2; x += dx){
        for (double y = -L0/2 ; y <= L0/2; y += dy){
            fprintf (fpc, "%g %g %g \n",
                     x, y,   interpolate (c, x, y));}
        fprintf (fpc,"\n");}
    fclose(fpc);
    fprintf(stdout," end\n");
    }

##Run Then compile and run:

_{bash qcc -O2 -Wall -o silo silo.c -lm; ./silo} ##Results

 set hidden3d
sp 'Fcsilo.txt'

3D plot of analycal solution and numerical solution (script)

 reset
 set pm3d map
 set size square
 set palette rgbformulae 22,13,-31
 sp 'Fcsilo.txt'

with colors (script)

##Bibliography

Version 1: 4 december 2015/ 14:58