Solution of the potential flow though an orifice

    the problem

    solution of the “Basic problem” in 2D, potential incompressible flow \displaystyle u =\frac{\partial p}{\partial x}, \;\; v = \frac{\partial p}{\partial y},\;\;\; \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}=0 to be solved in the upper half space y>0 filled with a porous media so that velocity is 0 at infinity (but pressure is not zero to have a flow). At the bottom, there is a slit (-1<x<1, y=0) where pressure is constant: p=0. And there is no penetration for (|x|>1, y=0), so normal velocity is \frac{\partial p}{\partial y}=0

    Solution is proposed in Sneddon and in Lamb. It is x=cosh(p) cos(\psi) and y=sinh(p) sin (\psi)

    iso lines of pressure are the ellipses \displaystyle \frac{x^2}{cosh^2 p} + \frac{y^2}{sinh^2 p } =1


    #include "run.h"
    #include "poisson.h"
    #define MAXLEVEL 8
    scalar p[], source[];
    face vector beta[];
    mgstats mgp;

    far away, cosh(p)= sinh(p)= e^p/2, hence iso pressure are p \simeq log(2 \sqrt{x^2+y^2})

    Of course, this corresponds to the expected solution of a source far enough.

    as arcsinh(\sqrt{x^2+y^2}) \simeq log(2 \sqrt{x^2+y^2}) we write the approximate BC on the top right and left (and “mix” for fun).

    On the bottom, the mixed condition.

    p[right] = dirichlet(log(sqrt(4.*(x*x+y*y)))) ;  
    p[left]  = dirichlet(asinh(sqrt(x*x+y*y))) ;   
    p[top]   = dirichlet(log(sqrt(4.*(x*x+y*y)))) ;  
    p[bottom] = fabs(x)<= 1 ? dirichlet(0): neumann(0);

    domain is large

    int main()
        init_grid (1 << MAXLEVEL);

    coefficient of porosity is constant

    event init (i = 0) {
        foreach_face() {
            beta.x[] = 1; 

    no source

    event defaults (i = 0)
        p[] = source[] = 0.;
      boundary ({p});

    At every timestep, but after all the other events for this timestep have been processed (the ‘last’ keyword), we update the pressure field p by solving the Poisson equation with coefficient \beta.

    event pressure (i++, last)

    solve \nabla \cdot (\beta \nabla p )= s with

      mgp = poisson (p, source, beta);


    event logfile (i++)
        stats s = statsf (p);
        fprintf (stderr, "%d %g %d %g %g %g\n",
                 i, t, mgp.i, s.sum, s.min, s.max);

    Save in a file

    event sauve (i++,last)
    FILE *  fpc = fopen("pressure.txt", "w");
    output_field ({p}, fpc, linear = true);
    fprintf(stdout," end\n");


    Then compile and run:

     qcc -O2 -Wall -o darcyLambSneddon darcyLambSneddon.c -lm

    or better

     make darcyLambSneddon.tst;make darcyLambSneddon/plots; make darcyLambSneddon.c.html



    set pm3d map
    set palette rgbformulae 22,13,-31;
    unset colorbox
    set xlabel "x  iso p"
     set size (.5*1.6),1
    splot [][:] 'pressure.txt' u 1:2:3   not
    pressure (script)

    pressure (script)

    Figure to compare with Lamb 1932 Art 66 p 73 showing the ellipses of iso pressure, the stream-lines are

    set view map
    set size (.5*1.6),1
    unset key
    unset surface
     set contour base
     set cntrparam levels incremental 0,.1,log(2*L0)
     splot [][:] 'pressure.txt' u 1:2:3  w l not
    pressure (script)

    pressure (script)

    Plot along x=0 showing that analytical solution p(0,y) = arcsinh(y) is close to the log far away (source solution).

    set key bottom
      set logscale x
     plot [:][:] 'pressure.txt' u 2:(abs($1)<.01?($3):NaN)  t'num.',asinh(x),acosh(x),log(2*x)
    pressure (script)

    pressure (script)