src/test/gaussian.c

    Transcritical flow over a bump with multiple layers

    This test case is similar to transcritical.c but using simpler (and more physical) boundary conditions. Both hydrostatic and non-hydrostatic solutions are obtained. The non-hydrostatic solution is compared with a DNS solution using the Navier-Stokes/VOF solver.

    More details are given in Popinet (2020).

    #include "grid/multigrid1D.h"
    #if !ML
    # include "saint-venant.h"
    #else // ML
    # include "layered/hydro.h"
    # define phi q
    # if !HYDRO
    #   include "layered/nh.h"
    # endif
    # include "layered/remap.h"
    # include "layered/perfs.h"
    #endif // ML

    The primary parameters are the flow rate, water level, viscosity and bump amplitude.

    #define QL 1.
    #define HR 0.6
    #define NU 1e-2
    #define BA 0.4
    
    int main()
    {

    The domain is 30 meters long.

      L0 = 30.;
      G = 9.81;
      N = 512; // less damping with 1024

    The viscosity is set to NU and 20 layers are used to discretise vertically.

      nu = NU;
      nl = 20; // going to 30 changes very little
    #if ML
    #if NOMETRIC
      max_slope = 0.;
    #endif
    #if !HYDRO  
      NITERMIN = 2;
    #endif
    #endif
      run();
    }

    Initialisation and boundary conditions

    The inflow is a parabolic profile with a total flow rate Q. The function below computes the height zc of the middle of the layer and returns the corresponding velocity.

    #if !ML
    double uleft (Point point, scalar s, double Q)
    {
      double zc = zb[];
      for (int l = 0; l < s.l; l++)
        zc += layer[l]*h[];
      zc += layer[s.l]*h[]/2.;
      return 3./2.*Q/h[]*(1. - sq(zc/h[] - 1.));
    }
    #else

    For the multilayer solver, we need some trickery to define the inflow velocity profile as a function of z. We need to sum the layer thicknesses to get the total depth H and the height zc of the current layer (of index point.l).

    double uleft (Point point, scalar s, double Q)
    {
      double H = 0.;
      double zc = zb[];
      for (int l = - point.l; l < nl - point.l; l++) {
        H += h[0,0,l];
        if (l < 0)
          zc += h[0,0,l];
      }
      zc += h[]/2.;
      return 3./2.*Q/H*(1. - sq(zc/H - 1.));
    }
    #endif

    We initialise the topography and the initial thickness of each layer h.

    event init (i = 0)
    {
      foreach() {
        zb[] = BA*exp(- sq(x - 10.)/5.);
    #if !ML
        h[] = HR - zb[];
    #else
        foreach_layer()
          h[] = (HR - zb[])/nl;
    #endif
      }

    The height of the free-surface \eta is imposed on the right boundary.

      eta[right] = dirichlet(HR);

    Boundary conditions are set for the inflow velocity and the outflow of each layer.

    #if !ML
      for (vector u in ul) {
        u.n[left] = dirichlet (uleft (point, _s, QL*(t < 10. ? t/10. : 1.)));
        u.n[right] = neumann(0.);
      }
      h[right] = dirichlet(HR);
    #else
      u.n[left] = dirichlet (uleft (point, _s, QL*(t < 10. ? t/10. : 1.)));
      u.n[right] = neumann(0);
    #endif

    In the non-hydrostatic case, a boundary condition is required for the non-hydrostatic pressure \phi of each layer.

    #if !HYDRO && ML
      phi[right] = dirichlet(0.);
    #endif
    }

    We can optionally add horizontal viscosity.

    #if 0
    event viscous_term (i++)
    {
      // add horizontal viscosity (small influence)
    #if HYDRO
      scalar * list = {u.x};
    #else
      scalar * list = {u.x,w};
    #endif
      scalar d2u[];
      foreach_layer()
        for (scalar s in list) {
          foreach()
    	d2u[] = (u.x[1] + u.x[-1] - 2.*u.x[])/sq(Delta);
          foreach()
    	u.x[] += dt*nu*d2u[];
        }
      boundary (list);
    }
    #endif

    We check for convergence.

    scalar etac[];
    
    event logfile (t += 0.1; i <= 100000) {
      FILE * fp = fopen ("dh", "w");
      foreach()
        fprintf (fp, "%g %g\n", x, eta[] - etac[]);
      fclose (fp);
      double dh = change (eta, etac);
      if (i > 1 && dh < 5e-5)
        return 1;
    }

    Uncomment this part if you want on-the-fly animation.

    #if 0
    #include "plot_layers.h"
    #endif

    Outputs

    We save profiles at regular intervals.

    event profiles (t += 5)
    {
      foreach (serial) {
    #if !ML
        double H = h[];
    #else
        double H = 0.;
        foreach_layer()
          H += h[];
    #endif
        fprintf (stderr, "%g %g %g\n", x, zb[] + H, zb[]);
      }
      fprintf (stderr, "\n\n");
    }

    For the hydrostatic case, we compute a diagnostic vertical velocity field w. Note that this needs to be done within this event because it relies on the fluxes hu and face heights hf, which are only defined temporarily in the multilayer solver.

    #if HYDRO
    scalar w = {-1};
    
    event update_eta (i++)
    {
      if (w.i < 0)
        w = new scalar[nl];
      vertical_velocity (w, hu, hf);

    The layer interface values are averaged at the center of each layer.

      foreach() {
        double wm = 0.;
        foreach_layer() {
          double w1 = w[];
          w[] = (w1 + wm)/2.;
          wm = w1;
        }
      }
      boundary ({w});
    }
    #endif // HYDRO

    We also save the velocity and non-hydrostatic pressure fields.

    event output (t = end)
    {
      foreach (serial) {
        double z = zb[];
    #if HYDRO
        printf ("%g %g %g %g\n", x, z, u.x[], w[]);
        foreach_layer() {
          z += h[];
          printf ("%g %g %g %g\n", x, z, u.x[], w[]);
        }
    #elif !ML
        vector u = ul[0];
        scalar w = wl[0];
        printf ("%g %g %g %g\n", x, z, u.x[], w[]);
        int l = 0;
        for (u,w in ul,wl) {
          z += layer[l++]*h[];
          printf ("%g %g %g %g\n", x, z, u.x[], w[]);
        }
    #else // ML
        printf ("%g %g %g %g %g\n", x, z, u.x[], w[], phi[]);
        foreach_layer() {
          z += h[];
          printf ("%g %g %g %g %g\n", x, z, u.x[], w[], phi[]);
        }
    #endif // ML
        printf ("\n");
      }
    #if HYDRO
      delete ({w});
    #endif
      printf ("# end = %g\n", t);
    }

    Results

    We compare the hydrostatic results obtained with the layered solver and those obtained with the Saint-Venant multilayer code.

    set yr [0.5:1.2]
    set xlabel 'x'
    set ylabel 'z'
    plot '../gaussian-hydro/log' u 1:2 w l t 'Multilayer', \
         '../gaussian-stvt/log' u 1:2 w l t 'De Vita et al.'
    Evolution of the free surface (hydrostatic). (script)

    Evolution of the free surface (hydrostatic). (script)

    The results which follow are obtained with the non-hydrostatic solver.

    plot 'log' u 1:2 w l t ''
    Evolution of the free surface (non-hydrostatic) (script)

    Evolution of the free surface (non-hydrostatic) (script)

    We compare the results obtained with the layered solver to those obtained with the Navier-Stokes VOF solver (and without metric).

    plot 'log' w l t 'Multilayer', \
         '../../examples/gaussian-ns/log' w l t 'Navier-Stokes VOF'
    Evolution of the free surface for both solvers (script)

    Evolution of the free surface for both solvers (script)

    plot 'log' index 11 w l t 'Multilayer', \
         '../../examples/gaussian-ns/log' index 'prof70' w l t 'Navier-Stokes VOF', \
         '../gaussian-nometric/log' index 12 w l t 'no metric'
    Final free surface profiles (script)

    Final free surface profiles (script)

    set term PNG enhanced font ",15" size 2048,512
    set yr [0:1.2]
    set output 'vel.png'
    set pm3d
    set pm3d map interpolate 10,4
    unset key
    # jet colormap
    set palette defined ( 0 0 0 0.5647, 0.125 0 0.05882 1, 0.25 0 0.5647 1,	\
    0.375 0.05882 1 0.9333, 0.5 0.5647 1 0.4392, 0.625 1 0.9333 0, 0.75 1 0.4392 0,	\
    0.875 0.9333 0 0, 1 0.498 0 0 )
    set contour base
    set cntrparam levels discrete 0
    set cntrlabel onecolor
    set cntrparam bspline
    splot 'out' u 1:2:3 lt 3 lc rgb "#ffffff" lw 2
    Horizontal velocity field (script)

    Horizontal velocity field (script)

    set term PNG enhanced font ",15" size 2048,512
    set output 'w.png'
    # set cbrange [-0.1:0.1]
    unset contour
    splot 'out' u 1:2:4
    Vertical velocity field (script)

    Vertical velocity field (script)

    set term PNG enhanced font ",15" size 2048,512
    set output 'phi.png'
    # set cbrange [-0.2:0.35]
    splot 'out' u 1:2:5
    Non-hydrostatic pressure (script)

    Non-hydrostatic pressure (script)

    See also