Incompressible Navier–Stokes solver with Ghost Fluid Method for surface tension (centered formulation)

This extension of the centered.h Navier–Stokes equations solver implements the GFM according to the VOF-based model proposed by Vukvcevic et al. 2017.

We solve the incompressible Navier–Stokes equations:

\displaystyle \partial_t\mathbf{u}+\nabla\cdot(\mathbf{u}\otimes\mathbf{u}) = \frac{1}{\rho}\left[-\nabla p + \nabla\cdot(2\mu\mathbf{D})\right] + \mathbf{a} \displaystyle \nabla\cdot\mathbf{u} = 0 with the deformation tensor \mathbf{D}=[\nabla\mathbf{u} + (\nabla\mathbf{u})^T]/2.

The difference with respect to the classic centered formulation is that the surface tension is not added as an acceleration term in the momentum equation, but it is directly included in the discretization of the Poisson equation at the projection step.

Ghost Fluid Method Properties

We declare a structure containing properties which are useful for the GFM, such as the surface tension coefficient sigma, the volume fraction f, and possibly gravity G, which can be accounted for in the pressure jump of the GFM formulation. The gravity G must be provided multiplied by the density jump, due to the order of inclusions of the headers: \displaystyle \mathbf{G} = \mathbf{g}[\rho] = \mathbf{g}(\rho_1 - \rho_2) A constant curvature can be imposed simply providing a value for the attribute kappa of the following structure.

struct GhostFluidMethod {
scalar f;
double sigma;
double kappa;
coord G;
};

struct GhostFluidMethod gfm = {
.sigma = 0.,
.kappa = 0.,
.G = {0.,0.,0.},
};

Curvature Calculation

We compute the curvature using the height–functions, just like in curvature.h, but we need to overwrite the function interfacial(), because we want to compute the curvature on a wider stencil, according to the different definition of interfacial cell used here ad adapted from Vukvcevic et al. 2017.

#include "fractions.h"
#include "curvature.h"
#include "poisson-gfm.h"

scalar kappag[];

static inline int interfacial_ghost (Point point, scalar c) {
int res = 0;
foreach_dimension() {
if (intface.x[1] || intface.x[])
res++;
}
return res;
}

trace
cstats curvature_ghost (scalar c, scalar kappa,
double sigma = 1.[0], bool add = false)
{
int sh = 0, sf = 0, sa = 0, sc = 0;

On trees we set the prolongation and restriction functions for the curvature.

#if TREE
kappa.refine = kappa.prolongation = curvature_prolongation;
kappa.restriction = curvature_restriction;
#endif

#if dimension > 1

vector ch = c.height, h = automatic (ch);
if (!ch.x.i)
heights (c, h);

We first compute a temporary curvature k: a “clone” of \kappa.

  scalar k[];
scalar_clone (k, kappa);

foreach(reduction(+:sh) reduction(+:sf)) {

If we are not in an interfacial cell, we set \kappa to nodata.

    if (!interfacial_ghost (point, c))
k[] = nodata;

Otherwise we try the standard HF curvature calculation first, and the “mixed heights” HF curvature second.

    else if ((k[] = height_curvature (point, c, h)) != nodata)
sh++;
else if ((k[] = height_curvature_fit (point, c, h)) != nodata)
sf++;
}

foreach (reduction(+:sa) reduction(+:sc)) {

We then construct the final curvature field using either the computed temporary curvature…

    double kf;
if (k[] < nodata)
kf = k[];
else if (interfacial_ghost (point, c)) {

…or the average of the curvatures in the 3^{d} neighborhood of interfacial cells.

      double sk = 0., a = 0.;
foreach_neighbor(1)
if (k[] < nodata)
sk += k[], a++;
if (a > 0.)
kf = sk/a, sa++;
else

Empty neighborhood: we try centroids as a last resort.

  kf = centroids_curvature_fit (point, c), sc++;
}
else
kf = nodata;

We add or set kappa.

    if (kf == nodata)
kappa[] = nodata;
kappa[] += sigma*kf;
else
kappa[] = sigma*kf;
}

#else // dimension == 1
foreach() {
if (!interfacial_ghost (point, c))
kappa[] = nodata;
else {
double r = x + sign(c[-1] - c[1])*(clamp(c[],0.,1.) - 0.5)*Delta;
double p = r > 0. ? - 2.*sigma/r : 0.;
kappa[] += p;
else
kappa[] = p;
}
}
#endif // dimension == 1

return (cstats){sh, sf, sa, sc};
}

Ghost Fluid Method Discretization

According to Liu et al. 2000, the final form of the GFM discretization is similar to the standard 2nd order discretization of the Poisson equation. However, we have to modify the source terms according to the pressure jump at the interface:

\displaystyle \left[p\right]_\Gamma = \sigma\kappa

We consider that the jump of pressure gradient at the interface is null:

\displaystyle \left[\partial_n p\right] = 0

Therefore, the final discretization of the pressure equation in 1D reads:

\displaystyle \left[\beta_{i+1/2}\left(\dfrac{p_{i+1} - p_{i}}{\Delta}\right) - \beta_{i-1/2}\left(\dfrac{p_{i} - p_{i-1}}{\Delta}\right) \right]\dfrac{1}{\Delta} = f_i + \dfrac{\beta_{i-1/2}\left[p\right]_\Gamma}{\Delta^2} - \dfrac{\beta_{i+1/2}\left[p\right]_\Gamma}{\Delta^2}

Where the pressure jump \left[p\right]_\Gamma changes sign according to the side of the interface.

attribute {
double sigma;
}

vector ubackup[];
face vector pjump[];
face vector betac[];

trace
mgstats project_ghost (face vector uf, scalar p,
(const) face vector alpha = unityf,
double dt = 1.,
int nrelax = 4)
{

We recover the volume fraction field.

  scalar f = gfm.f;

We store the current centered velocity, this is required to avoid to make modifications to the centered solver.

  extern vector u;
foreach()
foreach_dimension()
ubackup.x[] = u.x[];

We compute interface properties.

  intfacefun (f);
intsidefun (f);
reldistfun (f);
absdistfun (f);

We compute the curvature field for the GFM: kappag.

  curvature_ghost (f, kappag, sigma = gfm.sigma);

if (gfm.kappa)
foreach()
kappag[] = gfm.kappa*gfm.sigma;

And we interpolate the curvature on the faces based on the relative position of the interface. Other interpolation methods can also be tested.

  face vector kappaf[];
foreach_face() {
if (intface.x[])
kappaf.x[] = reldist.x[]*kappag[] + (1. - reldist.x[])*kappag[-1];
else
kappaf.x[] = 0.;
}

We compute the pressure jump on the faces.

  foreach_face()
pjump.x[] = kappaf.x[];

We add the gravity contribution in the pressure jump. The term gfm.G.x must be multiplied by the density jump at the interface outside this module, in order to avoid problems with the variables rho1 and rho2 which are defined in two-phase.h and not always included.

  foreach_face() {
double Gdist = 0.;
foreach_dimension()
Gdist += gfm.G.x * absdist.x[];
pjump.x[] -= fm.x[]*Gdist;
}

The diffusion coefficient \beta=1/\rho is computed like the field alpha using an arithmetic average based on the volume fraction. This is different with respect to the approach described by Vukvcevic et al. 2017 but it seems to work better here.

  foreach_face()
betac.x[] = alpha.x[];

We allocate a local scalar field and compute the divergence of \mathbf{u}_f. The divergence is scaled by dt so that the pressure has the correct dimension.

  scalar div[];
foreach() {
div[] = 0.;
foreach_dimension()
div[] += uf.x[1] - uf.x[];
div[] /= dt*Delta;
}

We solve the Poisson problem. The tolerance (set with TOLERANCE) is the maximum relative change in volume of a cell (due to the divergence of the flow) during one timestep i.e. the non-dimensional quantity \displaystyle |\nabla\cdot\mathbf{u}_f|\Delta t Given the scaling of the divergence above, this gives

  mgstats mgp = poisson_ghost (p, div, f = f, ajump = pjump, alpha = betac,
tolerance = TOLERANCE/sq(dt), nrelax = nrelax);

We recover the face velocity values from the pressure gradient and the pressure jump.

  foreach_face() {
if (intface.x[]) {
if (intside.x[] > 0.)
uf.x[] -= dt*betac.x[]*(p[] - p[-1] + pjump.x[])/Delta;
else
uf.x[] -= dt*betac.x[]*(p[] - p[-1] - pjump.x[])/Delta;
}
else
uf.x[] -= dt*betac.x[]*(p[] - p[-1])/Delta;
}

return mgp;
}

In order to avoid code duplication, we overwrite the projection function in the centered solver, using the new projection defined here which exploits the GFM.

#define project(...) project_ghost(__VA_ARGS__)
#include "navier-stokes/centered.h"
#undef project

The stability condition is copied from tension.h.

event stability (i++)
{

We first compute the minimum and maximum values of \alpha/f_m = 1/\rho, as well as \Delta_{min}.

  double amin = HUGE, amax = -HUGE, dmin = HUGE;
foreach_face (reduction(min:amin) reduction(max:amax) reduction(min:dmin))
if (fm.x[] > 0.) {
if (alpha.x[]/fm.x[] > amax) amax = alpha.x[]/fm.x[];
if (alpha.x[]/fm.x[] < amin) amin = alpha.x[]/fm.x[];
if (Delta < dmin) dmin = Delta;
}
double rhom = (1./amin + 1./amax)/2.;

The maximum timestep is set using the sum of surface tension coefficients.

  double sigma = 0.;
//for (scalar c in interfaces)
//sigma += c.sigma;
sigma += gfm.sigma;
if (sigma) {
double dt = sqrt (rhom*cube(dmin)/(pi*sigma));
if (dt < dtmax)
dtmax = dt;
}
}

At the end of the time step we reconstruct the centered velocity field using thhe pressure gradient and the interface pressure jump.

event end_timestep (i++, last) {

We first compute a face field \mathbf{g}_f combining both acceleration and pressure gradient.

  face vector gf[];
foreach_face() {
if (intface.x[]) {
if (intside.x[] > 0.)
gf.x[] = fm.x[]*a.x[] - betac.x[]*(p[] - p[-1] + pjump.x[])/Delta;
else
gf.x[] = fm.x[]*a.x[] - betac.x[]*(p[] - p[-1] - pjump.x[])/Delta;
}
else
gf.x[] = fm.x[]*a.x[] - betac.x[]*(p[] - p[-1])/Delta;
}

We average these face values to obtain the centered, combined acceleration and pressure gradient field.

  trash ({g});
foreach()
foreach_dimension()
g.x[] = (gf.x[] + gf.x[1])/(fm.x[] + fm.x[1] + SEPS);

foreach()
foreach_dimension()
u.x[] = ubackup.x[] + dt*g.x[];
}

Notes and Improvements

1. The adaptive grid works only if the interfacial cells used by the GFM are never unrefined.

2. Reduce code duplication

3. Increasing the density ratio, a static circular droplet shows unphysical displacements after some simulation time. This problem disappears if the curvature is constant, and it seems to happen also with tension.h or with integral.h, therefore it is probably a known issue.

References

 [vukvcevic2017implementation] Vuko Vukčević, Hrvoje Jasak, and Inno Gatin. Implementation of the ghost fluid method for free surface flows in polyhedral finite volume framework. Computers & fluids, 153:1–19, 2017. [liu2000boundary] Xu-Dong Liu, Ronald P Fedkiw, and Myungjoo Kang. A boundary condition capturing method for poisson’s equation on irregular domains. Journal of computational Physics, 160(1):151–178, 2000.