# A compressible two-phase flow solver

This solves the two-phase compressible Navier-Stokes equations including the total energy equation. \displaystyle \frac{\partial (f \rho_i)}{\partial t } + \nabla \cdot (f \rho_i \mathbf{u}) = 0 \displaystyle \frac{\partial (\rho_i \mathbf{u})}{\partial t } + \nabla \cdot ( \rho_i \mathbf{u} \mathbf{u}) = -\nabla p + \nabla \cdot \tau_i' \displaystyle \frac{\partial [\rho_i (e_i + \mathbf{u}^2/2)]}{\partial t } + \nabla \cdot [ \rho_i \mathbf{u} (e_i + \mathbf{u}^2/2)] = -\nabla \cdot (\mathbf{u} p_i) + \nabla \cdot \left( \tau'_i \mathbf{u} \right) an advection equation for the volume fraction f \displaystyle \frac{\partial f}{\partial t} + \mathbf{u} \cdot \nabla f = 0 and an Equation Of State (EOS) that needs to be defined.

The method is described in detail in Fuster & Popinet, 2018 and relies on a time splitting technique were we first solve the advection step using the VOF method for f, f_i \rho_i, f_i \rho_i \mathbf{u} f_i \rho_i e_{tot,i} and then perform the projection step using the all-Mach solver.

#include "all-mach.h"

We transport VOF tracers without the one-dimensional compressive term.

#define NO_1D_COMPRESSION 1
#include "vof.h"

The primary fields are: \displaystyle \begin{aligned} \text{frho1} & = f \rho_1, \\ \text{frho2} & = (1-f) \rho_2, \\ \text{q} & = f \rho_1 \mathbf{u} + (1-f) \rho_2 \mathbf{u}, \\ \text{fE1} & = f \rho_1 (e_1 + \mathbf{u}^2/2), \\ \text{fE2} & = (1-f) \rho_2 (e_2 + \mathbf{u}^2/2) \end{aligned}

scalar f[], * interfaces = {f};
scalar frho1[], frho2[], fE1[], fE2[];

The timestep can be limited by a CFL based on the speed of sound.

double CFLac = HUGE;

The dynamic viscosities for each phase, as well as the volumetric viscosity coefficients.

double mu1 = 0., mu2 = 0.;
double lambdav1 = 0., lambdav2 = 0. ;

These functions are provided by the Equation Of State. See for example the Mie–Gruneisen Equation of State.

extern double sound_speed          (Point point);
extern double average_pressure     (Point point);
extern double bulk_compressibility (Point point);
extern double internal_energy      (Point point, double fc);

By default the Harmonic mean is used to compute the phase-averaged dynamic viscosity.

#ifndef mu
# define mu(f) (mu1*mu2/(clamp(f,0,1)*(mu2 - mu1) + mu1))
#endif

The volumetric viscosity uses arithmetic average by default.

// fixme: Include the term depending on $\nabla \cdot u$. It is tricky because this
// quantity can be very different upon the phase
#ifndef lambdav
# define lambdav(f)  (clamp(f,0,1)*(lambdav1 - lambdav2) + lambdav2)
// # define lambdav(f)  (clamp(f,0.,1.)*(lambdav1 - lambdav2 - 2./3*(mu1 - mu2)) + lambdav2 - 2./3*mu2)
#endif

## Auxilliary fields

Auxilliary fields need to be allocated. The quantity \rho c^2, the average density rhov and its inverse alphav.

scalar rhoc2v[];
face vector alphav[];
scalar rhov[];
const face vector lambdav0[] = {0,0,0};
(const) face vector lambdav = lambdav0;

## Time step restriction based on the speed of sound

event stability (i++)
{
if (CFLac < HUGE)
foreach (reduction (min:dtmax)) {
double c = sound_speed (point);
if (CFLac*Delta < c*dtmax)
dtmax = CFLac*Delta/c;
}
}

#if TREE

## Energy refinement function

The energy is refined from the refined pressures, momentum and densities, using the equation of state.

void fE_refine (Point point, scalar fE)
{
foreach_child() {
double Ek = 0.;
foreach_dimension()
Ek += sq(q.x[]);
Ek /= 2.*(frho1[] + frho2[]);
fE[] = fE.inverse ?
(internal_energy (point, 0.) + Ek)*(1. - f[]) :
(internal_energy (point, 1.) + Ek)*f[];
}
}
#endif // TREE

## Initialisation

We set the default values.

event defaults (i = 0)
{
alpha = alphav;
rho = rhov;

If the viscosity is non-zero, we need to allocate the face-centered viscosity field.

  if (mu1 || mu2) {
mu = new face vector;
lambdav = new face vector;
}

rhoc2 = rhoc2v;
foreach () {
frho1[] = 1., fE1[] = 1.;
frho2[] = 0., fE2[] = 0.;
p[] = 1.;
f[] = 1.;
}

We also initialize the list of tracers to be advected with the VOF function f (or its complementary function).

  f.tracers = list_copy ({frho1, frho2, fE1, fE2});
for (scalar s in {frho2, fE2})
s.inverse = true;

We set limiting.

  for (scalar s in {frho1, frho2, fE1, fE2, q}) {
#if TREE

On trees, we ensure that limiting is also applied to prolongation and refinement.

    s.prolongation = s.refine = refine_linear;
#endif
}

We add the interface and the density to the default display.

  display ("draw_vof (c = 'f');");
display ("squares (color = 'rhov', spread = -1);");
}

event init (i = 0)
{
trash ({uf});
foreach_face()
uf.x[] = fm.x[]*(q.x[]/(frho1[] + frho2[]) + q.x[-1]/(frho1[-1] + frho2[-1]))/2.;

We update the fluid properties.

  event ("properties");

We set the initial timestep (this is useful only when restoring from a previous run).

  dtmax = DT;
event ("stability");

  if (f.gradient)
for (scalar s in {frho1, frho2, fE1, fE2, q})
}

We overload the vof() event to transport consistently the volume fraction and the momentum of each phase.

static scalar * interfaces1 = NULL;

event vof (i++)
{

We split the total momentum q into its two components q1 and q2 associated with f and 1 - f respectively.

  vector q1 = q, q2[];
foreach()
foreach_dimension() {
double u = q.x[]/(frho1[] + frho2[]);
q1.x[] = frho1[]*u;
q2.x[] = frho2[]*u;
}

Momentum q2 is associated with 1 - f, so we set the inverse attribute to true. We use the same limiting for q1 and q2.

  foreach_dimension() {
q2.x.inverse = true;
}

#if TREE

The refinement function is modified by vof_advection(). To be able to restore it, we store its value.

  void (* refine) (Point, scalar) = q1.x.refine;
#endif

We associate the transport of q1 and q2 with f and transport all fields consistently using the VOF scheme.

  scalar * tracers = f.tracers;
f.tracers = list_concat (tracers, (scalar *){q1, q2});
free (f.tracers);
f.tracers = tracers;

We recover the total momentum.

  foreach() {
foreach_dimension()
q.x[] = q1.x[] + q2.x[];

We avoid negative densities and energies which may have been caused by round-off during VOF advection.

    if (f[] <= 0.){
frho1[] = 0.;
fE1[] = 0.;
}
if (f[] >= 1.){
frho2[] = 0.;
fE2[] = 0.;
}
}

#if TREE

We restore the refinement function for the total momentum.

  for (scalar s in {q}) {
s.refine = s.prolongation = refine;
s.dirty = true;
}

#if 0

This is switched off by default for now as the standard refinement in /src/vof.h#vof_concentration_refine seems to work fine.

  for (scalar s in {fE1,fE2}) {
s.refine = s.prolongation = fE_refine;
s.dirty = true;
}
#endif
#endif // TREE

We set the list of interfaces to NULL so that the default vof() event does nothing (otherwise we would transport f twice).

  interfaces1 = interfaces, interfaces = NULL;
}

We set the list of interfaces back to its default value.

event tracer_advection (i++) {
interfaces = interfaces1;
}

## Pressure and density

During the projection step we compute the provisional pressure from the EOS using the values after advection.

event properties (i++)
{
foreach() {
rhov[] = frho1[] + frho2[];
ps[] = average_pressure (point);

We also compute \rho c^2.

    rhoc2v[] = bulk_compressibility (point);
}

foreach_face() {

If viscosity is present we obtain the averaged viscosity at the cell faces.

    if (mu1 || mu2) {
face vector muv = mu, lambdavv = lambdav;
double ff = (f[] + f[-1])/2.;
muv.x[] = fm.x[]*mu(ff);
lambdavv.x[] = lambdav(ff);
}

We also compute the averaged density at the cell faces.

    alphav.x[] = 2.*fm.x[]/(rhov[] + rhov[-1]);

}

The all-Mach solver needs rho*cm.

  foreach()
rhov[] *= cm[];

/* I still miss a source term in the divergence related to viscous
* dissipation. Usually it is small and a little bit complicated to
* calculate. In the current form of allmach it cannot be added
* because it does not allow user-defined divergence sources. */
}

## Update of the total energy

After projection we update the values of the total energy adding the term missing from the projection step.

For a Newtonian fluid, the stress tensor comprises the pressure term, the viscous uncompressible term and the viscous compressible term,

\displaystyle \tau = -p \mathbf{I} + \mu (\nabla \mathbf{u} + \nabla \mathbf{u}^T) + \lambda_v (\nabla \cdot \mathbf{u}) \mathbf{I}

where \mathbf{I} is the unity tensor and \lambda_v = \mu_v - 2/3 \mu. The volumetric viscosity \mu_v is negligible in most cases.

event end_timestep (i++)
{
#if 0

We first compute the divergence of the velocity at cell centers using the cell face velocity. Note that the face vector uf incorporates the metric factor.

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

We compute explicitly the contribution of the compressible viscous term to the momentum equation and the total energy equation, \displaystyle \partial_t \mathbf{q} = \nabla \cdot [\lambda_v(\nabla \cdot \mathbf{u}) \mathbf{I}] \displaystyle \partial_t [\rho (e + \mathbf{u}^2/2)] = \nabla \cdot [\lambda_v (\nabla \cdot \mathbf{u}) \mathbf{u}]

The axysimmetric case allows some simplifications. Since \displaystyle \mathbf{S} = \lambda_v(\nabla \cdot \mathbf{u}) \mathbf{I} = \left( \begin{array}{ccc} S_{xx} & 0 & 0 \ \ 0 & S_{yy} & 0 \ \ 0 & 0 & S_{\theta \theta} \end{array} \right) with S = S_{xx} = S_{yy} = S_{\theta \theta}= \lambda_v \nabla \cdot \mathbf{u}. Hence, the radial component of \nabla \cdot \mathbf{S} is simply \displaystyle (\nabla \cdot \mathbf{S}) \cdot \mathbf{e}_y = \frac{1}{y} \frac{\partial(y S_{yy}) }{\partial y} - \frac{S_{\theta \theta}}{y} = \frac{\partial S }{\partial y} Also u_\theta = 0.

  foreach() {
double momentum = 0., energy = 0.;
foreach_dimension() {
double right = lambdav.x[1]*(divU[1]  + divU[])/2.;
double left = lambdav.x[]*(divU[-1] + divU[])/2.;
momentum += right - left;
energy += uf.x[1]*right - uf.x[]*left;
}
foreach_dimension ()
q.x[] += dt*momentum/Delta;
energy *= dt/(Delta*cm[]);
double fc = clamp(f[],0,1);
fE1[] += energy*fc;
fE2[] += energy*(1. - fc);
}
#endif

The contribution of the pressure to the energy of each phase is lacking.

  {
face vector upf[];
foreach_face()
upf.x[] = - uf.x[]*(p[] + p[-1])/2.;

foreach () {
double energy = 0.;
foreach_dimension()
energy += upf.x[1] - upf.x[];
energy *= dt/(Delta*cm[]);
double fc = clamp(f[],0,1);
fE1[] += energy*fc;
fE2[] += energy*(1. - fc);
}
}

This is the contribution of the incompressible viscous term.

  if (mu1 || mu2) {

The velocity \mathbf{u} is first obtained from the updated momentum \mathbf{q}.

    vector u = q;
foreach()
foreach_dimension()
u.x[] = q.x[]/(frho1[] + frho2[]);

We add the incompressible contribution of the \nabla \cdot (u_i \tau'_i) term. Note that the compressible contribution, which is typically small, is neglected.

    // fixme: add the compressible contribution (careful, $\nabla \cdot // u$ can be very different upon the phase and also very different
// from the value defined for the mixture with an averaged velocity
face vector ueijk[];
foreach_dimension() {
foreach_face(x)
ueijk.x[] = 2.*uf.x[]*(u.x[] - u.x[-1])/Delta;
#if dimension > 1
foreach_face(y)
ueijk.y[] = uf.y[]*(u.x[] - u.x[0,-1] +
(u.y[1,-1] + u.y[1,0])/4. -
(u.y[-1,-1] + u.y[-1,0])/4.)/Delta;
#endif
#if dimension > 2
foreach_face(z)
ueijk.z[] = uf.z[]*(u.x[] - u.x[0,0,-1] +
(u.z[1,0,-1] + u.z[1,0,0])/4. -
(u.z[-1,0,-1] + u.z[-1,0,0])/4.)/Delta;
#endif
foreach () {
double energy = 0.;
foreach_dimension()
energy += ueijk.x[1] - ueijk.x[];
energy *= dt/(Delta*cm[]);
double fc = clamp(f[],0,1);
fE1[] += mu1*energy*fc;
fE2[] += mu2*energy*(1. - fc);
}

// fixme: Formally, we need to include a correction term due to
// the normal viscous stress jump (to be done)
}

Finally we recover momentum.

    foreach()
foreach_dimension()
q.x[] *= frho1[] + frho2[];
}
}

At the end of the simulation we clean the tracer list.

event cleanup (i = end) {
free (f.tracers);
f.tracers = NULL;
}

## References

 [fuster2018] Daniel Fuster and Stéphane Popinet. An all-Mach method for the simulation of bubble dynamics problems in the presence of surface tension. Journal of Computational Physics, 374:752–768, December 2018. [ DOI | http | .pdf ]