# Momentum-conserving formulation for two-phase interfacial flows

The interface between the fluids is tracked with a Volume-Of-Fluid method. The volume fraction in fluid 1 is $f=1$ and $f=0$ in fluid 2. The densities and dynamic viscosities for fluid 1 and 2 are rho1, mu1, rho2, mu2, respectively.

``````#include "all-mach.h"
#include "vof.h"

scalar f[], * interfaces = {f};
double rho1 = 1., mu1 = 0., rho2 = 1., mu2 = 0.;``````

Auxilliary fields are necessary to define the (variable) specific volume $\alpha =1/\rho$ and average viscosity $\mu$ (on faces) as well as the cell-centered density.

``````face vector alphav[], muv[];
scalar rhov[];

event defaults (i = 0) {
α = alphav;
ρ = rhov;
μ = muv;
}``````

The density and viscosity are defined using arithmetic averages by default. The user can overload these definitions to use other types of averages (i.e. harmonic).

``````#ifndef ρ
# define rho(f) (clamp(f,0,1)*(rho1 - rho2) + rho2)
#endif
#ifndef μ
# define mu(f)  (clamp(f,0,1)*(mu1 - mu2) + mu2)
#endif

event properties (i++) {
// fixme: metric
foreach()
rhov[] = ρ(f[]);
boundary ({rhov});
foreach_face () {
alphav.x[] = 2./(rhov[] + rhov[-1]);
double ff = (f[] + f[-1])/2.;
muv.x[] = fm.x[]*μ(ff);
}
boundary ((scalar *){muv});
}``````

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[]/ρ(f[]);
q1.x[] = f[]*rho1*u;
q2.x[] = (1. - f[])*rho2*u;
}
boundary ((scalar *){q1,q2});``````

Momentum $q2$ is associated with $1-f$, so we set the inverse attribute to true. We use (strict) minmod slope limiting for all components.

``````  θ = 1.;
foreach_dimension() {
q2.x.inverse = true;
q1.x.gradient = q2.x.gradient = minmod2;
}``````

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

``````  f.tracers = (scalar *){q1,q2};
vof_advection ({f}, i);``````

We recover the total momentum.

``````  foreach()
foreach_dimension()
q.x[] = q1.x[] + q2.x[];
boundary ((scalar *){q});``````

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;
}``````