# src/momentum.h

# 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) {
alpha = alphav;
rho = rhov;
mu = muv;
```

We use (strict) minmod slope limiting for all components.

```
theta = 1.;
foreach_dimension()
q.x.gradient = minmod2;
}
```

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 rho
# define rho(f) (clamp(f,0,1)*(rho1 - rho2) + rho2)
#endif
#ifndef mu
# define mu(f) (clamp(f,0,1)*(mu1 - mu2) + mu2)
#endif
event properties (i++) {
foreach()
rhov[] = rho(f[])*cm[];
foreach_face () {
double ff = (f[] + f[-1])/2.;
alphav.x[] = fm.x[]/rho(ff);
muv.x[] = fm.x[]*mu(ff);
}
}
```

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

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[]/rho(f[]);
q1.x[] = f[]*rho1*u;
q2.x[] = (1. - f[])*rho2*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;
q2.x.gradient = q1.x.gradient;
}
#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});
vof_advection ({f}, i);
free (f.tracers);
f.tracers = tracers;
```

We recover the total momentum.

```
foreach()
foreach_dimension()
q.x[] = q1.x[] + q2.x[];
#if TREE
```

We restore the refinement function for the total momentum.

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