# 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) {
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;
}
```