Simulation parameters

The mesh

We used the software Gambit to mesh our geometry. This mesh is cartesian and quasi uniform: it is a little bit concentrated in the middle (factor 0.95) where the gradients are stronger.

The mesh has the following characteristics:

Type: Rectangular channel
Length: 2 m
Height: 0.5 m
One velocity inlet in the left side
One outflow outlet in the right side
Wall on the upper side
Wall on the lower side

The following picture represents the mesh

Physical parameters

The following parameters were taken in the Fluent 5 database.

We took the following constants for the gravitation and surfacic tension: g = 9.81 m/s² and s = 0.0735 N/m. As we did not find the value of s for these two fluids, we decided to keep the standard value given by Fluent (water/air).

Upper fluid: designated by the indice 1, fuel oil liquid (C₁₉H₃₀), density: 960 kg/m³, cinematic viscosity: 5E-05 m^-2.s^-1.

Lower fluid: designated by the indice 2, water liquid (H₂O), density: 998.2 kg/m³, cinematic viscosity: 1E-06 m^-2.s^-1.

All this parameters are constants for every simulations. The velocities of the two fluids depend on the studied case. The two fluids have close densities to avoid numerical problems in the calculations (divergence).

Numerical parameters

This section describes the numerical parameters used for the simulations.

We performed unstationnary simulations with a constant timestep of 0.01s, for each case. This timestep allows a good precision and a good resolution for the different cases.
The convergence criteria were set to 0.001 for the residuals, the X-velocity and the Y-velocity.

All the simulations were realized with the laminar model, to have the possibility to see the development of the instabilities.

The multiphase model used is the VOF (Volume Of Fluid) model with the following options:

Scheme: geo-reconstruct,
Courant number: 0.25,
Surface tension 0.0735 N/m

We choosed to use the standard schemes proposed by Fluent5 for the discretization:

Pressure: standard
Momentum: first order upwind
Pressure velocity coupling: simple

Sinusoidal oscillator

The first simulations realized showed that the instability was unable to appear in the normal conditions (with different velocities of the two fluids). In order to force the appearance of the Kelvin-Helmholtz instability, we placed a vertical oscillator in the inlet.

The aim of this oscillator is to simulate a vertical beater which makes the interface between the fuel oil liquid and the water oscillate, with a given frequency and a given amplitude. With this beater, we were able to make a frequential study of the Kelvin Helmholtz instability.

This oscillator was introduced in the calculation by means of an User Defined Function, written in the C++ language adapted to Fluent 5. The input parameters are defined as constants in the beginning of the program. The source of this UDF is given below.

#include "udf.h"

/* INPUT ZONE **********************************************/
#define xmin 0.0 /* point bas */
#define xmax 0.5 /* point haut */
#define freq 5. /*frequence des oscillations */
#define amp 0.005 /* amplitude de la perturbation */
#define v1 3.0 /* vitesse fluide sup */
#define v2 1.0 /* vitesse fluide inf */
#define alpha1 1.0 /* alpha fluide inf */
#define alpha2 0.0 /* alpha fluide inf */
/***********************************************************/

/* INITIALISATION DES CONSTANTES ***************************/
#define PI 3.14159

/* PROFIL DE VITESSES **************************************/
DEFINE_PROFILE(batteur_velocity,
thread,
nv)
{
/* variables */
face_t f; /*face en cours*/
real flow_time = RP_Get_Real("flow-time");
real x[ND_ND];
real x0;

/* calcul de l'oscillation du point milieu */
x0=amp*sin(2.*PI*freq*flow_time)+0.5*(xmax + xmin);

/* affectation du profil */
begin_f_loop (f, thread)
{
F_CENTROID(x, f, thread);
if (x[1] >= x0)
{
F_PROFILE(f, thread, nv) = v1;
}
else
{
F_PROFILE(f, thread, nv) = v2;
}
}
end_f_loop (f,thread)
}

/***********************************************************/

/* PROFIL DES DENSITES *************************************/
DEFINE_PROFILE(batteur_densite,
thread,
nv)
{
/* variables */
face_t f; /*face en cours*/
real flow_time = RP_Get_Real("flow-time");
real x[ND_ND];
real x0;

/* calcul de l'oscillation du point milieu */
x0=amp*sin(2*PI*freq*flow_time)+0.5*(xmax + xmin);

/* affectation du profil */
begin_f_loop (f, thread)
{
F_CENTROID(x, f, thread);
if (x[1] >= x0)
{
F_PROFILE(f, thread, nv) = alpha1;
}
else
{
F_PROFILE(f, thread, nv) = alpha2;
}
}
end_f_loop (f,thread)
}

The result of this beater is the creation of an oscillation of the interface between the fluids, which is convected and deformed by the flow.