Rayleigh Taylor Instability And Mixing

OBJECTIVE :

To perform the Rayleigh Taylor instability simulation between two different fluid medium with different grid size .

Discuss the effect of Atwood number & viscosity by observing the behavior of instabilities .

ANSYS, SpaceClaim - To prepare FV model.

ANSYS, Meshing - To mesh the given model.

ANSYS, FLUNET - To setup the boundary conditions and to solve the given model.

ANSYS, CFD Post - To perform Post Processing.

Rayleigh–Taylor (R–T) instability :

The Rayleigh Taylor instability is an instability between the interface of two fluids of different densities which occurs when the lighter fluid is pushing the heavier fluid and both subjected to earths gravity. If a parcel of heavier fluid displaced downwards with an equal volume of lighter fluid upwards, the potential energy of the configuration lower than the intial state. Thus the disturbance will grow further and lead to further release of the potential energy, as the more dense fluid move downwards with gravitational field. This was setup and studied by lord Rayleigh and G.I. Taylor.

Examples of the Rayleigh-Taylor instability :

The Rayleigh Taylor instability can occur in diverse situations.

A.Natural Phenomena :

1. The formation of high luminosity twin-exhaust jets in rotating gas clouds in an external gravitational potential.
2. The overturn of the outer portion of the collapsed core of a massive star.
3. Mushroom clouds like those from the volcanic eruptions and the atmospheric nuclear reactions.

B.Technological Application :

1. Laser implosion of deuterium-tritium fusion targets.
2. Electromagnetic implosion of a metal liner .

Stages of development and eventual evolution into turbulent mixing :

Some practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves :

1.Richtmyer -Tayolr Instability
The Richtmyer instability occurs when two fluids with different viscosity is impulsively accelerated. These effects can be seen in shock waves where initially the shock wave is developed small and grows linearly with time.

2.Saffman -Taylor Instability
The Saffman -Taylor instability occurs when a morphologically unstable fluids interfere in a porous medium. This is also called as Viscous fingering. This type of instability can be seen in drainage when a low viscosity fluid is pushed inside a pipe where there is a fluid with higher viscosity. Most of the research using this model is done in the Hell-Shaw cells, Boundary integral methods and phase field methods.
3.Potential flow model:
This model was developed by Layzer and it provide info on the movement of fluids in a circular tube. A constant supporting pressure (Atwood number =1) is present, competition for both the Rayleigh Taylor and the Richtmyer-Meshkov instabilities.

4.Kelvin-Helmholtz Instability or Rayleigh Taylor Hybrid Model
This type of instability occurs when there is velocity shear in a continuous medium or when the velocity layer between two fluids are different. A classic example of this instability can be seen when atmospheric air is blow over a water. Also, this instability can be seen in sun's corona, clouds and Saturn’s bands.

5.Plateau-Rayleigh Instability
This instability model is used to explain how a falling water stream breaks into small droplets with same volume but with less surface area. The fluid thread breaks up in fluid dynamics branch involves most of its study using the plateau-Rayleigh instability. An example to demonstrate this effect is using the falling of water droplets from a faucet/tap. When the faucet is closed or when the flow is interrupted a long stretching of fluids occur at the neck of the faucet and if the faucet diameter is bigger the fluid is not sucked back and breaks down into smaller water droplets.

PROCEDURE :

STEP 1 : GEOMETRY USED

Two regions have been created for the fluid domain with the one domain over the other with a sizing of 20 mm x 20 mm.

Share topology to get conformal meshing .

AIR - WATER OIL - WATER

DIMENSIONS :

STEP 2 : MESHING

To study the case, experiment is done by considering different mesh sizes and different fluids.

1) BASE LINE MESH _ 0.5 mm 2) INTERMEDIATE MESH _ 0.3 mm 3) REFINED MESH _ 0.1 mm

Case B : Water - Oil instability test

case I : Refined mesh with mesh size 0.2 mm

STEP 3 : SET UP & SOLUTION

The Following are the various details of the mesh and Time steps used in this simulation run :

An Over View :

STEP 4 : RESULTS

CASE A : AIR - WATER

CASE 1 :

Residual Plot :

Rayleigh–Taylor instability with course mesh_0.5 mm Animation vedio :

CASE 2 :

Residual Plot :

Rayleigh–Taylor instability with intermediate mesh_0.3 mm animation vedio :

CASE 3 :

Residual Plot :

Rayleigh–Taylor instability with fine mesh_0.1mm animation vedio :

CASE B : OIL - WATER

CASE :

Residual Plot :

Rayleigh–Taylor instability with fine mesh_0.2mm_(OIL-WATER) :

Atwood number: In fluid dynamics Atwood number is used in the study of hydrodynamic instabilities in density stratified flows (i.e. to characterize the density difference between two fluids). The Atwood number (A) is a dimensionless number .

Where,
ρ₁ – Density of heavier fluid
ρ₂ – Density of lighter fluid

-For big density differences At-wood number tends to 1.

-For small differences at tends to zero.

-Also with Atwood number the penetration distance can be defined by Agt²

Effect of Atwood number on instability

If A is nearer to 1, R-T instability takes place in the form of fingers
If A is nearer to 0, R-T instability takes place in the form of bubble-like plumes.

CASE A : AIR - WATER

ρ₁ – Density of heavier fluid (water) – 998.2 kg/m³
ρ₂ – Density of lighter fluid (air) – 1.225 kg/m³

Atwood number for this case :

Since , Atwood number is nearer to 1 we can see symmetric "fingers" in this case.

CASE B : OIL - WATER

ρ₁ – Density of heavier fluid (water) – 998.2 kg/m³
ρ₂ – Density of lighter fluid (OIL) – 400 kg/m³

Atwood number for this case :

Since , Atwood number is nearer to 0 and less then 0.5 ,we could see there are spikes and bubbles formed.