Rayleigh Taylor instability

RAYLEIGH-TAYLOR INSTABILITY

OBJECTIVE:

To perform Rayleigh Taylor instability simulation for different mesh sizes using FLUENT and to compare the results by providing the animations.Also to discuss he effect of Atwood Number and viscosity on the Instabilty criterion.

DESCRIPTION:

The Rayleigh Taylor instability is a fingering instability on the interface between two fluids of different densities,which is occuring whwn the light fluid is pushing the heavy fluid.The simple criterion for the onset of the Taylor instability at the interface between two fluids of different densities can be given below:

1. If the heavy fluid pushes the light fluid, the interface is stable.
2. If the light fluid pushes the heavy fluid,the interface is unstable

There is a complex phenomenology associated with the evolution of a Rayleigh Taylor unstable interface.This includes the formation of the spikes,curtains and bubbles.

Examples of the Rayleigh-Taylor instablility:

The Rayleigh Taylor instability can occur in diverse situations.

A.Natural Phenomena

1. The formation of high luminosity rwin-exhaust jets in rotating gas clouds in an external gravitational potential
2. The overturn of the outerportion 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

Some of the Practical CFD models that are based on the mathematical anlysis of Rayleigh Taylor waves:

1. Kevin-Helmholtz Instability:

The theory predicts the onset of instability and transition to turbulent flow in fluids of different densities moving at various speeds.The instability occurs where there is a velocity shear or velocity difference between the interface of two fluids

2. Richtmyer-Meshkov Instability :

This type of instability occurs when there is an impulsive acceleration between two fluids.This occurs when there is a shock at greater speeds.The development of the instability begins with the small amplitude perturbations which initially grow linearly with the time.This is followed by a nonlinear regime with bubbles appearing in the case of a fluid penetrating a heavy fluid,and with spikes appearing a heavy fluid penetrating a light fluid.A chaotic regime eventually is reached and the two fluids mix.This instability can be comsidered the impulsive acceleration limit of the Rayleigh instability.

3.Plateau-Rayleigh Instability :

The Plateau-Rayleigh instability , just often called the Rayleigh Instability ,explains why and how a falling stream of fluid breaks up into smaller packets with the same volume but lesser surface area.It is related to the Rayleigh taylor Instability and is a greater part of the fluid dynamucs which is concerned with the fluid thread breakup.This fluid instability is exploited in the design of a particular type of ink jet technology,whereby a jet of a liquid is perturbed into a strady stream of droplets.The driving force of the Plateau-Rayleigh Instability is that the liquids,by virtue of their surface tensions ,tends to minimise their surface areas.

NUMERICAL SIMULATION:

• The simultion has been carries out in two cases:

CASE 1. Rayleigh Taylor instability simulation for the water and air interface

CASE 2. Rayleigh Taylor instability simulation for the water and oil interface

1.GEOMETRY:

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.

• Geometry has been same for both the cases and in the first case the fluid domain in the bottom is having air and in the next case ,the domain being for oil.In both these cases ,the heavier liquid,ie water remains at the top domain.

2.MESH:

• The meshing has been done for two different cases of simulations as discussed below:

CASE 1) Here the simulations has been carried out for 3 different element sizes

1.1 Element Size 0.5 mm

• Element Size: 0.5 mm
• Type of Elements : Quadrilateral
• Number of Elements : 3200
• Number of Nodes : 3321

1.2 Element Size : 0.25 mm

• Element Size : 0.25 mm'
• Number of Elements : 12,800
• Number of Nodes : 13,041

1.3 Element Size : 0.1 mm (Refined Meshing )

• Element Size : 0.1 mm
• Orthogonal Quality : 0.99
• Number of Elements : 80,000
• Number of Nodes : 80,601

CASE.2) Refined Meshing

• In this case ,the meshing profile used is the same with the element size of 0.1 mm

3.SETUP:

Solver - Type : Pressure-Based

Time : Transient

Velocity Formulation : Absolute

Gravity : -`9.81 m/s^2`along y direction

Model - Multiphase ON

- Volume of Fluid

- Number of Eulerian Phases : 2

- Implicit Solver

- Laminar Model

- Energy OFF

Materials - Fluid : Air ( for CASE 1)

water-liquid

oil ( for CASE 2 )

List Of Phases : CASE 1. Primary Phase - air

Secondary Phase - water-liquid

CASE 2. Primary Phase - oil

Secondary Phase - water -liquid

Initialization : Hybrid Initialization

• Patching has been done in both the cases.

Method : SIMPLE Method

Calculation : The size of the timestep chosen has been different for different cases.

CASE 1. Element size 0.5 mm

• Adapative time step has been used
• Maximum number of iteration/time-step : 20
• Maximum time-step size : 10
• Minimum time-step size : 1e-05
• Number of time-step : 1000

Element size 0.25 mm

• Number of time-step : 1500
• Time-step size : 0.005
• Maximum iteration/time-step : 20

Element size 0.1 mm

• Number of time-step : 2000
• Time-step size : 0.0005
• Maximum iterations/time-step : 20

CASE 2. Refined Mesh

• Number of time-step : 1700
• Time-step size : 0.005
• Maximum iterations/time-step : 20

4.SOLUTION & RESULTS

CASE 1. Rayleigh Taylor instability simulation with water-air interface

Element Size : 0.5 mm

1. Residual Plot:

• The solution is converged for a time-step number of 400

2. volume fraction contour for water:

• After the interface has been deformed, we could observe that there is a rapid turbulent mixing between the two fluid, ie water and air

3. velocity contour Plot:

4.Animation for the Rayleigh Taylor instability is provided below:

Element size 0.25 mm

1. Residual Plot:

2.volume fraction contour plot for water:

3.velocity contour plot:

4.Animation for the instablity is given below:

Element size 0.1 mm

1. Residual Plot;

2.Volume fraction contour plot for water:

3.velocity conotur Plot:

4.Animation for the instability has been given below:

ATWOOD NUMBER CALCULATION:

Atwood number is a dimensionless qauntity used in the fluid mechanics used to study the hydrodynamic instability.It is defined as:

Atwood Number , A = `(rho1-rho2)/(rho1+rho2)`

where `rho1`=density of heavier fluid

`rho2`=density of lighter fluid

Thus, the Atwood Number for the air-water is given as: A = (1000-1.225)/(1000+1.225)

= 0.997

CASE 2. RAYLEIGH TAYLOR INSTABILITY SIMULATION WITH WATER-OIL INTERFACE:

1.Residual Plot:

2. Volume fraction contour plot for water:

3.volume fraction contour plot for oil:

4.velocity contour plot:

5.Animation for the instability is provided below;

ATWOOD NUMBER CALCULATION:

Atwood Number , A = `(rho1-rho2)/(rho1+rho2)`

= (1000-400)/(1000+400)

= 0.428

EFFECT OF ATWOOD NUMBER ON RAYLEIGH-TAYLOR INSTABILITY:

• The Atwood number for the water-air interface and water-oil interface Rayleigh taylor instabilities were found to be A = 0.997 and A=0.428

• If A`<=`1 , which is the case we are observing for the instability of water-air , it is said that the light fluid will move into the heavy fluid in the form of round topped bubbles with circular cross-section.The heavy fluid will form spikes and wall or curtains between the bubbles,so that the horizontal section will show a honeycomb pattern,which is exaclty we are able to visualise in the volume fraction contour for the water-air Rayleigh taylor instability.

• If A`>=`0, which can be the case we can observe for the water-oil instability, where the Atwood number was found to be 0.428 ,it is said that we will find a pattern more like two sets of interpenetrating bubbles.

Furthermore , a steady state approach was not used for the Rayleigh taylor instability simulation because , the steady state approach is independent on the time progressed and it would benefit us only if we are interested in the end results only.But here, in this simualation , we have to observe the pattern of the instability wave formed form time to time from the starting time of the simulation.This could only be achieved once we use a transient simulation.

CONCLUSIONS:

• Transient state simulation is necessary for solving the instability problems
• Atwood number governs the behaviour of the instability interface during the Rayleigh Taykor instability and the value of the Atwood number ranges from 0`<=`A`<=1`

Thank You

Arjun Vs