Rayleigth - Taylor Instability

Rayleigth-Taylor :-

• The Rayleigth-Taylor instability is an instability of interface between two fluid of different densities which occur when the lighter fluid is pushing the heavier fluid.

Example :- As shown in above figure the supernova explosion in which expanding core gas is accelerated into denser shell gas instablities in plasma fusion reactor and inertial confinment fussion.

• The equilibrium here is unstable to any perturbation or disturbance of the interface.
• If a parcel of heavier fluid is displaced downward with an equal volume of higher fluid displaced upward the Potential Energy of the configuration is lower than the initial state.
• Thus the disturbance will grow and lead to further release of Potential Energy as more dense material moves down under the gravitational field and the less dense material is further upward.

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

1. Kelvin - Helmholtz Instability
2. Richtimyer - Meshkov Instability
3. Plateau - Rayleigh Instability

1. Kelvin - Helmholtz Instability:-

• The theory predict the onset of instability and transition to turbulent flow in fluid of different densities moving at various speeds.

Deep Oceanic Kelvin - Helmholtz Instability

• For some short enough wavelength, if the surface tension is ignored two fluid in parallel motion with different velocities and densities yeild an interface that is unstable for all speed.
• Surface tension stabiliser the short wavelength instability however and theory predict stability until a velocity threshold is reached.
• The theory with surface tension is included, broadly predict the oneset of wave formation in the important case of wind over water.

2. Richtmyer - Meshkov Instability:-

• The Richtmyer - Meshkov Instability occur when two fluid of different density are impulsively. Normally this is by the passage of a shock wave.

• The development of the instability being with small amplitude perturbation which initially grow linearly with time.
• This is followed by a non linear regime with bubbles appearing in the case of a light fluid penetrating a heavy fluid and with spikes appearing in the case of heavy penetrating a light fluid.

3. Plateau - Rayleigh Instability:-

• The Plateau - Rayleigh Instability, often just called the Rayleigh instability explain why and how a falling steam of fluid break up into a smaller pockets with some volume but less surface area.
• It related to the Rayleigh - Taylor instability and is part of a greater branch of fluid dynamic concered with fluid thread breakup.

Rain Drop from roof it is an example of Plateau - Rayleigh Instability

• The fluid instablity is exploited in the design of a particular type of ink jet technology where by a jet of liquid is pertubed into a steady stream of droplet.

SIMULATION :-

In this case study we going to solve with Volume of Fluid method in Ansys Fluent.

GEOMETRY:-

In upper section the water is an fluid where as in lower section air is an fluid.

MESH:-

In this we are going to simulate in three different mesh size.

SET UP:-

General:-

1. Solver :- Pressure based Absolute Solver.
2. State :- Transient State

Model:-

1. Multiphase :- Volume of fluid (IMPLICIT METHOD)
2. Viscous:- Laminar

Material and Phases:-

1. Air :- Primary Phase
2. Liquid Water :- Secondary phase

Method:-

1. Scheme - Simple
2. Initialization :- Standard

Before running the calculation we must patch it. Such in this case water is assigned to value as 1 and air as 0

Case 1:- Baseline method

MESH:-

Size of Mesh :- 0.5 mm

Number of element :- 3200

SOVLING:-

Number of Iteration :- 1000

Size of time step :- 0.005

RESULT:-

RESIDUAL PLOT:-

Residual Plot for base line mesh

Velocity Contour Plot:-

image of the simulation of the Baseline mesh

ANIMATION LINK:-

https://youtu.be/nWyf1EN_P04

Case 2:- Refined Mesh 1

MESH:-

Element Size :- 0.3 mm

Number of element :- 8978

SOLVING:-

Number of iteration :- 1300

Size of Time Step :- 0.003

RESULT:-

RESIDUAL PLOT:-

Residual Plot for first fined mesh

VELOCITY CONTOUR PLOT :-

Image of simulation of first refined mesh

Animation Link:-

https://youtu.be/GdIbug8WSWk

Case 3:- Second Refined Mesh

MESH :-

Size of element :- 0.1 mm

Number of element :- 80000

SOLVING:-

Number of iteration :- 1500

Size of time step :- 0.002

RESULT :-

RESIDUAL PLOT:-

Residual Plot for second refined mesh

VELOCITY CONTOUR PLOT:-

Image of velocity contour plot

Animation Link:-

https://youtu.be/Wq_7Nz8PWns

ATWOOD NUMBER :-

• Atwood Number is a dimensionless number in Fluid Dynamics used to study of hydrodynamic instability instabilities in density stratified flow
• Atwood Number is defined as the ratio for difference between the heavier and lighter density fluid to the sum of the heavier and lighter fluid.
• Atwood Number can be given as,

`A=frac{rho_1-rho_2}{rho_1+rho_2}`

`rho_1`= density of heavier fliud in our case is water density is 998.2 kg/m^3

`rho_2`=density of lighter fluid in our case is air density is 1.225 kg/m^3.

A= Atwood Number

In our case Atwood Number , `A=frac{998.2-1.225}{998.2+1.225}`

A = 0.9975

Hence, Atwood Number is 0.9975

• For Atwood Number close to zero, Rayleigth Taylor instability flow take the form of symmetric \'fingers\' of fluid, for Atwood Number close 1, the much lighter fluid below the heavier fluid take the form of larger bubbles like plumes.
• For now, the calculate Atwood number is 0.9975 which very close to 1.
• The Atwood Number as close as to 1, the much air bellow the water takes the form if larger bubbles.
• From the simulation result, it is shown that when the water is poured on the air with gravity, the air bubbles are formed and moved to upward in the region in the form of waves.
• Some air bubbles get trapped below, they moved to upward afterward
• At last, both the phase get trapped and diffusivity of both fluid has been left in between them.