Week 5 - Rayleigh Taylor Instability

Objectives:

1. What are some practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves? In your own words, explain how these mathematical models have been adapted for CFD calculations.
2. Perform the Rayleigh Taylor instability simulation for 2 different mesh sizes with the base mesh being 0.5 mm. Compare the results by showing the animations( Attach them from your google drive). Observe the differences in the animation and explain the effect of mesh size on simulation. Also, explain why a steady-state approach might not be suitable for these types of simulation.[ Choose Air and Water for this simulation]
3. Run one more simulation with water and user-defined material (density = 400 kg/m³, viscosity = 0.001 kg/m-s) for refined mesh.
4. Define the Atwood Number. Find out the Atwood number for both the cases and explain how the variation in Atwood number in the above two cases affects the behavior of the instability.

Introduction:

Rayleigh Taylor Instability :

The Rayleigh–Taylor instability is an instability of an interface between two fluids of different densities which occurs when the lighter fluid is pushing the heavier fluid. Examples include the behaviour of water suspended above oil in the gravity of Earth ,mushroom clouds like those from volcanic eruptions and atmospheric nuclear explosion, supernova explosions in which expanding core gas is accelerated into denser shell gas, instabilities in plasma fusion reactors and inertial confinement fusion.

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

1. Kelvin Helmholtz instability Model:

This instability can occur when there is velocity shear in a continuous fluid, or where there is a velocity difference across the interface between two fluids. An example is wind blowing over the water. other examples showing this instability are clouds,oceans,saturns bands.

2. Richtmeyer meshkow instability :

This instability occurs when a shock wave interacts with an interface separating two different fluids or in other words when two fluids of different densities are impulsively accelerated.The development of instability begins with small amplitude pertubations which initially grow linear with time.This instability can be considered the impulsive acceleration limit of the Rayleigh Taylor Instability.

3. Plateau Rayleigh instability :

This instability explains why and how a falling stream of fluid breaks up into smaller packets with the same volume but different or small surface area . It is often called as Rayleigh instability.The driving force of this instability is that the liquids by virtue of their surface tensions ,tends to minimize their surface area .

4.Rayleigh Taylor Instability :

The Rayleigh–Taylor instability is an instability of an interface between two fluids of different densities which occurs when the lighter fluid is pushing the heavier fluid.

Procedure:

Case1:

2 Squares of 20 X 20 MM is created one above the other and share topology is used in order to capture the interface between 2 squares as shown below.

0.5 MM mesh is used as the base mesh.

Prssure based solver and transient state simulation is used as we are going to simulate w.r.t time.

Laminar viscous model is used.

Multiphase model is used as we are going to simulate the interface between the 2 fluids.

Air and water is used as the materials

Under phases respective phases is selected .

Standard initizilization is done.

Under patch top region is patched with water and Bottom region is patched with air.

Contours and animations are created.

Case 2:

Mesh size of 0.25 MM is used and the results are shown below.

Case 3:

User defined material is used in place of air.(density = 400 kg/m³, viscosity = 0.001 kg/m-s).

The results are shown below.

Atwood Number:

A dimensionless number that describes the density difference between two adjacent fluids with a common interface

It is denoted by letter A

A=ρ1-ρ2ρ1+ρ2">A=ρ1−ρ2/ρ1+ρ2 →1">→1

where,

ρ1">ρ1=Density of heavier fluid `

ρ2">ρ2=Density of lighter fluid

Calculate the Atwood Number for Air-water RT instability

ρ1">ρ1=998.2 kgm3">kg/m3 (water density)

ρ2">ρ2=1.225 kgm3">kg/m3(Air density)

put this value in equation 1 we get A=0.9975

Calculate the Atwood Number for Water-UD material RT instability

ρ1=">ρ1=998.2 kgm3">kg/m3 (Water density)

ρ2">ρ2=400 kgm3">kg/m3 (User defined material density)

put this value in equation 1 we get A=0.4278

The Atwood number close to 1 then the lighter fluid below the heavier fluid which forms a larger bubbles which is less symmetrical and also it occurs very rapidly.

The Atwood number close to 0 then the lighter fluid below the heavier fluid which forms a bubbles which is relatively symmetrical and the pertubations are linear for larger time.

Observations:

Finer mesh gives better results than the coarser mesh as it capture the better details.

Files are attached in link given below

https://drive.google.com/file/d/1-BSfTcu-dxGvqc62FoVwu9yqlibmfYMp/view?usp=share_link