Rayleigh Taylor Instability Challenge

Q1. 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 .

The Rayleigh–Taylor instability, or RT 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 behavior of water suspended above oil in the gravity of Earth, mushroom clouds like those from volcanic eruptions and atmospheric nuclear explosions, supernova explosions in which expanding core gas is accelerated into denser shell gas, instabilities in plasma fusion reactors and inertial confinement fusion.

CFD Models based on Rayleigh Taylor waves:

1. Kelvin Helmholtz Instability - This can occur when there is velocity shear in single continous fluid or when there is velocity difference across the interface between two fluids. Example of this is when wind blows over water causing wave formation.
2. K-L Turbulence model - The method is used to simulate the self-similar growth of the RT instability. This model captures the dominant eddies with the parameters K and L.
3. Plateau Rayleigh instability - This model explains why and how a falling stream of fluid breaks into smaller pockets with same volume but less surface area.

Q2. Perform the Rayleigh Taylor instability simulation for 3 different mesh sizes with the base mesh being 0.5 mm. Compare the results by showing the animations( Attach them from your google drive). Also, explain why a steady state approach might not be suitable for these types of simulation.

• Objective :

1. To perform the Rayleigh Taylor instability simulation for 3 cases with different mesh sizes.
2. Compare results and animations of three cases.
3. Explain why steady state is not suitable for this simulation.

• Geometry :

The geometry is fairly simple in this case as two square blocks of different fluids are placed one above other with same dimensions of 20mm * 20mm. The total length being 40 mm and width 20 mm. The upper portion is water and lower portion is air in our case.

• General Case Setup :

1. Solver - Pressure based transient solver.
2. Viscous Model - Laminar.
3. Gravity - -9.81 m/s in -ve y direction.
4. Multiphase - Volume of fraction (implicit solver).
5. Materials - Air and water.
6. Courant Number - 0.25.

• Case I :

Mesh size = 0.5 mm.

Number of elements = 3200.

Number of Nodes = 3321.

Time step = 0.005.

Iterations = 1500.

Mesh with 0.5 mm size.

Plot of residuals.

Video of simulation for case 1.

• Case II :

Mesh size = 0.3 mm.

Number of elements = 8978.

Number of Nodes = 9180.

Time step = 0.005.

Iterations = 1500.

Mesh with 0.3 mm size.

Plot of residuals.

Video of simulation for case 2.

• Case III :

Mesh size = 0.1 mm.

Number of elements = 80000.

Number of Nodes = 80601.

Time step = 0.0005.

Iterations = 2500.

Mesh with size 0.1 mm.

Plot of residuals.

Video of simulation for case 3.

• Observations :

1. With finer mesh we get a visualization of the instability between two layers.
2. Time step is needed to be decreased as the mesh gets finer or the results start diverging.
3. It takes more computational time as the mesh gets finer.

Steady state approach is not useful in this simulation as steady state approach focuses on end result instead of path taken to achieve those results.

Q3. Define the Atwood Number. Find out the Atwood number for this case and explain how the variation in Atwood number affects the behaviour of the instability. How can we use the Atwood number to validate our simulation result?

The Atwood number (A) is a dimensionless number in fluid dynamics used in the study of hydodynamic instabilities in density stratified flows. It is a dimensionless density ratio defined as

A = `(rho_1-rho_2)/(rho_1+rho_2)`

`rho_1` = density of heavier fluid (water)

`rho_2`= density of lighter fluid (air)

A = `(rho_1-rho_2)/(rho_1+rho_2)` = `(998.2-1.225)/(998.2+1.225)`= 0.9975

When A is closer to 1 the lighter fluid forms bubbles while rising up, so looking at our results we can say that the simulation is correct.