Rayleigh Taylor Instability

Objective: Simulate Rayleigh Taylor Instability problem using Ansys Fluent and write a detailed report explaining the results obtained.

Question:

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.

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.

Rayleigh-Taylor instability :

What are some practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves?

Richtmyer–Meshkov instability :

The Richtmyer–Meshkov instability (RMI) occurs when two fluids of different density are impulsively accelerated. Normally this is by the passage of a shock wave. The development of the instability begins with small amplitude perturbations which initially grow linearly with time. This is followed by a nonlinear regime with bubbles appearing in the case of a light fluid penetrating a heavy fluid, and with spikes appearing in the case of a heavy fluid penetrating a light fluid. A chaotic regime eventually is reached and the two fluids mix. This instability can be considered the impulsive-acceleration limit of the Rayleigh-Taylor instability.

Kelvin–Helmholtz instability :

The Kelvin–Helmholtz instability can occur when there is velocity shear in a single continuous fluid, or where there is a velocity difference across the interface between two fluids. An example is a wind blowing over water: The instability manifests in waves on the water surface.

Plateau–Rayleigh instability :

The Plateau–Rayleigh instability, often just called the Rayleigh instability, explains why and how a falling stream of fluid breaks up into smaller packets with the same volume but less surface area. It is related to the Rayleigh-Taylor instability and is part of a greater branch of fluid dynamics concerned with fluid thread breakup.

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]

Case Setup and Execution:

Geometry: Geometry is modelled in ANSYS Space Claim.

Blueline is the partition line.

Top Square Dimensions (water) - 20 * 20 mm

Bottom Square Dimensions (air/user defined material)- 20 * 20 mm

Topology has been shared. Because the interfaces of both fluids are going to be disturbed with the gravitational force.

Solver

Type: Pressure base

Time: Transient

Velocity: Absolute

Gravity:-9.81m/s along y-direction

Model: Multiphase

Volume of Fluid

Number of Eulerian Phases :2

Implicit Solver

Phases:

Primary Phase -air

Secondary Phase-water

Water

Density = 998.2 kg/m^3

Viscosity = 0.001003 kg/ms

Air

Density = 1.225 kg/m^3

Viscosity = 1.7894e-05 kg/ms

Viscous: Laminar Model

Methods:

Scheme - SIMPLE

Initialization - Standard

For solution we use patch command in initialization

Phase - Water

Variable - Volume fraction

Value 0 for Air surface and 1 for Water surface.

CASE-2A:

Mesh: Element Size - 0.5mm

Nodes: 3321

Elements: 3200

Run calculation :

Time stepping method = Fixed

Time step size = 0.005 s

Number of time steps = 1500

Maximum iteration/time steps = 20

Results:

Residuals:

Volume Fraction: We can observe how the Air has moved upwards while the water settles down. The process in between appears to be still in mixture of state.

CASE-2B:

Mesh: Element Size - 0.2mm

Nodes: 20301

Elements: 20000

Run calculation :

Time stepping method = Fixed

Time step size = 0.003 s

Number of time steps = 1500

Maximum iteration/time steps = 20

Results:

Residuals:

Volume Fraction: A more visually appealing result is obtained as the mesh size is decreased, it took a long time for simulation to reach appropriate results.

Conclusion for Case-2:

Mesh refinement helps us to visualize the irregularities at the interface of two fluids more smoothly and clearly.

At first the fluids behave linearly & we can observe small linearity sinking going on between them, slowly the bubbles formation can be observed. In the end, two fluids get separated. There we can see some diffusivity between the fluids which is called the Volume Fraction (VOF) of air and water.

Steady is not used in this case because at steady state we will get to the behavior of fluid at independent of time but in case we need to study instability at interface from instant when it starts which help to get better idea how heavy dense get mixed and lighter density fluid rise above with respect to time domain.

3. Run one more simulation with water and user-defined material(density = 400 kg/m³, viscosity = 0.001 kg/m-s) for refined mesh.

CASE-3:

All settings and geometry are same as Case-2B, just replacing air with user defined material.

User Defined material:

density = 400 kg/m³

viscosity = 0.001 kg/m-s

Results:

Residuals:

Volume Fraction: We observe how the user defined material has moved upwards while the water settles down. The process in between appears to be still in mixture of state and when compared with CASE-2B, we can clearly observe bubbles and spikes formation still present after the completion of simulation. This formation is because of the increase in density of user defined material.

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.

Atwood Number:

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

For Atwood number less than 1, RT instability flows take the form of symmetric i.e the bubble and spike penetrations are very nearly the same;

For Atwood number greater than 1, the much lighter fluid “below” the heavier fluid takes the form of larger bubble-like plumes, the spike penetration is greater than that of the bubble owing to a narrowing of the spike, and a consequent reduction of drag on the head of the spike.

Atwood number for above cases

CASE-2:

`A=(998.2-1.225)/(998.2+1.225)=996.975/999.425=0.9975`

CASE-3:

`A=(998.2-400)/(998.2+400)=598.2/1398.2=0.4278`

Therefore comparing the above value of Atwood number and observation from the simulations i.e formation of bubbles when water comes down and air/user defined material rises above; we can say that the simulation result is validated`

Smaller Atwood numbers result in longer delay distances. Higher Atwood numbers shorten the delay and its associated distance, providing a longer development period for the R–T mixing.

For the case of small Atwood number, the density difference between the two sides of the interface is small, and the density distribution of the upper and lower layers is nearly symmetrical. The initial density stratification plays a dominant role, and the expansion-compression effect has little influence. With the increase in the Atwood number, the stabilization effect of initial density stratification decreases, and the instability caused by the expansion-compression effect becomes more significant. The flow structures of bubbles and spikes are quite different at medium Atwood number. The effect of compressibility on the bubble velocity is strong at large A. The bubble height is approximately a quadratic function of time at potential flow growth stage. The average bubble acceleration is nearly proportional to the square of Mach number at A = 0.9.

Reference:

https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.2010.0007