RAYLEIGH-TAYLOR INSTABILITY SIMULATION USING ANSYS FLUENT

OBJECTIVE:

The aim of this project is to simulate the phenomenon of Rayleigh taylor instability phenomenon in two dimension. Also to do it using 3 different sizes of mesh and study the behaviour accordingly. Finally, the Atwood number will be calculated for this simulation and the significance of Atwood number will be explained.

RAYLEIGH TAYLOR INSTABILITY:

The Rayleigh–Taylor instability is an instability of an interface between two fluids of different densities which occurs typically when the heavy fluid is supported by the lighter one.

CFD models based on Rayleigh waves:

1. Structure model:

Rayleigh waves are highly useful in geophysics where the effect of seismic waves on the structures like buildings, bridges other infrastructures can be simulated.

2. Acoustic model:

Ultrasonic waves for non-destructive testing, piezoelectric devices and other sound related problems can be solved. Also Rayleigh waves are used by animals to communicate. This can also be simulated.

3. Multiphase model:

In multiphase models also, it is useful in simulating ocean waves.

SIMULATION PROCEDURE:

Modelling:

The first step of the process is to create a 2D model in Spaceclaim. It is build as a rectangle shaped domain. A 20 mm X 20 mm air domain is created. And then, on top it. a water domain created of the same dimensions. Then their topology is shared in order to create a shared edge. It is as below.

Fig-1:2D CAD model for RT instability simulation

Meshing:

A simple meshing will be carried without any special methods. But it will be done three different sizes. They are

1. Baseline mesh setup.
2. Refined mesh setup.
3. High-refined mesh setup.

Baseline mesh setup:

• Mesh element size - 0.5 mm
• No. of nodes - 3321
• No. of elements - 3200
• Shape of element - Quads

Fig-2:RT instability baseline mesh model

Refined mesh setup:

• Mesh element size - 0.3 mm
• No. of nodes - 9180
• No. of elements - 8978
• Shape of element - Quads

Fig-3:RT instability refined mesh model

High-refined mesh setup:

• Mesh element size - 0.2 mm
• No. of nodes - 20301
• No. of elements - 20000
• Shape of element - Quads

Fig-4:RT instability highly refined mesh model

Solving in Fluent:

The corresponding mesh models are imported and solved in Fluent with appropriate physics values specified. The physics setup is the same for all the three types of mesh models. It is as follows.

1. Solver - Pressure based, transient solver

2. Gravity - Enabled in - y direction with acceleration of 9.81 m/s^2

3. Physical models - Viscous and Multiphase model

a. Viscous model - Laminar

b. Multiphase model - Volume Of Fluid model with Implicit formulation.

4. List of Phases - Air and Water

5. Initialization - Standard

6. Timestep size - 0.005 s

7. No.of time steps - 1000

One of the main processes in a Multiphase problem is patching. It has to be done after the initialization is done. Here in this simulation, in the phase Water, for the variable, the values are assigned as below.

• Value 1 : Water surface
• Value 0 : Air surface

After these values are set, the calculation is done for 1000 time steps. And the residuals plots are as below.

Fig-5:RT instability baseline setup residuals

Fig-6:RT instability refined setup residuals

Fig-7:RT instability high refined setup residuals

Comparing the above residual plots, the plot of baseline setup is a bit unrealistic. Because the amplitude of variations changes rapidly at some areas. From 7000 to 7200 and from 7500 to 8000, there is no change at all and takes a sudden series of spikes after that. But on the other two setups, the variation is pretty gradual.

RESULTS:

Baseline setup:

Fig-8:Volume fraction of water in baseline setup

Anime-1:Animation of VF contour in baseline setup

Fig-9:Volume fraction of water in refined setup

Anime-2:Animation of VF contour in refined setup

Fig-10:Volume fraction of water in high-refined setup

Anime-3:Animation of VF contour in high-refined setup

Inference from the contours and the animations:

The contour depicting the volume of fraction in baseline setup is very coarse as expected. The baseline animation is also coarse. There is also formation of vortices in all the three setups. But it is evidently accurate and fine in the high refined setup. The formation of vortices is due to the pressure gradient in different directions.

Fig-11:Vorticity in RT instability

The pressure differences produces a baroclinic torque that in turn produces vortices with an angular velocity \'`omega`\'. Eventhough it is seen in all three setups, it is very clear in the third setup. The independent bubble formation is also finer and is seen more accurate in the high refined setup.

Simulating the RT instability using steady state solver:

When the same simulation is done using a steady state solver, the results are very unrealistic. The residuals are behaving in an inappropriate manner as below.

Fig-12:RT instability simulation residuals using steady state solver

It can only be done if the pseudo transient approach is enabled and the P-V coupling is changed to Coupled scheme. It makes the case stable and approach to convergence will be faster.

ATWOOD NUMBER:

It is a dimensionless number that has a major application in RT instability analysis which is defined as the ratio of the difference of densities of the fluids involved to sum of their densities.

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

where `rho_1`= density of heavier fluid

`rho_2`= density of lighter fluid

Significance:

The penetration distance of heavier fluid into lighter fluid is directly proportional to the Atwood number.

Atwood number of this case:

Density of water = 998.2 kg/m^3

Density of air = 1.225 kg/m^3

`A = (998.2 -1.225)/(998.2 + 1.225)`

A = 0.997

Since the Atwood number is almost 1 in this case, the penetration distance of water into air is the highest as possible as it reaches the bottom end of the domain..

CONCLUSION:

Comparing the all three mesh models, the high-refined setup with a mesh element size of 0.2 mm accurately depicted the RT instability phenomenon. As explained, this phenomenon cannot be simulated in steady state without a pseudo transient solver. But it is better to solve using a transient solver to obtain the accurate results.