Week 5 - Rayleigh Taylor Instability
AIM: - TO PERFORM RAYLEIGH TAYLOR CFD SIMULATION.
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 behaviour of the instability.
The Rayleigh – Taylor Instability (RTI) can be defined as the phenomena which occurs at the interface of two fluids of different densities. The lighter fluid pushes the heavier fluid and displaces it in order to achieve minimum potential energy of the system.
The daily life example can be water placed atop an oil column. The oil being lighter than water moves the water from top and water settles at bottom.
The RTI occurs in 4 stages: In first stage, the initial perturbation amplitudes are less than their wavelengths and equation of motion can be linear. The linear behaviour converts into non-linear behaviour and formation of mushroom and spikes are seen.
The mushrooms growth continues in the second stage and can be modelled with the help of buoyancy drag models.
In third stage, the bubbles merge with others. Small bubbles are enveloped by bigger unsaturated bubbles. This results in the turbulent mixing, which is the final stage of the RTI.
Some practical CFD models that have been based on Rayleigh – Taylor waves:
RICHTMYER – MESHKOV INSTABILITY (RMI):
This is seen when two fluids of different density are impulsively accelerated. Generally, occurs due to the passage of a shock wave. The perturbations initially grow linearly, followed by non linear regions having bubbles formed when the light fluid pass through heavier fluid and spikes when heavier fluid pass through the light fluid.
KELVIN – HELMONTZ INSTABILTY:
This occurs when the velocity difference is present in either single fluid system or in two fluid system. The cloud formation in earth sky, red spot-on Jupiter are few examples of results of this phenomenon.
PLATEAU – RAYLEIGH INSTABILITY:
This phenomenon explains the formation of droplets. How falling liquid breaks into smaller drops and why.
MULTI – PHASE:
It is the simultaneous flow of multiple materials having different states or phases. It is flow of material having same phase but different chemical properties.
There are two types of multi-phase schemes:
Separate Phase, wherein there are one or more immiscible fluids in a continuous phase separated by an interface.
Mixed phase, wherein two or more miscible fluids mixes together. Both mixed and separate phases are present in this case.
The Volume of Flow method is an example of mixing of two or more immiscible liquids.
ATWOOD NUMBER:
Atwood number is a dimensionless number used in study of hydrodynamic instabilities in density stratified flows. Mathematically, Atwood number is the ratio of the difference of the mass densities and the sum of mass densities.
Higher the value of Atwood number faster the instabilities travel, lower the value, lower is the speed of the shock wave or bubbles or the mixing.
Simulation Setup:
1. Geometry:
The geometry consists of two square plates each having dimension of 20 mm stacked one over the other sharing one common edge. As shown in the figure below:
The lower square represents the air column and the upper box is representing water column.
2. MESHING:
The meshing for the given problem is done taking two different mesh sizes:
0.5 mm (base model)
0.2 mm (refined model)
The mesh for base model is shown in the figure below:
3. ANALYSIS SETUP:
The analysis setup is done in ANSYS fluent, mentioned below:
The analysis is done under transient condition with gravity being acting as the driving force.
The flow is considered to be Laminar Flow.
Select the multiphase option and check the Implicit formation for the Volume formulation Parameters.
Add Air and Water as materials for the base model as shown below:
For refined model, a user defined material must be used which should have the following properties:
density = 400 kg/m3
viscosity = 0.001 kg/m-s
for the base model, assign the materials to the respective phases in the mesh using the multiphase option.
Now, initialize the model using Standard initialization method. Then next thing is to Patch the phases. To patch the phases, select the water as the Phase and take Volume Fraction as the required variable.
Enter value as 1 and select the water surface (fff water in present case) and hit patch button. Similarly, enter value 0 and select air surface (fff air in present case) and hit patch button.
To verify if done correctly, plot a Phases contour, as shown below:
The above figure shows the setup is done correctly. Now take 150 timesteps and each timestep being of 0.01s. Hit run to begin the analysis. The results of the analysis can be seen in figures below:
Similarly, the results for the refined case setup with Air and Water components are shown below:
The third case is where the user defined material is coupled with water.
The results obtained for this case are as follows:
CONCLUSIONS:
The Atwood number for first two cases is same and is equal to 1.
For the third case, the Atwood number is equal to:
At = {(998.2 – 400) / (998.2 + 400)} = 0.4278.
For high values of Atwood number, the instabilities are moving very rapidly and interactions are also quickly stabilizing. However, for lower values of Atwood number, the instabilities are travelling slower and interactions are also taking more time.
So, it can be said that for higher Atwood number the instability travel rapidly and mixing occur quickly as compared to that for lower values of Atwood number.