Rayleigh Taylor Instability Simulation

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.

Answer :

Rayleigh Taylor Instability :

RT instability, is an instability of an interface between two fluids of different dnsities which occurs when the lighter fluid is pushing the heavier fluid.

CFD models based on mathematical analysis of Rayleigh Taylor Instability

• Bouyancy Drag model : It is a model which uses ordinary differential equations to evolve the width of the mixing layer. These models cannot model multiple mixing interfaces, also do not address demixing known as counter-gradient transport, i.e., reduction of total fluid masses within the mixing zone.

• BHR Model : Based on the evolution equations arising from second-order correlations and gradient-diffusion approximations which Uses a mass weighted averaged decomposition, includes full transport equations for the Reynolds stresses, turbulent mass-flux velocity, density fluctuations and the turbulence kinetic energy dissipation rate is the Besnard-Harlow-Rauenzahn also known as advanced version of sinfle fluid models.

• K-L model : The K-L model was proposed by Dimonte and Tipton for describing the turbulent self-similar regime of RT and RM induced mixing.
• K-L-a model : The three-equation model of Morgan and Wickett was developed as an extension to the two-equation K-L model of Dimonte and Tipton by including a third equation for the turbulent mass-flux velocity.
• K-L-a-b model : An extension of the K-L-a model by adding an evolution equation for the density-specific volume covariance which reflect the effects of the changes in the density fluctuations.

Simulation Procedure :-

Geometry :

1. Open the geometry section.
2. select the plane and create the geometry according to the givern values.
3. First create the lower voume and extract.
4. Then create the upper volume and extract it.
5. Name both the volumes, upper volume as water and lower one as air.
6. Finally, select the shareprep from workbench for entire volume so they share the interface edge.

Mesh :

1. Open the mesh section.
2. Go to mesh and edit the elment size according to requirement and mesh the geometry.

Setup :

1. Perform Mesh check.
2. Select the transient state approach and add gravity to the problem.
3. Set the viscous model as laminar.
4. Add Volume of fluid as multiphase model and set implicit formulation.
5. Then add materials according to the requirement (we are going to use air and fluid with standard densities and vicosities for first two simulation and water and user defined material for third simulation).
6. Now name both the phases as air and water respectively.
7. Initialize the problem with standard initialization as we are using transient state approach.
8. finally add contour and solution animation and run the simulation for some iteration after setting correct timestep( we are going to run it at 0.005 timestep size and for 1000 timesteps).

Result :

1. Make visible both the upper and lower volume and display variables on both volume.
2. Now create timestep animation and observe the variation with respect to time.

We are going to perform simulation on below geometry starting from baseline mesh to refined mesh and refined mesh with user-defined fluid properties.

• Geometry

############################# { Baseline Mesh } #############################

Element size	Elements	Nodes
0.0005 m	3200	3321

	Air	Water
Density (kg/m3)	1.225	998.2
Viscosity (kg/m-s)	1.7894e-5	0.001003

• Mesh

• Animation

############################# { Refined Mesh } #############################

Element size	Elements	Nodes
0.0002 m	20000	20301

	Air	Water
Density (kg/m3)	1.225	998.2
Viscosity (kg/m-s)	1.7894e-5	0.001003

• Mesh

• Animation

############################# { Refined Mesh with user defined fluid } #############################

Element size	Elements	Nodes
0.0002 m	20000	20301

	User-defined material	Water
Density (kg/m3)	400	998.2
Viscosity (kg/m-s)	0.001	0.001003

• Mesh

• Animation

############################# { Observations } #############################

• Looking at above videos we can say that refined mesh gives more smooter results.
• Also we can say that both the fluids settle down faster in coarser mesh and it takes much more time in finer mesh cases.
• providing correct timestep is important at it gives floating point error if wrong timestep is chosen.

Transient state over steady state :

• We cannot use steady state approach as there is a lot of turbulence and sudden transition at very smaller time variations which needs to be calculated with respect to time which is not possible with the steady state approach.
• Also in our case we need to observe results with respect to time i.e. at smaller time variation which will only be possible with transient state approach.

Atwood number :-

Atwood number is a dimensionless number used to describe density difference between two adjacent fluids sharing same interface.

atwood number = (ρ1−ρ2)/(ρ1+ρ2)

where, ρ1 = density of heavier fluid

ρ2 = density of lighter fluid

Atwood number for Baseline and refined case = (998.2-1.225)/(998.2+1.225)

= 0.9975

Atwood number for refined case with user defined fluid properties = (998.2-400)/(998.2+400)

= 0.4278

Effect of variation in atwood number on above cases :

• In First two videos we can clearly see that, the 1st two cases with standard densities of air and fluid has bubble type formation at upper part and spike type formation at lower part at inteference after collapsing and trying to mix.
• In third video in which refined case has user defined fluid with different density than standard one, forms almost similar bubble like formation on both upper and lower part of interference after collapsing.
• So we can say that cases with higher atwood number produce bubble type formation at upper part and spike type formation at lower part of inteference.
• Same way, the case with lower atwood number forms almost similar bubble like formation on both upper and lower part of interference.
• Yet the results vary with different atwood numbers, so we can refer our results from published research papers where the simulation is done for various atwood numbers.