Week 5 - Rayleigh Taylor Instability

Rayleigh-Taylor Instability

AIM: Simulation of Rayleigh-Taylor Instibility.

OBJECTIVES:

Some practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves, and explain how these mathematical models have been adapted for CFD calculations.

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]

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

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.

INTRODUCTION:

The Rayleigh–Taylor instability is the 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 a volcanic eruption and atmospheric nuclear explosion, supernova explosions in which expanding core gas is accelerated into denser shell gas, instabilities in plasma fusion reactors, and inertial confinement fusion.

Rayleigh Taylor Instability:

The Rayleigh–Taylor instability is an instability of an interface between two fluids of different densities which occurs when the lighter fluid is pushing the heavier fluid.

Kelvin Helmholtz instability Model:

This instability can occur when there is velocity shear in a continuous fluid, or where there is a velocity difference across the interface between two fluids. An example is wind blowing over the water. other examples showing this instability are clouds, oceans, Saturn’s bands.

Richtmyer meshkov instability:

This instability occurs when a shock wave interacts with an interface separating two different fluids or in other words when two fluids of different densities are impulsively accelerated. The development of instability begins with small amplitude perturbations which initially grow linear with time. This instability can be considered the impulsive acceleration limit of the Rayleigh Taylor Instability.

Plateau Rayleigh instability:

This instability explains why and how a falling stream of fluid breaks up into smaller packets with the same volume but different or small surface area. It is often called as Rayleigh instability. The driving force of this instability is that the liquids by virtue of their surface tensions, tends to minimize their surface area.

PROCEDURE/ STEPS TO BE CONSIDERED FOR SIMULATION:

1. CREATING 2D GEOMETRY:

Sketch 20x20 dimension square box with named file of water and air one other over another.

Design the created dimensions by converting into surfaces. Then share it by share topology to overlap one on another and simulate in a single mode of model or to remove the center edge which is dividing the both boxes.

2. MESHING THE 2D GEOMETRY:

Meshing the geometry by mesh size of 0.0005m to perform in good mesh and assume it as baseline mesh. Selecting separate box to apply them a name to particular box.

Baseline mesh size of 0.0005m with 3200 elements and 3321 nodes.

2D geometry with the element quality is below

3. SETUP FOR SIMULATION OF 2D GEOMETRY:

Setting up the 2D geometry for simulation by allowing pressure-based solver with absolute velocity for transient/unsteady state and enabling the gravity with applying negative 9.81 gravity in y-direction.

Setting viscous model as laminar and Multiphase model as volume of fluid(Vof) with implicit formulation for 2 phases. Before this select and allow the material of water and air.

Allowing standard boundary condition.

Initializing with standard simulation, after selecting patch as for water of volume fraction as 1 and air of volume fraction as 0.

Calculation performed for 800 iterations with Time step size of 0.005 seconds.

Then the outputs are as below

RESIDUAL PLOT:

4. Post-Processing:

Flow simulation of baseline mesh:

https://drive.google.com/file/d/1TclCO_sNfDjICdrJA7DtrE-AXGGZzAyh/view?usp=sharing

CASE 1:

Meshing:

Refining the geometry by mesh size of 0.0002m to perform in good mesh. Selecting separate box to apply them a name to particular box.

Refined mesh size of 0.0002m with 20000 elements and 20301 nodes.

Mesh with element quality

SETUP:

For this Refined simulation we kept the material as air and water with standard density and viscosity of air and water.

The calculation is calculated for 500 timestps/iteratiopns,

RESIDUAL PLOT:

Simulation of Refined mesh size:

https://drive.google.com/file/d/1UWoX2SB3SV4BkoAbwbS7OfWKbsrp-FPY/view?usp=share_link

Observations considered between baseline and refined mesh simulation:

By comparing the baseline and case 1 simulation we observe that the number of elements is increased 3200 to 20000.

The comparison of baseline and case1 simulation we say that, In baseline simulation we saw that the lighter density fluid is settled first for some timesteps and flow of heavier fluid is will settled down by creating the large bubbles.

As in case 1 the lighter density fluid will reach it destination first and here the flow of heavier fluid is smooth as compared to baseline method and here the bubble sizes are small comparing to baseline simulation.

The case1 mesh i.e., refined mesh captures instability better than baseline mesh, By taking longer time to settled down the heavier fluid.

The computational time is increased to for case1 refined mesh.

The steady state simulation is performed if we are concerned more about the final state results or the equilibrium state. That is to see the performance of simulation of fluids to how the interface collapsing as function of time with each other in negative gravity to heavier fluid in y-direction. So, in this performance test the transient state is more suitable than steady state.

CASE 2:

Meshing:

Refined mesh size of 0.0002m with 20000 elements and 20301 nodes.

SETUP:

We have changed the material air to user-defined material in simulation and for water we kept it same as water.

For the user-defined material the density is considered as 400 kg/m^3 and the viscosity is considered as 0.001 kg/m s. calculation is made for 500 timesteps/iterations.

RESIDUAL PLOT:

Simulation of Refined mesh size with chenge of material:

https://drive.google.com/file/d/12_7LcXF-gaYqBgSBVrFa9tUcE0D-y3NY/view?usp=share_link

COMPLETE SIMULATION WITH TOTAL 800 TIMESTEPS:

https://drive.google.com/file/d/1l9aOjZ7N6eYiURxR0ovG5WezrC7GII3s/view?usp=share_link

Atwood Number:

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

A= (ρ1- ρ2)/(ρ1+ ρ2)

Where,

A – Atwood Number

ρ1 – Higher density fluid

ρ2 - Lower density fluid

For case 1 we the fluid and air as the material for simulation

ρ1 = 998.2 kg/m^3 (density of water)

ρ2 = 1.225 kg/m^3 (density of air)

A = (998.2-1.225)/(988.2+1.225) = 0.9975

For case 2 we the fluid and user-defined material as the material for simulation

ρ1 = 998.2 kg/m^3 (density of water)

ρ2 = 400 kg/m^3 (density of user-defined material)

A = (998.2-400)/(998.2+400) = 0.4278

From the value of the Atwood Number seen from simulation is that the air forms larger bubbles when the water is poured downside then the air strikes the water to escape from it to top layer, in this time the air will create large bubbles. Therefore, the results can be validated by obtaining the Atwood Number.

If the Atwood number close to 1 then the lighter fluid will be below to the heavier fluid which forms a larger bubble, and it will be having less symmetrical and also it occurs very rapidly.

If the Atwood number close to 0 then the lighter fluid will be below to the heavier fluid which forms the bubbles which are relatively symmetrical and the perturbations are linear for larger time.

CONCLUSION:

From the above cases, we observe that finer mesh captures the instability better than the coarse mesh, but it takes more time for the heavier fluid to settle down. As computational time is long but produces results with better accuracy.

The steady state simulation is performed if we are concerned more about the final state results or the equilibrium state. That is to see the performance of simulation of fluids to how the interface collapsing as function of time with each other in negative gravity to heavier fluid in y-direction. So, in this performance test the transient state is more suitable than steady state.

The Atwood no validates the CFD simulation as it is observed. In the case of water and air mixing, bubble formation takes place, whereas in the other case no bubble takes place.