Week 5 - Rayleigh Taylor Instability

AIM:- To Study Rayleigh Taylor Instability and Perform Multiphase Simulation.

OBJECTIVE:-

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.

INTRODUCTION:-

• Rayleigh Taylor Instability:-

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.

Examples include the behavior of water suspended above oil in the gravity of Earth, mushroom clouds like those from volcanic eruptions and atmospheric nuclear explosions, supernova explosions in which expanding core gas is accelerated into denser shell gas, instabilities in plasma fusion reactors and inertial confinement fusion.

In Above Fig we can see the Red Color Fluid is More densed than Blue color fluid. so Denser Fluid tries to Push the Lighter fluid due to Gravity.

• Some practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves are:

1. 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,saturns bands.

2.Plateau Rayleigh instability :

In 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.

3.Richtmeyer meshkow instability :

In 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 pertubations which initially grow linear with time.

4. 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.

MODELING APPROCH:-

CASE-1 (Base Mesh):-

• GEOMETRY:-

Geometry Creation:-

we are created the two square of size 20X20mm one is above the other and we give the name of top square as water and bottom square as air. and use share topology due to that the interface edge is shared, by using Workbench >> Share >> Apply. after that we can get the Pink Edges we can confirm the Share Topology by it. The created geometry is shown as below:

• MESHING:-

Creating Named Selection:-

1. Water
2. Air

Base Mesh:-

Here we used base Mesh (0.5 mm) For Meshing. after that we are getting Mesh as below.

Mesh Element Quality:-

Total No. of Elements:-

We are getting 3200 Total Elements for CASE-1

• SOLVER - SET-UP:-

We are Using Set-up as below:

Solver:-

Here we are going to On the Gravity to capture the Density Effect of water.

Viscous Model:-

Multiphase Model:-

Fluid Materials:-

Method of initialization: Standard

After initialization we required to patch the two phases of the fluid :

Then We have to Create the Report for Volume Fraction Of Water.

Then We have to Create the Contour for Volume Fraction Of Water.

Then Run The Simulation For

Time step size (sec)= 0.01

No of Time step : 1100

RESULTS:-

Residual Plot:-

Area Weighted Avg VOF Report:-

Animation Link: Mesh size : 0.5mm

https://drive.google.com/file/d/1POdFha6ZU3KapQ0WZT-fvo3Jtt89A3TS/view?usp=sharing

CASE-2 (Refined Mesh):-

GEOMETRY:-

Geometry Used Same as Case-1

MESHING:-

Refined Mesh:-

Here we used base Mesh (0.2 mm) For Meshing. after that we are getting Mesh as below.

Mesh Element Quality:-

Total No. of Elements:-

We are getting 20000 Total Elements for CASE-1

SOLVER - SET-UP:-

We are Using Same Set-up as Case-1. only Time Step size and No. of time step Changed. because we have to Avoid the Floating Point Error.

Then Run The Simulation For

Time step size (sec)= 0.005

No of Time step : 900

RESULTS:-

Residual Plot:-

Area Weighted Avg VOF Report:-

Animation Link: Mesh size : 0.2 mm:-

https://drive.google.com/file/d/1caKPLDpnygTYTTP8PeuMVthrhXa2HbPj/view?usp=sharing

CASE-3 (User Defined Material with Refined Mesh):-

GEOMETRY:-

Geometry Used Same as Case-1

MESHING:-

Refined Mesh:-

Here we used base Mesh (0.2 mm) For Meshing.

SOLVER - SET-UP:-

We are Using Same Set-up as Case-1. onlyUser Defined Material Used for Simulation and Time Step size and No. of time step also Changed. because we have to Avoid the Floating Point Error.

Then Run The Simulation For

Time step size (sec)= 0.005

No of Time step : 1100

RESULTS:-

Residual Plot:-

Area Weighted Avg VOF Report:-

Animation Link (User Defined Material with Refined Mesh):-

https://drive.google.com/file/d/1mH1LJ2DmNDtdenHyxX20br5fmIgEAKMb/view?usp=sharing

Atwood Number:

It is a dimensionless number that is used to study hydrodynamic instabilities in density stratified flows.

It is denoted by letter `alpha`

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

where,

`rho_1=`Density of heavier fluid

`rho_2 =`Density of lighter fluid

Calculate the Atwood Number for Air-water instability

`rho_1=998.2 (kg)/(m^3)` ..........density of water

`rho_2=1.225 (kg)/(m^3)`..........density of water

put this value in above equation we are getting `alpha`=0.9975

Calculate the Atwood Number for Water and User Defined material RT instability

`rho_1=998.2 (kg)/(m^3)` ..........density of water

`rho_2=400 (kg)/(m^3)`..........density of User defined material

put this value in above equation we are getting `alpha`=0.4278

The Atwood number close to 1 then the lighter fluid below the heavier fluid which forms a larger bubbles.

The Atwood number close to 0 then the lighter fluid below the heavier fluid which forms a bubbles.

CONCLUSION:-

• Simulation will be performed as multiphase as two fluids are presents. For the multiphase we are going to use Volume of Fluid method, here we used Air and Water as a Materials.

• The Refined mesh captures the instability better than the Base mesh (Course Mesh). but it takes more time for the heavier fluid to settle down. Means We are getting Correct Results.

• The result obtained by Refined Mesh having better accuracy.

• Transient solver better captures the flow and can be used to run at smaller time steps.

• For case 1 it was very evident that lack of mesh elements results in very fast mixing and quick gaining of equilibrium position by both of the liquids. In second case we can see that the mixing is slowed down and number of iteration to complete the simulation has been increased as well.

• From the result of volume fraction graph with respect to time we can see that drop of area weighted average of volume fraction for water is earlier in Base mesh setup and as the mesh size is decreasing the drop get moving forward in time, this shows that as the No. of elements are increased we are getting better capture of flow.