Rayleigh Taylor Instability Analysis

Rayleigh Taylor Instability

Rayleigh-Taylor instability, or RT instability is an instability of an interface between two fluids of different densities which occurs when the lighter fluid is continuously pushing the heavier fluid. RT instability is a buoyancy-driven instabilty, where the fluids are initially at hydrostatic equilibrium. There is no shear motion between the two fluids. Some of the practical applications of RT phenomenon are :

1. RTI in a magnetic field:RTI is observed to play a role in many astrophysical systems. The magnetic RTI may be thought of as a competition between two forces: the gravitational potential, pulling the higher
   density fluid through the lower (and vice versa), mixing the two; and the magnetic tension, preventing deformation of the field lines and hence suppressing the mixing. A dimensionless parameter J has been defined, where A system with J `gtgt` 1 is likely to be dominated by the magnetic forces, and one with J `ltlt`1 is likely to be dominated by the gravitational forces. On performing simulations it has been observed that increasing the strength of the magnetic field leads to a decrease in non-linear growth rate of the rising bubbles of the instability.
   
2. Nuclear mushroom clouds:A nuclear/atomic explosion creates a mushroom cloud because the detonating bomb suddenly releases a great deal of heat rapidly, which interacts with the cooler surrounding air and makes it less dense. The giant fireball is extremely hot, which rises rapidly in the air, creating a vacuum that is then rapidly filled by the surrounding air, forming a mushroom cloud in the process. A particularly powerful explosion is accompanied by a sudden release of a great deal of heat. This heat interacts with the surrounding air, making it hotter and less dense, resulting Rayleigh–Taylor instability
3. Aerosol bulk motion due to RT instability:
   Rayleigh-Taylor instability occurs when a dense aerosol layer overlies a layer of clean air. This instability is characterized by abrupt breakthrough of the aerosol into the clean air region at discrete points with periodic spacing. Spacing b =etween breakthrough points, L, is given by Plesset and Whipple equation:
   L= 4.4`pi((2eta)/(rhoa(rhoc - rhoa)g))^(1/3)`, where `rhoa`is density of free air particles and `rhoc`is density of aerosol particles.

RAYLEIGH TAYLOR SIMULATION ON WATER-AIR INTERFACE

Here we are performing a multiphase simulation in 2D to study the effect of Rayleigh Taylor instability on a water-air mixture. The geometry for the same has been shown below:

This is a 20 mm by 40 mm geometry where the top square region in blue is water and the bottom square region is air.

• Solver Time- Transient
• Solver Type- Pressure based
• No of time steps- 1500
• Size of each time step - 0.005

We have run 3 cases of different mesh sizes to make a comparison as shown below:

CASE 1

Mesh size- 0.5 mm

No. of nodes- 3321

No of elements- 3200

A graph showing the initial water volume fraction at t=0 s is shown below:

A graph showing the final water volume fraction at t=1500 s is shown below:

An animation of the mixing of fluids is shown below:

CASE 2

Mesh size- 0.15 mm

No. of nodes- 35780

No of elements- 35380

A graph showing the initial water volume fraction at t=0 s is shown below:

A graph showing the final water volume fraction at t=1500 s is shown below:

An animation of the mixing of fluids is shown below:

CASE 3

Mesh size- 0.05 mm

No. of nodes- 321201

No of elements- 320000

As the meshing is very fine, hence image appears dark

A graph showing the initial water volume fraction at t=0 s is shown below:

A graph showing the final water volume fraction at t=1500 s is shown below:

An animation of the mixing of fluids is shown below:

From the above comparisons, the following things can be concluded:

1. The graph showing the initial condition of water volume fraction shows a broader transition area between water and air if a coarser mesh is used. For a finer mesh, there appears to be almost no gradient in the volume fraction of water in between both the regions.
2. In Case 1 where mesh is the coarsest, the final graph shows that the mixing is almost completed at the end of 1500s and the solution has almost reached a steady state and water is fully settled at the bottom.
3. In Case 2 and Case 3 which has a finer mesh, the mixing of fluids does not seem to have completed at the end of 1500s. The finer the meshing, the more number of time steps are required for the fluids to mix fully.

A steady state solution is not feasible in this case as we want to check the varience of fluid volume fraction and determine how two fluids mix per unit time. A steady state solution would give the final state but not the intermediatary results which occur at a given time.

Atwood number (A) is a dimensionless used in the study of hydrodynamic instabilities in density stratified flows. It is mathematically denoted as:

A = `(rho1 - rho2)/(rho1+rho2)`

where `rho1`= density of heavier fluid and `rho2`= density of lighter fluid.

A high value of Atwood number denotes that Rayleigh Taylor instability will form faster.

In our case `rho1 = 1000 (kg)/m^3` and `rho2= 1.225 (kg)/m^3`

Hence Atwood number = 0.9

Thus, for our simulation RTI should form faster.