Rayleigh-Taylor Instability

Rayleigh-Taylor Instability

It is an instability of an inteface between two fluids of different densites which occurs when the lighter fluid is pushing the heavier fluid. The equilibrium is unstable to any perturbations or disturbances of the interface: if a parcel of heavier fluid is displaced downward with an equal volume of lighter fluid displaced upwards, the potential energy of the configuration is lower than the initial state. Thus the disturbance will grow and lead to a further release of potential energy, as the more dense material is further dispalced upwards. Examples includes:

• Behavior of water suspended above oil the Gravity of earth
• Mushroom clouds like those from volcanic erruptions and atmospheric nuclear explosions
• Supernova explosions in which expanding core gases is accelerated into denser shell gas
• Instabilities in plasma fusion reactors and inertial confinement fusion

Some of the CFD models that have been used extensively based on RT instability

Kelvin-Helmholtz (KH) instability

It is the model that describes the instability can occur when there is velocity shear in a single continuous fluid or where there is a velocity difference across the interface between two fluids. An example is wind blowing over water. The instability manifests in waves on the water surface. More generally cluods, the oceans, Saturn\'s bands, Jupiter\'s Red Spot, and the sun\'s corona show this instability.

Plateau-Rayleigh Instability

It is part of a greater brach of fluid dynamics concerned with fluid thread breakup. It expalains why and how a falling system of fluid breaks up into smaller packets with the same volume but less surface area. This fluid instability is exploitted in the design of a particular type of ink jet technology whereby a jet of liquid is perturbed into a steady stream of deoplets.

Richtmyer-Meshkov Instability

RMI occurs when two fluids of different density are impulsively accelerated. Normally this is by the passage of a shock wave. The development of the instability begins with small amplitude perturbations which initially grow linearly with time. This is followed by a nonlinear regime with bubbles appearing in the case of a light penetrating a heavy fluid, and with spikes appearing in the case of a heavy fluid penetrating a light fluid. A chaotic regime eventually is reached and the two fluids mix.

In additions to these there are other several models which are being adapted for CFD calculations such as The Taylor analogy Breakup (TAB), Hybrid models like Pelloni and Bianchi, Lattice Boltzman model, Mushroom cloud, etc

Simulations procedure and setups:

Two 20x20 mm size squares are created and shares their topologies in spaceclaim. For different sizes of meshes Implicit multiphase for water and air volumes are used and then its propertiess has been patched 1 and 0 respectively. The time step has been choosen carefully so that avoid floating point exception error (smallar time steps preffered).

• Baseline Mesh (0.5mm elemnet size)

Animation:

https://drive.google.com/file/d/1cmuNSfCye7vVZFS3Aie7St9is_jfJWD9/view?usp=sharing

• Element size 0.3mm Mesh

Animation:

https://drive.google.com/file/d/1dla1wzapY2lV-XdSACSG9pVodSlQbged/view?usp=sharing

• Element size 0.2mm Mesh

Animation:

https://drive.google.com/file/d/1n5aQQxiyRks48xLNyXggB9HK3aBOjIcQ/view?usp=sharing

• Element size 0.1mm Mesh

Animation:

https://drive.google.com/file/d/1XTE_q1jc366HS85Bj7dK-uloIyBu5c5Y/view?usp=sharing

Why a steady state approach might not be suitable for these types of simulation?

• Steady state don\'t account for time, so in simple terms when time reaches infinity. The transient solver solves each time steps, so the results will have reached a finite time, so this simulations will not long enough to reach the equilibrium state.
• Steady state solution takes a few hundred to few thousand iterations to converge and has one answer: a transient solution takes thousands of timesteps each of 15-20 iterations so will take significantly longer time to run, Hence we need to understand the instabilities in small time steps for this case, transient is convenient to get the accurate results.

Atwood Number

The Atwood number (A) is a dimensionless number in fluid dynamics used in the study of hydrodynamic instabilities in density stratified flows. It is a dimensionless density ratio defiend as

`A = (rho_1 - rho_1 )/(rho_1 +rho_1 )`

where

rho_1 = density of heavier fluid = 997 kg/m^3 for water

rho_2 = density of lighter fluid = 1.225 kg/m^3 for air

Substituting and calculating we get

Atwood number = 0.9975

When Atwood number is close to 0, the instability flows take the form asymmetric \'finger\' of fluid, when it is around 1, the much lighter fluid below the heavier fluid takes the form of larger bubble like plumes. This literally means the higher the Atwood number the instability will be higher.

Since we calculated the Atwood number close to 1 in our case, the way our instabilty took place validate our result. As seen from animations as well, water is poured onto the comparatively really low dense air beneath, the bubble spikes generates which travelled upward region in the form of waves and instability pictured as it went further.