CFD analysis of Rayleigh Taylor instability

Practical CFD models based on the mathematical analysis of Rayleigh Taylor waves are Richtmyer–Meshkov instability (RMI), Kelvin–Helmholtz instability and Plateau–Rayleigh instability.

The 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 fluid 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.

The Kelvin–Helmholtz instability can occur 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, clouds, the ocean, Saturn\'s bands, and the sun\'s corona show this instability.

The Plateau–Rayleigh instability explains why and how a falling stream of fluid breaks up into smaller packets with the same volume but less surface area. It is a part of a greater branch of fluid dynamics concerned with fluid thread breakup. The driving force of the Plateau–Rayleigh instability is that liquids, by virtue of their surface tensions, tend to minimize their surface area.

In this project, Rayleigh Taylor instability simulations are performed for 3 different mesh sizes.

Case 1:

Case 1 will set the baseline for the other two cases and hence, the results can be compared.

Setting mesh size to 0.5mm, the number of elements are 3200.

Practically, the gravity is acting downwards. Hence, the value of gravity is set on Y-axis as -9.81m/s^2.

2D mesh is divided exactly in half. Water flows over the upper half of the mesh where as air flows over the below part of the mesh. Air being lighter than water, it tends to move upwards and simulataneously water is being flown in the downward direction.

Case 2:

Mesh size is reduced to 0.3mm. The number of elements are 8978.

Case 3:

Mesh size is reduced further to obtain 80000 elements.

As observed from the cases above, as the element size decreases the bubble formation along the model increases.

The collapse of the interface with respect to time is observed in these cases. Hence, transient state analysis is being used as compared to the steady state analysis.

Atwood number is a dimensionless number that describes the density difference between two adjacent fluids with a common interface.

Here, Atwood number,

A=ρ1−ρ2/ρ1+ρ2=1000-1.25/1000+1.25=0.9975

Where:

A = Atwood number

ρ1 = density of heavier fluid

ρ2 = density of lighter fluid

For Atwood number close to 0, Rayleigh Taylor instability flows take the form of symmetric \"fingers\" of fluid. For Atwood number close to 1, the much lighter fluid \"below\" the heavier fluid takes the form of larger bubble-like plumes.