Week 5 - Rayleigh Taylor Instability

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.

The Rayleigh–Taylor instability, or RT instability (after Lord Rayleigh and G. I. Taylor), 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.

CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves are as follow:

• Kelvin–Helmholtz instability

Kelvin–Helmholtz 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. It typically occurs when there is velocity shear in a single continuous fluid, or additionally where there is a velocity difference across the interface between two fluids. A common example is seen when wind blows across a water surface; the instability constant manifests as waves.

• Richtmyer–Meshkov instability

The Richtmyer–Meshkov instability occurs when two fluids of different density are impulsively accelerated. Normally this is by the passage of a shock wave. This fluid instability is exploited in the design of a particular type of inkjet technology whereby a jet of liquid is perturbed into a steady stream of droplets.

• Plateau–Rayleigh instability

The Plateau–Rayleigh instability, often just called the 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 related to the Rayleigh-Taylor instability and is part of a greater branch of fluid dynamics concerned with fluid thread breakup. This fluid instability is exploited in the design of a particular type of ink jet technology whereby a jet of liquid is perturbed into a steady stream of droplets

• Mushroom cloud

A mushroom cloud is a distinctive mushroom-shaped flammagenitus cloud of debris, smoke and usually condensed water vapor resulting from a large explosion. The effect is most commonly associated with a nuclear explosion, but any sufficiently energetic detonation or deflagration will produce the same effect. Mushroom clouds result from the sudden formation of a large volume of lower-density gases at any altitude, causing Rayleigh–Taylor instability. The buoyant mass of gas rises rapidly, resulting in turbulent vortices curling downward around its edges, forming a temporary vortex ring that draws up a central column, possibly with smoke, debris, and/or condensed water vapor to form the "mushroom stem".

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.

Geometry:

Meshing:

Baseline Simulation Element size of 0.5mm:

Setup:

After updating the mesh open the fluent, here we set the general setting to pressure-based solver with transient time simulation because we have to see the interface collapse with the time. You also have to enable gravity with a value of

-9.81 m/s^2.

In physics, we choose a laminar flow viscous model and enable multiphase model as we are dealing with the mixing of two fluids i.e air, water.

Air is the primary phase and water is the secondary phase.

In the patch function, use value of one for water surface and value of zero for air surface.

Timesteps: 100 timestep size: 0.025

Results:

Animation: https://drive.google.com/file/d/1eGgulBNCW5Wo-e2iHi6jixH9SMrAXqoQ/view?usp=sharing

The 2^nd simulation will run with a mesh size of 0.4mm

Animation file: https://drive.google.com/file/d/1XSxpH5G0ibpBPf-RVINzBb7uFfvMAKkm/view?usp=sharing

The 3^rd simulation will run with a mesh size of 0.35mm

Animation file: https://drive.google.com/file/d/1hcYKbgn6bN5-BlsClFvfb5eE09QVXWov/view?usp=sharing

User Defined Simulation:

density = 400 kg/m

viscosity = 0.001 kg/m-s

mesh = 0.35mm

Define the Atwood Number. Find out the Atwood number for both the cases and explain how variations in atwood number in the above two cases affects the behavior of the instability.

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 defined as

For Case 1,

A1 = 998.2-1.225/998.2+1.225 = 0.9975

As A close to 1, the much lighter fluid "below" the heavier fluid takes the form of larger bubble-like plumes

For Case 2,

A2 = 998.2-400/998.2+400 = 0.427

For A close to 0, RT instability flows take the form of symmetric "fingers" of fluid.