Objectives :

• To describe in brief some practical CFD models he mathematical analysis of Rayleigh Taylor waves and to explain how these mathematical models have been adapted for CFD calculations.
• To Perform Rayleigh Taylor instability simulation for mesh sizes of 0.5mm , 0.3mm, 0.2 mm and to compare the results by showing the animations
• To define Atwood number and to find the atwood number for this project and to explain How the variation in Atwood number affects the behaviour of the instability. How can we use the Atwood number to validate our simulation result?

some practical CFD models:

1. The Richtmyer–Meshkov instability (RMI): 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.

2. The Kelvin–Helmholtz instability: This 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.

3. Potential flow models of Rayleigh–Taylor and Richtmyer–Meshkov bubble front:A potential flow model of Rayleigh–Taylor and Richtmyer–Meshkov bubbles on an interface between an incompressible fluid and a constant supporting pressure (Atwood number A=1) is presented

Rayleigh Taylor Instability:

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, supernova explosions in which expanding core gas is accelerated into denser shell gas.

Water suspended atop oil is an everyday example of Rayleigh–Taylor instability, and it may be modeled by two completely plane-parallel layers of immiscible fluid, the more dense on top of the less dense one and both subject to the Earth\'s gravity. The equilibrium here 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 moves down under the (effective) gravitational field, and the less dense material is further displaced upwards.

Case 1 : Mesh size 0.5mm

Animation:

Case 2 : Mesh size 0.3mm

Animation:

Case 3: Mesh size 0.2mm

Animation:

As we go from Case 1 to Case 3 the detailing in the simulation like formation of bubbles increases

Atwood\'s Number (A):

It is a dimensionless number used in the study of instabilities at the interface for fluids with different densities.

It is a dimensionless density ratio defined as A = (rho1-rho2)/(rho1+rho2)

where,

rho1=Density of the heavier fluid (water)

rho2=Density of lighter fluid (air)

A = (997 kg/m^3 - 1.225 kg/m^3)/(997 kg/m^3 + 1.225 kg/m^3) = 0.9975 .