STUDYING RAYLEIGH TAYLOR INSTABILITIES

Rayleigh Taylor Instabilities Report

Overview

• What is Rayleigh Taylor Instability?

The Rayleigh Taylor Instability is the Instability of an interface between two fluids of different densities which occurs when the lighter fluid is pushing the heavier fluid. It is a dynamic process in which two fluids tend to reduce their potential energy. 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 gravitational field, and the less dense material is further displaced upwards.

• Some CFD models based upon the Mathematical analysis of Rayleigh Taylor waves and their adaption for CFD Calculations:

1. The Plateau-Rayleigh 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 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. The driving force of the Plateau-Rayleigh instability is that liquids, by virtue of their surface tensions, tend to minimize their surface area. A considerable amount of work has been done recently on the final pinching profile by attacking it with self-similar solutions.

2. The Richtmyer–Meshkov instability (RMI) - It 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. This instability can be considered the impulsive-acceleration limit of the Rayleigh Taylor Instability.
3. The Kelvin–Helmholtz instability - It occurs 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 a wind blowing over water: The instability manifests in waves on the water surface. Numerically, the KH instability is simulated in a temporal or a spatial approach. In the temporal approach, experimenters consider the flow in a periodic (cyclic) box \"moving\" at mean speed (absolute instability). In the spatial approach, experimenters simulate a lab experiment with natural inlet and outlet conditions (convective instability).

• In this CFD Simulation analysis has been done for three cases of the geometry having different mesh sizes which are discussed below.

Case 1 - Fine Mesh

Geometry - Two squares of 20 mm x 20 mm have been placed over one another in the same plane.

Meshing (Element Size - 0.5 mm)

Rayleigh Taylor Instabilities Solution

Case 2 - Finer Mesh

Geometry - Two squares of 20 mm x 20 mm have been placed over one another in the same plane.

Meshing (Element Size - 0.3 mm)

Rayleigh Taylor Instabilities Solution

Case 3 - Finest Mesh

Geometry - Two squares of 20 mm x 20 mm have been placed over one another in the same plane.

Meshing (Element Size - 0.1 mm)

Rayleigh Taylor Instabilities Solution

Meshing & Solution Comparison

Element No. of Time Step No. of Time

Size Elements Size Steps

Case 1 0.5mm 3200 0.005 2000

Case 2 0.3 mm 8978 0.004 2000

Case 3 0.1mm 80000 0.002 2000

Observation: It has been observed that when the mesh was refined, the time step size was also required to be decreased as otherwise floating-point error was generating in Ansys Fluent.

Results and conclusions:

• Why the Transient state approach and not a steady-state approach?

In this Simulation, we have opted for the Transient State Analysis instead of the Steady-state analysis. This is done because in this case the results are varying with the increment in the time and are therefore time-dependent. Also, the Steady-state analysis is done to achieve final results and therefore any intermediate results achieved would not be time accurate. So, to capture the instabilities with the time, we performed Transient State Analysis.

• What is 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 defined as :

where

= density of the heavier fluid

= density of the lighter fluid

Here, the density of the heavier fluid (water) is 997 Kg/m3 and that of lighter fluid (air) is 1.225 Kg/m3. Therefore,

A = (997 - 1.225) / (997 + 1.225)

A = 0.9975

• Effects of variation in Atwood\'s number?

The rate of disturbance of the interface depends upon the value of Atwood\'s Number.

1. For Atwood\'s no. near zero (Smaller values) - The mixing of two different fluids occurs slowly and the bubbles and spikes formed are quite similar in shape.
2. For Atwood\'s no. near to unity (Larger Values) - The mixing of two fluids occurs at a higher rate and the lighter fluid below the heavier fluids tends to take the form of a larger bubble-like plume.

• Validating The results obtained using Atwood\'s Number?

The results achieved by these simulations and the value of Atwood\'s Number can be verified with the help of past experiments and research works. Many scientists and physicists have worked upon the Rayleigh Taylor Instabilities for different fluids and have observed the values of Atwood\'s Number. So, the results obtained by us can be verified by referring to these past observations.

• The Rayleigh Taylor Instabilities were studied and Simulated.