Rayleigh Taylor Instability Challenge

Objective :

We need to perform the Rayleigh Taylor instability simulation for 3 different mesh sizes with the base mesh being 0.5 mm. and also explaining why a steady-state approach might not be suitable for these types of simulation.

Q1.Some practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves

Practical CFD models that are based on mathematical analysis of rayleigh Taylor waves:

1.Kelvin–Helmholtz instability.

2.Richtmyer–Meshkov instability.

3.Plateau–Rayleigh instability.

1.Kelvin–Helmholtz instability:

The 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.

An example is a wind blowing over water: The instability manifests in waves on the water surface.

2.Richtmyer–Meshkov 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. This instability can be considered the impulsive-acceleration limit of the Richtmyer–Meshkov instability.

3.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.

Q2. Performing the Rayleigh Taylor instability simulation for 3 different mesh sizes with the base mesh being 0.5 mm

Base mesh - 0.5:

Refined mesh - 0.3:

Refined mesh - 0.2:

Mesh Independency Study:

The Simulation is carried out by using three different grid sizes. the base mesh is 0.5it is further refined to 0.3 and 0.2 mm respectively as shown

Simulation for different grid size:

Size Nodes Elements

0.5 3321 3200

0.3 9180 8978

0.2 20301 200000

CASE-1

CASE-2

CASE-3

1. The animations of the three cases are shown above.

2. In this case, the steady-state simulations are not beneficial, as we already know the end result at a steady-state that is, the denser Fluid gets to settle down and the lighter density fluid rises above.

3. We are interested in the mixing with a time of heavy dense fluid with less dense fluids and we therefore need a transient simulation of such situations.

Q3. Define the Atwood Number. Find out the Atwood number for this case and explain how the variation in Atwood number affects the behavior of the instability.

Atwood Number:

The development of the R-t mixing region was characterized as a three-step process.

1. Originally, an exponential increase in infinitesimal fluctuations leading to linear stability analysis.

2. At a wavelength of roughly one-half, The instability is concentrated and the wavelength was longer(that have yet to reach saturation) take over. Emmons et al. (1960) also coined this regime\'s term \' bubble competition.

3. Ultimately, a self-similar R-T mixing layer with a scalar similarity is formed by nonlinear mode interaction and successive wavelength saturation gt^2.