Rayleigh Taylor Instability Challenge

OBJECTIVE

1. Describe in brief about CFD models that are based on mathematical analysis of Rayleigh Taylor instabilities
2. To perform Rayleigh instability simulation on different refined mesh and compare the results
3. To find out the Atwood number and finds its effects on instability by variation and how could we validate simulation using Atwood number

THEORY

The Rayleigh–Taylor instability, or RT instability, is the 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 cloud-like those from a volcanic eruption and atmospheric nuclear explosion, supernova explosions in which expanding core gas is accelerated into denser shell gas, instabilities in plasma fusion reactors and inertial confinement fusion.

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 denser on top of the less dense one and both subject to the Earth\'s gravity. The equilibrium here is unstable to any perturbation 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.

As the RT instability develops, the initial perturbations progress from a linear growth phase into a non-linear growth phase, eventually developing \"plumes\" flowing upwards (in the gravitational buoyancy sense) and \"spikes\" falling downwards

Based on the Rayleigh Taylor instability the CFD models are

Richtmyer Taylor instability

The Richtmyer instability occurs when two fluids with different viscosity are impulsively accelerated. These effects can be seen in shock waves where initially the shock waves are developed small and grow linearly with time

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 inkjet technology whereby a jet of liquid is perturbed into a steady stream of droplets

Kelvin Helmholtz instability

Kelvin Helmholtz instability occurs when there is velocity shear in a continuous medium or when the velocity layer between two fluids is different. A classic example of this instability can be seen when atmospheric air is blow over a water body. Also, this instability can be seen in sun corona clouds and Saturn brands.

Ansys simulation

Geometry

The geometry consists of two regions in the shape of a square placed one above the other with both having the same dimension length and breath 20mmx20mm.

◌.Use the pull tool to enable the surface on both geometries and designate lower layer as air region and bottom layer as water region ie dense fluid

◌.Enable share topology and use share option to so to obtain conformal mesh in the meshing process

MESH

As the objective is to find results from three refined cases of mesh, we will therefore, take three cases of mesh

Case I

Element size:0.5mm

No of elements:3200

No of nodes:3321

Case III

Element size:0.25mm

No of elements:6498

No of nodes:6670

Case III

Element size:0.25mm

No of elements:12800

No of nodes:13400

SETUP

Solver

Type: Pressure base

Time: Transient

Velocity: Absolute

Gravity:-9.81m/s along y-direction

Model: Multiphase

Volume of Fluid

Number of Eulerian Phases :2

Implicit Solver

Phases: Primary Phase -air

Secondary Phase-water

Viscous: Laminar Model

Material: Air and Water and for part II oil and water

Initialize it and then go to patch to setup

RESULTS

CASE I

Residual

CASE II

Residual

CASE III

Residual

OBSERVATION

Increment in the refinement of the mesh results in greater visualization of instabilities.

The computational time increases from coarser to the refiner

The Atwood number is a dimensional number in fluid dynamics used in the study of hydrodynamic instabilities in density stratified flows. It is a dimensionless density ratio defined as

A = ρ 1 − ρ 2 ρ 1 + ρ 2 {\\displaystyle \\mathrm {A} ={\\frac {\\rho _{1}-\\rho _{2}}{\\rho _{1}+\\rho _{2}}}}

where

ρ 1 {\\displaystyle \\rho _{1}} = density of heavier fluid

ρ 2 {\\displaystyle \\rho _{2}} = density of the lighter fluid

The density of water =997 kg/m^3

The density of air =1.255 kg/m^3

Atwood number=0.9976

PART II

Rayleigh Taylor instability simulation with Oil-Water interface

with having Density=400kg/m and viscosity =

Residual

Here, in this case, the Atwood number is 0.4273

The effect that can be drawn from the above parts of the simulation that

if A≥1 result in rapid penetration of fluids into each other region, the lower denser fluid forms bubble and moves upwards and the higher dense fluids forms spikes and gets into the lower dense region,

if A<1 which is the simulation of oil and water interface results in a slow process of penetration and at the end formation of two swapped regions are seen

The simulation can be done with steady-state conditions because it is time-independent and for this Rayleigh Taylor instabilities simulation is where we need to capture the physical phenomena at each every time so instead, we use transient simulation which is time-dependent