Rayleigh Taylor Instability Challenge

Objective

To perfom Rayleigh Taylor instability simulation for 2 different mesh sizes and explain the effect of mesh size on simulation.

Introduction

Rayleigh Taylor instability is an instability of an interface between two fluids that occurs whenever a light fluid pushes on a heavy fluid. It is a dynamic process whereby the two fluids seek to reduce their combined potential energy.

Some examples of Rayleigh Taylor instability are:-

1. Salt dome formed when a thick bed of evaporite minerals (mainly salt, or halite) found at depth intrudes vertically into surrounding rock strata forming a diapir.
2. Normally, air temperature decreases with an increase in altitude. During weather inversion, warmer air is held above cooler air and the normal temperature profile with altitude is inverted.
3. RT instability structure is evident in the Crab Nebula, in which the expanding pulsar wind nebula powered by the Crab pulsar is sweeping up ejected material from the supernova explosion 1000 years ago.

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

Kelvin-Helmholtz Instability

It is a hydrodynamic instability in which immiscible, incompressible, and inviscid fluids are in relative and irrotational motion. It typically occurs when there is velocity shear in a single continuous fluid and where there is a velocity difference across the interface between two fluids. When two fluids separated by a surface move with different velocities, their interface could be unstable due to inertia. When the shear velocity exceeds the critical value, surface waves arise and can lead into the intense mixing of both fluids.

In nature the phenomenon can be observed in sudden wind over the calm sea surface or in a single-phase atmospheric motions.

Plateau-Rayleigh Instability

The Plateau-Rayleigh Instability describes the phenomenon where a falling jet or cylinder of fuid at one point ceases to be a jet and breaks into multiple droplets of smaller total surface area. The driving force of the Plateau–Rayleigh instability in liquids is due to surface tensions by virtue of which they tend to minimize their surface area.

A special case of this is the formation of small droplets when water is dripping from a tap.

Richtmyer Meshkov Instability

It is the fluid flow phenomenon that occurs when a shock wave impinges (in the normal direction) upon an interface separating two gases having different densities. 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. Although the shock interaction process is inherently compressible, Richtmyer–Meshkov instability can also be produced by the impulsive acceleration of two incompressible fluids. In this sense,Richtmyer–Meshkov instability is related to the better known Rayleigh–Taylor instability, and is often referred to as impulsive Rayleigh–Taylor instability.

When a heavy fluid is resting on top of a light fluid in a gravitational field, the interface separating the two fluids is unstable with perturbations initially growing exponentially in time. A wide variety of fluid motions result in order to lower the potential energy of thesystem. The same configuration with light fluid on top of heavy fluid is stable with perturbations on the interface remaining bounded. This instability is commonly referred to as Rayleigh-Taylor (RT) instability after its discoverers. In contrast, if
the interface separating the two fluids is impulsively accelerated (say by a shock wave), the interface again goes unstable and the resulting instability is referred to as Richtmyer-Meshkov (RM) instability.

Geometry

2-D surface is generated with upper surface named as water and lowe surface as air.

Meshing

Mesh size=0.5mm

No. of nodes=3,321

No. of element=3,200

Simulation Setup

The model is simulated using transient laminar flow. Volume of fluid is method is applied because it is multi-phase simulation. Gravitaional force is acting in negative Y-direction.

Top fluid is water with volume fraction equal to 1, and

Bottom fluid is air with volume fraction equal to 0.

Result

Residual Plot

Volume Fraction Contour Plot at 7.5 sec

Animations for different mesh size

1. mesh size=0.5mm

2.mesh size=0.3mm

3. mesh size=0.2mm

Google drive link https://drive.google.com/drive/folders/1CKm6v0g5gwg3WzxvPa7gFm1qNLA8xbKT?usp=sharing

Comparision of results for different mesh size

More number of buubles and vortices are visible as the mesh size reduces. The vortices on the finest grid have a tightly wound spiral, where as on the coarsest grid there is very little roll-up. Outside the vortex the interfaces differ less, although the position of the vortices and the amplitude of the spike show a weak dependence on resolution.

Reason why a steady-state approach might not be suitable for these types of simulation

Rayleigh Taylor instability occurs due to sudden change in energy of system. Horizontal plane between two fluid get collapsed and shock is generated. This results in formation of spikes and bubbles. All these process are time dependent. Hence, steady state solver cannot work for time dependent process and depict the actual motion of fluid.

Simulation with water and user-defined material(density = 400 kg/m³, viscosity = 0.001 kg/m-s) for refined mesh size 0.3mm

Residual Plot

Volume Fraction contour plot at 7.5 sec

Atwood Number (A)

Atwood number is a dimensionless number in fluid dynamics used in the study of hydrodynamic instabilities in density stratified flows. It describes density difference between two adjacent fluids with a common interface.

`A={rho_1-rho_2}/{rho_1+rho_2}`

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

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

Density of Air`=1.2754{ kg}/{m³}`

Density of Water`=997 {kg}/{m³}`

User defined Material`=400{kg}/{m^3}`

Atwood number for Air and water

`A={997-1.2754}/{997+1.2754}`

`=0.9974`

Atwood number for user defined material and water

`A={997-400}/{997+400}`

`=0.4273`

Effects of Atwood number in above two cases

For first case, the two interacting fluids are air and water. Atwood number for this case is close to 1. The process is quiet rigorous and bubbles are large like plumes.

For second case, Atwood number is close to 0.42. The process is slower and vortex formed is more evident than first case.