AIM:- Perform the Rayleigh Taylor instability simulation for 2 different mesh sizes with the base mesh being 0.5 mm. Compare the results by showing the animations and Run one more simulation with water and user-defined material(density = 400 kg/m³, viscosity = 0.001 kg/m-s) for refined mesh.

Find out the Atwood number for the above cases and explain how the variation in Atwood number in the above cases affects the behavior of the instability.

Introduction Rayleigh Taylor Instability

The Rayleigh–Taylor instability, or RT instability, 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, mushroom clouds like those from volcanic eruptions and atmospheric nuclear explosions, 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 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. This was the set-up as studied by Lord Rayleigh. The important insight by G. I. Taylor was his realisation that this situation is equivalent to the situation when the fluids are accelerated, with the less dense fluid accelerating into the more dense fluid. This occurs deep underwater on the surface of an expanding bubble and in a nuclear explosion.

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

Kelvin–Helmholtz instability:

It generally occurs when velocity shear in a single continuous fluid, or additionally where there is a velocity difference across the interface between two fluids. A common example is seen with wind blowing over water.

Plateau-Rayleigh instability

It explain how a falling stream of fluid breaks up into smaller packets with the same volume but less surface area.

Karman vortex street

It is a repeating pattern of swirling vortices, caused by a process known as vortex sheeding, which is responsible for the unsteady seperation of flow of a fluid around blunt bodies.

Salt fingering

Salt fingering is a mixing process that occurs when relatively warm, salty water overlies relatively colder, fresher water. It is driven by the fact that heated water diffuses more readily than salty water.

Above example are the mixture of the two fluid which shows the rayleigh instability .

Rayleigh Taylor Instability Simulation

In this simulation air and water is used as the fluid ,water is taken as the above ,air is taken below the water surface. Density of water and air is taken as 998.02 and 1.225 kg/m^3 respectively. For conformal mesh between air and water surface use share topology.

Case -1

Meshing

Here three different mesh size is used

Mesh	Elements	Nodes
0.5mm	3200	3321
0.3mm	8978	9180
0.1mm	80000	80601

Snap shot for 0.5mm mesh

Snap shot for 0.3mm mesh

Snap shot for 0.1mm mesh

Setup

Solver Type-pressure-based Velocity formulation- Absolute Time- Transient 2D space-planer Gravity -9.81m/s^ in y direction

Setting of physics Viscous model-Laminar Multi phase model-Volume of fluid Courant number-0.25

solution Initilization- standard Time step size-0.0005 No of time step-1500

Video linc for animation of 0.5mm mesh size

https://drive.google.com/open?id=1peV2VeT-sKWmNJ2Dheh589GVmdSMioI7

https://youtu.be/lzu60WxQU-s

Video linc for animation of 0.3mm mesh size

https://drive.google.com/open?id=1MjeKSlSzu3vmQAJGjfesHIgf5DQaiM0C

https://www.youtube.com/watch?v=2VMNGmqAB04

Video linc for animation of 0.1mm mesh size

https://drive.google.com/open?id=1qBy4x7WSz8zE-_LGVwc2dsXF1HfxMzMt

https://youtu.be/yIY8uPafx34

Video linc for animation of 0.1mm mesh size user defined material

https://drive.google.com/open?id=10edYOJgHwjJPJF8Rh7TY3NxejO2rKcmT

https://youtu.be/Omjp0COFXwA

The above simulation can not run in steady state condition because in each and every iteration is vary with respect to time due to gravity (heavy fluid is above the lower fluid).

Conclusion

1. For 0.5 mm mesh size large disturbances is found on the water air interface its forces in to high dense fluid like a shock wave.

2.For 0.3mm mesh size less disturbances is found on the water air interface, pushing through the high dense fluid mushroom-like structure is formed and the formation of bubbles can be seen around the mushroom structure.

3. For 0.1 mm mesh size disturbances found slowly and more clearly visible air bubbles on the interface and below.

4.For 0.1mm mesh size user defined material and water mushroom like are clearly visible and stability is with respect to all above cases.

Atwood Number

The Atwood number (A) is a dimensionless number in fluid dunamic used in the study of hydrodynamic instability in density stratified flows. It is a dimensionless density ratio defined as

where

= density of heavier fluid

= density of lighter fluid

For case 1

`A=(998.02-1.225)/(998.02+1.225)`

0.9975

CAse-2

`A=(998.02-400)/(998.02+400)`

=0.427

• When the Atwood number is so high the more instability takes place in the fluid and measuring the spikes of the instability is very difficult.
• When the Atwood number is low the more stable is the fluid mixture and measuring the spikes of the instability can be done to some extent because it expresses linearity to some extent.