Week 5 - Rayleigh Taylor Instability

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 denser fluid 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 denser 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 denser fluid. This occurs deep underwater on the surface of an expanding bubble and in a nuclear explosion.

Types of instability and some practical CFD models that have been based on the mathematical analysis of Rayleigh–Taylor waves:

• Richtmyer–Meshkov instability : 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.
• 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.
• Kelvin–Helmholtz instability : occurs when there is velocity shear in a single continuous fluid or a velocity difference across the interface between two fluids. Kelvin-Helmholtz instabilities are visible in the atmospheres of planets and moons, such as in cloud formations on Earth or the Red Spot on Jupiter .

Geometry Setup

• Open spaceclaim from Ansys workbench.
• Choose a plane and create a square of 20mm and pull it to make surface.
• Rename it as air and create another square on top of it and then pull it and rename as water.
• Select both geometry and click on share to create share topology.
• The geometry will look like

Meshing

After creating geometry create a mesh of size 0.5mm

Setup

After updating mesh click on setup and follwoing steps has been done

• Solver type : Pressure based
• Time : Transient
• gravity g=-9.81 m/s^2
• Viscous model :laminar
• Add material of water using fluent database
• Change phase names to air and water
• Choose multiphase model

• Initialize using standard and patch the water and air zones with water volume fraction.
• Create a contour for volume fraction of water and create a anime.
• Set the time step as 0.025 and 300 iteration of time step .

Result:

Residual

Final contour plot of volume fraction

The video link for this base case

Case_2

Now refine the mesh with element size of 0.2 mm and follow the same procedure

Choose the time step as 0.005 and run for 300 iteration

Residual

Final time contour

The video link for this case

Case_3

Now refine the mesh with element size of 0.2 mm and follow the same procedure

First create a user defined material and set the proporties as per given.

Choose the time step as 0.005 and run for 300 iteration

Residual

Final time contour

The video link for this case

Conclusion

From the above result we can observe the hydrostatic equilibrium and Rayleigh-Taylor instability. The lower density fluid pushes the higher density fluid thus disturbance at the interface. As we refine the mesh simulation results get smoother about the irregularities. In case-3 the solution looks more smooth. Since the sudden collapse at the layer is a time dependent phenomenan steady state solver will not work.

Atwood number (A)

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

`A={\frac {\rho _{1}-\rho _{2}}{\rho _{1}+\rho _{2}}}`

where

`rho_1` = density of heavier fluid

`rho_2` = density of lighter fluid

For air and water A=0.9975

For Case_3 A=0.427

If A is closer to 0 then RT instability takes the form of symmetric finger of fluid . If A is closer to 1 then heavier fluid will takes the form of larger bubble.