Rayleigh Taylor Stability Analysis

Project title: Rayleigh Taylor Stability Analysis

Objective: 1. Study the rayleigh Taylor stability.

2. Study the VOF model.

3. Study the Atwood number(At)

theory:

Rayleigh Taylor Stability

Rayleigh Taylor Stability 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 behaviour 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 a top oil is an everyday example of Rayleigh–Taylor instability, and it may be modelled by two completely plane-parallel layers of immersible fluid, the more denser on top of the less dense one and both subject to the Earth's gravity. The equilibrium here is unstable to any retribution 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.

Atwood Number

Atwood Number is a dimensionless number that described the density difference between two fluids that shear common interface. It is very important to study hydrodynamic instabilities.

Atwood number = (ρ1−ρ2)÷(ρ1+ρ2)

ρ1 = heavier fluid

ρ2 = lighter fluid

The At number is increased beyond 0.5. the mixing symmetry developed with spikes forming on the lighter fluid and bubbles on the heavier fluid. The falling bubbles and rising spikes are not readily apparent till At>0.5

VOF modelling

In CFD, the volume of fluid (VOF) method is a free surface modelling technique, i.e. a numerical technique for tracking and locating the free surface (or fluid-fluid interface). It belongs to the class of Eulerian methods which are characterized by a mesh that is either stationary or is moving in a certain prescribed manner to accommodate the evolving shape of the interface. As such, VOF is an advection scheme—a numerical recipe that allows the programmer to track the shape and position of the interface, but it is not a standalone flow solving algorithm. The N-S equations describing the motion of the flow have to be solved separately. The same applies to all other advection algorithms.

Model

Geometry dimensions are 20 by 20 mm square blocks of air and water.

Meshing

Setup

1. The transient state model is considered to study the time variation.

2. Gravity turns on in y-direction.

3. laminar flow condition is considered.

4. to study the Rayleigh Taylor Stability VOF model is turn on.

5. An implicit method is used.

6. for VOF patching is done between air and water.

7. for 0.5mm mesh size time sep size 0.005 is considered and for fin mesh 0.00125 considered.

8. In the second case, I use oil with water

Results

Convergence

This is a transient simulation

Videos

Water with air simulation

mesh size used 0.5mm

the time required for water to break the surface tension of air is less compaered to oil

Atwood number is around 0.98 for water and air combination

Water with air simulation

mesh size used 0.4mm

As the mesh size reduce more clear flow path can be observed

Water with air simulation

mesh size used 0.4mm

As the mesh size reduce more clear flow path can be observed

Water and oil combination

mesh size used 0.5mm

the time required for water to break the surface tension of oil is more compared to air

Atwood number is around 0.42 for water and oil combination