RAYLEIGH TAYLOR INSTABILITY

Rayleigh Taylor instability:

From the Newtons law of viscosity, we know that even the fluid cannot resist the slightest shear force. Density is one of the main property of the fluid which the shear force can incorporate with it.

Depending upon the density we have lighter fluid like water etc and denser fluid like mercury, oil, honey etc.

Lord Rayleigh observed that when two liquids i.e lighter and heavier fluids are mixed together where the deser fluid is top on the lighter fluid, due to the gravity and the density variation the interface of the liquid is not stable. (i.e) water suspended on the oil .

So, the instability of the interface between the different densities of fluid when the lighter fluid is accelerated towards the heavier fluid is knowns as RAYLEIGH TAYLOR INSTABILITY.

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

RITCHMYER-MESHKOV INSTABILITY:

The Richtmyer-Meshkov instability arises when a shock wave interacts with an interface separating two different fluids. It combines compressible phenomena, such as shock interaction and refraction, with hydrodynamic instability, including nonlinear growth and subsequent transition to turbulence, across a wide range of Mach numbers.

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. This fluid instability is exploited in the design of a particular type of ink jet technology whereby a jet of liquid is perturbed into a steady stream of droplets.

Kelvin–Helmholtz instability:

can occur when there is velocity shear in a single continuos fluid, or where there is a velocity difference across the interface between two fluids. An example is a wind blowing over water: The instability manifests in waves on the water surface. More generally, clouds, the ocean, Saturn\'s bands, and the sun\'s corona show this instability.

CFD MODELS:

1.Fluid thread breakup models

2.Vorticity

3.late-time behaviour

4.Turbulent models.

1.Fluid thread breakup models

This is the process by which a single mass of fluid breaks into several smaller fluid masses. The process is characterized by the elongation of the fluid mass forming thin, thread-like regions between larger nodules of fluid. The thread-like regions continue to thin until they break, forming individual droplets of fluid.

2.Vorticity

The RT instability can be seen as the result of baroclinic torque created by the misalignment of the pressure and density gradients at the perturbed interface, as described by the two-dimensional inviscid viscosity equation, D ω D t = 1 ρ 2 ∇ ρ × ∇ p {\\displaystyle {\\frac {D\\omega }{Dt}}={\\frac {1}{\\rho ^{2}}}\\nabla \\rho \\times \\nabla p} , where ω is vorticity, ρ density and p is the pressure. In this case, the dominant pressure gradient is hydrostatic, resulting from the acceleration.

3.Late time behaviour.

It was noted in Taylor (1950) that two superposed fluids with an interface normal to the acceleration field could exist in a state of unstable equilibrium. However, if small corrugations were to exist on the interface, a growing standing wave pattern could be expected. By considering an unbounded two-dimensional domain, incompressible potential flow, and a sinusoidal interface, the velocity potentials φ in upper and lower layers are given by

4.Turbulent models:

It depends upon velocity that is the fluid accelerated by the other fluid.

Rayleigh Taylor Simulation.

For this simulation 3 different size mesh are used. VOF multiphase model is used to run the simulation.

Geometry:

This is the geometry for the rayleigh taylor instability. Where the lighter fluid air is in bottom and denser fluid water is in the top.

1.Base mesh 0.5mm

Velocity contour

console

residual

2. 0.25mm meshing

velocity contour

console

residual

3 0.1mm mesh

velocity contour

console

residual

Comparision by Showing animation for the different mesh

I attached the zip file from the google drive which contain these animation for the three cases.

Since it is a instability analysis, it depends upon the time. Transient simulation is done by using the time marching solution. It gives the result more accurate than the steady state which ignore some cross terms.

since the end result we know that in this case, but this is a time dependent problem which it figured out about the mixing of the interface which not able to get appropriate in the steady state.

ATWOOD NUMBER:

It is a dimensionless number used in the study of instabilities at the interface for fluids with different densities. It is the ratio of density difference.

It is a dimensionless density ratio defined as A = (P1-P2)/(P1+P2)

where,

P1=Density of the heavier fluid (water)

P2=Density of lighter fluid (air)

A = (997 kg/m^3 - 1.225 kg/m^3)/(997 kg/m^3 + 1.225 kg/m^3) = 0.9975 .

The value of atwood number indicates that whether the bubble or spikes or formed in the interface. It shows the effect in bouyancy flow drives. It affect the turbulent mix parameters.

Small atwood number (less than 1)

high atwood number (more than 1) it implies a symmetry to the flow that is contrary to the familiar bubbles and spikes of water falling out of a glass.

In this case the atwood number is less than one which it shows it have effect of buoyancy.