Rayleigh Taylor Instability Challenge
AIM: To study the Rayleigh-Taylor instability
Theoritical Background:
Rayleigh-Taylor Instability
The Rayleigh–Taylor instability is an instability of an interface between two fluids of different densities which occurs when the lighter fluid is pushing the heavier fluid. 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 equilibuium here is unstable to any 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 gravitational field, and the less dense material is further displaced upwards.
CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves
1. Kelvin-Helmholtz instability
Kelvin–Helmholtz instability occurs in the atmosphere as a sporadic, but widespread, phenomenon. It is usually invisible and so can be dangerous as clear air turbulance .The Kelvin–Helmholtz instability typically occurs when there is 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, the instability constant is able to manifest itself through waves on a water surface. The Kelvin-Helmholtz instability is not only restricted to a water surface as clouds, but is evident through other natural phenomena as the ocean, Saturn's bands and the sun's corona. The theory predicts the onset of instability and transition to turbulent flow within fluids of different densities moving at various speeds.
Helmholtz studied the dynamics of two fluids of different densities when a small disturbance, such as a wave, was introduced at the boundary connecting the fluids. Kelvin-Helmholtz instability can thus be characterized as unstable small scale motions occurring vertically and laterally.
2. Plateau-Rayleigh instability
The Plateau–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. 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.
The driving force of the Plateau–Rayleigh instability is that liquids, by virtue of their surface tensions, tend to minimize their surface area. A considerable amount of work has been done recently on the final pinching profile by attacking it with self-similar solutions. Liquid dispensers, inkjet printers, and similar spotting systems are designed to induced perturbations (either by means of mechanical oscillations or thermally) in order to ensure correct decomposition of a fluid jet into droplets.3. Richtmyer meshkov instability
The Richtmyer–Meshkov instability (RMI) 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. This instability can be considered the impulsive-acceleration limit of the Rayleigh-Taylor instability.
During the implosion of an inertial confinement fusion target, the hot shell material surrounding the cold D-T fuel layer is shock-accelerated. This instability is also seen in Magnetized target fusion. Mixing of the shell material and fuel is not desired and efforts are made to minimize any tiny imperfections or irregularities which will be magnified by Richtmyer meshkov instability.
Case1
Pre-Processing:
The geometry is established in the spaceclaim by using the share topology tool.
Meshing:
We generate the conformal mesh between water and Air surfaces.
Case 1: baseline mesh
element size = 0.5mm Number of nodes = 3321, Number of elements = 3200
Fluent Setup:
Solver: Steady state
Type: Pressure based
Model: Viscous laminar model , Multiphase: Volume of Fluid and Implicit formulation used
Material: Air: Density = 1.225 kg/m3, Dynamic viscosity = 1.7894e-05 kg-m/s, Water: Density = 998.2 kg/m3, Dynamic viscosity = 0.001003 kg-m/s
Residual plots:
Contour of Volume fraction of water
Case 2:
element size = 0.3mm Number of nodes = 9180, Number of elements = 8978
Meshing:
Residual plots:
Contour of Volume fraction of water
Case 3:
element size = 0.1mm Number of nodes = 80,601, Number of elements = 80,000
Meshing:
Residual plots:
Contour of Volume fraction of water
Special case:
Here along with water we have considered air as the the user defined material and perform the calculation
Residual plots:
Contours of Volume fraction of water
Observations:
Rayleigh-Taylor instability was observed at the interface of the fluids with different densities. We have dense fluid at the top and light fluid at the bottom. The interface is going to be unstable when the gravity is present. The lower dense fluid pushes the higer dense fluid above it. The heavier fluid which is water comes at the bottom and replaces the air at that position. The air bubble formation take place and it start moving upwards.
Better simulation of Rayleigh-Taylor instability can be obtained by refining the mesh. All the instabilities and the irregularities that take place at the interface are better visualized when we refine the mesh. As we refine the mesh we get greater possibility of accurate flow prediction.
Volume of Fluid:
It is a numerical technique that tracks or locates the surfaces or interfaces. It is used to model 2 or more immisible fluids.
The steady state analysis is used when we are more concerned about the end results. But in Rayleigh-Taylor instability we are focussing on observing the changes that occurs at the interface and the fluid region. Here we are doing transient simulation, because we want to see how the interface collapses as the function of time but the steady state analysis is independent of time. Hence Transient simulation is more suitable over steady state approach.
Atwood number:
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 ratio that describes density difference between two adjacent fluids with a common interface
For value of A close to 0 Rayleigh Taylor instability takes the form of symmetric fingers of the fluid.
For value of A close to 1 much lighter fluid below the heavier fluid takes the form of large bubble like plumes
In this case the the less denser fluid air rises up and takes the form of bubble like plumes, since the atwood number is close to 1.