Rayleigh Taylor Instability Challenge
Aim
1.To discuss practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves
2.Perform the Rayleigh Taylor instability simulation for 2 different mesh sizes with the base mesh being 0.5 mm. Observe the differences in the animation and explain the effect of mesh size on simulation. Explain why a steady-state approach might not be suitable for these types of simulation
3.Run one more simulation with water and user-defined material(density = 400 kg/m3, viscosity = 0.001 kg/m-s) for refined mesh.
4.Define the Atwood Number. Find out the Atwood number for both the cases and explain how the variation in Atwood number in the above two cases affects the behavior of the instability.
Solution:
Rayleigh Taylor Instability
The Rayleigh–Taylor instability, or RT instability (after Lord Rayleigh and G. I. Taylor), 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 supernova explosions in which expanding core gas is accelerated into denser shell gas.
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 pertubations 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 , 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.
As the RT instability develops, the initial perturbations progress from a linear growth phase into a non-linear growth phase, eventually developing "plumes" flowing upwards (in the gravitational buoyancy sense) and "spikes" falling downwards. In the linear phase, the fluid movement can be closely approximated by linear equations, and the amplitude of perturbations is growing exponentially with time. In the non-linear phase, perturbation amplitude is too large for a linear approximation, and non linear equations are required to describe fluid motions. In general, the density disparity between the fluids determines the structure of the subsequent non-linear RT instability flows (assuming other variables such as surface tension and viscosity are negligible here). The difference in the fluid densities divided by their sum is defined as the atwood number , A. For A close to 0, RT instability flows take the form of symmetric "fingers" of fluid; for A close to 1, the much lighter fluid "below" the heavier fluid takes the form of larger bubble-like plumes.
Practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves:
1. Kelvin–Helmholtz instability
It typically occurs when there is velocity shear in a single continous 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.
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. At times, the small scale instabilities can be limited through the prescience of a boundary. The boundaries are evident in the vertical direction, through an upper and lower boundary.
2.Richtmyer–Meshkov instability (RMI)
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.
3.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 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 .
Rayleigh Taylor Instability Simulation
Now we will be simulating the following surfaces with rayleight taylor instability having two different fluids in them .
Case 1 : Element size = 0.5 mm
Case 2 : Element size = 0.25mm
Differences in the animation and explain the effect of mesh size on simulation.
As we have refined the mesh to 0.25 mm from 0.5 mm we can see that as the accuracy increases the no of air particles are more diffused into each other .Hence the effect of mesh sizing is that it increases the accuracy of the results as it becomes more turbulent as seen in the case 2 video.
Steady-state approach might not be suitable for these types of simulation because in steady state the flow is mostly is laminar and linear but in this case we are using transient state because the flow changes from laminar to turbulent and hence the equations used to define them become non linear .
3.Simulation with water and user-defined material(density = 400 kg/m3, viscosity = 0.001 kg/m-s)
Atwood Number :
The Atwood number (A) is a dimensionless number in fluid dynamics used in the study of hydrodynamics instablities in density stratified flows. It is a dimensionless density ratio defined as
= density of heavier fluid
= density of lighter fluid