Simulation of Rayleigh Taylor instability using ANSYS Fluent
Simulation of Rayleigh Taylor instability using ANSYS Fluent
I. Overview
The destabilization of interface between two fluids of different densities, which occurs when the heavier fluid is placed on top of the lighter fluid under the effect of gravitational field, is termed as Rayleigh Taylor instability. Some examples of RT instability include: mushroom-shaped clouds formed during volcanic eruptions and nuclear explosions, behaviour of water placed above oil, acceleration of core gas (lighter) into shell gas (heavier) during supernova explosions etc.
The evolution of RT instability can be divided into four stages:
(i) Initial growth stage: During this stage, the ampitude of perturbations at the interface are small as compared to their wavelengths and linearized equations of motion can be used to describe the growth of instability.
(ii) Formation and growth of mushroom-shaped spikes and bubbles: During this stage, the non-linear effects can no longer be ignored and the instability grows to produce mushroom-shaped spikes (heavier fluid structures growing inside the lighter fluid) and bubbles (lighter fluid structures growing inside the heavier fluid).
(iii) Interaction between structures: During this stage, the individually growing structures (spikes and bubbles) interact with each other and form new structures. The instability continues to grow.
(iv) Turbulent mixing: During this stage, the instability develops into a region of turbulent mixing.
An important dimensionless number used in the study of hydrodynamic instabilities, such as RT instability, is Atwood number. It is defined as the ratio of difference between the two fluid densities to their sum. Hence:
`A=(rho_1-rho_2)/(rho_1+rho_2)`
where:
`rho_1`: Density of heavier fluid
`rho_2`: Density of lighter fluid
For the current simulation we are using water and air as the two fluids. Hence:
`rho_1=998.2Kg//m^3` and `rho_2=1.225Kg//m^3`
`A=(rho_1-rho_2)/(rho_1+rho_2)=0.9975`
Significance of Atwood number in RT instability studies
Atwood number is a measure of the disparity between the two fluid densities. If the densities are similar then Atwood number is closer to zero whereas if the densities are far apart numerically then the Atwood number is closer to one. Atwood number dictates the type fluid structures formed during the evolution of RT instability.
(a) A~0: Symmetric fingers of the two fluids are formed.
(b) A~1: Large bubble-like plumes of the lighter fluid are formed inside the heavier fluid.
For the current case A~1 and hence bubble-like plumes of air are formed inside water.
Atwood number characterizes RT instability flows and hence can be used for validating the numerical simulations. RT instability flows have been studied extensively with different combinations of fluids (different Atwood numbers). The results from such studies are published in literature. For validating numerical simulations of RT instability, the simulation results for a particular Atwood number are compared with the results available in literature for the same Atwood number.
Some CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves are as follows:
(i) Plateau Rayleigh instability: It is an interfacial instability which appears on the surface of a liquid jet and causes it (jet) to break-up into spherical droplets if the perturbation wavelength on its (jet\'s) surface exceeds its (jet\'s) circumference. An example of Plateau Rayleigh instability is the formation of spherical droplets when water is dripping from a tap.
(ii) Kelvin Helmholtz instability: It appears on the interface between two fluid layers of different densities when there is a finite relative velocity between them. An example of Kelvin Helmholtz instability is the formation of waves on the water surface when air blows parallel to it.
(iii) Richtmyer Meshkov instability: It appears on the interface between two fluids of different densities when they are impulsively accelerated.
II. Simulation results
The RT instability is simulated using transient solver in ANSYS Fluent for three different cases:
(i) Case 1: Mesh size of 0.5mm and time step of 0.005s is adopted. The simulation is run for 400 time steps.
(ii) Case 2: Mesh size of 0.25mm and time step of 0.0025s is adopted.The simulation is run for 800 time steps.
(iii) Case 3: Mesh size of 0.125mm and time step of 0.00125s is adopted. The simulation is run for 1600 time steps.
The purpose of simulating RT instability for three different mesh sizes and time steps is to analyze their effects on the results (if any).
For modeling the two phases (air and water) volume of fluid method is employed. All the simulations are run for a physical time of 2s.
(i) Case 1: Mesh size=0.5mm
(ii) Case 2: Mesh size=0.25mm
(iii) Case 3: Mesh size=0.125mm
Observations:
(i) Fine mesh captures the flow features during turbulent mixing better than a relatively coarse mesh.
(ii) In case of fine mesh, the captured interface is sharper due to the availability of more mesh points.
The aim of current simulation is to capture the evolution of RT instability over time. Hence transient simulations are performed. Had our goal been to simulate the final steady state achieved by the configuration, steady state approach would have been useful. Furthermore for the current configuration steady state is already known: air (lighter fluid) would rest above water (heavier fluid) with a stable interface between the two.