Rayleigh-Taylor instability modelling with air and water
Rayleigh-Taylor instability is the instability that occurs when a less density fluid pushes a high density fluid. Since there is a instablity small inhomogeneities in the interface develop into turbulent structures when the fluids start mixing. Some practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor instability include modelling of behaviour of water when placed above oil with effects of gravity taken into account, modelling of spray atomization, modelling of turbulent mixing in shock tube flows, modelling of instabilities in plasma fusion reactors etc.
The RT instability follows three stages each having its own unique mathematical modelling adopted for CFD calculations. In the first stage the equations of the motion are linearized as perturbation amplitudes are small when compared to their wavelengths and sinusoidal. After the end of this first stage mushroom-shaped spikes and bubbles begin to form which are better modelled by using buoyancy drag models and here the growth rate is approximately constant in time. The bubbles and spikes formed in the before stage begin mixing which leads to turbulence better modelled by turbulence mixing model adopted for the last stage.
Rayleigh Taylor instability simulation is done by taking a simple geometry of 40mm x 20mm. The materials considered here are air and water(liquid form). A transient simulation is conducted here so as to observe how the simulation changed with respect to time. Multiphase option is selected with laminar flow. Energy equation is not taken into account in the simulation. The water is assumed to be in the top square of the geometry with air being in the lower square of the geometry at time equals zero. The simultion is carried out implicitly and the animation is also attached below. Since the geometry is a simple one the minimum mesh quality considering all meshes reported by mesh metrics for element quality is 0.9773.
Geometry:
Base mesh:
Nodes: 3321
Elements: 3200
Google drive link:
Refined mesh 1:
Nodes: 13041
Elements: 12800
Google drive link:
Refined mesh 2:
Nodes: 20301
Elements: 20000
Google drive link:
Atwood number:
Atwood number is the ratio of the difference of the densities of both material considered here that is air and water to the sum of the densities of both the materials. It is given by
`A_t= (rho_w-rho_(air))/(rho_w+rho_(air))`
Atwood number for the simulation done here is 0.9975. Atwood number determines the behaviour of two materials undergoing Rayleigh Taylor instability. As Atwood number reaches closer to 0 the pattern obtained from Rayleigh Taylor instability simulation will look like fingers of fluid while for Atwood number close to 1, the much lighter fluid which is below the heavier fluid here takes the form of larger bubble-like shapes as in the the case of our simulation which also validates the simultation.