Rayleigh Taylor Instability Challenge

Objectives:

• some practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves
• explain how these mathematical models have been adapted for CFD calculations.
• Perform the Rayleigh Taylor instability simulation for 2 different mesh sizes with the base mesh being 0.5 mm. Compare the results by showing the animations
• explain why a steady-state approach might not be suitable for these types of simulation.
• Run one more simulation with water and user-defined material(density = 400 kg/m³, viscosity = 0.001 kg/m-s) for refined mesh.
• 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.

Theory:

The Rayleigh–Taylor instability, or RT instability, 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 the behavior of water suspended above oil in the gravity of Earth, mushroom clouds like those from volcanic eruptions and atmospheric nuclear explosions, supernova explosions in which expanding core gas is accelerated into denser shell gas, instabilities in plasma fusion reactors and inertial confinement fusion.

The following are a few CFD models that are based on Rayleigh-Taylor instability calculations:

• Kelvin-Helmholtz Instability: This phenomenon occurs when there is a velocity difference across the interface between two fluids or a velocity shear in a continuous fluid. Two fluids streams travelling at two different speeds will lead to this phenomenon.
• Richtmyer-Meshkov Instability: This phenomenon occurs when a shock-wave separates two different fluids. It combines compressible phenomena, such as shock interaction and refraction, with hydrodynamic instability, including non linear growth and subsequent transition to turbulence, across a wide range of Mach numbers.
• Plateau-Rayleigh Instability: This phenomenon, explains how and why a falling system of fluids breaks into smaller packets of fluids with same volume but different surface area. It is related to Rayleigh-Taylor instability, and is a part of a greater branch of fluid dynamics concerned with fluid thread breakup. This fluid is exploited in the design of a particular type of inkjet technology where a jet of liquid is perturbated in to a steady stream of droplets.
• Potential Flow model: This model was developed by Layzer and it provides information on the movement of fluids in a circular tube when a constant supporting pressure is present.

Atwood number: In fluid dynamics, Atwood number is used in the study of hydrodynamic instabilities in density stratified flows. The Atwood number (A) is a dimensionless number.

where

= density of heavier fluid

= density of lighter fluid

Atwood number is an important parameter in the study of Rayleigh-Taylor instability. For Atwood number close to 0, RT instability flows take the form asymmetric “finger” of fluid; for Atwood number close to 1, the much lighter fluid “below” the heavier fluid takes the form of larger bubble-like plumes.

The calculated Atwood number is close to 1 and from the simulation results, it is found that when high dense fluid i.e. water poured upon low dense fluid i.e. air under gravity, the formation of air bubble-like plumes takes place which travels towards upward region in the form of waves and some gets trapped at the lower regions during the initial stages, which afterward try to move towards upper region and at the end, both phases gets separated with some diffusivity left at the middle portion between them.

Procedure :-

Case 1: Simulation done using two different mesh sizes for water and air

• Meshing

(a) Element size : 0.5mm.

Residual plot :

Animation video :

(b) Element size :0.3mm

Residual plot :

Animation video :

Atwood number(A)=0.996

Case 2: Simulation done using finest mesh for water and user-define material(density = 400 kg/m³, viscosity = 0.001 kg/m-s) .

• Meshing

(c) Element size :0.2mm

Residual plot :

Animation video :

Atwood number(A)=0.427

• variation in Atwood number in the above two cases affects the behavior of the instability.
• For Atwood number close to 0, RT instability flows take the form of symmetric fingurs of fluid.
• When Atwood number close to 1, the much lighter fluid below the heavier fluid takes the form of larger bubble like plumes.
• Therefore, from the value of Atwood number it is seen from the simulations that air forms larger bubbles when water is poured down the air pushes through and forms large bubbles. Therefore, the results can be validated by obtaining the value of Atwood number.
• Atwood number is an important parameter in the body of Rayleigh taylor instability and Richtmyer-Meshkov instability. In Rayleigh-Taylor instability, the penetration distance of heavy fluid bubbles into the light fluid is a function of acceleration time scale. A*g*t^2, where 'g' is the acceleration due to gravity and 't' is the time.

Conclusion:

1. Atwood number of Case 1 is much higher compared to Case 2. Therefore, the Rayleigh-Taylor instability in Case 2 is slower than that of Case 1.
2. The surface area of formed of bubbles and spikes in Case 2 are lower than that of Case 1.
3. User defined material is higher due to which the growth rate of instability is much slower that that of air as it has much lower viscosity.