Week 5 - Rayleigh Taylor Instability

Aim: The aim is to understand the Rayleigh Taylor Instability phenomena and how it takes place.

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Q1. What is Rayleigh Taylor Instability and list some CFD models that have been based on the mathematical analysis of Rayleigh Taylor Instability.

• It is an instability of an interface between two fluids of different density which occurs when lighter fluid is pushing heavier fluid. Example, whater suspended above oil.
• Water suspended atop oil is an everyday example of Rayleigh–Taylor instability, and it may be modeled by two completely plane-parallel layers of immscible fluid, the denser fluid on top of the less dense one and both subject to the Earth's gravity.
• The equilibirum here is unstable to any perturbation 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 energy, as the denser material moves down under the (effective) gravitational field, and the less dense material is further displaced upwards.
• 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.

CFD Models that are based on mathematical models of Rayleigh Taylor Instability:

1. Kelvin-Helmholtz Instability:

• The Kelvin–Helmholtz instability (after Lord Kelvin and Hermann von Helmholtz) is a fluid instability that occurs when there is velocity shear in a single continuous fluid or a velocity difference across the interface between two fluids.
• Kelvin-Helmholtz instabilities are visible in the atmospheres of planets and moons, such as in cloud formations on Earth or the Red Spot on Jupiter, and the atmospheres of the Sun and other stars.

2. 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.

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–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.

Reference 1 : Rayleigh-Taylor Instability

Reference 2 : Kelvin-Helmholtz Instability

Reference 3 : Richtmyer-Meshkov Instability

Reference 4 : Plataeu Rayleigh Instability

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Simulation Procedure:

Create Geometry:

• Create new component and name it air.
• Draw a square of lenght 20 mm x 20 mm and then select pull tool.
• Now again create new component and name it water. Draw a similar square with same dimensions on the top of air component.
• Do not forget to enable share topology.

• Upper square is water and lower square is air.

Create Meshing:

• This straight forward mesh.
• Give an element size of 0.5 mm for refined case give 0.2 mm and update.

Setting up the Physics:

• Select 'Transient' from general tab.
• Enable Gravity. Specify -9.81 in y direction.

• Go to physics tab, select 'Laminar' in viscous tab.
• Go to 'Materials Create/Edit', and select fluent workbench and search for 'water-liquid' and then hit copy.
• Enable the 'Multiphase' option. Select Implicit. Select Phases and name the phases as per problem statement.

• Name = air; Phase Material = air. Name = water; Phase Material = Water
• Standard Initialize.
• Then Select 'Patch' (Solution --> In the section of Initialization). For phase select water then select volume fraction, then select water present in the right hand column and assign the value of 1 for water and patch. Deselect water, select air assign value of 0 and patch.

• Time Step size = 0.005 s.
• Number of Time Step = 300 and then run.

Results:

Three cases have been run for this setup.

Case 1. Base Mesh; size = 0.5 mm.

Case 2. Refined Mesh; size = 0.2 mm.

Case 3. Refined with user define material; size = 0.2 mm and (Density = 400, Viscosity = 0.001)

• Contour Plot for Case 1:

• Video Animation for Case 1:

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• Contour Plot for Case 2:

• Video Animation for Case 2:

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• Contour Plot for Case 3:

• Video animation for Case 3:

Drive Link

Observation:

• With higher mesh resolution, the solution becomes unstable and one has to reduce the time step size.
• Finner Mesh produces better flow pattern.
• For the user defined material case, one must run the solution for more time to settle down. There are more bubbles and spikes in the first case (coarse and refine) case when compared to third case, because the desity difference is less, therefore the Atwood number is low and the penetration is also low for the third case.
• Steady-State simulation gives the end result of the solution and we are interested in the intermediate solution as well, so Transient solution is preffered.

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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 density ratio defined as:

`A = (rho_(1) - rho_(2))/(rho_(1) + rho_(2))`

`rho_1` = Density of Heavier Fluid

`rho_2` = Density of Lighter Fluid

• 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.
• Atwood number is an important parameter in the study 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 gravitational acceleration and t is the time.

For Case 1:

Atwood Number = (998.2 - 1.225)/(998.2 + 1.225) = 0.9975

For Case 2:

Atwood Number = (998.2 - 400)/(998.2 + 400) = 0.4278

Therefore, Penetration distance of heavy bubble (`A*g*t^2`) in case 1 would be higher, as it is directly proportional to 'A' (Atwood Number).

Reference : Atwood Number