Week 5 - Rayleigh Taylor Instability

Rayleigh Taylor Instability Challenge:-

1. What are some practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves? In your own words, explain how these mathematical models have been adapted for CFD calculations.
2. 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( Attach them from your google drive). Observe the differences in the animation and explain the effect of mesh size on simulation. Also, explain why a steady-state approach might not be suitable for these types of simulation.[ Choose Air and Water for this 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.

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1. What are some practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves? In your own words, explain how these mathematical models have been adapted for CFD calculations.

RT instability is the instability of an interface between two fluids of different densities which occurs when the lighter fluid is pushing the heavier fluid. 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 perturbations 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 more dense material moves down under the (effective) gravitational field, and the less dense material is further displaced upwards.

Practical CFD models that have been based on mathematical analysis of RT waves...

1. Kelvin-Helmholtz instability
2. Richtmyer-Meshkov instability
3. Plateau-Rayleigh instability

Kelvin-Helmholtz instability- This instability typically occurs when there is velocity shear in a single continuous fluid, or additionally where there is a velocity difference across the interface between two fluids. A common example is seen when wind blows across a water surface; the instability constant is able to manifest itself through waves on water surface. The Kelvin-Helmholtz instability is not only restricted to a water surface as clouds, but is evident through other natural phenomena as ocean and the sun's corona.

Richtmyer-Meshkov instability- This 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.

Plateau-Rayleigh instability- This 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 RTI 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 tension, tend to minimize their surface area.

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2. 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( Attach them from your google drive). Observe the differences in the animation and explain the effect of mesh size on simulation. Also, explain why a steady-state approach might not be suitable for these types of simulation.[ Choose Air and Water for this simulation]

Steps in simulation: Open Ansys workbench, drag & drop fluid flow (Fluent) onto project schematic. below are steps we need to follow to solve a particular engineering system from CFD standpoint.

1. Geometry: Spaceclaim, it is used for import the geometry & other CAD related functions like geometry clean-up etc.
2. Mesh: used to generate mesh & other mesh control option there, also assign name which helps during BCs setup.
3. Setup: used to define physics of problem, like BCs, material & type of solving method, iterations ect.
4. Solution: where runtime results visualization done.
5. Results: CFD-Post, where post processing of results will be done.

Geometry:

For this simulation, we create 2 components of Air & water.

• Create 2 rectangles of size 20x20 mm, lower one of air & upper one for water.
• Create conformal mesh to have common interface between 2 interface & to share information.

Mesh:

Baseline mesh: Create a baseline mesh of 0.5 mm.

Refined mesh: one refined mesh of 0.3 mm.

2nd refined mesh of 0.2 mm.

Fluent Setup:

setting up physics: Laminar model

Solver: pressure-based, assign time as transient (non-steady state)

Material selection: 1) water & air

Properties of Material:

for air: density= 1.22 kg/m^3

viscosity= 0.0000181 kg/m.s

for water: density= 997 kg/m^3

viscosity= 0.0000890 kg/m.s

Boundary condition:

Enable gravity= -9.81 m/s^2

select volume of fluid method & number of phases are 2, one is water & one is air.

select implicit method.

Rename phases as air & water.

Standard initialization.

Do patch

Create phase contour

Create animation

assign time step size 0.005s & number of time steps 1000.

Animation:

for mesh size 0.5 mm (air & water)

for mesh size 0.3 mm (air & water)

for mesh size 0.2 mm (air & water)

Comparasion:

Element size (mm)	Cell counts	Nodes
0.5	3200	3321
0.3	8978	9180
0.2	20000	20301

why a steady-state approach might not be suitable for these type of simulation?

There is unsettleness of interface generates wrt time due to penetration of heavier & lighter fluid into one another.

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3. Run one more simulation with water and user-defined material(density = 400 kg/m³, viscosity = 0.001 kg/m-s) for refined mesh.

all boundary condition & setup is same as above.

Animation:

For mesh element size 0.2 mm, ( water & user-defined material (density = 400 kg/m³, viscosity = 0.001 kg/m-s)

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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.

The Atwood number 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= (rho1-rho2)/(rho1+rho2)

where, rho1 = desnity of heavier fluid

rho2= density of lighter fluid

Application- Atwood number is an important parameter in the study of rayleigh-tayor instability and richmyer-mechkow instability. In RTI, the penetration distance of heavy fluid bubbles into the light fluid is a function of acceleration time scale, where g is the gravitational acceleration & t is the time.

For air & water, A= 0.9975

For water & user-deifned material, A= 0.4273

Conclusion:

• As mesh become finer, the mixing becomes much clear & visible.
• The computation time also increases with increases in finer of mesh.