Week 5 - Rayleigh Taylor Instability

Rayleigh Taylor Instability

Rayleigh Taylor Instability (RT instability):-

The Rayleigh–Taylor instability is the 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.

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 denser fluid 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 denser material moves down under the (effective) gravitational field, and the less dense material is further displaced upwards.

1) What are some practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves?

Richtmyer–Meshkov instability (RMI):-

The Richtmyer–Meshkov instability occurs when two fluids of different densities 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.

Kelvin–Helmholtz instability:-

The Kelvin–Helmholtz instability occurs when there is velocity shear in a single continuous field 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.

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 inkjet 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.

2) Perform the Rayleigh Taylor instability simulation for 2 different mesh sizes with the base mesh being 0.5 mm.

Procedure:

1. Geometry
2. Meshing
3. Setup
4. Solution
5. Results

Open Ansys Workbench and create Fluent Standalone system.

Geometry:-

• Open SpaceClaim for creating Geometry.
• 
• Component 1 (Gas)
• Sketch 20*20 mm quadrilateral on XY-plane where the bottom left point merges with origin.
• use the pull tool to create a 2-D plane and name it Air
• 
• Component 2 (Liquid)
• Again draw 20*20 mm quadrilateral on XY-plane where the bottom side merges with existing air component.
• use the pull tool to create a 2-D plane and name it Water
• 
• Use share commend to create connectivity between components.
• 
• The above blue line indicates that Topology shared

Meshing:-

• Creating base mesh with element size = 0.5mm
• Refining mesh to 0.2,0.1 mm
• 

• Mesh size	Nodes	Elements
  0.5	3321	3200
  0.2	20301	20000
  0.1	80601	80000

3 & 4. SetUp & Solution:-

• Double Precision, Solve processers = 4.
• Solver = Pressure Based, Velocity Formulation = Absolute, Time= Transient
• Gravity = on `Y = -9.81 m/s^2`
• Fluid Material => Default Air & Water from Fluent database.
• 
• Multiphase Model:-
• Models => Volume of Fluid => Implicit => Number of Eulerian Phases = 2
• Phases => Primary phase = Air(Name) => Phase Material = air
• Phases => Secondary phase = water(Name) => Phase Material = water
• 
• 
• 
• Solution methods & Solution Methods :-
  • Try to avoid higher-order methods & further reduce Under-Relaxation factors to avoid floating point error
• 
• Initialization:- Standard Initialization
• Patching zones:-
• 
• 
• Phase Contour:-
• 
• Run calculation :-
• 300*0.01 = 3 second's
• 

Results and Animations:-

Mesh Element size = 0.5 mm

https://drive.google.com/file/d/1yqwBdcieEb-zLdS8ZUeAuSnMd4eoewBq/view?usp=sharing

Mesh Element size = 0.2 mm

https://drive.google.com/file/d/1FCy08dr_w0EZAApNv-8Xh6wyQlFUwB5e/view?usp=sharing

Mesh Element size = 0.1 mm

https://drive.google.com/file/d/1Aen0pXNXV2FOT7UHuFqSQaIKS0ZPMgCy/view?usp=sharing

Conclusion:-

All case simulations are started at the hydrostatic equilibrium state, where heavier fluid is on top of lighter fluid. Due to the earth's gravity heavier fluid gets attracted downwards having more potential energy acting on lighter fluid. Due to the high pressure from the top, this light fluid wants to escape upwards. Here we can observe Rayleigh–Taylor instability at the interface between two fluids shock waves are created and lighter fluid pushes the heavier fluid.

We can clearly observe the difference in solutions of different mesh sizes as the mesh is getting refined from 0.5 to 0.2 mm quality in the solutions improved a lot. whenever mesh gets refined computation time and cost will increase but as a return, we will get more accurate results. Refined mesh captures the finest details level of details depending on the quality of the mesh.

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

The process is the same as the above mentioned except change air properties to Use defined material properties.

https://drive.google.com/file/d/1MZOX1fDczu2-VPjvBeoOWubXJu_tVY80/view?usp=sharing

when we compare Water-Air & Water-User material simulations we can clearly distinguish the difference between the two cases. In the water-air case to a larger difference in density, we can observe the formation of larger bubbles. for in Water-user material case density difference is smaller that's why we see symmetric fingers only.

4) Define the 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)`

where

`rho_1 text( = Density of heavier fluid)`

`rho_2 text( = Density of lighter fluid)`

Atwood number is an important parameter in the study of Rayleigh–Taylor instability the penetration distance of heavy fluid bubbles into the lighter fluid is a function of acceleration time scale, `text(Ag)t^2`

where

A = Atwood number

g = gravitational acceleration

t = time

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.

Case 1:- Air and Water

`text(Density of heavier fluid water ) rho_1 = 998.2 text( kg)/m^3`

`text(Density of lighter fluid air ) rho_2 = 1.225 text( kg)/m^3`

`A = (998.9-1.225)/(998.9+1.225)= 0.997`

Case 2:- Air and user defined material

`text(Density of heavier fluid water ) rho_1 = 998.2 text( kg)/m^3`

`text(Density of lighter fluid user material ) rho_2 = 400 text( kg)/m^3`

`A = (998.9-400)/(998.9+400)= 0.428`

If the two fluids have a larger density difference between them then the Atwood number is closer to '1'. in this case, the potential energy of the heavier fluid leads to generating more upward pushing force in the lighter fluid. This leads to the formation of larger bubble-like plumes.

If the two fluids have a smaller Density difference between them then the Atwood number is closer to '0'. In this case, the potential energy of the heavier fluid can't produce that much pushing force in the lighter fluid. ie., flows take the form of symmetric "fingers" of fluid.

5) Explain why a steady-state approach might not be suitable for these types of simulation.

we use a steady-state approach when we are concerned about the final solution, but in the case of Rayleigh–Taylor instability instabilities are started at the beginning of the simulation and progress with time. To observe these instabilities with respect to time we have to use a transient approach.