Rayleigh Taylor Instability Challenge

                                                  Reyleigh  Taylor Instability

Aim

• To perform the Reyleigh Taylor instability Simulation for 2 different mesh sizes and compare the results.Thereby observing the changes and coming up with a conclusion.
• To find the Atwood number for the required cases.

Introduction

The Rayleigh–Taylor instability named after Lord Reyleigh and G.I.Taylor can be explained as 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 erruptions and Nuclear explosions, Supernova explosions in which expanding core gas is accelerated into denser shell gas.

                          Source:https://en.wikipedia.org/wiki/Rayleigh%E2%80%93Taylor_instability

Water suspended atop oil is an everyday example of Rayleigh–Taylor instability, and it can be modeled by two completely plane-parallel layers of Immicible 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 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 more dense material moves down under the (effective) gravitational field, and the less dense material is further displaced upwards. This was the set-up as studied by Lord Rayleigh. The important insight by G. I. Taylor was his realisation that this situation is equivalent to the situation when the fluids are Accelerated, with the less dense fluid accelerating into the more dense fluid.This occurs deep underwater on the surface of an expanding bubble and in a nuclear explosion.

Theory

The theory of the Rayleigh-Taylor instability of accelerated fluid layers is systematically developed from basic fluid equations. Starting with the classical potential flow theory for moving contact surfaces, the discussion extends to various fluid systems describing inhomogeneous, viscous, compressible, and isobaric flows. Thereby an overview on the major stability issues under a broad variety of physical conditions can be given.

   Source:http://rebloggy.com/post/science-fluid-dynamics-instability-physics-droplets-splashes-fluids-as-art-vorte

In particular, the stability analysis is addressed to layered materials in plane and spherical geometries under various dynamical conditions, to inhomogeneous media with variable gradients and different boundary conditions, to viscous boundary layers, compressible atmospheres, and to stationary ablation fronts in laser-driven plasma experiments. The stability theory is further extended to the nonlinear stage of the Rayleigh-Taylor instability and to a discussion of bubble dynamics in two and three dimensions for closed and open bubble domains. For this purpose simple flow models are studied that can describe essential features of bubble rise and bubble growth in buoyancy-driven mixing layers.

practical CFD Models that have been based on mathematical analysis of Reyleigh Taylor waves are given below:

• The Saffman–Taylor instability, also known as viscous fingering, is the formation of patterns in a morphologically unstable interface between two fluids in a porous medium. This situation is most often encountered during drainage processes through media such as soils.It occurs when a less viscous fluid is injected, displacing a more viscous fluid. Essentially the same effect occurs driven by gravity (without injection) if the interface is horizontal and separates two fluids of different densities, the heavier one being above the other: this is known as the Reileigh Taylor Instability.                                                                Source:https://en.wikipedia.org/wiki/Saffman%E2%80%93Taylor_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.

          RMI                                                                                               Reyleigh Taylor instability v/s RMI

                 Source:https://en.wikipedia.org/wiki/Richtmyer%E2%80%93Meshkov_instability

• The Kelvin–Helmholtz instability (afterLord kelvin and Hermann Von Helmholtz) typically occurs when there is velocity shear in a single continous fluid, or additionally where there is a velocity difference across the interface between two fluids. A common example is seen with wind blowing over water, the instability constant is able to manifest itself through waves on a water surface. The Kelvin-Helmholtz instability is not only restricted to a water surface as clouds, but is evident through other natural phenomena as the ocean, Saturn's bands and the sun's corona.

         Cont:  https://www.youtube.com/watch?v=nVZzwcsyHWY

            Source:https://en.wikipedia.org/wiki/Kelvin%E2%80%93Helmholtz_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-siimilar solutions.

                               Source:https://en.wikipedia.org/wiki/Plateau%E2%80%93Rayleigh_instability,

                                                                                             https://www.princeton.edu/~stonelab/Teaching/Oren%20Breslouer%20559%20Final%20Report.pdf

these are some of the few practical CFD models based on Reyleigh Taylor Instability.

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

where

 = density of heavier fluid

 = density of lighter fluid

In Rayleigh–Taylor instability, the penetration distance of heavy fluid bubbles into the light fluid is a function of acceleration time scale, 

where g is the gravitational acceleration and t is the time.

• If Atwood number is closer to 1, Reyleigh Taylor instability takes place in form of fingers.
• If Atwood number is closer to 0, Reyleigh Taylor instability takes place in form of Bubble like plumes.

Solving and Modelling Approach

Geometry:

Upper Zone represents Water and the Lower portion As Air.The Topology is shared between the two fluids,As it should be a Conformal mesh environment.

Patching:

Multi phase model:

Solution methods:

Viscous Model:

volume Fraction:

the Analysis is Splited to 3 cases :

1. Case1, With Base mesh (mesh size as 0.5mm ),(Air and Water)
2. Case2, Refined Mesh (mesh size as ),(Air and Water)
3. case 3,using the same mesh size of Refined mesh(mesh size as ),But with user defined material(Water and user defined material)

Case1: Base Mesh (0.5mm)(Air and Water)

Mesh:(0.5mm)

Residuals plot

Animation: https://drive.google.com/drive/u/1/folders/1_QNbIG8KJ2aY0b08-jsSpKAA1-XXsfZg

Case 2:Refined Mesh(1.5mm)(Air and Water)

Mesh:

Residual Plot:

Animation:https://drive.google.com/drive/u/1/folders/1_QNbIG8KJ2aY0b08-jsSpKAA1-XXsfZg

Case 3: Refined mesh (1.5mm) (Water and user specified material)

Mesh:

Residual plot:

Animation:https://drive.google.com/drive/u/1/folders/1_QNbIG8KJ2aY0b08-jsSpKAA1-XXsfZg

Results

• Steady state deals with frozen time Space i.e is independent of time.In order to find whats happenning in instability we need it to be time dependent .tha t's why we need transient state.

Atwood numbers of the cases:

• case1:Base Mesh :A=(998.2-1.225)/(998.2+1.225)

                       therefore A=0.997 that is closer to 1

• Case 3 :Refined Mesh with user defined Material: A=(998.2-400)/(998.2+400)  

                        therefore A=0.428

Conclusion

Comparing the Refined mesh Simulation and base line mesh Simulation we can conclude as a fact that more refine the mesh the better results and Closer to the real experimental values.From the Refined simulation Animations we can see the formation of mushroom flow in detail.