Week 5 - Rayleigh Taylor Instability

Aim - To conduct the Rayleigh Taylor CFD simulation in diffrent cases like water and air instability and a user defined material with water.

Introduction - 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.

Theory - Initially the system is in a state of hydrostatic equilibrium, in which the heavier fluid(ex: water) sits atop a lighter fluid(ex: air) in a constant gravitational field. As the RTI develops, the initial perturbations progress from a linear growth phase, eventually developing "plumes" flowing upwards(in the gravitational buoyancy sense) and "spikes" falling downwards. In the linear phase, equations can be linearized and the amplitude of perturbations is growing exponentially with time. In the non-linear phase, the perturbation amplitude is too large for the non-linear terms to be neglected. 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).

Examples include supernova, water suspended atop oil, mushroom clouds formed due to volcanic eruptions.

Practical CFD models based on mathematical analysis of Rayleigh Taylor waves:

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.

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.

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.

Modeling Approach:

Pre-Processing:

Step 1: Preparation of geometry

The geometry is essentially two squares , one on top of another of side 20 mm. The top square has been named 'water' and bottom square 'air' . The common edge between the domains was shared using share tool in Workbench. The created geometry is depicted below:

Step 2: Meshing

2.I. Baseline mesh

Mesh properties:

Element size = 0.5 mm

Statistics:

The mesh created is depicted below:

Step 3: Solving using Fluent

Solver - Pressure based , transient solver with Absolute velocity formulation with Gravity of value 9.81 ms2">ms2ms2 enabled

Viscous model - Laminar

Material properties of air, water and user-defined material:

Multi-phase model properties:

Solution Methods:

After Standard Initialization ,we have to patch both the domains to the respective materials, wherein we provide a volume-fraction value of 1 to water and 0 to air and user-defined material.

After solving for 1000 Time steps with 5e-3 dt (time step size), the following solution/animation was obtained:

Case 1: Baseline mesh(Air and water):

please go through with the given link.

Case 2:Refined mesh(Air and water):

please go through with the given link.

Case 3:Refined mesh(Water and user-defined material):

please go through with the given link.

Observation:

Finer mesh captures the instability better than the coarse mesh , but it takes more time for the heavier fluid to settle down. So it is computationally expensive but produces results with better accuracy.

Since the simulation involves capturing instabilities which are changing in an unpredictable manner as a function of time , transient approach is preferred over steady state approach.

Atwood Number

It is a dimensionless number that is used to study hydrodynamic instabilities in density stratified flows.

A=ρ1-ρ2ρ1+ρ2">A=ρ1−ρ2ρ1+ρ2A=ρ1-ρ2ρ1+ρ2

Where,

ρ1">ρ1ρ1 is density of heavier fluid

ρ2">ρ2ρ2 is density of lighter fluid

For Air-water RT instability,

ρ1">ρ1ρ1 = 998.2 kgm3">kgm3kgm3 [Water density]

ρ2">ρ2ρ2 = 1.225 kgm3">kgm3kgm3 [Air density]

A = (998.2-1.225)/(998.2+1.225)

A = 0.998

For water- UD material RT instability,

ρ1">ρ1ρ1 = 998.2 kgm3">kgm3kgm3 [Water density]

ρ2">ρ2ρ2 = 1.225 kgm3">kgm3kgm3 [UD material density]

A = (998.2-400)/(998.2+400)

A = 0.428

Behaviour of RT Instability on variation of Atwood number:

RTI evolves in three distinct stages: Linear stability , mushroom head(falling) or bubble(rising) formation , long term evolution due to bubble merging and mixing. For low Atwood number, bubble and mushroom head shape is relatively symmetrical, the disturbances or perturbations are linear for a larger time. In contrast, for large Atwood number, bubble and mushroom head shape is less symmetrical, also it occurs very rapidly at an accelerating rate which leads to a non linear growth rate.

Conclusion:

Atwood number forms a basis to compare and study research papers based on RT Instabilty. Thus we can use these papers to validate our results for beetter accuracy.