Week 5 - Rayleigh Taylor Instability

Objectives:

• some practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves
• explain how these mathematical models have been adapted for CFD calculations.
• 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
• explain why a steady-state approach might not be suitable for these types of simulation.
• Run one more simulation with water and user-defined material(density = 400 kg/m³, viscosity = 0.001 kg/m-s) for refined mesh. 
• 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. 

Rayleigh Taylor instability:

From the Newtons law of viscosity, we know that even the fluid cannot resist the slightest shear force. Density is one of the main property of the fluid which the shear force can incorporate with it.  

Depending upon the density we have lighter fluid like water etc and denser fluid like mercury, oil, honey etc.

Lord Rayleigh observed that when two liquids i.e lighter and heavier fluids are mixed together where the deser fluid is top on the lighter fluid, due to the gravity and the density variation the interface of the liquid is not stable. (i.e) water suspended on the oil .

So, the instability of the interface between the different densities of fluid when the lighter fluid is accelerated towards the heavier fluid is knowns as RAYLEIGH TAYLOR INSTABILITY.

Types of Instability and some practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves:

Ritchmyer-Meshkov instability:

The Richtmyer-Meshkov instability arises when a shock wave interacts with an interface separating two different fluids. It combines compressible phenomena, such as shock interaction and refraction, with hydrodynamic instability, including nonlinear growth and subsequent transition to turbulence, across a wide range of Mach numbers.

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. 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.

Kelvin–Helmholtz instability:

can occur when there is velocity shear in a single continuos fluid, or where there is a velocity difference across the interface between two fluids. An example is a wind blowing over water: The instability manifests in waves on the water surface. More generally, clouds, the ocean, Saturn's bands, and the sun's corona show this instability.

Practical CFD models:

SPH method(Smoother particle hydrodynaics)

Smoothed-particle hydrodynamics (SPH) is a computational method used for simulating the mechanics of continuum media, such as solid mechanics and fluid flows.Smoothed-particle hydrodynamics is being increasingly used to model fluid motion as well. This is due to several benefits over traditional grid-based techniques. First, SPH guarantees conservation of mass without extra computation since the particles themselves represent mass. Second, SPH computes pressure from weighted contributions of neighboring particles rather than by solving linear systems of equations. Finally, unlike grid-based techniques, which must track fluid boundaries, SPH creates a free surface for two-phase interacting fluids directly since the particles represent the denser fluid  and empty space represents the lighter fluid.

SINGLE-FLUID MODEL

A typical approach used for the analysis of two-phase flows is a mixture model, i.e. the individual fluid phases are assumed to behave as a flowing mixture described in terms of the mixture properties. The applied single-fluid model is a five-equation model consisting of the mass, momentum and energy equations for a vapor/liquid mixture, and two equations describing the formation and growth of the liquid phase.

TWO-FLUID MODEL

In the two-fluid model, separate sets of the governing equation for the vapor and liquid phases have been used. The interaction between the droplets and the heat exchange between the liquid phase and the solid boundary are not modelled here as well. Additionally, the velocity slip between vapor and the liquid phase is in this model taken into account.

Turbulence model

Turbulence models are needed to predict the average mixing behaviour in flows that are on average one- or two-dimensional. The approach to the construction of tile turbulence model is guided by the experimental behaviour.The equations governing turbulent flows can only be solved directly for simple cases of flow. For most real life turbulent flows, CFD simulation use turbulent models to predict the evolution of turbulence. These turbulence models are simplified constitutive equations that predict the statistical evolution of turbulent flows.

 Rayleigh Taylor instability simulation:

The idea is, the dense fluid will be at the top and lighter fluid will be at the bottom and due to gravity, there will be instability at their interface, which is going to be calculated by using VOF multiphase method in ANSYS Fluent.

hared 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:

2.II. Refined mesh

Mesh properties:

Element size = 0.2 mm

Statistics: 

The mesh created is depicted belo

Step 3: Solving using Fluent

Solver - Pressure based , transient solver with Absolute velocity formulation with Gravity of value 9.81 $\frac{m}{s^2}$ 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: Mesh element size: 0.5mm:

Mesh:

Residuals:

Simulation animation:

Case:2: Mesh element size: 0.3mm:

Mesh:

Residuals:

Simulation animation:

 Case:3: Mesh element size: 0.2mm:

Mesh:

Residuals:

Simulation animation: