Week 5 - Rayleigh Taylor Instability

AIM: To perform the Rayleigh Taylor CFD simulation and understand the phenomena.

OBJECTIVE:

• To mention some practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves and explain how these mathematical models have been adapted for CFD calculations.
• To perform the Rayleigh Taylor instability simulation for 2 different mesh sizes with the base mesh being 0.5 mm and compare the results by showing the animations.
• To run one more simulation with water and user-defined material (density = 400 kg/m³, viscosity = 0.001 kg/m-s) for refined mesh.
• To define the Atwood Number and find out the Atwood number for both cases and explain how the variation in the Atwood number in the above two cases affects the behavior of the instability.

THEORY:

Rayleigh Taylor Instability is an instability of the interface between the two fluids of different densities which occurs when the lighter fluid pushes the heavier fluid. It is clear from Newton's law of viscosity, fluid cannot resist the slightest shear force and Density is one of the main properties of the fluid to incorporate the shear force. Based on the density, when lighter and heavier fluids are mixed, denser fluid comes to the bottom of the lighter fluid, due to the gravity and the density variation at the interface of the liquid is not stable. This instability of the interface between the different densities of fluid when the lighter fluid is accelerated towards the heavier fluid is called RAYLEIGH TAYLOR INSTABILITY.

Examples:

• Formation of high luminosity twin exhaust jets in rotating gas clouds in an external gravitational potential,
• Instabilities in plasma fusion reactors and inertial confinement fusion,
• Electromagnetic implosion of the metal liner
• Behaviour of water that is suspended above oil in the gravity of Earth

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 inkjet technology whereby a jet of liquid is perturbed into a steady stream of droplets.

Kelvin–Helmholtz instability:

Kelvin–Helmholtz instability can occur when there is velocity shear in a single continuous 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 hydrodynamics)

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 the 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 is not modelled here as well. Additionally, the velocity slip between the vapor and the liquid phase is in this model taken into account.

Turbulence model

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

Simulation Procedure:

Create Geometry:

• Create a new component and name it air.
• Draw a square of length 20 mm x 20 mm and then select pull tool.
• Now again create a new component and name it water. Draw a similar square with the same dimensions on the top of air component.
• Do not forget to enable share topology.
• Upper square is water and the lower square is air.

Creating meshing :

• Give an element size of 0.5mm and generate the mesh.

Fluent setup:

• After updating the mesh open the fluent, here we set the general setting to pressure-based solver with transient time simulation because we have to see the interface collapse with the time. And select the gravity with a y coordinate -9.81m/s^s
• Under physics Use laminar in viscous flow and the multiphase select volume of the fluid model because here we designed to find out the position of the interface between the two or more immiscible fluids

• Add material water using a fluent database and then assign the air as primary and water as a secondary phase
• Initialize using standard initialization, then use the patch option to give the alpha value for water as 1 and 0 for air.
• create a contour for the phases and also the animation for the volume fraction of water
• Now simulate with a time step size of 0.005 and 500 no. of time steps.

CFD post:

Use the post to see the fluid movement between the 2 phases.

Case 1: Basemesh with an elemental size of 0.5mm and having the fluid materials air and water.

No. of Elements =3200

timestep size=0.005

no. of timestep=500

Mesh

Case 1: Animation:

https://drive.google.com/file/d/1oA1DxD-C7viBE2zILhS398gqhi88N8LZ/view?usp=sharing

Case:2: Mesh element size: 0.2mm

refined mesh with an elemental size of 0.2mm and having the user material and water.

No. of Elements=20000

timestep size=0.005

no. of timestep=600

CFD Post:

Video Animation:

https://drive.google.com/file/d/14HJ1GGH1joUsEAq8lQaDy-_ODo3nBYFt/view?usp=sharing

Case:3: User Defined Material and Water with Mesh element size: 0.2mm

refined mesh with an elemental size of 0.2mm and having the user material and water.

No. of Elements=20000

timestep size=0.005

no. of timestep=600

Video Animation:

https://drive.google.com/file/d/1sviql0YOM8Iv0y00p5wrzKHh1OtmVN08/view?usp=sharing

Observation:

In the above cases, the simulation starts with the state of hydrostatic equilibrium, and the Rayleigh Taylor instability is observed at the interference when a lower density fluid pushes a higher density fluid due to which formation of shock waves at the interface takes place. Formation of air bubbles starts taking place, which compresses the heavy fluid around it, due to which shock waves of multidimensional fashion generates and get stronger as it moves upward.

It is observed that the more we refine the mesh, the more the simulation results get smoother about the irregularities that take place at the interface of the two fluids. As higher-density fluid replaces lower-density fluid, the formation of air bubbles and vortex takes place, which travels towards the upward region with time. In case 2, we can see more detailed results about the formation of waves and the formation of some air bubbles that get trapped at the lower region during the initial stage and then travel towards the upper region, generating shock waves. At the end of the simulation, it is to be observed that the two phases get separated from each other though, we can see some diffusivity between them which is a volume fraction of air and water.

why a steady-state approach might not be suitable for the above types of simulation?

The difference between the steady and transient is that you can't see the small-time variation of instability. The steady-state simulation is performed if we are concerned more about the final state results or the equilibrium state. In RT-Instability CFD models, we are more concerned to learn about the transition of the irregularities that starts developing when we pour highly dense fluid upon low dense fluid under gravity effect so, by using a transient solver along with a refined mesh of the model, we can compute the smooth transition of irregularities that takes place at the interface of the fluids. The final state results for both the steady-state and transient-state will be the same.

In this problem we are observing the instabilities when it is occurring so we are not concerned about the final answer because the steady state is more into capturing the final results but by transient, we can see the behavior of the solution and every instant such that capturing bubbles, Vortec and shockwaves. so this is the reason why the transient is more suitable than the steady-state model

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 = (1000 - 1.25) / (1000 + 1.25) = 0.9975

Atwood number is an important parameter in the study of Rayleigh-Taylor instability. For an Atwood number close to 0, RT instability flows take the form asymmetric “finger” of fluid; for an Atwood number close to 1, the much lighter fluid “below” the heavier fluid takes the form of larger bubble-like plumes.

The calculated Atwood number is close to 1 and from the simulation results, it is found that when high-dense fluid i.e. water poured upon low dense fluid i.e. air under gravity, the formation of air bubble-like plumes takes place which travels towards the upward region in the form of waves and some gets trapped at the lower regions during the initial stages, which afterward try to move towards the upper region and at the end, both phases gets separated with some diffusivity left at the middle portion between them. Thus, the calculated Atwood number is validated for our simulation results.

For above cases

case 1 &2:

heavier fluid(water)

density 1=998.2 kg/m^3

lighter fluid(air)

density 2=1.225 kg/m^3

Atwood number(A) =(density 1-density2)/(density 1+density2)

Atwood number(A)=0.996

case 3:

heavier fluid(water)

density 1=998.2 kg/m^3

lighter fluid(user-defined material)

density 2=400 kg/m^3

Atwood number(A) =(density 1-density2)/(density 1+density2)

Atwood number(A)=0.427

variation in the Atwood number in the above two cases affects the behavior of the instability.

For an Atwood number close to 0, RT instability flows take the form of symmetric fingers of fluid.

When the Atwood number is close to 1, the much lighter fluid below the heavier fluid takes the form of larger bubbles like plumes.

Therefore, from the value of the Atwood number, it is seen from the simulations that air forms larger bubbles when water is poured down the air pushes through and forms large bubbles. Therefore, the results can be validated by obtaining the value of the Atwood number.

Atwood number is an important parameter in the body of Rayleigh Taylor instability and Richtmyer-Meshkov instability.