Week 5 - Rayleigh Taylor Instability

Aim:

• To study the concept of Rayleigh Taylor Instability and some practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves.
• To study the Rayleigh Taylor Instability simulation for 2 different mesh sizes in addition to baseline and compare the results.
• To study the the effect of Atwood number variation on Rayleigh Taylor Instability for all the cases.

Given:

• Baseline mesh size= 0.5mm or 0.0005m
• Materials selection - Air, water and user-defined material(density = 400 kg/m³, viscosity = 0.001 kg/m-s)

Introduction:

Rayleigh Taylor Instability is an instability of the interface betweeen the two fluids of different densities which occurs when the lighter fluid is pushing the heavier fluid. It is clear from Newtons law of viscosity,fluid cannot resist the slightest shear force and Density is one of the main property of the fluid to incorporate the shear force. Based on the density, when lighter and heavier fluids are mixed together, denser fluid comes to top 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 intertial confinement fusion,
• Electromagnetic implosion of metal liner
• Behaviour of water that is suspended above oil in gravity of Earth

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

Model	Description
Ritchmyer-Meshkov instability:	The Richtmyer-Meshkov instability arises to the instability of an interface between the two fluids that is impulsively accelerated, generally by a shock wave. When a shock wave interacts with an interface separating two different fluids hydrodynamic instability occurs, including nonlinear growth and subsequent transition to turbulence, across a wide range of Mach numbers. Example: Supersonic combustion in a scramjet, Combustion, supernova, hypersonic air-breathing engines In combustion, Instability begins with a very small amplitude perturbations which initially grow linearly with respect to time. The instability can be considered the implusive-acceleration limit of the Rayleigh Taylor Instability. During the implosion of an inertia confinement fusion target , the hot shell material surrounding the cold D-T fuel layer is shock accelerated. Such instability is visualised in Magnetic target fusion.
Turbulence model	It is required to predict the average mixing behaviour in flows that are on average one- or two-dimensional. The governing equations of turbulent flows can be solved directly for simple cases of flow. For turbulent flows, CFD simulation uses turbulent models to predict the evolution of turbulence and are simplified constitutive equations that predict the statistical evolution of turbulent flows.
Plateau–Rayleigh instability:	It is also known as Rayleigh instability, and explains why the 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:	This instability is observed when there is velocity shear in a single continuous fluid or additionally where there is a velocity difference across the interface between the two fluids. Kelvin-Helmholtz studied the dynamics of two fluids of different densities subjected to a small disturbance such as wave was introduced at the boundary connecting both the fluids. This phenomena is mostly responsible for natures basic structures. It 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. Examples: wind blowing over water, clouds, the ocean, Saturn's bands, and the sun's corona
Smoother particle hydrodynaics (SPH) method	It is a computational method used for simulating the mechanics of continuum media, such as solid mechanics and fluid flows. SPH is increasingly used to model fluid motion and it has several benefits over traditional grid-based techniques. It guarantees conservation of mass without extra computation as the particles themselves represent mass. SPH also computes pressure from weighted contributions of neighboring particles rather than by solving linear systems of equations. Furthermore, 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	The analysis of two-phase flows (mixture model) is done using single fluid model. In this model individual fluid phases are assumed to behave as a flowing mixture described in terms of the mixture properties. It 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	This model separates the sets of governing equation for the vapor and liquid phases but the interaction between the droplets and heat exchange between the liquid phase and the solid boundary are not modelled. In addition the velocity slip between vapor and the liquid phase is in this model taken into account.
Descrete Boltzman Model	It is used for premixied, non-premixed or partially pri-mixed n0n equilibrium reactive flows. It is suitable for both subsonic and supersonic flows with/without chemical reaction and external force. It is developed to investigate the hydrodynamic and thermodynamic non-equilibrium effects in phase seperation processes. Based on the thermodynamic non-equilibrium features, its estimates the deviation amplitude from the thermodynamic equilibrium.

1. Geometry preparation

• OPEN IN SPACE CLAIM
• 
• Create a square of 20mm*20mm.Use Pull tool in order to create a surface.
• Move the generated component to the New Component. Name it as Air. Now click on design and create a new component --> Water. Repeat sketch just above the Air with the similar dimensions as mentioned.
• Check if both the components i.e Air and Water are created.
• Workbench > click on Share > Ok

2. Mesh

• OPEN MESH --> Generate the mesh --> baseline mesh element size 0.0005m(0.5mm).

3. Set Up

• DOUBLE CLICK SETUP -- > FLUENT LAUNCHER -->DOUBLE PRECISION.
• Ansys Fluent is opened.
• In Graphics window --> GEOMETRY AND CONSOLE window is displayed.
• SETTING UP DOMAIN -->MESH --> select CHECK. Then in CONSOLE WINDOW --> it shows 'CHECKING MESH…. DONE'
• To Change the Units, SETTING UP DOMAIN --> click Units
• SETTING UP PHYSICS --> SOLVER --> select Time = Transient, Type = PRESSURE BASED & Velocity Formation = ABSOLUTE.
• VISCOUS MODEL --> Laminar--> OK.
• Multiphase--> Model=Volume of Fluid, select Implict Formulation, -->Number of Eulerian Process is 2.
• Phases --> primary phase=Air(phase-1) and secondary phase=Water(phase-2).
• Material selection. Materials -->Air and Water.
• Setting up Domain ---> go to Display --> check for all the zones.
• Solving -->Method=standard --> Patch--> Phase=Water, Variable=Volume Fraction --> select water_surface=1 --> click Patch. Similarly for air_surface=0 --> click Patch.
• Postprocessing --> create contours --> contour of= Phases, Phase=water, select all the surfaces --> Save/Display. Now contour is generated showing water at top and air at bottom.
• Solving --> select Method=standard > go to Patch --> Phase=Water, Variable=Volume Fraction --> select water_surface=1 > click Patch. Similarly select air_surface=0 > click Patch.
• 
• Run-->Calculation > Type=Fixed Time STep, Time step Size=0.005, Number of iterations=1000 --> click Calculate.

4. Solution

• SOLVING --> CONTROLS, REPORTS, INITIALIZATION, RUN CALCULATION.
• REPORTS-- > DEFINITIONS --> NEW --> SURFACE REPORT -->AREA WEIGHTED AVERAGE. Similarly, select STANDARD DEVIATION. For both the cases, a 'SURFACE REPORT DEFINITION WINDOW'pops up > select the FIELD VARIABLE = Temperature
• CREATE --> select REPORT FILE and REPORT PLOT.
• INITIALIZATION --> select METHOD = HYBRID --> click t=0 INITIALIZE
• RUN CALCULATION --> NO. OF ITERATIONS =150 and press CALCULATE.

5. Results and Discussion:

CASE 1.1

Animation Link

https://youtu.be/aJyAsHdmP5o

Inference:

It is envisaged from the contour that lower density fluid (Air) is setteled on top and the higher density fluid (bottom) is settled at the bottom. This is formed due to the instability. This growing instabilities are known as Rayleigh Taylor Instability. We could see the formation of bubbles whose position is later replaced by water. Simulation begins with fluids of two phases at hydrostatic equilibrium state and then Rayleigh Taylor instability is seen at the interference when the lower density fluid pushes a higher density fluid due to which formation of shock waves occur at interface.The disturbance at the interface ends with lighter fluid settling up and the heavier fluid settles down.

CASE1.2

Animation Link:

https://youtu.be/xR3aAH-r1XA

CASE1.3

Inference:

It is clear from the results that instabilities that are occuring at the interface ie, formation of bubbles and vortices are explicitly seen as we refine the mesh quality or finer mesh.

Steady State Approach: It is used when we are only interested in end result and the intermittent behaviour is not important. Whereas in this case we are interested in the real-time changes at the interface as well as to see the changes in the fluid region. This real time observations can only be studied in Transient state Analysis. We are interested in the behaviour at interface when the fluids of various densities collapses as a function of time and this time step is too small to capture explicit behaviour the Instabilitty effect known as Rayleigh Taylor Instability. Thus Transient approach was selected.

Animation Link

https://youtu.be/Rh1jTRVwhYw

CASE 2: USER DEFINED MATERIAL AND WATER

Inference: It is envisaged from the results that simulation begins with fluids of two phases at hydrostatic equilibrium state and then Rayleigh Taylor instability is seen at the interference when the lower density fluid pushes a higher density fluid due to which formation of shock waves occur at interface. This in turn lead to formation of bubble like but later it becomes assymetric, which compresses the heavy fluid around it due to shock waves that pushes heavy fluid updwards. Later disturbance at the interface ends with lighter fluid settling up and the heavier fluid settles down.

https://youtu.be/xTVl2hz_BsU

Atwood Number: It 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:

Iti is paramount in the study of Rayleigh-Taylor instability.

Atwood number close to 0, RT instability flows take the form asymmetric “finger” of fluid;

Atwood number close to 1, the much lighter fluid “below” the heavier fluid takes the form of larger bubble-like plumes.

Inference: It is observed from the simulations that air forms larger bubbles when water is burst down the air and forms large bubbles. The Atwood number calculated in first case is 1 (approx.) and other case is 0.42. Rayleigh Taylor instability shows three distinct stages namely Linear stability , bubble(rising) formation and long term evolution due to bubble merging and mixing.

For low Atwood number, bubble and mushroom head shape is relatively symmetrical, the disturbances are linear for a larger time. In contrast, for large Atwood number, bubble and mushroom head shape is assymetrical and occurs rapidly at an accelerating rate leading to non linear growth rate.

Hence simulation results and Atwood calculation results are similar, thus validating the results.

CONCLUSION:

Rayleigh Taylor Instability simulation was studied at interface of both fluids having different densities under the influence of gravity and the following observations were made:

• Finer mesh shows formation of bubbles and vortices explicity
• Transient state analysis is preffered instead of steady state analysis as we are interested at real-time changes at the interface
• The Atwood number depends on the density of the fluid and ranges from 0 to 1.
• Both cases have larger Atwood number showing the Rayleigh Taylor Instability. Thus matching the simulation results.