Rayleigh Taylor Instability analysis using Ansys fluent

Abstract:

In this project work, volume of fluid (vof) method is applied to understand the Rayleigh-Taylor instability effect between two different fluid. Here a baseline mesh is simulated first followed by refined case set up. With addition a new setup with a user material property has been simulated to understand the instability phenomenon. Atwood no computation is also carried out to predict the corresponding CFD solutions.

Objective:

1. What are some practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves? In your own words, explain how these mathematical models have been adapted for CFD calculations.
2. 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( Attach them from your google drive). Observe the differences in the animation and explain the effect of mesh size on simulation. Also, explain why a steady-state approach might not be suitable for these types of simulation.[ Choose Air and Water for this simulation]
3. Run one more simulation with water and user-defined material(density = 400 kg/m³, viscosity = 0.001 kg/m-s) for refined mesh.
4. 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.

Theory:

The Rayleigh–Taylor instability, or RT instability , is an instability of interface between two fluidsof different densitieswhich occurs when the lighter fluid is pushing the heavier fluid. Examples include the behavior of water suspended above oil in the gravity of earth and mushroom clouds like those from volcanic eruptions and atmospheric nuclear explosions in which expanding core gas is accelerated into denser shell gas instabilities in plasma fusion reactors and inertial confinement fusion.

Water suspended atop oil is an everyday example of Rayleigh–Taylor instability, and it may be modeled by two completely plane-parallel layers of immiscible 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 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.

As the RT instability develops, the initial perturbations progress from a linear growth phase into a non-linear growth phase, eventually developing "plumes" flowing upwards (in the gravitational buoyancy sense) and "spikes" falling downwards. In the linear phase, the fluid movement can be closely approximated by llinear equation and the amplitude of perturbations is growing exponentially with time. In the non-linear phase, perturbation amplitude is too large for a linear approximation, andnon linear equations are required to describe fluid motions. 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). The difference in the fluid densities divided by their sum is defined as the Atwood no A. For A close to 0, RT instability flows take the form of symmetric "fingers" of fluid; for A close to 1, the much lighter fluid "below" the heavier fluid takes the form of larger bubble-like plumes.

Some popular models of Rayleigh-Taylor instability are as follows:

• 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 R-T instability.
• The Kelvin–Helmholtz instability 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,Jupiters red spot, and the sun's corona.The Kelvin-Helmholtz instability phenomena is an all encompassing occurrence of fluid flow seen time and time again in nature. From the waves of the ocean to the clouds above in the sky, the Kelvin-Helmholtz instability is responsible for some of nature's most basic structures. Further analysis and modeling of the Kelvin-Helmholtz instability can result in an understanding of the worlds natural phenomena and more.

• 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 theR-T instability and is part of a greater branch of fluid dynamics concerned with fluid breakup. This fluid instability is exploited in the design of a particular type of ink 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 similar solutions.

Modelling and problem solving approach:

Geometry creation:

1. Open Ansys workbench>> click on geometry.
2. Ansys space claim design domain opens up.
3. Select a plane to create the domain of air. Select the rectangular tool. Put the corner point of the rectangle at the coordinate origin. Draw a rectangle of 20x20mm.
4. Select the pull tool to convert to surface domain. Rename the component as air.
5. Now for creating the second component which is water, move to a new component.
6. Now to create the water domain, we select the same plane over which we drew the air component. Select the rectangular tool and draw the profile above the air surface.
7. Now for checking the topological sharing. Go to workbench>> select share prep option>>click on ok. The geometry gets shared.

Meshing:

1. Open the meshing console.
2. Click on the mesh and select generate mesh option.
3. Now a basic mesh pattern will be created.
4. Now we will run a baseline mesh having size of 0.5mm. We go to mesh size and put the desired value>> click on generate mesh tab.
5. Now on the next step we will refine the mesh size by 0.3mm and 0.1mm respectively.

Pre-processing physics setup:

• Open the Fluent setup console.
• We will be performing a transient simulation with gravity force acting in the negative y direction.
• 
• Now we choose the viscous model as laminar flow model and will be performing a multiphase simulation.
• 
• Now to define the material,>> go to material section>> select water from fluent database>> copy and close.
• 
• On choosing multiphase simulation>> select volume of fluid method.
• 
• Select the implicit scheme and select create option.
• Now open the phase profile>> select the phases separately>> choose primary phase as air and secondary phase as water.
• 
• Now we will perform standard initialization approach.
• 
• Now after initialization we will create patch.
• Select patch>> select – as water>> choose volume of fluid>> select air surface and provide value of 0 and then select water and provide a value of 1.
• 
• We will create contour plot>> select contour>> choose phases>> select volume of fraction>> material as water>> select all the surfaces and select save and display.
• 
• Click on solution animation to create animation of the contour plots.
• Set the time interval of 500>> step size of 0.001.
• 
• Click on calculate. The iteration and contour plot will begin to udate at each step of iterations.

CASE1:

BASELINE MESH SETUP

MESHING SIZE:0.5mm

FLUID: air and water

AIR DENSITY: 1.22

WATER DENSITY: 1000

ITERATIONS TO CONVERGE

POST PROCESSING CONTOUR PLOTS

ANIMATION PLOTS

CASE2:

REFINED MESH SETUP

MESHING SIZE:0.3mm

FLUID: air and water

AIR DENSITY: 1.22

WATER DENSITY: 1000

ITERATIONS TO CONVERGE

POST PROCESSING CONTOUR PLOTS

ANIMATION PLOTS

CASE3:

REFINED MESH SETUP

MESHING SIZE: 0.1mm

FLUID: air and water

AIR DENSITY: 1.22

WATER DENSITY: 1000

ITERATIONS TO CONVERGE

POST PROCESSING CONTOUR PLOTS

ANIMATION PLOTS

CASE4:

REFINED MESH SETUP (user-material)

MESHING

FLUID:

USER MATERIAL DENSITY: 400

WATER DENSITY:1000

ITERATIONS TO CONVERGE

POST PROCESSING CONTOUR PLOTS

ANIMATION PLOTS

Transient simulation over steady state simulation:

In RT instability, due to sudden collapse of horizontal plane between dense and lighter fluid, shock wave generation takes place. And due to this reason bubble and spike formation takes place., which is an time dependent parameter.. hence using steady state approach will not be fruitful to obtain the results, where we are more interested to know about the flow field variables changing wrt time.

Atwood no:

It is a dimensionless no in fluid dynamicsused in the study of hydrodynamics instabilities in density stratified field. It is defined as

A=`(rho1+rho2)/(rho1-rho2)`

`rho1= density of fluid 1`

`rho2= density of fluid 2`

computation of Atwood no for case1:

A=`(1000-1.22)/(1000+1.22)=0.9975`

computation of Atwood no for case4:

A=`(1000-400)/(1000+400)=0.42857`

Atwood no close to 1 indicates the bubble formation, and value close to 0 indicates wake like formation without any bubble.

Conclusions

• The finer the mesh, the more correct the observation of bubble formation can be captured.
• Incorrect time steps or large courant no can lead to floating point errors. So required relaxation factors should be given accordingly.
• With increase in mesh refinement, the computational time increases.
• The Atwood no validates the CFD simulation as it is observed. In case of water and air mixing, bubble formation takes place, whereas for the other case no bubble takes place.