Rayleigh Taylor Instability Challenge
Rayleigh Taylor Instability
Objective:
- To understand the concept of Rayleigh Taylor Instability and also some cfd models that are based on the mathematical analysis of Rayleigh Taylor waves.
- To perform the Rayleigh Taylor Instability simulation for different mesh sizes and comparing the results.
- To find out the the effect of Atwood number variation on Rayleigh Taylor Instability for all the cases.
Given data:
- Base mesh size= 0.5mm.
- Materials selection - water and user-defined material(density = 400 kg/m3, viscosity = 0.001 kg/m-s).
Description:
Rayleigh Taylor Instability is the instability of the interface betweeen the two fluids of different densities which occurs when the lighter fluid is pushing the heavier fluid. RTI has a o]profound effect on th earths climate. Winds off the Greenland and iceland drive evaporation of the ocean surface making the upper layers saltier and heavier. This generates descending plumes that drive the abyssal ocean circulation, which in turn acts as a heat pump transporting warm equatorial waters up north. There are several examples that are related to Rayleigh Taylor Instability which include the behaviour of water that is suspended above oil in gravity of Earth, The 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 etc.
Practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves are:
- Richtmyer-Meshkov Instability - This refers to the instability of an interface between the two fluids that is impulsively accelerated, generally by a shock wave. It occurs over a wide ranges of length and time scales in technological applications and natural phenomina such as combustion, supernova, hypersonic air-breathing engines and the internal confinement fusion,a promising approach for fusion energy generation. The growth of theinstability 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. This kind of instability can also be visualised in Magnetic target fusion. Supersonic combustion in a scramjet is benifited by RMI as the fuel oxidents interface is enhanced by the breakup of the fuel into fine droplets.
- Kelvin-Helmholtz Instability - The Kelvin-Helmholtz Instability occurs 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 studed the dynamics of the two fluids of different densities when a small disturbance such as wave was introduced at the boundary connecting both the fluids. This phenomena is mostly responsible for natures basic structures.
- Plateau-Rayleigh Instability - Also called as the Rayleigh Instability i.e. similar to the Rayleigh Taylor Instability which is also a part of fluid dynamics related to the fluid thread breakup. The fluid instability is exploited in the design of a particular type of ink jet technology whereas a jet of the liquid is perturbed into a steady stream of droplets. The driving force of the Plateau Rayleigh Instability is the liquids by virtue of the surface tensions tend to minimize their surface area.
- Potential flow models of Rayleigh Taylor and Richtmyer Meshkov for bubble fronts - Potential flow models of Rayleigh Taylor and Richtmyer Meshkov for bubbles on a interface between an incompressible fluid and a constant supporting pressure(Atwood number(A)=1)is presented. they use ODE's for the bubble heights and the curvatures are obtained by considering the potential flow equations near the tips of bubble formation. The model is applied to the two dimensional single-mode evolution as wellas two bubble competition, for both RT and the Richtmyer-Meshkov Instabilities.
- Descrete Boltzman Model - Multi-component DBModel for premixied, non-premixed or partially pri-mixed n0n equilibrium reactive flows. This 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.
- K-L turbulence Model - This model is a two equation turbulence model based on Abdol-Hamid's closure and Menter's modification to Rotta's two equation model. Rotta show that the reliable transport equation can be formed fro the turbulent length scaleL and the turbulent kinetic energy k.
- Buoyancy Drag Model for evolution of the Rayleigh Taylor Instabilities - The growth of single mode perturbqtion is described by buoyancy drag equationwhich describes all instability stages at time dependent atwood number and the acceleration profile. The evolution of a multimode spectrum of perturbations from a short wavelength random disturbance is described using a single characteristic wavelength. The temporal evolution of this wavelength allows the description of both the linear stage and the late time selfsimilar behaviour.
Procedure:
In this case we consider dense fluid is on the top and light fluid in the bottom. Here the analysis is carried out in 2-dimensional.
- Initially, Go to SpaceClaim > create a square of 20mm*20mm. Go to Pull tool in order to create a surface.
- Once the surface is created , move the generated component to the New Component. This is called as Air. Now click on design and create a new component.This is called as Water. For water create a new sketch just above the Air with the similar dimensions as mentioned. Once the sketch is complete, using pull tool create a surface.
- Check if both the components i.e Air and Water are created.
- To create a conformal mesh for the created geometry, select both the components and go to Workbench > click on Share > Ok. We can see that one edge between both is shared. To check if the components are shared, go to preview box > check for the modified icons. Automaticaly in properties we can see share topology=share.
- Close the spaceClaim and open the Meshing. Initially generate the mesh without changing the parameters. For baseline mesh in this case create a mesh of element size 0.0005m(0.5mm).
- Check if the mesh is updated or not. Next open setup. Here we use Solver Type = Pressure-Based, Time= Transient, Velocity Formulation= Absolute. Also unable gravity where in 2Dsimulation gravity is in y-direction i.e. x=0,z=0,y=-9.81.
- Go to Setting up Physics > select Viscous Model= Laminar. Go to Multiphase > select Model=Volume of Fluid, select Implict Formulation, also check the Courant Number and see that Number of Eulerian Process is 2. Here time step cannot be too high. Now shift to Phases > primary phase=Air(phase-1) and secondary phase=Water(phase-2).
- Move to Material selection. Materials >initially we can see Air in materials so go to Fluent Database and select Water.
- Go to Setting up Domain > go to Display > check for all the zones.
- Then next open 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.
- Now check if the initialize setup is done perfectly. Here go to 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.
- In this case consider a Time step size is 0.025 and number of iterations as 100. When we run thhe calculation for these values of time step we notice the solution starts diverging. Which says that selected time step is too big. Now reduce the time step size.
- Now reinitiallize and open 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.
- Now double click on RunCalculation > Type=Fixed Time STep, Time step Size=0.005, Number of iterations=1000 > Hit Calculate.
- Finally we can see instabilities which are called as Rayleigh Taylor Instability.
Mesh refinement:
- During the mesh refinement all the process remains same only element size keeps changing i.e go to Meshing > Element size =0.0005 is reduced so as to get refined meshes. Also for the finer meshes water and user material (Density=400kg/m^3 and viscosity=0.001kg/sec2) is used.
- For setting up user-materials > go to Materials > Fluid Data Base materials > select Air > Copy > type user-material > Ok. Now go to F.F.Materials we can see 2 options in which air is selected in default. Here we should tell fluent that dont use Air as standard material instead use user-material which is created by us. Next go to Zones > cell Zones > click fff-surface > change material name to user-material > click OK. Again go back to Materials > go to F.Fluid Materials > click Air > Delete. Go to Properties and change the values i.e. Density=400kg/m^3 and viscosity=0.001kg/sec2 > press Change/Create > Close.
- Check in materials if the air is deleted and we can see only user-material and water. Hence during patch just consider water_surface=1 and user-material=0. After assigning the parameters, now run the calculations.
Geometry:
Dimensions of the geometry:
Observations:
Case- Baseline Mesh
Mesh:
E.size- 0.0005m
No.of Nodes- 3321
No.of Elements - 3200
Residuals:
Contours:
Animation:
Case-1 (Using Water & Air)
Mesh:
E.size- 0.0004m
No.of Nodes- 5151
No.of Elements - 5000
Residuals:
Contours:
Animation:
Case-2 (Using Water & Air)
Mesh:
E.size- 0.0002m
No.of Nodes- 20301
No.of Elements - 20000
Residuals:
Contours:
Animation:
Case- Baseline Mesh
Mesh:
E.size- 0.0005m
No.of Nodes- 3321
No.of Elements - 3200
Residuals:
Contours:
Animation:
Case-1 (Using Water & User-defined materials)
Mesh:
E.size- 0.0003m
No.of Nodes- 9180
No.of Elements - 8978
Residuals:
Contours:
Animation:
Case-2 (Using Water & User-defined materials)
Mesh:
E.size- 0.0002m
No.of Nodes- 20301
No.of Elements - 20000
Residuals:
Contours:
Animation:
Result:
- On running multiple simulations Rayleigh Taylor Instability was observed at the interface of the fluids with different densities under the influence of the gravity.
- It is clear that the lower density fluid is setteled on top and the higher density fluid is settled at the bottom. This is formed because of the instability.Slowly these instabilities start growing which is calle as Rayleigh Taylor Instability.
- Also observed that the instabilites are slowly forming into individual bubbles, which replaces the position of fluids.
- By performing different set of mesh refinement it is clear that on refining to the fine mesh the simulation is much better when compared with the baseline mesh.
- The instabilities that are happening at the interface is clearly visualized as we refined the mesh quality.
- Compared to initial simulations, in the last case the irregularities are more clear in the form of bubbles and the formation of vortices are also clear in this case.
- Due to this disturbance at the interface the lighter fluid moves up and the heavier fluid settles down.
- As the grid siize is much fine , predicting the flow is more accurate and clear for visualization.
- In the initial stages , we noticed that time step affecting the flow formation. Hence selecting the time step is also important to save the computation time.
A steady state approach is bascially used when we are intrested with the end result. But here in this case we want to find the real-time changes near the interface as well as to observe the changes of the fluid region. This real time observations can be carried out in Transient state Analysis. Also In our case we want to see how the interface and the fluids of various densities collapses as a function of time and this time step was too small and also need to capture clearly to have a clear understanding of this Instabilitty effect. Hence Transient state analysis is more suitable in this case.
Atwood Number - Atwood number is the dimensionless number in the fluid dynamics which is used in the study of hydrodynamics instabilities. Its an important parameter in the study of the Rayleigh Taylor Instability.Its defined by the density ratio. Denoted by-
i.e. A= (Density of heavier fluid - Density of lighter fluid) / (Density of heavier fluid + Density of lighter fluid)
Calculating Atwood number :
Atwood number (Air -water) = (998.2-1.225)/ (998.2+1.225) = 0.9975.
Atwood number (User defined material -water) = (400-1.225)/ (400+1.225) = 0.9938.
As in our solution we see that the lighter fluid rises up and bubble formation takes place. The atwood number calculated in our case s almost close to 1. Hence it signifies that both obtained simulation result and the atwood calculation results are same , thus validates the results.
Conclusions:
- On performing the several simulation , Rayleigh Tayalor Instability was observed at the interface of the both fluids with different densities under the influence of gravity. Also it is clear how the mesh size affects the RTI by maintaining the computational time.
- The above simulations also show why the transient state analysis is preffered instead of steady state analysis.
- The Atwood number depends on the density of the fluid and ranges from 0 to 1.
- Mixing of both fluid has the Atwood number close to 1 which shows the Rayleigh Taylor Instability. Thus the Atwood number matches the simulations results. Hence it validates the obtaoined results.
Google drive links -
https://drive.google.com/drive/folders/1Bd-zTiQ8XcZvkLzOkM7KolxSUQKOx4Bk?usp=sharing
https://drive.google.com/drive/folders/1Bd-zTiQ8XcZvkLzOkM7KolxSUQKOx4Bk?usp=sharing
Youtube links -
https://www.youtube.com/watch?v=N9Zz-YLG57c
https://www.youtube.com/watch?v=bt2aM6bLHYE
https://www.youtube.com/watch?v=LQJYXhAdE60