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. And then we have to perform the Rayleigh Taylor Instability simulation for 2 different mesh sizes and three different baseline and compare the results, then to study the effect of Atwood number variation on Rayleigh Taylor Instability for all the cases.
Objective:
In this challenge they are given some objectives so we have to follow the below given as.
- 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.
- 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]
- 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 behaviour of the instability.
Theory:
Rayleigh Taylor instability:
Rayleigh Taylor instability is nothing but the Rayleigh–Taylor instability, or RT instability (after Lord Rayleigh and G. I. Taylor), is an instability of an interface between two fluids of different densities which occurs when the lighter fluid is pushing the heavier fluid.
Introduction The Rayleigh-Taylor instability is a fingering instability of an interface between two fluids of different densities, which occurs when the light fluid is pushing the heavy fluid [1, 2].
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 linear equations, and the amplitude of perturbations is growing exponentially with time. In the non-linear phase, perturbation amplitude is too large for a linear approximation, and non-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 number, 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.
Reference image:
Atwood number (A):
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} ={\frac {\rho _{1}-\rho _{2}}{\rho _{1}+\rho _{2}}}`
where
`{rho _{1}}` = density of heavier fluid
`{rho _{2}}` = density of lighter fluid
Atwood number is an important parameter in the study of Rayleigh Taylor instability and Richtmyler-Meshkov instability. In Rayleigh–Taylor instability, the penetration distance of heavy fluid bubbles into the light fluid is a function of acceleration time scale, `{A g t^{2}}`where g is the gravitational acceleration and t is the time.
Challenge approach:
In this challenge we have to solve 3 cases by using ANSYS Fluent solver. We will be using ANSYS Fluent solver to create the geometry and perform structured quads CFD mesh, setup the physics of fluid flow and run the iterations to get solutions, using post processing we will get the plots, contours, vectors, streamlines etc.,
In this challenge we have to use the gravity on of -9.81 m/s^2 and laminar viscous model we shall study grid independence for Rayleigh Taylor instability study. When we say grid independence study, we shall study for 2 cases as follows
Cases Baseline element size (mm) Denser medium Lighter medium
Case 1 0.5 Water Air
Case 2 0.1 Water Air
Case 3 0.1 Water User-defined
As usual in this challenge also using five major steps those are geometry, meshing, set-up, solution and results.
Challenege case 3 referance image:
Step 1: Geometry
In this challenge the first step is create geometry so first we have to right click on the geometry and select New Space Claim Geometry. The space claim application opens, select new sketch and moving the mouse pointer to the top right, and then we can select the XY plane. We can create the geometry by using sketch option the geometry we should create a square shape with side 20 mm, pull tool or return to 3D should make the 2D sketch into a plane. We shall create another plane like this with new component. In this challenge we should create two new components one is for air and another one is for water. Once we created the new components after that we shall use SHARE TOPOLOGY so that the nodes are shared during meshing process. This challenge the geometry is same for three cases. Once the geometry is completing after that we can able to move next meshing process.
Three cases geometry:
Share topology image:
Step 2: Meshing
In this challenge the step 2 is meshing process so we can right click and open the Mesh. In the mesh tab the geometry is already preloaded for meshing process so we have to click the GENERATE MESH option it should create a mesh automatically. In this process we have to give 0.5 mm element size, we can directly change the mesh size in mesh details tab. We shall name the air and water domains surfaces such as Air and water by selecting the faces and using the ‘N’ button and name them. In this challenge the mesh size is change for 2nd and 3rd case is 0.1mm mesh element size. Once we done this meshing process, we can see the green tick symbol nearby Mesh option. So, we can able to move the next set-up process.
Case 1 meshing images:
Case 2 and 3 meshing image:
Step 3: Set-up
In this challenge the 3rd process is set-up process so we have to right click and open the Setup tab. Once opening the set-up tab, there are some selections to be done to define the simulation definitions. In the set-up process we can able to select the Type: Pressure based, Velocity Formulation: Absolute, Time: Transient.
Next, we go to physics tab, we can turn on viscous and select laminar option. And then we can turn on gravity option and set -9.81 m/s^2 in Y direction.
Next, we have to Click on the Multiphase option > select VOF method, implicit, and select phases air as primary phase. In the materials option we shall select air as our default material with material type as fluid.
We can add the water by using Fluent database option. In the cell zone conditions option we should select the air for air domain and water for water domain. In order to start the simulation for the above set up physics, we need to initialize the entire flow field by clicking on the t=0 initialize icon. For Multiphase problems, we need to do Patch, select air and input as 0, and select water and input as 1.
In the Run Calculation we can input the timestep as 0.002 s and no of time step as 2500, so that we get a 5 second by using Autosave saves option this is used to create an animation in post processing, and clicking on the Calculate should start the iterations. After iterations, we get scaled residuals phase contour Results > Graphics > Contours > New phases for phase contour. Once complete the set-up process, we can able to move next option in solution.
Step 4: Solution
Some solutions are calculated in set-up option itself. So almost completes the solution process also so we can able to move next process of the results option.
Case 1 contour image:
Case 2 contour image:
Step 5: Result
In this challenge we are going to see the final step of result so we have to right click it is also called Post processor. The result tab default view is a wireframe model, we can create a plane in XY to view contours by using location option. The solution, all contours and plots can be created here, in addition to that we can create an animation of the velocity contour by clicking on AUTOSAVE during the solution and clicking on ANIMATE select make movie and save in desired location. In this challenge for Case 2, we shall repeat the same except for changing the grid size in meshing step. We can simply right click and select DUPLICATE option get duplicate same procedure we have to follow for third case also. In that set-up process for under materials we shall input user defined material with density 400 Kg/m^3 and viscosity 0.001 Kg/ms for case three. Finally, we completed the all steps.
Case 1 animation gif:
Case 2 animation gif:
Case 3 animation gif:
Conclusion:
Finally, we can compare the three cases between cases 1 and 2 clearly shows that finer mesh, instable mushroom and spike formation is very good. In this challenge the case 3, density difference between the two medias decreases and also the mixing is much easier and mushroom and spike formations are clearly visible but the settling time is longer.
Inferences:
In this challenge we can able to see that the density difference and the Atwood number is close to 1 for lower density differences Atwood number is close to 0. In this case we can able to see the rapidness of the instability, and the time taken is to settle after the mixing for lower. In case 3 the Atwood number is 0.428 so we can see the more of mushroom and spikes, many vortices and turbulence of mixing and it will take longer time to settle, whereas in cases 1 and 2, we see large mushroom and spike and few vortices as well. So, this is the inferences of overall challenge we concluded.
Reference:
https://www.researchgate.net/publication/260542299_The_Rayleigh-Taylor_instability