Week 5 - Rayleigh Taylor Instability
AIM: The aim is to understand the Rayleigh taylors instability phenomena and how it takes place.
OBJECTIVES:
- Some practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves.
- 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
- Explain why a steady-state approach might not be suitable for these types of 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 behavior of the instability.
THEORY:
Rayleigh Taylor instability :
- From the Newtons law of viscosity, we know that even the fluid cannot resist the slightest shear force. Density is one of the main property of the fluid which the shear force can incorporate with it.
- Depending upon the density we have lighter fluid like water etc and denser fluid like mercury, oil, honey etc.
- Lord Rayleigh observed that when two liquids i.e lighter and heavier fluids are mixed together where the deser fluid is top on the lighter fluid, due to the gravity and the density variation the interface of the liquid is not stable. (i.e) water suspended on the oil .
- So, the instability of the interface between the different densities of fluid when the lighter fluid is accelerated towards the heavier fluid is knowns as rayleigh taylors instability
Practical CFD models:
SPH method(Smoother particle hydrodynaics)
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 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 mode :
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 mode :
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 are not modelled here as well. Additionally, the velocity slip between vapor and the liquid phase is in this model taken into account.
Turbulence model :
Turbulence models are needed to predict the average mixing behaviour in flows that are on average one- or two-dimensional. The approach to the construction of tile turbulence model is guided by the experimental behaviour.The equations governing turbulent flows can only be solved directly for simple cases of flow. For most real life turbulent flows, CFD simulation use turbulent models to predict the evolution of turbulence. These turbulence models are simplified constitutive equations that predict the statistical evolution of turbulent flows.
GEOMETERY:
Create a component name air . Draw a sqaure with dimesnion of 20mm from the origin . Use pull command to convert tha sketch into a surface . Now create a new component named water and sketch a new sqaure over air and convert that into surface. Use share edge command to generate a shared topology in the geometry.
MESHING:
In Meshing two different meshes were generated , one being the ase line mesh with element size of 0.5mm and other mesh with element size of 0.25mm to generate a finer one.
For 0.5mm:
For 0.25mm:
SETUP AND RESULTS:
During case setup in gneral conditions select the transient condition and inout a value of -9.81 as gravity along y direction. Here steady state simulation is not used as we the intrested in the movement of fluid through out the simulation but not only on the end result.
Select VOF method for multiphase simualtions and select the implicit method to the aviod the issue of bigger time steps.
Now create air and water in material section and then allocate the primary and secondary phases to air and water respectively and for new material it is water and new material during simulation.
water:
new material:
air:
Intialize the simulation and use patch option to define water and air in the language that fluent understand. the alpha value for water was given a 1 and for air was given a 0.
water:
air:
Now create contour for volume fraction of water and use this contour in generating an
animation file to oberve the movement of fluid through out the simulation.
Case-1
There are two sub cases in Case-1 with two different meshes to undertand the effect of mesh on the flow pattern
For baseline_mesh (0.5mm):
The animation of the simulation is provided below with drive link
https://drive.google.com/file/d/1WZQz1eLTnR3oWwpINiqh7O-P2udQgEza/view?usp=drive_link
For fine_mesh (0.25mm):
The animation of the simulation is provided below with drive link
https://drive.google.com/file/d/1EgiKfXkL6uJTFGXvVbyquM3fxNoknSyC/view?usp=drive_link
From both the simulation we can observe that the both the simulation end result was same but the path they followed or the movement of the flow to reach that end point is different . baseline mesh took less number of iterations compared to fine mesh to reach the end result .
Case-2
In Case-2 the only chnage was that the material air is replaced by a user defined material with density of 400kg/m3 and viscocity of 0.001kg/m-s. For this simulation base line mesh was used and from the simulation u can observe that the intial movement of the fluid is same in 0.5mm air-water simulation and case-2 but the only difference was the depth it entered due to the density difference.
From the atwood number given below we can observe that as the denity difference between air and water is more compared to other case the distnace covered by heavy bubbles is high in the formaer than in the latter. This simulation took more number of iteration compared to other two to obtain the end result.
The animation of the simulation is provided below with drive link
https://drive.google.com/file/d/1nB323KJToXw0CXTSvc7le8wObTRTW_Yn/view?usp=drive_link
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 = (rho_1-rho_2)/(rho_1+rho_2)`
`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 Richtmyer–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 accleration and t is the time.
- For case 1 : Atwood number = (998.2-1.225)/(998.2+1.225) = 0.9975
- For case 2 : Atwood number = (998.2-400)/(998.2+400) = 0.4278.
CONCLUSIONS:
- The movement of the fluid changes with chnage in the mesh , however the end result is same. We used transient for the very same to observe the movement of fluid throughout the simulation.
- The penetration distance fo heavy bubble `A.g.t^2`in case 1 would be higher,as it is directly proportional to A (Atwood number).