Week 5 - Rayleigh Taylor Instability
Aim : The aim is to understand the Rayleigh taylors instability phenomena and how it takes place.
Objective :
- 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.
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.
Ritchmyer-Meshkov instability:
The Richtmyer-Meshkov instability arises when a shock wave interacts with an interface separating two different fluids. It combines compressible phenomena, such as shock interaction and refraction, with hydrodynamic instability, including nonlinear growth and subsequent transition to turbulence, across a wide range of Mach numbers.
Plateau–Rayleigh instability:
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. 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:
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. An example is a wind blowing over water: The instability manifests in waves on the water surface. More generally, clouds, the ocean, Saturn's bands, and the sun's corona show this 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.
Simulation Procedure:
Create Geometry:
- Create new component and name it air.
- Draw a square of lenght 20 mm x 20 mm and then select pull tool.
- Now again create new component and name it water. Draw a similar square with same dimensions on the top of air component.
- Do not forget to enable share topology.
- Upper square is water and lower square is air.
Creating meshing :
- Give element size of 0.5mm and generate the mesh.
Setting up physics :
- Select 'Transient' from general tab.
- Enable Gravity. Specify -9.81 in y direction.
- Go to physics tab, select 'Laminar' in viscous tab.
- Go to 'Materials Create/Edit', and select fluent workbench and search for 'water-liquid' and then hit copy.
- Enable the 'Multiphase' option. Select Implicit. Select Phases and name the phases as per problem statement.
- Name = air; Phase Material = air. Name = water; Phase Material = Water
- Standard Initialize.
- Then Select 'Patch' (Solution --> In the section of Initialization). For phase select water then select volume fraction, then select water present in the right hand column and assign the value of 1 for water and patch. Deselect water, select air assign value of 0 and patch.
Results :
Case 1 :
- Mesh size = 0.5mm
- Time steps size = 0.005 sec
- Number of time steps = 500 and then calculate.
Residual plot :
Video animation file :
https://drive.google.com/file/d/1-S8x1REY-fUuY4UBqxuDk_j1ELJtzPTQ/view?usp=sharing
Case 2 :
- Mesh size = 0.2mm (refined mesh)
- Time steps size = 0.005 sec
- Number of time steps = 500 and then calculate.
Residual plot :
Video animation file :
https://drive.google.com/file/d/1UeGZyi_6h4priWaSO3ez4WOZvgB2W-J4/view?usp=sharing
Case 3 :
- Mesh size = 0.2mm (refined mesh)
- Time steps size = 0.005 sec
- Number of time steps = 500 and then calculate.
- Materials = water and user defined.
- With density of 400 kg/m^2
- And with viscosity of 0.001kg/m-s.
Residual plot :
Video animation file :
https://drive.google.com/file/d/16rvOLlktSwdUCzAnYM4I3L4pZb6hVqc5/view?usp=sharing
Observation:
- With higher mesh resolution, the solution becomes unstable and one has to reduce the time step size.
- Finner Mesh produces better flow pattern.
- For the user defined material case, one must run the solution for more time to settle down. There are more bubbles and spikes in the first case (coarse and refine) case when compared to third case, because the desity difference is less, therefore the Atwood number is low and the penetration is also low for the third case.
- Steady-State simulation gives the end result of the solution and we are interested in the intermediate solution as well, so Transient solution is preffered.
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.
Conclusion :
- 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.
- 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.
- Therefore, penetration distance fo heavy bubble `A*g*t^2` in case 1 would be higher,as it is directly proportional to A (Atwood number).