Week 5 - Rayleigh Taylor Instability
Objective:
- 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 behavior of the instability.
1.
The Rayleigh–Taylor instability, or RT instability, is the instability of an interface between two fluids of different densities which occurs when the lighter fluid is pushing the heavier fluid. Examples include the behavior of water suspended above oil in the gravity of earth, mushroom clouds like those from a volcanic eruption and atmospheric nuclear explosion, supernova 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 denser on top of the less dense one and both subject to the Earth's gravity. The equilibrium here is unstable to any perturbations or 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 realization 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
Types of Instability and some practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves:
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 inkjet technology whereby a jet of liquid is perturbed into a steady stream of droplets.
Kelvin–Helmholtz instability:
Kelvin–Helmholtz instability can occur when there is velocity shear in a single continuous 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
2.
Geometry setup:
- Choose Fluent from the workbench setup, edit SpaceClaim to create geometry.
- Choose the XY plane to sketch 2Dgeometry, create a square of 20mm, and select pull tool to make it a surface of that.
- Move the geometry to a new component and rename it as Air.
- Now create new geometry under the main design as water and create a square of 20mm right on top of the air square which creates an interface between the 2 boxes.
- Select the entire geometry for share prep under the workbench and select share so that the interface can face information and conformal meshing can be achieved between the 2 phases.
Mesh setup:
After creating the geometry open the mesh and generate the mesh with an elemental size 0f 0.5mm
Fluent setup:
- After updating the mesh open the fluent, here we set the general setting to pressure-based solver with transient time simulation because we have to see the interface collapse with the time. And select the gravity with a y coordinate -9.81m/s^s
- Under physics Use laminar in viscous flow and in the multiphase select volume of the fluid model because here we designed to find out the position of the interface between the two or more immiscible fluids
- Add material water using fluent database and then assign the air as primary and water as secondary phase
- Initialize using standard initialization, then use the patch option to give the alpha value for water as 1 and 0 for air.
- create a contour for the phases and also the animation for the volume fraction of water
- Now run the simulation with a time step size of 0.002 and 500 no. of time steps.
CFD post:
Use the post to see the fluid movement between the 2 phases.
Case 1: Basemesh with an elemental size of 0.5mm and having the fluid materials air and water.
elements = 3200
timestep size=0.002
no. of timestep = 500
Mesh
residual plot:
CFD post:
Use the post to see the fluid movement between the 2 phases.
Case 1: Basemesh with an elemental size of 0.5mm and having the fluid materials air and water.
elements=3200
timestep size=0.025
no. of timestep=100
Mesh
residual plot:
CFD post:
Use the post to see the fluid movement between the 2 phases.
Case 1: Basemesh with an elemental size of 0.5mm and having the fluid materials air and water.
elements=3200
timestep size=0.025
no. of timestep=100
Mesh
residual plot:
volume-contour:-
Animation Link of the upper setup:-
Case 2: refined mesh with an elemental size of 0.25mm and having the fluid materials air and water.
timestep size = 0.002
no. of timestep = 500
Mesh
residual plot:-
Animation for this setup:-
Case 3: refined mesh with an elemental size of 0.125mm and having the user material and water.
Density of User material = 400kg/m^3 (as per the problem statement)
Viscosity = 0.001kg/m-s (given in the problem statement)
timestep size = 0.001
no. of timestep = 1000
residual plot:-
volume-contour:-
Animation for the above case :-
Conclusions:
In the above challenge, the simulation starts with the state of hydrostatic equilibrium, and the Rayleigh Taylor instability is observed at the interference when a lower density fluid pushes a higher density fluid due to which formation of shock waves at the interface takes place. Formation of air bubbles starts taking place, which compresses the heavy fluid around it, due to which shock waves of multidimensional fashion generates and it gets more stronger as it moves upward.
It is observed that the more we refine the mesh, the more the simulation results get smoother about the irregularities that take place at the interface of the two fluids. As higher density fluid replaces lower density fluid, the formation of air bubbles and vortex takes place, which travels towards the upward region with time.
In case-3, we can see the more detail results about the formation of waves and the formation of some air bubbles that get trapped at the lower region during the initial stage and then they travel towards the upper region, generating shock waves. At the end of the simulation, it is to be observed that the two phases get separated from each other though, we can see some diffusivity between them which is a volume fraction of air and water.
In the Rayleigh- Taylor Instability, due to a sudden collapse in the layer between the denser and lighter fluid spike generation takes place which is obviously time dependent,hence using a steady state solver will not work as we are interested in the flow parameters with respect to time.
4. 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
where
= density of heavier fluid
= density of lighter fluid
Case 1:For fluid materials air and water
A = 0.9975
Case 2:For fluid materials user defined and water
A=0.427
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.