Week 5 - Rayleigh Taylor Instability
Aim:
· To explain the some practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves? and how these mathematical models have been adapted for CFD calculations.
· To Perform the Rayleigh Taylor instability simulation for 2 different mesh sizes with the base mesh being 0.5 mm. To 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]
· To Run one more simulation with water and user-defined material (density = 400 kg/m3, viscosity = 0.001 kg/m-s) for refined mesh.
· To 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.
Introduction:
Multiphase Modeling
The simultaneous movement of several phases, such as liquids, gases, or solids, within a system is called multiphase flow. In a multiphase flow, two or more phases coexist and communicate continuously or dispersedly.
Multiphase modeling is a type of numerical analysis that deals with the modeling of two or more phases of matter interacting with one another in a domain. For instance, take the example of oil dispersed in water. The two are immiscible fluids, so the interaction of oil with water can be tracked numerically as there will be an interface region between the two.
In many industrial cases, two or more miscible or immiscible fluids interact, which has several implications for studying various properties. Problems for mixing come into the picture in a lot of cases, where we are interested in understanding what’s the ideal time for mixing two fluids and how this can be enhanced for better quality output.
CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves
The Rayleigh–Taylor instability
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. Examples include the behaviors of water suspended above oil in the gravity of Earth, mushroom clouds like those from volcanic eruptions 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.
Richtmyer–Meshkov instability
The Richtmyer–Meshkov instability occurs when two fluids of different density are impulsively accelerated. Normally this is done by the passage of a shock wave. The development of the instability begins with small amplitude perturbations which initially grow linearly with time. This is followed by a nonlinear regime with bubbles appearing in the case of a light fluid penetrating a heavy fluid, and with spikes appearing in the case of a heavy fluid penetrating a light fluid.
Kelvin Helmholtz instability Model:
This instability can occur when there is velocity shear in a continuous fluid, or where there is a velocity difference across the interface between two fluids. An example is wind blowing over the water. other examples showing this instability are clouds, oceans, Saturn’s bands.
Plateau Rayleigh instability:
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. It is related to the Rayleigh–Taylor instability and is part of a greater branch of fluid dynamics concerned with fluid thread breakup. 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. This instability explains why and how a falling stream of fluid breaks up into smaller packets with the same volume but different or small surface area. It is often called as Rayleigh instability. The driving force of this instability is that the liquids by virtue of their surface tensions, tends to minimize their surface area.
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
Where, ρ1= density of heavier fluid
ρ2ρ2 = 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, Agt2
where g is the gravitational, A acceleration and t is the time.
Calculate the Atwood Number for Air-water RT instability
ρ1=998.2 kgm3 (water density)
ρ2=1.225 kgm3 (Air density)
put this value in equation 1 we get A=0.9975
Calculate the Atwood Number for Water-UD material RT instability
ρ1=998.2 kgm3 (Water density)
ρ2=1.225 kgm3 (User defined material density)
put this value in equation 1 we get A=0.4278
Physics /Mesh Setup
Geometry
This 2D simulation so we have created one 20mmx20mm box on above the 20mmx20mm Box.
By pull option we have created the surface. Use the share Topology to due to that the interface edge is shared.
Mesh
Case1
Mesh size is 0.5mm
Case2
Element size 0.1mm.
Setting up of Physics,
Material Properties
Multiphase Model
standard Initialization selected.
After initialization we required to patch the two phases of the fluid.
Analysis of all the cases
Case 1
Mesh size is 0.5mm
Fluids are Air and Water
1.2 Line plot of Residual versus the iteration
Animation
Case 2
2.1 Description of the case 2
Mesh size is 0.1mm
Fluids are Air and Water
2.2 Line plot of Residual versus the iteration.
Animation
Case 3
3.1 Description of the case 3
Mesh size is 0.1mm
Fluids are user defined fluid and Water
3.2 Line plot of Residual versus the iteration
Animation
Conclusion
· The finer mesh captures the instability better than the coarse mesh. But it takes more time for the heavier fluid to settle down.
· The Atwood number close to 1 then the lighter fluid below the heavier fluid which forms a larger bubble which is less symmetrical and also it occurs very rapidly.
· The Atwood number close to 0 then the lighter fluid below the heavier fluid which forms a bubble which is relatively symmetrical and the perturbations are linear for larger time.
Why not steady-state calculation?
The steady state calculations are useful when the study is focused on to calculate / simulate the end results. Here in this study, the instabilities information is sensitive and completely time-dependent. At each and every time step, the instabilities grow exponentially. Though steady state simulation is capable to capture those instabilities but the results would not be time-accurate. Anyways transient state calculation with reduced time-steps can capture these instabilities at each and every time-steps and sensible time dependent results can be obtained.