Rayleigh Taylor Instability Challenge
Rayleigh- Taylor instability is an instability which occurs due to density difference between two fluids. The heavier fluid resides over lighter fluid and under any force experienced, the interface gets unstable and starts detiorating. Some examples of the following are water on top of oil experiencing graavitational acceleration. The equilibrium state is unstable under the effect of any perturbation which results into moving of heavier fluid downwards and lighter fluid upwards resulting into spreading of potential energy.
Pic courtsey:-https://upload.wikimedia.org/wikipedia/en/f/f1/Evolution_of_the_Rayleigh_Taylor_Instability.xcf
Some of the practical CFD models that are based on mathematical analysis of RT wave are :-
- Kelvin-Helmholtz instability
- Plateau- Rayleigh Instability
- Saffman- Taylor Instability
- Richtmyer- Meshkov instability
Kelvin-Helmholtz instability :- This generally occurs during shearing of two fluid flowing relative to each other. Due to extremely small perturbation there occurs a varying force due to bernouille's lift. This can be seen in everyday life in nature when wind flows over water or when patterns are seen in clouds. Such patterns are common in nature.
Plateau- Rayleigh Instability :- This explains why falling of liquid breaks into smaller packets with same volume but less surface area. The driving force for this is surface tension and daily life application can be seen in ink jet technology printer. This basically happens because whenever any fluid flows there is small perturbation in it no matter how small and when it is resolved into components some of the perturbation increase with time while some decay with time.In 1873, Plateau found experimentally that a vertically falling stream of water will break up into drops if its wavelength is greater than about 3.13 to 3.18 times its diameter, which he noted is close to pi. Later, Rayleigh showed theoretically that a vertically falling column of non-viscous liquid with a circular cross-section should break up into drops if its wavelength exceeded its circumference, which is indeed pi times its diameter
Saffman- Taylor Instability:- The Saffman–Taylor instability is also known as viscous fingering, which is the formation of patterns in a morphologically unstable interface between two fluids in a porous medium.
Richtmyer- Meshkov instability :- The Richtmyer–Meshkov instability (RMI) occurs when two fluids of different density are subjected to impulsive acceleration. Normally this is done by shock wave. The development 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.
METHODOLGY:- A 2-d model was made in space claim with the dimensions as shown in figure , each of the phase represents two different phase namely air and water. The phases were modeled such that the heavier fluid is always above the lighter fluid.
The meshing was performed for three different mesh sizes with base mesh being 0.5 mm followed by 0.35 and 0.25 mm. The details are shown in table
| Mesh Size | Total Nodes | Total Elements |
| 0.5 mm | 3321 | 3200 |
| 0.35 mm | 6670 | 6498 |
| 0.25 mm | 13041 | 12800 |
Solver:- Transient analysis was performed in order to see the instability occuring. The interface breaks down depending on the density of the fluid and the Atwood number it has. Atwood number is a dimensionless number which is used in studying hydrodynamic instabilities in density stratified flow.
- If Atwood number is closer to 1 then bubbles are found during breaking of interface.For Atwood number close to 0, RT instability flows take the form of symmetric fingers of fluid.
- A = ρ(heavy)-ρ(light)/ρ(heavy)+ρ(light)
- A=(997-1.044) / (997+ 1.044)
- A = 0.9799
- Multiphase flow analysis was considered for the process with water being the primary fluid and air being the lighter fluid. Patching was performed which was an important step for transient analysis.
For Case 1:- Mesh Size 0.5 mm
For Case 2:- Mesh Size of 0.35 mm
For Case 2:- Mesh Size of 0.25 mm
- A steady state analysis is not useful for such case since we already know the final result i.e water is going to be setting up at the bottom because its heavier than air. Since we are much more interested in looking at the breaking of the interface between it and what kind of pattern it follows afterwards.
- It can be seen from the animations that with reducing mesh size the bubbles formation gets finer and hence better pattern is seen. This can be seen after slowering the speed of the simulation. The photos have been attached for the reference.
Mesh size 0.5 mm
Mesh size 0.35 mm
Mesh Size 0.25 mm
- Run one more simulation with water and user-defined material(density = 400 kg/m3, viscosity = 0.001 kg/m-s) for refined mesh.
The simulation was done for the following fluids, Atwood number was calculated as
- A= (997-400) / (997 + 400)
- A= 0.42734
By this the Atwood number is lesser compaared to previous analysis, Animation has been attached for the reference.