Rayleigh Taylor Instability Challenge
Objectives:
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To understand the Rayleigh-Taylor instability
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To carry out the transient simulation mixing of different density fluid using Multiphase VOF method
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To understand the different CFD models based on RT instability
- To realize the effects of Mesh size and Atwood Number on RT instability
Rayleigh–Taylor (R–T) Instability:
The Rayleigh–Taylor instability occurs when a high-density (ρ1) fluid is placed over a lower-density (ρ2) fluid in a gravitational field. When the density interface is disturbed away from horizontal, the hydrostatic pressure acts to drive the heavier fluid into the lighter one, with the disturbance amplitude initially growing exponentially, but shortly afterward saturating and forming a characteristic mushroom shape, at least for some time. Eventually, either other wavelengths become important and accelerate the development of the instability, or mixing and dilution of the density contrast will lead to a deceleration.
Such naturally occurring R–T instabilities are quite common e.g. in oceans, supernovae and oil-trapping salt domes, Behavior of water suspended above oil in the gravity of Earth, Mushroom clouds, Volcanic eruptions, Atmospheric nuclear explosions, Supernova explosions in which expanding core gas is accelerated into denser shell gas, Instabilities in plasma fusion reactors, Inertial confinement fusion.
The RT instability is widely used in the CFD experiments in terms of several models and some of the widely used models are:
1.Saffman-Taylor Instability:
The Saffman-Taylor instability occurs when morphologically unstable fluids interfere in a porous medium. This is also called as Viscous fingering. This type of instability can be seen in drainage when a low viscosity fluid is pushed inside a pipe where there is a fluid with higher viscosity. Most of the research using this model is done in the Hell-Shaw cells, Boundary integral methods, and phase-field methods.
2.Richtmyer-Taylor Instability:
The Richtmyer instability occurs when two fluids with different viscosity are impulsively accelerated. These effects can be seen in shock waves where initially the shock wave is developed small and grows linearly with time.
3.Kelvin-Helmholtz Instability or Rayleigh Taylor Hybrid Model:
This type of instability occurs when there is velocity shear in a continuous medium or when the velocity layer between two fluids is different. A classic example of this instability can be seen when atmospheric air is blow over water. Also, this instability can be seen in the sun's corona, clouds, and Saturn’s bands.
4.Plateau-Rayleigh Instability:
This instability model is used to explain how a falling water stream breaks into small droplets with the same volume but with less surface area. The fluid thread breaks up in the fluid dynamics branch involves most of its study using the plateau-Rayleigh instability. An example to demonstrate this effect is using the falling of water droplets from a faucet/tap. When the faucet is closed or when the flow is interrupted a long stretching of fluids occurs at the neck of the faucet and if the faucet diameter is bigger the fluid is not sucked back and breaks down into smaller water droplets.
5.Potential flow model:
This model was developed by Layzer and it provides info on the movement of fluids in a circular tube. A constant supporting pressure (Atwood number =1) is present, competition for both the Rayleigh Taylor and the Richtmyer-Meshkov instabilities.
Stages of development and eventual evolution into turbulent mixing :
GEOMETRY:
Open fluid flows fluent, then space-claim was opened and a sketch was created on the X-Y plane (x-horizontal axis) (y- vertical axis). The rectangle was created on the sketching module with a 20x20mm dimension and the surface was created from that sketch and using the pull command it is created into surface which is moved into new component Air. Again, a new component is created under this new sketch is created the same as air, one above the other water with the same dimension, and using pull command it is also converted to surface.
Now for these two surfaces, we have to share topology so that we can generate a conformal mesh, in the meshing module. only a single edge was shared from the surface which is connecting both the components.
A portion of water is suspended above a portion of air and is allowed to mix under the influence of gravity.
Due to the difference in the densities of water and air, there will be instability developing at the interface which will settle finally when the higher density water settles below the lower density air.
After share topology:
CaseA:
Case-A1:
Meshing:
Element Size:0.5mm
No of Elements:3200
SetUp and Solution:
Solver type: pressure based
Time state: Transient
Enable gravity with -9.81m/s on y-axis and 0 on x-axis
Model: Multiphase model with Volume of fluid method
Formulation method: implicit
No boundary conditions
Materials used: water as heavy fluid
Air as light fluid
set the required fluid for primary and secondary phases.
Initialization: Standard
Patch: phase - water
variable- volume fraction
heavier fluid value as 1 and lighter fluid as 0
*patching should be done individually for different fluids and should be done every time after initialization
Use appropriate time step based the case
Results:
Time Step size:0.005
No of Time Steps:1000
Iterations per Time Step:20
Residual plots:
The contour of Volume Fraction:
Animation:
Case-A2:
Meshing:
Element Size:0.3mm
No of Elements:8978
SetUp and Solution:
Solver type: pressure based
Time state: Transient
Enable gravity with -9.81m/s on y-axis and 0 on x-axis
Model: Multiphase model with Volume of fluid method
Formulation method: implicit
No boundary conditions
Materials used: water as heavy fluid
Air as light fluid
set the required fluid for primary and secondary phases.
Initialization: Standard
Patch: phase - water
variable- volume fraction
heavier fluid value as 1 and lighter fluid as 0
*patching should be done individually for different fluids and should be done every time after initialization
Use appropriate time step based the case
Results:
Time Step size:0.005
No of Time Steps:1200
Iterations per Time Step:20
Residual plots:
The contour of Volume Fraction:
Animation:
Case-A3:
Meshing:
Element Size:0.2mm
No of Elements:20000
SetUp and Solution:
Solver type: pressure based
Time state: Transient
Enable gravity with -9.81m/s on y-axis and 0 on x-axis
Model: Multiphase model with Volume of fluid method
Formulation method: implicit
No boundary conditions
Materials used: water as heavy fluid
Air as light fluid
set the required fluid for primary and secondary phases.
Initialization: Standard
Patch: phase - water
variable- volume fraction
heavier fluid value as 1 and lighter fluid as 0
*patching should be done individually for different fluids and should be done every time after initialization
Use appropriate time step based the case
Results:
Time Step size:0.005
No of Time Steps:1500
Iterations per Time Step:20
Residual plots:
The contour of Volume Fraction:
Animation:
Conclusion on Different Mesh Sizes:
- The formation of air bubbles takes place which moves upward.
- In all the cases the lower densed fluid pierces into the high-density fluid as it seems to be the propagation of shock wave.
- In case A1 at the interface, we can see that some disturbances are formed as time passed the large disturbance is formed its forces into the high dense fluid like a shock wave.
- In case A2 at the interface, we can see the disturbances which are more than case 1 pushing through the high dense fluid mushroom-like structure is formed and the formation of bubbles can be seen around the mushroom structure.
- In case A3 at the interface disturbances are observed in more detail in the form of many bubbles and the formation of vortices are can be clearly seen.
- After performing the 3 case setups with respective base grid sizes the case-A2 & A3 we can see that there were only minor changes in the post-processing results showing us that the case has converged. The finer the mesh the greater the possibility of accurate flow prediction. Here Case A3 has the finest mesh of all. So can we take data from case A3 since it will have an accurate simulation result.
Case-B: User-Defined Material and Water
Run one more simulation with water and user-defined material(density = 400 kg/m3, viscosity = 0.001 kg/m-s) for mesh size of 0.2mm.
Results:
Time Step size:0.005
No of Time Steps:1500
Iterations per Time Step:20
Residual plots:
The contour of Volume Fraction:
Animation:
Atwood number:
In fluid dynamics, Atwood number is used in the study of hydrodynamic instabilities in density stratified flows (i.e. to characterize the density difference between two fluids). The Atwood number (A) is a dimensionless number
ρ1=Density of heavier fluid
ρ2=Densityof light fluid
For big density differences At tends to 1. For small differences, At tends to zero.
Also with Atwood number, the penetration distance can be defined by Agt2
Effect of Atwood number on the instability
If A is nearer to 1, R-T instability takes place in the form of fingers
If A is nearer to 0, R-T instability takes place in the form of bubble-like plumes.
Atwood no. for CaseA- instability of water-air refined mesh
ρ1 – Density of heavier fluid (water) – 998.2 kg/m3
ρ2 – Density of lighter fluid (air) – 1.225 kg/m3
A= 0.997
Atwood no. for CaseB- instability of water-oil:
ρ1 – Density of heavier fluid (water) – 998.2 kg/m3
ρ2 – Density of lighter fluid (oil) – 400 kg/m3
A= 0.428
- When the Atwood number is so high the more instability takes place in the fluid and measuring the spikes of the instability is very difficult.
- When the Atwood number is low the more stable is the fluid mixture and measuring the spikes of the instability can be done to some extent because it expresses linearity to some extent.
Why Transient state simulation?
The steady-state analysis is used when we are more concerned with the end result. But for Rayleigh Taylor instability we are more interested in observing the real-time changes of the interface and the fluid region. The real-time observation is possible in Transient State analysis.
The difference between the steady and transient is that you can't see the small-time variation of instability. The steady-state simulation is performed if we are concerned more about the final state results or the equilibrium state. In RT-Instability CFD models, we are more concerned to learn about the transition of the irregularities that starts developing when we pour high dense fluid upon low dense fluid under gravity effect so, by using transient solver along with refined mesh of the model, we can compute the smooth transition of irregularities that takes place at the interface of the fluids. The final state results for both the steady-state and transient state will be the same.
This simulation cannot be run under steady-state conditions because it changes from time to time iterations so we cant achieve this simulation through this condition.
Conclusions:
- The mesh refinement leads to better simulation of the Rayleigh Taylor instability. All the instability and the irregularities that take place at the interface are better visualized as we refined the mesh.
- Rayleigh Taylor instability was observed at the interface of the fluids with different density under the influence of gravity.
- The higher density fluid pushes the lower density fluid above it. The heavier fluid which is water, in this case, comes at the bottom and replaces the air from the bottom.