Rayleigh-Taylor Instability:

The objective of this project is to get a clarity on the phenomenon of Rayleigh-Taylor instability by running simulations where two immiscible fluids, denser fluid being on top of the lighter one, are subjected to gravity and to understand the relation between instability characteristics and Atwood number.

The instability experienced at the interface of a heavier fluid supported by a lighter fluid is known as Rayleigh-Taylor or RT instability. Two immiscible fluids with different densities, under the presence of gravity or equivalently, under an acceleration towards each other will result in RT instability. Some of the practical CFD models based on Rayleigh-Taylor instability are,

1) Saffman–Taylor instability: Also known as viscous fingering, it is the formation of patterns at the interface between two fluids with different viscousities. It occurs when a less viscous fluid is injected, displacing a more viscous fluid.The phenomenon is observed in petroleum recovery where water is often used to drive oil from the reservoir.

2) Richtmyer–Meshkov instability: RMI instability occurs when two fluids of different densities are accelerated by the passage of shock waves which causes perturbations on the interface to grow, producing vortical structures that results in the formation of a turbulent mixing layer.

3) Kelvin–Helmholtz instability: The Kevlvin-Helmholtz instability occurs when there is a difference in velocity across the interface of two fluids or a velocity shear in a single continous fluid. It can be observed in nature as the formation of waves over the sea as wind blows over it.

Atwood number:

It is a dimensionless number used to describe the density difference between two fluids sharing an interface. It is an important parameter in the study of Rayleigh-Taylor instability. The Atwood number is given by,

A = `(rho1 -rho2)/(rho1 +rho2)`

Where, `rho1` = density of heavier fluid

`rho2` = density of lighter fluid

Rayleigh Taylor instability simulation:

Here we conduct two simulations of the Rayleigh Taylor instability on a 2D plane with the fluids being,

1) water & air

density of water = 997 kg/m3

density of air = 1.225 kg/m3

Atwood number= `(997-1.225)/(997+1.225)` = 0.997

2)water & User-defined fluid(`rho = 400`kg/m3, `mu = 0.001`kg/m-s)

Atwood number = `(997-400)/(997+400)` = 0.427

• Case 1(Water & air):

Geometry:

The geometry of the domain is basically two 2D square planes kept on top of the other with the denser fluid being on top. The dimensions of both planes being 20*20 mm. Share topology had been enabled to exchange data between two domains.

Mesh:

The geometry being a simple 2D rectangular plane, the meshing was fairly simple. With an initial baseline mesh of element size 0.5mm, the simulation was carried out to check the results. The total element count of 3200 with all the elements being quadrilateral, the baseline mesh is shown below;

A slightly refined mesh for further simulation, with element size of 0.2mm and total element count of 13000 is shown below,

Element metrics for refined mesh;

The Setup:

Pressure based, absolute transient solver was used to run the simulation.Gravity was enabled in the negative y direction. Here, a steady state solver is not desirable due to the following reasons;

• Steady-state simulation computes the fully developed solution that does not change in time. When steady-state used the mean values are computed. But here, we are interested in obaserving the development and propagation of the instability.
• Transient simulation computes the instantaneous values in each time for each quantity and the results aremusch more comprehensive.

The viscous model was set to laminar and energy equations were not enabled. Also, Multiphase, volume of fluid model had been enabled with two phases for this case being water and air. The calculation had been set to implicit method. The solution was initialized using standard mode. After patching the domains with respect to volume fraction of water, a contour plot was created.

• Initial contour plot(Baseline mesh):

The top red region indicates presence of water and the bottom blue region indicates zero presence of water which implies it is air.

• Initial contour plot(Refined mesh):

Finally, with a maximum iterations per time step of 20 and time step size of 0.25, the calculation was run.

Solution:

• The link for solution animation video of Baseline Mesh simulation is given below;

https://youtu.be/yEHaObKzQrE

• The link for solution animation video of Refined Mesh simulation is given below;

https://youtu.be/QNBuCnKTEsQ

Results:

• Baseline mesh residual plot:

• Refined mesh residuals plot:

• Baseline mesh, Final contour:

• Refined mesh, Final contour:
  
  

With the refinement of the mesh, a much more visually clear representation of the instability and propagation of petrubations was observed.

• Case 2(Water & User defined fluid):

Here the same setup as before was duplicated for running another simulation with water and a user defined fluid as fluids. The properties of the user defined fluid are shown below;

Density = 400kg/m3

Viscosity = 0.001 kg/m-s

The viscosity of the fluid is close to that of water but the density is less than half of water density.

Solution:

• The link for solution animation video of Baseline Mesh simulation is given below;

https://youtu.be/DknD37pKjKU

Results:

• Residuals plot:

• Final contour plot:

Observations:

• In the water and air case, initially it was observed that air took the form of large bubble like shapes and moved upwards while water was moving down in sharp,knife like streams in between these bubbles. But eventually the whole of water volume pushed aside the air bubbles and got settled at the bottom.
• In case of water and the user-defined fluid, the instability began to form slowly than compared to the previous case. Here both the fluids took roughly globular form while the instability propagated across the surface although, water broke the pattern quickly and forced itself down. Many vortices where formed within the domain due to the opposing direction of flow of the two fluids but eventually both the fluids settled down with water at the bottom.

Conclusions:

• When the Atwood number is close to 1, the propagation of the instablity is seemingly quicker. In case of water and air, the atwood number was close to 1 and the petrubations were formed instantaneously and violently.
• When Atwood number is lower, like in case of water and user defined fluid simulation, where the value was 0.4, the petrubations propagated much slowly and the instability took much more time to end and reach equilibrium.