Rayleigh Taylor Instability Challenge

Objective:

1. To perform Rayleigh-Taylor instability simulation for different mesh sizes on instability for water and air, And refined mesh for water and oil with given parameters.
2. To find Atwood number for the simulations performed and to determine the effects of Atwood number on the instability model.

Theory:

The Rayleigh–Taylor instability, or RT instability, 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 behavior of water suspended above oil in the gravity of Earth, mushroom clouds like those from volcanic eruptions and atmospheric nuclear explosions, supernova explosions in which expanding core gas is accelerated into denser shell gas, instabilities in plasma fusion reactors and inertial confinement fusion.

The following are a few CFD models that are based on Rayleigh-Taylor instability calculations:

• Kelvin-Helmholtz Instability: This phenomenon occurs when there is a velocity difference across the interface between two fluids or a velocity shear in a continuous fluid. Two fluids streams travelling at two different speeds will lead to this phenomenon.
• Richtmyer-Meshkov Instability: This phenomenon occurs when a shock-wave separates two different fluids. It combines compressible phenomena, such as shock interaction and refraction, with hydrodynamic instability, including non linear growth and subsequent transition to turbulence, across a wide range of Mach numbers.
• Plateau-Rayleigh Instability: This phenomenon, explains how and why a falling system of fluids breaks into smaller packets of fluids with same volume but different surface area. It is related to Rayleigh-Taylor instability, and is a part of a greater branch of fluid dynamics concerned with fluid thread breakup. This fluid is exploited in the design of a particular type of inkjet technology where a jet of liquid is perturbated in to a steady stream of droplets.
• Potential Flow model: This model was developed by Layzer and it provides information on the movement of fluids in a circular tube when a constant supporting pressure is present.

Atwood number: In fluid dynamics, Atwood number is used in the study of hydrodynamic instabilities in density stratified flows. The Atwood number (A) is a dimensionless number.

 

            (b) Element size: 0.3 mm.

         

Case 2: The mesh with element size of 0.15mm is used for this case.

Step 3: Setup & Solution:

Case 1:

1. Solver type: Pressure based.
2. Time: Transient.
3. Gravity: -9.81 m/s².
4. Model: Multiphase.
   1. -Volume of fluid.
   2. -No. of Eulerian phases = 2.

• -Implicit solver type.

5. Viscous model: Laminar.
6. Energy- off.
7. Materials:
   1. Air (Phase 1) (Case 1)
   2. Oil (Phase 1) (Case 2)
   3. Water (Phase 2) (Case 1&2).

8. Initialization: Standard.
9. Calculation: No. of time steps: 1000(Case 1) & 1300(Case 2).

                    No. of iterations for each time step: 20.

                    Time step size: 0.005.

Step 4: Results:

 Case 1: Rayleigh-Taylor instability for water and air.

 Case 1(a):

Residuals plot:

Case 1(b):

Residuals plot:

 

Case 1(c):

Residuals plot:

 

Animation video:

Case 2:

Residuals plot:

 

Animation video:

Atwood number:

 Case 1: A = 0.9975

 Case 2: A = 0.4278

Conclusion:

1. Atwood number of Case 1 is much higher compared to Case 2. Therefore, the Rayleigh-Taylor instability in Case 2 is slower than that of Case 1.
2. The surface area of formed of bubbles and spikes in Case 2 are lower than that of Case 1.
3. Viscosity of oil is higher due to which the growth rate of instability is much slower that that of air as it has much lower viscosity.