Rayleigh Taylor Instability Simulation at an Interface between two fluids of different densities

RAYLEIGH TAYLOR INSTABILITY SIMULATION AT AN INTERFACE BETWEEN TWO FLUIDS OF DIFFERENT DENSITIES USING ANSYS FLUENT                                                      

    1. AIM

Our aim is to simulate Rayleigh Taylor Instability at an interface between two fluids of different densities in transient state with different mesh configurations using ANSYS FLUENT.

2. THEORY/EQUATIONS/FORMULAE USED

ANSYS FLUENT academic version CFD package is used to carry out the simulation. It is a user friendly interface which provides high productivity and easy-to-use workflows. Workbench contains all workflow needed for solving a problem such as pre-processing, solving and post-processing.

Rayleigh Taylor Instability

It is instability of an interface between two immiscible fluids of different densities under the gravitational pull. The instability is driven by the density stratification. A heavy liquid on top of the lighter liquid under the gravity is fundamentally unstable.

Typical examples are water suspended on top of oil, mushroom clouds from volcanic eruption, and atmospheric nuclear explosions, and etc. 

As the equilibrium here is unstable for any disturbance and there will always be the tendency for the heavy fluid to come downwards and the lighter fluid to move upwards, this disturbance will keep growing. This activity was studied by Lord Rayleigh and the crucial point of realizing that the instability is equivalent to the situation of accelerating fluid, less dense fluid into the heavier fluid was done by G.I. Taylor. This instability also occurs in deep underwater on the surface of expanding bubble.

Some Practical CFD Models based on mathematical analysis of Rayleigh Taylor waves

1. Kelvin-Helmholtz Instability
2. Richtmyer - Meshkov Instability
3. Plateau – Rayleigh Instability

Phases of Rayleigh Taylor (RT) Instability

There are two phases of the Rayleigh Taylor Instability.

Linear Phase - Initially the disturbances are small as shown the below figure part (a) and instability can be approximated using linear equation. The amplitude of the disturbance grows exponentially with time.       

Non-Linear Phase – As the disturbances keeps growing the instability moves from linear to non-linear phase where the instability can’t be approximated using linear equation as shown the below figure part (b). The amplitude of the disturbances becomes huge with time.

                                                                                               Fig. Evolution of Rayleigh Taylor (RT) Instability

As a result it starts forming bubbles flowing upwards and spikes falling downwards and then these bubbles starts to merge as shown in the above figure part (c). At last it develops into a mixing region as shown in the above figure part (d).

This non-linear RT instability flow structure is mainly determined by the density variation of the fluids where the surface tension and viscosity of the fluids are negligible. The density variation of the fluid system can be studied by using a dimensionless number, called Atwood Number.       

Atwood Number (A)

It is a dimensionless number used to study the hydrodynamic instability in density stratified flow in fluid dynamics. It is the ratio of difference in densities to the summation of densities of the two fluids under the gravitational pull.  

                                                                                                              A = (ρ₁ - ρ₂)/ (ρ₁+ρ₂) 

                                                                                 where ρ₁ = Density of heavier fluid, ρ₂ = Density of lighter fluid

As the A gets closer to 0, the RT instability flow structure takes the form of “symmetric fingers” of the fluids and for A closer to 1, the lighter fluid takes a form of bubble-like structure. 

Problem – RT Instability

Transient state simulation of two phases to study the perturbation at the interface for density stratified fluids.

Calculation

Transient State Flow Simulation

Different Fluids chosen for the problem – Air, Water, User Defined Material

Courant number maintained = 0.2

Simulation details regarding each case are tabulated below.

    3. PROCEDURE

• In the process, firstly the flow domain for 2-phase is created in SpaceClaim and made sure that the interface is created between the 2 phases using share topology feature.
• Baseline meshing is done to understand the problem and then to know how much and where the refinement is required in order to study the flow and all the boundaries are named respectively. Also checked the mesh quality is good enough to carry out the simulation.
• Material, solver type, the initial and boundary condition, suitable turbulence model are defined as per the problem. In the current problem 4 cases are defined with changes in fluid material and the number of elements.
• Contour for phases with volume fraction of water on the flow domain is initiated in order to visualize the perturbations. Finally the animation is also added to study the overall behavior after the simulation is converged.

    4. NUMERICAL ANALYSIS (Software used – ANSYS 2018 R1)

• Geometric Model

The 2D geometry of flow domain of 2 phases of fluid is created in SpaceClaim and the cleanup is done as per the figure given below.

2D Geometry of flow domain of 2 phases

• Mesh

                                                                    Fig: Baseline Mesh

                                                                    Fig: Element Quality

• Solver Set-up

1. The mesh file is imported into the FLUENT software for the analysis of the modeled surface.
2. Once the mesh file is loaded and displayed, it is checked for the units, scale, and mesh.
3. There are two types of numerical solver methods pressure –based and density-based solver. The pressure based solver was used rather than the density based solver as the simulation is for lower Mach number. Absolute velocity formulation, Transient state was used for the analysis as only then the evolution of the instability can be visualized and understand how the stability is achieved at the end.
4. Gravity is checked on with -9.81 on the Y-axis as density stratified fluids are under the gravitational pull.

     5. Energy equation was switched off for the analysis as we are not interested in temperature of the system.

     6. Laminar turbulence model was used for the analysis.

     7. The fluid material chosen is air, Water Liquid and UDM.

     8. Multiphase Model chosen – Volume of Fluid with 2 numbers of Eulerian Phases and Implicit formulations for Volume Fraction parameters.

    9.  Phases need to be specified into primary phase as air and secondary phase as water.

   10. Solution methods – SIMPLE Scheme used for Pressure-Velocity coupling and the methods for Spatial Discretization are as per the below image.

   11. Use Mesh Display option to check whether the phases are specified as per the requirement.

    12. Standard initialization is done and then zones are patched accordingly.

    13. Contour for phases with volume fraction of water on the flow domain is initiated in order to visualize the perturbations during run time and also animations are added for every time-step.

    14. Run the iteration with appropriate time-step size and the number of time-steps.

    5. RESULTS

• CASE 1 - Baseline mesh – 0.5 mm At = 0.99
• Video Link - https://youtu.be/sH9hg_xiAT0

                                                                            1                                     2                                     3                                       4

                                                                           5                                     6                                      7                                        8

• CASE 1A - Mesh – 0.3 mm At = 0.99
• Video Link - https://youtu.be/_9QSGI84l3E

                                                                           1                                     2                                     3                                        4

                                                                           5                                     6                                      7                                        8

• CASE 1B - Mesh – 0.1 mm At = 0.99
• Video Link - https://youtu.be/wbsybi20CTk

                                                                         1                                     2                                   3                                    4

                                                                         5                                     6                                    7                                     8

• CASE 2 At = 0.43
• Video Link - https://youtu.be/p1zs625TwGs 

                                                                        1                                     2                                     3                                     4

                                                                       5                                     6                                    7                                     8

   6. CONCLUSION

1. The transient state flow simulation of two phases to study the perturbation at the interface for density stratified fluids is carried out with different time step size and number of time steps in all 4 cases.
2. In order to visualize such phenomena where we are interested to see the changes happening at the interface and both the phases throughout before getting stabilized, transient state simulations is the best suited option.
3. We can see the evolution of instability at the interface of density stratified fluids in all the cases as it starts forming bubbles flowing upwards and spikes falling downwards and then these bubbles starts to merge as per the theory.
4. We can also see the fluids getting reversed i.e., higher density fluid on bottom and less denser fluid on top at the end finally achieving the steady state by running the simulation for large number of time steps such as 6000 with time step size of 0.001.
5. Contours of phase from all the cases shows that the evolution of instability can be seen clearly in finer mesh configuration compared to coarse mesh.
6. The animation of all the cases shows that as the value of A gets closer to 0 (Case-2), the initial RT instability flow structure at the interface takes the form of bubbles and spikes of the fluids and for A closer to 1 (Case-1, 1A, 1B), the lighter fluid takes a form of bubble-like structure.
7. Hence the results are in good agreement and the underlying physics is visible clearly.

   7. REFERENCES

1. https://en.wikipedia.org/wiki/Rayleigh%E2%80%93Taylor_instability
2. https://www.researchgate.net/figure/Numerical-simulation-of-Rayleigh-Taylor-instability-with-single-fluid-model-without_fig1_230885678
3. https://www.preprints.org/manuscript/202007.0385/v1