Transient state analysis of Rayleigh Taylor instability in ANSYS FLUENT.

Objective

• Discuss some practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves. And explain how these mathematical models have been adapted for CFD calculations.
• Perform the Rayleigh Taylor instability simulation for 3 different mesh sizes with the base mesh being 0.5 mm. Compare the results by showing the animations.
• Define the Atwood Number. Find out the Atwood number for this case and explain how the variation in Atwood number affects the behavior of the instability. How can we use the Atwood number to validate our simulation result?

Rayleigh Taylor Instability

The Rayleigh–Taylor instability or RT instability is 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.

Some practical CFD models based on mathematical analysis of Rayleigh Taylor waves.

• Turbulent Mixing Modelling by Rayleigh-Taylor Instability

Rayleigh-Taylor instability occurs when a perturbed interface between two fluids of different density is subjected to a normal pressure gradient, Taylor. If the pressure is higher in the light fluid than in the dense fluid the differential acceleration produced causes the two fluids to mix.

Direct two-dimensional numerical simulation and experiments, in which small rocket motors accelerate a tank containing two fluids, have been used to investigate turbulent mixing by Rayleigh-Taylor instability at a wide range of density ratios. The experimental data obtained so far has been used to calibrate an empirical model of the mixing process which is needed to make predictions for complex applications. The model devised, which is a form of turbulence model, is based on the equations of multiphase flow. These equations describe velocity separation arising from the action of a pressure gradient on fluid fragments of different density. The dissipation arising from the drag between the fluid fragments is treated as a source of turbulence kinetic energy which is then used to define turbulent diffusion coefficients. Gradient diffusion processes are thereby included in the model.

• A Lattice Boltzmann Scheme for Incompressible Multiphase Flow and Its Application in Simulation of Rayleigh–Taylor Instability.

The new lattice Boltzmann scheme for simulation of multiphase flow in the nearly incompressible limit. The new scheme simulates fluid flows based on distribution functions. The interfacial dynamics, such as phase segregation and surface tension, are modeled by incorporating molecular interactions. The lattice Boltzmann equations are derived from the continuous Boltzmann equation with appropriate approximations suitable for incompressible flow. The numerical stability is improved by reducing the effect of numerical errors in the calculation of molecular interactions. An index function is used to track interfaces between different phases. Simulations of the two-dimensional Rayleigh–Taylor instability yield satisfactory results. The interface thickness is maintained at 3–4 grid spacings throughout simulations without artificial reconstruction steps.

The past several years have witnessed numerous efforts to apply the lattice Boltzmann method to multiphase flows. Two fundamental interfacial dynamics, the Laplace law, and the dispersion law in capillary waves have been verified. Other applications include simulations of the spinodal decomposition and multiphase flows through porous media. These works either focused on simple problems or lacked quantitative comparisons with benchmark studies. The accuracy and efficiency of the LBM models in the simulation of multiphase flows remain to be explored. We will use the Rayleigh–Taylor instability as our test case. There are several reasons for us to choose the Rayleigh–Taylor instability as our benchmark problem. First, the Rayleigh–Taylor instability is of great significance in both fundamental research and practical applications. At late stages, the flow involves turbulent mixing—a ubiquitous but poorly understood phenomenon. Our study will provide more insight into this classical multiphase flow problem. Second, the Rayleigh–Taylor instability provides enough complexities to challenge the capability of our scheme. Beyond the initial stage, the Rayleigh–Taylor instability exhibits strong non-linearity which is associated with the growth of the secondary Kelvin–Helmholtz instability. The instability evolving from a random initial perturbation exhibits an even more complicated pattern. The success in simulating such a complicated multiphase flow problem will bring great confidence for future applications of the lattice Boltzmann method. Finally, there are copious theoretical studies and numerical simulations on the Rayleigh–Taylor instability in literature. We can use these data to quantify the accuracy of our scheme.

• Tilted Rayleigh-Taylor for 2-D Mixing Studies

The significance of this test problem is that, unlike planar RT experiments such as the Rocket-Rig, Linear Electric Motor - LEM, or the Water Tunnel, the Tilted-Rig is a unique two-dimensional RT mixing experiment that has experimental data and now (in this TP) Direct Numerical Simulation data from Livescu and Wei. The availability of DNS data for the tilted-rig has made this TP viable as it provides detailed results for comparison purposes. The purpose of the test problem is to provide 3D simulation results, validated by comparison with the experiment, which can be used for the development and validation of 2D RANS models. When such models are applied to 2D flows, various physics issues are raised such as double counting, combined buoyancy and shear, and 2-D strain, which have not yet been adequately addressed. The current objective of the test problem is to compare key results, which are needed for RANS model validation, obtained from high-Reynolds number DNS, high-resolution ILES or LES with explicit sub-grid-scale models. The experiment is incompressible and so is directly suitable for algorithms that are designed for incompressible flows (e.g. pressure correction algorithms with multi-grid); however, we have extended the TP so that compressible algorithms, run at low Mach number, may also be used if careful consideration is given to initial pressure fields. Thus, this TP serves as a useful tool for incompressible and compressible simulation codes, and mathematical models.

                                                                      Case 1

1. Geometry

• We have to create two 20 mm square 2D plane with a common interface as shown in the above image.
• By using pull feature we can generate a surface from our sketch in SpaceClaim.
• Now we have to name the surfaces. The above surface is water and the bottom surface is air.
• The shared topology is enabled for the two blocks.

2. Meshing

• In case 1 we will generate a baseline mesh of element size.
• Element size: 0.5 mm.
• Total nodes: 3321.
• Total elements: 3200.

Generated mesh

3. Setup

• Transient-state analysis.
• Solver: pressure based.
• Velocity formulation: absolute.
• Solution method used: SIMPLE.
• Viscous model: Laminar.
• Material: Primary fluid - Air, Secondary fluid - liquid water
• Solution method: SIMPLE.
• Multiphase enabled and volume of fluid method selected. Solving implicitly.
• Initialization method: Standard.
• Gravity is enabled: -9.81 m/s^2
• Time step size: 0.005 seconds.
• Number of Time steps: 1500.

After standard initialization, we have to define the patch. This is done by opening the patch window and selecting the water from the drop-down menu. Then defining 1 in the water area and entering 0 in the air region.

Fluid       Top-block       bottom-block

Air               0                          1

Water           1                          0

4. Result

Residuals

Rayleigh Table instability animation

                                                                      Case 2

1. Geometry

The same geometry used in case 1 will be used in case 2. In this case, we will refine the mesh to see how the solution changes with mesh refinement.

2. Meshing

• In case 2 we will generate a refined mesh.
• Element size: 0.25 mm.
• Total nodes: 13041.
• Total elements: 12800.

Generated mesh

3. Setup

• Transient-state analysis.
• Solver: pressure based.
• Velocity formulation: absolute.
• Solution method used: SIMPLE.
• Viscous model: Laminar.
• Material: Primary fluid - Air, Secondary fluid - liquid water
• Solution method: SIMPLE.
• Multiphase enabled and volume of fluid method selected. Solving implicitly.
• Initialization method: Standard.
• Gravity is enabled: -9.81 m/s^2
• Time step size: 0.005 seconds.
• Number of Time steps: 1500.

After standard initialization, we have to define the patch. This is done by opening the patch window and selecting the water from the drop-down menu. Then defining 1 in the water area and entering 0 in the air region.

Fluid       Top-block       bottom-block

Air               0                          1

Water           1                          0

4. Result

Residuals

Rayleigh Table instability animation

                                                                      Case 3

1. Geometry

The same geometry used in case 1 will be used in case 3. In this case, we will further refine the mesh to see how the solution changes with more refined mesh than case 2.

2. Meshing

• In case 3 we will further reduce the element size to generate a more refined mesh.
• Element size: 0.15 mm.
• Total nodes: 35780.
• Total elements: 35380.

Generated mesh

3. Setup

• Transient-state analysis.
• Solver: pressure based.
• Velocity formulation: absolute.
• Solution method used: SIMPLE.
• Viscous model: Laminar.
• Material: Primary fluid - Air, Secondary fluid - liquid water
• Solution method: SIMPLE.
• Multiphase enabled and volume of fluid method selected. Solving implicitly.
• Initialization method: Standard.
• Gravity is enabled: -9.81 m/s^2
• Time step size: 0.005 seconds.
• Number of Time steps: 1500.

After standard initialization, we have to define the patch. This is done by opening the patch window and selecting the water from the drop-down menu. Then defining 1 in the water area and entering 0 in the air region.

Fluid       Top-block       bottom-block

Air               0                          1

Water           1                          0

4. Result

Residuals

Rayleigh Table instability animation

Conclusion

• Rayleigh Taylor instability was observed at the interface of the fluids with different density under the influence of gravity.
• The lower density fluid pushes the higher density fluid above it. The heavier fluid which is water, in this case, comes at the bottom and replaces the air from the bottom.
• The formation of air bubbles takes place which moves upward.
• 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.

Reason for using Transient State analysis

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.

Atwood Number

The Atwood number (A) is a dimensionless number in fluid dynamics used in the study of hydrodynamic instabilities in density stratified flows. Atwood number is an important parameter in the study of Rayleigh–Taylor instability and Richtmyer–Meshkov instability. It is a dimensionless density ratio defined as:

`A = (rho_1 - rho_2)/(rho_1 + rho_2)`

where

`rho_1` = density of heavier fluid.

`rho_2` = density of lighter fluid. 

For Atwood number close to 0, the fluid flow in Rayleigh Taylor instability flows take the form asymmetric fingers of fluid; for Atwood number close to 1, the lighter fluid below the heavier fluid takes the form of larger bubble-like plumes.

Calculation Atwood number for the above-performed simulation

A = (1000 - 1.25)/(1000 + 1.25) = 0.9975

In our simulation, we see that the lighter fluid (air) rises up and bubble-like plumes take place. The Atwood number calculated for our result is close to 1. The Atwood number close to 1 signifies the same phenomenon. And hence validates our result.