Studying the Rayleigh -Taylor instability of an interface between air and water using ANSYS Fluent

INTRODUCTION:

Rayleigh-Taylor instability occurs at the interface of two fluids of different densities when there is mixing due to the lighter fluid pushing the heavier fluid under the influence of gravity. Examples include water suspended in oil, mushroom clouds formed by volcanic eruptions or nuclear explosions, instabilities in plasma core reactors, etc. Various practical CFD models have been based on the mathematical analysis of the Rayleigh-Taylor waves based on different phenomena. Some of them are:

1. Kelvin-Helmholtz instability

This instability predicts the transition to turbulent flow in fluids of different densities moving at various speeds on top of each other. The discontinuity of the flow at the interface of the fluids generates shear stress at the interface inducing an unstable vortex sheet that propagates through the flow as a spiral. An example includes wind blowing over water which manifests in waves on the water surface.

2. Richtmyer-Meshkov instability

This instability occurs when two fluids of different densities are impulsively accelerated, normally by the passage of shock waves. The development of this instability begins with small perturbations which initially grow linearly with time. This is followed by a non-linear regime with bubbles appearing in case of light fluid penetrating a heavy fluid and spikes appearing in case of heavy fluid penetrating the lighter fluid.

3. Plateau-Rayleigh instability

This instability explains why and how a falling stream of fluid breaks up into smaller packets with the same volume but less surface area. This driving force of this instability is that liquids, by virtue of their surface tension, tend to minimize their surface area. A liquid jet falling vertically breaks up into smaller droplets when the length of the falling liquid reaches a critical value. This instability is used in the development of a particular type of inkjet technology whereby a jet of liquid is perturbed into a steady stream of droplets.

COMPUTATIONAL GEOMETRY AND MESH:

In this project, we will study the Rayleigh-Taylor instability at the interface of water and air, where 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 density of water and air, there will be an instability developing at the interface which will finally settle when the water with a higher density is below the air with a lower density. The simulation is a 2-D simulation with two blocks, one for each fluid, stacked on top of one another. These two blocks have a shared interface where the instability is initiated.

The simulation is studied for three different mesh sizes starting from the baseline mesh and consequently making the mesh finer and finer. The entire domain is meshed using quadrilaterals with the proper sizes. The size and number of elements in the mesh are listed in the table below:

`{:(, ,ul{text{Mesh Size (mm.)}},,ul{text{No. of Elements}}),(text{Mesh 1},,0.5,,3200),(text{Mesh 2},,0.25,,12800),(text{Mesh 3},,0.15,,35378):}`

The figure below shows the three meshes used for the simulation.

SETUP OF THE FLOW PHYSICS:

The simulation is a pressure-based transient 2-D simulation. The acceleration due to gravity is activated for this problem with a value of `g = -9.81` `m"/"s^2` in the Y-direction. Without gravity, there will be no mixing. The flow is a laminar flow since there is no acceleration/velocity of the fluid as an input. An implicit Volume of Fluid Multiphase model with 2 Eulerian phases is also activated so that both air and water can be defined. The properties of air and water are defined below:

Air

Density: `rho = 1.225` `kg"/"m^3`

Dynamic Viscosity: `mu = 1.7894 times 10^{-5}` `kg"/"m-s`

Water

Density: `rho = 998.2` `kg"/"m^3`

Dynamic Viscosity: `mu = 0.001003` `kg"/"m-s`

The difference in the density of the two materials combined with gravity initiates the mixing of the two fluids. Both the top and the bottom blocks are assigned a mixture cell zone. The solution is initiated using hybrid initialization, after which the two blocks are patched with the corresponding material (water/air). The patch is done by defining the initial volume fraction of the fluid in each block.

Top block

Volume Fraction of Water `= 1`

Volume Fraction of Air `=0`

Bottom block

Volume Fraction of Water `=0`

Volume Fraction of Air `=1`

Hence, initially both the blocks fully consist of either air or water as shown below:

A monitor for the volume fraction of water in the bottom block is defined so as to check the convergence of the solution. Mesh 1 and 2 are simulated with a time-step of `0.01` `sec` for a total of `2000` time-steps (`20` `sec`), whereas Mesh 3 is simulated with a time-step of `0.005` `sec` for a total of `4000` time-steps (`20` `sec`).

RESULTS AND DISCUSSION:

The convergence plots for the three meshes are shown below which show that the solution is converged. The movies for all the three meshes are also shown below.

Mesh 1

Mesh 2

Mesh 3

Steady-state simulation is not used for this challenge as we want to study and visualize the mixing stage of the process. In steady-state results, we will directly see the end result without ever seeing the instabilities and patterns which are formed in the initial stages of the mixing.

During the mixing process, we can see the bubbles of air rising through the water and eventually, the water settles down because of its relatively higher density than air. The refined meshes can effectively show the finer dynamics of the mixing process than the coarser meshes.

ATWOOD NUMBER:

The Atwood number is a dimensionless quantity in fluid dynamics which is used to study the hydrodynamic instabilities in density stratified flows. It is defined as:

`text{A} = {rho_1 - rho_2}/{rho_1 + rho_2}`

Here, `rho_1` is the density of the heavier fluid and `rho_2` is the density of the lighter fluid. So for this scenario, `rho_1 = 998.2` `kg"/"m^3` and `rho_2 = 1.225` `kg"/"m^3`.

`text{A} = {998.2 - 1.225}/{998.2 + 1.225} = 0.9975`

The Atwood number is useful in the prediction of the penetration distance of the heavy fluid into the lighter fluid with time as `text{A}g t^2`, where `g` is the acceleration due to gravity and `t` is the time. For low Atwood number (close to `0`), the lighter fluid rises in the form of symmetrical fingers of fluid since the densities are similar. For high Atwood number (close to `1`), the lighter fluid rises rapidly in the form of huge bubble-like plumes because of the greater difference in the densities.