Rayleigh Taylor Instability Simulation

OBJECTIVE

         The aim of the project is to perform Rayleigh Taylor instability for two fluids one dense and light fluid at the simulation is done to check the variation at the interface when gravity is present.

TASK

          Perform a grid dependence test on the 2D simulation with the base mesh being 0.5mm and explain what type of approach to be used for the simulation.

Practical CFD models based on Mathematical Analysis of RT-waves.

1. Kelvin-Helmholtz instability can occur when there is velocity shear in a single continuous fluid or where there is a velocity difference across the interface between two fluids. eg- wind blowing over the water (i.e) instability manifests in the waves on the water surface.

2. Richtmyer-Meshkov instability occurs when two fluids of different densities are impulsively accelerated. The penetration distance of heavy fluid bubbles into the lighter fluid is a function of the acceleration time scale. The development of the instability begins with small amplitude perturbations that initially grow linearly with time. This is followed by a nonlinear regime with bubbles appearing in the case of a light fluid penetrating a light fluid. A chaotic regime eventually is reached and the two fluids mix. 

3. Plateau-Rayleigh instability occurs in a falling stream of fluid breaks up into smaller packets with the same volume bt less surface area. It is related to RT instability and part of a greater branch of fluid dynamics concerned with fluid thread breakup. This fluid instability is exploited in the design of a particular type of injector technology where a jet of liquid is perturbed into a steady stream of droplets. The driving force of the plateau rayleigh instability is that liquids, by virtue of their surface tensions tend to minimize their surface area. 

PROCEDURE

• Create a two rectangle of size 20 * 20 and name the component as air & water. Workbench --> share prep option is used to share the edge of the two rectangles.
• Create a baseline mesh of size 0.5mm

                  a) Geometry                                                       b) Mesh

• Open the setup Solver: Transient pressure based.                                                                                                                                                                                                             Gravity: X = 0 m/s^2.                                                                                                          Y = -9.81 m/s^2.                                                                                                                                                                                                                     Viscous type: Laminar.                                                                                                                                                                                                                           Model type: Multiphase.                                                                                                                                                                                                                           Method: Implicit.                                                                                                                                                                                                                                   Material: Air & water-Liq.                                                                                                                                                                                                                         Time step size: 0.005.                                                                                                                                                                                                                             No of time steps: 1500.

• Initialize the solver & click on patch i) Phase- water, variable- volume fraction, value-1.                                                      ii) Phase - Air, variable - volume fraction, value -0.

• The patch is used to initialize the data (i.e) L = 1 where water is present & @ L = 0 water is not present. Create a new contour for all phase & volume fraction for which the animation file is created.

Scale Residuals

Contour Plot

Line plot

Line Chart

CASE 2:

The element size in this case has been reduced to 0.25

Scale residuals

Contour plot

Line plot

Line chart

CASE 3:

The element size in this case has been reduced to 0.1

Scale residual

                        Contour plot                                                        Line plot

Line Chart

INFERENCE

In the above cases, the Rayleigh Taylor instability is observed at the interface when a lower density fluid pushes a higher density fluid due to which formation of shock waves at the interface takes place where the formation of air bubbles occurs. by refining the mesh the simulation results get smoother at the irregularities that take place at the interface of the two fluids. The vortex takes place at the interface which travels towards the upward region with time. In the third case, the solution has diverged due to the mesh size due to which the CFL number has become more than the critical so the solution fails. CFL number depends on two factors time step & mesh size (i.e) CFL no = dt/dx & for diffusivity, the CFL number must be less than 0.25. So the time step of the simulation is to be reduced from 0.005 to 0.0001 in order to achieve the condition of the CFL diffusivity. 

why a steady-state approach might not be suitable for these types of simulation

In both steady & transient solver, the end results obtained are the same, so the difference between steady-state & transient is that type of results calculated from them. In steady-state simulation only the end state results are considered whereas the path of the function is considered. Since in RT-instability, we are more concerned to learn about the transition of the irregularities that occurs when dense fluid poured into a lighter fluid under the effect of gravity. So the steady-state approach is not suitable for the above simulation

The Atwood number (A) is a dimensionless number in the fluid dynamics used in the study of hydrodynamic instabilities in density stratified flows. It is a dimensionless density ratio defined as

where

 = density of heavier fluid

 = density of lighter fluid

       A = 1000 - 1.225                                                                                                                    1000 + 1.225

          = 0.9975.

Atwood number is an important parameter in the study of Rayleigh-Taylor instability. For A close to 0, RT instability flows take the form of symmetric fingers of fluid, for A close to 1, the much lighter fluid below the heavier fluid takes the form of larger bubble-like plumes.

The calculated Atwood number is close to 1 and from the simulation results, it is found that when high dense fluid poured on low dense fluid under gravity the formation of bubble-like plumes takes place which travels towards the upward region in the form of waves and some gets trapped at the bottom during initial stages, which afterward try to move to upper region which gets separated with diffusivity process.