Rayleigh Taylor Instability Validation Case Study

Q.1 Practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves:

1. K-L turbulence model for the self-similar growth of the Rayleigh-Taylor and Richtmyer-Meshkov instabilities.

K-epsilon (k-ε) turbulence model is the most common model used in Computational Fluid Dynamics (CFD) to simulate mean flow characteristics for turbulent flow conditions. It is a two equation model that gives a general description of turbulence by means of two transport equations (PDEs).

2. Potential flow models of Rayleigh–Taylor and Richtmyer–Meshkov for bubble fronts.

A potential flow model of Rayleigh–Taylor and Richtmyer–Meshkov bubbles on an interface between an incompressible fluid and a constant supporting pressure (Atwood number A=1) is presented. competition, for both the Rayleigh–Taylor (RT) and the Richtmyer–Meshkov (RM) instabilities

3.Kelvin-Helmholtz or Rayleigh Taylor Hybrid Model.

This is particularly used for modelling spray atomization. Example modelling of fuel spray in diesel engines.

4.Rayleigh-Taylor Instability with Two-Fluid Model and Interface Sharpening.

The free surface flows are successfully modelled with one of the existing free surface models, such as: level set method, volume of fluid method, front tracking method, two-fluid model (two momentum equations) with modified interphase force and some others.

5.Discrete Boltzmann model (DBM).

We use a discrete Boltzmann model (DBM) to simulate the multi-mode Rayleigh-Taylor instability (RTI) in a compressible flow. This DBM is physically equivalent to a Navier-Stokes model supplemented by a coarse-grained model for thermodynamic non equilibrium behaviour.

Q2. Simulation of Rayleigh Taylor instability using 3 different mesh sizes :

Three different cases were set up using 3 different element sizes starting from a 0.5mm onwards.

Case 1 - Element size = 0.5mm Case 2 - Element size = 0.25mm Case 3 - Element size = 0.1 mm

The cases were solved using a multiphase VOF model using implicit formulation to capture the instabilities and track the position of the interface. A time step of 0.005 was chosen and 1000 iterations were run.

Case 1:

Case 2:

Case 3:

Observations:

When a lower density pushes a higher density fluid, Rayleigh Taylor instablility is observed at the interference.

• In the first case we can observe that as higher density fluid replaces the lower density fluid bubbles and vortexes are formed. And eventually these bubbles travel upwards.
• For second case we can observe that as higher density fluid replaces the lower density fluid bubbles and vortexes formed in more detail. Also there are small packets of trapped air which travel upwards.
• For third case we can clearly observe irregularities clearly forming at the intereference and individual bubbles are forming. We can also observe the whirls forming and how liquid collapses at the end.

Q3. Atwood number and its signficance:

It is a dimensionless number which is defined by the difference in the fluid densities divided by their sum.

Atwood Number for above case:

`A = (998-1.225)/(998+1.225)`

`A = 0.9975

For an atwood number close to 1, the lighter fluid below the heavier form bubble like plumes which can be observed in case number 3 validating the obtained results.