Rayleigh Taylor Instability

1,

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

1. KELVIN-HELMHOLTZ OR RAYLEIGH TAYLOR HYBRID MODEL:

• This model was used to predict the primary break up of the intact liquid core of the jet and the Raleigh Taylor model determines the effects of acceleration of the different phases.
• This model predicts the shape of the spray considering the temperature at different phaces.

2.POTENTIAL FLOW MODELS

• In the model, which extends the work of Layzer , ordinary differential equations for the bubble heights and curvatures are obtained by considering the potential flow equations near the bubble tips. The model is applied to two‐dimensional single‐mode evolution as well as two‐bubble competition, for both the Rayleigh–Taylor (RT) and the Richtmyer–Meshkov (RM) instabilities, the latter treated in an impulse approximation.
• the model is applied to the evolution of three‐dimensional modes with an initial rectangular geometry. The model yields the aspect ratio dependence of the early nonlinear stages, in agreement with third‐order perturbation theory.

3.GRANULAR RAYLEIGH-TAYLOR INSTABILITY:

• A granular instability driven by gravity is studied experimentally and numerically. The instability arises as grains fall in a closed Hele-Shaw cell where a layer of dense granular material is positioned above a layer of air.
• The initially flat front defined by the grains subsequently develops into a pattern of falling granular fingers separated by rising bubbles of air. A transient coarsening of the front is observed right from the start by a finger merging process. The coarsening is later stabilized by new fingers growing from the center of the rising bubbles.

4. LATTICE BOLTZMANN MODEL:

• It is developed for two dimensional combustion simulations.
• It is an efficient numerical method.
• They are suitable for incompressible flows.

5. K L TURBULENCE MODEL:

• Used to describe the self similar growth of rayleigh -Taylor and Richtmyer-Meshkov instabilities
• The model describes the dominant eddies in the mixing zone with evolutionary equations for their characteristic dimension (L) and energy per unit mass (K=(v^2)/2).
• The equations are based on buoyancy-drag models for RT and RM flows.

2,

CASE 1

Element size = 0.5mm

Residuals Graph:

RAYLEIGH TAYLOR INSTABILITY:

Animation:

https://drive.google.com/open?id=1Y2gj-YTGllpg19gPUG82P2NDdMTRZxEn

CASE 2

Element size = 0.2mm

Residuals Graph:

RAYLEIGH TAYLOR INSTABILITY:

Animation:

https://drive.google.com/open?id=1phTyzh87uWHOV1Eos2jG_9KjindY1zRB

CASE 3

Element size = 0.15mm

Residuals Graph:

RAYLEIGH TAYLOR INSTABILITY:

Animation:

https://drive.google.com/open?id=1ecf1jRYxEJie8AAzPYH_9kcO5rAh0XjN

Why a steady state approach might not be suitable for these types of simulation.

• Transient solver better capture the flows.
• Transient solver can be used to run at a smaller time step, it can also run at varying time step.
• Smaller time steps are required to understand the instabilities which is only possible with transient solver.
• we expect more accuracy from transient simulation because it is time marching solution.
• Steady state simulation ignores many of the cross terms and higher order terms dealing with time.

3, ATWOOD NUMBER:

• The difference in the fluid densities divided by their sum is defined as Atwood number.
• It is a dimensionless number used in the study of hydrodynamic instabilities in density stratified flows.

= density of heavier fluid.(i.e, the density of water =997kg/m^3)

= density of lighter fluid.(i.e, the density of air=1.225 kg/m^3)

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