Week 5 - Rayleigh Taylor Instability

Aim: To perform Rayleigh-Taylor instability CFD simulation.

Objectives:

1. To develop a numerical case set-up for Rayleigh-Taylor instability problem in Ansys Fluent.

2. To conduct grid dependency test to understand the variation in RT instabilites.

3. To understand the effect on RT instabilities due to variation in Atwood number.   

Introduction:

Rayleigh-Taylor Instability:

• It is an instability which occur at the fluid interface due to density stratification under the influence of gravity.
• The two fluids must be immiscible.
• The instability is primarily driven by density differences.

                                                                                   Fig (a):  Rayleigh-Taylor Instability

In fig (a),

The fluid in blue color represents lighter fluid

The fluid in red color represents heavier fluid

• Intially at time t= 0, the system of two fluids is in hydrostatic equlibrium. The dense fluid (e.g. water) sits atop of lighter fluid (can be air) in a constant gravitational field. The pressure gradients are balanced and fluid is in static equilibrium. 
• As a function of time advancement, perturbation develops at the interface. These perturbations grow rapidly as the denser fluid accelerates in downward direction. Near boundaries lighter fluid pushes the heavier fluid and strives to move in upward direction.
• This cause pressure imbalances on either side which drives the flow. The flow is generally driven by pressure gradients below and top of the fluid. 
• The instabilities results in plumes flowing upwards and spikes falling downwards. 
• The initial linear phase with small wavelengths and amplitudes completely transforms into non-linearity with larger irregular wavelengths and amplitudes.
• In general, the density disparity between two fluids determines the subsequent non-linear Rayleigh-Taylor instabilities. 
• Assuming the effect of surface tension and viscosity to be negligible.

To examine the hydrodynamic instabilities due to density stratification, a dimensionless density ratio called as Atwood number is used.

Mathematically Atwood number is given by,

`A = (rho2-rho1)/(rho2+rho1)`

where, 

`rho2` = density of the heavier fluid

`rho1` = density of the lighter fluid

There are some CFD models that are based on mathematical analysis of Rayleigh-Taylor waves:

1. Mass diffusion model

The mass diffusion equation describes the transport of mass and energy due to model viscosity and hence drives actual mixing in the fluids. Even if any configuration is Rayleigh-Taylor unstable, the diffusion process could be responsible for significant mixing if the viscosity is sufficiently large. 

2. Single fluid model

2.1 Potential flow model:

This model is valid for all instability changes from the very linear stage through the early non-linear but limited to Atwood number of 1. In this model, the single mode perturabtion expands, the flow equation to second order around the bubble tips, hence obtaining an ordinary differential equation for the bubble velocity. 

2.2 Buoyancy drag model:

It is valid for every Atwood number but limited to non-linear stages of the instability. It uses ordinary differential equations to evolve the width of the mixing layer. The bubble, spikes or amplitudes are described by balancing the inertia, buoyancy and drag forces. This model cannot deal with multiple mixing interfaces; cannot be easily extend to two or three dimensions, and as a rule, do not address demixing, also known as counter gradient transport i.e. reduction of total fluid masses within the mixing zone. To address the above problems, two-fluid (multi-fluid) models has to be used. 

2.3 Besnard-Harlow-Rauenzahn (BHR):

It is more advanced version of the single-fluid model. The BHR model is based upon the evolution equations arising from second-order correlations and gradient-diffusion approximations. Using a mass weighted-averaged decomposition, the original BHR model includes the full transport equation for the Reynold's stresses, turbulent mass-flux velocity, density fluctuations and turbulent kinetic energy dissipation rate.  

3. Multi-fluid models:

These models use a separate set of equation for each fluid in addition to the main flow eqautions and provide an accurate modeling framework for demixing by correctly capturing the relative motion of different fluid fragments. 

3.1 Two equation K-L turbuence model:

The K-L turbulence model inolves equation for turbulent kinetic energy (K) and turbulent length scale (L). The starting point for deriving the model quations are the buoyancy-drag models for the self similar growth for RT instabilities.

3.2 Three equation K-L-a turbulence model:

The three equation model was developed as an extension to the two equation K-L model by including a third equation for the turbulent mass-flux velocity. 

3.3 Four equation K-L-a-b turbulence model:

The four equation model was developed as an extension for three equation K-L-a model by including a fourth equation for density-specific volume covariance. The model can accurately reflect the effect of  changes in the density fluctuations. 

Case studies:

Case study 1: Effect of mesh refinement over CFD results.

In this study, the results obtained from Case 1 and Case 2 will be evaluated on the basis of mesh refinement.

Case study 2: Effect of Atwood number over Rayleigh-Taylor instabilities. 

For same mesh characteristics the two case set-up's (Case 2 and Case 3) with different Atwood number will be evaluated to understand the effect on Rayleigh-Taylor instabilities.

Case set-up:

• Solver- Pressure based and absolute velocity is used.
• Transient state calculation.
• Gravity is enabled in -ve Y-axis.
• Multiphase model is turned on along with volume of fluid and number of Eulerian phases is chosen as 2. Accordingly, implicit scheme is selected.
• Viscous model- Laminar
• Patching of the volume of the two phases is performed before start of the simulation. Phase used for patching is water and volume fraction of "1" is taken for water surface and "0" for air surface.

Why not steady-state calculation?

The steady state calculations are useful when the study is focused on to calculate / simulate the end results. Here in this study, the instabilities information is sensitive and completely time-dependent. At each and every time step, the instabilities grow exponentially. Though steady state simulation is capable to capture those instabilities but the results would not be time-accurate. Anyways transient state calculation with reduced time-steps can capture these instabilities at each and every time-steps and sensible time dependent results can be obtained.  

[A] Pre-Processing:

Geometry:

• Geometry is created in "SpaceClaim" with two different components (surfaces) named as "air" and "water".
• The dimensions of the constraint domain is 20 x 20 mm for each surface. Both surfaces are equal in dimension.
• The interface between the surface is shared using "share" tool under workbench to deploy conformal mesh and share information through the interface.

Case 1

1.1 Details:

1.2 Mesh:

1.3 Enlarged view:

Case 2:

2.1 Details:

2.2 Mesh:

2.3 Enlarged view:

Case 3

3.1 Details:

Note: Same mesh is deployed as of Case 2. 

[B] Post-Processing :

I. Case (1):

II. Case (2):

III. Case (3):

Technical Discussion (Case study 1):

• In the first case study, baseline (coarse) mesh produced initial results and gave confirmation about the case set-up. The results obtained using coarse mesh lack in resolution of the numerical solution. It seems like most of the instabilities are averaged out and constructed as an advancement of time-step. The initial perturbation at the interface is asymmetrical and pre-maturely much unstable. The interface is bit thicker because of coarser mesh, which has cell-centroids spaced bit away from the interface. As a function of time advancement the interface gets smeared out due to numerical difusion within the interfaces. This causes inappropriate capture of instabilities at the interface. 
• The case with finer mesh resulted in good resolution of the numerical solution due to small/finer interface. This is because more number of cells capture the volume of fluid and provides fluidity to numerical solution. Whereas in case 1 numerical solution was comparatively stiff. Furthermore the instabilities grow exponentially and the simulated results look quite realistic as compared to the results obtained in case 1. 

Technical Discussion (Case study 2):

• In this study, the two case set-up's with different Atwood number are evaluated. The larger density ratio results in stronger force to develop the interface instability. 
• For the case with smaller Atwood number, (Case 3) the density disparity between the two sides of the interface is small and the density distribution of the upper and lower layer is nearly symmetrical. The initial density stratification plays a dominant role and the expansion compression effect has a little influence.
• With increase in Atwood number close to 1 (Case 2) the stabilization effect of the initial density stratification decreases and the instability caused by expansion compression effect becomes more significant. 
• The evolution of interface grows faster with increase in Atwood number.
• Larger Atwood number results in larger amplitude of perturbation.  

Conclusion:

In the present study, the transient simulation approach was implemented to study the Rayleigh-Taylor instabilities. For the grid dependency, the interface thickness has significant impact on numerical diffusion. The interface thickness reduces with mesh refinement and produces better results at less interface thickness. Additionally, the variation of Atwood number has direct impact on the perturbation amplitude and evolution of interface growth. The implementation of finer mesh and smaller time-steps gave promising visualization of Rayleigh-Taylor instability. 

References: 

 1. Indrashis saha, (2020). Study of Rayleigh Taylor Instability with the help of CFD simulation.

 2. http://www.scholarpedia.org/article/Rayleigh-Taylor_instability_and_mixing

 3. https://eng.libretexts.org/Bookshelves/Civil_Engineering/Book%3A_Fluid_Mechanics_(Bar-Meir)/04%3A_Fluids_Statics/4.7%3A_Rayleigh%E2%80%93Taylor_Instability