Week 5 - Rayleigh Taylor Instability

Aim : -

• To conduct the Rayleigh Taylor Instability in CFD simulation.

Introduction : -

• Rayleigh Taylor Instability (RTI) is fluid instability that occurs when lighter fluid accelerate over the heavier fluid. It is a fundamental instability that occurs into the interface between two fluids of different densities . It is a dynamic process where the two fluids interpenetrate and mix to reduce their combined the potential energy.

Theory : -

• In this Rayleigh Taylor Instability (RTI), we have two phases in hydrostatic equilibrium, in which the heavier fluid(ex: water) sits atop a lighter fluid(ex: air) in a constant gravitational field. As the RTI develops, the initial perturbations progress from a linear growth phase, eventually developing "plumes" flowing upwards(in the gravitational buoyancy sense) and "spikes" falling downwards. In the linear phase, equations can be linearized and the amplitude of perturbations is growing exponentially with time. In the non-linear phase, the perturbation amplitude is too large for the non-linear terms to be neglected. In general, the density disparity between the fluids determines the structure of the subsequent non-linear RT instability flows (assuming other variables such as surface tension and viscosity are negligible here).
• Examples include supernova, water suspended atop oil, mushroom clouds formed due to volcanic eruptions.

CFD models based on mathematical analysis of Rayleigh Taylor waves : -

1 . Mass diffusion model :

• Rayleigh-Taylor unstable, the diffusion process could be responsible for significant mixing if the viscosity is sufficiently large. 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 linear stage through the early non-linear to the asymptotic but limited to Atwood number = 1. In this model, single-mode perturbation expands the flow equations 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 the non-linear stages of the instability. It uses ordinary differential equations to evolve the width of the mixing layer. The bubble, or spike, amplitudes are described by balancing the inertia, buoyancy and drag forces. These models cannot model multiple mixing interfaces; cannot be easily extended to 2 or 3 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(or multi-fluid)models have to be used.

2.3. Besnard - Harlow - Rauenzahn (BHR)

• It is a 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 full transport equations for the Reynold stresses, turbulent mass-flux velocity, density fluctuations, and the turbulent kinetic energy dissipation rate.

3. Multi-Fluid model :

• These models use a separate set of equations for each fluid in addition to the main flow equations and provide an accurate modeling framework for demixing by correctly capturing the relative motion of the different fluid fragments.
• The first step for the development of turbulence models is to perform Reynold averaging of the governing equations and resultant terms.

3.1. Two equation K-L turbulence model

• The K-L turbulence model involves equations for turbulent kinetic energy K and turbulent length scale L. The starting point for deriving the model equations are the buoyancy-drag models for the self-similar growth of 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 to the three-equation K-L model by including a fourth equation for density-specific volume covariance. The model can accurately reflect the effects of the changes in the density fluctuations.

Pre-processing : -

step-1 :

• We have go inside geometry in Ansys fluent.
• We have to select XY plane the create a square which length and height is 20mm.
• We have click the pull tool by using this tool we can create a surface .
• In same XY plane , we have to create a another square with same measurement over the created surface.
• Again we have to click the pull tool by using this tool we can create a surface .

• After we created geometry then we have to click on share tool by using this tool this both plane can shared information .

step-2 :

Case-1

• We have to go inside the mesh in Ansys fluent.
• We have to generate the mesh with element size is 0.5mm.

• We will got the element count after meshing.

Case-2

• We have make it refined mesh.
• We have to give the element size is 0.2.

• We will got the element count after meshing.

step-3 :

• We have to go inside setup in Ansys fluent.
• We are going to click on transient and also we have give the value of -9.81 in Y-axis inside the gravitational acceleration.

• We are going inside the multiphase option, Inside this we have to click on volume of fluid.
• we have to select on Implicit and apply.

• After that we have to go inside the martials then we can create the air (Lighter fluid) ,Water-liquid (Havier fluid) and User defined material.

• We have to go inside phase.
• In phases , we are set the primary phase which is air and secondary phase which is Water.

• We are going click initialize then check nothing going to happen inside console.
• Then we have to patch.

• We have to create a contour plot.

• We have to go inside run calculation , Inside this we have to take the time step size is 0.005.
• We have run the calculation 1000 time then only we will got results.

Baseline mesh :

https://drive.google.com/file/d/1Odq8kSzN_xFh8Vb7qYjQirceG0QozWmw/view?usp=drive_link

Refined mesh :

https://drive.google.com/file/d/1LT1_m7lhTGQ_pvp9U5k-OmrfA65FIt7N/view?usp=drive_link

Refined mesh (Water with User defined material):

https://drive.google.com/file/d/10NmcFCYpUkjh2r6ZLCTuDC-Z95vqzGct/view?usp=drive_link

Observation :

• Finer mesh captures the instability better than the baseline mesh , but it takes more time for the heavier fluid to settle down. So it is produces results with better accuracy.
• The simulation involves capturing instabilities which are changing in an unpredictable manner as a function of time , transient approach is preferred over steady state approach.

Atwood Number :

It is a dimensionless number that is used to study hydrodynamic instabilities in density stratified flows.

A=ρ1-ρ2/ ρ1+ρ2

Where,

ρ1${\displaystyle {\rho }_1}$ is density of heavier fluid

ρ2${\displaystyle {\rho }_2}$ is density of lighter fluid

For Air-water RT instability,

ρ1${\displaystyle {\rho }_1}$ = 998.2 kg/m3 [Water density]

ρ2${\displaystyle {\rho }_2}$ = 1.225 kg/m3 [Air density]

A = (998.2-1.225)/(998.2+1.225)

A = 0.998

For water- UD material RT instability,

ρ1${\displaystyle {\rho }_1}$ = 998.2 kg/m3 [Water density]

ρ2${\displaystyle {\rho }_2}$ = 1.225 kg/m3 [UD material density]

A = (998.2-400)/(998.2+400)

A = 0.428

Behavior of RT Instability on variation of Atwood number:

• RTI evolves in three distinct stages: Linear stability , mushroom head(falling) or bubble(rising) formation , long term evolution due to bubble merging and mixing. For low Atwood number, bubble and mushroom head shape is relatively symmetrical, the disturbances or perturbations are linear for a larger time. In contrast, for large Atwood number, bubble and mushroom head shape is less symmetrical, also it occurs very rapidly at an accelerating rate which leads to a non linear growth rate.

Conclusion:

• Atwood number forms a basis to compare and study research papers based on RT Instability. Thus we can use these papers to validate our results for better accuracy.