Week 5 - Rayleigh Taylor Instability

Aim: - To discuss CFD models based on mathematical analysis of Rayleigh-Taylor wave and simulate the Rayleigh-Taylor instability on Ansys Fluent and show the effect of variation in Atwood Number on behavior of Instability.

Objective: -

1. List and discuss some CFD models based on Mathematical analysis of Rayleigh-Taylor wave.
2. Perform Rayleigh-Taylor Instability simulation with base (0.5 mm) and Refined mesh size using Air and Water fluid medium and compare results. Explain, why steady state simulation is not suitable for such simulation.
3. Perform Rayleigh-Taylor Instability simulation with refined Mesh with Water and User-Defined material (density = 400 kg/m³, viscosity = 0.001 kg/m-s).
4. Define Atwood Number and its effect on behavior of instability.

Theory: -

Rayleigh-Taylor Instability –

It is the interpenetration of materials that occurs whenever a light fluid pushes on a heavy fluid. It is a dynamic process whereby the two fluids seek to reduce their combined potential energy. In other words, it is an instability of an interface between two fluids of different densities which occurs when the lighter fluid is pushing the heavier fluid. This process is evident in variety of Phenomena such as weather inversions, salt dome formations, astrophysics (such as expanding interstellar galactic clouds, nebula), electrohydrodynamic etc.

Models based on mathematical analysis of RT Instability –

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. 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.

3. 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.

4. 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.

5. Two/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 modelling 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.

6. 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.

7. 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.

8. 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.

Atwood Number –

It is a dimensionless number in fluid dynamics used in the study of hydrodynamic instabilities in density stratified flows. It describes the density difference between two adjacent fluids with a common interface. It is a density ratio defined as

`A = (ρ_h- ρ_l)/(ρ_h+ ρ_l )`

where,

`ρ_h` = density of heavier fluid and `ρ_l` = density of lighter fluid

For Atwood number close to 0, RT instability flows take the form of symmetric “fingers” of fluid; for Atwood number close to 1, the much lighter fluid “below” the heavier fluid takes the form of larger bubble-like plumes.

Case Setup: -

Geometry – Two component made of fluid body (Air and Water respectively) as shown below

Mesh – There is three mesh sizes are used i.e., 0.5 mm (base mesh), 0.2 mm and 0.1 mm (refined).

Case 1:

Base mesh (0.5 mm) 0.2 mm Refined mesh (0.1 mm)

No. of Nodes = 3321 No. of Nodes = 20301 No. of Nodes = 80601

No. of Elements =3200 No. of Elements = 20000 No. of Elements = 80601

Fluent setup:

• Pressure based solver and Absolute velocity is used.
• Transient-state Calculation is performed.
• Compressible fluid flow is chosen.
• 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 and Implicit scheme is selected.
• Viscous model-Laminar is used.
• Patching of the volume fraction of the two phases is performed before the start of simulation. Phase used for patching was water and volume fraction of 1 is taken for water surface and 0 for air surface

Residue Plot: -

Base mesh (0.5 mm) 0.2 mm Refined mesh (0.1 mm)

Simulation Animation: -

Base mesh (0.5 mm)

0.2 mm

Refined mesh (0.1 mm)

Case 2:

RT instability simulation is run for water and User-defined material (density = 400 kg/m³, viscosity = 0.001 kg/m-s).

It is run on refined mesh (grid size = 0.1 mm)

Residue Plot: -

Animation: -

Calculating Atwood Number –

case 1 - `A = (ρ_w- ρ_a)/(ρ_w+ ρ_a )` = `(998.2-1.225)/(998.2+1.225)` = 0.9975

case 2 - `A = (ρ_w- ρ_u)/(ρ_w+ ρ_u )` = `(998.2-400)/(998.2+400)` = 0.4278

Observation: -

1. Finer mesh captures Rayleigh-Taylor Instability better as compared to coarser mesh.
2. For fluids air and water, the RT instability starts takes the form of Bubbles as Atwood number is close to 1.
3. For fluids User-material and water, the RT instability starts takes the form of fingers as Atwood number is close to 0.

References: -

1. https://en.wikipedia.org/wiki/Rayleigh%E2%80%93Taylor_instability
2. http://www.scholarpedia.org/article/Rayleigh-Taylor_instability_and_mixing
3. https://www.researchgate.net/publication/230885678_Numerical_Simulation_of_Rayleigh-Taylor_Instability_With_Two-Fluid_Model_and_Interface_Sharpening
4. https://www.researchgate.net/publication/330266850_Modeling_of_Rayleigh-Taylor_mixing_using_single-fluid_models