Rayleigh Taylor Instability Challenge
Rayleigh-Taylor Instability Simulation using Multiphase VOF method
Objectives:
- To understand the Rayleigh-Taylor instability
- To carry out the transient simulation mixing of different density fluid using Multiphase VOF method
- To understand the different CFD models based on RT instability
- To realize the effects of Mesh size and Atwood Number on RT instability
Introduction:
Rayleigh–Taylor instability (RTI) occurs whenever fluids of different density are subjected to acceleration in a direction opposite that of the density gradient. It is a dynamic process whereby the two fluids seek to reduce their combined potential energy. The ensuing turbulence and mixing have far-reaching consequences in many natural and man-made flows.
Beginning from a state of hydrostatic equilibrium, in which a heavy fluid rests at top and light fluid at bottom in a constant gravitational field, Rayleigh–Taylor instability at the interface will launch a shock wave into the upper fluid. Rising bubbles of light fluid act like a piston, compressing the heavy fluid ahead of the fronts and generating shocks. These shocks coalesce in multidimensional fashion into a strong normal shock, which increases in strength as it propagates upwards. RTI is encountered in a variety of contexts, such as combustion, rotating machinery, inertial-confinement fusion, supernovae explosions, and geophysics.
CFD models based on mathematical analysis of Rayleigh Taylor waves:
- Mass diffusion model:
The mass diffusion equation describes the transport of mass and energy due to a model viscosity and hence drives actual mixing in the fluids. Even if any configuration is Rayleigh–Taylor unstable, diffusion processes could be responsible for significant mixing if the viscosity is sufficiently large.
- Single Fluid Model:
2.1 Potential flow model
Used by Layzer (1955) and Hecht et al. (1994), this model is valid for all instability stages from the linear stage through the early nonlinear to the asymptotic but limited to A =1. In this model,single-mode perturbation and expand 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
Buoyancy–drag models valid for every Atwood numbers, but limited to the nonlinear 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 two and 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 (or multi-fluid) models have to be used.
2.3 Besnard-Harlow-Rauenzahn (BHR)
It is a more advanced version of a single fluid model. The BHR model is based on 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 Reynolds stresses, turbulent mass-flux velocity, density fluctuations, and the turbulence kinetic energy dissipation rate.
- 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 Reynolds averaging of the governing equations and resultant terms.
3.1 Two-equation K-L turbulence model:
The k–L turbulence model evolves equations for the turbulent kinetic energy k and the turbulent length scale L. The K-L model was proposed by Dimonte and Tipton for describing the turbulent self-similar regime of RTI induced mixing. 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 of Morgan and Wickett was developed as an extension to the two-equation K-L model of Dimonte and Tipton by including a third equation for the turbulent mass-flux velocity.
3.3 Four-equation K-L-a-b turbulence model:
It is an extension of a K-L-a model by adding an evolution equation for the density-specific volume covariance. This model can accurately reflect the effects of the changes in the density fluctuations.
Procedure:
Step: 1 Geometry
Open the Spaceclaim and make 2D geometry
First select the XY plane and make square of size 20 mm. Then, use pull tool to make surface from this square. Similarly, make another square of same size having one same side and use pull tool to make surface. After doing this, use share option to share data on common side of surfaces.
Step: 2 Meshing
In this geometry, meshing is very simple which is given below:
Baseline mesh Mesh size 0.25 mm Mesh size 0.1 mm
In this case study, there are three cases considered which are shown in table:
|
Type |
Mesh size |
No. of Elements |
|
Baseline
|
0.5 mm |
3200 |
|
Refinement 1 |
0.25 mm |
12800 |
|
Refinement 2 |
0.1 mm |
80000 |
Step 3 : Setup of Simulation
In this study, transient simulation is used because there is density between two fluids and mixing takes because of gravity is very less which is necessary to capture so that the transient simulation is best to use.
In this simulation, the following parameters are selected:
Type: Transient simulation
Solver: Pressure based
Velocity Formulation: Absolute Velocity
Energy Equation: Unselected
Turbulence model: Multiphase VOF method with Implicit Formulation
Laminar Model
Gravity Acceleration: -9.81 m/s^2 (Y-direction)
Primary Phase: Air
Secondary Phase: Water
Method: Simple
Initialization: Standard
Patch: Volume of fraction > water > value 1 . Volume of fraction >air> value 0
Case 1(a): Baseline mesh
Primary phase - Air
Secondary phase - Water
Residual plot:
Volume fraction water contour:
Simulation animation:
Case 1(b): Mesh 0.25 mm
Primary phase - Air
Secondary phase - Water
Residual plot:
Volume fraction water contour:
Simulation animation:
Case 1(c): Mesh 0.1 mm
Primary phase - Air
Secondary phase - Water
Residual plot:
Volume fraction water contour:
Simulation animation:
Patch: Volume of fraction > water > value 1 . Volume of fraction >oil> value 0
Case 2: Mesh 0.1 mm
Primary phase - Oil
Secondary phase - Water
Residual plot:
Volume fraction water contour:
Simulation animation:
Effect of Mesh Size:
From the above animations, we can conclude that the mesh size is directly affects our simulation. In this case study, when the mesh size is reduced from 0.5 mm to 0.1 then it shows good distribution in this instability and vortices are also captured very well. This will increase the compute time also.
Disadvantage of Steady-State simulation:
In the RT instability case, there are very less time where high and low dense fluids are undergo this instability effect which is very necessary to capture in detail to understand this effect. This can not be possible with steady state simulation because it deals with steady flows only.
Atwood number and its significance
Atwood number is a dimensionless number that describes the density difference between two adjacent fluids with common interface and is used to study the hydrodynamic instabilities. Mathematically Atwood number is given as:
A=`(rho1 - rho2)/(rho1 + rho2)`
where ρ1= density of heavier fluid
and ρ2= density of lighter fluid
Atwood number is an important parameter in the study of Rayleigh Taylor instability. The penetration of heavier fluid is a function of Atwood number.
In case 1
ρ1 = 1000
ρ2= 1.225
A = 0.997
In case 2
ρ1 = 1000
ρ2= 400
A = 0.428
After comparing the two cases we can see that the penetration of heavier fluid i.e. water is maximum in case 1 and this can be used to validate the simulation.
Conclusion:
Rayleigh Taylor instability with water and other fluid being air and oil in respective case is studied. It can be said that with increase in mesh size it takes longer time for the two fluids to reach a steady state. Also viscosity plays an important role as can be seen in case of oil the surface takes longer to deform.
References
https://www.researchgate.net/publication/330266850_Modeling_of_Rayleigh-Taylor_mixing_using_single-fluid_models