Analysis of Rayleigh Taylor instability
Rayleigh–Taylor instability:-
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 atop a light fluid 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:-
1. 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.
2. 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.
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 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.
CFD Analysis of Rayleigh Taylor Instabilities:
The analysis is done for 3 different cases by increasing the number of cells. As its multi-fluid analysis, VOF method in fluent used, and implicit solver with active implicit body force chosen. Choosing the accurate time step is more important to capture the instabilities at the interface as well as in the fluid zones.
Simulation I:
Mesh size - 0.05mm, No. of Elements - 3200, Time step -0.01, No. of time steps - 300, Flow time - 3 sec.
Simulation II:
Mesh size - 0.2mm, No. of Elements - 20000, Time step -0.0075, No. of time steps - 400, Flow time - 3 sec.
Simulation III:
Mesh size - 0.1mm, No. of Elements - 80000, Time step -0.0025, No. of time steps - 1200, Flow time - 3 sec.
Observations:-
1. Choosing the wrong timestep leading to floating point error.
2. The smaller timestep is not giving a solution at all. So, proper selection of the time step is important.
3. Instabilities are well captured when the mesh size is smaller but, it certainly needs more time to settle down the heavy fluid at the bottom.
Atwood Number(A):-
A basic parameter characterizing a Rayleigh-Taylor unstable interface is the density Atwood number, defined as
A = (ρ1−ρ2)/(ρ1+ρ2)
ρ1 = Density of heavier fluid.
ρ2 = Density of lighter fluid.
For this case:
ρ1 = 998.2 Kg/m3
ρ2 = 1.225 Kg/m3
A = (998.2 - 1.225)/ (998.2 + 1.225)
A = 0.998
Effect of variation in Atwood number on Instability:-
RTI evolves in three distinct stages, namely linear stability, mushroom-head formation, and long-term evolution.
For Boussinesq flows, small density variations relative to the total density, the growth rate of rising structures (bubbles) and falling structures (spikes) are equal and instability development in a non-linear. Larger Atwood number system develops into a
much less symmetrical mushroom pattern that has more elongated mushroom caps. This process occurs very rapidly and thus suggests that it may be the result of a transition to turbulence in the flow. Larger Atwood number leads to larger acceleration and which makes the growth rate nonlinear.
For low Atwood no., the bubble and spike shape is relatively symmetrical.
Atwood number near 1, approaches a constant velocity at late times. Both bubble and spike portions approach constant velocities when the fluid densities are similar.
The shapes of bubbles and spikes differ noticeably between the two extremes in the Atwood number. Low Atwood number bubbles and spikes show the presence of vortices which are absent in the A=1 instability.
How Atwood number can be used to verify CFD result?
We can use past experiments or CFD data to verify our results because many researchers performed experiments with different Atwood number and different combination of fluids. The graphical representation of Atwood number with other parameters like dimensionless numbers, time, instabilities, etc. is available. We can compare our plots with them to verify our results.
Experiments performed by Read and young(1983), Read(1984), Burrows et al. (1984), Smeeton and Youngs(1984), Dimonte and Schneider(1996), Smeeton and Youngs(1988), Andrews(1986), Andrews and Spalding(1990), Voropayev et al. (1993), Linden and Rodondo(1991), Debacq et al.(2001), Holford et al. (2003), Banarjee and Andrews(2006), Dalziel et al. (2008) etc. can certainly helps to compare our CFD results.
Few References:
1. Abarzhi, S. I., Nishihara, K. & Glimm, J. 2003 Rayleigh–Taylor and Richtmyer–Meshkov
instabilities for fluids with a finite density ratio. Phys. Lett. A 317, 470.
2. Abarzhi, S. I., Gauthier, S. & Rosner, R. 2008 Turbulent mixing and beyond. Physica Scr. T132,011001.
3. Andrews, M. J. 1986 Turbulent mixing by Rayleigh–Taylor instability. Ph.D. dissertation, London University, UK.
4. Andrews, M. J. & Spalding, D. B. 1990 A simple experiment to investigate two-dimensional mixing by Rayleigh–Taylor instability. Phys. Fluids A 2, 922–927.
5. Banerjee, A. & Andrews, M. J. 2006 Statistically steady measurements of Rayleigh–Taylor mixing in a gas channel. Phys. Fluids 18, 035107.
6. Burrows, K. D., Smeeton, V. S. & Youngs, D. L. 1984 Experimental investigation of turbulent mixing by Rayleigh–Taylor instability, II. AWRE Report no. 022/84.
7. Dalziel, S. B. 1993 Rayleigh–Taylor instability: experiments with image analysis. Dyn. Atmos. Oceans 20, 127–153.
8. F. F. Grinstein, L. G. Margolin, and W. J. Rider, Implicit Large Eddy Simulation
(Cambridge University Press, New York, 2007).
9. Small Atwood number Rayleigh Taylor experiments, Malcolm J. Andrews and Stuart B. Dalziel, Phil. Trans. R. Soc. 2010 368.
10. Youngs, D. L. 1994 Numerical simulation of mixing by Rayleigh-Taylor and Richtmyer-Meshkov instabilities. Lasers and Particle Beams 12, 725.