Rayleigh Taylor Instability Challenge
AIM
The aim of this simulation is to simulate the Rayleigh Taylor Instability.
OBJECTIVES
The objectives are as follows
- The importance of Rayleigh Taylor Instability in CFD.
- Simulate the Rayleigh Taylor Instability with two different mesh sizes.
- Define an user defined fluid for this simulation
- Significance of Atwood number on this simulation.
RAYLEIGH TAYLOR INSTABILITY
We observe the development of the RTI whenever two matters of different densities are accelerated against the density gradients. Extensive interfacial material mixing ensues with time. The turbulent mixing plays a key role in a wide variety of natural phenomena, spanning astrophysical to micro-scales, as well as in technological applications. It influences the formation of the ‘hot spot’ in inertial confinement fusion (ICF) , limits radial compression of imploding Z-pinches and controls non-equilibrium heat transfer induced by ultra-fast high-power lasers in solids . Rayleigh–Taylor (RT) turbulent mixing governs transports of mass, momentum and energy in mantle–lithosphere tectonics, ocean and atmosphere, interstellar molecular clouds, flashes of supernovae, and in flames and fires . Our ability to control the unsteady turbulent mixing is crucial for industrial applications in aerodynamics and aeronautics, and in free-space optical telecommunications. RTI is essential to occur under extreme conditions of high-energy density, but one can still easily observe it in everyday life when looking at how salted and fresh water mix or how wine flows out from an overturned glass.
Mathematical aspects of the dynamics of RTI and turbulent mixing reveal intellectual richness of the theoretical problem . For instance, at atomistic and mesoscales, RT-induced material transport may depart from a standard scenario of Gibbs’ ensemble and quasi-static Boltzmann equation. On the other hand, at macroscopic scales singular character and similarity of the non-equilibrium dynamics are interplayed with the fundamental properties of three-dimensional Euler or Navier–Stokes equations and with the problem sensitivity to the boundary conditions at discontinuities and the initial conditions. Furthermore, turbulent mixing is a multi-scale, anisotropic, non-local and statistically unsteady process, and its fundamental invariant properties depart dramatically from those in the classical Kolmogorov turbulence. As in any turbulent process, contribution of all scales into the flow dynamics induces randomness of RT mixing. Yet, because of statistical unsteadiness and non-Gaussian character of fluctuations, capturing this randomness is a challenging task that necessitates further advancements of the methods of stochastic modelling.
On the side of the numerical simulations, modelling RT mixing is a severe test, as the numerical solutions appear sensitive to the initial and boundary conditions as well as to the influence of unresolved small-scale structures on the large-scale dynamics and to the anomalous character of energy transport . Addressing these difficulties brings up the issue of predictive capability of the numerical models, their verification and validation as well as quantification of uncertainty of the solutions obtained. Synergy between theory, experiment and simulations thus appears crucial for obtaining an integrated description of the phenomenon and for grasping its universality features. On the side of the experiments, RT flows are hard to implement and systematically study in a well-controlled laboratory environment, as their sensitivity and transient character of the dynamics impose high requirements on accuracy, spatio-temporal resolution and dynamic range of the flow quantities as well as on data rate acquisition. One of the primary concerns for RT experiments is the influence of an observer on the results of observation and data interpretation. To provide reliable and repeatable datasets, experiments on the RTI and turbulent mixing in fluids and plasmas employ the most advanced diagnostic approaches.
There are many applications of Rayleigh Taylor Instability in the field of CFD
- Kelvin-Helmholtz instability -The theory predicts the onset of instability and transition to turbulent flow within fluids of different densities moving at various speeds. Helmholtz studied the dynamics of two fluids of different densities when a small disturbance, such as a wave, was introduced at the boundary connecting the fluids. Kelvin-Helmholtz instability can thus be characterized as unstable small scale motions occurring vertically and laterally. At times, the small scale instabilities can be limited through the prescience of a boundary. The boundaries are evident in the vertical direction, through an upper and lower boundary. The upper boundary can be seen as through examples as the free surface of an ocean and lower boundary as a wave breaking on a coast. On a lateral scale, diffusion and viscosity are the main factors of considerations as both affect small-scale instabilities. Through the aforementioned definition of the Kelvin-Helmholtz instability, the distinction between Kelvin-Helmholtz instability and small scale turbulence can be difficult. Although the two are not inherently inseparable, Kelvin-Helmholtz is viewed as a two dimensional phenomena compared to turbulence occurring in three dimensions.
- Richtmyer–Meshkov instability (RMI) occurs when two fluids of different density are impulsively accelerated. Normally this is by the passage of a shock wave. The development of the instability begins with small amplitude perturbations which initially grow linearly with time. This is followed by a nonlinear regime with bubbles appearing in the case of a light fluid penetrating a heavy fluid, and with spikes appearing in the case of a heavy fluid penetrating a light fluid. A chaotic regime eventually is reached and the two fluids mix. This instability can be considered the impulsive-acceleration limit of the Rayleigh Taylor Instability.
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The Plateau–Rayleigh instability, often just called the Rayleigh instability, explains why and how a falling stream of fluid breaks up into smaller packets with the same volume but less surface area. It is related to the Rayleigh Taylor Instability. and is part of a greater branch of fluid dynamics concerned with fluid jet breakup. This fluid instability is exploited in the design of a particular type of ink jet technology whereby a jet of liquid is perturbed into a steady stream of droplets.The driving force of the Plateau–Rayleigh instability is that liquids, by virtue of their surface tensions, tend to minimize their surface area.
RAYLEIGH TAYLOR INSTABILITY SIMULATION(BASE LINE MESH)
The simulation has the following stages for the seeting up of the instability criteria
- Geometry
- Mesh
- Setup
- Solution
- Results
Geometry
In the geometry stage we have to first sketch the air column using the rectangle command and use the pull tool to convert it into a surface. Then we create a new component for the water column we again sketch the water column and then use pull tool to create the surface out of it. Then for the fluids to interact with each other we have to enable the share topology. In order to create the shared topology in workbench just using on share, select the whole geometry and share is enabled.
Mesh
Now in the mesh module we create the base line mesh with element size of 0.0005m. We are creating a total of 3321 elements in the meshing zone.
Setup
For the setup stage we create these multiple parameters
- Solver - Pressure Based Solver
- Solver state - Transient state
- Gravity enabled in Y-axis - 9.81 m/s^2
- Model - Laminar Flow
- Materials used - water and air
- Multi phase - Volume of Fluid(VOF)
- solution - patch
For this particular case we ran the simulation for 1500 iterations with a time step of 0.005 seconds.
ATWOOD NUMBER FOR CASE 1
The Atwood number (A) is a dimensionless number in fluid dynamics used in the study of hydrodynamic instabilities in density stratified flows. It is a dimensionless density ratio defined as
`A=frac[rho_1-rho_2][rho_1+rho_2`
where
= density of heavier fluid
= density of lighter fluid
Atwood number is an important parameter in the study of Rayleigh Taylor stability. In Rayleigh–Taylor instability, the penetration distance of heavy fluid bubbles into the light fluid is a function of acceleration time scale, 
. where g is the gravitational acceleration and t is the time.
In this case the atwood number is calculated as follows
`rho _1= 998.2 (kg)/m^3`
`rho_2= 1.225(kg)/m^3`
`A = frac[998.2-1.225][998.2+1.225]`
A = 0.9975
Results
Now in the above animation we observe the animation which was created for a mesh size of 0.5mm
RAYLEIGH TAYLOR INSTABILITY SIMULATION(FINE MESH)
The simulation has the following stages for the setting up of the instability criteria
- Geometry
- Mesh
- Setup
- Solution
- Results
Geometry
In the geometry stage we have to first sketch the air column using the rectangle command and use the pull tool to convert it into a surface. Then we create a new component for the water column we again sketch the water column and then use pull tool to create the surface out of it. Then for the fluids to interact with each other we have to enable the share topology. In order to create the shared topology in workbench just using on share, select the whole geometry and share is enabled.
Mesh
Now in the mesh module we create the base line mesh with element size of 0.00025m. We are creating a total of 12800 elements in the meshing zone.
Setup
For the setup stage we create these multiple parameters
- Solver - Pressure Based Solver
- Solver state - Transient state
- Gravity enabled in Y-axis - 9.81 m/s^2
- Model - Laminar Flow
- Materials used - water and user defined
- Multi phase - Volume of Fluid(VOF)
- solution - patch
For this particular case we ran the simulation for 1500 iterations with a time step of 0.005 seconds.
ATWOOD NUMBER FOR CASE 2
The Atwood number (A) is a dimensionless number in fluid dynamics used in the study of hydrodynamic instabilities in density stratified flows. It is a dimensionless density ratio defined as
`A=frac[rho_1-rho_2][rho_1+rho_2`
where
- = density of heavier fluid
- = density of lighter fluid
Atwood number is an important parameter in the study of Rayleigh Taylor stability. In Rayleigh–Taylor instability, the penetration distance of heavy fluid bubbles into the light fluid is a function of acceleration time scale, . where g is the gravitational acceleration and t is the time.
In this case the atwood number is calculated as follows
`rho _1= 998.2 (kg)/m^3`
`rho_2= 400(kg)/m^3`
`A = frac[998.2-400][998.2+400]`
A = 0.4275
Results
Looking at the both the animations as we refine the mesh the gradients start to appear in a much smooth manner, the heavier fluid will settle at the top and as we go down the grid we observe that the water content decreases. We cannot run this simulation in steady state because as we know from fluid dynamics, the fluid with more density will remain at the bottom forming layers and the fluid with less density settles on the top. The transcient simulaton will show how the column collapses and how the well the two fluid mixes as we observe the instabilities. As far as the instability is concerned we see that when we refine the mesh the gradients of mixing spread out in a more smooth manner and mixing process can be acessed using the volume of fluid method.
CONCLUSION
- The simulation for the rayleigh taylor instability was run for two diffrent mesh sizes.
- The volume of fluid can capture the effect of mixing for rayleigh taylor instability.
- As we refine the mesh we see better mixing of the two fluids.
- Now in the mixing efficiency, as we refine the mesh we see mixing gradients in much smooth manner.
- This stability could be useful to study the interaction of two fluids with good accuracy.