Rayleigh taylor Instability Transient state simulation
Introduction
Practical CFD models that have been based on the Mathematical Analysis of Rayleigh Taylor Waves are:-
1.KELVIN-HELMHOLTZ OR RAYLEIGH TAYLOR HYBRID MODEL:
This model was used to predict the primary breakup of the intact liquid core of the jet and the Rayleigh Taylor model determines the effects of acceleration of different phases.This model predicts the shape of the spray considering the temperature of the different phases.The theory predicts the onset of instability and transition to turbulent flow in fluids of different densities moving at various speeds.This instability occurs when there is a velocity shear or velocity difference between the interface of the two fluids.
2.RICHTMYER-MESHKOV INSTABILITY:
This type of instability occurs when there is an impulsive acceleration between the two fluids. This type of instability can occur when there is a shock at greater speeds. The development of the instability begins with small aptitude perturbations which initially grows linearly with time.This is followed by a non linear regime with bubbles appearing in the case of a light fluid penetrating a heavy fluid, and with spikes appearing in the case of aheavy fluid penetrating a light fluid.A chaotic regime eventually is reached and the two fluids nix. The instability can be considered the impulsive-acceleration limit of the Rayleigh-Taylor instability.
3.GRANULAR RAYLEIGH TAYLOR INSTABILITY:
A granular instability is driven by gravity and is studied experimentally and numerically.The instability arises as grains fall in a closed Hele-Shaw cell where a layer of dense granular material is positioned above a layer of air.The initially flat front defined by the grains develops into a pattern of falling granular fingers separated by rising bubbles of air. A transient coarsening of the front is observed right from the start by a finger merging process. The coarsening is later stabilized by new fingers growing from the center of the rising bubbles.
4.PLATEAU-RAYLEIGH INSTABILITY:
It is often 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.It is concerned with fluid thread breakup. the 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 tends to minimizew their surface area.
5. POTENTIAL FLOW MODELS
In this model, which extends the work of Layzer, ordinary differential equations for the bubble heights and curvatures are obtained by considering the potential flow equations near the bubble tips.This model is applied to 2D single mode evolution as well as two-bubble competition, for both the R-T and R-M instabilities, the later traeted in an impulse approximation.This model is applied to the 3D modes with an initial rectangular geometry. The model yields the aspect ratio dependance of the early non-linear stages in agreement with third order Perturbation theory.
6.LATTICE BOLTZMANN MODEL
It is developed for 2D combustion simulations and is highly efficient numerical method.They are suited for incompressible flows.
7.K L TURBULENCE MODEL
This model is used to describe the self similar growth of Rayleigh Taylor and Richtmyer-Meshkov instabilities and describes the dominant eddies in the mixing zones with evolutionary equations for their characteristic dimension(L) and energy per unit mass(K).The equations are based on buoyancy-drag models for RT and RM flows.This model is used to simulate mean flow characteristics for turbulent flow conditions. It is a two equation model that gives a general description of turbulence by the means of transport equations(PDEs)
RAYLEIGH - TAYLOR INSTABILITY
The Rayleigh-Taylor instability is an instability of an interface between two fluids of different densities which occurs when the lighter fluid is pushing the heavier fluid in agravitational field.When the density interface is disturbed away from horizontal, the hydrostatic pressure acts to drive the heavier fluid into the lighter one, with the disturbance amplitude initially growing exponentially but shortly afterwards saturating and forming a characteristic mushroom shape, at least for some time.
Water suspended a top air is an everyday example of Rayleigh-Taylor instability, and it may be modelled by two completely plane-parallel layers of immiscible fluid, the denser on top of the less dense one and both subjects to the Earth\'s gravity.The equilibrium is unstable to any perturbations or disturbances of the interface.If a parcel of a heavier fluid is displaced downward with an equal volume of lighter fluid displaced upwards, the potential energy of the configuration is lower than the initial state. thus the disturbance will grow amd lead to a further release of potential energy, as the denser material moves down under the gravitational field and the less dense material is further displaced upwards.
Either some wavelengths become important and accelerate the development of instability, or mixing and dilution of the density contrast will lead to deacceleration.As the R-T instability develops, the initial perturbations progress from a linear growth phase into a non-linear growth phase, eventually developing \"plumes\" flowing upwards and \"spikes\" falling downwards. In the linear phase, the fluid movement can be closely approximated by linear equations.In the non-linear phase perturbation amplitude is too large for a linear approximation, and non-linear equations are required to describe fluid motions.In general the density disparity between the fluids determines the structure of the subsequent non-linear R-T flows(assuming other variables such as surface tension and viscosity negligible.)The difference of the fluid densities divied by their sum is defined as the Atwood number, A. For \'A\' close to 0, R-T instability flows take the form of symmetric \"fingers\" of fluid. For \'A\' close to 1, the much lighter fluid \"below\" the heavier fluid takes the form of larger bubble like plumes.
Stages of development of Rayleigh-Taylor instability and eventually evpolution into turbulent mixing
It follows four main stages.
In the first stage, the perturbation amplitudes are small compared to their wavelengths, the equations of motion can be linearlized, resulting in exponential instability growth. At the initial portion of this stage, a sinusoidal initial perturbations retains its sinusoidal shape. However at the end of this stage one can observe the evolution of mushroom shaped spikes(fluidf structures of heavy fluid growing into light fluid) and bubbles(fluid structures of light fluid growing into heavy fluid). this happens when non-linear effects starts to appear.
The growth of the mushroom structures continues in the second stage and can be modelled using buoyancy drag models, resulting in a growth rate that is approximately constant in time. At this stage non-linear terms in the equations can no longer be ignored.The spikes and bubbles then begin to interacrt with one another in the third stage.
In the third stage, bubble merging takes place where the non-linear interactopn of mode coupling acts to combine smaller spikes and bubbles to produce larger ones.Also, bubble competition takes places, where spikes and bubbles of smaller wavelengths that have become saturated are enveloped by larger ones that have not yet saturated.This eventually develops into a region of turbulent mixing, which is the fourth and final stage in the evolution. It is generally assumed that the mixing region that finally develops is self-similar and turbulent, provided that the Reynolds Number is sufficiently large.
Fig-1 Rayleigh Taylor instability
Geometry meshing
Geometry is created as square boxes and converted to surface using Pull tool.
Share topology is turned on.
Case A. Mesh
Baseline mesh- 0.5mm Element size
Elements - 3200
Nodes - 3321
Case B. Mesh
Mesh - 0.18mm Element size.
Elements - 24642
Nodes - 24976
Case C. Mesh
Mesh - 0.1 mm Element size
Elements - 80000
Nodes - 80601
Case Setup
- Pressure based solver
- Transient condition
- Gravity : -ve Y ax (-9.81)
- Multiphase VOF
- Viscous model-Laminar
- Cell zone : Fluid-air;water-surface:water-liquid
- Patching done.
Post Processing Results
Case A.
Contour before solving the case.The red part is water and blue is air.
Transient case is ran for 1500 Time steps with a time step size of 0.005.The solution converged at 12187 iterations as observed from the Residual plot below.However further variation is possible if case is ran more.
Residual plot:
Water Volume Fraction:We can observe how the air has moved upwards while the water settles down.
Case B.
Contour before solving the case.
This case was run for 1000 time steps with a Time step size of 0.005.The solution converges at 10000 iterations as observed from the residual plot below.
A more clearer settled fluids are observed.
Case C.
Contour before solving the case.
Case ran for 2000 time steps with a time step size of 0.0005.The solution converges at 37974 iterations as observed from the Residual plot below.
As the mesh size is increased we get more defined results which we can see from the below plot and it took a long time for the simulation to reach appropriate results.
Animation of the 3 cases.
Case A.
Case B.
Case C.
Why transient simulation should be used in this case:
Steady state solution determines the fully developed solution(it does not changes with time).For Transient simulation, compared to steady state has higher demand in terms of computational time.Transient solver better captures the flow.Transient solver can be used to run at smaller time steps which is seen from the 3rd case above with a time step size of 0.0005.It can also run at varying time step.Smaller time steps are required to observe the instabilities and can be only possible through transient simulation. Since it is a time marching solution, we expect more accuracy from transient simulation.
Atwood Number
The Atwood Number is a dimensionless number in Fluid Dynamics. It is used in the study of instabilities of the dynamics of the fluid flow.It gives the relation between density of higher fluid and the density of lighter fluid.
`A = (rho_1-rho_2)/(rho_1+rho_2)`
where `rho_1` = density of heavier fluid(kg/m^3) and `rho_2` = density of lighter fluid(kg/m^3)
`A = (997-1.225)/(997+1.225)= 0.998`
For Atwood no. close to 0, RT instability flows takes the form of symmetric \"fingers\" of fluid. For Atwood No. close to 1, the much lighter fluid below the heavier fluid takes the form of larger bubble like plumes.
Since in our simulation has Atwood Number close to 1, heavier fluid takes bubble form.thus helping in validating that our simulation is correct.