Rayleigh Taylor Instability Challenge

RAYLEIGH TAYLOR INSTABILITY

Aim: To simulate Rayleigh Taylor instability occurring between Water and air. Also, for a user-defined fluid with water.

Introduction: Rayleigh Taylor's instability or RT instability is a phenomenon that is observed when two fluids of different densities are placed one over the other particularly when the heavier one is placed over the lighter one. Due to the action of gravity, the heavier fluid tends to move downwards and this pushes the lighter fluid to the top and thereby pushing the lighter fluid to the top and this creates an instability at the interface. This is also observed in the cases where the lighter fluid accelerates to the top via a medium containing the heavier fluid and this leads to instability. A series of spikes and plumes are created due to the interaction of fluids. The plumes are created by the lighter fluids which are being pushed by the heavier fluid. Whereas, the heavier fluid creates a spike which progresses downward. This is a dynamic phenomenon.

In this simulation, turbulence is not considered (laminar flow is assumed). But, the phenomenon is entirely turbulent and is as complex as the mixing of any other turbulent regime.

Following are few practical CFD models that have been based on the mathematical analysis of Rayleigh Taylor waves:

(1) Potential flow model:

Rayleigh Taylor instability can be given in terms of potential flow theory. This method isolates interfacial perturbations and is based on a non-linear approach. Basic governing equations are of potential flows with moving boundaries is considered. The mathematical model is specialized for linear perturbations about planar and spherical coordinates [1].

(2) K-L turbulence model for self-similar growth of the Rayleigh Taylor and Richtmyer- Meshkov instabilities

The instabilities rise due to higher Reynolds number (Re) and they become non-linear when the duration is large. As time progresses the flow becomes turbulent and evolves with a scale length of 'L'. For Re >10^6, turbulence models are required to describe the transport [2].

(3) 3- Equation model for self-similar growth of Rayleigh Taylor and Richtmyer-Meskov instabilities

The two-equation model K-L model is extended by a third equation for mass flux velocity. This model is applicable for one dimensional Rayleigh Taylor and Richtmeyer-Meshkov test problems.

The module used in this simulation include VOF (Volume of fluid). This is particularly used when there are two or multiple phases involved [3].

Practical applications

Some of the practical applications and physical phenomena include

• Aftermath of a supernova
• Underwater explosions
• Formation of mushroom clouds after the atomic explosion
• Fluidized beds
• Inertial phenomenon

All of the above phenomena include heavy and complex mixing of turbulent regimes and hence can be used to study the turbulence.

Modeling approach

In this simulation, we make use of the Volume of fluid approach. As we are dealing with two different phases. Also, we are interested in the mixing phenomenon of the fluids which is a dynamic event, so we opt for a transient simulation.

We make use of Space claim for the geometry and create a 2D geometry where the top portion is filled with a fluid having a density of higher magnitude followed by the fluid at the bottom of lower density. This approach is followed to create instability at the interface.

The reason for choosing a time-dependent approach is that we are interested in finding the way how the gases mix. In order to observe the discontinuity propagation, this approach is suitable. Steady-state simulation just gives us the final state of the model which is not so significant for the analysis which we intend to make.

Preprocessing and solver settings

Geometry of the model:

As mentioned in the approach, the 2D approach is used. Two squares each of the 20 mm sides are placed on top of each other.

Fig 1: Geometry for simulation

The top square is filled with a fluid of higher density, whereas the bottom is filled with a density of the lower magnitude. This is done to ensure that there exists an instability at the interface.

Meshing of the model

The above model should be meshed properly for further simulations. As a baseline mesh, mesh size of 0.5 mm is chosen and a further 2 steps of meshing are done. The following images depict the mesh sizes.

Fig 2: Mesh size of a size 0.5 mm

Fig 3: Mesh of a size 0.25 mm

Fig 4: Mesh size of 0.1 mm

These three meshes have been used and the mesh size of 0.1 mm has been considered as the refined mesh and the same is used for simulation where a user-defined fluid along with water is considered for another simulation.

Solver requirements of the problem

• Type of simulation: Transient, Implicit method ( this avoids the courant number input and offers a better stability, but a bit time consuming)
• Turbulence: Laminar flow is considered.
• Method used: Volume of fluid (VOF) with 2 eulerian phases

Apart from this, the gravity is being set as per the coordinate system taken into consideration. Standard initialization is considered for solving by applying a patch where the volume fraction of water being considered to be 1 at the top.

An animation has been created which records the flow at every possible time step.

Choosing an appropriate timestep for a given mesh is an important criterion. Incompatible time step size leads to a floating-point exception.

So, the time step size is varied accordingly as per the mesh size and the flow time is adjusted to 5 seconds for every simulation. Three simulations of Water and air were run and a simulation with 0.1 mm mesh size is run with a user-defined fluid of density 400 kg/m^3 and viscosity 0.001 kg/m-s is used.

Table 1: Mesh statistics

Fig 5: General settings in Fluent

Fig 6: Multiphase model settings

Results

Fig 7: Water volume fractions at the 0th timestep for 0.5mm size mesh,

Time step size= 0.005 sec, flow time= 5 seconds

Fig 8: Water volume fractions at the final time for 0.5mm size mesh,

Time step size= 0.005 sec, flow time= 5 seconds

Fig 9: Water volume fraction at the 0th time step for 0.25 mm mesh

Time step size= 0.005 sec, flow time= 5 seconds

Fig 10: Water volume fraction at 1000th-time step for 0.25 mm mesh

Time step size= 0.005 sec, flow time = 5 seconds

Fig 11: Water volume fraction at 0th-time step for 0.1 mm mesh

Time step size= 0.025 sec, flow time= 5 seconds

Fig 12: Water volume fraction at 2000th-time step for 0.1 mm mesh

Time step size= 0.025 sec, flow time= 5 seconds

Fig 13: Water volume fraction at 0th-time step for 0.1 mm mesh

(Water+ user-defined fluid)

Time step size= 0.025 sec, flow time = 5 seconds

Fig 14: Water volume fraction at 2000th-time step for 0.1 mm mesh

(Water+ user-defined fluid)

Time step size= 0.025 sec, flow time = 5 seconds

Atwood number: It is defined as a dimensionless number which is given by the ratio of the difference in fluid densities to that of the sum of the densities. It is denoted by A.

Atwood number =(ρ1 - ρ2/ρ1 +ρ2)

ρ1- density of heavier fluid

ρ2- density of lighter fluid

Case 1( Water and air ): Density of water = 1000 kg/m^3, Density of air = 1.2 kg/m^3

Atwood number = 0.9976

Case 2 (Water and user-defined fluid): Density of fluid = 400 kg/m^3

Atwood number = 0.4285

For Atwood number approximate to 0, RT instability flows appear like symmetric fingers of fluid

For Atwood number near to 1, RT instability appears to be like a heavier fluid which appears to be a larger bubble-like plume.

Figure 15: Mixing of air and water which appears like a larger bubble due to higher Atwood number

Figure 16: Mixing of water and user-defined fluid which appear to be like a symmetric finger-like

structure due to a relatively lower Atwood number

The animations of the simulations are attached here

Observations

• It could be observed in figures 8, 10, and 12. The mesh size affects the final result very much.
  
• The time step size is related to mesh size. The same step size is not valid for every mesh. Incompatibility leads to a floating-point exception.
• The fluid of higher density stakes down gradually due to the pull of gravity and buoyancy. Turbulence is not considered in this simulation.
• If we observe in the case of the user-defined fluid and water, the mixing pattern is quite different and certainly, the flow time of 5 seconds is not sufficient. Owing to computational requirements flow time has been restricted to 5 seconds.
• Spikes and plumes are evident which goes in hand with the experiments.
  

Gaps in the challenge

• This mixing pattern is restricted to 2D.
  

References

[1] H.J.Kull, Theory of Rayleigh-Taylor instability, Physics Reports (Review Section of Physics Letters) 206, No.5 (1991), 197-325.

[2] Guy Demonte and Robert Tipton, K-L turbulence model for self-similar growth of the Rayleigh Taylor and Richtmyer- Meshkov instabilities(2006), Physics of Fluids, 18.

[3] Brandon E. Morgan and Michael E. Wickett, Three-equation model for the self-similar growth of Rayleigh-Taylor and Richtmyer-Meskov instabilities(2015).