Week 5 - Rayleigh Taylor Instability

Aim :

To Perform the Rayleigh Taylor instability simulation for 2 different mesh sizes with the base mesh being 0.5 mm.

Objectives :

1. The objective is to simulate a two-dimensional Rayleigh-Taylor instability problem using the classical method of Finite Element analysis of a multiphase model using ANSYS FLUENT.

2. To develop a numerical case set-up for Rayleigh-Taylor instability problem in Ansys Fluent.

3. To conduct grid dependency test to understand the variation in RT instabilites.

4. To understand the effect on RT instabilities due to variation in Atwood number.

Introduction :

Rayleigh-Taylor Instability :

• In Multiphase flow, the dynamic variables like the velocity, viscosity, pressure and density are generally used to explain the movement of fluids under the gravitational field.
• Multiphase flows takes place in nuclear geothermal power plants, food processing industries and many others. One of the basic example is Rayleigh Taylor Instability.

Fig (a): Rayleigh-Taylor Instability

In fig (a),

The fluid in blue color represents lighter fluid

The fluid in red color represents heavier fluid

• The Rayleigh Taylor Instability is a hydrodynamic instability arises when a high density fluid is placed over a low density fluid in a gravitational field and the two fluids must be immiscible.
• Intially at time t= 0, the system of two fluids is in hydrostatic equlibrium. The pressure gradients are balanced and fluid is in static equilibrium.
• When the density interface is disturbed, the hydrostatic pressure is generated, so the heavier fluid moves downwards and mixes with the lighter one, with the amplitude of perturbation initially growing exponentially but after sometime saturates and forms characteristic mushroom shape.
• When a heavier fluid is displaced downward with an equal volume of lighter fluid displaced upwards, it results in decrease of potential energy than it’s initial state.
• This RTI has been applied to a wide range of problems including the mushroom clouds like volcanic eruptions, nuclear explosions and plasma fusion reactors instability.

Atwood Number :

Dealing with mixtures of fluid one uses the Atwood number to characterize the density difference between two fluids.

Assume that `rho_2` is the density of the heavier fluid.
The definition is: At = `(rho_2 - rho_1) / (rho_2 + rho_1)`.

• For big density differences At tends to 1.
• For small differences At tends to zero.

Literature Studies made :

• The RTI phenomenon was initially introduced by Rayleigh 1883 and later was observed for all accelerated fluids by Taylor 1950.
• Budiana et al. 2020 in their study observed that the evolution of the interface grows faster with the increase in Atwood Number for which in this study the Atwood Number is keptas high as possible in Case 1A. It was seen that with larger Atwood Number, larger amplitude of perturbations are observed. It is due to the larger density ratio of the two fluids, the stronger is the force to develop the interface instability.
• volume of fluid methods(Hirt and Nichols 1981, Gopala and BGMvan 2008, Raessi et al. 2010) and phase field methods(Dinget al. 2007, Chiu and Lin 2011, Lee et al. 2011, Kim 2005).
• Gou et al. 2018 used the multiphase moving particle semi-implicit(MMPS) with periodic boundary conditions to simulate RTI of the immiscible two phase incompressible fluid. The effect of initial perturbation wavelength, surface tension and Atwood Number was investigated in the single mode RTI problem and it was observed that the numerical results were in good agreement with the analytical results.
• Indrashis Saha 2020 study of Rayleigh Taylor Instability with the help of CFD simulation and was well validated with experimental results.

Case studies :

Case 1 :

Subcase 1 :

Effect of mesh refinement over CFD results.

Subcase 2 :

In this study, the results obtained from Subcase 1 and Subcase 2 will be evaluated on the basis of mesh refinement.

Case 2 : Effect of Atwood number over Rayleigh-Taylor instabilities.

For same mesh characteristics the two case set-up's (Subcase 2 and case 2) with different Atwood number will be evaluated to understand the effect on Rayleigh-Taylor instabilities.

Case set-up :

• Solver- Pressure based and absolute velocity is used.
• Transient state calculation.
• Gravity is enabled in -ve Y-axis.
• Multiphase model is turned on along with volume of fluid and number of Eulerian phases is chosen as 2. Accordingly, implicit scheme is selected.
• Viscous model- Laminar is used
• Patching of the volume of the two phases is performed before start of the simulation.
• Air and Water are the two phase used.
• Then patch the surfaces. volume fraction of "1" is taken for water surface and "0" for air surface.

Why not steady-state calculation?

The steady state calculations are useful when the study is focused on to calculate / simulate the end results. Here in this study, the instabilities information is sensitive and completely time-dependent. At each and every time step, the instabilities grow exponentially. Though steady state simulation is capable to capture those instabilities but the results would not be time-accurate. Anyways transient state calculation with reduced time-steps can capture these instabilities at each and every time-steps and sensible time dependent results can be obtained.

[A] Pre-Processing :

Geometry :

• Geometry is created in spaceclaim with a dimension of 20x20.
• Two squares has been created in such a way that it is placed one over the other.
• After that pull tool is used to convert a sketch into surfaces.
• Then share the topology between two surfaces for the effective transfer of information and to connect a mesh between two interfaces.

Mesh :

Case 1 :

Subcase 1 :

1.1.1 Details :

Mesh	Coarse
Element size	0.5 mm
Number of elements	3200
Density of primary fluid	Air (1.225 kg/`m^3`)
Density of secondary fluid	Water (998.2 kg/`m^3`)
Atwood number	0.9975

Mesh quality :

STEP 3 : SETUP

SETUP
Solver Type	Pressure-Based
Velocity Formulation	Absolute
Time	Unsteady
Models	Viscous model : laminar Multiphase : Volume of fluid : Implicit formulation : 2 phases
Material	Fluid type : Air : water liquid Solid-type : Aluminium
Cell zones	Fluid type : Air : water
Boundaries	Standard
Reference values	Standard
Initialization	Standard
Patch	Water-surface (1 value of volume fraction of water) Air-surface (0 value of volume fraction of water)

Note : The above table is for case 1 .

Residual plot :

Animation :

Subcase 2 :

1.2.1 Details :

Mesh	Refined
Element size	0.18 mm
Number of elements	24642
Density of primary fluid	Air (1.225 kg/`m^3`)
Density of secondary fluid	Water (998.2 kg/`m^3`)
Atwood number	0.9975

Mesh quality :

Residuals :

Animation :

Case 2 :

2.1 Details :

Mesh	Refined
Element size	0.18 mm
Number of elements	24642
Density of primary fluid	User-defined material (400 kg/`m^3`)
Density of secondary fluid	Water (998.2 kg/`m^3`)
Atwood number	0.4278

STEP 3 :

SETUP
Solver Type	Pressure-Based
Velocity Formulation	Absolute
Time	Un-Steady
Models	Viscous model : laminar Multiphase : Volume of fluid : Implicit formulation : 2 phases
Material	Fluid type : user-defined : water liquid Solid-type : Aluminium
Cell zones	Fluid type : Air : water
Boundaries	Standard
Reference values	Standard
Initialization	Standard
Patch	Water-surface (1 value of volume fraction of water) Air-surface (0 value of volume fraction of water)

Residual :

Animation :

Conclusion :

• Steady state solution determines the fully developed solution.This numerical simulation was validated with the observations obtained by previous researchers and this was done by the variation in Atwood Number where we obtained the same observation. In lower Atwood numbers, spikes and globules were still present at the end of the simulation which was not there in Atwood Number of 0.9 in Case 1.
• From the grid dependency we also observed that the interface thickness reduces with the refined mesh and when run with smaller time steps. For Transient simulation, compared to steady state, has higher demand in terms of computational time. Transient solver better captures the flow and can be used to run at smaller time steps which is seen from the Subcase 2 and case 2 with a time step size of 0.0005. 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 for which Transient simuation is the ideal setup to run the calculation for Rayleigh Taylor Instability.

References :

1. Indrashis Saha 2020 study of Rayleigh Taylor Instability with the help of CFD simulation.
2. Budiana, E. P. (2020) Meshless numerical model based on radial basis function (RBF) method to simulate the Rayleigh–Taylor instability (RTI). Computers & Fluids, 201, 104472.
3. Hirt, C. W., & Nichols, B. D. (1981) Volume of fluid (VOF) method for the dynamics of free boundaries. Journal of computational physics, 39(1), 201
4. Gopala, V. R., & van Wachem, B. G. (2008) Volume of fluid methods for immiscible-fluid and free-surface flows. Chemical Engineering Journal, 141(1-3), 204-221.
5. Kim, J. (2005) A diffuse-interface model for axisymmetric immiscible two-phase flow. Applied mathematics and computation, 160(2), 589-606.
6. Ding, H., Spelt, P. D., & Shu, C. (2007) Diffuse interface model for incompressible two-phase flows with large density ratios. Journal of Computational Physics, 226(2), 2078-2095.