Project

Understanding and Simulation of Rayleigh Taylor Instability using VOF Method

BBandi Sai Mukesh15 Jul 2019

INTRODUCTION

Flows with the free surface are important in many industrial systems including nuclear power plants, oil industry and other fields of process engineering[1]. Appropriate modeling of the free surface flows is important also for the understanding of some basic phenomena in fluid dynamics like Rayleigh-Taylor instability, Kelvin-Helmholtz instability, Richtmyer-Meshkov instability, and Plateau-Rayleigh instability.

Rayleigh-Taylor instability is instability between two free surfaces of fluids having different densities. This instability occurs at the interface of both fluids with one pushing another due to gravity. Examples of Rayleigh-Taylor instability are water suspended over oil or water over the air in the gravity of Earth, atmospheric nuclear explosions, etc. Rayleigh-Taylor instability is named after the two scientists, Lord Rayleigh and G.I. Taylor who worked on this instability[2].

Kelvin-Helmholtz instability is named after the scientists, Lord Kelvin and Hermann von Helmholtz who worked on this instability. This instability occurs when there is a velocity difference between the interface of two fluids. Example of Kelvin-Helmholtz instability is wind blowing over the water which is observed on the surface of oceans where due to this instability waves are formed on the surface of the water. This instability is also observed in clouds, Saturn\'s bands, Jupiter\'s Red Spot and the Sun\'s corona[3].

Richtmyer-Meshkov instability is named after the scientists, R.D. Richtmyer who provided the theoretical prediction and E.E. Meshkov who provided the experimental verification. This instability occurs when two fluids of different density are impulsively accelerated by the passage of a shock wave. The development of this 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. This instability is seen in magnetized target fusion, Scramjet Combustor where supersonic combustion occurs, etc.[4]

Plateau-Rayleigh instability is named after the scientists Joseph Plateau and Lord Rayleigh who worked on this instability. This instability makes us understand how a falling stream of fluid breaks up into smaller packets with the same volume but less surface area which is also related to the Rayleigh-Taylor instability. The driving force of the Plateau-Rayleigh instability is that liquids, by virtue of their surface tension, tend to minimize their surface area. Examples of Plateau-Rayleigh instability are water dripping from a tap, urination, etc.[5]

Modeling of these instabilities is important to understand the physics involved in these free surface flows. Many different approaches have been developed to model free surface flows. Marker and Cell (MAC) method [6] is one of the first numerical methods for time-dependent free surface flow simulations. This method is based on a fixed Eulerian grid of control volumes. The location of fluid within the grid is determined by a set of marker particles that move with the fluid but otherwise have no volume and mass. Lagrangian grid methods use grid that moves with the fluid and because the grid and fluid move together, the grid automatically follows free surface. The principal limitation of these methods is that they cannot track surfaces that break apart or intersect[7,8]. Volume of fluid (VOF) method [9] introduces a continuity equation for the volume of fluid. In later implementations the numerical diffusion of the interface is eliminated with the reconstruction of the interface from the volume fraction in computational cells with the geometrical elements. There are many reconstruction techniques, which use different reconstruction segments such as simple line interface calculation - SLIC [10], flux line-segment for advection and interface reconstruction – FLAIR [11], and least square method [12]. VOF method (or single fluid method) is commonly used for many free surface problems. Level set method [13] uses level set function, which is positive in the space occupied by the first fluid and negative in the space of the other fluid. The level set function is defined as the minimum distance to the interface and the interface position is defined with the zero value of level set function. All methods mentioned above are excellent for simulations of free surface flows, but when the flow beside free surface also involves dispersed phase with the structures smaller than the grid size they are not appropriate. For dispersed flows two-fluid model is used, but the same model can also be used for simulations of flows with free surface. The numerical diffusion of the interface is a big disadvantage of two-fluid model used for simulation of free surface flows. Several techniques for reduction of numerical diffusion of the interface have been implemented in the volume of fluid (VOF) model (single- fluid model) and are based on modified numerical schemes for advection of volume fraction near interface. The same approach could be used also for two-fluid method, but from the experience of Štrubelj and Tiselj[1], better reduction of numerical diffusion of the interface can be achieved with interface sharpening which is also called conservative level set method [14]. The basic conservative level set method with one momentum equation (single-fluid) was previously validated on rising bubble, oscillating droplet, dam break, and Rayleigh-Taylor instability test case[1].

These mathematical models for modeling free surface flows have been developed and adapted for appropriate CFD calculations. From Mark Cell Method to the two fluid methods, each method has its limits and with advancing from one to other model made the mathematical models adapting to the required CFD calculations for modeling the free surface flows. As the Rayleigh-Taylor instability test case is already validated using VOF method (Single fluid method), in this report the Raleigh-Taylor instability is modeled using VOF Model. Here the simulation of water above the air is done using the VOF model to understand the Raleigh-Taylor instability.

The non-dimensional number used for analyzing the Rayleigh-Taylor instability is Atwood Number. 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

`\\rho _{1}` = density of heavier fluid

`\\rho _{2}` = density of lighter fluid

NUMERICAL MODEL

(i)Geometry:

The geometry for modeling Rayleigh-Taylor Instability is simple as shown in the below figure. As shown in below figure the low-density fluid, the air is below the high-density fluid, water to observe the Rayleigh-Taylor instability.

(ii)Mesh:

(a)Baseline

As the geometry is simple, the mesh is made of quadrilaterals as shown in below figure.

The mesh parameters are shown in below table:

Element Type	Quadrilateral
Element size	0.5 mm
Cell Count	3200

The mesh quality is shown in the below figure. The quality of the mesh is good which is above 98% and can proceed for further steps.

(b)Fine Mesh

As the geometry is simple, the mesh is made of quadrilaterals as shown in below figure.

The mesh parameters are shown in below table:

Element Type	Quadrilateral
Element size	0.25 mm
Cell Count	12800

The mesh quality is shown in the below figure. The quality of the mesh is good which is above 98% and can proceed for further steps.

As the geometry is simple, the mesh is made of quadrilaterals as shown in below figure.

The mesh parameters are shown in below table:

Element Type	Quadrilateral
Element size	0.125 mm
Cell Count	51198

The mesh quality is shown in the below figure. The quality of the mesh is good which is above 56% and can proceed for further steps.

(iii) Setup:

The properties of fluids used are shown in below table:

property

Value

Density

Water: 998.2 kg/m³

Air: 1.225 kg/m³

Viscosity

Water: 0.001003 kg/m-s

Air: 1.7894e-05 kg/m-s

The equations solved are the continuity and momentum equations. The Laminar model used in the simulations. The computations are performed using ANSYS FLUENT with unsteady, double precision and a pressure based solver is used. The SIMPLE algorithm is selected for the velocity-pressure coupling. The second-order upwind scheme is used for the space discretization for higher accuracy and first-order implicit is used for time discretization. The VOF multiphase model with 2 phases is used with Implicit formulation and the sharpening interface modeling.

The operating pressure used for the simulations is 1 atm or 101325 Pa.

The time step used for the simulations is 0.0025 s.

RESULTS and DISCUSSIONS

(a) Baseline mesh:

The below figure shows the residuals of the simulation. The residuals look good as they are under limits.

The Rayleigh-Taylor instability is observed in the below video for the baseline mesh where the contour of volume fraction of water is shown with time.

The below figure shows the plot of volume fraction of water with time at a point (10mm,5mm). As it is observed the instability became steady after four seconds of time.

The contour of volume fraction of water after achieving steady state is shown in the below figure.

(b) Fine Mesh:

The below figure shows the residuals of the simulation. The residuals look good as they are under limits.

The Rayleigh-Taylor instability is observed in the below video for the fine mesh where the contour of volume fraction of water is shown with time.

The below figure shows the plot of volume fraction of water with time at a point (10mm,5mm). As it is observed the instability became steady after four seconds of time.

The contour of volume fraction of water after achieving steady state is shown in the below figure.

The below figure shows the residuals of the simulation. The residuals look good as they are under limits.

The Rayleigh-Taylor instability is observed in the below video for the finer mesh where the contour of volume fraction of water is shown with time.

The below figure shows the plot of volume fraction of water with time at a point (10mm,5mm). As it is observed the instability became steady after four seconds of time.

The contour of volume fraction of water after achieving steady state is shown in the below figure.

As we increase the number of elements, the Rayleigh-Taylor instability is observed with more clarity at interfaces and the variation of steady result for all the three cases can also be observed.

The Atwood number for this simulation is 0.997. For A close to 0, Rayleigh-Taylor instability flows take the form of symmetric fingers of fluid and for A close to 1, the much lighter fluid below the heavier fluid takes the form of larger bubble-like plumes. As in our case, the Atwood number is close to 1, the plumes are observed in the flow which is shown in the above videos. This validates our model using Atwood Number as it is close to one and corresponding results are obtained.

CONCLUSIONS

The hydrodynamic instabilities involved with the free surface flows are discussed and the various mathematical models used for modeling of free surface flows are also discussed. Understanding of Rayleigh-Taylor instability and it\'s numerical modeling using the VOF method is performed.

REFERENCES

[1] Štrubelj L, Tiselj I. Numerical Simulation of Rayleigh-Taylor Instability With Two-Fluid Model and Interface Sharpening. ASME. Fluids Engineering Division Summer Meeting, Volume 1: Symposia, Parts A and B pp. 33-42. doi:10.1115/FEDSM2008-55063.

[2] Rayleigh–Taylor instability, https://en.wikipedia.org/w/index.php?title=Rayleigh%E2%80%93Taylor_instability&oldid=897115583

[3] Kelvin–Helmholtz instability, https://en.wikipedia.org/w/index.php?title=Kelvin%E2%80%93Helmholtz_instability&oldid=886501730

[4] Richtmyer–Meshkov instability, https://en.wikipedia.org/w/index.php?title=Richtmyer%E2%80%93Meshkov_instability&oldid=896465599

[5] Plateau-Rayleigh instability, https://en.wikipedia.org/w/index.php?title=Plateau%E2%80%93Rayleigh_instability&oldid=884588960

[6] Harlow, F.H., Welch, J. E, 1965, Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface, Phys. Fluid, Vol. 8, pp. 2182-2189.

[7] Hirt, C.W, Amsden, A.A., Cook, J.L., 1974, An Arbitrary Lagrangian-Eulerian Computing Method for All flow speeds, J. Com Phys. Vol. 14, pp. 227-253.

[8] Hyman, L.M., 1984, Numerical Methods for Tracking Interfaces, Physica, Vol. 12D, pp. 396-407.

[9] Hirt, C. W., Nichols, B. D., 1981, Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries, J. Comp. Phys., Vol. 39, pp- 201-225.

[10] Noh W. F. and. Woodward P. R, 1976, SLIC (simple line interface calculation), in Proceedings of the Fifth International Conference on Numerical Methods in Fluid Dynamics, edited by A. I. Van de Vooren and P. J. Zandbergen (Springer-Verlag, New York), Lecture Notes in Physics 59, pp. 330-340.

[11] Ashgriz, N., Poo, J.Y, FLAIR: Flux line segment model for advection and interface reconstruction, 1991, J. Comp. Phys., Vol. 93, pp. 922-927.

[12] Pilliod, J.E, Puckett, E.G., 1998 Second-Order Accurate Volume-of-Fluid Algorithms for Tracking Material Interfaces, Vol. 199, pp. 465-502.

[13] Osher, S., Sethian, J.A., 1988, Front Propagating with Curvature-Dependent Speed: Algorithms Based on Hamilton-Jacobi Formulations, J. Comp. Phys., Vol. 79, pp. 12-49.

[14] Olsson E., Kreiss G., Zahedi S., 2007, A conservative level set method for two phase flow II, Journal of Computational Physics, vol. 225, pp. 785–807.

Understanding and Simulation of Rayleigh Taylor Instability using VOF Method

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