In the geophysical context, there are a wide variety of mechanisms which may lead to the formation of unstable density stratification, leading in turn to the development of the Rayleigh-Taylor instability and, more generally, interfacial gravity-driven instabilities, which involves moving boundaries and interfaces. The purpose of this work is to study the level set method and to apply the process to study the Rayleigh-Taylor instability experimentally and numerically. With the help of a simple, inexpensive experimental arrangement, the R-T instability has been visualized with moderate accuracy for real fluids. The same physical phenomenon has been investigated numerically to track the interface of two fluids of different densities to observe the gravitational instability with the application of level set method coupled with volume of fraction replacing the Heaviside function. Good agreement between theory and experimental results was found and growth of instability for both of the methods has been plotted.

The Rayleigh-Taylor instability is instability of an interface of two fluids of different densities which occurs when the interface between the two fluids is subjected to a normal pressure gradient with direction such that the pressure is higher in the light fluid than in the dense fluid. This is the case with an interstellar cloud and shock system. A similar situation occurs when gravity is acting on two fluids of different density—with the denser fluid above a fluid of lesser density—such as water balancing on light oil. Considering two completely plane-parallel layers of immiscible fluid, the heavier on top of the light one and both subject to the Earth’s gravity, the equilibrium here is unstable to certain perturbations or disturbances. An unstable disturbance will grow and direct to a release of potential energy, as the heavier material moves down under the gravitational field and the lighter material is displaced upwards. Such instability can be observed in many situations including technological applications as laser implosion of deuterium-tritium fusion targets, electromagnetic implosion of a metal liner and natural phenomena as overturn of the outer portion of the collapsed core of a massive star, and the formation of high luminosity twin-exhaust jets in rotating gas clouds in an external gravitational potential.

Various numerical and experimental works have been done by many researchers concentrating on the growth of single wavelength perturbations as well as considering different wavelength modes. Sharp [

The volume of fluid (VOF) technique has been presented by Hirt and Nichols [

Sethian [

Various numerical methods were developed to study the propagating interfaces. Osher and Sethian [

Theory and algorithms of level set method were reviewed by Sethian [

Kaliakatos and Tsangaris [

Sethian and Smereka [

The Rayleigh-Taylor instability is a gravity driven instability of a contact surface and this growth of this instability is sensitive to numerical or physical mass diffusion. Li et al. [

In this present work, the level set methodology has been applied to visualize theoretically the RT instability using a triangular distribution of initial disturbance. The fraction of volume in the interface control volumes has been successfully incorporated for identifying the interface very accurately. The topological changes with time have been captured accurately and this has been matched effectively with the experimental results. The instability growth rate which is predicted by the theory is confirmed by the experimentation with the initial incipience of linear distribution of disturbances as already stated. This is a positive contribution along with the theoretical topological visualization of the RT effects.

On the other hand, the merging and consequent breaking up of the interfaces has been captured while the RT instability growth takes place. These results are important as they provides the probable trapping, merging, and consequent breaking of the oil and natural gas pools trapped between the formation of salt domes and overlying sedimentary rocks. These effects of the geothermal RT instabilities and deformation of the rocks above the salt domes are important as they provide the possibility of exploration of oil and gas pools, thus coagulated and subsequent fragmented in huge mass under the earth for million of years. These results are encouraging and can bridge our knowledge of RT to apply to the oil and gas industry.

The geometry of the problem is shown in Figure

Geometrical presentation of analysis of Rayleigh-Taylor instability of a dense fluid overlying a lighter fluid.

The experimental setup consists of a closed rectangular box made of Perspex of 20.4 cm × 10.2 cm × 15 cm dimension. There are two openings at the top surface with valve arrangement for the purpose of filling the box with the required liquids. The two side handles are provided for convenience turning of the setup to upside down or vice versa in quick time. The setup is placed on a preleveled surface and lower half of the box is filled with glucose solution and upper half is filled with colored refined soya bean oil, with the help of funnels. The viscosities of both the liquids were measured in the laboratory at room temperature by “Falling Sphere method” and density of the fluids was measured by simply measuring their mass and volume (see Figure

Diagram of the experimental setup.

The viscosity and specific gravity of the liquids have been measured as follows:

Viscosity of glucose syrup = 350 Pa-S,

Viscosity of oil = 0.0791 Pa-S,

Specific Gravity of Glucose syrup = 1.4,

Specific Gravity of oil = 0.92.

In the experiment, first the setup rests at position 3 where the light fluid lies over the heavy one. In this position, it is totally balanced and stable. Then the setup is turned upside down quickly so that heavy liquid lies in the upper half and thus instability is initiated. The instability can also be initiated by keeping the setup at position 2 where the heavy and light liquids stand vertically side by side in an unbalanced and unstable condition. Naturally all these configurations want to return to position 3 to minimize the potential energy and to gain a stable and balanced position. The whole process is captured to track the moving interface and to study the growth rate of the instability with time (see Figure

Illustration of different types of stability and experimental procedure.

The term two-phase flow refers to the motion of two different interacting fluids or with fluids that are in different phases. In the present analysis, only two immiscible incompressible fluids have been considered and a low enough Reynolds number is assumed so that the flow can be considered as laminar flow. Level set method may be applied to track the interface efficiently in case of incompressible, immiscible fluids in which steep gradient in viscosity and density existed across the interface. In these problems, the role of surface tension is crucial and formed an important part of the algorithm.

For mathematical analysis, we assume a system of two-fluid phases constituting a two-dimensional domain. The individual fluid phases are assumed to be incompressible but deformable in shape on account of shear stresses prevailing between various fluid layers as well as fluid-solid interfaces. We assume the flow field to be two dimensional and laminar.

Navier-Stokes equation is given as

The surface tension term acts normal to the fluid interface and is proportional to the curvature, due to balance of force argument between the pressure on each side of the interface. This leads to the relation,

Now replacing normal

Thus the equation of motion become

Consider

Consider

A scalar variable, level set function is used to identify the interface between two fluids and also acts as a distance function. The equation transporting the interface can be written as

But at all instant of times

Equation (

The reinitialization process is iteration of (

It is evident that pseudo-steady-state value of

The equation for the one-dimensional volume fraction is given by

and for two-dimensional volume fraction the concept has been taken from [

At the solid boundary, the Neumann boundary condition for the level set function has been utilized.

The governing differential equations, coupled with appropriate boundary conditions, are solved using a pressure based finite volume method, as per the SIMPLER algorithm [

The location of the interface at time

The temporal term of the momentum equation has been discretized as follows. Equation (

(a) Control volume for the two-dimensional situation. (b) Control volume of

The source term is linearized in the usual manner anticipating negative slope while the unsteady terms

Similarly the pressure gradient term is discretized considering the staggered control volume as:

This is for the

If the box is rotated in the

Illustration of the experimental technique.

From theoretical analysis of the problem, the growth rate is given by

Here it has been assumed that the flow is laminar owing to the fact that the viscosities are of very high order in a two-dimensional, incompressible flow.

The solution of this equation is

Now, it can be seen from the growth equation that growth rate varies linearly with displacement of that point at a particular time.

Now, for comparison purpose, a point at a distance of 6.1 cm, that is, approximately

The displacement of the considered point is measured at different times from the undisturbed interface by proper measurement in the series of snapshots presented in Figure

Development of growth of instability.

It can be seen that the best fitted curve is unbounded exponential in nature, which agrees very much with the theory demanding exponential growth of the instability. The curve is of the form

Theoretically, the value of

where ^{3} kg/m^{3}, ^{3} kg/m^{3},

Here, ^{2}, and

Now, from the experimental study, the initial

So the theoretical growth equation becomes,

It can be observed from the above two expressions that the characteristics of the development of growth of instability are quite similar, with a slight difference in the growth time. The growth time is slightly higher in case of experimental observation than the numerical investigation.

Figure

Development of growth of instability as found by theoretical modeling.

Comparison of the development of the growth of instability.

Comparison of the growth Rate of the instability.

The same problem is numerically analyzed considering a rectangular two-dimensional domain. Two arrays of

Distribution of initial instability triggered for the numerical analysis.

Comparison between experimental and numerical results.

The variation of the propagating interface with time has been shown. Both the experimental and numerical results are presented here (see Figure

It can be observed from the above figures that the experimental results are in good agreement with the numerical results. The interface between the two fluids shows similar pattern during the study for both experimental and numerical analysis. However, three-dimensional features are observed to affect the results as seen in the experimental study. Figure

Comparison of the growth rate of instability (numerical result versus linear theory).

In Figure

Merging and consequent breaking of two bubbles in a heavier matrix.

The nature of the development of instability was experimentally found as a function of sine curve as predicted by theoretical model. A numerical methodology was devised and validated with experimental results so that the methodology can handle any gravitational interfacial instability. It was found that, in the early stages of the growth of instability, the growth rate is proportional to the instantaneous growth in a particular position, that is, growth rate varies linearly with growth at that moment at a particular point on the interface. But, at the later stage of development of instability, substantial deviation from the linear theory was observed. The pictorial views of the interface between the two fluids have been studied both theoretically and experimentally and they have matched satisfactorily.

Atwood number (−)

Height of one fluid layer (cm)

Speed function (m/s)

Acceleration due to gravity (m/s^{2})

Heaviside function (−)

Pressure (N/m^{2})

Time (s)

Velocity (m/s)

Growth at a particular time (cm)

Initial growth (cm)

Small time step (s).

Level set function (m)

Dummy variable for level set function (m)

Dirac delta function (−)

Fluid properties such as density and viscosity (−)

Surface tension coefficient (N/m)

Coefficient of viscosity (Pa-s)

Curvature (m^{−1})

Density of the liquid (kg/m^{3})

Growth time (s)

Wavelength of the perturbation (cm).

This work is supported by the Council of Scientific and Industrial Research (CSIR), Government of India.