Introduction

Contents

Sr noTitlePage no

iList of figures

iiList of tables

1Introduction

2Analysis

3Phenomenology

4Some factors affecting development of rt instability

5Refernces

i.

LIST OF FIGURES

Sr NoTitlePage no

Fig.1Layer of water above air

Fig.2co-ordinate system

Fig.3Different stages of development

LIST OF TABLESSr No

Table 1Factors affecting development of RT instability

LIST OF SYMBOLS

Density

tTime

UX component of velocity

VY component of velocity

WZcomponent of velocity

U0Initital velocity in X direction

V0Initial velocity in Y dirction

W0Initial velocity in Z direction

FBodyBoby forces

FSurfaceSurface forces

Viscous stress tensor

AAmplitude

KWave number

1.0 INTRODUCTIONThe Rayleigh-Taylor instability is a fingeringInstability of an interface between two fluids of different densities, which occurs when the heavy fluid is pushing the light fluid, is called as Rayleigh Taylor instability [1]. It can occur both under the action of gravity or by the action of fluid acceleration of the fluid in the direction of the denser fluid [9].Consider a container existing on earth with normal gravitational effects. Suppose this container contains a volume of air, which is supporting a uniform layer of water that lies on the top. The configuration can be seen in Figure1. Since gravity is present, the effective acceleration is in the positive z-direction (upwards). Therefore, the air (light fluid) pushes on the water (heavy fluid). As long as the interface between fluids remains uniform, that is completely horizontal and perpendicular to the effective acceleration, the air has sufficient pressure to hold the water. However, small deviations are bound to occur at the interface. When the perturbations occur, portions of the interface lie higher than the average and therefore feel more pressure from the air than is necessary to support the water. Therefore, the interface continues to rise in the spots. In the portions where the interface has dropped by small amounts below the average, more pressure is needed for support, and the interface drops further. Even the slightest perturbation will cause the instability. If the situation were reversed, and the water layer was on the floor supporting the volume of air, the pressure gradients reverse. In the portions of the interface that have risen, the water will lower back to average, while the lowered portions will rise again. This qualitative analysis leads to the conclusive statement that the interface is stable if the heavy fluid pushes the light fluid and unstable if the light fluid pushes the heavy fluid. The unstable case is Rayleigh-Taylor instability.

Figure.1 1.1ANALYSIS

To begin we will consider a large box filled with a fluid with the coordinate system as shown in Fig 1.2. The coordinate system has x as the horizontal, z as the vertical, and y as going into the page. The acceleration of gravity, g, is in the negative vertical direction. The Navier- Stokes equations govern the motion in fluid dynamics and are a set of five coupled, conservation equations: mass, momentum (one equation for each of the three principal directions), and energy.

Figure 2

1.1.1 CONSERVATION OF MASS

The continuity, or conservation of mass, equation on a per volume basis is: + =01

Where, is density, t is time, is the vector differential operator, and V is the velocity vector. The continuity equation is expanded as:+

2

+

3

Where U, V, and W are the velocity components in the x, y, and z directions. Now making the problem in 2-dimensional space, neglect any changes in the y-direction and this makes any derivatives with respect to y equal to zero (the third term in the above equation). The above equation reduces to:

+

4

Now expanding this equation:

The next assumption is the one of incompressibility for a given layer of fluid, that is, a horizontal plane of fluid at a given z coordinate will have a constant density. Thus, changes in density (with respect to time and space) are zero meaning the temporal and spatial derivatives are zero. The above equation reduces to:

Now, dividing through by density,

The final step is to write the velocity in terms of a perturbed quantity. Consider that the fluid is initially at a constant state and only consider small changes from this state:

U=U0 +u

V=V0+v

W=W0+w

Where the subscript 0 indicates the constant initial value and the lower-case letters are small velocity deviations. The deviations in velocity are noted to be functions of both time and space. Substituting into the above solenoidal equation:

Noting that derivatives of constant values are zero, the final form of the continuity equation is:

1.1.2 CONSERVATION OF MOMENTUM

The derivative operator D/Dt, sometimes called the material, substantial, or total derivative is defined as:

Where q may be a scalar, such as density, or a vector such as velocity. We employ the total derivative when using the Eulerian point of view (as opposed to the Lagrangian).From this perspective we analyse a given point in space, perhaps an infinitesimally small cube of dimensions dx,dy,dz, and the first term is how much q is changing in time and the last three terms are how much q is being convected through this small volume. If q is the velocity, then we have the Eulerian acceleration.

Newtons second law of motion (force equals mass times acceleration) on a per volume basis is:

= FBody Force + Fsurface

Where the body force we will consider is due to gravity, and the surface forces are due to pressure and shear. The momentum equation then becomes:

= +

Where the acceleration vector is g = - gk , p is the pressure, and is the viscous stress tensor. We will employ the in viscid assumption so we will neglect the viscous stress tensor term. The momentum equation becomes:

=

The x-momentum equation is the first of the three momentum equations

=

and noting that gx=0 and expanding the total derivative

-

The pressure and density will be rewritten in terms of average values (r and p) with an additional, small, perturbed quantity, dr and dp. The average values or r and p are constant within a given fluid layer at a constant z, however, due to the hydrostatic pressure gradient, they are both functions of height. The perturbed density dr is considered incompressible so that as a small element of increased density fluid moves through the fluid, this small elements density will not change. Therefore, r=r(z) and p=p(z) and dr=dr(x,y,z,t) and dp=dp(x,y,z,t). Substituting these into the x-momentum equation along with the decomposed velocities gives:

-

Expanding the velocity terms and neglecting derivatives with respect to y and substituting zero for derivatives of constant terms-

At this point, another assumption is made regarding the initial velocity condition. In the box of fluid, assume that the initial velocity is zero for Rayleigh-Taylor analysis. Hence x-momentum equation becomes:

-

Expanding the density and pressure terms, this equation becomes:

= -

The next step is to linearize the equation. The justification is: changes in small quantities are initially themselves, small; therefore, a small quantity multiplied by a small quantity is much smaller than the initially small quantities and may therefore be neglected. Finally, using this assumption and the fact that p is only a function of z, the x-momentum equation becomes:

We will neglect the analysis of the y-momentum equation for this two dimensional study. The z-momentum equation is:

=

Conducting a similar step by step analysis of this equation, and noting that our acceleration, g = -gk , the z-momentum equation reduces to:

=

The hydrostatic pressure, p(z), at any z-location is due to the weight of fluid above it and is related to the density by p/z = -rg . Using this in the above equation after expanding density and pressure terms gives the final form of the z-momentum equation:

The last of the Navier-Stokes equations is the energy equation. We will neglect this equation by saying that there is no heat or work produced by, or transferred to, the system, there are no dissipative processes, and the system is isothermal. Hence we have the continuity, and x- and z-momentum equations:

We have three equations and four unknowns: u, w, dp, and dr. For our fourth equation, once again employ the continuity equation but with the density perturbation, where previously we used the density, (and use the assumption that the initial velocity is zero):

+ +=0

On expanding and neglecting the non-linear terms and derivatives with respect to y, we get

++ = 0

Noting the term in parenthesis is zero from the first analysis of the continuity equation and that = (z) (not t or x), the final form of our fourth equation is

= -w

Review list of assumptions:1. 2-dimensional, i.e., /y=0.2. Incompressibility for a given layer of fluid.3. Velocity rewritten in terms of initial constant values plus small deviations.4. Inviscid.5. Initial velocities are zero.6. Momentum equation is linearized.7. Neglect energy equation (isothermal, etc.)

1.1.3 NORMAL MODE ANALYSIS

Single-Mode Analysis of Rayleigh-Taylor instability [2]

For a mathematical analysis of Rayleigh-Taylor instability, two incompressible fluids are considered. The configuration with a single-mode perturbation is shown in Figure 5. The amplitude of the distrubance is described by the following formula:

A(x,z,t) = Ak(z,t)eikx

where solutions for Ak(Z,T) are in the form

Ak(z,t) = Ak(z)ent

where n is an eigenvalue corresponding to the wave number k. Therefore

A(x,z,t)= Ak(z)eikx + nt

Using this relation with velocities, density, and pressure, that is

u(x,z,t)= uk(z)eikx + nt

w(x,z,t)= wk(z)eikx + nt

(x,z,t)= k(z)eikx + nt

(x,z,t)= k(z)eikx + nt

and applying them to the simplified Navier-Stokes equations, that is

=

=

= -w

(2w = 0

The following is the governing differential equation

(2w = -wg

Letting the fluids be incompressible simplifies the result even further:

Letting the boundary conditions be that at large distance above or below the interface the velocity is zero, the following is the solution

w2=w0e-kz

w1=w0e-kz

Applying the governing equation at the interface, by integrating over an infinitesimal distance dz across the interface, the following results

- 2wdz = -2d

) = -wg()2

-(2+1)= -g(k/n2) (2-1)

Solving for the eigenvalue n, the following is the result:

n = =

Where,

A=

is called the Atwood number. For a positive Atwood number, the heavy fluid lies on top of the light fluid in the gravitational field and the eigen value is real. Therefore, the system is unstable and Rayleigh-Taylor instability occurs. For a negative Atwood number, the light fluid lies on top of the heavy fluid and the n is imaginary. For this situation, the system is stable[4].

The initial growth would be linear with viscosity setting the maximal growth rate m to length scale of the order of(2/(Ag)1/3, the associated time scale is (/A2g2)1/3 the mean kinematic viscosity. The extent of the mixing region has been assumed to follow h=Agt2 [6] where h represent the penetration length, and has been introduced as a constant of proportionality[6], called the acceleration constant. Recent experimental investigations, for 3D cases, give .03 [8].

1.2 PHENOMENOLOGY

There is a complex phenomenology associated with the evolution of a Taylor unstable interface[3]. This includes the formation of spikes, curtains and bubbles, the development of Helmholtz instability on the side of the spikes, competition among bubbles leading to their amalgamation, formation of droplets, entrainment and turbulent mixing, and a possible chaotic limit with a fractalized interface. It is helpful to organize a description of the growth of the instability into a number of stages. This can be done as follows

Figure 3

STAGE 1

If the initial perturbations in the interface or velocity field are extremely small, the early stages in the growth of the instability can be analysed using the linearized form of the dynamical equations for the fluid. The result is that small amplitude perturbations of wavelength will grow exponentially with time. It sees a liner growth of the perturbations until instability amplitudes grow to the order of 0.1 to 0.4, where is the perturbation wavelength (if multi-mode perturbations are considered, then represents the most unstable wavelength). At this point, substantial deviations are observed from the linear dynamical equations for the fluid.

STAGE 2

During the second stage, while the amplitude of the perturbation grows non linearly to a size of order 2, the development is strongly influenced by three-dimensional effects and the value of the density ratio, or Atwood number, the light fluid moves into the heavy fluid in the form of round topped bubbles with circular cross sections. If the Atwood number is large (close to 1), then the light fluid moves into the heavy fluid as round topped bubbles with circular cross sections. Also, the heavy fluid penetrates the light fluid as spikes and curtains between the bubbles, so that a horizontal section would show a honeycomb pattern.. These structures would form so that a horizontal section cut out would look like a honeycomb pattern. If the Atwood number is small (close to 0), then the pattern would instead show two sets of interpenetrating bubbles. The formation of these structures is also greatly affected by three-dimensional effects. Note that "two dimensional" plane bubbles are unstable to perturbations along the axis perpendicular to the plane of the bubble, and a trough having a plane bubble as a cross section will break up into three dimensional bubbles.

STAGE 3

This stag hosts the evolution of the structures formed in Stage 2.This stage is characterized by the development of structure on the spikes and interactions among the bubbles. These phenomena can originate from several sources. There is a non linear interaction among initial perturbations of different frequency. Also Helmholtz instability along the side of the spike can cause it to mushroom, increasing the effect of drag forces on the spike. This effect is more pronounced at low density ratios. If the perturbation was multi mode, there is a non linear interaction among the initial perturbations of differing wavelengths. If the initial perturbation level is low enough, structures evolve from the non linear interaction between the smaller structures. When the amplitude reaches about 10.6 Kelvin-Helmholtz instability now forms on the side of the spikes, as seen in the figure as the cat-eye formations along the side of the spike. The Kelvin-Helmholtz instability is also the reason for the mushroom structure at the tip of the spike, since there are now tangential velocity variations. This increases the drag forces on the spike and causes the mushrooming. There are some experimental evidence for bubble amalgamation, a process in which large bubbles absorb smaller ones, with the result that large bubbles grow larger and move faster. The presence of heterogeneities in various physical quantities can modify the shape and speed of bubbles and spikes to a degree which depends on the strength and length scale of the heterogeneity.

STAGE 4

In the final stage, we encounter the break up of the spike by various mechanisms, the penetration of a bubble through a slab of fluid of finite thickness and other complicated behavior that leads to a regime of turbulent or chaotic mixing of the two fluids.

1.3 SOME FACTORS INFLUENCING THE DEVELOPMENT OF RAYLEIGH-TAYLOR INSTABILITY

Numerous factors influence the development of Taylor instability in a simple fluid[3]. These include surface tension, viscosity, compressibility, effects of converging geometry, three-dimensional effects, the time dependence of the driving acceleration, shocks, and a variety of forms of heterogeneity. An assessment of some of these factors is given in table I

Table1

FactorRelative size of effect (Dimensionles parameter)Effect on growth of instability

Density RatioAtwood numberAtwood number is the most important factoe governing the growth rate of Rayleigh taylor instability for small amplitude perturbations of wavelength

Surface tensionWeber numer

Viscosity

Compressility

Heterogeneity

In natural phenomena and technological applications where Taylor instability occurs, there are many other factors that can play an important role. For example, material properties and the equation of state of the fluids may be important. The fluids may conduct heat or diffuse mass. The material may change phase or consist of several components. Radiation often couples to hydrodynamics. Here we restrict the discussion to a few of the factors which effect the behavior of simple fluids as it is difficult to scientifically consider with the whole range of factors that can influence Rayleigh-Taylor instability.

REFERENCES

1. Chandrasekhar, S., Hydrodynamic and Hydromagnetic stability, Dover, 1981,first published by Oxford University Press, 1961.2. Dr. Kassoy ;Development and Applications of Important Interfacial InstabilitiesScott Reckinger Fluid Dynamics Term Paper Fall 20063. Sharp, D.H. 1984. An overview of Rayleigh-Taylor instability. Physica D 12:3-18. 4. A. Sedaghat*,Numerical Simulation ofRayleigh-Taylor Instability, Advanced Design and Manufacturing Technology, Vol. 6/ No. 1/ March 20135 Jean-Paul JEFFRAI; Internal structure in Rayleigh-Taylor instability ;Centre for Mathematical Science; May - July 20036. By Andrew W Cook, William Cabot and Paul L Miller; The mixing transition in RayleighTaylor instability; J. Fluid Mech. (2004), vol. 511, pp. 333362.7. H.J. Kull; Theory of theRayleigh-taylor instability; physics reports North-holland ,206, no. 5 (1991) 197-325. 8. A. Sedaghat, S. Mokhtarian; Numerical Simulation ofRayleigh-Taylor Instability; Advanced Design and Manufacturing Technology, 2013 IAU, Majlesi, Vol. 6/ No. 1/ March 20139.Nishihara Katsunobu and Ikegawa Tadashi; weakly nonlinear theory of Rayleigh-Taylor instability; j. plasma fusion res. series, vol.2 (1999) 536-540.

At sufficiently late times, the extent of the mixing region has historically beenassumed to follow Agt2 (Anuchina et al. 1978; Youngs 1984), where is adimensionless coefficient, A (2 1)/(2 + 1) is the Atwood number (with 1and 2 being the densities of the light and heavy fluids, respectively), g is theacceleration and t is time. This similarity solution results from dimensional analysis ifthe following conditions are met: (I) all memory of initial conditions is lost, (II) thereare no boundary effects, and (III) viscosity and diffusivity (or surface tension) are notimportant. Satisfying all three of these requirements has proved extremely difficult,both experimentally and computationally.

An initial single-mode perturbation on an interfacebetween two uids of densities and ( > )is the simplest case and its linear growth has beenwidely analyzed.[1] The heavier uid is superposedover the lighter uid in a gravitational _eld, ^,where is the acceleration.

The initial growth ofa single-mode perturbation of wavelength is ex-ponential, 2/2 = 2 with 2 = , where = 2/ is the wave number, is the amplitude and = ( )/( + ) is the Atwood number.

We consider the Rayleigh-Taylor instability [14] which occurs when an initially perturbedinterface between a heavier uid which is on top of a lighter uid is allowed to grow underthe inuence of gravity. The ngers of lighter uid penetrate the heavier uid in what arereferred to as `bubbles', while `spikes' of heavier uid move into the lighter uid, as shown inFigure 1. With time, the bubbles and spikes, which are initially distinct, continue to evolve.In the process, they may grow, split, merge with surrounding bubbles (spikes), or shrink insize relative to other bubbles (spikes) which grow and overtake them.

Linear and non-linear growth of the mixingzoneGenerally, the growth of the RTI mixing zone, evolving from an interfacewhich is initially flat apart from infinetesimal disturbances, between viscous andmiscible fluids can be described by a linear and nonlinear theory as the flow becomesrapidly turbulent.1. A good summary of the linear theory can be found in Chandrasekhar. The initial growth would be linear with viscosity setting the maximalgrowth rate m to length scale of the order of(2/(Ag)1/3, the associatedtime scale is (/A2g2)1/3 where A is the Atwood number and the meankinematic viscosity:A = 1 21 + 2 = 1 + 21 + 2 When a perturbation characteristics for m reaches the amplitude of0.5 m, its growth rate slows down and larger structures appear. Whenthe dominant wavelength exceeds about 10m the memory of the initialconditions is lost.Note: For the flow considered in this paper these scales correspond to alength scale of the order of 1mm and time scale of 0.1s. That means thatthis linear growth appears in the very earlier stages of the experiment.2. Due to the length of the dominant wave growing, viscosity now has littleeffect on the growth of large scale structure. This leads to the result that thewidth of the mixing region between the two layers only depends on 1, 2, gand time t. Then for A 1 (as in the experiments), the penetration of thelower (upper) layer into the upper (lower) layer of the tank should follow.h = Agt2 (1)where is a dimensionless constant. It is convenient then to use this resultto determine a time scale for the flow given by T = (H/gA)1/2( 5s in our experiments). By non-dimensionalizing the penetration h bythe depth of the tank H, we obtain = 2with = t/T .9The equation h = Agt2 results of a self-similar behaviour of the flow whereall memory of the initial conditions has been lost, and the only relevant scale isgt2. While this suggests that is a universal constant, experiments and numericalsimulations have produced differing values.In the next section we will investigate the motion of the flow in the tank and seethat, due to the presence of a plume introduced by the removal of barrier, thegrowth rate of the mixing zone doesnt fit exactly to (1).

The three-dimensional RTI between two incompressible and immiscible fluids was studied using a phase field. Duringthe early stages, the interface grows nearly symmetrically up and down and remains rather simple. During the late stages,the heavy fluid rolls up at both the saddle point and the spike tip due to the KelvinHelmholtz instability. As a result, weobserved the two-layer roll-up phenomenon of the heavy fluid, which does not occur in the two-dimensional case. And westudied the positions of the bubble front, spike tip, and saddle point to investigate the effect of the Reynolds and Atwoodnumbers on the three-dimensional RTI. We showed that a decrease in the Reynolds or Atwood numbers delays the developmentof the RTI. Note that the three-dimensional RTI exhibits a stronger dependence on the Atwood number than on theReynolds number. Finally, owing to the pressure boundary treatment, we were able to perform long time three-dimensionalsimulation resulting in an equilibrium state.

The effect of the Atwood numberTo study the effect of the Atwood number on the three-dimensional RTI, we perform numerical simulations at a numberof Atwood numbers with the Reynolds number fixed at 1024. We use h = 1/128 (a 128128512 grid),1t = 0.001, . =0.01, and Pe = 1/.. Fig. 6 shows the positions of the bubble front, spike tip, and saddle point at different Atwood numbers.When the Atwood number is small (at At = 0.2), the position change of the bubble front is almost the same as that ofthe spike tip. This means that the interface grows nearly symmetrically up and down. However, as the Atwood numberincreases, the position of the spike tip changes much more quickly than that of the bubble front and the symmetry of theinitial structure becomes lost. From the results of Fig. 6, we see that the three-dimensional RTI exhibits a strong dependenceon the Atwood number.

Extensive analysis of RayleighTaylor instability with linearinviscid theory can be found in the book of Chandrasekar (1961).Flow of two fluids in our case is isothermal and incompressible andis simulated in two dimensions. Beside density ratio and viscosityratio, a dimensionless parameter, known as the Atwood numberAt = (1 2/1 + 2) defines the system.

All simulations are performed in a closed box (L = 4m H =1m) containing two immiscible fluids. The fluid with higher densityis located above the lighter one and the interface between themis disturbed with a random noise of amplitude < 1mm. Densityof fluids in our simulations are _2 = 3 kg/m3 and _1 = 1 kg/m3.Dynamic viscosities of fluids are_1 = 0.03 Pa s and_2 = 0.01 Pas,respectively. The corresponding Atwood number is At = 0.5. Thedevelopment of RayleighTaylor instability with various surfacetension coefficients _ is analyzed. Fluid 1 initially occupies 50% of total volume (H1 = 0.5m). The gravity in the system is g =10 m/s2. The surface tension _ increases the most unstable wavelength_ = 2_/k.Amplitude growth in RayleighTaylor instability is shown inFig. 5. It can be seen that the theoretical prediction based onlinear analysis predicts exponential amplitude growth. In simulationswith high values of surface tension coefficient (_ =0.16 N/m)the theory is valid, as the amplitude growth id near theoreticalone (Fig. 5). Simulation with smaller value of surface tension(_ =0.01 N/m) shows that exponential growth of the amplitudepractically does not exist, thus, the disagreement between the theoryand simulation is more pronounced.Theoretically predicted trend of increasing the most unstablewavelength with increasing surface tension is evident in numericalsimulations (Table 1). In numerical simulations with small surfacetension coefficient, the most unstable wavelength is alwayslarger than the theoretical one (Table 1); for larger surface tensioncoefficients the range of wavelengths that appear in simulationis near the theoretical one. In Table 1 the range of most unstablewavelengths in simulation is presented, since several wavelengthsappears as the most unstable.