e final form of the continuity equation is:

∂ u∂ x

+ ∂ w∂ z

=0

1.1.2 CONSERVATION OF MOMENTUM

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

defined as:

dqdt

=¿ ∂ q∂ t

+U ∂ q∂ x

+V ∂q∂ y

+W ∂ q∂ z

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.

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

ρ dVdt = 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:

ρ dVdt = ρ g−∇ ρ+∇ τ

Where the acceleration vector is g = - gkˆ , 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:

ρ dVdt = ρ g−∇ ρ

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

ρ dUdt = ρ gx−

∂ p∂ x

and noting that gx=0 and expanding the total derivative

ρ( ∂ U∂ t

+U ∂ U∂ x

+V ∂ U∂ y

+W ∂ U∂ z )=¿-∂ p

∂ x

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 element’s 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:

(ρ+∂ ρ)¿-∂( p+∂ p)∂ x

Expanding the velocity terms and neglecting derivatives with respect to y and substituting zero

for derivatives of constant terms

(ρ+∂ ρ)( ∂ u∂ t

+U o∂u∂ x

+u ∂ U∂ x

+W o∂ u∂ z

+w ∂ u∂ z )=¿

-∂( p+∂ p)∂ x

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:

(ρ+∂ ρ)( ∂ u∂ t

+u ∂U∂ x

+w ∂u∂ z )=¿-∂( p+∂ p)

∂ x

Expanding the density and pressure terms, this equation becomes:

ρ ∂u∂ t

+ρ u ∂u∂ x

+ ρ w ∂u∂ z

+∂ p ∂ u∂t

+∂ p u ∂ u∂ x

+∂ pw ∂u∂ z

= -∂ p∂ x

−∂ δ p∂ x

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:

ρ ∂u∂ t

=−∂ δ p∂ x

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

The z-momentum equation is:

ρ dWdt

= ρ gz−∂ p∂ z

Conducting a similar step by step analysis of this equation, and noting that our acceleration, g = -

gkˆ , the z-momentum equation reduces to:

ρ dWdt = (ρ+∂ ρ)gz−

∂(p+∂ p)∂ z

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:

ρ ∂ w∂t

=−∂ ρ g−∂ δ p∂ x

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:

∂ u∂ x

+ ∂ w∂ z

=0

ρ ∂u∂ t

=−∂ δ p∂ x

ρ ∂ w∂t

=−∂ ρ g−∂ δ p∂z

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):

∂(ρ+∂ ρ)∂ t

+ ∂∂ x

(( ρ+∂ ρ ) u)+∂

∂ y ( ( ρ+∂ ρ ) v )+ ∂∂ z

( ( ρ+∂ ρ ) w)=0

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

∂ ρ∂t +

∂ δ p∂ x

+u ∂ ρ∂ t +ρ( ∂u

∂ x+ ∂ w

∂ z )+w ∂ ρ∂ z = 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

∂ δ ρ∂ t

= -w∂ ρ∂ z

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

∂ ρ (x,z,t)= ∂ ρ k(z)eikx + nt

δ p (x,z,t)= δ p k(z)eikx + nt

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

∂ u∂ x

+ ∂ w∂ z

=0

ρ ∂u∂ t = −∂ δ p

∂ z

ρ ∂ w∂t = −∂ ρ g−∂ δ p

∂ z

∂ δ ρ∂ t = -wρ

ddz

(ρ ∂ w∂z

¿−ρ k2w = 0

The following is the governing differential equation

ddz(ρ dw

dz¿−ρ k2w = -wgk2

n2d ρd z

Letting the fluids be incompressible simplifies the result even further:

d2 yd z2 −k2w=0

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

∫ ddz

(ρ ∂ w∂ z

) - ∫ ρk2wdz = -∫wg(¿ k

n)¿

2dρ

∆ ¿) = -wg(kn)2∆ ρ

-(ρ2+ρ1)= -g(k/n2) (ρ2-ρ1)

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

n = √ gkρ2−ρ1

ρ2+ρ1 =√ gkA

Where,

A= ρ2−ρ1

ρ2+ρ1

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 λm to length

scale of the order of(ν2/(Ag)1/3, the associated time scale is (ν/A2g2)1/3 ‘ν’ the mean kinematic

viscosity. The extent of the mixing region has been assumed to follow h=αAgt2 [6] where ‘h’

represent the penetration length, and ‘α ’ has been introduced as a constant of proportionality[6],

called the acceleration constant. Recent experimental investigations, for 3D cases, give α ≈ .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 0.1λ to 0.4λ, where λ is the perturbation wavelength (if multi-mode

perturbations are considered, then λ 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 10λ.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

Factor Relative size of effect

(Dimensionles parameter)

Effect on growth of instability

Density Ratio Atwood number Atwood number is the most important

factoe governing the growth rate of

Rayleigh taylor instability for small

amplitude perturbations of wavelength

Surface tension Weber 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 Instabilities

Scott Reckinger Fluid Dynamics Term Paper Fall 2006

3. 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 – 2013

5 Jean-Paul JEFFRAI; Internal structure in Rayleigh-Taylor instability ;Centre for Mathematical

Science; May - July 2003

6. By Andrew W Cook, William Cabot and Paul L Miller; The mixing transition in Rayleigh–

Taylor instability; J. Fluid Mech. (2004), vol. 511, pp. 333–362.

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 – 2013

9.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 αAgt2 (Anuchina et al. 1978; Youngs 1984), where α is adimensionless coefficient, A ≡ (ρ2 − ρ1)/(ρ2 + ρ1) is the Atwood number (with ρ1and ρ2 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 𝜌𝐴 and 𝜌𝐵(𝜌𝐴 > 𝜌𝐵)is the simplest case and its linear growth has beenwidely analyzed.[1] The heavier uid is superposedover the lighter uid in a gravitational _eld, −𝑔^𝑦,where 𝑔 is the acceleration.

The initial growth of

a single-mode perturbation of wavelength 𝜆 is ex-

ponential, 𝑑2𝜁/𝑑𝑡2 = 𝛾2𝜁 with 𝛾2 = 𝐴𝑇 𝑘𝑔, where𝑘 = 2𝜋/𝜆 is the wave number, 𝜁 is the amplitude and𝐴𝑇 = (𝜌𝐴 − 𝜌𝐵)/(𝜌𝐴 + 𝜌𝐵) is the Atwood number.

We consider the Rayleigh-Taylor instability [14] which occurs when an initially perturbed

interface between a heavier

uid which is on top of a lighter

uid is allowed to grow under

the in

uence of gravity. The

ngers of lighter

uid penetrate the heavier

uid in what are

referred to as `bubbles', while `spikes' of heavier

uid move into the lighter

uid, as shown in

Figure 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 in

size relative to other bubbles (spikes) which grow and overtake them.

Linear and non-linear growth of the mixing

zone

Generally, the growth of the RTI mixing zone, evolving from an interface

which is initially flat apart from infinetesimal disturbances, between viscous and

miscible fluids can be described by a linear and nonlinear theory as the flow becomes

’rapidly’ turbulent.

1. A good summary of the linear theory can be found in Chandrasekhar.

• The initial growth would be linear with viscosity setting the maximal

growth rate λm to length scale of the order of(ν2/(Ag)1/3, the associated

time scale is (ν/A2g2)1/3 where A is the Atwood number and ν the mean

kinematic viscosity:

A = ρ1 − ρ2

ρ1 + ρ2

ν = μ1 + μ2

ρ1 + ρ2

• When a perturbation characteristics for λm reaches the amplitude of

0.5 λm, its growth rate slows down and larger structures appear. When

the dominant wavelength exceeds about 10λm the memory of the initial

conditions is lost.

Note: For the flow considered in this paper these scales correspond to a

length scale of the order of 1mm and time scale of 0.1s. That means that

this linear growth appears in the very earlier stages of the experiment.

2. Due to the length of the dominant wave growing, viscosity now has little

effect on the growth of large scale structure. This leads to the result that the

width of the mixing region between the two layers only depends on ρ1, ρ2, g

and time t. Then for A 1 (as in the experiments), the penetration of the

lower (upper) layer into the upper (lower) layer of the tank should follow.

h = αAgt2 (1)

where α is a dimensionless constant. It is convenient then to use this result

to determine a time scale for the flow given by T = (H/gA)1/2

(≈ 5s in our experiments). By non-dimensionalizing the penetration h by

the depth of the tank H, we obtain

δ = ατ2

with τ = t/T .

9

The equation h = αAgt2 results of a self-similar behaviour of the flow where

all memory of the initial conditions has been lost, and the only relevant scale is

gt2. While this suggests that α is a universal constant, experiments and numerical

simulations have produced differing values.

In the next section we will investigate the motion of the flow in the tank and see

that, due to the presence of a plume introduced by the removal of barrier, the

growth rate of the mixing zone doesn’t fit exactly to (1).

The three-dimensional RTI between two incompressible and immiscible fluids was studied using

a phase field. During

the 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 Kelvin–Helmholtz

instability. As a result, we

observed the two-layer roll-up phenomenon of the heavy fluid, which does not occur in the two-

dimensional case. And we

studied the positions of the bubble front, spike tip, and saddle point to investigate the effect of

the Reynolds and Atwood

numbers on the three-dimensional RTI. We showed that a decrease in the Reynolds or Atwood

numbers delays the development

of the RTI. Note that the three-dimensional RTI exhibits a stronger dependence on the Atwood

number than on the

Reynolds number. Finally, owing to the pressure boundary treatment, we were able to perform

long time three-dimensional

simulation resulting in an equilibrium state.

The effect of the Atwood number

To study the effect of the Atwood number on the three-dimensional RTI, we perform numerical

simulations at a number

of Atwood numbers with the Reynolds number fixed at 1024. We use h = 1/128 (a 128×128×512

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 of

the spike tip. This means that the interface grows nearly symmetrically up and down. However,

as the Atwood number

increases, the position of the spike tip changes much more quickly than that of the bubble front

and the symmetry of the

initial structure becomes lost. From the results of Fig. 6, we see that the three-dimensional RTI

exhibits a strong dependence

on the Atwood number.

Extensive analysis of Rayleigh–Taylor instability with linear

inviscid theory can be found in the book of Chandrasekar (1961).

Flow of two fluids in our case is isothermal and incompressible and

is simulated in two dimensions. Beside density ratio and viscosity

ratio, a dimensionless parameter, known as the Atwood number

At = (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 ı < 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 Pa·s,respectively. The corresponding Atwood number is At = 0.5. Thedevelopment of Rayleigh–Taylor 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 Rayleigh–Taylor 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.