# Compressibility effects on the Rayleigh-Taylor instability growth between immiscible fluids

###### Abstract

The linearized Navier-Stokes equations for a system of superposed immiscible compressible ideal fluids are analyzed. The results of the analysis reconcile the stabilizing and destabilizing effects of compressibility reported in the literature. It is shown that the growth rate obtained for an inviscid, compressible flow in an infinite domain is bounded by the growth rates obtained for the corresponding incompressible flows with uniform and exponentially varying density. As the equilibrium pressure at the interface increases (less compressible flow), increases towards the uniform density result, while as the ratio of specific heats increases (less compressible fluid), decreases towards the exponentially varying density incompressible flow result. This remains valid in the presence of surface tension or for viscous fluids and the validity of the results is also discussed for finite size domains. The critical wavenumber imposed by the presence of surface tension is unaffected by compressibility. However, the results show that the surface tension modifies the sensitivity of the growth rate to a differential change in for the lower and upper fluids. For the viscous case, the linearized equations are solved numerically for different values of and . It is found that the largest differences compared with the incompressible cases are obtained at small Atwood numbers. The most unstable mode for the compressible case is also bounded by the most unstable modes corresponding to the two limiting incompressible cases.

###### pacs:

47.20.-k,47.20.Bp## I Introduction

The Rayleigh-Taylor instability, which occurs due to the gravitational instability of a heavy fluid overlying a lighter fluid, CH81 ; Taylor50 is of fundamental importance in a multitude of applications ranging from the turbulent mixing in Inertial Confinement FusionK94 ; Knauer00 to astrophysical phenomena. R94 ; FL00 Small perturbations of the interface between the two fluids grow to large amplitudes. At early times, for small enough initial perturbations, the flow can be described by the linearized equations and the amplitude grows exponentially. Later, the interface evolves into bubbles of lighter fluid and spikes of heavier fluid penetrating the opposed fluid. If the initial interface is randomly perturbed then bubbles and spikes of different sizes are generated. Mathematically, the Rayleigh-Taylor instability is an ill-posed problem and the dependence on initial conditions is still an interesting question. Sharp84

The theory for the linear stage for incompressible fluids agrees well with the experiments. CH81 ; WNJ01 In the absence of surface tension and viscosity, the growth rate increases indefinitely with the wavenumber. This trend is changed by the presence of viscosity, in which case the growth rate has a peak value and decreases towards zero for large wavenumbers. On the other hand, the presence of surface tension stabilizes perturbations with the wavenumber larger than a critical value.

The role of compressibility on the development of the Rayleigh-Taylor instability between inviscid fluids has been studied by several authors, Sharp84 ; Vandervort61 ; PP82 ; Baker83 ; BB83 ; LP89 ; YZ93 ; Amala95 ; Turner02 however its effect compared with the incompressible case is still under debate. The earlier studies of the linear stage dealing with ideal fluids Vandervort61 ; PP82 introduced simplifying assumptions and are strictly valid only when . Bernstein and Book BB83 and Turner Turner02 removed these assumptions and studied the effects of compressibility as a function of . They found that these effects are more important at small wavenumbers and the rate of growth increases as decreases. They concluded that compressibility has a destabilizing effect. The same conclusion is obtained for a multilayer system by Yang and Zhang YZ93 by comparing the compressible growth rate with that corresponding to an incompressible system obtained as . The increase in the growth rate as decreases can also be explained using the energy principle, as a special case of the comparison theorem in the calculus of variations. Newcomb83

On the other hand, Sharp Sharp84 finds a stabilizing effect of compressibility. Moreover, numerical results for late time growth seem to indicate that (the constant of proportionality in the quadratic law for the rate of growth) increases with the speed of sound, Li93 so that compressibility would have a stabilizing effect. Baker Baker83 found both stabilizing and destabilizing effects of compressibility in the linear regime, however his results were based on previously derived formulas using different assumptions than ideal gas. The role of compressibility on the instability growth is thus not yet settled. Moreover, to the best of our knowledge no study of the effects of compressibility in the linear regime for viscous fluids has been performed. Additionally, there is no systematic study of the effects of surface tension and finite size domain in the compressible case.

In this paper we resolve the apparent contradiction between the stabilizing and destabilizing character of compressibility for ideal fluids. We show that compressibility can be characterized by two parameters, and the speed of sound, with opposing influence on the instability growth. Moreover, as or the speed of sound (varied by changing the equilibrium pressure at the interface while keeping the interface equilibrium density constant) increase to , the limiting incompressible flows (and the bases of comparison) are different. The compressible growth rate is bounded by the growth rates obtained for these two incompressible flows which have exponentially varying and constant density, respectively.

A physical explanation for the decrease in the growth rate as decreases can be formulated based on the influence of on the local Atwood number. As the interface develops, the heavier fluid reaches regions of larger and larger densities of the lighter fluid while the lighter fluid reaches regions of smaller densities of the heavier fluid. The local Atwood number in these regions away from the initial position of the interface depends on , since the equilibrium density profile depends on . Using the equilibrium density provided below, it can be shown that the local Atwood number is lower for the points on the interface above the initial position, while it is higher below, compared to a system at higher (less compressible). However, the decrease in leads to a larger change in the local Atwood number above the initial position of the interface. The overall effect would be a decrease in the average Atwood number, thus offering an intuitive argument for the decrease of the growth rate with decreasing . Nevertheless this argument might break for small domain sizes and the validity of the results for finite size domains is discussed. The influence of surface tension and viscosity on the growth rate are also considered.

## Ii Linearized Equations

The case of two superposed compressible ideal fluids separated by an interface at is considered. The fluids are subject to a constant gravitational acceleration . For each fluid, the motion is governed by the continuity, momentum transport, and energy transport equations

(1) |

(2) |

(3) |

where is the density, the velocity in direction, the pressure, the specific internal energy, and the temperature. The viscous stress is assumed Newtonian,

(4) |

### ii.1 “0”-order equations

The two fluids are assumed initially at rest and the primary variables are written as small perturbations around the equilibrium (“0”) state, with . Then the governing equations reduce to

(5) | |||||

(6) | |||||

(7) |

Moreover, the “0”-order variables are assumed to be in steady state, so that and and are continuous across the interface. For infinite domain or finite size domain in direction with adiabatic walls, the energy equation yields . Consequently, the equilibrium state is given by

(8) | |||||

(9) | |||||

(10) |

where is the unperturbed pressure at the interface () and denotes the material or , with material chosen to be above material .

### ii.2 First-order equations

The interface between the two fluids is perturbed with an and dependent perturbation. The location of the interface is described by the function , with . Moreover, a surface tension is added at the interface between the two fluids. The first order linearized equations are obtained as

(11) |

(12) | |||||

(13) |

(14) |

(15) |

where is the dilatation and denotes . Equations 11-15 should be supplemented by boundary conditions and jump conditions across the interface. These conditions will be discussed at length in the next sections.

By examining equations 11-15 it should be noted that in the absence of heat diffusion, the limit of incompressible flow () can be obtained either by letting (as considered by the previous authors) or by letting (note that since the equilibrium density at the interface should not be affected by the change in pressure, the equation of state implies in this case). In the latter case the equilibrium density becomes , while in the former the exponential variation is still allowed. Therefore the incompressible limit (and the base of comparison for the rate of growth) is different in the two cases. In the next sections and are considered as independent parameters, both affecting the compressibility of the flow. The inviscid, infinite domain and no surface tension case will be examined first. Then the influence of finite size domain, surface tension and viscosity on the results obtained will be investigated. The non-zero heat diffusivity case was also considered, but the results were very close to the non-diffusive case and are not presented here.

## Iii Inviscid case

In general, the linearized equations do not admit analytical solutions. However, it is possible to obtain an analytical solution in the absence of viscosity and heat diffusion. Following the usual approach (e.g. Ref. CH81 ), we seek solutions whose dependence on , , and time have the form

(16) |

(17) | |||||

(18) | |||||

(19) | |||||

(20) | |||||

(21) |

where and . After eliminating , , , and from the above equations, an equation for is obtained as

(22) | |||||

where is the speed of sound. The jump condition at the interface can be obtained by integrating equation 22 over an infinitesimal element of which includes the interface

(23) |

where , with , , is the jump of a quantity f across the interface. The subscript denotes the value which a quantity, continuous at the interface, takes at .

On each side of the interface, and 22 becomes are constant, so that equation

(24) |

with the solution of the form , where

(25) |

Formula 25 for was also obtained by Amala. Amala95 The coefficients and can be determined to a multiplying constant from the conditions that vanishes at the rigid boundaries located at and and that it is continuous over the interface. After replacing in the jump condition 23, a dispersion relation can be obtained as

(26) | |||||

where the nondimensional quantities are defined as , , , ,

Relation 26 represents a generalization of the dispersion relations obtained by previous authors. It includes the effect of a finite size domain and surface tension, and makes no isentropic assumption. For the dispersion relation becomes

(27) |

As explained above, the incompressible flow limit () can be obtained either by letting or . As , the flow approaches incompressible flow with constant density, for which the nondimensional rate of growth is given by

(28) |

which for infinite domain becomes . CH81 On the other hand, as , the equilibrium flow has still exponential varying density, with the dispersion relation

(29) | |||||

where

(30) |

and it is easy to show that for and as .

In the dispersion formula for the compressible case (equation 26) the decrease in is equivalent to an increase in at constant. Thus, for the compressible case, represents a measure of the compressibility effects on the rate of growth. On the other hand, is proportional to the ratio between the wavelength of the initial perturbation and the density exponential change length scale. For small values of , this lengthscale is much larger than the wavelength of the initial perturbation, and the rate of growth approaches the incompressible, constant density result.

For infinite domain, an approximate relation for , valid to order for small values of is

(31) | |||||

For the relation derived by Plesset and Prosperetti PP82 is recovered. In general, for , it can be shown that relation 31 implies for any combination of parameters, with the constraint . Moreover, Fig. 1 shows that decreases as decreases. It should be noted that the same effects of decreasing on the nondimensional growth rate can be obtained either by increasing or decreasing , with all other parameters kept constant. In other words, compressibility effects are more important at larger values of and lower wavenumbers, which is supported by the numerical solutions of the dispersion relation presented in Fig. 1.

For large and no surface tension, it follows from the dispersion relation that

(32) |

A further reduction in the rate of growth can be obtained by increasing the adiabatic exponents. At the limit, when , the nondimensional rate of growth becomes

(33) |

which is the same as that obtained from formula 30 for large by Bernstein and Book BB83 and Turner. Turner02 In general, for finite values of it can be shown BB83 ; Turner02 that increasing the ratio of specific heats leads to a decrease in the rate of growth, also supported by Fig. 1. However, the rate of growth has different sensitivities to the change of and . Thus, as Fig. 1 shows, the change in the ratio of specific heats of the lower fluid leads to a larger change of . Therefore, the rate of growth is more sensitive to the change in compressibility of the lower fluid. However, as Fig. 1 indicates, these results are also sensitive to the value of the Atwood number. For large values of the Atwood number the results obtained for the compressible cases show little sensitivity to changes in the ratios of specific heats and the rate of growth obtained for the compressible case is close to the incompressible variable density result. Nevertheless, at small Atwood numbers and for large , the values of and become important in determining the rate of growth. In addition, the relative difference between the growth rates obtained for the two limiting incompressible flows increases as the Atwood number decreases, so that compressibility effects are larger at small Atwood numbers.

In conclusion, the instability growth rate for a compressible flow () in the inviscid, infinite domain and no surface tension case is bounded by the growth rates of the corresponding incompressible flows obtained for uniform () and exponentially varying density (), so that . As increases (so that the flow becomes less compressible), increases towards , while as increases (so that the fluid becomes less compressible), decreases towards .

### iii.1 Influence of Finite Size Domain

For the case in which the domain is bounded by rigid boundaries located at and , the growth rates obtained for constant density incompressible flow and incompressible flow with exponentially varying density are given by equations 28 and 29, respectively. Figure 2(a) presents the growth rate as a function of the nondimensional parameter for different domain sizes. For domain sizes not very small compared to the wavelength of the perturbation, the nondimensional growth rate is still bounded by and and the rate of growth decreases as the domain size decreases. However, similar to the incompressible case, the decrease in is less significant when is decreased. Therefore, for the growth rate varies more when is changed, so it is more sensitive to the change in compressibility.

For the extreme case when (domain size small compared to the wavelength of the initial perturbation) and it is possible, as Fig. 2(b) shows, that the compressible growth rate becomes larger than the constant density incompressible growth rate for smaller than a critical value. Numerical solutions of the dispersion relation 26 for a large range of parameters indicate that the curve can intersect only once the line . Therefore, an analytical condition for the existence of the overshoot can be found by letting (in which case the dispersion relation simplifies considerably) and imposing . After some algebra one obtains

(34) |

where varies from to as increases from to . Consistent with the numerical results, condition 34 can be fulfilled only for small values of the domain size of the upper fluid and ratio of specific heats close to for the lower fluid.

### iii.2 Influence of Surface Tension

The presence of surface tension tends to inhibit the growth of the instability. Moreover, for the incompressible case there is a critical wavenumber , CH81 so that the arrangement is stable for . By imposing in the dispersion relation 26 it can be seen that the critical wavenumber remains the same as in the incompressible case. However, for , the wavenumber appears as an explicit parameter in the dispersion relation 26 so the variations of and are no longer equivalent. Nevertheless, at each , the nondimensional compressible rate of growth obtained for infinite domain is still bounded by and , as Fig. 3 shows. However, the lower limit is approached differently as or increase to . Thus, the variation in the compressibility of the lower fluid is more important at lower wavenumbers, while the variation in the compressibility of the upper fluid is more important at higher wavenumbers.

## Iv Effect of Viscosity

Consider the case of two viscous fluids, bounded by two rigid surfaces at and . Following the previous procedure, the linearized equations can be reduced to a single fourth order ordinary differential equation in of the form

(35) |

where the coefficients are given in the Appendix for the compressible case and the two incompressible limiting cases. However, only for the uniform density incompressible case this equation has constant coefficients and an easily derived analytical solution. The boundary conditions for equation 35 are at and , and tangential viscous stresses continuous at the interface, and a jump condition found from the integration of the governing equation over the interface. For the compressible case, the condition that tangential velocities vanish at the rigid boundary can be written as at and , while the continuity of and at the interface leads to the continuity of . The divergence of the velocity fluctuations is given by

(36) |

with the coefficients given in the Appendix. The continuity of the tangential viscous stress over the interface can be written as

(37) |

An expression for in terms of the derivatives of is provided in the Appendix. The jump condition at the interface can be found by eliminating , , and from the momentum equation and integrating the resulting equation over an infinitesimal element of which includes the interface

(38) | |||||

For condition 38 reduces to the condition derived for the incompressible case in Ref. CH81 In the Appendix an expression for is provided. Since can be found only to a multiplying constant, the boundary conditions are supplemented with the specification of or one of its derivatives at one point inside the domain. Then equation 35 together with the boundary conditions described above form a closed set of equations from which on each side of the interface and the rate of growth can be determined. For all cases considered and are large compared to the wavelength of the initial perturbation so the configuration is close to the infinite domain case. Equation 35 was integrated on each side of the domain using a fourth order Runge-Kutta scheme. In order to determine and from the matching conditions at the interface, a multidimensional secant method (Broyden’s method) was employed.

Figure 4 presents numerical solutions of the viscous linearized equations for different Atwood numbers. Consistent with the previous results, the compressible rate of growth is bounded by the incompressible rates of growth obtained for uniform density and exponentially varying density cases. Moreover, its behavior is similar to the well known incompressible constant density result. It has a peak at some critical wavenumber and decreases towards zero as the wavenumber becomes large. However, the location of the peak is different compared to the incompressible case, with the highest difference at small Atwood numbers. Here we should note a qualitative difference between the results obtained for the constant density incompressible case and those obtained for the compressible and variable density incompressible cases. At small Atwood numbers, the critical wavenumber decreases with Atwood number for the constant density incompressible case, while it increases for the other two cases. Moreover, as shown in the previous sections, the largest differences in the inviscid rate of growth compared to the constant density incompressible case are obtained at small wavenumbers and small Atwood numbers. It is expected then that, for the viscous case at small Atwood numbers, the relative difference between the rate of growth obtained for the compressible and incompressible cases considered will be largest, also confirmed by Figs. 4(a) and 4(b). Although this difference should become very small as approaches zero, Fig. 4(b) shows that, for small Atwood numbers, it persists at smaller values of .

## V Conclusions

The effects of compressibility on the growth rate of Rayleigh-Taylor instability between two immiscible ideal fluids are examined in the linear regime. The results distinguish between the stabilizing and destabilizing character of compressibility. For infinite domains, the growth rate obtained for the compressible case is bounded by the growth rates obtained for the corresponding incompressible flows with constant and exponentially varying density, and this result is not affected by the presence of surface tension or viscosity. For ideal gases with zero heat diffusivity, the limiting incompressible flow (defined by ) can be attained either by increasing the ratio of specific heats, , or the speed of sound (varied by changing the equilibrium pressure at the interface at constant equilibrium density at the interface). The equilibrium density distribution for the limiting incompressible flow is different in the two cases. Moreover, the two parameters have opposing influence on the rate of growth. As the speed of sound is increased, the rate of growth increases towards the value obtained for the corresponding constant density incompressible flow, while as increases, decreases towards the value obtained for the corresponding incompressible flow with exponentially varying density. The presence of heat diffusion was also considered, but the results were very close to those obtained for the nondiffusive case and were not presented here.

The equilibrium density for a compressible flow varies exponentially with and depends on . Therefore, the local Atwood number changes as the interface moves away from the original position. Compared to a flow with a higher value of (less compressible), the local Atwood number decreases for the points on the interface situated above the initial position, while it increases for the points on the interface situated below the initial position. However, the change in the local Atwood number is larger above the initial position of the interface, so that the overall effect would be a decrease of the average local Atwood number. This offers an intuitive argument for the decrease of the growth rate as decreases. Moreover, this argument suggests that the bubble velocity decreases, while the spike velocity increases for more compressible flows. On the other hand, as decreases the fluids are more compressible, however the equilibrium density and pressure do not change. Therefore, as the heavier fluid moves towards regions of higher pressures, its volume decreases and the volume change is larger for more compressible fluids, so that the spike velocity decreases. Similarly, for more compressible fluids the bubble velocity increases. If the two fluids have different values for , it is shown that the growth rate is more sensitive to the change in the ratio of the specific heats of the lower fluid. However, at large Atwood numbers the rate of growth is little influenced by the values of and and becomes the main compressibility parameter. In addition, it is shown that compressibility effects are more important at small Atwood numbers.

For domains bounded by rigid surfaces, the compressible growth rate is still bounded by the two incompressible growth rates described above, except for the extreme case when the domain size of the upper fluid is small compared to the wavelength of the initial perturbation and for the lower fluid. In this case, the compressible growth rate can become larger than the growth rate obtained for the corresponding constant density incompressible flow for values of the compressibility parameter smaller than a critical value. An analytical condition for the existence of this overshoot is provided. In general, the results show that the compressible growth rate varies more when the rigid boundary of the lower fluid is closer to the interface than the rigid boundary of the upper fluid, so that it is more sensitive to the change in compressibility.

The presence of surface tension tends to inhibit the growth rate of the instability and for the incompressible case there is a critical wavenumber above which the configuration becomes stable. It is shown that the value of this critical wavenumber is not affected by compressibility. For wavenumbers below this critical value the general result presented above remains valid. However, the presence of surface tension modifies the sensitivity of the growth rate to a differential change in the value of for the two fluids. At smaller wavenumbers, the change in for the lower fluid is more important for the variation of , while the opposite holds true at higher wavenumbers.

Numerical solutions of the linearized equations show that for viscous compressible fluids, the growth rate behaves in a manner analogous to the incompressible growth rate. It has a most unstable wavenumber and decreases towards zero at larger wavenumbers. Moreover, both the growth rate and the most unstable mode are bounded by the values obtained for the corresponding constant and variable density incompressible flows. For the constant density incompressible flow it is known that the most unstable mode moves to small wavenumbers as the Atwood number is decreased. The inviscid results presented in this paper show that the effects of compressibility are more important at small wavenumbers and small Atwood numbers. Consistent with these results, it is found that for viscous fluids compressibility becomes more important at small Atwood numbers. For small enough Atwood numbers, the difference between the compressible and incompressible growth rates will remain sizable at larger values of the equilibrium pressure.

An interesting question raised by the results presented in this paper is if they remain valid in the nonlinear regime for single and/or multimode initial perturbation. Our preliminary numerical results seem to indicate a similar influence of and on the growth rate (and on the spike and bubble velocities) to that found in the linear regime. Moreover, even for large values of the equilibrium pressure so that the early time results are close to the incompressible flow results, the late time spike and bubble velocities become different than in the incompressible case. Another interesting question is about the range of the amplitudes of the perturbation for which the growth rate agrees with the linear theory prediction. Again, our preliminary numerical results seem to indicate that the range of validity of the linear assumption remains approximately the same as in the incompressible case. These results will be published elsewhere.

This study was concerned with the effects of compressibility on the instability growth between immiscible fluids with uniform equilibrium temperature. It does not cover many of the configurations of practical interest, for example the presence of an equilibrium temperature gradient, a more general equation of state or diffuse interfaces, which might be important in certain applications. DHH62 ; K94 However this study offers a systematic approach for examining the effects of compressibility which could represent a starting point for analyzing different or more complex configurations.

###### Acknowledgements.

The author would like to thank Dr. Timothy T. Clark and Dr. Francis H. Harlow for their advice on the preparation of this article and many helpful and insightful discussions. Computational resources were provided through the Institutional Computing Project, Los Alamos National Laboratory. This work was supported by the U.S. Department of Energy.## Appendix A Equations for the viscous case

Following the usual procedure for the incompressible, constant density case, the variables are nondimensionalized using as time scale and as lengthscale. The compressibility parameter is defined by . For simplicity the kinematic viscous coefficient is considered continuous over the interface, so that , with and constant on each side of the interface. The value of the kinematic viscous coefficient at the interface is denoted by .

The scaled equations for and on each side of the interface can be written as

(39) | |||||

(40) |

where the coefficients (with the index denoting the side of the interface suppressed for simplicity) are given by

(41) | |||||

(42) | |||||

(43) | |||||

(44) | |||||

(45) | |||||

(46) | |||||

(47) | |||||

(48) | |||||

(49) |

(50) | |||||

(51) | |||||

(52) | |||||

(53) | |||||