1 Introduction

Rayleigh-Taylor instability (RTI) happens when a light fluid supports a heavy fluid in a gravity field^[1-2]. RTI has been extensively studied analytically,numerically,and experimentally due to its importance in geophysics^[3-5],astrophysics^[6],and inertial confinement fusion^[7-9]. The linear RTI of viscous fluids was first studied by Harrison^[10],developed by Bellman and Pennington^[11] and Chandrasekhar^[12],summarized by Chandrasekhar^[13],and extended to stratified fluids^[14-16] and fixed boundary conditions^[17]. The implicit dispersion relation of the viscous RTI was simplified by Bellman and Pennington^[11] to obtain an approximate but analytical solution of the linear growth rate,where the viscous effect was shown explicitly. Recently,the approximate solution of viscous RTI has been widely used in numerical and theoretical studies of the accel-decel-accel profile^[18],multiphase flow^[19],and the Knudsen-number dependence^[20] because of its concept clarity and ease of use. However,it has been illustrated that Bellman's approximate solution deviated from the exact value has a maximum error of 12$\%$^[21-24]. In addition,though the dispersion relation for fluids with the same viscosity was verified with the Lattice Boltzmann simulations^[25-26],clear discrepancies between the molecular dynamics simulations and the linear theory for different viscosities were observed at the low wave number region^[27] and at high wave numbers^[28].

The approximate analytical dispersion relation of viscous RTI including the concentration-diffusion effect was offered by Duff et al.^[29] in the whole wave number range for different viscosities,Schmidt numbers ($Sc$),and Atwood numbers ($A$). Later on,the implicit dispersion relation for fluids with the same viscosity and small $A$ was studied by Batchelor and Nitsche^[30] with the Fourier transform method and by Kurowski et al.^[31] with the normal mode method. The approximate model of Duff et al.^[29] has been widely used in the studies of the Richtmyer-Meshkov instability^[32],the compressible RTI^[33],the supercritical pure fluids[34-35], the direct numerical simulation models[36-37],and the late-time growth of single mode^[38]. It should be noted that Duff's solution inherits the error (about 12$\%$) from Bellman's approximate result. The main motivation of this paper is to propose a new explicit solution of the implicit dispersion relation with better accuracy and verify it for fluids with different viscosities in the whole wave number range.

2 Theoretical and numerical results 2.1 Theoretical result of viscous RTI

We consider the classical RTI,in which a heavy fluid of the density $\rho_{2}$ lies on a light fluid of the density $\rho_{1}$ in a gravity field of $ {g}$ pointing to the opposite direction of the $y$-coordinate. The $x$-coordinate lies on the undisturbed interface. The two fluids are regarded as incompressible and of infinite extent. The control equations of this system are given by

(1)

(2)

where $\rho$ and $\nu$ are the fluid density and the kinematic viscosity,respectively. The undisturbed density and dynamic viscosity profiles are assumed to be

(3)

(4)

where the subscripts 1 and 2 correspond to Fluid 1 and Fluid 2, respectively. $\mathrm {sgn}(y)=-1$ at $y<0$,$\mathrm {sgn}(y)=0$ at $y=0$,and $\mathrm {sgn}(y)=1$ at $y>0$.

After introducing the normal mode $ {u}'=\hat{ {u}}(y)\mathrm e^{\mathrm ikx+nt}$,the implicit dispersion relation is obtained as follows:

(5)

(6)

where $k$ and $n$ are the wave number and the linear growth rate of normal mode,respectively. $A=\frac{\rho_{2}-\rho_{1}}{\rho_{2}+\rho_{1}}$ is the Atwood number. By assuming $\rho_{1}n\ll\mu_{1}k^2$ and $\rho_{2}n\ll\mu_{2}k^2$,Bellman and Pennington^[11] simplified the above equation to an explicit manner and obtained its analytical solution $N$ (an approximate value of $n$),

(7)

(8)

where $\nu=(\mu_{1}+\mu_{2})/(\rho_{1}+\rho_{2})$ is the density-weighted average kinematic viscosity. $N$ is an upper bound of the positive root of Eq.(5) for arbitrary $A$ and $\nu$. The viscosity tends to erase the velocity gradients. Hence,at small $k$ or large wavelength,the viscous effect is weak,and the solution is close to the inviscid value $(gkA)^{1/2}$. The perturbations at high wave numbers are suppressed by the viscosity. Hence,Bellman's assumptions are proper,and the growth rate decreases with $k$. It is found that $N$ is a good approximation of $n$ at low and high wave numbers,but leads to a maximum error of 12$\%$ near the maximum growth rate^[21-24].

In order to improve the accuracy of this solution,the contribution of $n$ to $M$ is not ignored completely but estimated by replacing $n$ with $N$. Then,we get

(9)

(10)

The analytical positive root of this explicit equation is

(11)

(12)

2-2 Numerical result of viscous RTI

The two-dimensional incompressible Navier-Stokes equations are solved by a projection method in Cartesian coordinates to examine this new solution. The volume of fluid and the multi-grid method are used to track the interface and solve the pressure Poisson equation, respectively^[39-40]. In all the simulations,the width of the domain is $L$,and the height of the domain is $H(H=4L)$. The uniform mesh $1\;024\times4\;096$ is found to be fine enough to calculate the initial growth stage. The periodic boundary conditions are used in the spanwise direction,and the no-slip conditions are imposed at the top and bottom boundaries. RTI is seeded by adding a one-wavelength linear inviscid mode around the interface lying at the mid-height of the domain.

Though the implicit dispersion relation (see Eq.(5)) can be solved numerically now,Bellman's explicit solution is more convenient to use and has been applied recently to validate the numerical schemes and results^[18-19]. Furthermore,the analytical manner of Bellman's solution is easy to include more physical processes. For example,Duff et al.^[29] extended Bellman's solution to consider the diffusion effect,and their solution has been widely used to validate RTI-related numerical simulations^{[20, 37-38]},though it inherits the approximation error from Bellman's solution. Therefore,the accuracy of the proposed solution and Bellman's solution will be compared next.

In order to compare the dispersion relations more conveniently,the parameters are non-dimensionalized by the characteristic length and time scales $(\nu^{2}/g)^{1/3}$ and $(\nu/g^{2})^{1/3}$. The dimensionless wave number and growth rate are

(13)

(14)

The unstable interface will penetrate into the light fluid and form spikes. Figure 1(a) shows the time evolution of the height of the spike or the penetration. Initially,the spike grows exponentially as $\mathrm e^{n^*t^*}$. Then,the growth rate becomes less than that for the linear solution due to the nonlinear effect. For convenience of comparison,the exact solution of the implicit dispersion relation (see Eq.(5)) is solved numerically and shown by dashed line in Fig. 1. It is shown that at the initial stage, the present solution agrees very well with the numerical simulation results and the exact solution,while Bellman's approximate solution overestimates the growth rate.

It is shown in Fig. 1(b) that the previous approximate solution is consistent with the simulation data and the exact solution at low and high wave numbers,but has a maximum error around the maximum growth rate. On the contrary,the present solution almost coincides with the exact solution in the whole wave number range.

The effects of the Atwood number and the viscosity ratio on the growth rate are shown in Fig. 2. Larger $A$ leads to larger growth rate,the same trend as predicted by the inviscid theory. It is illustrated that at different viscosity ratios and Atwood numbers, Eq.(8) has a maximum error (about 12$\%$) near the maximum growth rate,while the present analytical solutions (see Eq.(11)) are consistent with the numerical simulations in the whole wave number range with a relative error less than 1$\%$. In addition,it is the first time to the authors' knowledge that the dispersion relation of RTI for fluids with different viscosities is verified successfully.

2.3 Theoretical result of viscous RTI with diffusion effect

When the concentration-diffusion effect is included,Duff et al.^[29] developed an approximate theory based on Eq.(8) by assuming that $\nu\epsilon$ was a high-order small term and then the viscous and diffusive effects were independent,where $\epsilon=2(Dt)^{\frac{1}{2}}$,a length scale at the diffusivity $D$ and the elapsed time $t$. The diffusion effect includes two parts: (i) the dynamic diffusion effect which broadens the density transition zone between the fluids,decreases the effective Atwood number,and then reduces the linear growth rate; and (ii) the static diffusion effect which reduces the amplitude of the mean density perturbation. Based on the above ideas,Duff et al.^[29] derived the following approximate growth rate:

(15)

where $\psi$ is a function of $a$ and $A$,reflecting the dynamic diffusion effect. $-D k^{2}$ represents the effect of static diffusion. Obviously,this solution inherits the maximum error (about 12$\%$) from Bellman's solution $N$. Therefore,it is convenient to improve its accuracy by replacing the contribution of the viscous part with the proposed solution as follows:

(16)

It can be seen from Eqs.(15) and (16) that both the momentum diffusion (viscosity) and the concentration diffusion tend to stabilize RTI. The coupled Navier-Stokes equations and the concentration equation are solved with a pseudo-spectral solver (SIMSON)^[41] at different Schmidt numbers ($\nu/D$),and it is confirmed that the growth rates predicted by Eq.(16) agree with the simulation results better than those of Eq.(15),especially near the most unstable modes.

3 Conclusions

} The implicit dispersion relation of viscous RTI is solved with an approximate method to obtain an analytical solution of the growth rate,which is at least one order of magnitude more accurate than Bellman's solution. According to the numerical simulations,the maximum relative error is reduced from 12$\%$ to less than 1$\%$. In addition,for fluids with different viscosities,the dispersion relation is verified in the whole wave number range with numerical simulations. This solution is also successfully extended to include the concentration-diffusion effect. Because of their explicit manner and ease of use,these solutions are expected to be used to validate numerical schemes and to include more effects,e.g.,the electromagnetic effect and the thermal diffusion.