2.1. Governing equations

DNS has been performed on the RTI of cylindrical geometry for verifying the theoretical model and analysing the relevant behaviours. Considering the RTI of cylindrical geometry described by the cylindrical coordinates ($r, \theta$), the initial pressure $p_I$ and density $\rho _I=(\rho _h+\rho _l)/2$ at the heavier/lighter fluid interface are chosen as the characteristic scales, where $\rho _h$ and $\rho _l$ are the heavier and lighter fluid density. Then the characteristic velocity and temperature are described as $U_I=\sqrt {p_I/\rho _I}$ and $T_I=p_I/(R\rho _I)$ with the perfect gas constant, $R$, respectively. The radius of the unperturbed interface, $r_0$, is used as the characteristic length. Thus, the non-dimensionalized governing equations are given as

(2.1)\begin{gather} \frac{\partial \rho}{\partial t}+\boldsymbol{\nabla} \boldsymbol{\cdot}(\rho \boldsymbol{u})=0, \end{gather}

(2.2)\begin{gather}\frac{\partial (\rho \boldsymbol{u})}{\partial t}+\boldsymbol{\nabla} \boldsymbol{\cdot}(\rho\boldsymbol{u}\boldsymbol{u})=-\boldsymbol{\nabla} p+\frac{1}{Re}\boldsymbol{\nabla} \boldsymbol{\cdot}\boldsymbol{\tau}-\frac{\rho}{Fr} \widehat{\boldsymbol{e}}_r, \end{gather}

(2.3)\begin{gather}\frac{\partial (\rho E)}{\partial t}+\boldsymbol{\nabla} \boldsymbol{\cdot}[(\rho E+p)\boldsymbol{u}]=\frac{1}{Re}\boldsymbol{\nabla} \boldsymbol{\cdot}(\boldsymbol{\tau}\boldsymbol{\cdot}\boldsymbol{u})+\frac{1}{Re Pr}\boldsymbol{\nabla} \boldsymbol{\cdot}(\boldsymbol{\nabla} T)-\frac{\rho}{Fr}\boldsymbol{u}\boldsymbol{\cdot}\widehat{\boldsymbol{e}}_r, \end{gather}

where $t$, $\rho$, $\boldsymbol {u}=[u_r, u_\theta ]$, $p$, $T$ and $E=T/(\Gamma -1)+\boldsymbol {u}\boldsymbol {\cdot }\boldsymbol {u}/2$ denote the time, density, velocity, pressure, temperature and total energy, respectively, where $\Gamma =1.4$ is the specific heat ratio. Here, $\,\widehat {\boldsymbol {e}}_r$ is the unit vector in the radial direction. The stress tensor is obtained as

(2.4)\begin{equation} \boldsymbol{\tau}=2\mu\boldsymbol{{S}}-\frac{2}{3}\mu(\boldsymbol{\nabla} \boldsymbol{\cdot}\boldsymbol{u})\boldsymbol{\delta}, \end{equation}

where $\boldsymbol{{S}}$ is the strain-rate tensor, $\mu =T^{3/2}(1+c)/(T+c)$ is the viscosity computed by the Sutherland law, with $c=110/T_r$ and the reference temperature $T_r$ and $\boldsymbol {\delta }$ is the unit tensor. The above governing equations are closed with the equation of state of ideal gas, i.e. $p=\rho T$.

The non-dimensional parameters in (2.1)–(2.3) are the Reynolds, Froude and Prandtl numbers defined, respectively, as

(2.5a–c)\begin{equation} Re=\frac{\rho_{I}U_{I}r_{0}}{\mu_{I}}, \quad Fr=\frac{U_{I}^{2}}{r_{0} g} \quad \textrm{and} \quad Pr=C_{p}\frac{\mu_{I}}{\kappa}, \end{equation}

where $g$ is the external acceleration, $C_p$ the constant-pressure specific heat and $\kappa$ the thermal conductivity.

2.2. Numerical method and validation

The present DNS of cylindrical RTI is performed based on a numerical algorithm composed of high-order accuracy schemes to solve the compressible Navier–Stokes equations in cylindrical coordinates (Li et al. Reference Li, He, Zhang and Tian2019). Specifically, the seventh-order finite difference weighted essentially non-oscillatory (WENO) scheme is implemented to discretize the convective terms (Jiang & Shu Reference Jiang and Shu1996) and the eighth-order central difference scheme to discretize the viscous terms in the governing equations (2.1)–(2.3). The time derivative is approximated by the standard third-order Runge–Kutta method.

To validate our DNS algorithm, we typically employ the multicomponent Riemann problem in cylindrical geometry (Zhang & Graham Reference Zhang and Graham1998), i.e. an incident circular shock colliding with a circular material interface. This problem also involves dynamical evolution of cylindrical interfaces, similar to the RTI problem considered here. However, it is much more complicated as the material interface evolves due to its interaction with moving shocks, and thus contains both density and pressure discontinuities. The test calculation is set following Zhang & Graham (Reference Zhang and Graham1998), namely, an incident shock with shock Mach number $1.2$ propagates inward from the air towards the SF$_6$ (sulphur hexafluoride). Figure 1(a) shows the positions of the material interface, transmitted shock and reflected shock, which agree well with those of Zhang & Graham (Reference Zhang and Graham1998). The velocity of the material interface in figure 1(b) is also in good agreement with the data of Zhang & Graham (Reference Zhang and Graham1998). The relevant extensive tests ensure that the present DNS is reliable for capturing interfacial dynamics.

Figure 1. (a) Positions of the material interface, transmitted shock and reflected shock versus time. (b) Velocity of the material interface versus time. The lines represent the results of our DNS, and the circles represent those of Zhang & Graham (Reference Zhang and Graham1998).

2.3. Problem statement and numerical results for cylindrical RTI

To avoid the instability suppression due to background stratification, the uniform density field is initialized as $\rho _h=1+A_T$ and $\rho _l=1-A_T$ on each side of the interface (Bian et al. Reference Bian, Aluie, Zhao, Zhang and Livescu2020), where the Atwood number $A_{T}=(\rho _h-\rho _l)/(\rho _h+\rho _l)$. The initial velocity field is set to be zero. The hydrostatic equilibrium requires that $\mathrm {d} p/\mathrm {d} r=-\rho _i/Fr (i=l,h)$ away from the interface, thus the pressure field can be determined by assuming $p=1$ at the interface. The initial temperature field can be obtained by the equation of state. The interface at $r_0=1$ is perturbed by the cosine wave $\eta _0(0)\cos (n \theta )$, where $\eta _0(0)$ is the initial amplitude and $n$ is the number of perturbation waves. The velocity, pressure and density at the far boundary are fixed at their initial values to ensure hydrostatic equilibrium (Hu et al. Reference Hu, Zhang, Tian, He and Li2019).

Although a uniform density field is initialized to avoid the background stratification, the dynamic compressibility of flow should be noticed. Here, we present the relevant analysis. The reference Mach number, $Ma=u_{\infty }/c_{\infty }$, is used to consider the dynamic compressibility, where the reference velocity for single-mode RTI is $u_{\infty }=\sqrt {A_T g\lambda /(1+A_T)}$ (Bian et al. Reference Bian, Aluie, Zhao, Zhang and Livescu2020) with $\lambda$ being the perturbation wavelength, and the reference sound speed is $c_\infty =\sqrt {\Gamma p_I/\rho _I}$. Thus, the reference Mach number reaches $Ma=\sqrt {A_T/(1+A_T)(2{\rm \pi} /nFr\Gamma )}$. In the present study, the Froude number is selected as $Fr=10$ for the convergent case and as $Fr=-10$ for the divergent case, the Atwood number is $A_T=0.3$ and $0.6$, the numbers of perturbation waves are $n=4$ and $8$, the Prandtl number is $Pr=0.72$ for air and the Reynolds number is $Re = 40\ 000$. Based on these parameters, the typical case with $A_T=0.6$ and $n=4$ used in our simulations has the maximum reference Mach number 0.205. Therefore, we can reasonably determine that the compressibility effect on the RTI simulations is weak.

In the present simulation, to avoid a pole singularity at the centre of cylindrical coordinates, the inner boundary radius $r_{min}=0.05$ has been used based on the previous treatment (e.g. Annamalai et al. Reference Annamalai, Parmar, Ling and Balachandar2014; Lombardini, Pullin & Meiron Reference Lombardini, Pullin and Meiron2014). To verify that the spatial resolutions are enough to predict properly the RTI evolution, we present some typical results calculated by three sets of the grid resolution, i.e. $M=500$, $750$ and $1000$ in the radial direction; $N=800$, $1200$ and $1600$ in the circumferential direction, respectively. The radial grids are set as $r_{i+1}=r_i(1+{\rm \Delta} \theta )$, where $r_{i}$ is the radial position of the $i$th grid and the width of uniform circumferential grids ${\rm \Delta} \theta =2{\rm \pi} /N$. Figure 2 shows the location of the bubble tip with different grid resolutions for a typical case $A_T=0.6$ and $n=4$. The location of the bubble is determined as the point of the bubble tip with $\rho =\rho _I$. It is seen that the results collapse together for the different grid resolutions. To make the prediction accurate, the results given in the following were calculated by $M=1000$ and $N=1600$, which are enough to properly resolve the hydrodynamic scales.

Figure 2. Location of the bubble tip for $A_T=0.6$ and $n=4$ with different grid resolutions.

The reliability of simulations for cylindrical RTI is also verified by comparing the positions of the bubble tips with the exponential growth (Wang et al. Reference Wang, Wu, Ye, Zhang and He2013; Guo et al. Reference Guo, Wang, Ye, Wu and Zhang2018) that $\eta _0=\eta _0(0)\cosh (\gamma t)$ for inviscid incompressible RTI in the linear regime. As shown in figure 3, it is seen that the numerical simulation results are in good agreement with the linear theory, indicating that the simulation results are reliable and the effects of compressibility and viscosity can be reasonably ignored. When $\gamma t>1.4$, the simulation results with $n=8$ somewhat deviate from the linear theory, which is related to the fact that the RTI begins to enter the nonlinear stage.

Figure 3. Comparisons of radial positions of bubble tips for the convergent (a) and divergent (b) cases with linear theory.

Further, figure 4 shows our numerical results on the bubble velocity behaviours for the convergent and divergent cases in the nonlinear regime. The bubble velocity is determined at the point of the bubble tip with $\rho =\rho _I$. It is revealed that the bubble velocity of the divergent case when $\gamma t\leq 4.6$ prevails, and when $\gamma t>4.6$ becomes overridden by that of the convergent case. Of particular interest, no evident saturation of bubble velocity is realized as in the planar RTI. Therefore, to understand the bubble dynamics under the cylindrical geometry effect, an analytical model for the nonlinear evolution of cylindrical RTI is presented as follows.

Figure 4. Bubble velocity growth of cylindrical RTI at $A_{T}=0.6$ obtained by DNS.

3.1. Model of nonlinear bubble evolution

We consider two inviscid, irrotational, incompressible fluids that have a cylindrical perturbed interface $r_{I}(\theta ,t)=r_{0}+\eta (\theta ,t)$, where $\eta (\theta ,t)$ is the perturbation displacement. The external acceleration is imposed as $\boldsymbol {g}=-g\widehat {\boldsymbol {e}}_{r}$ for the convergent case and as $\boldsymbol {g}=g\widehat {\boldsymbol {e}}_{r}$ for the divergent case. The velocity potentials of the lighter ($\varphi _{l}$) and heavier ($\varphi _{h}$) obey the Laplace equations

(3.1)\begin{equation} {\rm \Delta}\varphi_{l}={\rm \Delta}\varphi_{h}=0. \end{equation}

At the interface ($r=r_{I}$), the following kinematic and dynamic boundary conditions must be satisfied (Goncharov Reference Goncharov2002; Wang et al. Reference Wang, Wu, Ye, Zhang and He2013):

(3.2)\begin{gather} \frac{\partial\eta}{\partial t}=\frac{ \partial\varphi_{l}}{\partial r}-\frac{1}{r^{2}}\frac{\partial\eta}{\partial\theta}\frac{\partial\varphi_{l}}{\partial\theta} =\frac{ \partial\varphi_{h}}{\partial r}-\frac{1}{r^{2}}\frac{\partial\eta}{\partial\theta}\frac{\partial\varphi_{h}}{\partial\theta}, \end{gather}

(3.3)\begin{gather}\rho_{l}\left(\frac{\partial\varphi_{l}}{\partial t}+\frac{|\boldsymbol{\nabla}\varphi_{l}|^{2}}{2}+\Psi\right)-\rho_{h}\left(\frac{\partial\varphi_{h}}{\partial t}+\frac{|\boldsymbol{\nabla}\varphi_{h}|^{2}}{2}+\Psi\right)=f(t), \end{gather}

where $f(t)$ is an arbitrary function of time, the potential of external force $\Psi =gr$ and $-gr$ for the convergent and divergent cases, respectively. In reality, (3.2) and (3.3) represent the continuity of the velocity component normal to the interface, and pressure at the interface. Following Layzer's treatment (Layzer Reference Layzer1955), we expand the perturbation of the bubble tip localized at the point $\{r,\theta \}=\{r_{I}(0,t),0\}$ to the second order in $\theta$, i.e. $\eta =\eta _{0}(t)+\eta _{2}(t)\theta ^{2}$. A similar expanding procedure applied to (3.2) and (3.3) yields six equations for zero- to second-order of $\theta$, which are well posed by six unknown functions of time. As three of these six functions have been given as $\eta _{0}(t)$, $\eta _{2}(t)$ and $f(t)$, the other three are required to be given by the velocity potentials. Inspired by Goncharov's model (Goncharov Reference Goncharov2002), by solving the corresponding eigenvalue problem for the velocity potentials, the expressions of velocity potentials near the bubble tip are presumed as

(3.4)\begin{gather} \varphi_{h}=a_{1}(t)\cos(n\theta)\left(\frac{r}{r_{0}+\eta_{0}(t)}\right)^{S_h n}, \end{gather}

(3.5)\begin{gather}\varphi_{l}=b_{1}(t)\cos(n\theta)\left(\frac{r}{r_{0}+\eta_{0}(t)}\right)^{S_l n}+b_{2}(t)\ln\frac{r}{r_{0}+\eta_{0}(t)}, \end{gather}

where $S_h=-1$ and $S_l=+1$ corresponding to the convergent case, and $S_h=+1$ and $S_l=-1$ to the divergent case. Note that $\varphi _{l}$ is valid only near the bubble tip which cannot take the limit of $r\rightarrow 0$ and $r\rightarrow +\infty$ in (3.5). Thus only the first term of the Fourier series is kept in (3.4) and (3.5), which can obtain an excellent approximation at the bubble tip as verified by Goncharov (Reference Goncharov2002). Then the models for both the convergent and divergent cases are given as follows, respectively.

For the convergent case, using (3.4) and (3.5) and expanding (3.2) and (3.3) near the bubble tip, we have the following relations:

(3.6)\begin{equation} \dot{\eta}_{2}=-\frac{(3n+1)\dot{\eta}_{0}}{r_{0}+\eta_{0}}\eta_{2}-\frac{n^{2}}{2}\dot{\eta}_{0}, \end{equation}

(3.7)\begin{align} &\frac{n^{2}-4A_{T}Hn-12A_{T}H^{2}}{2(6H-n)(r_{0}+\eta_{0})}\frac{\mathrm{d} [(r_{0}+\eta_{0})\dot{\eta}_{0}]}{\mathrm{d} t}+\frac{A_{T}H\dot{\eta}_{0}^{2}}{r_{0}+\eta_{0}}\nonumber\\ &\quad-\frac{(4A_{T}-3)n^{2}+6(3A_{T}-5)Hn+36A_{T}H^{2}+12(A_{T}-1)H}{2(6H-n)^{2}(r_{0}+\eta_{0})} n^{2}\dot{\eta}_{0}^{2}=A_{T}gH, \end{align}

where $H=\eta _{2}/(r_{0}+\eta _{0})$. Here, we assume the initial perturbation as $\eta _{0}(0)\cos (n\theta )$ and expand as $\eta _{0}(0)(1-n^2\theta ^2/2)$ at the bubble tip. Then, by integrating (3.6), it reads

(3.8)\begin{equation} H=\left[\frac{n^{2}}{6n+4}-\frac{n^{2}\eta_{0}(0)}{2(r_{0}+\eta_{0}(0))}\right] \left(\frac{r_{0}+\eta_{0}(0)}{r_{0}+\eta_{0}}\right)^{3n+2}-\frac{n^{2}}{6n+4}. \end{equation}

Further, by substituting (3.8) into (3.7) and integrating it, an analytic relation for the bubble amplitude can be obtained. In reality, such a relation is difficult to formulate explicitly because (3.7) is nonlinear. Thus the bubble amplitude can be determined by means of a numerical solution of (3.7).

Similarly, for the divergent case, an analytical model is given as

(3.9)\begin{align} &-\frac{n^{2}+4A_{T}Hn-12A_{T}H^{2}}{2(6H+n)(r_{0}+\eta_{0})}\frac{\mathrm{d} [(r_{0}+\eta_{0})\dot{\eta}_{0}]}{\mathrm{d} t}-\frac{A_{T}H\dot{\eta}_{0}^{2}}{r_{0}+\eta_{0}}\nonumber\\ &\quad+\frac{(4A_{T}-3)n^{2}-6(3A_{T}-5)Hn+36A_{T}H^{2}+12(A_{T}-1)H}{2(6H+n)^{2}(r_{0}+\eta_{0})} n^{2}\dot{\eta}_{0}^{2}=A_{T}gH, \end{align}

(3.10)\begin{align} &H=\left[-\frac{n^{2}}{6n-4}-\frac{n^{2}\eta_{0}(0)}{2(r_{0}+\eta_{0}(0))}\right] \left(\frac{r_{0}+\eta_{0}}{r_{0}+\eta_{0}(0)}\right)^{3n-2}+\frac{n^{2}}{6n-4}. \end{align}

To verify the above theoretical solutions, we employ our high-fidelity DNS on the evolution of the cylindrical RTI, especially in the nonlinear regime. Figure 5 shows the positions of the bubble tip for the convergent and divergent cases, where the Atwood number is chosen as $A_{T}=0.3$ and $0.6$, and the number of perturbation waves as $n=4$ and $8$ (coloured by red and blue, respectively) with an initial perturbation amplitude $\eta _{0}(0)=0.02r_{0}$. It is identified that the results of the analytical models (3.7) and (3.9) are in good agreement with data from DNS. Further, it is found that the bubbles grow exponentially (see the black solid lines in figure 5) in the linear regime ($n\eta _{0}/r_{0}\ll 1$), which is consistent with the linear theory (Wang et al. Reference Wang, Wu, Ye, Zhang and He2013), and a significant influence of the cylindrical geometry occurs in the nonlinear regime ($n\eta _{0}/r_{0}>1$).

Figure 5. Radial positions of the bubble tip for the convergent (a) and divergent (b) cases at $A_{T}=0.3$ (blue/red solid lines and squares) and $A_{T}=0.6$ (blue/red dashed lines and circles) obtained by the analytical models (blue/red solid and dashed lines) and DNS (blue/red squares and circles). For comparison, the profiles are also plotted for the exponential growth (black solid thin line) with $\eta _0=\eta _0(0)\cosh (\gamma t)$ in the linear regime given by Wang et al. (Reference Wang, Wu, Ye, Zhang and He2013), and for quadratic-in-time growth (green solid thin lines) with $r_0+\eta _0\sim 1/2a_{b}t^{2}$ in the nonlinear regime obtained by the present study.

3.2. Model of nonlinear bubble saturation

The effect of cylindrical geometry on the nonlinear evolution of RTI can be evidently revealed by the asymptotic solutions of (3.7) and (3.9). As shown in figure 5, taking the limit of $t\rightarrow \infty$, the asymptotic tendency is obtained as $(r_{0}+\eta _{0}(0))/(r_{0}+\eta _{0})\rightarrow 0$ for the convergent case and $(r_{0}+\eta _{0})/(r_{0}+\eta _{0}(0))\rightarrow 0$ for the divergent case. Then substituting the above limits into (3.8) and (3.10), we have $H\rightarrow -n^{2}/(6n+4)$ and $H\rightarrow n^{2}/(6n-4)$ for the convergent and divergent cases, respectively. Finally, (3.7) and (3.9) are reduced as the following asymptotic ordinary differential equations:

(3.11)\begin{gather} \frac{1}{r_{0}+\eta_{0}}\frac{\mathrm{d} [(r_{0}+\eta_{0})\dot{\eta}_{0}]}{\mathrm{d} t} +C\frac{\dot{\eta}_{0}^{2}}{r_{0}+\eta_{0}}=D, \end{gather}

(3.12)\begin{gather}\frac{1}{r_{0}+\eta_{0}}\frac{\mathrm{d} [(r_{0}+\eta_{0})\dot{\eta}_{0}]}{\mathrm{d} t} +E\frac{\dot{\eta}_{0}^{2}}{r_{0}+\eta_{0}}=F, \end{gather}

where the constants $C-F$ can be obtained based on (3.7) and (3.9) and the asymptotic expression $H$. The solutions of (3.11) and (3.12) are derived as $\dot {\eta }_{0}/\sqrt {r_{0}+\eta _{0}}=\sqrt {2D/(3+2C)}$ and $\dot {\eta }_{0}/\sqrt {r_{0}+\eta _{0}}=-\sqrt {2F/(3+2E)}$, respectively.

Different from planar geometry, the bubble velocity $V_{b}=\dot {\eta }_{0}$ cannot saturate to a constant value in cylindrical geometry. Of special interest, $\dot {\eta }_{0}/\sqrt {r_{0}+\eta _{0}}$ approaches a constant value as $t\rightarrow \infty$, which is given as for the convergent and divergent cases, respectively,

(3.13)\begin{align} &\frac{\dot{\eta}_{0}}{\sqrt{r_{0}+\eta_{0}}}\notag\\ &\quad\rightarrow \left({\frac{4n(3n+1)^{2}A_{T}g}{54(A_{T}+1)n^4+9(15A_{T}+17)n^3 +3(35A_{T}+53)n^{2}+24(A_{T}+3)n+12}}\right)^{\frac{1}{2}}, \end{align}

(3.14)\begin{align}&\frac{\dot{\eta}_{0}}{\sqrt{r_{0}+\eta_{0}}}\notag\\ &\quad\rightarrow -\left({\frac{4n(3n-1)^{2}A_{T}g}{54(A_{T}+1)n^4-9(15A_{T}+17)n^3 +3(35A_{T}+53)n^{2}-24(A_{T}+3)n+12}}\right)^{\frac{1}{2}}. \end{align}

Further, the saturation of $V_{b}/\sqrt {r_{0}+\eta _{0}}$ in the cylindrical RTI has been confirmed by our DNS results for $A_{T}=0.3$ and $0.6$ in figure 6. It is clearly identified that the normalized $|V_{b}|/\sqrt {r_{0}+\eta _{0}}$ reaches approximately the expected quasi-steady value at $\gamma t\approx 4.0$ when the nonlinear bubble evolution is realized. The nonlinear bubble growth is quadratic-in-time, i.e. $r_{0}+\eta _{0}\sim 1/2a_{b}t^{2}$ (see the green solid thin lines in figure 5), where the bubble acceleration is obtained analytically as $a_{b}=D/(3+2C)$ for the convergent case and as $a_{b}=F/(3+2E)$ for the divergent case. It reveals that the bubbles grow asymptotically with a motion of uniform acceleration in the nonlinear regime for the cylindrical RTI.

Figure 6. Normalized $|V_{b}|/\sqrt {r_{0}+\eta _{0}}$ in convergent (a) and divergent (b) cases at $A_{T}=0.3$ and $0.6$ calculated by our DNS. Here, the dashed line of value 1 is plotted to illustrate the tendency of asymptotic analytical solution.

Based on the preceding analysis, when the cylindrical RTI grows from the linear into the nonlinear regime, the bubble evolution changes from the exponential to the quadratic-in-time growth. We could call such a transition ‘nonlinear saturation’ for the cylindrical RTI; it means that the bubble acceleration (or $V_{b}/\sqrt {r_{0}+\eta _{0}}$) tends to be a constant value. As the bubble tip has $r_{0}+\eta _{0}\rightarrow \infty$ for the convergent case and $r_{0}+\eta _{0}\rightarrow 0$ for the divergent case, the bubble velocity is bound to have $V_{b}\rightarrow \infty$ and $V_{b}\rightarrow 0$ when the bubbles evolve in the nonlinear regime. Thus, it is rational that the bubble velocity of the divergent case is smaller than that of the convergent case at a certain moment, such as $\gamma t\geq 4.6$ in figure 4.

Furthermore, we can verify that the nonlinear saturation of cylindrical RTI (quadratic-in-time growth) could be reduced to that of planar RTI (linear-in-time growth), when the effect of cylindrical geometry becomes vanishingly small. The RTI in cylindrical geometry differs from that in plane geometry in two typical aspects. One is that the curvature effect $1/r_0$ occurs at the cylindrical interface, the other is that the external acceleration direction is not parallel, with the maximum angle $2{\rm \pi} /n$ at single perturbation wave. Thus, the effect of cylindrical geometry will become negligibly weak as $r_0\rightarrow \infty$ and $n\rightarrow \infty$. Further, the nonlinear saturation regime appears as $n\eta _{0}/r_{0}\sim 1$, thus $r_{0}\gg \eta _{0}$ lies in this regime as a result of the reduction of cylindrical geometry to a planar one. As $r_{0}\gg \eta _{0}$ and $n\rightarrow \infty$, the right-hand sides of (3.13) and (3.14) are simplified as $\pm \sqrt {2A_{T}/(1+A_{T})(g/3n)}$ as $n\rightarrow \infty$ and $\sqrt {r_{0}+\eta _{0}}\sim \sqrt {r_{0}}$ as $r_{0}\gg \eta _{0}$. Then, (3.13) and (3.14) are reduced as $\dot {\eta }_{0}\rightarrow \pm \sqrt {A_{T}g\lambda /(1+A_{T})(1/3{\rm \pi} )}$ with the perturbation wavelength $\lambda =2{\rm \pi} r_{0}/n$. The reduced bubble velocity is the same as that in the planar RTI (Goncharov Reference Goncharov2002; Mikaelian Reference Mikaelian2003; Sohn Reference Sohn2003). This analysis also ensures that the nonlinear saturation models (3.13) and (3.14) are valid for the cylindrical RTI.