⇦Chinese Physics Letters, 2020, Vol. 37, No. 5, Article code 055201 Simulation of the Weakly Nonlinear Rayleigh–Taylor Instability in Spherical Geometry * Yun-Peng Yang (杨云鹏)1,2, Jing Zhang (张靖)3, Zhi-Yuan Li (李志远)3, Li-Feng Wang (王立锋)2,3, Jun-Feng Wu (吴俊峰)3, Wen-Hua Ye (叶文华)2,3**, Xian-Tu He (贺贤土)2,3 Affiliations 1School of Physics, Peking University, Beijing 100871 2Center for Applied Physics and Technology, HEDPS, Peking University, Beijing 100871 3Institute of Applied Physics and Computational Mathematics, Beijing 100094 Received 30 December 2019, online 25 April 2020 *Supported by the National Natural Science Foundation of China under Grant Nos. 11575033, 11675026, and 11975053.
**Corresponding author. Email: ye_wenhua@iapcm.ac.cn Citation Text: Yang Y P, Zhang J, Li Z Y, Wang L F and Wu J F et al 2020 Chin. Phys. Lett. 37 055201    Abstract The Rayleigh–Taylor instability at the weakly nonlinear (WN) stage in spherical geometry is studied by numerical simulation. The mode coupling processes are revealed. The results are consistent with the WN model based on parameter expansion, while higher order effects are found to be non-negligible. For Legendre mode perturbation $P_n(\cos\theta)$, the nonlinear saturation amplitude (NSA) of the fundamental mode decreases with the mode number $n$. When $n$ is large, the spherical NSA is lower than the corresponding planar one. However, for large $n$, the planar NSA can be recovered by applying Fourier transformation to the bubble/spike near the equator and calculating the NSA of the converted trigonometric harmonic. DOI:10.1088/0256-307X/37/5/055201 PACS:52.57.Fg, 47.20.Ma, 52.35.Py © 2020 Chinese Physics Society Article Text The Rayleigh–Taylor instability (RTI) occurs when a heavy fluid is supported or accelerated by a light fluid.[1,2] The RTI is a key factor in inertial confinement fusion (ICF)[3–5] and many astrophysical phenomena.[6] For a light fluid of density $\rho_{\rm l}$ and a heavy fluid of density $\rho_{\rm h}$, the linear RTI growth is given by $\eta_{\rm L}(t)\equiv\eta_0 e^{\gamma t}$, where $\eta_0$ is the initial amplitude, $\gamma=\sqrt{A_{\rm T} kg}$ is the growth rate, $A_{\rm T}=(\rho_{\rm h}-\rho_{\rm l})/(\rho_{\rm h}+\rho_{\rm l})$ is the Atwood number, $k=2\pi/\lambda$ is the perturbation wave number, and $g$ is the gravity. When the perturbation amplitude becomes comparable to the wavelength, the RTI grows into a weakly nonlinear (WN) regime. Theories based on the parameter expansion method are used to describe the WN growth.[7–9] These theories are useful when predicting the harmonic coupling processes. The second harmonic generated through the second-order coupling breaks the symmetry of the interface. The third-order coupling feeds back to the fundamental harmonic. This feedback slows down the growth of the fundamental harmonic and leads to nonlinear saturation. The nonlinear saturation amplitude (NSA) $\eta_{\rm s}$ is defined as the linear amplitude at the saturation time $t_{\rm s}$, when the actual amplitude of the fundamental harmonic $\eta_{\rm f}(t)$ is 90% of the theoretical linear amplitude $\eta_{\rm L}(t)$, i.e., $\eta_{\rm s}=\eta_{\rm L}(t_{\rm s})$, s.t. $\eta_{\rm f}(t_{\rm s})=0.9\eta_{\rm L}(t_{\rm s})$. In multimode RTI, NSA is one of the factors that decide the dominant modes.[10] After the WN regime, the RTI develops into the fully nonlinear stage and the interface deforms into bubbles and spikes. The RTI in ICF or astrophysics concerns nonplanar geometry. To describe the nonplanar WN RTI, Wang et al.[11] extended the WN model to cylindrical geometry, and Zhang et al.[12] extended the WN model to spherical geometry. The spherical WN model shows that mode coupling processes are more complicated than those in planar cases. In this Letter, we present a simulation study on the WN stage of the 2D single-mode spherical RTI. We firstly introduce the theoretical framework and our numerical setup, then some results and discoveries are drawn from the simulations. We consider two incompressible, irrotational and immiscible ideal fluids in spherical coordinates. Initially they are separated by a sphere $r(\theta,t=0)=R_0+\eta(\theta,t=0)$, where $R_0$ is the unperturbed interface radius, and $\eta(\theta,t)$ is the perturbation. The external gravity is $\boldsymbol{g}=-g \mathbf{e}_{r}$. If we denote the physical variables of the interior fluid ($r < R_{0}+\eta$) by the superscript 'in', and those of the exterior fluid ($R_{0}+\eta < r < \infty$) by the superscript 'ex', the governing equations will be $$\begin{alignat}{1} & {\nabla^{2} \phi^{\mathrm{ex}}=\nabla^{2} \phi^{\mathrm{in}}=0},~~ \tag {1} \end{alignat} $$ $$\begin{alignat}{1} & {\frac{\partial \eta}{\partial t}+\frac{1}{r^{2}} \frac{\partial \eta}{\partial \theta} \frac{\partial \phi^{\mathrm{ex}}}{\partial \theta}-\frac{\partial \phi^{\mathrm{ex}}}{\partial r}=0~~\mathrm{ at }~~ r=R_{0}+\eta},~~ \tag {2} \end{alignat} $$ $$\begin{alignat}{1} & {\frac{\partial \eta}{\partial t}+\frac{1}{r^{2}} \frac{\partial \eta}{\partial \theta} \frac{\partial \phi^{\mathrm{in}}}{\partial \theta}-\frac{\partial \phi^{\mathrm{in}}}{\partial r}=0 ~~\mathrm{at}~~ r=R_{0}+\eta},~~ \tag {3} \end{alignat} $$ $$\begin{alignat}{1} & \rho^{\mathrm{ex}}\Big\{\frac{\partial \phi^{\mathrm{ex}}}{\partial t}+\frac{1}{2}\Big[\Big(\frac{\partial \phi^{\mathrm{ex}}}{\partial r}\Big)^{2}+\frac{1}{r^{2}}\Big(\frac{\partial \phi^{\mathrm{ex}}}{\partial \theta}\Big)^{2}\Big]+g r\Big\} \\ & \;\;\;\; -\!\rho^{\mathrm{in}}\Big\{\frac{\partial \phi^{\mathrm{in}}}{\partial t}\!+\!\frac{1}{2}\Big[\Big(\frac{\partial \phi^{\mathrm{in}}}{\partial r}\Big)^{2}\!+\!\frac{1}{r^{2}}\Big(\frac{\partial \phi^{\mathrm{in}}}{\partial \theta}\Big)^{2}\Big]\!+\!g r\Big\} \\ & = f(t) ~~\mathrm{at}~~ r=R_{0}+\eta,~~ \tag {4} \end{alignat} $$ where $\rho$ is the density, $\phi$ is the velocity potential defined as ${\boldsymbol v}=\nabla \phi$, and $f(t)$ is an arbitrary function of time. We can expand the perturbation and the velocity potentials into Legendre modes, $$\begin{align} & \eta(\theta, t) = \displaystyle\sum_{l} a_{l}(t) P_{l}(\cos \theta),~~ \tag {5} \end{align} $$ $$\begin{alignat}{1} & \phi^{\mathrm{ex}}(r, \theta, t) = \displaystyle\sum_{l} b^{\mathrm{ex}}_{l}(t)\Big(\frac{r}{R_{0}}\Big)^{-(l+1)} P_{l}(\cos \theta),~~ \tag {6} \end{alignat} $$ $$\begin{alignat}{1} & \phi^{\mathrm{in}}(r, \theta, t) = \displaystyle\sum_{l} b^{\mathrm{in}}_{l}(t)\Big(\frac{r}{R_{0}}\Big)^{l} P_{l}(\cos \theta),~~ \tag {7} \end{alignat} $$ where $a_{\rm l}$, $b^{\mathrm{ex}}_{\rm l}$, and $b^{\mathrm{in}}_{\rm l}$ are the temporal amplitudes of mode $l$. In the scenario of the WN model,[7] these amplitudes can be further expanded into power series of a formal parameter $\epsilon$ as $$\begin{alignat}{1} & a_{l} = \epsilon a_{l}^{(1)}+\epsilon^{2} a_{l}^{(2)}+\epsilon^{3} a_{l}^{(3)}+\cdots,~~ \tag {8} \end{alignat} $$ $$\begin{alignat}{1} & b_{l}^{\mathrm{ex}} = \epsilon b_{l}^{\mathrm{ex}(1)}+\epsilon^{2} b_{l}^{\mathrm{ex}(2)}+\epsilon^{3} b_{l}^{\mathrm{ex}(3)}+\cdots,~~ \tag {9} \end{alignat} $$ $$\begin{alignat}{1} & b_{l}^{\mathrm{in}} = \epsilon b_{l}^{\mathrm{in}(1)}+\epsilon^{2} b_{l}^{\mathrm{in}(2)}+\epsilon^{3} b_{l}^{\mathrm{in}(3)}+\cdots,~~ \tag {10} \end{alignat} $$ where $a_{l}^{(j)}$, $b_{l}^{\mathrm{ex}(j)}$, and $b_{l}^{\mathrm{in}(j)}$ are the temporal amplitudes for Legendre mode $l$ at order $\epsilon^j$. Substituting Eqs. (5)-(10) to Eqs. (1)-(4), and collecting terms of the same order in $\epsilon$, we can obtain a set of equations about $a_{l}^{(j)}$, $b_{l}^{\mathrm{ex}(j)}$, and $b_{l}^{\mathrm{in}(j)}$ for each order $O(\epsilon^j)$. The series of equation sets can then be solved from the lower order to higher order.
cpl-37-5-055201-fig1.png
Fig. 1. Density pseudocolor images for $P_n(\cos\theta)$ at $\gamma_n t=5.5$ with (a) $n=6,A_{\rm T}=0.5$, (b) $n=6,A_{\rm T}=0.8$, (c) $n=9,A_{\rm T}=0.5$, and (d) $n=9,A_{\rm T}=0.8$. The $y$ axis points to the polar direction ($\theta=0$). The origin locates at the sphere center.
For a single-mode perturbation $\eta(\theta,t=0)=\eta_0 P_n(\cos\theta)$, where $\eta_0$ is the initial amplitude of the Legendre mode $P_n(\cos\theta)$, the solution at the first order $O(\epsilon)$ involves only the fundamental mode, and gives its linear growth rate $$\begin{align} \begin{aligned} \gamma_{n}=\sqrt{\frac{n(n+1)\left(\rho^{\mathrm{ex}}-\rho^{\mathrm{in}}\right) g} {\left[(n+1) \rho^{\mathrm{in}}+n \rho^{\mathrm{ex}}\right] R_{0}}}. \end{aligned}~~ \tag {11} \end{align} $$ It should be noted that the Atwood number $A_{\rm T}=(\rho^{\mathrm{ex}}-\rho^{\mathrm{in}})/(\rho^{\mathrm{ex}}+\rho^{\mathrm{in}})$ can be negative if $\rho^{\mathrm{ex}} < \rho^{\mathrm{in}}$. If we define the wave number to be the square root of the Laplacian eigenvalue of Legendre polynomial $P_n(\cos\theta)$, i.e., $k_{n}=2\pi/\lambda_{n}=\sqrt{n(n+1)}/R_{0}$, and denote the equivalent Atwood number as $A_{T}'=(\rho^{\mathrm{ex}}-\rho^{\mathrm{in}}) /(\sqrt{(n+1)/n}\rho^{\mathrm{in}}+\sqrt{n/(n+1)}\rho^{\mathrm{ex}})$, we again obtain $\gamma_{n}=\sqrt{A_{T}^{\prime} g k_{n}}$. If the equations are solved to the third order[12], mode coupling processes at the second and the third order will be recovered, which generate modes besides $P_n$. From here on, we use $P_n$ to denote the initial perturbation mode, and use $P_{\rm l}$ to denote the modes involved in the coupling processes. The modes generated by the second-order coupling $O(\epsilon^2)$ include $l=2i$, $i=0,1, \ldots, n$, and the modes generated by the third-order coupling $O(\epsilon^3)$ include $l=3n-2i$, $i=0,1, \ldots,\lfloor 3 n / 2\rfloor$, where $\lfloor x \rfloor$ denotes the maximum integer less than or equal to $x$.
cpl-37-5-055201-fig2.png
Fig. 2. Interface spectra up to the $4n$th mode ($l=0,\ldots,4n$) at $\gamma_n t=6.0$ for (a) $n=6$, $A_{\rm T}=0.5$, (b) $n=9$, $A_{\rm T}=0.5$, (c) $n=6$, $A_{\rm T}=0.8$, and (d) $n=9$, $A_{\rm T}=0.8$.
A set of simulations are conducted to validate the model and to study the spherical WN stage in detail. In our numerical simulation (NS), the hydrodynamical code FLASH[13] is used. A directionally split piecewise-parabolic method (PPM) solver and a 2D spherical geometry are employed. The radius $r$ takes a range $50\,\mathrm{µ m} < r < 950\,\mathrm{µ m}$. A reflective boundary condition is set for the inner boundary, and a hydrostatic boundary condition is set for the outer boundary. The interface separating two fluids is initially placed at $R_0 = 410\,\mathrm{µ m}$. The polar angle $\theta$ takes a range of $0 < \theta < \pi$ for odd perturbation modes. For even modes, as they are symmetric about the equatorial plane, the polar angle takes a range of $0 < \theta < \pi/2$. Reflective boundary conditions are set for the poloidal direction. The initial amplitude of the perturbation is set as $\eta_0 = 0.002\lambda_n$. The gravity points to the sphere center. The pressure at the outer radial boundary is set to be 80% of that at the inner boundary. The ideal gas equation of state (EOS) is used. To reduce compressibility effects, the adiabatic index $\gamma_{h}$ is set to $5.0$ unless specified otherwise. A transition layer is employed at the density discontinuity place. The initial 1D density profile is in the form $\rho_0(r')=\left(\rho_{\mathrm{in}}+\rho_{\mathrm{ex}}\right) / 2 -\left[\left(\rho_{\mathrm{in}}-\rho_{\mathrm{ex}}\right) / 2\right] \tanh \left(r' / L\right)$, where $r'$ is the radial distance to the interface, and $L$ is the typical half width of the density transition layer, which satisfies $\min[\rho_0/(\mathrm{d}\rho_0/\mathrm{d}r')] = L/A_{\rm T}$. In the NS, for perturbation mode $P_n$, $L$ is set to be $0.1/k_n$. Here $192n$ zones are employed in $r$ direction. In $\theta$ direction, $256n$ zones are employed if $n$ is even, and $512n$ zones are employed if $n$ is odd. To check mesh convergence on this problem, we have tested different zonings (1/4 zones, half zones and double zones in both directions), and the results agree well with each other. Figure 1 shows the density pseudocolor images at $\gamma_n t=5.5$ for perturbation mode $P_6$/$P_9$ and Atwood number $0.5$/$0.8$.
cpl-37-5-055201-fig3.png
Fig. 3. Logarithmic plots of mode amplitudes versus $\gamma_n t$ for (a) $n=6$, $A_{\rm T}=0.5$, (b) $n=9$, $A_{\rm T}=0.5$, (c) $n=6$, $A_{\rm T}=0.8$, and (d) $n=9$, $A_{\rm T}=0.8$. The $0$th, the $n$th, the $2n$th and the $3n$th modes, as well as the fundamental feedback $\eta_{\rm fd}$ are shown.
The perturbation interface is determined from the NS data by density contour $\rho=(\rho^\mathrm{in}+\rho^\mathrm{ex})/2$. For even perturbation modes, we extend the interface range from $0 < \theta < \pi/2$ to $0 < \theta < \pi$ by mirroring the data about the equatorial plane. Mode components are extracted from the interface through Legendre polynomial fitting. Figure 2 shows the spectra of the interface up to the $4n$th mode at $\gamma_n t=6.0$ for perturbation mode $P_6$/$P_9$ and Atwood number $0.5$/$0.8$. It can be seen that all modes will emerge from the mode coupling processes (for even perturbation modes, all even modes emerge). The $n$th, $2n$th, and $3n$th modes dominate the first $3n$ modes. These results are consistent with the 3rd-order WN theory. However, it shall be noticed that the modes from the $(3n+1)$th to the $4n$th are non-negligible, and also for $P_9$, the even modes from the $(2n+1)$th to the $3n$th (i.e., 20th, 22th, 24th, 26th) are non-negligible. These modes can not emerge from the 2nd- and the 3rd-order coupling theory,[12] which indicates that higher order coupling processes can not be neglected. Figure 3 shows the temporal growth of the $0$th, $n$th, $2n$th and $3n$th modes, as well as the feedback to the fundamental mode $\eta_{\rm fd}$. The fundamental feedback $\eta_{\rm fd}$ is calculated by subtracting the linear amplitude $\eta_{\rm L}$ from the actual fundamental mode amplitude $\eta_n$, where $\eta_{\rm L}$ is retrieved by numerical exponential fitting out of $\eta_n$. The figure clearly shows that these modes experience a period of exponential growth. The growth rates $\gamma_{\rm l}'$ can be numerically fitted out, as shown in Fig. 4. In the scenario of single-mode WN model, the mode generated through the $p$th order coupling contains terms whose exponential growth rates are equal to or less than $p\gamma_n$. This means that, if the initial perturbation is small ($\eta_0 \ll \lambda_n$) and if higher order coupling is negligible, the mode at the $p$th order coupling will experience a period of exponential growth with growth rate $p\gamma_n$. As is seen from Fig. 4, $\gamma_0'$ and $\gamma_{2n}'$ are approximately twice as much as $\gamma_n'$, indicating that the $0$th and $2n$th modes are dominated by the 2nd-order coupling. The $0$th mode will cause the averaged interface position to move along the radial direction, which is a phenomenon showing up only in convergent geometry due to mass conservation. Here $\gamma_{3n}'$ and the growth rate of fundamental feedback $\gamma_{\rm fd}'$ are approximately equal to or slightly lower than $3\gamma_n'$, indicating that they are mainly the results of the 3rd-order coupling. It is worth noticing that, according to the 3rd-order WN solution, for even modes, the 2nd-order coupling also generates fundamental feedback, but the feedback generated at the 3rd-order coupling dominates.
cpl-37-5-055201-fig4.png
Fig. 4. The fitted mode growth rates $\gamma_{\rm l}'$ normalized by $\gamma_n$ for (a) $n=6$, $A_{\rm T}=0.5$, (b) $n=9$, $A_{\rm T}=0.5$, (c) $n=6$, $A_{\rm T}=0.8$, and (d) $n=9$, $A_{\rm T}=0.8$.
With the fundamental mode amplitude $\eta_n$ and the fitted linear amplitude $\eta_{\rm L}$, one can determine the nonlinear saturation time $t_{\rm s}$, s.t. $\eta_n = 0.9\eta_{\rm L}$, and the nonlinear saturation amplitude $\eta_{\rm s} = \eta_{\rm L}(t_{\rm s})$. The amplitude of a Legendre mode represents the height of the bubble/spike at the pole ($\theta=0/\pi$), but we are more interested in the bubble/spike near the equator for it is more similar to that in the 2D planar cases. Thus, for the convenience of comparison with the planar cases, we multiply the spherical NSAs by the height ratio of the bubble/spike nearest to the equator to the bubble/spike at the pole. For even modes, this height ratio is in fact the value $P_n(\cos\frac{\pi}{2})$. For odd modes, this ratio is the value $P_n(\cos\theta')$, where $\theta'$ is the extremum point of $P_n(\cos\theta)$ nearest to $\frac{\pi}{2}$. Figure 5 plots the normalized NSA versus the perturbation mode $n$. The NSAs from the 3rd-order WN model, and the NSAs from planar simulations are also plotted. It is shown that there is a small discrepancy between the NS and the model. Figure 5 also shows the NSAs from another group of simulations with a much higher adiabatic index $\gamma_{\rm h}=20.0$. The results change little from their corresponding ones with lower $\gamma_{\rm h}$. Thus the sound speed is high enough and compressibility is not the main source of the difference between the NS and the model. We believe that this discrepancy comes from two aspects. The first is the influence of the transition layer, which slows down the growth of modes generated by the 3rd-order coupling. The second is the higher order ($\geqslant 4$) effects, which are non-negligible but are ignored in the 3rd-order theory.
cpl-37-5-055201-fig5.png
Fig. 5. Normalized NSA versus perturbation mode $n$ for (a) $A_{\rm T}=0.2$, (b) $A_{\rm T}=0.5$, (c) $A_{\rm T}=0.8$. Blue circles are the NSAs of the fundamental modes. Yellow lozenges are the NSAs predicted by the spherical 3rd-order WN model. Green triangles are the NSAs of the trigonometric fundamental harmonics retrieved by applying Fourier transformation to the equatorial bubble/spike. The dotted lines are the planar NSAs from planar simulations with corresponding Atwood number. The 8 red crosses are the NSAs of the fundamental modes from another group of simulations, where the adiabatic index is set to be $\gamma_{\rm h}=20.0$ and all other parameters are the same.
As can be seen from Fig. 5, for moderate Atwood numbers, NSA decreases with $n$. NSA is larger than the corresponding planar NSA when $n$ is small, and is lower than the planar NSA when $n$ is large.[14] However, one shall expect the spherical NSA to approach the planar NSA when $n$ is sufficiently large. Here we will show this is indeed the case. For even $n$, $\theta=\frac{\pi}{2}$ is one of the extremum points of $P_n(\cos\theta)$, and one can find the two extremum points nearest to $\pi/2$, say $\theta_1$ and $\theta_2$. For odd $n$, $\theta=\frac{\pi}{2}$ is one of the zero points of $P_n(\cos\theta)$, and one can find the two zero points nearest to $\pi/2$, also denoted as $\theta_1$ and $\theta_2$. In the range of $\theta_1 < \theta < \theta_2$, there is exactly one bubble and one spike. The behavior of this bubble/spike should approach the behavior of the bubble/spike in the planar case when $n$ is large. Thus, we apply Fourier transformation method to the perturbation interface within the range of $\theta_1 < \theta < \theta_2$. In this way, the $1$st, $2$nd and $3$rd trigonometric harmonics and the feedback to the $1$st harmonic are extracted. Then, the NSA of the $1$st harmonic can be calculated. The results of these NSAs are shown in Fig. 5 as green triangles, which approach the planar cases as expected. Trigonometric functions, which are the basis functions of Fourier transformation, are not the eigen functions in spherical geometry. The amplitude of each converted trigonometric harmonic is contributed by multiple Legendre modes. Thus the saturation at the equator involves multiple Legendre modes other than only the fundamental mode $P_n$ itself. In conclusion, the Rayleigh–Taylor instability at the weakly nonlinear (WN) stage in spherical geometry is studied by numerical simulation. The results reveal the mode coupling processes. The zeroth mode mainly arises from the 2nd-order coupling processes, and the feedback to the fundamental mode mainly arises from the 3rd-order coupling processes. These results are consistent with the 3rd-order WN model. Higher order effects are found non-negligible. One may tend to get a higher-order model through parameter expansion to recover the NS results. The parameter expansion method requires that higher-order components shall be much smaller than lower-order components. However, our NS shows that this requirement may be broken for $P_{3n}$ and higher modes. Thus, a higher-order model may not be practical. The nonlinear saturation amplitude (NSA) is calculated. For perturbation mode $P_n(\cos\theta)$, the NSA of the fundamental mode decreases with $n$. When $n$ is large, this NSA is lower than the NSA in planar geometry with the corresponding Atwood number. To compare with planar NSA for large $n$, a Fourier transformation is applied to the bubble/spike near the equator and the NSA of the converted trigonometric fundamental harmonic is calculated, which recovers the value of the planar NSA. References Investigation of the Character of the Equilibrium of an Incompressible Heavy Fluid of Variable DensityThe Instability of Liquid Surfaces when Accelerated in a Direction Perpendicular to their Planes. IThe physics basis for ignition using indirect-drive targets on the National Ignition FacilityTheoretical and simulation research of hydrodynamic instabilities in inertial-confinement fusion implosionsA hybrid-drive nonisobaric-ignition scheme for inertial confinement fusionModeling Astrophysical Phenomena in the Laboratory with Intense LasersThree-dimensional Rayleigh-Taylor instability Part 1. Weakly nonlinear theoryTwo-Dimensional Rayleigh?Taylor Instability in Incompressible Fluids at Arbitrary Atwood NumbersWeakly Nonlinear Rayleigh–Taylor Instability in Incompressible Fluids with Surface TensionDependence of turbulent Rayleigh-Taylor instability on initial perturbationsWeakly nonlinear incompressible Rayleigh-Taylor instability growth at cylindrically convergent interfacesWeakly nonlinear incompressible Rayleigh-Taylor instability in spherical geometryFLASH: An Adaptive Mesh Hydrodynamics Code for Modeling Astrophysical Thermonuclear FlashesWeakly nonlinear incompressible Rayleigh-Taylor instability in spherical and planar geometries
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