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Guo Hong-Yu, Wang Li-Feng, Ye Wen-Hua, Wu Jun-Feng, Li Ying-Jun, Zhang Wei-Yan. Rayleigh-Taylor instability at spherical interfaces of incompressible fluids . Chinese Physics B, 2018, 27(2): 025206  
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Rayleigh-Taylor instability at spherical interfaces of incompressible fluids
Guo Hong-Yu1, 2, Wang Li-Feng2, 3, Ye Wen-Hua2, 3, †, Wu Jun-Feng2, Li Ying-Jun4, ‡, Zhang Wei-Yan2, 3
Graduate School, China Academy of Engineering Physics, Beijing 100088, China
Institute of Applied Physics and Computational Mathematics, Beijing 100094, China
HEDPS, Center for Applied Physics and Technology, Peking University, Beijing 100871, China
State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Beijing 100083, China

 

† Corresponding author. E-mail: ye_wenhua@iapcm.ac.cn lyj@aphy.iphy.ac.cn

Project supported by the National Natural Science Foundation of China (Grant Nos. 11275031, 11475034, 11575033, 11574390, and 11274026) and the National Basic Research Program of China (Grant Nos. 2013CB834100 and 2013CBA01504).

Abstract

Rayleigh-Taylor instability (RTI) of three incompressible fluids with two interfaces in spherical geometry is derived analytically. The growth rate on the two interfaces and the perturbation feedthrough coefficients between two spherical interfaces are derived. For low-mode perturbation, the feedthrough effect from outer interface to inner interface is much more severe than the corresponding planar case, while the feedback from inner interface to the outer interface is smaller than that in planar geometry. The low-mode perturbations lead to the pronounced RTI growth on the inner interface of a spherical shell that are larger than the cylindrical and planar results. It is the low-mode perturbation that results in the difference between the RTI growth in spherical and cylindrical geometry. When the mode number of the perturbation is large enough, the results in cylindrical geometry are recovered.

PACS: 52.57.Fg;47.20.Ma;52.35.Py
Keyword:Rayleigh-Taylor instability;spherical geometry;inertial confinement fusion
1. Introduction

The Rayleigh–Taylor instability (RTI) occurs at a perturbed fluid interface where a light fluid supports another heavy fluid or accelerates a heavier one.[1,2] This instability plays a critical role in many areas, such as magnetized liner inertial fusion (MagLIF) experiment[3] and inertial confinement fusion (ICF) implosions.[4,5] In typically central ignition ICF implosion experiments, the spherical implosion target consisting of a high density ablator-fuel shell filled with a low density deuterium-tritium gas is irradiated directly by lasers or irradiated indirectly by x rays. During the acceleration stage, the dense shell is accelerated by the ablative plasmas, resulting in the instability that occurs on the outer interface of the shell. In the deceleration stage, the dense shell is decelerated by the central low density hot-spot, and the shell’s inner interface is subject to the RTI. The RTI growth can cause the mixing of cold fuel with the hot-spot, leading to the large asymmetric deformation of a hot-spot, causing ICF implosion performance degradation and even resulting in ignition failure. So far, the Rayleigh–Taylor instability is still a critical issue in the ICF community.

The Rayleigh–Taylor instability has been investigated analytically,[6–18] numerically,[19–21] and experimentally[22,23] by many researchers. The classical RTI is first given by Rayleigh,[1] where two semi-infinite fluids are separated by a perturbed interface. The linear growth rate is obtained as , where A is the Atwood number at the interface, k is the perturbed wave number, and g is the acceleration. Based on this classical configuration, RTI has been extensively studied by considering different factors like surface tension,[7] mass ablation,[9] density gradients,[10] and magnetic field,[11] etc. The linear growth of RTI for arbitrary stratified fluid layers is investigated by Mikaelian.[12] The interface coupling phenomenon is shown in Mikaelian’s linear model.[13] The RTI growth in two finite-thickness fluids is studied in Ref. [8], it is found that the finite-thickness effect reduces the linear growth rate of RTI. The weakly nonlinear (WN) RTI in a finite-thickness slab is developed by Wang,[14] the feedthrough effect and the perturbation evolution are discussed in some detail. These analytical models mentioned above are limited to planar geometry. However, RTI always occurs in non-Cartesian geometries such as supernova explosions or ICF implosions. Yu[15] studied the RTI growth in cylindrical geometry (CG) for compressible fluid. The RTI growth at the spherical interface between viscous fluids is analyzed by Terrones.[16] The growth rate and the most unstable modes are illustrated analytically. Zhang[17] has developed a WN model for RTI growth at a single interface in spherical geometry (SG). It is found the mode-coupling behavior in SG is different from that in CG. Recently, the RTI in multi-fluids is analyzed in CG.[18] The perturbation growth and the feedback effect between two cylindrical interfaces are discussed. In the spherical ICF implosion experiments, the source of perturbations on the inner interface of the imploded shell can be either initial perturbations due to the defects of the shell or the perturbations feedthrough from the outer interface. The feedthrough effect is a severe problem in ICF implosions that results in significant perturbation growth on the inner interface.[24,25] In this paper, we investigate the RTI growth of multi-fluid layers for the low-mode perturbations in SG.

This paper is organized as follows. Section 2 describes the theoretical model of the spherical RTI growth. The behavior of spherical RTI and the low-mode RTI growth are discussed in Section 3. Section 4 summarizes the major results and gives some conclusions.

2. Theoretical model

In this section, we will show the governing equations of the theoretical model. The coupling solutions for the RTI growth are also derived. The two-dimensional spherical coordinate system is introduced, where the r and θ are the radial direction and the polar direction, respectively. We consider three spherical fluid layers separated by an inner interface of radius R1 and an outer interface of radius R2 (R2 = R1 + ΔR), where Δ R is the thickness of the middle layer. The three fluids are immersed in a gravitational field g = −g er (i.e., during the deceleration stage of the ICF implosion). The two perturbed fluid interfaces can be expressed as r(t, θ) = R1 + η1(t, θ) and r(t, θ) = R2 + η2 (t, θ), where the η1(t, θ) and η2(t,θ) are small perturbations at the inner and outer interfaces, respectively. In the following derivation, the physical quantities of the interior fluid (0 < r < R1), the middle fluid (R1 < r < R2), and the exterior fluid (R2 < r < ∞) will be denoted by the subscript “i”, “m”, and “e”, respectively, unless otherwise stated. The governing equations for the present inviscid, irrotational, and incompressible fluids are

where ρi, ρm, and ρe are densities (ρi < ρm > ρe). The ϕi, φm, and φe are the perturbed velocity potentials relating to the fluid velocities, f1(t) and f2(t) are arbitrary functions of time. Equations (1) and (3) represent the conditions that the normal velocity is continuous across the inner and outer interface, respectively. The Bernoulli equations (2) and (4) describe the pressure continuity condition across the inner and outer interface, respectively. Considering a single-mode spherical Legendre perturbation at two interfaces, the perturbation displacements η1(t,θ) and η2(t,θ) are expressed as
where Pl(cosθ) is the Legendre polynomial and l is the mode number. The ηi and ηe are the amplitudes of the perturbations. The perturbed velocity ϕn(t, r, θ) (n = i, m, and e) can be written as
where aj(t) (j = 1, 2, 3, 4) are the amplitudes of the corresponding perturbed velocity potentials. As can be seen in Eqs. (7)–(9), the perturbed velocity potentials ϕn(t, r, θ) have satisfied the Laplace equation ∇2 ϕn = 0 and the boundary conditions ∇ ϕi|r → 0 = 0 (l ≥ 2) and ∇ϕe|r → ∞ = 0. We substitute Eqs. (5)–(9) into Eqs. (1)–(4) and expand the equations at the perturbed interfaces. Eliminating the time-dependent parameters aj(t), the governing equations for the evolutions of ηi(t) and ηe(t) are obtained as
where χ1 = 1 + αl – A1/(1 + l), χ2 = 1 + αl − A2/(1 + l), and αl = l/(1 + l). The ζ = R2/R1 represents the ratio of the outer and the inner interface radius. The dot in the equations denotes the time derivative. As can be seen in Eqs. (10) and (11), the coupled second-order ordinary differential equations (ODEs) explicitly depend on the Legendre mode number l and Atwood numbers A1 and A2, where A1 = (ρm − ρi)/(ρm + ρi) and A2 = (ρe − ρm)/(ρe + ρm) are the Atwood numbers at the inner interface and the outer interface, respectively. In the limit of large mode number perturbation l ≫ 1, the approximate expressions αl ∼ 1, χ1 ∼ 2, and χ2 ∼ 2 are obtained. Then, the governing equations in SG are reduced to those in CG as follows:[18]
as can be seen above, the low-mode perturbation would result in the significant difference between SG and CG, which will be discussed in the following.

Considering that the two interfaces are initially perturbed with interface displacements, namely, ηi(t = 0) = εi, , ηe(t = 0) = εe, and . The ODEs for the amplitude of perturbations ηi(t) and ηe(t) are solved as

the perturbation growth on the two interfaces are coupled as illustrated in Eqs. (14) and (15). The coupling behaviors of the spherical RTI are similar to the results in cylindrical geometry.[18] However, the growth rates of spherical RTI and the feedback effects between two interfaces are different from that in CG. The coefficient S21 represents the feedback factor of the perturbation from the outer interface to the inner interface and the S12 denotes the feedback coefficient from the inner interface to the outer interface. The specific feedthrough coefficients S21 and S12 are
as can be seen, the feedthrough coefficients are influenced by the shell thickness Δ R (represented by ζ), mode number l, and the Atwood numbers A1 and A2. The growth rates γi at the inner interface and γe at the outer interface are expressed as
the coefficients s1, s2, and s3 are the functions of the Atwood numbers A1, A2, mode number l, and the ratio of radius ζ, which are written as

As illustrated in Mikaelian’s planar results,[12] the growth rates are invariant under the condition ρm → ρi ρe/ρm (exchanging the value of A1 and A2). However, it is shown in Eqs. (18)–(22) that this symmetrical property is lost in spherical RTI. Considering short wavelength perturbations (the wavelength is much smaller than the shell thickness Δ R), we can obtain

which is identical to the linear results for RTI at a single spherical interface.[16,17] As can be seen, the two interfaces are decoupled for short wavelength perturbations. In the limit of long wavelength perturbations (Δ R → 0), the growth rates reduce to
where R1 = R2 ≡ R. Equation (25) is the well-known linear growth rate of spherical RTI. When the mode number l → ∞, the growth rate of the planar case is also recovered.[2]

It is interesting to note that the RTI growth in SG [Eqs. (14)–(22)] is approximately identical to that in CG[18] for large mode number perturbation (l ≫ 1). For quite short wavelength perturbations compared to the radius, namely, , the spherical results in this paper should reduce to that planar case. In the present work, we define the equivalent wavenumber .[26] Considering a special case for an infinity radius of R1 and finite shell thickness Δ R, namely, Δ R ≪ R1 → ∞, then we have

the approximation in Eq. (26) is derived with the formula limx → ∞ (1 + 1/x)x! ∼ e. Mikaelian’s planar results[12,13] are recovered with the approximations of Eqs. (26) and (27). That is to say, the behavior of RTI in the spherical case is similar to that in the planar case in the limit of large radius. Thus, the spherically convergence effect plays an important role for the low-mode RTI perturbations, which will be discussed below.

The above analytical results can be directly applied to the case in the presence of g = g er by substituting A1 → −A1, A2 → −A2, and g → −g, which will be of great interest in the acceleration stage of ICF implosion where the ablation surface is unstable and the hot-spot interface is stable.

3. Results and discussion

In this section, the growth rates of the RTI and the feedback effect between two interfaces in SG is discussed. The difference among the RTI growth in spherical, cylindrical, and planar geometry is also illustrated.

The dependencies of normalized growth rates and on R2/R1 are demonstrated in Fig. 1. For the case of A1 = 0.9 and A2 = −0.9, the inner interface of the middle shell is subject to the RTI. Thus, the value of is positive. With the increasing R2/R1, the increases gradually to a maximum value. The normalized growth rate is negative as the outer interface is stable. The absolute value of increases with increasing R2/R1, reaching to a maximum value at last. That means the thin shell effect (small R2/R1) will reduce the linear growth rate of RTI, which is consistent with our previous planar result.[8] The perturbations at two spherical interfaces are decoupled for large R2/R1, resulting in the individual growth rate for inner and outer interface, respectively. As can be seen in Fig. 1(a), the growth rate at the inner interface in SG is larger than that in the cylindrical case for fixed R2/R1. However, the growth rate at the outer interface in two geometries are approximately identical, which are shown in Fig. 1(b).

Fig. 1. (color online) Normalized growth rates (a) and (b) versus R2/R1 for mode number l = 6. The Atwood numbers are A1 = 0.9 and A2 = −0.9. The cylindrical results[18] are also displayed for comparison.

In Fig. 2, the effect of mode number l on the normalized growth rate and is displayed. The growth rate decreases with increasing l. The in spherical RTI is larger than that in the cylindrical case for low-mode number perturbation (l < 20), which is also shown in Fig. 1(a). The absolute value increases with increasing l and the spherical result is a little larger than the cylindrical result for l < 5. When l is large enough, the growth rates and in SG are completely equal to that in CG. That means the low-mode perturbation will cause much severe RTI growth in spherical implosions. This may be consistent with the recent simulation results of high-foot implosions on the National Ignition Facility (NIF)[27] where the low-mode perturbations at the inner interface have a large growth, leading to the large deformation of the hot-spot.

Fig. 2. (color online) The normalized growth rates (a) and (b) versus mode number l. The parameters are A1 = 0.9, A2 = −0.9, and ζ = 1.2. The cylindrical results[18] are plotted together.

Figure 3 displays the relationship between the feed-through coefficients S21, S12, and mode number l for different Atwood numbers. The coefficients S21 and S12 decrease to zero with increasing mode number l. That is to say, the two interfaces are decoupled for large mode number perturbation. It is because the perturbed wavelength is small when compared to the shell thickness for large l. The thick shell will lead to the enhanced feedthrough damping. The feedthrough coefficient S21 denotes the perturbation feedback from the outer interface to the inner interface that is influenced by A2 at the outer interface. The S21 decreases quickly with the decreasing absolute value of A2 for fixed l. When A2 tends to −1.0, the approximation S21 ∼ (R2/R1)−l + 1 is obtained that can be seen in Fig. 3(a). As shown in Fig. 3(b), the S12 (feedback from inner interface to outer interface) increases with increasing A1 at the inner interface. When A1 approaches to 1.0, we can find S12 ∼ (R2/R1)−l−2. Obviously, S21 is larger than S12 under the same conditions. However, the feedback coefficient between two interfaces of a planar slab is always e −k d, where k is the wave number and d is the fluid thickness in the planar case.[7] It is the spherically convergence effect that leads to this difference between S21 and S12.

Fig. 3. (color online) The feedthrough coefficients S21 (a) and S12 (b) versus mode number l. The Atwood numbers are A1 = 0.8, A2 = −0.1, −0.5, −1.0 for S21 and A1 = 0.1, 0.5, 0.9, A2 = −0.8 for S12, respectively. The radius ratio is ζ = 1.2.

Figure 4 describes the dependence of coefficients S21 and S12 on normalized thickness for various mode number l. The feedback effects become weak with the increasing , as the perturbations decay very fast when the shell becomes thick. The coefficient S21 is larger than the corresponding planar case for small l. When the mode number l is large enough (l > 30), Mikaelian’s result is recovered.[13] As can be seen in Fig. 4(a), for the case of extremely low-mode perturbation (l = 2), the S21 ∼ 0.2 in the spherical case is obtained for , whereas the corresponding value is ∼ 0 in the planar case. In other words, the planar interfaces are decoupled for . However, the feedback effect from the outer interface to the inner interface still exists in SG for low-mode perturbation. Thus, it is worth pointing out that special attention must be given to the control of low-mode RTI growth for the ICF ignition target design. As displayed in Fig. 4(b), the feedthrough coefficient S12 is smaller than that in the planar case for small mode number perturbations.

Fig. 4. (color online) The coefficients S21 (a) and S12 (b) versus the normalized shell thickness for mode number l = 2, 5, 30. The Atwood numbers are A1 = 0.9 and A2 = −0.9. Mikaelian’s planar result[13] is plotted for comparison.

Temporal evolutions of the normalized amplitudes of perturbations and initiated by inner interface perturbation alone and by outer interface perturbation alone are shown in Fig. 5. It can be clearly seen in Figs. 5(c) and 5(d), the phase of perturbations on the two interfaces initiated only by outer interface perturbation is shifted by π compared to that in Figs. 5(a) and 5(b) where only the inner interface is perturbed initially with the same amplitude. In Fig. 5(a), the amplitude in spherical RTI is larger than that in cylindrical and planar cases with the growing time, as the growth rate of spherical RTI is larger than the cylindrical and planar RTI for low-mode l. The amplitude on the outer interface in SG is a little larger than the other two cases, as the feedback coefficient S12 is smaller than that in the other two geometries, which is shown in Fig. 5(b). When the outer interface is initially perturbed alone, the RTI growth on the inner interface grows quickly with a negative phase due to the negative feedback effect that is displayed in Fig. 5(c). Although the inner interface is initially undisturbed, the amplitude of perturbation on the inner interface still has a large growth especially for low-mode perturbation in SG. In Fig. 5(d), the perturbations on the outer interface oscillate at the beginning, and then grow rapidly with time because of the positive feedback effect from the inner interface. In short, the RTI growth in SG is very severe for low-mode perturbations. The high-foot implosions on the NIF indicate that the instability at the ablation surface has been effectively controlled but the low-mode perturbation growth at the inner interface is still very serious,[27] resulting in the severe deformation of hot-spot interface. Therefore, the low-mode perturbations of RTI must be strictly controlled in ICF implosions.

Fig. 5. (color online) The temporal evolutions of normalized amplitudes of perturbations [panels (a) and (c)] and [panels (b) and (d)] initiated by only the inner interface perturbation and by only the outer interface perturbation for normalized thickness , respectively. Namely, , for the left column [panels (a) and (b)] and , for the right column [panels (c) and (d)]. The Atwood numbers are A1 = 0.9 and A2 = −0.9. The mode number is l = 3. The spherical, cylindrical, and planar results are plotted together.
4. Conclusion

In this research, the linear growth of RTI on two interfaces of three fluids in SG is investigated analytically. The two spherical interfaces are decoupled with increasing R2/R1. When the Atwood numbers A1 → and A2 → −1, the feedthrough coefficient S21 from the outer interface to the inner interface is approximate to ∼(R2/R1)−l + 1 and S12 from the inner interface to the outer interface is approximately equal to ∼(R2/R1)− l − 2. We have compared the linear growth of the RTI in SG with that in cylindrical and planar cases. The growth rate in SG is larger than that in CG for low-mode perturbations. In the limit of R1 → ∞ and Δ R ≪ R1, Mikaelian’s planar result[13] is recovered. The perturbation growth is very severe in SG when compared to the corresponding cases in cylindrical and planar geometry, especially for the small mode number perturbations. For the low-mode perturbations, even though the initial perturbation is only imposed on the outer interface and the inner interface is kept unperturbed, there still exists a large RTI growth on the inner interface with a negative phase (compared to the initial perturbation) due to the negative feedback effect from the outer interface. In ICF implosions, the feedthrough effect can be a severe problem that contributes obvious perturbation growth to the inner interface.[4] Our model can be used to evaluate how much perturbations on the outer interface feed through to the inner interface. Furthermore, the present work could be valuable for verification of numerical codes.

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