Study on preheating ablative effects of two-mode Rayleigh-Taylor instability
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摘要: 针对双模扰动下的烧蚀瑞利-泰勒不稳定性增长问题,采用高精度的数值计算方法,研究了不同预热程度下模耦合产生的多个高次谐波幅值的发展和演化问题。研究表明,三种预热烧蚀条件下,当扰动基模满足长波与短波耦合方式时,谐波中的长波模态占主导,而短波模发展明显受到抑制;当满足短波与短波耦合时,耦合结果带来了许多新的增长较快的长波模态,此时短波模增长呈现小幅震荡形式。比较两种耦合方式可以发现,长波结构在烧蚀瑞利-泰勒不稳定性弱非线性阶段都占主导地位,尤其是短波与短波耦合中气泡与尖钉表现出不同于两个基模的长波模结构。进一步分析预热效应对模耦合增长的影响,发现预热程度越强就越能削弱耦合谐波的增长,这说明预热对烧蚀瑞利-泰勒不稳定性具有致稳作用,这对惯性约束聚变工程中控制烧蚀瑞利-泰勒不稳定性发展具有重要意义。Abstract: Aiming at the growth of ablative Rayleigh-Taylor instability with two perturbations, the evolutions of the amplitudes of high-order harmonics excited by two-mode coupling under different preheating conditions are studied by using a high-precision numerical simulation method. When the fundamental modes are a long-wavelength and a short-wavelength mode, the long-wavelength modes of the excited harmonics are dominant, while the development of short-wavelength modes are obviously suppressed; when the fundamental modes are two short-wavelength modes, many fast-growing and long-wavelength modes are excited, and the growth of short-wavelength modes are in the form of small oscillation. By comparing the two different two-mode coupling cases, it is found that the long-wavelength structures are dominant in the weakly nonlinear stage. Especially, in the two short-wavelength modes coupling case, the bubbles and spikes show long-wavelength structures which are different from the two fundamental modes. By further comparing the three preheating ablative effects, it is found that the higher the preheat degree is, the more the coupled harmonics growth will be weakened. It is of great significance to control the development of ablative Rayleigh-Taylor instability in inertial confinement fusion engineering.
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Key words:
- two-mode perturbations /
- preheating /
- Rayleigh-Taylor instability /
- high-order harmonics /
- bubbles /
- spikes
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当轻流体加速重流体时,流体交界面处会发生瑞利-泰勒不稳定性(RTI)[1-4]。惯性约束聚变(ICF)[5-8]内爆过程中,靶丸外表面高温高压低密度等离子体加速压缩内层高密度流体时,流体交界面会发生烧蚀瑞利-泰勒不稳定性(ARTI)[9-12],并且由于激光强度分布不均、靶丸表面粗糙程度等因素,ARTI的发展会被进一步放大,从而破坏内爆壳层的对称性,导致靶丸[5]破裂,甚至影响点火热斑的形成。因此,ARTI的研究对聚变靶的内爆压缩及点火烧蚀具有重要意义。通常利用数值模拟方法研究ARTI,首先在无扰动下对一维流场进行稳定性分析,得到一个加速度基本保持不变的平衡流场,然后在烧蚀面处引入物理量的小扰动,追踪流场发展获得进一步的不稳定性分析。以往的单模扰动[13-18]研究表明,ARTI的演化大致可以分为线性阶段与非线性阶段,其增长会受到烧蚀面附近物质的对流[7]、烧蚀面密度梯度[6]和预热[19-21]等因素的影响,其中密度梯度效应对ARTI的线性增长和非线性阶段模耦合发展具有显著影响[22]。
由于引起界面小扰动的因素很多,因此在烧蚀面附近可能引发多模扰动,从而激发许多新的谐波[23-25],这些耦合谐波会进一步影响ARTI的增长。本文研究的主要内容是双模扰动下带有预热的ARTI发展问题,这里的预热是指较低能的电子预热,所以能量是沉积在烧蚀面附近,因此如果预热程度越高,意味着烧蚀面密度梯度越大,烧蚀面宽则烧蚀程度强。由于预热会引起烧蚀面宽度(密度梯度效应)变化,数值模拟中设置强、中等和弱三种不同预热条件,通过一维稳定性分析,首先得到三种不同预热条件下的稳态流场,并在此基础上引入波长不同的两个速度扰动[26]。研究中将数值模拟分为两组:第一组是长波与短波分别在强、中和弱三种预热条件下的耦合;第二组是短波与短波分别在强、中和弱三种预热条件下的耦合。通过数值计算给出高次谐波幅值随时间演化的曲线和ARTI演化一段时间后的气泡尖钉结构图,比较分析两种不同耦合方式下预热强度对谐波演化的影响可以发现,随着预热强度的增大,谐波增长趋势变缓。
1. 物理模型与控制方程
1.1 一维稳态流场
我们采用的物理模型为
200μm 厚的CH平面靶,初始密度为1g/cm3 ,激光功率线性上升至4 ns后达到峰值强度1014W/cm3 然后维持峰值不变,临界密度为0.03g/cm3 。将等离子体视为理想气体,考虑电子热传导,守恒型控制方程如下[27]∂ρ∂t+∇⋅ρv=0 (1) ∂(ρv)∂t+∇⋅(ρvv)+∇⋅p=0 (2) ∂∂t(12ρv⋅v+ε)+∇⋅[(12ρv⋅v+ε+p)v]=∇⋅(κ∇T)+S (3) 式中:
ρ ,v ,T 和ε=cvρT 分别为密度、速度、温度和内能,其中cv=Γ/(γh−1) 是定容比热,γh=5/3 是CH材料中气体绝热指数;κ 为电子热传导系数;S 为激光能量。对于电子热传导,我们采用的模型为
κ=κSHf(T)h (4) 其中
κSH=kST5/2 为经典电子热传导系数,h 取1,f(T)=c/T3/2+b/T+1 为预热函数[19]。数值模拟中可以通过改变b 和c 的值来改变预热程度。我们选取强预热(strong preheating,SP)、中等预热(moderate preheating,MP)和弱预热(weak preheating,WP)所对应的b 和c 的值,如表1所示[28-30]。表 1 三种预热条件的参数设置Table 1. Parameter settings of strong, medium and weak preheating conditionscase b c strong preheating (SP) 8.6 1.6 moderate preheating (MP) 2 0.4 weak preheating (WP) 0.86 0.24 一维计算到8 ns左右,烧蚀面附近流场加速度基本保持不变,达到稳态。8 ns时三种预热情况下的一维平衡流场如图1所示。以最小密度梯度定标长度
Lm=min[|ρ(dρ/dx)−1|] 来表征烧蚀面的宽度,SP,MP和WP三种情况下的一维平衡流场对应的Lm 分别为1.83,0.41 和0.23μm ,这表明三种模型中,预热程度越大,烧蚀面越宽。1.2 线性增长率
设一维平衡流场以加速度g作匀加速运动,在加速运动的非惯性系坐标系下,二维形式的控制方程可以写为
∂ρ∂t+∂ρu∂x+∂ρv∂y=0 (5) ∂ρu∂t+∂(ρu2+p)∂x+ρ∂uv∂x=ρg (6) ∂ρv∂t+∂ρuv∂x+∂(ρv2+p)∂y=0 (7) ∂ρε∂t+∂(ρε+p)u∂x+∂(ρε+p)v∂y=∂∂xκ∂T∂x+∂∂y(κ∂T∂y)+ρug (8) 式中:
u 是x 方向速度;v 是y 方向速度。该物理模型中烧蚀面距离激光吸收位置(临界密度附近)较远,因此可以忽略激光能量项S ,能量靠电子热传导传输。引入一个初始速度扰动u′=∑iuicos(kiy+ϕi)exp(−ki|x−x0|) (9) v′=∑ivisin(kiy+ϕi)exp(−ki|x−x0|) (10) 其中
ki=2π/λ 表示波数,ϕi 表示相位,本文不考虑相位变化,即ϕi=0 。通过数值计算得到各个波长下对应的线性扰动增长率如图2所示,WP情况下线性增长率最大,MP次之,SP最小。WP,MP和SP情况下对应的截止波长分别为λWP=4μm ,λMP=4.2μm 和λSP=5μm ,这表明随着预热增强,截止波长在变大。三种预热情况下,扰动波长λ=18μm 左右具有最大线性增长率。接下来为了研究预热烧蚀效应对不同模耦合发展的影响,我们计算两组对比数值模拟。第一组扰动选在截止波长附近
λ1=10μm 的短波与远离截止波长λ2=40μm 的长波,第二组在截止波长附近λ1=10μm 与λ2=12μm 的两个短波。不同波长初始幅值相同,为满足扰动周期性特点,y 方向计算域为120μm 。2. 分析与讨论
2.1 长波与短波耦合
随着不稳定性的增长进入弱非线性阶段,两个初始扰动模的相互作用开始变得明显,将产生和频谐波与差频谐波及介于两者之间的其他谐波[31]。
长波与短波耦合所得到谐波幅值
η 随时间的演化如图3所示。图3(a)为SP情况,图3(b)为MP情况,图3(c)为WP情况。比较发现,在三种预热条件下都产生了波长为40μm 和20μm 的幅值涨幅较大的长波模,以及波长为13 ,10μm 的幅值涨幅较小的短波模。在5 ns左右,SP情况下长波模幅值迅速增长,短波模幅值增长被抑制,随后长波模一直占据上风,到7.5 ns左右,40μm 和20μm 的长波模幅值达到第一个极值点30μm 左右,随后开始以震荡形式衰减,短波模幅值同时震荡衰减。在MP与WP情况下,5 ns左右长波模迅速增长,短波模幅值增长被抑制,长波模一直占据上风,到7.5 ns左右40μm 长波模幅值达到30μm 左右,而20μm 的长波模幅值达到了70μm 左右,随后长波模与短波模开始迅速衰减。这表明长波与短波耦合情况,在三种不同预热程度下,长波模在弱非线性阶段占主导地位,而短波模受到抑制。在预热程度相对较少的MP与WP情况下,20μm 的长波模幅值达到的峰值要高很多,这表明预热可以抑制谐波的增长。我们给出了波长为
λ1=10μm 与λ2=40μm 的扰动下t=7.5 ns时的密度等值线,如图4所示。图4(a)为SP情况,图4(b)为MP情况,图4(c)为WP情况。可以看到此时气泡展现出两种不同的波长的结构,其波长大致为40μm 和20μm ,这是因为模耦合产生的和频与差频谐波(8μm 和13μm ,前者增长不明显,文中没有给出其增长曲线)都为短波,耦合谐波的增长不大,演化中主要增长的模式为基模中的长波模(40μm )以及其二次谐波(20μm )。随着预热程度减小,这两种结构更加明显。这进一步说明在长波与短波的耦合情况下,长波结构占主导地位,且预热程度越高,气泡尖钉增长越慢。2.2 短波与短波耦合
短波与短波耦合所得到谐波幅值随时间的演化如图5所示。图5(a)为SP情况,图5(b)为MP情况,图5(c)为WP情况。在三种预热条件下都产生了波长为
60 ,30 ,20 和15μm 的长波模,以及波长为12μm 和10μm 的幅值涨幅较小的短波模。在SP情况下,5.5 ns左右波长为60μm 和30μm 的长波模的幅值迅速增长,短波模的幅值出现震荡,长波模一直占据上风,随后在8 ns左右,波长为60μm 的长波模达到峰值,波长为30μm 的长波模继续增长。在MP与WP情况下,5 ns左右波长为60μm 和30μm 的长波模迅速增长,长波模同样占据上风,而短波的幅值在整个过程中都处于小幅值震荡状态。MP和WP情况下,长波模幅值达到的最大幅值的时间比SP情况要早5 ns左右。这表明在短波与短波耦合中,会激发原本基模扰动中没有的长波模(该模态的波长满足两个短波波长的最小公倍数值),并且在弱非线性阶段长波模占据主导,在预热较多的SP情况下,弱非线性阶段谐波幅值达到峰值的时间要更长。波长为
λ1=10μm 与λ2=12μm 的扰动下,t =7.5 ns时的密度等值线如图6所示。图6(a)为SP情况,图6(b)为MP情况,图6(c)为WP情况。此时气泡展现出多种不同的波长的结构,其中最为明显的是大约60μm 和30μm 的长波结构。在MP和WP情况下,60μm 的伞状尖钉结构更加明显。这也进一步说明了在短波与短波的耦合情况下,依然是长波结构占主导地位,且预热程度越小,气泡尖钉增长越快。3. 结 论
本文采用数值求解流体方程的方法研究了不同预热条件下的双模扰动烧蚀-瑞利泰勒不稳定性的演化规律。通过一维稳态分析得到了强、中等和弱预热条件下的平衡流场,并在此基础上引入两个速度扰动,获得了双模耦合谐波幅值随时间的演化。在三种预热条件下,不论是长波与短波耦合还是短波与短波耦合,都会激发出许多新的高次谐波,并且谐波中的长波总是占主导。在我们设置的预热程度范围内,预热程度越大,长波结构增长越缓慢,这表明预热会抑制双模耦合高次谐波的增长,对烧蚀瑞利-泰勒不稳定性具有一定的致稳作用。
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表 1 三种预热条件的参数设置
Table 1. Parameter settings of strong, medium and weak preheating conditions
case b c strong preheating (SP) 8.6 1.6 moderate preheating (MP) 2 0.4 weak preheating (WP) 0.86 0.24 -
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