激光烧蚀瑞利—泰勒不稳定性研究
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摘要
本文采用流动稳定性的线性和弱非线性理论,对预热条件下激光烧蚀面上的瑞利-泰勒不稳定性问题进行了研究。在进行稳定性分析时,基本流选用数值求解的定常基本流,且将烧蚀面附近流场看作是连续的,从而充分包含了预热条件下烧蚀面变宽的因素。本文重点考察了激光烧蚀瑞利-泰勒不稳定性扰动分布的特征和幅值的演化规律。
首先利用空间二阶精度的直接数值模拟对激光驱动下CH靶的运动进行了分析,CH靶的厚度为200μm,激光功率强度线性增长4 ns后达到峰值强度1014 W/cm2,然后维持不变,得到了在一段时间内CH靶做匀加速运动的平衡解。根据平衡解的烧蚀参数,本文给出了在烧蚀面的加速运动坐标系下的定常解,并与直接数值模拟的结果进行了比较。用该定常解做基本流,利用流动稳定性的线性理论和弱非线性理论对烧蚀瑞利-泰勒不稳定性进行了研究,并通过空间精度为四阶的直接数值模拟对稳定性理论的结果进行了验证,得出以下结论:
1.激光烧蚀面附近流场匀加速运动阶段的平衡解可以用烧蚀面匀加速运动坐标系下、具有相同烧蚀参数的定常解描述。
2.利用流动稳定性的线性理论研究了烧蚀瑞利-泰勒不稳定性的线性增长规律。给出了不稳定波各扰动量的特征函数,分析了特征函数的特点。理论分析给出的增长率曲线与修正的Lindl公式给出的结果基本符合,与直接数值模拟的结果几乎完全相同。
3.在弱非线性理论中,对具有较大增长率的扰动,提出了新的确定Landau系数方法,即基模产生项的形状函数应该与相应的伴随特征值问题的特征函数双线性正交,该方法能方便得用于高阶展开的情况。
4.利用Stuart弱非线性理论的方法对烧蚀瑞利-泰勒不稳定性的弱非线性增长规律进行了研究。给出了各次谐波的形状函数,并分析了形状函数的特点。弱非线性理论出了Landau系数曲线,波数较大时,Landau系数为负数,而当波数较小时,Landau系数则为正数。弱非线性理论分析的结果还表明:一次谐波的非线性修正中,Landau系数起的作用较小,基模产生项的非线性修正起主要作用。
5.在弱非线性理论的展开中,提出了将基模扰动的线性幅值作为小参数进行展开的方法,与Stuart的弱非线性理论相比,该方法具有如下优点:幅值方程中不再包含未知的Landau系数,不需要补充其它条件来确定Landau系数,方程可直接求解,使展开变得简单,有利于进行高阶展开。与DNS的结果对比表明:当非线性作用相对较强时,改进方法的高阶展开,在一定程度上能更准确的描述烧蚀瑞利-泰勒不稳定性的演化规律。
In this paper, hydrodynamic linear stability theory and weakly nonlinear theory are adopted to study the Rayleigh-Taylor instability at the laser ablation front in the preheat case. In the stability analysis, the numerically solved steady state flow field is used as the basic flow, and the ablation front is treated as a continuous flow field so as to take into account the broaden thickness of the ablation front. In this paper, amplitude distribution and amplitude evolvement of the laser ablative Rayleigh-Taylor instability is the main focus.
First, the behavior of the laser-driven CH ablation target is analyzed using a direct numerical simulation which is second order in space. The thickness of the CH target is 200μm, the laser intensity linearly grows 4 ns to it maximum of 1014 W/cm2, and then keeps on. The constantly accelerating equilibrium of the ablation CH target is obtained. Based on the ablation parameters, the steady state flow field at the reference frame of the constantly accelerating ablation front is given and compared with the result of direct numerical simulation. The hydrodynamic linear stability theory and weakly nonlinear theory are used to study the ablative Rayleigh-Taylor instability based on the steady state flow field, and the results are tested through comparison with a direct numerical simulation which is fourth order accurate in space. And the following conclusions can be drawn:
1. The constantly accelerating equilibrium of the laser ablation front can be represented by the corresponding steady state flow field at the reference frame of the constantly accelerating ablation front which is obtained using the same ablation parameters.
2. The linear growth of the ablative Rayleigh-Taylor instability is studied using the linear stability theory. The Eigen-Functions of the instable perturbations are presented, and the characteristics of the Eigen-Functions are analyzed. The growth rate of the linear stability theory is in agreement with the modified Lindl formula, and is almost identical to that of direct numerical simulation.
3. In the weakly nonlinear theory, a new method is proposed to determine the Landau constant for the perturbations with relatively larger growth rates. The method is that the shape function of the regenerated fundamental mode is bi-orthogonal to the Eigen-function of the ad-joint linear stability problem. This method can be easily applied for higher order expansion (in the weakly nonlinear theory).
4. The weakly nonlinear growth of ablative Rayleigh-Taylor instability is studied using the weakly nonlinear theory by Stuart. Shape functions of each harmonic are presented and their characteristics are analyzed. The Landau constant curve is given by the weakly nonlinear theory. It is negative when the wave number is small and positive when the wave number is large. The weakly nonlinear theory shows that the Landau constant has relatively smaller correction to the first harmonic, whereas the regenerated fundamental mode plays the dominant role.
5. In the weakly nonlinear theory, a new expansion method with respect to the linear amplitude of the fundamental mode is proposed. Compared with the expansion method by Stuart, the new method has the following advantages: the amplitude equations no longer contains the Landau constant, no extra condition is needed to determine the Landau constant, and the equations can be solved directly. The expansion method is simpler, and is convenient for higher order expansion. The compassion with direct numerical simulation shows that: when the nonlinear effect is relatively stronger, the new method can describe the ablative Rayleigh-Taylor instability more accurately in a certain sense.
首先利用空间二阶精度的直接数值模拟对激光驱动下CH靶的运动进行了分析,CH靶的厚度为200μm,激光功率强度线性增长4 ns后达到峰值强度1014 W/cm2,然后维持不变,得到了在一段时间内CH靶做匀加速运动的平衡解。根据平衡解的烧蚀参数,本文给出了在烧蚀面的加速运动坐标系下的定常解,并与直接数值模拟的结果进行了比较。用该定常解做基本流,利用流动稳定性的线性理论和弱非线性理论对烧蚀瑞利-泰勒不稳定性进行了研究,并通过空间精度为四阶的直接数值模拟对稳定性理论的结果进行了验证,得出以下结论:
1.激光烧蚀面附近流场匀加速运动阶段的平衡解可以用烧蚀面匀加速运动坐标系下、具有相同烧蚀参数的定常解描述。
2.利用流动稳定性的线性理论研究了烧蚀瑞利-泰勒不稳定性的线性增长规律。给出了不稳定波各扰动量的特征函数,分析了特征函数的特点。理论分析给出的增长率曲线与修正的Lindl公式给出的结果基本符合,与直接数值模拟的结果几乎完全相同。
3.在弱非线性理论中,对具有较大增长率的扰动,提出了新的确定Landau系数方法,即基模产生项的形状函数应该与相应的伴随特征值问题的特征函数双线性正交,该方法能方便得用于高阶展开的情况。
4.利用Stuart弱非线性理论的方法对烧蚀瑞利-泰勒不稳定性的弱非线性增长规律进行了研究。给出了各次谐波的形状函数,并分析了形状函数的特点。弱非线性理论出了Landau系数曲线,波数较大时,Landau系数为负数,而当波数较小时,Landau系数则为正数。弱非线性理论分析的结果还表明:一次谐波的非线性修正中,Landau系数起的作用较小,基模产生项的非线性修正起主要作用。
5.在弱非线性理论的展开中,提出了将基模扰动的线性幅值作为小参数进行展开的方法,与Stuart的弱非线性理论相比,该方法具有如下优点:幅值方程中不再包含未知的Landau系数,不需要补充其它条件来确定Landau系数,方程可直接求解,使展开变得简单,有利于进行高阶展开。与DNS的结果对比表明:当非线性作用相对较强时,改进方法的高阶展开,在一定程度上能更准确的描述烧蚀瑞利-泰勒不稳定性的演化规律。
In this paper, hydrodynamic linear stability theory and weakly nonlinear theory are adopted to study the Rayleigh-Taylor instability at the laser ablation front in the preheat case. In the stability analysis, the numerically solved steady state flow field is used as the basic flow, and the ablation front is treated as a continuous flow field so as to take into account the broaden thickness of the ablation front. In this paper, amplitude distribution and amplitude evolvement of the laser ablative Rayleigh-Taylor instability is the main focus.
First, the behavior of the laser-driven CH ablation target is analyzed using a direct numerical simulation which is second order in space. The thickness of the CH target is 200μm, the laser intensity linearly grows 4 ns to it maximum of 1014 W/cm2, and then keeps on. The constantly accelerating equilibrium of the ablation CH target is obtained. Based on the ablation parameters, the steady state flow field at the reference frame of the constantly accelerating ablation front is given and compared with the result of direct numerical simulation. The hydrodynamic linear stability theory and weakly nonlinear theory are used to study the ablative Rayleigh-Taylor instability based on the steady state flow field, and the results are tested through comparison with a direct numerical simulation which is fourth order accurate in space. And the following conclusions can be drawn:
1. The constantly accelerating equilibrium of the laser ablation front can be represented by the corresponding steady state flow field at the reference frame of the constantly accelerating ablation front which is obtained using the same ablation parameters.
2. The linear growth of the ablative Rayleigh-Taylor instability is studied using the linear stability theory. The Eigen-Functions of the instable perturbations are presented, and the characteristics of the Eigen-Functions are analyzed. The growth rate of the linear stability theory is in agreement with the modified Lindl formula, and is almost identical to that of direct numerical simulation.
3. In the weakly nonlinear theory, a new method is proposed to determine the Landau constant for the perturbations with relatively larger growth rates. The method is that the shape function of the regenerated fundamental mode is bi-orthogonal to the Eigen-function of the ad-joint linear stability problem. This method can be easily applied for higher order expansion (in the weakly nonlinear theory).
4. The weakly nonlinear growth of ablative Rayleigh-Taylor instability is studied using the weakly nonlinear theory by Stuart. Shape functions of each harmonic are presented and their characteristics are analyzed. The Landau constant curve is given by the weakly nonlinear theory. It is negative when the wave number is small and positive when the wave number is large. The weakly nonlinear theory shows that the Landau constant has relatively smaller correction to the first harmonic, whereas the regenerated fundamental mode plays the dominant role.
5. In the weakly nonlinear theory, a new expansion method with respect to the linear amplitude of the fundamental mode is proposed. Compared with the expansion method by Stuart, the new method has the following advantages: the amplitude equations no longer contains the Landau constant, no extra condition is needed to determine the Landau constant, and the equations can be solved directly. The expansion method is simpler, and is convenient for higher order expansion. The compassion with direct numerical simulation shows that: when the nonlinear effect is relatively stronger, the new method can describe the ablative Rayleigh-Taylor instability more accurately in a certain sense.
引文
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