高超声速钝锥边界层稳定性特征
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摘要
本论文利用一般正交曲线坐标系下的抛物化稳定性方程(PSE)和高精度激波装配法,开展了高超声速钝锥边界层稳定性特征的研究工作,主要研究成果及创新如下：
(1)对高精度激波装配法提出了一种改进的并行算法,提高了该方法的并行求解效率,扩大了应用范围,同时构造了七阶紧致迎风有限差分格式用于逼近对流项,以提高方法的分辨率.通过对钝锥绕流问题的验证计算表明,本文的计算结果与他人结果[87,108]吻合的很好,因此,当前的数值方法有效、可靠,能够用于模拟来流中10-5-10-4量级的小扰动演化问题.
(2)发展了基于一般正交曲线坐标系的抛物化稳定性方程及其计算软件,该软件包括线性PSE和非线性PSE两个求解器。与直接数值模拟(DNS)相比,该方法不仅便于对边界层内扰动波的演化特征进行分析,而且降低了至少一个数量级的计算时间。算例表明,该方法的结果与DNS以及线性稳定性理论(LST)的结果吻合甚好,验证了其可靠性.
(3)细致地分析了壁温对球头半径为3.81×10-3米,半锥角为7°的高超声速钝锥边界层稳定性的影响,得到如下有意义的结果：
ⅰ)DNS结果表明,壁温变化显著地影响高超声速钝锥边界层内广义速度拐点沿壁面法向的数量和位置；LST分析表明,降低壁温加速第二模态不稳定波的增长,同时抑制了第一模态扰动的增长.向下游推进中,当地最不稳定的第二模态扰动频率逐渐降低,且增长率持续增大；当地最不稳定的第一模态频率先升高,然后又有所降低,且增长率也持续增大.壁温越低中性曲线越靠后,且第一模态推后的比第二模态更严重.由此可见,壁温对高超声速边界层稳定性有显著的影响.
ⅱ)当来流中加入10-4的小扰动时,DNS表明下游290头半径左右的壁面扰动幅值与壁温之间没有单调变化关系,对于冷却壁存一个临界温度Tc∈(280,375K),当壁温低于Tc后,冷却壁时的扰动幅值大于绝热壁的情况,反之,结论相反。这一结论与Stetson[79]的实验基本一致。
ⅲ)在中性曲线下分支处,壁面压力扰动的感受性系数的分布表明,壁温越低,感受性系数也越小.低频扰动波的感受性系数大约在(0.1,0.2)之间,而高频扰动波的情况大约在(0.7,0.8)之间.
ⅳ)用线性PSE分析了不同壁温情况下小扰动的演化特征,结果表明,壁温变化显著影响流向波数、扰动增长率和扰动幅值的变化特征.在壁面特定位置以前,壁温越低波数越小,当大于这个值以后,结论相反；降低壁温使第二模态增长率增大；且壁温越低最终扰动幅值越大.
ⅴ)用非线性PSE模拟边界层中有限振幅扰动演化表明,边界层内扰动演化足够长时间后高次谐波得以快速增长,且降低壁温加速了高次谐波的增长,特别是三维高次谐波具有更大的增长率,高次谐波促进了边界层不稳定的发展.
This thesis investigates the stability characteristics of boundary layers in hypersonic flows around a blunt cone by using a kind of parabolized stability equations (PSE) in a general orthogonal curvilinear system of coordinates, to-gether with a shock-fitting method. The main contributions in this work are as follows:
(1) The parallel algorithm of the original shock-fitting method is consider-ably improved, and hence the efficiency of solving problems in parallel with the method is appreciably raised and its applicability scope enlarged. Additionally, a seventh-order upwind compact finite difference scheme is developed for improving resolution of the present method. The numerical tests indicate that the presented numerical method and code are effective and reliable, and could be employed to simulate the receptivity in the boundary layer of super/hypersonic flows around blunt cones, even to reproduce the time evolution of the small disturbances of order 10-5 - 10-4 in free stream.
(2) A new PSE (parabolized stability equation) software system is devel-oped in the general orthogonal curvilinear coordinates, consisting of two solvers respectively for linear and nonlinear PSEs. Compared to DNS, the PSE is not only convenient to analyze the evolution of disturbance in the boundary layer, but also reduce the computing time at least by one order of magnitude. The numerical examples show that the presented PSE solvers could give simulated results agreeing well with the DNS results and LST predictions, providing an efficient and reliable computing tool for analyzing the stability characteristics of super/hypersonic boundary layer.
(3)The effects of wall temperature on the boundary layer stability for the case of a spherical nose with the radius of 3.81 x 10-3m and a half cone angle of 7°at the Mach number of 7.99 are meticulously analyzed and the following meaningful results are obtained.
i) The DNS results indicate that the wall temperature remakably impact the number and positions of GIPs (general inflection points) along the wall normal direction in the boundary layer of hypersonic flow around the blunt cone. The LST results show that decreasing the wall temperature accelerates the growth of the second-mode disturbance and restrains the growth of the first-mode dis-turbance. The frequency of the most unstable second-mode local disturbance decreases monotonically along the streamwise direction. However, the frequency of the most unstable first-mode local disturbance does not monotonically change, which first increases and then decreases. The growth rates of the first mode and second mode keep increasing along the streamwise direction. But the growth rate of the second mode is much larger than that of the first mode. The neutral curve moves toward downstream with decreasing wall temperature. Therefore, the wall temperature has significant effects on the stability of hypersonic boundary layers.
ii) When the free-stream disturbances are of the order 10-4, the DNS re-sults imply that the amplitude of disturbance at about 290 times of nose radius downstream along the surface does not change monotonically with the wall tem-perature. There exists a critical wall temperature Tc∈(280,375K) for cold wall. If the wall temperature is lower than Tc, then the amplitude of disturbance for the cold wall is larger than that for the adiabatic wall, and vice versa. Such a conclusion agrees with that drawn by Stetson [79].
iii) The receptivity analysis shows that the receptivity coefficient decreases with decreasing wall temperature. The receptivity of lower frequency disturbance waves falls into the interval (0.1,0.2), and that of higher ones into the interval (0.7,0.8). Hence the strength of the receptivity for the second modes is larger than that of the first ones.
iv) The results of linear PSE show that decreasing the wall temperature ap-preciably affects the streamwise wave number, growth rate, and amplitude of 2D and 3D small disturbances. The streamwise wavenumber decreases with decreas-ing wall temperature at the front of a given point on the wall surface, but the situation is in contrast behind this point, which depends on the wall temperature. The amplitude of disturbance increases with decreasing wall temperature.
v) The nonlinear PSE simulations indicate that lots of high harmonics are enhanced quickly after sufficiently long time evolution of finite amplitude distur-bances. Moreover, decreasing the surface temperature promotes the growth of the higher harmonics, and the 3D harmonic modes have larger growth rate than the 2D ones.
(1)对高精度激波装配法提出了一种改进的并行算法,提高了该方法的并行求解效率,扩大了应用范围,同时构造了七阶紧致迎风有限差分格式用于逼近对流项,以提高方法的分辨率.通过对钝锥绕流问题的验证计算表明,本文的计算结果与他人结果[87,108]吻合的很好,因此,当前的数值方法有效、可靠,能够用于模拟来流中10-5-10-4量级的小扰动演化问题.
(2)发展了基于一般正交曲线坐标系的抛物化稳定性方程及其计算软件,该软件包括线性PSE和非线性PSE两个求解器。与直接数值模拟(DNS)相比,该方法不仅便于对边界层内扰动波的演化特征进行分析,而且降低了至少一个数量级的计算时间。算例表明,该方法的结果与DNS以及线性稳定性理论(LST)的结果吻合甚好,验证了其可靠性.
(3)细致地分析了壁温对球头半径为3.81×10-3米,半锥角为7°的高超声速钝锥边界层稳定性的影响,得到如下有意义的结果：
ⅰ)DNS结果表明,壁温变化显著地影响高超声速钝锥边界层内广义速度拐点沿壁面法向的数量和位置；LST分析表明,降低壁温加速第二模态不稳定波的增长,同时抑制了第一模态扰动的增长.向下游推进中,当地最不稳定的第二模态扰动频率逐渐降低,且增长率持续增大；当地最不稳定的第一模态频率先升高,然后又有所降低,且增长率也持续增大.壁温越低中性曲线越靠后,且第一模态推后的比第二模态更严重.由此可见,壁温对高超声速边界层稳定性有显著的影响.
ⅱ)当来流中加入10-4的小扰动时,DNS表明下游290头半径左右的壁面扰动幅值与壁温之间没有单调变化关系,对于冷却壁存一个临界温度Tc∈(280,375K),当壁温低于Tc后,冷却壁时的扰动幅值大于绝热壁的情况,反之,结论相反。这一结论与Stetson[79]的实验基本一致。
ⅲ)在中性曲线下分支处,壁面压力扰动的感受性系数的分布表明,壁温越低,感受性系数也越小.低频扰动波的感受性系数大约在(0.1,0.2)之间,而高频扰动波的情况大约在(0.7,0.8)之间.
ⅳ)用线性PSE分析了不同壁温情况下小扰动的演化特征,结果表明,壁温变化显著影响流向波数、扰动增长率和扰动幅值的变化特征.在壁面特定位置以前,壁温越低波数越小,当大于这个值以后,结论相反；降低壁温使第二模态增长率增大；且壁温越低最终扰动幅值越大.
ⅴ)用非线性PSE模拟边界层中有限振幅扰动演化表明,边界层内扰动演化足够长时间后高次谐波得以快速增长,且降低壁温加速了高次谐波的增长,特别是三维高次谐波具有更大的增长率,高次谐波促进了边界层不稳定的发展.
This thesis investigates the stability characteristics of boundary layers in hypersonic flows around a blunt cone by using a kind of parabolized stability equations (PSE) in a general orthogonal curvilinear system of coordinates, to-gether with a shock-fitting method. The main contributions in this work are as follows:
(1) The parallel algorithm of the original shock-fitting method is consider-ably improved, and hence the efficiency of solving problems in parallel with the method is appreciably raised and its applicability scope enlarged. Additionally, a seventh-order upwind compact finite difference scheme is developed for improving resolution of the present method. The numerical tests indicate that the presented numerical method and code are effective and reliable, and could be employed to simulate the receptivity in the boundary layer of super/hypersonic flows around blunt cones, even to reproduce the time evolution of the small disturbances of order 10-5 - 10-4 in free stream.
(2) A new PSE (parabolized stability equation) software system is devel-oped in the general orthogonal curvilinear coordinates, consisting of two solvers respectively for linear and nonlinear PSEs. Compared to DNS, the PSE is not only convenient to analyze the evolution of disturbance in the boundary layer, but also reduce the computing time at least by one order of magnitude. The numerical examples show that the presented PSE solvers could give simulated results agreeing well with the DNS results and LST predictions, providing an efficient and reliable computing tool for analyzing the stability characteristics of super/hypersonic boundary layer.
(3)The effects of wall temperature on the boundary layer stability for the case of a spherical nose with the radius of 3.81 x 10-3m and a half cone angle of 7°at the Mach number of 7.99 are meticulously analyzed and the following meaningful results are obtained.
i) The DNS results indicate that the wall temperature remakably impact the number and positions of GIPs (general inflection points) along the wall normal direction in the boundary layer of hypersonic flow around the blunt cone. The LST results show that decreasing the wall temperature accelerates the growth of the second-mode disturbance and restrains the growth of the first-mode dis-turbance. The frequency of the most unstable second-mode local disturbance decreases monotonically along the streamwise direction. However, the frequency of the most unstable first-mode local disturbance does not monotonically change, which first increases and then decreases. The growth rates of the first mode and second mode keep increasing along the streamwise direction. But the growth rate of the second mode is much larger than that of the first mode. The neutral curve moves toward downstream with decreasing wall temperature. Therefore, the wall temperature has significant effects on the stability of hypersonic boundary layers.
ii) When the free-stream disturbances are of the order 10-4, the DNS re-sults imply that the amplitude of disturbance at about 290 times of nose radius downstream along the surface does not change monotonically with the wall tem-perature. There exists a critical wall temperature Tc∈(280,375K) for cold wall. If the wall temperature is lower than Tc, then the amplitude of disturbance for the cold wall is larger than that for the adiabatic wall, and vice versa. Such a conclusion agrees with that drawn by Stetson [79].
iii) The receptivity analysis shows that the receptivity coefficient decreases with decreasing wall temperature. The receptivity of lower frequency disturbance waves falls into the interval (0.1,0.2), and that of higher ones into the interval (0.7,0.8). Hence the strength of the receptivity for the second modes is larger than that of the first ones.
iv) The results of linear PSE show that decreasing the wall temperature ap-preciably affects the streamwise wave number, growth rate, and amplitude of 2D and 3D small disturbances. The streamwise wavenumber decreases with decreas-ing wall temperature at the front of a given point on the wall surface, but the situation is in contrast behind this point, which depends on the wall temperature. The amplitude of disturbance increases with decreasing wall temperature.
v) The nonlinear PSE simulations indicate that lots of high harmonics are enhanced quickly after sufficiently long time evolution of finite amplitude distur-bances. Moreover, decreasing the surface temperature promotes the growth of the higher harmonics, and the 3D harmonic modes have larger growth rate than the 2D ones.
引文
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