【作者】 涂国华；

【导师】 袁湘江；

【作者基本信息】 中国空气动力研究与发展中心 ， 流体力学， 2008， 博士

【摘要】 随着航空航天事业的发展,(高)超音速飞行器越来越多。层流和湍流的差异所导致的气动力、气动热以及气动噪声等的明显不同直接影响着飞行器的性能与安全,而目前对(高)超音速边界层的稳定性以及转捩的了解很少。(高)超音速边界层稳定性研究非常具有挑战意义,因为实验代价高且不易实现,受计算方法和计算机速度和内存的限制,计算也有一定困难,而理论研究在一百多年来还是进展缓慢。超音速与低速最大的不同之一便是激波,比如激波与边界层干扰,甚至在边界层失稳过程中还可能存在小激波,所以要求计算格式既能光滑无振荡地捕捉激波,又必须具有相当高精度以保证能精确描述边界层内不稳定因素的微小变化。本文首先从有限差分格式出发,发展出了基本无振荡的高阶激波捕捉格式,然后,我们采用抛物化稳定性研究方法以及直接数值模拟(DNS)方法等对超音速边界层的稳定性进行了研究。在计算格式方面:(1)给出了一种TVD格式的判据,TVD格式的构造方法以及相应的TVD限制器。这种方法可以很简单地把不能光滑捕捉激波的线性格式改装成能够基本光滑捕捉的非线性TVD格式。计算表明,这类TVD格式总体上能保证原始线性格式的精度,而且不会影响原格式的尺度分辨率。(2)讨论了线性紧致格式,列出了一阶至八阶线性紧致格式的系数所应该满足的条件,并采用Fourier方法对格式的精度和分辨率进行分析,结果表明紧致格式不仅具有较高精度,还具有优于传统单点显式差分的高分辨率(保频谱性)。(3)把紧致格式改写成重构函数的形式(表达在网格单元中心上),同时给出了一种紧致重构函数的边界处理方法,稳定性分析表明,这些格式是渐进稳定的。(4)发展出了适用于双曲系统的紧致TVD格式以及特征型紧致TVD格式。(5)通过大量算例着重考察了特征型紧致TVD格式的精度,并与文献中的高阶格式进行了一些比较,发现本文格式具有精度高,虚假波动小,对流场细小结构的刻画能力强等优点。本文还对格式精度、网格和CFD不确定度进行了简单考察。计算表明,若采用二阶格式,压力计算很容易达到两位真值准确,但是,除非网格极密,摩擦阻力很难达到两位真值准确。采用高阶格式可以提高计算结果的置信度,而且压力基本能达到三位真值准确。在摩阻和热流方面,计算达到两位真值准确是相对比较容易实现的,但是要想达到三位真值准确,高阶格式已经很难,低阶格式就更难。在稳定性研究方面,本文采用抛物化稳定性方程(PSE)研究了二维超音速边界层的线性和非线性稳定性,并对PSE的特征值进行了分析,发现了一些文献中没有发现的现象,比如PSE的主特征值除了一个为v/u外,其余都为零。PSE的次特征与流向波数α有关。若α=0,则次特征为抛物-双曲型。若α的实部不为零,则必然会出现复特征值,即当扰动在流向存在波动性时,必定会在稳定性方程中导致椭圆性。若α的虚部的绝对值超过某个临界值,仍然可能在稳定性方程中导致椭圆性。根据参考文献,目前主要有两种方法可以克服PSE的椭圆性,一种是对压力扰动采取修正,另一种是采用较大的空间推进步长。本文发展出了PSE的算法,采用PSE对Ma = 4.5的平板边界层的稳定性问题进行了研究,并用DNS验证了本文研究方法的正确性。通过与DNS和LST的比较,发现PSE的计算效率最高,而且计算精度基本与DNS相当。通过采用二维非线性PSE对边界层稳定性的研究,发现随着基本扰动的增大,会出现以下非线性现象:(1)稳定的高阶谐波会变得不稳定;(2)基频扰动幅值越大,失稳的高阶谐波越多,谐波的增长速度越快;(3)高阶谐波失稳的位置随着基频扰动幅值的增大而向上游移动;(4)高阶谐波增长有向饱和状态发展的趋势;(5)从涡量上看,大涡向小涡破碎,在临界层内的小涡更加丰富,大涡和小涡以及小涡和小涡之间相互作用,使得临界层内的涡量增长更快。本文发展出了三维PSE的算法,并采用三维线性PSE进行了稳定性研究,发现: (1)二维波可以自动调制出展向扰动,而且展向扰动的增长速度比其它扰动更快; (2)展向扰动呈现出两个极值点,一个在紧靠壁面的附近,另一个更大的极值点在临界层附近,临界层内的展向扰动等值线向前倾斜分布; (3)当采用数值方法进行稳定性分析时,网格密度应该与波矢方向上的波长相匹配。本文最后还采用DNS方法研究了(1)激波-边界层相互作用对小扰动传播的影响和(2)法向速度扰动在2度攻角高超音速钝锥中的演化。一个斜激波被入射到Ma=4.5的平板边界层上,与边界层相互作用诱导分离等复杂流动结构,并对小扰动的传播存在非常大的影响。本文发现分离区不仅对扰动波具有抑制作用,同时,还会激发出新的低频扰动;在分离区内扰动呈现出第二模式的特征;激波和边界层加厚会使扰动波的传播速度变慢;激波与分离对扰动波的影响并不局限在激波两侧和分离区内,都存在着相应的影响区域。通过对2度攻角高超音速钝锥的稳定性计算发现,由于攻角的存在,钝锥的稳定性特征与零攻角时有本质的差别,比如背风面的扰动比迎风面增长更快,但扰增长最慢的地方并不是迎风面,而是侧面的某个位置;又比如背风面主要是长波起作用,迎风面和侧面主要是短波起作用;斜模式不稳定在整个钝锥边界层中起最主要的作用。更多还原

【Abstract】 With the continuous development of aeronautic and astronautic technology, there are more and more supersonic and hypersonic aero/space crafts. The essential difference between laminar flow and turbulent flow causes large changes in aerodynamic force and heat as well as noise, which has vital effect on the performance and safety of the crafts. However, the achievements on supersonic/hypersonic boundary layer instability and laminar to turbulent transition are still limited. The study of the instability of supersonic/hypersonic flows is a tough task because of the following difficulties: it is very difficult to simulate this kind of flight environments in wind tunnels, and the experiment costs are expensive; direct numerical simulations (DNS) are restrained for the limitation of computing speed and RAM capability; theoretical studies about fluid instability were advancing very slowly for more than 100 years. The existence of shock waves in high-speed flows makes things much harder, such as shock-boundary interactions. Furthermore, small-structure shock waves may come into being during the instability process. Numerical algorithms should not only smoothly capture discontinuities, but also preserve favorable high-order accuracy in order to accurate simulating every trivial instability factors. A kind of nonoscillatory high-order shock-capturing finite difference schemes is formulated in this paper. Parabolized stability equations (PSE) and DNS are employed to study the instability of supersonic/hypersonic boundary layers.The following works about high-order schemes have been done: (1) Works about TVD schemes including a new TVD criterion, TVD limiters, and a new method transforming linear schemes into TVD ones. Numerical tests show that this kind of TVD schemes can generally hold the high-order accuracy of their original linear schemes, and the limiters have no effect on resolution properties. (2) Linear compact schemes are reviewed. A class of compact schemes from 1st order to 7th order is given. The accuracy and resolution power of some high-order schemes are studied by Fourier analysis method. The results show that compact schemes possess the advantage of high-order accuracy and high resolution compared with the traditional schemes. (3) Compact schemes are converted into reconstruction forms (at cell center). Boundary closure is given for compact reconstruction functions. Stability analysis shows that the schemes are linearly stable. (4) Compact-TVD schemes and characteristic-based compact-TVD schemes are devised for hyperbolic systems. (5) The properties of the characteristic-based compact-TVD schemes are checked on some benchmark problems in one, two and three space dimensions. Our results are compared with those of other high-order schemes, and the results show that our schemes are high-order accurate with high resolution and less oscillatory.The relations among CFD uncertainty, grid and numerical discrete accuracy are discussed. Our results indicate that pressure will easily reach two-digital virtual value accuracy even by lower order numerical schemes (2nd order). However, if the scheme is a 2nd order one, friction can hardly reach two-digital virtual value accuracy except that the grid is enough refined. High-order schemes (at less 3rd order) show advantages in improving the probability of accurate computing. If higher order schemes are used, pressure can even reach three-digital virtual value accuracy. Generally speaking, present numerical means can relatively easily reach two-digital virtual value accuracy, while three-digital virtual value accuracy is a tough goal for high-order schemes, and a much harder task for lower order schemes.Linear and nonlinear stability of 2D supersonic boundary layers is studied using PSE. Characteristics analysis shows some special aspects which are ignored in literatures in the references. The main characteristics of PSE are zero except v/u. The second characteristics of PSE are related to the complex wave numberα. Ifαis zero, the PSE with their second characteristics are parabolic-hyperbolic. If the real part ofαis nonzero, two or more complex characteristics must appear. If the ABS of the image part ofαis greater than a critical number, a pair of conjugate complex number would appear in the second characteristics. There are mainly two ways to suppress the ellipticity of PSE: by modifying the pressure-gradient term of the pressure disturbance or by employing large marching steps.The discretization formulas are given for PSE. PSE are employed to study the stability of a Ma = 4.5 planar plate boundary layer, and its correctness is confirmed by DNS. Compared with LST and DNS, linear and nonlinear PSE calculations yield results that are in good agreement with that of DNS, while the computational cost of PSE is the lowest among the three. The nonlinear PSE calculations show that the following nonlinear phenomena will emerge with every enhanced initial disturbance: (1) The stable harmonic waves will become unstable; (2) The larger the basic disturbance is, the more unstable harmonic waves there will be, and the faster the increasing speed of the unstable harmonic waves will become; (3) The critical positions where the harmonic waves become unstable are moving upwind with the increasing amplitude of the basic disturbance; (4) Harmonic waves are likely to reach a saturation state; (5) Large scale vortices will break down into small scale vortices, which makes the critical layer filled with abundant small scale vortices, and the mutually interaction of vortices results in that the vortices increase their amplitude more fast.The algorithms for solving 3D PSE are developed. 3D linear PSE is applied to study the stability performance of a boundary layer. The results show that: (1) 2D waves will self-modulate a transverse disturbance, and the transverse disturbance increases faster than the 2D waves; (2) Two extreme points are found on the transverse disturbance, the smaller one is closer to the wall, while the larger one is in the critical layer; (3) When numerical methods are applied to study fluids instabilities, the grid size shall be allocated in according to the wavewise wavelength instead of the streamwise wavelength.The stabilities of shock-boundary interaction and a hypersonic blunt cone at 2 degree angle of attack (AOA) are studied by DNS. An oblique shock wave is imposed on a two dimensional laminar Ma = 4.5 flow. A separation bubble forms between the shock wave and the solid plane, and the bubble causes compressing waves, expansion waves and a vortex. These complex flow structures have great influence on the stability. We find that the separation bubble can not only make the disturbance stable, but also modulate lower frequency waves. Second mode instability is found in the separation bubble. The travel speed of the disturbance is slowed down by the conjoint influence of the shock and the thickened boundary layer. The influence regions of the shock and the separation bubble are far more than the shock and the bubble themselves. Another DNS study is the evolutions of a vertical velocity disturbance in the boundary layer of the hypersonic cone at 2 degree AOA. The results show that the stability characters are quite different from cases without AOA. The leeward boundary is more unstable than the windward boundary. The most stable position is not in the windward, but in the sideward. The unstable waves in the leeward are mainly long waves, while the unstable waves in the windward and sideward are shorter ones. The main instability mechanism is an oblique model.更多还原