Stability Analysis Of Supersonic Boundary Layers And High Order Shock-Capturing Schemes

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.