A Variational Approach Via Bipotentials for a Class of Frictional Contact Problems

In the reference (Cui and Yin, Pacific J. Math. 233:257–289, 2007), under the assumptions that the supersonic incoming flow is isothermal and symmetrically perturbed with respect to a uniform supersonic constant state, the authors have shown the global existence and stability of a symmetric supersonic conic shock for such a supersonic flow past a circular cone. In this paper, we will remove all the symmetric assumptions in the previous paper and study the global existence problem on a really multidimensional shock wave. More concretely, we establish the global existence and stability of a three-dimensional supersonic conic shock wave for a perturbed steady supersonic isothermal flow past an infinitely long conic body.     

Analysis on linear stability of oblique shock waves in steady supersonic flow

An attached oblique shock wave is generated when a sharp solid projectile flies supersonically in the air. We study the linear stability of oblique shock waves in steady supersonic flow under three dimensional perturbation in the incoming flow. Euler system of equations for isentropic gas model is used. The linear stability is established for shock front with supersonic downstream flow, in addition to the usual entropy condition.     

Stability of transonic shocks for the full Euler system in supersonic flow past a wedge

We study the stability of transonic shocks in steady supersonic flow past a wedge. It is known that in generic case such a problem admits two possible locations of the shock front, connecting the flow ahead of it and behind it. They can be distinguished as supersonic–supersonic shock and supersonic–subsonic shock (or transonic shock). Both these possible shocks satisfy the Rankine–Hugoniot conditions and the entropy condition. We prove that the transonic shock is conditionally stable under perturbation of the upstream flow or perturbation of wedge boundary. Copyright © 2005 John Wiley & Sons, Ltd.     

Stability of a supersonic flow past a wedge with adjoint weak neutrally stable shock wave

We study the classical problem of a supersonic stationary flow of a nonviscous nonheat-conducting gas in local thermodynamic equilibrium past an infinite plane wedge. Under the Lopatinski? condition on the shock wave (neutral stability), we prove the well-posedness of the linearized mixed problem (the main solution is a weak shock wave), obtain a representation of the classical solution, where, in this case (in contrast to the case of the uniform Lopatinski? condition—an absolutely stable shock wave), plane waves additionally appear in the representation. If the initial data have compact support, the solution reaches the given regime in infinite time.     

Global transonic conic shock wave for the symmetrically perturbed supersonic flow past a cone

In this paper, we are concerned with the global existence and stability of a steady transonic conic shock wave for the symmetrically perturbed supersonic flow past an infinitely long conic body. The flow is assumed to be polytropic, isentropic and described by a steady potential equation. Theoretically, as indicated in [R. Courant, K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, Inc., New York, 1948], it follows from the Rankine-Hugoniot conditions and the entropy condition that there will appear a weak shock or a strong shock attached at the vertex of the sharp cone in terms of the different pressure states at infinity behind the shock surface, which correspond to the supersonic shock and the transonic shock respectively. In the references [Shuxing Chen, Zhouping Xin, Huicheng Yin, Global shock wave for the supersonic flow past a perturbed cone, Comm. Math. Phys. 228 (2002) 47-84; Dacheng Cui, Huicheng Yin, Global conic shock wave for the steady supersonic flow past a cone: Polytropic case, preprint, 2006; Dacheng Cui, Huicheng Yin, Global conic shock wave for the steady supersonic flow past a cone: Isothermal case, Pacific J. Math. 233 (2) (2007) 257-289] and [Zhouping Xin, Huicheng Yin, Global multidimensional shock wave for the steady supersonic flow past a three-dimensional curved cone, Anal. Appl. 4 (2) (2006) 101-132], the authors have established the global existence and stability of a supersonic shock for the perturbed hypersonic incoming flow past a sharp cone when the pressure at infinity is appropriately smaller than that of the incoming flow. At present, for the supersonic symmetric incoming flow, we will study the global transonic shock problem when the pressure at infinity is appropriately large.     

Stability analysis of supersonic entropy layers including shock effects

The prediction of the laminar/turbulent transition location in supersonic boundary layers plays an important role to accurately compute aerodynamic forces and heating rates for the aerodynamic design and control of hypersonic vehicles. The stability characteristics of supersonic boundary layers depend e.g. on nose bluntness, transverse curvature, wall temperature, shock waves, etc. Most parameters can be theoretically investigated by performing conventional stability calculations with vanishing or asymptotic perturbation conditions at the far field. In this approach the formation of a shock in front of the leading edge of a blunt body is ignored. However, to improve the understanding of the interaction between instability waves originating inside supersonic boundary layer with those coming from the inviscid entropy layer, the presence of the shock has to be taken into account. This paper presents a method, how shock effects can be physically consistently included in stability calculations. The outer free‐stream boundary conditions are replaced by appropriate shock conditions. The required perturbation equations can be derived from the linearized unsteady Rankine‐Hugoniot equations, accounting for the effect of shock oscillations due to perturbated waves which originate from the flow field windward of the shock.     

Existence and Nonlinear Stability of Stationary Solutions to the Viscous Two-Phase Flow Model in a Half Line

The outflow problem for the viscous two-phase flow model in a half line is investigated in the present paper. The existence and uniqueness of the stationary solution is shown for both supersonic state and sonic state at spatial far field, and the nonlinear time stability of the stationary solution is also established in the weighted Sobolev space with either the exponential time decay rate for supersonic flow or the algebraic time decay rate for sonic flow.     

超声速蒙皮颤振微分积分方程的特征值问题

根据二维线化理论讨论超声速薄钣的动力稳定性,导致一类新颖的数学物理问题:非自共轭Volterra型四阶微分积分方程的复特征值问题.求得这一气动弹性系统的严格解.与其它近似分析对比,本法的临界曲线与实验数据符合良好,在低超声速范围不存在发散问题.此外,在数学物理实质方面,发现:(1)颤振频谱与固有频谱有互为间隔现象;(2)临界Mach数有简并现象.指出本法可以推广应用于三维机翼模型和燃气轮中叶栅的超声速颤振问题.     

抛物化稳定性方程在可压缩边界层中应用的检验

用抛物化稳定性方程（PSE）,研究了可压缩边界层中扰动的演化,并与由直接数值模拟（DNS）所得进行比较．目的在检验PSE方法用于研究可压缩边界层中扰动演化的可靠性．结果显示,无论是亚音速还是超音速边界层,由PSE方法和由DNS方法所得结果都基本一致,而温度比速度吻合得更好．对超音速边界层,还计算了小扰动的中性曲线．与线性稳定性理论（LST）的结果相比,二者的关系和不可压边界层的情况相似．     

Stability of Functionally Graded Circular Cylindrical Shells under Combined Loading

Mechanics of Composite Materials - The stability of loaded circular cylindrical shells made of functionally graded materials and flowed round by a supersonic gas stream is investigated. The...     

平头一锥体无粘超音速绕流的数值计算

复杂外形飞行体的超音速绕流气动力计算,显然是重要的.仅无粘超音速绕流,国内外的工作就很多,如[1]—[5].本文介绍的方法是[1,2]方法的修改与发展.近来我们计算了好几种头部与身部形状的物体在不同来流马赫数与攻角下的超音速绕流的流     

Stability of oblique shock front

The stability of the weak planar oblique shock front with respect to the perturbation of the wall is discussed. By the analysis of the formation and the global construction of shock and its asymptotic behaviour for stationary supersonic flow along a smooth rigid wall we obtain the stability of the solution containing a weak planar shock front. The stability can be used to single out a physically reasonable solution together with the entropy condition     

Supersonic flow past a flat lattice of cylindrical rods

Two-dimensional supersonic laminar ideal gas flows past a regular flat lattice of identical circular cylinders lying in a plane perpendicular to the free-stream velocity are numerically simulated. The flows are computed by applying a multiblock numerical technique with local boundary-fitted curvilinear grids that have finite regions overlapping the global rectangular grid covering the entire computational domain. Viscous boundary layers are resolved on the local grids by applying the Navier–Stokes equations, while the aerodynamic interference of shock wave structures occurring between the lattice elements is described by the Euler equations. In the overlapping grid regions, the functions are interpolated to the grid interfaces. The regimes of supersonic lattice flow are classified. The parameter ranges in which the steady flow around the lattice is not unique are detected, and the mechanisms of hysteresis phenomena are examined.     

Stability of transonic shock for supersonic flow past a wedge

When steady supersonic flow hits a slim wedge, there may appear an oblique transonic shock attached to the vertex of the wedge, if the downstream pressure is rather large. This paper studies stability in certain weighted partial Hölder spaces of the oblique transonic shock attached to the vertex of a wedge, which is against steady supersonic flows, under perturbations of the upstream flow and the profile of the wedge. We show that under reasonable conditions on the upcoming supersonic flow and the slope of the wedge, such transonic shocks are structural stable. Mathematically, we solve an elliptic–hyperbolic mixed type in an unbounded domain, and the flow field is proved to be C¹. Copyright © 2015 John Wiley & Sons, Ltd.     

The existence and stability of conic shock waves

We study the existence of the nonsymmetrical conic shock wave produced by a supersonic flow past a distorted conic projectile. For the weak conic shock wave, we establish the existence and its linear stability using the mathematical model of an isentropic irrotational flow.     

Global Conic Shock Wave for the Steady Supersonic Flow Past a Large Curved Cone at Infinity

In this paper, we establish the global existence and stability of a steady symmetric shock wave for the constant supersonic flow past an infinitely long and large curved conic body. The flow is assumed to be polytropic, isentropic and described by a steady potential equation. Through looking for the suitable “dissipative” boundary conditions on the shock and the conic surface together with the special form of shock equation, we show that the conic shock attached at the vertex of the cone exists globally in the whole space when the speed of the supersonic incoming flow is appropriately large.     

Nonstationary interaction of supersonic inviscid gas flows

We consider the flow of supersonic homogeneous gas past a supersonic spherical source. This problem provides a gas-dynamic model of the interaction of interstellar wind with solar wind, and is thus also of independent interest. It is solved using an explicit through divergence scheme of third-order approximation. The analysis focuses on formation and stability of the structure of discontinuity surfaces and convergence to the stationary solution. The results are compared qualitatively and quantitatively with solutions obtained by other methods. Translated from Chislennye Metody v Matematicheskoi Fizike, Published by Moscow University, Moscow, 1996, pp. 125–128.     

Instabilities of a flexible surface in supersonic flow

The stability of a supersonic boundary layer above a flexiblesurface is considered in the limit of large Reynolds numberand for Mach numbers O(1). Asymptotic theory of viscous–inviscidinteraction has been used for this purpose. We found that fora simple elastic surface, for which deflections are proportionalto local pressure differences, the boundary-layer flow remainsstable as it is for a rigid wall. However, when either dampingor surface inertia is included the flow becomes unstable. Moreover,in a certain range of wave numbers the boundary layer developsmore then one unstable mode. It is interesting that these modesare connected to one another via saddle points in the complex-frequencyplane. A more complex Kramer-type surface is also analysed andin some parameter ranges is found to permit the evolution ofunstable Tollmien–Schlichting waves. The neutral curvesare found for a variety of situations related to the parametersassociated with the flexible surface.     

Global supersonic conic shock wave for the steady supersonic flow past a cone: Polytropic gas

In this paper, we establish the global existence and stability of a steady conic shock wave for the symmetrically perturbed supersonic flow past an infinitely long conic body as long as the vertex angle is less than a critical value. The flow is assumed to be polytropic, isentropic and described by a steady potential equation. Based on the delicate asymptotic expansion of the background solution, one can verify that the boundary conditions on the shock and the conic surface satisfy the “dissipative” property. From this property, by use of the reflected characteristics method and the special form of the shock equation, we show that the conic shock attached at the vertex of the cone exists globally in the whole space when the speed of the supersonic coming flow is appropriately large. On the other hand, we remove the smallness restriction on the sharp vertex angle in order to establish the global existence of a shock or a global weak solution, moreover, our proof approach is different from that in [Shuxing Chen, Zhouping Xin, Huicheng Yin, Global shock wave for the supersonic flow past a perturbed cone, Comm. Math. Phys. 228 (2002) 47-84] and [Zhouping Xin, Huicheng Yin, Global multidimensional shock wave for the steady supersonic flow past a three-dimensional curved cone, Anal. Appl. 4 (2) (2006) 101-132].     

The boundary layer transition at supersonic velocities

The transition to turbulent flow in a boundary layer at supersonic velocities, the study of which was started at the Institute of Theoretical and Applied Mechanics of the Siberian Branch of the Academy of Sciences of the USSR on the initiative of V. V. Struminskii is considered. It is shown that complex investigations into this problem, including the stability of the laminar boundary layer and structure of the perturbations in the operational part of a wind tunnel at supersonic velocities, enable the mechanism of the boundary layer transition on a flat plate to be established and demonstrate the decisive effect of the spectral composition of the external flow perturbations and the blunting of the leading edge of the model that enables one to determine the role of the unit Reynolds number.