后掠翼边界层定常横流涡的非线性演化

横流失稳是后掠翼边界层主要的失稳形式.实验和数值研究发现在后掠翼边界层转捩之前,有一段较长的非线性幅值饱和阶段,因此线性稳定性不能有效预测横流失稳转捩过程,所以研究横流涡的非线性演化过程就极为必要.以NLF(2)-0415翼型为研究模型,在来流Mach数为0.8、后掠角为45°、攻角为-4°的条件下,用扰动方程计算了定常横流涡非线性演化过程.结果显示非平行性起着更加不稳定的作用.当基本波的幅值到达0.1时,非线性作用开始明显.横流涡经历了非线性幅值饱和过程,涡的形状呈现半蘑菇状,涡的涡轴与边界层外缘无粘势流平行.饱和涡使得原有流场发生极大的扭曲,流向速度和展向剖面出现了拐点.     

On the Nonlinear Development of Görtler Vortices in a Compressible Boundary Layer

The nonlinear development of the Görtler instability in compressible boundary layers on curved walls is considered for vortices of asymptotically large wavenumber. The starting point for our calculations lies in the work of Hall and Lakin (Proc. Roy. Soc. London Ser. A 415:421–444), where the incompressible results were formulated. Without neglecting downstream partial derivatives, the initial development of a vortex from the point where it first starts to grow is calculated. It is shown how the same basic structure that occurs in incompressible flow exists, where the disturbance is confined to a core region bounded above and below by thin shear layers, but that the flow in the core region is of more complicated form than that for incompressible flow.     

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．     

On the Non-linear Evolution of Gortler Vortices in Non-parallel Boundary Layers

The non-linear development of finite amplitude Görtlervortices in a non-parallel boundary layer on a curved wall isinvestigated using perturbation methods based on the smallnessof e, the non-dimensional wavelength of the vortices. The crucialstage in the growth or decay of the vortices takes place inan interior viscous layer of thickness O(²) and length O().In this region the downstream velocity component of the perturbationcontains a mean flow correction of the same order of magnitudeas the fundamental which is driving it. Moreover, these functionssatisfy a pair of coupled non-linear partial differential equationswhich must be solved subject to some initial conditions imposedat a given downstream location. It is found that, dependingon whether the boundary layer is more or less unstable downstreamof this location, the initial disturbance either grows intoa finite amplitude Görtler vortex or decays to zero. Forthe Blasius boundary layer on a concave wall it is found thatGörtler vortices can only develop if the rate of increaseof curvature of the wall is sufficiently large. In this casethe finite amplitude solution which develops initially in an-neighbourhood of the position where the disturbance is introducedchanges its structure further downstream. This structure isinvestigated at a distance O() (with 0< <1) downstreamof the above -neighbourhood. In this régime the downstreamfundamental velocity component has an elliptical profile overmost of the flow field. However, in two thin boundary layerslocated symmetrically either side of the centre of the viscouslayer the fundamental velocity component decays exponentiallyto zero. The locations of these layers are determined by aneigenvalue problem associated with the one-dimensional diffusionequation. The mean flow correction persists both sides of theboundary layer and ultimately decays exponentially to zero. This large amplitude motion is not sensitive to the imposedinitial conditions and appears to be the ultimate state of anyinitial disturbance. However, in the initial stages of the growthof the vortex, some surprising flows are possible. For example,it is possible to set up a vortex flow similar to that observedby Wortmann (1969) which consists of a sequence of cells inclinedat an angle to the vertical.     

PSE在可压缩边界层转捩问题中的应用

提出了用抛物化稳定性方程（PSE）预测层流-湍流转捩的一种新的概念．它被试用于平板可压缩边界层转捩位置的预测问题中,并将结果与直接数值模拟（DNS）所得进行比较．结果二者符合情况令人满意．二者符合的原因在于PSE方法准确地再现了层流-湍流转捩中导致breakdown过程的机理,即平均流剖面的修正导致其稳定性特性的明显改变．     

The Existence of Görtler Vortices in Separated Boundary Layers

The linear stability properties of Görtler vortices within a general separated boundary layer flow are addressed. There has been little previous theoretical work directed toward this problem and here we are able to characterize the important features of vortices over the complete wavenumber spectrum. This investigation complements earlier studies of vortices within an attached flow which demonstrated that there are three distinctive wavenumber régimes which together describe the most relevant possibilities for vortex behavior. In the first of these, at relatively small wavenumbers, the mode is inviscid in character; as the vortex wavenumber increases so the spatial amplification rate of the vortices increases until viscous effects become significant and the growth rate begins to diminish. As the wavenumber increases yet further so the vortex is completely stabilized. Here we discuss the corresponding structures which may exist within a separated flow and the most significant result we find is that the maximum growth rate of a mode in this type of flow is actually greater than that which occurs when the flow has not separated. In addition, the inviscid modes are shown to have a far more complicated configuration than within an attached boundary layer and, indeed, their structure can only be completely determined by implementation of numerical procedures.     

转捩边界层中次生流向涡演化的数值研究

采用高精度直接数值模拟方法和高效的特征无反射边界条件,进行可压缩流转捩边界层中出现的次生流向涡演化的数值研究.精细的数值模拟结果清楚地揭示了转捩边界层的复杂流场中次生流向涡的形成和演化过程,探讨它对转捩至关重要的环状涡生成的影响,发现次生流向涡和主流向涡的共同作用形成环状涡的一种新机理.     

Influence of Finite Amplitude Disturbances on the Nonstationary Modes of a Compressible Boundary Layer Flow

A weakly nonlinear stability analysis is performed to search for the effects of compressibility on a mode of instability of the three-dimensional boundary layer flow due to a rotating disk. The motivation is to extend the stationary work of [ 1 ] (hereafter referred to as S90) to incorporate into the nonstationary mode so that it will be investigated whether the finite amplitude destabilization of the boundary layer is owing to this mode or the mode of S90. Therefore, the basic compressible flow obtained in the large Reynolds number limit is perturbed by disturbances that are nonlinear and also time dependent. In this connection, the effects of nonlinearity are explored allowing the finite amplitude growth of a disturbance close to the neutral location and thus, a finite amplitude equation governing the evolution of the nonlinear lower branch modes is obtained. The coefficients of this evolution equation clearly demonstrate that the nonlinearity is destabilizing for all the modes, the effect of which is higher for the nonstationary waves as compared to the stationary waves. Some modes particularly having positive frequency, regardless of the adiabatic or wall heating/cooling conditions, are always found to be unstable, which are apparently more important than those stationary modes determined in S90. The solution of the asymptotic amplitude equation reveals that compressibility as the local Mach number increases, has the influence of stabilization by requiring smaller initial amplitude of the disturbance for the laminar rotating disk boundary layer flow to become unstable. Apart from the already unstable positive frequency waves, perturbations with positive frequency are always seen to compete to lead the solution to unstable state before the negative frequency waves do. Also, cooling the surface of the disk will be apparently ineffective to suppress the instability mechanisms operating in this boundary layer flow.     

The Nonparallel Evolution of Nonlinear Short Waves in Buoyant Boundary Layers

Buoyant boundary-layer flows, typified by the flow over a heated flat plate, have the curious property that they can exhibit regions of "overshoot" in which the streamwise velocity exceeds its free-stream value. A consequence of this is the streamwise velocity develops a local maximum and is inflectional in nature. It is therefore inviscidly unstable, and the fastest growing wave mode is known to be one whose wavelength is short compared to the boundary-layer thickness. In this work we consider the nonparallel evolution of these short waves and show that they can be described in terms of the solution of a system of ordinary differential equations. Numerical and asymptotic studies enable us to explain the ultimate fate of the wave and show, depending on a key parameter which is a function of the underlying boundary layer, that two possibilities can arise. Nonparallelism may be sufficiently stabilizing so as to extinguish the linearly unstable waves or, in other cases, the mode may intensify but concentrate itself in a very thin zone surrounding the maximum in the streamwise velocity. These findings enable us to give some indication of the part these modes play in the transition to turbulence in buoyant boundary layers.     

Viscous Boundary Layers and Their Stability (I)

This paper is concerned with the asympootic limiting behavior of solutions to one-dimensional quasilinear scalar viscous equations for small viscosity in the presence of boundaries. We consider only non-characteristic boundary conditions. The main goals are to understand the evolution of viscous boundary layers, to construct the leading asymptotic ansatz which is uniformly valid up to the boundaries, and to obtain rigorously the uniform convergence to smooth solution of the associated inviscid hyperbolic equations away from the boundaries.     

Dynamics of Two Point Vortices in an External Compressible Shear Flow

This paper is concerned with a system of equations that describes the motion of two point vortices in a flow possessing constant uniform vorticity and perturbed by an acoustic wave. The system is shown to have both regular and chaotic regimes of motion. In addition, simple and chaotic attractors are found in the system. Attention is given to bifurcations of fixed points of a Poincaré map which lead to the appearance of these regimes. It is shown that, in the case where the total vortex strength changes, the “reversible pitch-fork” bifurcation is a typical scenario of emergence of asymptotically stable fixed and periodic points. As a result of this bifurcation, a saddle point, a stable and an unstable point of the same period emerge from an elliptic point of some period. By constructing and analyzing charts of dynamical regimes and bifurcation diagrams we show that a cascade of period-doubling bifurcations is a typical scenario of transition to chaos in the system under consideration.     

有限振幅T-S波在非平行边界层中的非线性演化研究

研究对非平行边界层稳定性有重要影响的非线性演化问题,导出与其相应的抛物化稳定性方程组,发展了求解有限振幅T-S波的非线性演化的高效数值方法。这一数值方法包括预估-校正迭代求解各模态非线性方程并避免模态间的耦合,采用高阶紧致差分格式,满足正规化条件,确定不同模态非线性项表和数值稳定地作空间推进。通过给出T-S波不同的初始幅值,研究其非线性演化。算例与全Navier-Stokes方程的直接数值模拟(DNS)的结果作了比较。     

Effects of Suction on the Nonstationary Lower Branch Modes of a Compressible Boundary Layer Flow Due to a Rotating Disk

In this paper, suction and injection effects are investigated theoretically on the structure of the lower branch neutral stability modes of three-dimensional small disturbances imposed on the compressible boundary layer flow due to a rotating disk. In a recent study [ 1 ], it was demonstrated that the short-wavelength stationary/nonstationary compressible crossflow vortex modes at sufficiently high Reynolds numbers with reasonably small scaled frequencies can be described by an asymptotic expansion procedure as set up in [ 2 ] for the incompressible stationary modes, which rigorously takes into account the nonparallel effects. Employing this rational asymptotic technique, it is shown here that the wavenumber and the orientation of the compressible lower branch modes are governed by an eigenrelation that is under the strong influence of a suction/injection parameter     , which, when set to zero, the relation turns out to be the one obtained previously by Turkyilmazoglu [ 1 ] for zero-suction compressible modes.
The boundary layer growth contributes in the way of destabilizing all the modes, in particular for the compressible modes, though the wall cooling in the case of suction and the wall insulation and heating in the case of injection are found to persist to the destabilization for the modes in the vicinity of the stationary mode. From a linear stability analysis point of view, suction is found to be stabilizing, whereas injection enhances the instability as compared to the no suction through the surface of the disk. In both cases, positive frequency waves are found to be highly destabilized as compared to the waves having negative frequencies. The findings of the work are also fully supported after a comparison between the numerical results obtained from directly solving the linearized compressible system with a usual parallel flow approximation and the asymptotic compressible data obtained at a high Reynolds number.     

Solving a Local Boundary Value Problem for a Nonlinear Nonstationary System in the Class of Feedback Controls

An algorithm convenient for numerical implementation is proposed for constructing differentiable control functions that transfer a wide class of nonlinear nonstationary systems of ordinary differential equations from an initial state to a given point of the phase space. Constructive sufficient conditions imposed on the right-hand side of the controlled system are obtained under which this transfer is possible. The control of a robotic manipulator is considered, and its numerical simulation is performed.     

Stability and Structure of Stretched Vortices

The radially symmetric Burgers vortex is an example of a solution to the Navier-Stokes equations in which the intensification of vorticity due to vortex stretching is balanced by the diffusion of vorticity through viscosity. The linear stability of the symmetric Burgers vortex to a class of two-dimensional perturbations is demonstrated. Analytical solutions obtained from a perturbation analysis and numerical computations are presented of nonsymmetric Burgers vortices in which the radial flow field in a plane perpendicular to the vorticity is nonsymmetric.     

The Effect of Profile Symmetry on the Nonlinear Stability of Mixing Layers

The nonlinear stability of arbitrary mixing-layer profiles in an incompressible, homogeneous fluid is studied in the high-Reynolds-number limit where the critical layer is linear and viscous. The type of bifurcation from the marginal state is found to depend crucially on the symmetry properties of the basic-state profile. When the vorticity profile of the mean flow is perfectly symmetric, the bifurcation is stationary. When the symmetry of the profile is broken, the bifurcation is Hopf. The nonsymmetry of the mixing layer also introduces some changes in the critical layer and the matching of flow quantities across it.     

Large Eddy Simulation of a Compressible Boundary Layer

A large eddy simulation of a compressible boundary layer is performed. To generate an appropriate inflow distribution the rescaling technique for compressible flows is discribed. In this method Morkovin's hypothesis in which the total temperature fluctuations are neglected compared with the static temperature fluctuations is applied to rescale and generate the temperature profile at inlet. This new technique is used for various large eddy simulations of subsonic and supersonic three‐dimensional boundary layers of a flat plate. Simulation results for the time‐averaged mean flow and Reynolds stresses are compared with numerical and analytical data to demonstrate the high quality of the method.     

Magnetohydrodynamic Boundary Layers in Czochralski Crystal Growth

The boundary-layer flow close to the crystal-melt interface,and at the free surface of the melt, in Czochralski crystalgrowth in the presence of an axial magnetic field, is examined.Particular attention is devoted to the effective segregationcoefficient and to the inhibiting effect of the magnetic fieldupon the forced convective radial flow.     

On the Stability of Compressible Swirling Flow

The linear stability to axisymmetric perturbations of compressible non-dissipative swirling flow is shown to be insured if a suitably defined “Richardson number” depending on the basic velocity, temperature and density fields everywhere exceeds 1/4. This result combines and generalizes in a natural way some known results for incompressible swirling flow and stratified parallel flow.     

Dynamic Stability and Instability of Pinned Fundamental Vortices

We study the dynamic stability and instability of pinned fundamental ±1 vortex solutions to the Ginzburg–Landau equations with external potential in ℝ². For sufficiently small external potentials, there exists a perturbed vortex solution centered near each non-degenerate critical point of the potential. With respect to both dissipative and Hamiltonian dynamics, we show that perturbed vortex solutions which are concentrated near local maxima (resp. minima) are orbitally stable (resp. unstable). In the dissipative case, the stability is in the stronger “asymptotic” sense. The research of S. Gustafson partially supported by a grant from NSERC. The research of F. Ting on this paper was supported by NSERC under grant N298724.