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.     

On the Generation of Mean Flows by the Interaction of Görtler Vortices and Tollmien-Schlichting Waves in Curved Channel Flows

There are many fluid flows where the onset of transition can be caused by different instability mechanisms which compete in the nonlinear regime. Here the interaction of a centrifugal instability mechanism with the viscous mechanism which causes Tollmien-Schlichting waves is discussed. The interaction between these modes can be strong enough to drive the mean state; here the interaction is investigated in the context of curved channel flows so as to avoid difficulties associated with boundary layer growth. Essentially it is found that the mean state adjusts itself so that any modes present are neutrally stable even at finite amplitude. In the first instance the mean state driven by a vortex of short wavelength in the absence of a Tollmien-Schlichting wave is considered. It is shown that for a given channel curvature and vortex wavelength there is an upper limit to the mass flow rate which the channel can support as the pressure gradient is increased. When Tollmien-Schlichting waves are present then the nonlinear differential equation to determine the mean state is modified. At sufficiently high Tollmien-Schlichting amplitudes it is found that the vortex flows are destroyed, but there is a range of amplitudes where a fully nonlinear mixed vortex-wave state exists and indeed drives a mean state having little similarity with the flow which occurs without the instability modes. The vortex and Tollmien-Schlichting wave structure in the nonlinear regime has viscous wall layers and internal shear layers; the thickness of the internal layers is found to be a function of the Tollmien-Schlichting wave amplitude.     

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.     

Nonlinear Stability of Nonstationary Cross-Flow Vortices in Compressible Boundary Layers

The nonlinear evolution of long-wavelength non stationary cross-flow vortices in a compressible boundary layer is investigated; the work extends that of Gajjar [1] to flows involving multiple critical layers. The basic flow profile considered in this paper is that appropriate for a fully three-dimensional boundary layer with O(1) Mach number and with wall heating or cooling. The governing equations for the evolution of the cross-flow vortex are obtained, and some special cases are discussed. One special case includes linear theory, where exact analytic expressions for the growth rate of the vortices are obtained. Another special case is a generalization of the Bassom and Gajjar [2] results for neutral waves to compressible flows. The viscous correction to the growth rate is derived, and it is shown how the unsteady nonlinear critical layer structure merges with that for a Haberman type of viscous critical layer.     

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.     

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.     

Boundary Layers Associated with Incompressible Navier-Stokes Equations: The Noncharacteristic Boundary Case

The goal of this article is to study the boundary layer of wall bounded flows in a channel at small viscosity when the boundaries are uniformly noncharacteristic, i.e., there is injection and/or suction everywhere at the boundary. Following earlier work on the boundary layer for linearized Navier-Stokes equations in the case where the boundaries are characteristic (no-slip at the boundary and non-permeable), we consider here the case where the boundary is permeable and thus noncharacteristic. The form of the boundary layer and convergence results are derived in two cases: linearized equation and full nonlinear equations. We prove that there exists a boundary layer at the outlet (downwind) of the form e^−Uz/ε where U is the speed of injection/suction at the boundary, z is the distance to the outlet of the channel, and ε is the kinematic viscosity. We improve an earlier result of S. N. Alekseenko (1994, Siberian Math. J.35, No. 2, 209-230) where the convergence in L² of the solutions of the Navier-Stokes equations to that of the Euler equations at vanishing viscosity was established. In the two dimensional case we are able to derive the physically relevant uniform in space (L^∞ norm) estimates of the boundary layer. The uniform in space estimate is derived by properly developing our previous idea of better control on the tangential derivative and the use of an anisotropic Sobolev imbedding. To the best of our knowledge this is the first rigorously proved result concerning boundary layers for the full (nonlinear) Navier-Stokes equations for incompressible fluids.     

A Sharp Existence Theorem for Vortices in the Theory of Branes

In this paper, we investigate the existence of solutions for a system of BPS vortex equations arising from the theory of multiple intersecting D-branes. Using a direct minimization method, we establish a sharp existence and uniqueness theorem for this system over a doubly periodic domain and over the full plane, respectively. In particular, we obtain an explicit necessary and sufficient condition, explicitly expressed in terms of the vortex numbers and the size of the domain, for the existence of a solution of the system in the doubly periodic domain case.     

奇异边值问题的正解存在性

该文应用不动点定理,研究了非线性二阶奇异边值问题正解的存在性和多个正解存在性,改进和推广了近期文献［６］的结果．     

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.     

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.     

Boundary Layers in Periodic Homogenization of Neumann Problems

This paper is concerned with a family of second‐order elliptic systems in divergence form with rapidly oscillating periodic coefficients. We initiate the study of homogenization and boundary layers for Neumann problems with first‐order oscillating boundary data. We identify the homogenized system and establish the sharp rate of convergence in L² in dimension three or higher. Regularity estimates are also obtained for the homogenized boundary data in both Dirichlet and Neumann problems. The results are used to establish a higher‐order convergence rate for Neumann problems with nonoscillating data. © 2018 Wiley Periodicals, Inc.     

两点边值问题解的存在性

本文讨论两点边值问题解的存在性,在没有孤立性条件下,获得了不动点定理,作为应用实例,给出了反应扩散方程稳态解的存在性证明。