二维不可压缩磁流体方程边界层分离的条件

该文研究了二维不可压缩磁流体方程的解,其中要求磁流体的速度满足Dirichlet边界条件、磁场在边界上的值与时间无关. 利用Taylor展开式和不可压缩流的结构分歧理论, 得到了磁流体方程发生边界层分离的条件, 它取决于外力、初值和磁场在边界上的取值, 并且该条件可以预测磁流体边界层分离发生的时间与地点.     

Equations of the boundary layer for a modified Navier–Stokes system

We consider a system of equations of the boundary layer derived from the hydrodynamical system for generalized Newtonian media. This modification of the Navier–Stokes system was proposed by O. A. Ladyzhenskaya in connection with the uniqueness of the solution of this system in general. We prove the existence and the uniqueness of a solution for the problem of continuation of the boundary layer and consider some questions connected with the separation of the boundary layer.     

Use of active control for elimination of flow separation with estimation of energy costs

Problems were posed and solved concerning the aerodynamic computation of the flow past an airfoil with an active boundary layer control device used to prevent flow separation. A moving wall, suction, or tangential blowing in the boundary layer was used as a flow control device. The turbulent boundary layer was computed by directly solving the boundary layer equations using an implicit difference scheme with adaptive grid generation and the determination of the computational domain size. A software code was developed, and numerical simulations were performed taking into account the energy costs related to the flow control device. The numerical results showed that the active flow control devices can be used to prevent flow separation.     

Boundary Layer Theory and the Zero-Viscosity Limit of the Navier-Stokes Equation

Abstract A central problem in the mathematical analysis of fluid dynamics is the asymptotic limit of the fluid flow as viscosity goes to zero. This is particularly important when boundaries are present since vorticity is typically generated at the boundary as a result of boundary layer separation. The boundary layer theory, developed by Prandtl about a hundred years ago, has become a standard tool in addressing these questions. Yet at the mathematical level, there is still a lack of fundamental understanding of these questions and the validity of the boundary layer theory. In this article, we review recent progresses on the analysis of Prandtl's equation and the related issue of the zero-viscosity limit for the solutions of the Navier-Stokes equation. We also discuss some directions where progress is expected in the near future. Also at Courant Institute, New York University     

Active Control of Separating Boundary Layer

Flow control refers to the ability to alter flows with the aim of achieving a desired effect: examples include drag reduction, noise attenuation, improved mixing or increased combustion efficiency among many other industrial applications. The reduction and control of the viscous drag force exerted on bodies moving in a fluid is of great technical interest. Several active and passive methods to achieve a delay of separation in the boundary layer have been developed and are being developed. In this paper we present a new concept for boundary layer separation control that is based on the synthetic jet concept which converts acoustic oscillations into mean fluid motions. We use synthetic jets as methods for Coanda effect's amplification. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Control of a boundary layer separation

The results of experimental study on a boundary layer separation control are given in the paper. The boundary layer on a flat wall is subjected to adverse pressure gradient. The active control strategy evolving a synthetic jet has been applied. The separation process is investigated using TR-PIV method. Dynamical aspects of the phenomenon are analyzed in details. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling of flow in a very small surface separation

When a fluid flows in a very small surface separation, the very thin boundary layer physically adhering to the solid surface will participate in the flow, while between the two boundary layers is a continuum fluid flow. An analysis is here presented for this multiscale flow. The continuum fluid is treated as Newtonian. The physical adsorbed boundary layer is treated as non-continuum across the layer thickness. The interfacial slippage can occur on the adsorbed layer-solid surface interface, while it is absent on the adsorbed layer-fluid interface. Three flow equations are derived respectively for the two adsorbed layers and the intermediate continuum fluid. They together govern the multiscale flow in such a small surface separation.     

Dynamics of flow near a separation point

Dynamical behavior of a boundary layer separation is studied using the experimental approach. The dynamical nature of the phenomenon is demonstrated on a simple case of a flat–plate boundary layer in adverse pressure gradient. The Time–Resolved PIV technique was utilized for monitoring instantaneous structure of the separation region and its time development. Distinctive coherent structures and their dynamical behavior were identified using the Proper Orthogonal Decomposition method. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Examples of boundary layers associated with the incompressible Navier-Stokes equations

The author surveys a few examples of boundary layers for which the Prandtl boundary layer theory can be rigorously validated. All of them are associated with the incompressible Navier-Stokes equations for Newtonian fluids equipped with various Dirichlet boundary conditions (specified velocity). These examples include a family of (nonlinear 3D) plane parallel flows, a family of (nonlinear) parallel pipe flows, as well as flows with uniform injection and suction at the boundary. We also identify a key ingredient in establishing the validity of the Prandtl type theory, i.e., a spectral constraint on the approximate solution to the Navier-Stokes system constructed by combining the inviscid solution and the solution to the Prandtl type system. This is an additional difficulty besides the wellknown issue related to the well-posedness of the Prandtl type system. It seems that the main obstruction to the verification of the spectral constraint condition is the possible separation of boundary layers. A common theme of these examples is the inhibition of separation of boundary layers either via suppressing the velocity normal to the boundary or by injection and suction at the boundary so that the spectral constraint can be verified. A meta theorem is then presented which covers all the cases considered here.     

Long Eddies in Sheared Flows

The characteristic feature of the wide variety of hydraulic shear flows analyzed in this study is that they all contain a critical level where some of the fluid is turned relative to the ambient flow. One example is the flow produced in a thin layer of fluid, contained between lateral boundaries, during the passage of a long eddy. The boundaries of the layer may be rigid, or flexible, or free; the fluid may be either compressible or incompressible. A further example is the flow produced when a shear layer separates from a rigid boundary producing a region of recirculating flow. The equations used in this study are those governing inviscid hydraulic shear flows. They are similar in form to the classical boundary layer equations with the viscous term omitted. The main result of the study is to show that when the hydraulic flow is steady and contained between lateral boundaries, the variation of vorticity ω(ψ) cannot be prescribed at any streamline which crosses the critical level. This variation is, in fact, determined by (1) the vorticity distribution at all streamlines which do not cross the critical level, by (2) the auxiliary conditions which must be satisfied at the boundaries of the fluid layer, and by (3) the dimensions of the region containing the turned flow. If at some instant the vorticity distribution is specified arbitrarily at all streamlines, generally the subsequent flow will be unsteady. In order to emphasize this point, a class of exact solutions describing unsteady hydraulic flows are derived. These are used to describe the flow produced by the passage of a long eddy which distorts as it is convected with the ambient flow. They are also used to describe the unsteady flow that is produced when a shear layer separates from a boundary. Examples are given both of flows in which the shear layer reattaches after separation and of flows in which the shear layer does not reattach. When the shear layer vorticity distribution has the form ωαyⁿ, where y is a distance measure across the layer, the steady flows are of Falkner-Skan type inside, and adjacent to, the separation region. The unsteady flows described in this paper are natural generalizations of these Falkner-Skan flows. One important result of the analysis is to show that if the unsteady flow inside the separation region is strongly sheared, then the boundary of the separation region moves upstream towards the point of separation, forming large transverse currents. Generally, the assumption of hydraulic flow becomes invalid in a finite time. On the other hand, if the flow inside the separation region is weakly sheared, this region is swept downstream and the flow becomes self-similar.     

Three‐dimensional boundary layer on wind turbine blades

This paper describes a three‐dimensional analysis of the laminar boundary layer that develops on the blades of an horizontalaxis wind turbine. The main aim was to investigate a fundamental phenomenon: the effect of rotation on the blade boundary layer of a wind turbine in conjunction with the widely observed phenomenon of stall‐delay. The separation position in retarded flows with pressure gradients is calculated and compared for the rotation and non‐rotation cases. It is concluded that the stall is linearly postponed due to the Coriolis force and the separation point is delayed as a result of increasing rotation speed or decreasing blade spanwise position. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solution of the Falkner–Skan equation for wedge by Adomian Decomposition Method

The Adomian Decomposition Method is employed in the solution of the two dimensional laminar boundary layer of Falkner–Skan equation for wedge. This work aims at the solution of momentum equation in the case of accelerated flow and decelerated flow with separation. The Adomian Decomposition Method is provided an analytical solution in the form of an infinite power series. The effect of Adomian polynomials terms is considered on accuracy of the results. The velocity profiles in boundary layer are obtained. Results show a good accuracy compared to the exact solution.     

Dual solutions in mixed convection boundary layer flow of micropolar fluids

The steady mixed convection boundary layer flow over a vertical surface immersed in an incompressible micropolar fluid is considered in this paper. Employing suitable similarity transformations, the governing partial differential equations are transformed into ordinary differential equations, and the transformed equations are solved numerically by the Keller-box method. Numerical results are obtained for the skin friction coefficient and the local Nusselt number as well as the velocity, angular velocity and temperature profiles. Both cases of assisting and opposing buoyant flows are considered. It is found that dual solutions exist for the assisting flow, besides that usually reported in the literature for the opposing flow. Moreover, in contrast to the classical boundary layer theory, the separation point of the boundary layer is found to be distinct from the point of vanishing skin friction.     

Boundary Layer Growth on a Spinning Sphere

The method of matched asymptotic expansions is employed forinvestigating the growth of the boundary layer on a sphere whichis impulsively started from rest in an incompressible viscousfluid of infinite extent. The sphere is supposed to move alongthe direction of a diameter with constant linear velocity andat the same time to rotate about this diameter with constantangular velocity. It is assumed that the Reynolds number islarge and the time of investigation is short. It is shown thatthe solution of the problem contains terms which are absentfrom the solution based on the boundary layer approximation.These additional terms depend on the Reynolds number, and delayflow separation.     

On the initial phase of the triple deck stage in marginally separated flows

The method of matched asymptotic expansions is used to investigate marginally separated boundary layer flows (laminar or alternatively transitional separation bubbles) at high Reynolds numbers. Typical examples include, among others, the flow past slender airfoils at small to moderate angels of attack and channel flows with suction. As is well-known, classical (hierarchical) boundary layer computations usually break down under the action of an adverse pressure gradient on the flow, a scenario associated with the appearance of the Goldstein separation singularity. If, however, the parameter controlling the strength of the pressure gradient (the angle of attack or the relative suction rate in the examples mentioned above) is adjusted accordingly, the application of a local viscous-inviscid interaction strategy is capable of describing localized boundary layer separation. Moreover, taking into account unsteady effects and flow control devices allows the investigation of the conditions leading to forced or self-sustained vortex generation and the subsequent evolution process culminating in bubble bursting. Within the asymptotic formulation of this stage bubble bursting is associated with the formation of finite time singularities in the solution of the underlying equations and a corresponding break down. The distinct blow-up structure gives rise to a fully non-linear triple deck interaction stage featuring shorter spatio-temporal scales characteristic of the successive vortex evolution process. The paper will focus on the numerical treatment of the initial phase of the latter stage. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Transonic Marginal Separation

Laminar boundary‐layer separation near the leading edge of a thin airfoil is one of the principal factors that limits the lift force acting on the airfoil. Marginal separation, in particular, denotes the onset of separation which is accompanied by the formation of a short separation bubble. Using asymptotic analysis this effect is studied in the limit of high Reynolds number and for transonic external flow conditions. It is assumed that the fluid under consideration is a perfect gas and the airfoil surface is taken to be thermally insulated. Results to be presented include the analytical investigation of the emerging three layer structure, the associated transonic far field and the calculation of representative wall shear stress distributions. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A stall-delay model for rotating blades

The paper is aimed at describing a fundamental phenomenon: the effect of rotation on the inboard blade boundary layer on a wind turbine. The three-dimensional incompressible steady momentum integral boundary-layer equations are used to predict the boundary-layer growth and limiting streamline angles on the blade surface for both attached and separating flow. The chordwise skin friction coefficient is used to identify boundary layer separation and shear layer reattachment locations. The nature of flow near the axis of rotation is discussed and the physical mechanism associated with 3-D and rotational effects is identified. A semi-empirical correction law for the lift coefficient based on 2-D airfoil data is established. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boundary layer analysis for the eigenvalues of hanging rod

The motion of a naturally straight inextensible flexible elastic hanging rod is formulated and then linearized about the straight solution. To solve this equation by separation of variables, an eigenvalue problem is derived. When the stiffness of the rod is small, the eigenvalue equation is a singular perturbation problem. This paper is devoted to solving this eigenvalue problem by boundary layer analysis when the stiffness is suitably small, especially on the analytic approximate solutions of the first several eigenvalues and eigenfunctions. The first three eigenvalues are also compared with the numerical results computed by a finite difference method. The excellent agreement shows the efficiency of the boundary layer analysis.