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.     

Nonlinear high-wavenumber Bnard convection

Weakly nonlinear two-dimensional roll cells in Bnard convectionare examined in the limit as the wavenumber a of the roll cellsbecomes large. In this limit the second harmonic contributionsto the solution become negligible, and a flow develops wherethe fundamental vortex terms and the correction to the meanare determined simultaneously, rather than sequentially as inthe weakly nonlinear case. Extension of this structure to Rayleighnumbers O(a³) above the neutral curve is shown to be possible,with the resulting flow field having a form very similar tothat for strongly nonlinear vortices in a centripetally unstableflow. The flow in this strongly nonlinear regime consists ofa core region, and boundary layers of thickness O(a^–1)at the walls. The core region occupies most of the thicknessof the fluid layer and only mean terms and cos az terms playa role in determining the flow; in the boundary layer all harmonicsof the vortex motion are present. Numerical solutions of thewall layer equations are presented and it is also shown thatthe heat transfer across the layer is significantly greaterthan in the conduction state.     

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.     

Phase-Averaged Vortex Train Flow Generated by Plasma DBD Actuator

It will be shown how plasma actuator can generate wall-jet-like flow or train of periodical vortices depending on the generator setting. For generation the high-frequency high-voltage AC is used. Low-frequency modulation of the supply voltage is required to generate vortices. Data acquisition will be performed using time-resolved PIV technique. Phase-averaging will be studied from two different perspectives. Firstly, sampling of phases will be ensured using trigger that is contained in the PIV software and, secondly, phase-averaged flow will be computed from two main modes of POD analysis. The generated flow patterns are to be applied for control of a boundary layer. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

A coastal ocean model intercomparison study for a three-dimensional idealised test case

Several coastal ocean models have been used to compute the circulation on the Northwest European Continental Shelf. Five of them, developed within the European Union, are compared in the scope of an idealised three-dimensional test case, dealing with the geostrophic adjustment of a freshwater cylinder. As the central eddy adjusts, unstable baroclinic vortices start to grow. All the models are able to produce such unstable vortices. However, two of them produce an order-two instability, which is in accordance with a previous laboratory experiment, while the others exhibit an order-four instability. Using a simple scaling analysis, it is seen that the azimuthal wavenumber depends on the ratio of the kinetic energy to the available potential energy. It appears that the discrepancy in the azimuthal wavenumber is mainly due to the effect of the discretisation of the horizontal advection of momentum which could produce significant decrease of the total kinetic energy.     

The compressible Gortler problem in two-dimensional boundary layers

In this paper the authors investigate the growth rates of Görtlervortices in a compressible flow in the inviscid limit of largeGörtler number. Numerical solutions are obtained for O(1)wavenumbers. The further limits of (i) large Mach number and(ii) large wavenumber with O(1) Mach number are considered.It is shown that two different types of disturbance mode canappear in this problem. The first is a wall layer mode, so namedas it has its eigenfunctions trapped in a thin layer near thewall. The other mode investigated is confined to a thin layeraway from the wall and termed a trapped-layer mode for largewavenumbers and an adjustment-layer mode for large Mach numbers,since then this mode has its eigenfunctions concentrated inthe temperature adjustment layer. It is possible to investigatethe near crossing of the modes which occurs in each of the limitsmentioned. The inviscid limit does not predict a fastest growingmode, but does enable a most dangerous mode to be identifiedfor O(1) Mach number. For hypersonic flow the most dangerousmode depends on the size of the Görtler number.     

A further note on the force discrepancy for wing theory in Euler flow

Uniform steady potential flow past a wing aligned at a small angle to the flow direction is considered. The standard approach is to model this by a vortex sheet, approximated by a finite distribution of horseshoe vortices. In the limit as the span of the horseshoe vortices tends to zero, an integral distribution of infinitesimal horseshoe vortices over the vortex sheet is obtained. The contribution to the force on the wing due to the presence of one of the infinitesimal horseshoe vortices in the distribution is focused upon. Most of the algebra in the force calculation is evaluated using Maple software and is given in the appendices. As in the two previous papers by the authors on wing theory in Euler flow [E Chadwick, A slender-wing theory in potential flow, Proc. R. Soc. A461 (2005) 415–432, and E Chadwick and A Hatam, The physical interpretation of the lift discrepancy in Lanchester-Prandtl lifting wing theory for Euler flow, leading to the proposal of an alternative model in Oseen flow, Proc. R. Soc. A463 (2007) 2257–2275], it is shown that the normal force is half that expected. In this further note, in addition it is demonstrated that the axial force is infinite. The implications and reasons for these results are discussed.     

Trapping waves by a submerged cylinder

For large values of the wavenumber k in the direction of cylinder, there is only one trapped wave. We construct asymptotics of these trapped modes and their frequencies as k → ∞ in the case of one submerged cylinder into a plane water layer by means of reducing the initial problem to three integral equations on the boundaries and then solving them using a method suggested by Zhevandrov and Merzon (Amer. Math. Soc. Translations (2) 208, 235–284 (2003)).     

Lower Branch Modes of the Compressible Boundary Layer Flow Due to a Rotating-Disk

In this article, a theoretical study is pursued to investigate 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. Special attention is focused on to the short-wavelength stationary/nonstationary compressible crossflow vortex modes at sufficiently high Reynolds numbers with reasonably small scaled frequencies. Following closely the asymptotic framework introduced in [ 1 ] for the incompressible stationary modes, it is demonstrated here that the compressible modes having sufficiently long time scale can also be described by an asymptotic expansion procedure based on the triple-deck approach. Making use of this rational asymptotic technique, which rigorously takes into account the nonparallel effects, the asymptotic structure of the nonstationary modes is shown to be adjusted by a balance between viscous and Coriolis forces, and resulted from the fact of vanishing shear stress at the disk surface, as in the incompressible Von Karman's flow. As a consequence of matching successive regions in the asymptotic procedure, it is found that the wavenumber and the orientation of the compressible lower branch modes are governed by an eigenrelation, which is akin to the one obtained previously in [ 1 ] for the incompressible stationary mode and in [ 2 ] for the compressible stationary modes. The nonparallel influences are toward destabilizing all the modes, though the wall insulation and heating are relatively stabilizing for the modes in the vicinity of the stationary mode, unlike the wall cooling. The asymptotic compressible data obtained at high Reynolds number limit compares fairly well with the numerical results generated directly solving the linearized compressible system with usual parallel flow approximation.     

Zero Modes of the Dirac Operator in Vortex-Type Gauge Fields

We consider the D4 Euclidean Dirac equation in a SU(2) gauge field with the form of intersecting central vortices and obtain its solutions corresponding to normalizable zero modes. In the context of the problem of constructing the mechanism of chiral symmetry breaking based on the model with randomly distributed central vortices, we study a possibility for splitting fermionic zero modes in the case of a superposition of many vortices with Pontryagin indices of different signs.     

Pseudospectral reduction of incompressible two-dimensional turbulence

Spectral reduction was originally formulated entirely in the wavenumber domain as a coarse-grained wavenumber convolution in which bins of modes interact with enhanced coupling coefficients. A Liouville theorem leads to inviscid equipartition solutions when each bin contains the same number of modes. A pseudospectral implementation of spectral reduction which enjoys the efficiency of the fast Fourier transform is described. The model compares well with full pseudospectral simulations of the two-dimensional forced-dissipative energy and enstrophy cascades.     

The effect of buoyancy on upper-branch Tollmien-Schlichting waves

We investigate the effect of buoyancy on the upper-branch linearstability characteristics of an accelerating boundary-layerflow. The presence of a large thermal buoyancy force significantlyalters the stability structure. As the factor G (which is relatedto the Grashof number of the flow, and defined in Section 2)becomes large and positive, the flow structure becomes two layeredand disturbances are governed by the Taylor-Goldstein equation.The resulting inviscid modes are unstable for a large componentof the wavenumber spectrum, with the result that buoyancy isstrongly destabilizing. Restabilization is encountered at sufficientlylarge wavenumbers. For G large and negative the flow structureis again two layered Disturbances to the basic flow are nowgoverned by the steady Taylor—Goldstein equation in themajority of the boundary layer, coupled with a viscous walllayer. The resulting eigenvalue problem is identical to thatfound for the corresponding case of lower-branch Tollmien—Schlichtingwaves, thus suggesting that the neutral curve eventually becomesclosed in this limit.     

On the dynamics in a transitional boundary layer

IntroductionIll 1883 Professor Osborne Reynolds published in Philosopl1ical Transactions of the RoyalSociety the outcomes of his flow visua1ization at Manchester. These had shown that whetherthe flow in a pipe was direct to sinuous (or, as nowadays we would say, laminar to turbulent)depended on its Reynolds number. Transition from Iaminar to turbuIent flow becomes animportant probIem i1l fluid mechanics, which has attracted the interest of investigators fOrmore than l00 years. The partic…     

Structure of a Linear Array of Uniform Vortices

The shape and properties of an infinite steady linear array of uniform vortices are calculated. A nonlinear singular integrodifferential equation is obtained for the shapes, which is solved numerically by Newton's method and Euler continuation to give a one parameter family of shapes as size over separation is varied. The kinematic properties and energy of the array are obtained. It is found that there exists an array of maximum area, for given separation, which also possesses minimum energy in accordance with a general argument of Kelvin. A simple model based on elliptical vortices is constructed, which reproduces the qualitative kinematic properties and is quantitatively quite accurate. Continuation of the numerical solution past the array of maximum area leads to a limit of finite, lens shaped, touching vortices. This array is also shown to be limit of a finite amplitude bifurcation of a vortex sheet of finite thickness. The stability of the array to two dimensional subharmonic and superharmonic disturbances is considered. General arguments, based on ideas of Kelvin, are given to show that the array is stable to superharmonic disturbances if the area is less than the maximum and otherwise unstable, and that it is always unstable to subharmonic disturbances, of which the pairing instability is a special case. It is verified by direct calculation in an Appendix that hollow vortices, whose shapes can be determined analytically in closed form, are unstable to the pairing instability whatever their size. Some speculations are made about the possible relevance of the results to the observed properties of organized structures in the turbulent mixing layer.     

On the resistanceless statement of the two-dimensional Neumann-Kelvin problem for a surface-piercing tandem

A new set of supplementary conditions is proposed for the two-dimensionalNeumann-Kelvin problem describing the steady-state forward motionof a surface-piercing tandem in an infinite-depth fluid. Thisproblem is shown to be uniquely solvable for almost every valueof the forward speed U. The velocity potential solving the problemcorresponds to a flow about the tandem providing no resistance(wave and spray resistance vanish simultaneously). On the otherhand, for exceptional values of U examples of non-uniqueness(trapped modes) are constructed using the inverse procedurerecently applied by McIver (J. Fluid Mech. 1996) to the problemof time-harmonic water waves. For the proposed statement ofthe Neumann-Kelvin problem the inverse method involves the investigationof streamlines generated by two vortices placed in the freesurface. The spacing of vortices delivering trapped modes dependson U.     

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.     

Barotropic Instability of the Bickley Jet at High Reynolds Numbers

We investigate the linear stability of the Bickley jet in the framework of the beta-plane approximation. Because singular inviscid neutral modes exist in the retrograde case     , it is necessary to add viscosity to interpret them. One of these modes was found in closed form by Howard and Drazin [1] . However, its critical point is at the center of the jet and it was therefore not possible for these authors to ascertain the relationship of this mode to the stability problem or to discuss how to continue the eigenfunction across the singularity.
The viscous critical layer problem associated with this singularity is considerably more difficult than the usual one (which leads to integrals of the Airy function) because     and, consequently, a second-order turning point is involved. Our analysis shows that the Howard–Drazin mode is degenerate in the domain where it is valid as a limit of the viscous problem (wavenumber  α²≤ 9/2  ), that is, it corresponds to both an odd and an even mode. This conclusion is confirmed by direct numerical solution of the Orr–Sommerfeld equation which shows, in addition, that viscosity is destabilizing along portions of the stability boundary. For a retrograde jet, instability is found to occur beyond the inviscid critical value of β, that is, in the region where the flow would be stable according to the Rayleigh–Kuo condition.     

细长体截面绕流的稳定性及扰动的影响

应用拓扑结构的稳定性理论,分析了细长旋成体截面绕流的结构稳定性．在分析时取极限流线作为流场的内边界,并证明极限流线的鞍点-鞍点连接是拓扑结构稳定的A·D2通过分析发现,由于旋成体背涡的发展,导致截面流场拓扑结构变化,由稳定对称旋涡流态变成不稳定对称旋涡流态．此时流场中存在空间的鞍点-鞍点连接的不稳定拓扑结构,在小扰动下出现分叉,变成稳定非对称旋涡流态,形成非对称背涡．并应用开折理论分析了扰动对流场结构的影响．