PSE在超音速边界层二次失稳问题中的应用

用抛物化稳定性方程（PSE）研究超音速边界层中的二次失稳问题．结果显示无论二维基本扰动是第一模态还是第二模态的T-S波,二次失稳机制都起作用．三维亚谐波的放大率随其展向波数和二维基本波幅值的变化关系与不可压缩边界层中所得类似．但是,即使二维波的幅值大到2％的量级,三维亚谐波的最大放大率仍远小于最不稳定的第二模态二维T-S波的放大率．因此,二次失稳应该不是导致超音速边界层转捩的主要因素．     

Analysis of ratio-dependent food chain model

In this paper, a food chain model with ratio-dependent functional response is studied under homogeneous Neumann boundary conditions. The large time behavior of all non-negative equilibria in the time-dependent system is investigated, i.e., conditions for the stability at equilibria are found. Moreover, non-constant positive steady-states are studied in terms of diffusion effects, namely, Turing patterns arising from diffusion-driven instability (Turing instability) are demonstrated. The employed methods are comparison principle for parabolic problems and Leray-Schauder Theorem.     

An Analytic Approach for Calculating Absolutely Unstable Inviscid Modes of the Boundary Layer on a Rotating Disk

An analytical treatment of inviscidly absolutely unstable modes is pursued using the long-wavelength asymptotic approach. It is shown using the inviscid Rayleigh scalings in conjunction with the linear critical layer theory that the rotating-disk boundary layer flow undergoes a region of absolute instability for some small azimuthal wave numbers. The analytically calculated branch points for the absolute instability are found to be in good agreement with those obtained via a numerical solution of the inviscid Rayleigh equation.     

Spatial eigenmodes of a boundary layer with a triple-deck velocity field

The beforehand unclear relation between the viscous-inviscid interaction and the instability of viscous gas flows is illustrated using three-dimensional boundary-layer perturbations in the case of sub- and supersonic outer flows. The assumptions are considered under which asymptotic boundary layer equations with self-induced pressure are derived and the excitation mechanisms of eigenmodes (i.e., Tollmien-Schlichting waves) are described. The resulting dispersion relations are analyzed. The boundary layer in a supersonic flow is found to be stable with respect to two-dimensional perturbations, whereas, in the three-dimensional case, the modes become unstable. The increment of growth is investigated as a function of the Mach number and the orientation of the front of a three-dimensional Tollmien-Schlichting wave.     

The Propagation of Sound in Cylindrical Ducts with Mean Flow and Bulk-reacting Lining II. Bifurcated Ducts

The transmission and reflexion of sound in a bifurcated coaxialcylindrical duct is investigated. The inner tube carries a uniformflow and consists of two semi-infinite tubes: one is hard andthe other is perforated. The space between the coaxial cylindersis filled with a sound-absorbing material. Transmission andreflexion matrices are calculated for the causal solution. Itis found that causality and the boundary conditions requirean instability wave when the perforated tube is downstream itsjunction to the hard tube. When the perforate is situated upstreamthe junction the analysis permits incident waves that are unstable.This is important for applications to multiple reflexions. Itfollows from the analysis that, in addition to giving rise tothe instability wave, the gas flow has several other importanteffects on the acoustic properties of the junction.     

Investigation of secondary flow and the related instability on an infinite yawed cylinder with Schubauer’s ellipse as section

This paper presents the principal results of a theoretical investigation of the secondary flow and the related instability performed in the laminar incompressible boundary layer on an infinite uniform yawed solid cylinder with Schubauer’s ellipse of axial ratio 2·96:1 as the section normal to the leading edge. The secondary flow profiles and the value of the instability criterion are obtained at different points of the wing section and for various angles of sweepback. It is found that in favourable pressure gradients and at pressure minimum, the secondary flow profiles have negative values. In regions of adverse pressure gradients after the pressure minimum the secondary flow changes sign from negative to positive values and have points of inflexion. The change of sign starts from the surface and extends to the edge of the boundary layer downstream. At some points in adverse pressure gradients the secondary flow profiles have double points of inflexion and values of both signs simultaneously. It is found that an adverse pressure gradient produces more powerful secondary flow than a favourable pressure gradient of the same strength. It is also found that the values of the instability criterion increase with the increasing sweepback whether the pressure gradient is favourable or adverse. At every point of the wing section, there are two values of the criterion for a given sweepback. The effect of adverse pressure gradient on the variation of the criterion is much more pronounced than that of a favourable pressure gradient. It is also seen that the flow is intermittently laminar and turbulent for low values of the chordwise free stream Reynolds number and for low values of sweepback and consists of an irregular sequence of laminar and turbulent regions.     

On spatial instability of electrically forced axisymmetric jets with variable applied field

This paper considers the problem of instability of electrically forced axisymmetric jets with respect to spatially growing disturbances and in the presence of a variable applied electric field. A mathematical model, which is developed for the dependent variables of such disturbances, is based on the relevant approximated versions of the equations of the electrohydrodynamics for an electrically forced jet flow. The approximations include the assumptions that the length scale along the axial direction of the jet is much larger than that in the radial direction of the jet and the disturbances are axisymmetric and infinitesimal in amplitude. For neutral temporal stability boundary, we find, in particular, two new spatial modes of instabilities under certain conditions. Both modes are found to be enhanced with increasing the strength of the field. The more dominant instability mode is found to exist for a wider range of values of the wave number in the axial direction. The effect of variable applied electric field is found to increase the growth rates of the disturbances but operate over a more restricted domain in the axial wave number.     

Nonlinear Spatial Evolution of Small Disturbances from Boundary Conditions in Flat Inclined-Channel Flow

The asymptotic behavior of small disturbances as they evolve spatially from boundary conditions in a flat inclined channel is determined. These small disturbances develop into traveling waves called roll waves, first discussed by Dressler in 1949. Roll waves exist if the Froude number F exceeds 2, which consist of a periodic pattern of bores, or discontinuities. After confirming the instability condition for   F > 2  for the linearized equations in the boundary value case, the nonlinear boundary value problem for the weakly unstable region of F slightly larger than 2 is studied. Multiple scales and the Fredholm alternative theorem are applied to determine the evolution of the solution in space. It is found that the solution is dominated by the evolution of the disturbance along one characteristic. The shock conditions governing the asymptotic solution are determined and these conditions are used to determine the approximate shape of the resulting traveling wave from the solution. Both asymptotic and numerical results for periodic disturbances are presented.     

Spatial pattern formations in diffusive predator-prey systems with non-homogeneous Dirichlet boundary conditions

A reaction-diffusion predator-prey system with non-homogeneous Dirichlet boundary conditions describes the persistence of predator and prey species on the boundary. Compared with homogeneous Neumann boundary conditions, the former conditions may prompt or prevent the spatial patterns produced through diffusion-induced instability. The spatial pattern formation induced by non-homogeneous Dirichlet boundary conditions is characterized by the Turing type linear instability of homogeneous state and bifurcation theory. Furthermore, transient spatiotemporal behaviors are observed through numerical simulations.     

Coexistence and stability of a competition—diffusion system in population dynamics

The coexistence and stability of the population densities of two competing species in a bounded habitat are investigated in the present paper, where the effect of dispersion (transportation) is taken into consideration. The mathematical problem involves a coupled system of Lotka-Volterra-type reaction-diffusion equations together with some initial and boundary conditions, including the Dirichlet, Neumann and third type. Necessary and sufficient conditions for the coexistence and competitive exclusion are established and the effect of diffusion is explicitly given. For the stability problem, general criteria for the stability and instability of a steady-state solution are established and then applied to various situations depending on the relative magnitude among the physical parameters. Also given are necessary and sufficient conditions for the existence of multiple steady-state solutions and the stability or instability of each of these solutions. Special attention is given to the Neumann boundary condition with respect to which some threshold results for the coexistence and stability or instability of the four uniform steady states are characterized. It is shown in this situation that only one of the four constant steady states is asymptotically stable while the remaining three are unstable. The stability or instability of these states depends solely on the relative magnitude among the various rate constants and is independent of the diffusion coefficients.     

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.     

Long-wave oscillatory Marangoni instability in a horizontal liquid layer

The long-wave instability in the problem of thermocapillary convection in a horizontal layer with a free deformable boundary and a solid bottom is investigated. The transcendental equation for the main asymptotic term of the spectral parameter is written in explicit form. The main attention is paid to investigating oscillatory instability. For the frequency of neutral oscillations, simple transcendental equations are obtained that contain the Prandtl and Biot numbers. In a number of cases, exact solutions are indicated. Explicit formulae are given for the main asymptotic term of the Marangoni number. In the case of a non-heat-conducting solid wall, the relation between the critical values of the parameters for inverse Prandtl numbers is found. It is shown that, for different Prandtl numbers, the asymptotic values are in good agreement with the numerical values.     

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.     

Investigation of secondary flow instability on an infinite yawed cylinder with 15 per cent. Thick joukowski profile as section

This paper contains a theoretical investigation of the secondary flow instability in the incompressible boundary layer on an infinite yawed cylinder with chordwise section as Joukowski profile of 15 per cent. thickness at zero incidence and with homogeneous suction, the suction mass flow coefficient being equal to 0·2085. Values of the instability criterion are obtained at different points of the wing section and for various angles of sweepback. It is found that the values of the criterion increase with the increasing sweepback whether the pressure gradient is favourable or adverse. The effect of adverse pressure gradient on the variation of the criterion is more pronounced than that of a favourable pressure gradient. At some points in adverse pressure gradients, there are two values of the criterion for a given sweepback. It is also found that the flow is intermittently laminar and turbulent for low values of the chordwise free stream Reynolds number and consists of an irregular sequence of laminar and turbulent regions.     

Asymptotic calculation of inviscidly absolutely unstable modes of the compressible boundary layer on a rotating disk

In this work a long-wavelength asymptotic approach is used to analyze the region of absolute instability in the compressible rotating disk boundary layer flow. Theoretically determined values of branch points for the occurrence of absolute instability in the compressible flow are shown to match onto the ones which are obtained via a numerical solution of the linear inviscid compressible Rayleigh equations.     

Primary elements in formal modules

The stability of an elastic plate in a supersonic gas flow is considered in the presence of a boundary layer formed on the surface of the plate. The problem is solved in two statements. In the first statement, the plate is of large but finite length, and a coupled-mode type of flutter is examined (the effect of the boundary layer on another, single-mode, type of flutter has been studied earlier). In the second statement, the plate is assumed to be infinite, and the character of its instability (absolute or convective) is analyzed. In both cases, the instability is determined by a branch point of the roots of the dispersion equation, and the mathematical analysis is the same. It is proved that instability in a uniform gas flow is weakened by a boundary layer but cannot be suppressed completely, while in the case of a stable plate in a uniform flow the boundary layer leads to the destabilization of the plate.     

Thermal convection problem of micropolar fluid subjected to hall current

Thermal instability of a micropolar fluid layer heated from below in the presence of hall currents is investigated. Using the appropriate boundary conditions on the boundary surfaces of the fluid layer, the frequency equation is derived and then critical Rayleigh number is determined. It is found that hall current parameter has destabilizing effect on the system. For specific values of parameters, oscillatory convection in observed in the system. The behavior of Rayleigh number with wavenumber is also computed for different values of various parameters. The results of some earlier workers have been reduced as a special case from the present problem.     

Linear and non‐linear stability thresholds for thermal convection in a box

We analyse the problem of finding instability thresholds and global non‐linear stability bounds for thermal convection in a linearly viscous fluid in a finite box. The vertical walls are maintained at different temperatures which gives rise to a non‐uniform temperature field in steady state. This problem was previously analysed by Georgescu and Mansutti (Int. J. Non‐Linear Mech. 1999; 34 :603–613). In our work we determine the linear instability threshold to be well above the global stability boundary found by an energy method. Since the perturbed system is not symmetric we expect this to be the case, and our analysis yields a parameter region where possible sub‐critical instabilities may be found. Copyright © 2006 John Wiley & Sons, Ltd.     

Flow Past a Swept Wing with a Compliant Surface: Stabilizing the Attachment-Line Boundary Layer

Many aquatic species such as dolphins and whales have fins, which can be modeled as swept wings. Some of these fins, such as the dorsal fin of a dolphin, are semi-rigid and therefore can be modeled as a rigid swept wing with a compliant surface. An understanding of the hydrodynamics of the flow past swept compliant surfaces is of great interest for understanding potential drag reduction mechanisms, especially since swept wings are widely used in hydrodynamic and aerodynamic design. In this paper, the flow past a swept wing with a compliant surface is modeled by an attachment-line boundary layer flow, which is an exact similarity solution of the Navier–Stokes equations, flowing past a compliant surface modeled as an elastic plate. The hydrodynamic stability of the coupled problem is studied using a new numerical framework based on exterior algebra. The basic instability of the attachment line boundary layer on a rigid surface is a traveling wave instability that propagates along the attachment line, and numerical results show that the compliance results in a substantial reduction in the instability region. Moreover, the results show that, although the flow-field is three-dimensional, the qualitative nature of the instability suppression is very similar to the qualitative reduction of instability of the two-dimensional Tollmien–Schlichting modes in the classical boundary-layer flow past a compliant surface.