The Nonlinear Critical Layer in a Slightly Stratified Shear Flow

An attempt is made in this paper to extend the nonlinear critical layer analysis, as developed for homogeneous shear flows by Benney and Bergeron [1] and Davis [2], to the case of a stratified shear flow. Although the analysis is restricted to small values of the Richardson number evaluated at the edge of the critical layer, it is definitely shown that buoyancy leads to the formation within the critical layer region of thin velocity and thermal boundary layers which tend to reduce the local Richardson number. We suggest that this result has considerable relevance to the phenomenon of clear air turbulence. As in the homogeneous case, no phase change of the disturbance takes place across the nonlinear critical layer.     

The Small Vorticity Nonlinear Critical Layer for Kelvin Modes on a Vortex

We consider in this paper the propagation of neutral modes along a vortex with velocity profile being the radial coordinate. In the linear stability theory governing such flows, the boundary in parameter space separating stable and unstable regions is usually comprised of modes that are singular at some value of r denoted r_c , the critical point. The singularity can be dealt with by adding viscous and/or nonlinear effects within a thin critical layer centered on the critical point. At high Reynolds numbers, the case of most interest in applications, nonlinearity is essential, but it develops that viscosity, treated here as a small perturbation, still plays a subtle role. After first presenting the scaling for the general case, we formulate a nonlinear critical layer theory valid when the critical point occurs far enough from the center of the vortex so that the vorticity there is small. Solutions are found having no phase change across the critical layer thus permitting the existence of modes not possible in a linear theory. It is found that both the axial and azimuthal mean vorticity are different on either side of the critical layer as a result of the wave–mean flow interaction. A long wave analysis with O (1) vorticity leads to similar conclusions.     

Subcritical Rossby Waves in Zonal Shear Flows with Nonlinear Critical Layers

It was shown by Benney and Bergeron [ 1 ] that singular neutral modes with nonlinear critical layers are mathematically possible in a variety of shear flows. These are usually subcritical modes; i.e., they occur at values of the flow parameters where their linear, viscous counterparts would be damped. One question raised then is how such modes might be generated.
This article treats the problem of Rossby waves propagating in a mixing layer with velocity profile ū = tanh y . The beta parameter, which is a measure of the stabilizing Coriolis force, is taken to be large enough so that linear instability cannot occur. First, computed dispersion curves are presented for singular modes with nonlinear critical layers. Then, full numerical simulations are employed to illustrate how these modes can be generated by resonant interaction with conventional nonsingular Rossby waves, even when the singular mode is absent initially.     

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.     

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.     

On the stability of space-periodic convective flows in a vertical layer with sinuous boundaries : PMM vol. 43, no. 6, 1979, pp. 998–1007

The problem of convection in a vertical layer with harmonically distorted boundaries is examined by perturbation theory methods for a small amplitude of sinuosity. The solutions obtained are applicable both in the stability region as well as in the supercritical region of the plane-parallel flow. The stability of the solutions found is investigated with respect to a certain class of space-bounded perturbations that are not necessarily space-periodic. The method of amplitude functions [1], generalized to the case of curved boundaries, is used. The Grashof critical number is found as a function of the period of sinuosity and the form of the neutral curve for the space-periodic motions and their stability region are obtained. It is established that if the deformation period of the boundaries is close to the wavelength of the critical perturbation for the plane-parallel flow or is twice as great, then as the Grashof number grows stability loss does not occur and the motion's amplitude changes continuously (cf. [2 — 4]). A comparison is made with the results of the numerical calculation in [5], An attempt was made in [6] to construct a stationary periodic motion in a layer with weakly-deformed boundaries, in the form of series in powers of a small sinuosity amplitude. However, the solution obtained diverges in a neighborhood of the neutral curve of the plane-parallel flow and approximates unstable motion in the supercritical region of the unperturbed problem. Flows under a finite sinuosity amplitude are calculated by the net method in [5] wherein the stability of the flows was investigated as well, but only with respect to perturbations with wave numbers that are multiples of 2π/l, where l is the length of the calculated region.     

二维气固两相混合层中固粒对流场影响的研究

采用双向耦合模型对含有固粒的二维气固两相混合层流场进行了研究。流场采用拟谱方法直接数值模拟,固粒采用颗粒-轨道模型,在考虑流场对固粒作用的同时,考虑固粒对流场的反作用。结果发现固粒的浓度和Stokes数对流场影响明显。固粒的作用使涡量扩散加快,并阻碍流场的变化,减弱了流场中拟序结构的强度,缩短涡的生存期;固粒在流场中的分布规律与单相耦合所得结果相似。     

Rossby Solitary Waves in the Presence of a Critical Layer

This study considers the evolution of weakly nonlinear long Rossby waves in a horizontally sheared zonal current. We consider a stable flow so that the nonlinear time scale is long. These assumptions enable the flow to organize itself into a large‐scale coherent structure in the régime where a competition sets in between weak nonlinearity and weak dispersion. This balance is often described by a Korteweg‐de‐Vries equation. The traditional assumption of a weak amplitude breaks down when the wave speed equals the mean flow velocity at a certain latitude, due to the appearance of a singularity in the leading‐order equation, which strongly modifies the flow in a critical layer. Here, nonlinear effects are invoked to resolve this singularity, because the relevant geophysical flows have high Reynolds numbers. Viscosity is introduced in order to render the nonlinear‐critical‐layer solution unique, but the inviscid limit is eventually taken. By the method of matched asymptotic expansions, this inner flow is matched at the edges of the critical layer with the outer flow. We will show that the critical‐layer–induced flow leads to a strong rearrangement of the related streamlines and consequently of the potential‐vorticity contours, particularly in the neighborhood of the separatrices between the open and closed streamlines. The symmetry of the critical layer vis‐à‐vis the critical level is also broken. This theory is relevant for the phenomenon of Rossby wave breaking and eventual saturation into a nonlinear wave. Spatially localized solutions are described by a Korteweg‐de‐Vries equation, modified by new nonlinear terms; depending on the critical‐layer shape, this leads to depression or elevation waves. The additional terms are made necessary at a certain order of the asymptotic expansion while matching the inner flow on the dividing streamlines. The new evolution equation supports a family of solitary waves. In this paper we describe in detail the case of a depression wave, and postpone for further discussion the more complex case of an elevation wave.     

The Bernoulli Integral for a Certain Class of Non‐Stationary Viscous Vortical Flows of Incompressible Fluid

It has been shown in our paper [1] that there is a wide class of 3D motions of incompressible viscous fluid which can be described by one scalar function dabbed the quasi‐potential. This class of fluid flows is characterized by three‐component velocity field having two‐component vorticity field; both these fields can depend of all three spatial variables and time, in general. Governing equations for the quasi‐potential have been derived and simple illustrative example of 3D flow has been presented. Here, we derive the Bernoulli integral for that class of flows and compare it against the known Bernoulli integrals for the potential flows or 2D stationary vortical flows of inviscid fluid. We show that the Bernoulli integral for this class of fluid motion possesses unusual features: it is valid for the vortical nonstationary motions of a viscous incompressible fluid. We present a new very nontrivial analytical example of 3D flow with two‐component vorticity which hardly can be obtained by any of known methods. In the last section, we suggest a generalization of the developed concept which allows one to describe a certain class of 3D flows with the 3D vorticity.     

Stability of flow of a liquid crystal layer on an inclined plane : PMM vol. 38, n 6, 1974, pp. 1031–1040

The framework of the linear mechanics of liquid crystal media [1] is used to study propagation of waves in a layer of a nematic liquid crystal (NLC) on an inclined plane, in a magnetic field, for three different cases of orientation of the anisotropy axis, namely orthogonal to the inclined plane, parallel to the inclined plane and orthogonal to the plane of flow. Such orientations of the anisotropy axis are realized in practice in the course of special machining of solid surfaces [2]. Exact solutions of the equations of motion are obtained describing the steady flow of the layer, and the behavior of small plane perturbations is studied. It is shown that two types of plane waves can propagate in a layer of the nematic mesophase, namely, the surface and the orientational waves. In the case of long surface waves the formulas for the critical Reynolds number are obtained. For the orientational waves a sufficient criterion of stability of the flow in the layer is obtained for two cases. The influence of the magnetic field and of the rheological parameters of NLC on the character of propagation of the first and second type waves is investigated.From amongst the papers dealing with wave propagation in NLC, we draw the readers' attention to [3] which deals with the longitudinal, shear and torsional waves in a liquid crystal domain and obtains the corresponding dispersion relationships.     

Meshless solution of two-dimensional incompressible flow problems using the radial basis integral equation method

The two-dimensional incompressible fluid flow problems governed by the velocity–vorticity formulation of the Navier–Stokes equations were solved using the radial basis integral (RBIE) equation method. The RBIE is a meshless method based on the multi-domain boundary element method with overlapping subdomains. It solves at each node for the potential and its spatial derivatives. This feature of the RBIE is advantageous in solving the velocity–vorticity formulation of the Navier–Stokes equations since the calculated velocity gradients can be used to compute the vorticity that is prescribed as a boundary condition to the vorticity transport equation. The accuracy of the numerical solution was examined by solving the test problem with known analytical solution. Two benchmark problems, i.e. the lid driven cavity flow and the thermally driven cavity flow were also solved. The numerical results obtained using the RBIE showed very good agreement with the benchmark solutions.     

Magnetohydrodynamics flow of a Burgers’ fluid in an orthogonal rheometer

The magnetohydrodynamics flow of an electrically conducting, incompressible Burgers’ fluid in an orthogonal rheometer is investigated. An exact solution is obtained. The effects of various dimensionless parameters existing in the model on the velocity field, vorticity and traction are studied graphically. It is noted that boundary layers form for a variety of reasons. It form as the Reynolds number increases. Also, as the Weissenberg number increases a distinct boundary layer formation is observed. It can develop at low Reynolds number provided the Weissenberg number is sufficiently high, however, it is not possible in the case of a Newtonian fluid. It is shown that no torque is exerted by the fluid on one of the disks. Results are compared with Oldroyd-B fluid.     

The Evolution of a Nonlinear Critical Level

The evolution of inviscid forced Rossby waves on a parallel flow in the presence of a critical layer is discussed. It is shown that the transient critical layer becomes nonlinear after sufficient time has elapsed and that the “? π” logarithmic phase shift is modified accordingly. All harmonics are found to be important in the nonlinear range, even outside the critical layer.     

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.     

Dispersion relations for rotational gravity water flows having two jumps in the vorticity distribution

We derive the dispersion relation for periodic traveling waves propagating at the surface of water with a layer of constant non-zero vorticity situated between two layers of irrotational flow. Due to the complicated nature of the dispersion relation – a fourth order algebraic equation with intricate coefficients – we also give an estimate of a very simple form involving only the levels at which the vorticity has jumps. Our formula generalizes a corresponding one from [5].     

Amplitude-dependent Stability of Boundary-layer Flow with a Strongly Non-linear Critical Layer

The amplitude-dependent neutral stability properties, mainlyof an accelerating boundary-layer flow, are studied theoreticallyfor large Reynolds numbers when the disturbance size is sufficientlylarge to provoke a strongly non-linear critical layer withinthe flow field. The theory has a rational basis aimed at a detailedunderstanding of the delicate physical balances controllingstability. It shows that when the fundamental disturbance size rises to O(R^-1/3, where R is the Reynolds number based on theboundary-layer thickness, the neutral wavelength shortens andthe wavespeed increases in such a way that they become comparablewith the typical thickness and speed, respectively, of the basicflow. In this Rayleigh-like situation a new (previously negligible)feature emerges, that of a substantial pressure variation acrossthe critical layer, which strongly affects the jump conditionson the Rayleigh solutions holding outside the critical layer.As a result of the strong non-linearity the total velocity jumpis affected non-linearly by the critical layer vorticity, whilein contrast the phase shift remains linearly dependent on thevorticity. Furthermore, it is shown that the phase shift, notthe total velocity jump, dictates the neutral stability criteria. Also, flow reversal occurs near the wall where the disturbanceis greater than the basic flow. The link between the viscouseffects in the wall layers and in the critical layer fixes theamplitude-dependence of the neutral modes throughout. As thedisturbance amplitude increases the critical layer with vorticitytrapped within it moves toward the edge of the boundary layerand is forced to leave the boundary layer when exceeds O(R^-1/3,if neutral stability is to be maintained. This departure israther abrupt, involving a dependence on (scaled amplitude)^–12.A study of the more practical application to temporally growingdisturbances should be interesting.     

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 Equation of Motion of a Thin Layer of Uniform Vorticity

A higher order extension to Moore's equation governing the evolution of a thin layer of uniform vorticity in two dimensions is obtained. The equation, in fact, governs the motion of the center line of the layer and is valid for consideration of motion whereby the layer thickness is uniformly small compared with the local radius of curvature of the center line. It extends Birkoff's equation for a vortex sheet. The equation is used to examine the growth of disturbances on a straight, steady layer of uniform vorticity. The growth rate for long waves is in good agreement with the exact result of Rayleigh, as required. Further, the growth of waves with length in a certain range is shown to be suppressed by making this approximate allowance for finite thickness. However, it is found that very short waves, which are quite outside the range of validity of the equation but which are likely to be excited in a numerical integration of the equation, are spuriously amplified as in the case of Moore's equation. Thus, numerical integration of the equation will require use of smoothing techniques to suppress this spurious growth of short wave disturbances.     

Dispersion relations for steady periodic water waves with an isolated layer of vorticity at the surface

In this article we obtain the dispersion relation for small-amplitude steady periodic water waves, which propagate over a flat bed with a specified and fixed mean depth, and where the underlying flow has a discontinuous vorticity distribution. This discontinuity takes the form of an isolated layer of constant non-zero vorticity at the surface of the flow, with purely irrotational flow beneath this layer.