Initial Development of Diffusion in Poiseuille Flow

The combined effect of diffusion, and of convection by Poiseuilleflow, on the distribution of a small quantity of miscible additiveinjected into a tube of radius a, is to spread it longitudinallywith a Taylor "effective diffusion coefficient", to an approximationthat is good at times greater than about 0.5a²/D (Bailey &Gogarty, 1962), where D is the molecular diffusion coefficient.The present theory, complementary to the Taylor theory, determinesthe initial action of diffusion on the front of the concentrationdistribution, to an approximation that is good at times t lessthan about 0.1a²/D. The theory is exact wherever the added substancedoes not yet interact with the tube wall, and predicts thatthe spread in the front due to diffusion extends (Fig. 2) overa distance of order DUt²/a², where U is the velocity on theaxis of the tube. The transition between distributions characteristicof the two theories is illustrated (Fig. 4); and the introductionindicates the relevance of the new theory to work (Caro, 1966)on tracers used in study of the blood circulation.     

A Family of Wave-Mean Shear Interactions and Their Instability to Longitudinal Vortex Form

The inviscid instability of O(ε) two-dimensional periodic flows to spanwiseperiodic longitudinal vortex modes in parallel O(1) shear flows of the form ū = ± |z|^q is considered. Here the mean velocity ū is relative to the wave and q is a constant. Such shear flows admit neutral Rayleigh waves with amplitudes that either diminish or diverge with |αz|; both are considered. Of particular interest are streamwise α and spanwise l wavenumbers in the range l² ? α², α = O(1), as it is here that the most analytical progress can be made. A generalized Lagrangian-mean formulation is used to describe the effect of fluctuations upon the mean state and, because the developing mean flow acts to distort the waves, a further equation, the Rayleigh-Craik equation, is employed to complete the specification. It is shown that instability to longitudinal vortex form is likely for both classes of waves in many physically interesting situations, from simple mixing layers to atmospheric boundary layers over undulating surfaces.     

Flow Invariance on Stratified Domains

This paper studies conditions for invariance of dynamical systems on stratified domains as originally introduced by Bressan and Hong. We establish Hamiltonian conditions for both weak and strong invariance of trajectories on systems with non-Lipschitz data. This is done via the identification of a new multifunction, the essential velocity multifunction. Properties of this multifunction are investigated and used to establish the relevant invariance criteria.     

The KdV Equation in Stratified Fluid Flow

This paper describes a method for the formal derivation of the equation governing long waves on the surface of a shallow fluid which is stably and continuously stratified. By using a three variable expansion procedure involving the normal time scale together with two slow time scales suggested as a result of the formulation of the problem the governing equation is shown to be the ordinary Korteweg-de Vries equation.

AMS(MOS) CLASSIFICATION: 76D30, 35Q20, 35C20.     

Weakly Nonlinear Analysis of a Localized Disturbance in Poiseuille Flow

In this article, we investigate, via a perturbation analysis, some important nonlinear features related to the process of transition to turbulence in a wall-bounded flow subject to a spatially localized disturbance that is harmonic in time. We show that the perturbation expansion, truncated at second order, is able to capture the generation of streamwise vorticity as a weakly nonlinear effect. The results of the perturbation approach are discussed in comparison with direct numerical simulation data for a sample case by extracting the contribution of the different orders. The main aim is to provide a tool to select the most effective nonlinear interactions to enlighten the essential features of the transitional process.     

Stability of the Poiseuille Flow in a Channel with Comb Grooves

The stability of the Poiseuille flow in a channel with longitudinal comb grooves on the lower wall is studied numerically. Dependences of the linear and energy critical Reynolds numbers on the groove spacing and height are obtained and analyzed. The results are compared with data available for wavy grooves, which tend to comb grooves as one of the groove parameters approaches infinity.     

Long-Wave Instability in a Three-Layer Stratified Shear Flow

In inviscid fluid flows, instability can occur because of a resonance between two wave modes. For the case when the modes remain distinct at the critical point where the wave phase speeds coincide, then in the weakly nonlinear, long-wave limit, there is an expectation that the generic outcome is a model consisting of two coupled Korteweg–deVries equations. This situation is examined for a certain three-layer stratified shear flow.     

Initial Algebraic Growth of Small Angular Dependent Disturbances in Pipe Poiseuille Flow

The evolution of small, angular dependent velocity disturbances in laminar pipe flow is studied. In particular, streamwise independent perturbations are considered. To fully describe the flow field, two equations are required, one for the radial and the other for the streamwise velocity perturbation. Whereas the former is homogeneous, the latter has the radial velocity component as a forcing term. First, the normal modes of the system are determined and analytical solutions for eigenfunctions, damping rates, and phase velocities are calculated. As the azimuthal wave number (n) increases, the damping rate increases and the phase velocities decrease. Particularly interesting are results showing that the phase velocities associated with the streamwise eigenfunctions are independent of the radial mode index when n = 1, and when n = 5 the same is obtained for phase velocities associated with the eigenfunctions of the radial component. Then, the initial value problem is treated and the time development of the disturbances is determined. The radial and the azimuthal velocity components always decay but, owing to the forcing, the streamwise component shows an initial algebraic growth, followed by a decay. The kinetic energy density is used to characterize the induced streamwise disturbance. Its dependence on the Reynolds number, the radial mode, and the azimuthal wave number is investigated. With a normalized initial disturbance, n = 1 gives the largest amplification, followed by n = 2 etc. However, for small times, higher values of n are associated with the largest energy density. As n increases, the distribution of the streamwise velocity perturbation becomes more concentrated to the region near the pipe wall.     

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.     

Shock Interactions in Nonequilibrium Hypersonic Flow

A shock interaction problem is solved with finite difference methods for a hypersonic flow of air with chemical reactions. If a body has two concave corners, a secondary shock is formed in the shock layer and it meets the main shock later. As the two shocks meet, the flow becomes singular at the interaction point, and a new main shock, a contact discontinuity and an expansion wave appear as a result of interaction between the two shocks. Therefore, the problem is very complicated. Using proper combinations of implicit and explicit finite difference schemes according to the property of the equations and the boundary conditions, we compute the flow behind the interaction point successfully.     

Resonant Long–Short Wave Interactions in an Unbounded Rotating Stratified Fluid

A theoretical study is made of finite-amplitude modulated internal wavetrains and the attendant nonlinear interaction with the mean flow induced by the modulations, in an unbounded uniformly stratified Boussinesq fluid rotating around the vertical axis. When the rotation is relatively weak, in particular, 'flat' wavetrains, that feature stronger vertical than horizontal modulations, are resonantly coupled with the mean flow in a manner analogous to the resonant long–short wave interaction between gravity and capillary waves on the surface of deep water. A coupled set of evolution equations for the vertical wavenumber, the wave amplitude, and the mean flow is derived under resonant conditions, and is used to examine the propagation of locally confined wavetrains with initially uniform wavenumber and no pre-existing mean flow. The resonant interaction causes radiation of energy away from a flat wavetrain by means of the induced mean flow which forms a trailing wake; this furnishes a possible mechanism for generating low-frequency inertial–gravity waves in the atmosphere, as suggested by field observations. Moreover, owing to refraction by the mean flow, a finite-amplitude wavetrain may experience rapid wavenumber variations in certain locations, consistent with prior numerical simulations. Eventually, in these regions, the wavenumber tends to become multi-valued, suggesting the formation of caustics.     

On the Initial Value Problem for Rotating Stratified Flow

The time-dependent properties of the flow generated by a small disturbance of a rigidly rotating, compressible, non-dissipative fluid in a gravitational field are derived, without restricting the spacial dependence of the gravitational field or the shape of the rigid container. The conserved quantities are used to characterize in a new way the time-independent portion of the flow in terms of data from the initial disturbance.     

On Stratified Shear Flow in Sea Straits of Arbitrary Cross Section

Equations and theorems governing the flow of an inviscid, incompressible, continuously-stratified fluid in a gradually varying channel with an arbitrary cross section are developed. The stratification and longitudinal velocity are assumed to be uniform in the transverse direction, an assumption that is supported under the assumption of gradual topographic variations. Extended forms of Long's model and the Taylor–Goldstein equation are developed. Interestingly, the presence of topographic variation does not alter the necessary condition for instability (Richardson number     ) nor the bounds on unstable eigenvalues (the semi-circle theorem). The former can be proved using a new technique introduced herein. For the special case of homogeneous shear flow, generalized versions of the theorems of Rayleigh and Fjørtoft do depend on the form of the topography, though no general tendency toward stabilization or destabilization is apparent. Previous results on the bounds and enumeration of neutral modes are also extended. The results should be of use in the hydraulic interpretation of exchange flow in sea straits.     

On the Nonlinear Development of Small Three-Dimensional Perturbations in Plane Poiseuille Flow: Single-Mode Approach

Nonlinear aspects of developing three-dimensional perturbations in plane Poiseuille flow have been elucidated at the primary, instead of the conventional secondary, level. Three-dimensional perturbation velocities generate normal vorticity by stretching and tilting the basic-flow vorticity. The amplitude of the induced normal vorticity, and hence that of the streamwise perturbation velocity, can grow temporally to significant peak values before the exponential decay predicted by the linear theory sets in. These growths, according to the linear theory, do not influence the amplitudes of the normal perturbation velocity that are monotonically decaying with time. It is shown in this study that the normal velocity continues to be oblivious to the development of induced normal vorticity, even in the nonlinear regime, if the perturbation velocities are described by waves traveling in a single oblique direction. Also, the Reynolds number dependence of the amplitude of the normal vorticity is discussed.     

Existence of Neutral Rayleigh Waves in the Hagen–Poiseuille Flow Through a Pipe of Circular Cross Section

The inviscid neutral stability of Hagen–Poiseuille flow through a circular pipe is studied using both analytical and numerical techniques. A zero phase shift is applied across the critical surface to represent the effects of strong nonlinearity. Using a form of Sturm's comparison theorem it is possible to prove that no neutral solutions exist if a combination of the axial and azimuthal wave numbers of the perturbation exceeds a critical value. As a consequence, the physical problem admits only neutral solutions for an azimuthal wave number of unity.     

Viscous-Inviscid Interactions in a Boundary-Layer Flow Induced by a Vortex Array

In this paper, we investigate the asymptotic validity of boundary-layer theory. For a flow induced by a periodic row of point-vortices, we compare Prandtl’s boundary-layer solution to Navier-Stokes solutions with different Reynolds numbers. We show how Prandtl’s solution develops a finite-time separation singularity. On the other hand, the Navier-Stokes solutions are characterized by the presence of two distinct types of viscous-inviscid interactions that can be detected by the analysis of the enstrophy and of the pressure gradient on the wall. Moreover, we apply the complex singularity-tracking method to Prandtl and Navier-Stokes solutions and analyze the previous interactions from a different perspective.     

L-fuzzy层保序算子空间中的层连通性

在[6]中引入ωα-闭集的基础上,引入ωα-连通,讨论其主要的拓扑性质,说明在层保序算子空间中樊矶定理仍然成立。另外,文中建立了两种不同层保序算子之间的关系,给出两种定义的等价条件。     

L-fuzzy层拟一致结构理论

在L-拓扑空间中提出α-远域族和α-拟一致结构的概念,讨论它的一些基本性质。在此基础上证明了每一个α-拟一致结构都可以诱导出一个α-层拓扑,每个α-层拓扑空间都可以α-拟一致化。由此,进一步分析α-拟一致连续与层连续之间的关系。     

Existence of a Nonstationary Poiseuille Solution

The nonstationary Poiseuille solution describing the flow of a viscous incompressible fluid in an infinite cylinder is defined as a solution of the inverse problem for the heat equation. The existence and uniqueness of such nonstationary Poiseuille solution with the prescribed flux F(t) of the velocity field is studied. It is proved that under some compatibility conditions on the initial data and flux F(t) the corresponding inverse problem has a unique solution in Holder spaces.Original Russian Text Copyright © 2005 Pileckas K. and Keblikas V.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 649–662, May–June, 2005.