Cross-stream Scale Research Articles

Nonlinear Dynamics of Anisotropic Disturbances in Plane Couette Flow

The dynamics of disturbances to a two-dimensional inviscid Couette flow is examined under the assumption that the disturbances have a cross-stream scale which is asymptotically smaller than their streamwise scale. Such anisotropic disturbances emerge naturally from isotropic perturbations after long times because of the shearing effect of the basic flow. An asymptotic equation describing the nonlinear evolution of anisotropic disturbances is derived using aregular-perturbation technique. The close relationship between this equation and those found in critical-layer problems is discussed. The case of doubly periodic disturbances is examined in detail, since it leads to a remarkable dynamics: formally, the evolution is discontinuous in time and is given by a sequence of jumps which take place when the time is a rational number (multiplied by a fixed geometric factor). The interpretation of such a dynamical system, in a sense intermediate between discrete and continuous systems, is discussed. The asymptotic model is used to study simple interactions between two sheared (Fourier) modes and to investigate the hydrodynamic echo effect. Analytical results are complemented by results of direct numerical simulations.

Simple Curvilinear Method for Numerical Methods of Open Channels

Three-dimensional finite-difference or finite-volume models of sinuous open channels (e.g., narrow rivers, estuaries, and reservoirs) generally require boundary-fitted grids and curvilinear flow solution. Cartesian models with square grid cells are simpler to apply, but require a larger number of cells, as the cell size is determined by cross-stream resolution. This paper presents a simplified curvilinear approach suitable for systems where the along-stream length scale is larger than the cross-stream scale. The curvilinear Navier-Stokes equations are manipulated so the left-hand side is identical to the Cartesian momentum equations. The right-hand side then consists of grid-stretching curvature terms. These terms are written as functions of a perturbation parameter, so the first-order curvilinear effects are obtained with the lowest-order perturbation terms. As the Cartesian equations' form is preserved, we can readily adapt a Cartesian model to this perturbation curvilinear approach by adding the small curvilinear terms as explicit momentum sources.

A Numerical Study of Stratified Airflow over Mesoscale Heat Sources with Application to Carolina Coastal Frontogenesis

Abstract This paper presents the results from a numerical investigation of the responses of stratified airflow to prescribed near-surface mesoscale axisymmetric (circular) and elongated (elliptical) heat sources under uniform basic wind conditions using a simple three-dimensional model. Model results indicate that the structure of the response depends on the Froude number (Fr = V/NH) and the Rossby number (Ro = V/fL) associated with an axisymmetric thermal forcing, where H and L are the vertical and horizontal scales of the heat source, V the wind speed, N the Brunt-Vaisala frequency, and f the Coriolis parameter. However, the response to an elongated thermal forcing depends not only on the Froude and Rossby numbers, but also on the ratio of the alongstream scale (Ls) to the cross-stream scale (Ln) of the heat source, which are determined by the horizontal shape of the heat source and the direction of the basic flow. When Ls/Ln is less than 1, that is, the heat source is elongated in the cross-stream dire...

Resonant growth of three-dimensional modes in trnsitioning Blasius boundary layers

By carefully controlled phase-coupled input of simultaneous two- and three-dimensional disturbances, the nonlinear evolution and breakdown of the laminar flow in a boundary layer was examined. This involved the generation of plane Tollmien–Schlichting waves and pairs of oblique waves so as to promote nearresonance conditions which have been theoretically shown to lead to the rapid development of three-dimensionality in unstble boundary layers. Special emphasis is placed on the two prominent mechanisms, namely resonant-triads of Orr–Sommerfeld modes and the secondary instability of the streamwise periodic flow to spanwise periodic three-dimensional disturbances. The sensitivity of these mechanisms on the amplitudes and wavenumbers of the input disturbances was of special focus.The simultaneous two- and three-dimensional wave generation was accomplished using a spanwise array of line heaters suspended just above the wall at the approximate height of the critical layer in the laminar boundary layer. These were operated to produce, through local heating, time-periodic spanwise-phase-varying velocity perturbations. Of primary emphasis in this paper are conditions obtained by the combined forcing of fundamental plane waves with wavenumbers (α, 0) and pairs of subharmonic oblique waves (½α, ± β). The reslults document resonant growth of energy in the subharmonic modes, the formation of staggered lambda vortex patterns with a cross-stream scale commensurate with the seeded ± β condition, and their subsequet transition to turbulence. Complete documentation of the flow field at these various stages is presented using smoke-wire flow visualization and through phase-conditioned hot-wire surveys measuring all three velocity components in three space dimensions.

On the theory of the East African Low Level Jet Stream

Several theoretical models for the East African Low Level Jet Stream are described. They all share the notion that the northward advection of planetary vorticity across the equator, coupled with the presence of a north—south mountain barrier, leads to the formation of a low-level western boundary current (akin to the Gulf-Stream) along the equatorial east coast of Africa. They differ in the manner in which the planetary vorticity advection is balanced to obtain a quasi-steady state. A purely inertial model predicts the correct cross-stream scale of the jet, but does not reproduce the observed inner shear layer which reduces the jet velocity to zero inland near the highlands. The lateral friction model can produce a realistic jet profile if the horizontal eddy viscosity (appearing as a free parameter) is chosen appropriately. However this solution shows a recirculation, i.e., northerly flow, off the coast that has not yet been observed. Finally, a model that includes bottom friction over variable topography also can give realistic jet profiles. If one accepts that the mountains, the Beta effect, and some form of inertial or frictional acceleration act together to produce the cross-equatorial low level jet stream, then one can formulate the types of observations needed to distinguish between the various theories.

On the dynamics of finite-amplitude baroclinic waves as a function of supercriticality

A finite-amplitude model of baroclinic instability is studied in the case where the cross-stream scale is large compared with the Rossby deformation radius and the dissipative and advective time scales are of the same order. A theory is developed that describes the nature of the wave field as the shear supercriticality increases beyond the stability threshold of the most unstable cross-stream mode and penetrates regions of higher supercriticality. The set of possible steady nonlinear modes is found analytically. It is shown that the steady cross-stream structure of each finite-amplitude mode is a function of the supercriticality.Integrations of initial-value problems show, in each case, that the final state realized is the state characterized by the finite-amplitude mode with the largest equilibrium amplitude. The approach to this steady state is oscillatory (nonmonotonic). Further, each steady-state mode is a well-defined mixture of linear cross-stream modes.