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Volume 6, Issue 3, March 1994

Reduction in the dimensionality of turbulence due to a strong rotation
View Description Hide DescriptionThree‐dimensional incompressible hydrodynamicturbulence is driven to a statistically steady state. A strong uniform rotation is then turned on. It is shown that the turbulence reduces to an approximate two‐dimensional state. Furthermore, the energy inverse cascades to longer length scales.

Streamwise structures in a turbulent supersonic boundary layer
View Description Hide DescriptionFlow visualizations in a high Reynolds number,Mach 3, fully developed turbulent boundary layer indicate that the upper half of the boundary layer is populated with elongated longitudinal structures. These structures are robust with considerable streamwise but very limited spanwise extent, and are randomly distributed in space and time. Possible mechanisms for the generation of these structures are discussed.

Self‐similar, surfactant‐driven flows
View Description Hide DescriptionConsider a dilute, insoluble surfactantmonolayer on the free surface of a thin viscous film. A gradient in surfactant concentration generates a gradient in surface tension, driving a flow that redistributes the surfactant so that these gradients decay. The nonlinear evolution equations governing such flows, derived using lubricationtheory, have previously been shown to admit a set of simple similarity solutions representing the spreading of a monolayer over an uncontaminated interface. Here, a much more general class of similarity solutions is considered, and a transformation is identified reducing the governing partial differential equations to a set of nonlinear ordinary differential equations, the solutions of which correspond to integral curves in a two‐dimensional phase plane. This allows the construction of solutions to a wide range of problems. Many new solutions are revealed, including one that cannot be determined by simpler techniques, namely the closing of an axisymmetric hole in a monolayer, the radius of which is shown to be proportional to (−t)^{δ} as t→0−, where δ≊0.807 41; this solution corresponds to a heteroclinic orbit in the phase plane.

Asymmetric, oscillatory motion of a finite‐length cylinder: The macroscopic effect of particle edges
View Description Hide DescriptionThe oscillatory motion of a finite‐length, circular cylinder perpendicular to its symmetry axis in an incompressible, viscous fluid is described by the unsteady Stokes equations. Numerical calculations are performed using a first‐kind, boundary‐integral formulation for particle oscillation periods comparable to the viscousrelaxation time. For high‐frequency oscillations, a two‐term, boundary layer solution is implemented that involves two, sequentially solved, second‐kind integral equations. Good agreement is obtained between the boundary layer solution and fully numerical solutions at moderate oscillation frequencies. At the edges, where the base joins the side of the cylinder, the pressure and both components of tangential stress exhibit distinct, singular behaviors that are characteristic of steady, two‐dimensional, viscousflow. Numerical calculations accurately capture the theoretically predicted singular behavior. The unsteady flow reversal process is initiated by a complex near‐field flow reversal process that is inferred from the tangential stress distribution. A qualitative picture is constructed that involves the formation of three viscouseddies during the decelerating portion of the oscillation cycle: two attached to the ends of a finite‐length cylinder, and a third that wraps around the cylinder centerline; the picture is similar to the results for axisymmetric flow. As deceleration proceeds, the eddies grow and coalesce at the cylinder edges to form a single eddy that encloses the entire particle. The remainder of the oscillatoryflow cycle is insensitive to particle geometry and orientation. The macroscopic effect of the sharp edges is illustrated by considering ultrasonic, viscous dissipation in a dilute suspension. For a fixed particle‐to‐fluid density ratio, four different frequency regimes are identified. Four distinct viscous dissipation spectra are shown for different particle‐to‐fluid density ratios. The results indicate that particle geometry is important only for particles considerably less dense than the suspending fluid. The effect of edges is most apparent for disk‐ and rod‐shaped particles.

Measurements of the collision properties of small spheres
View Description Hide DescriptionAn experiment to measure the properties of the collisions between two small spheres or between a small sphere and a semi‐infinite flat wall are described. The apparatus releases the particles in a free‐fall without initial spin. The impacts are modeled in terms of three coefficients. The first is the coefficient of normal restitution. The second represents the frictional properties of the contact surfaces. The last characterizes the restitution of the tangential components of the velocity of the contact point for impacts that do not involve sliding. The coefficients are calculated from stroboscopic photographs of the ballistics of the particles near the collision. The results establish that the collision model provides an accurate description of the dynamics of the impacts.

The accumulation and dispersion of heavy particles in forced two‐dimensional mixing layers. I. The fundamental and subharmonic cases
View Description Hide DescriptionThis paper presents detailed computational results for the dispersion of heavy particles in transitional mixing layers forced at both the fundamental and subharmonic frequencies. The results confirm earlier observations of particle streaks forming in the braid region between successive vortices. A scaling argument based on the idealization of the spatially periodic mixing layer as a row of point vortices shows that the formation of these concentrated particle streaks proceeds with optimum efficiency for St≂1. It thereby provides a quantitative basis for experimental and numerical observations of preferential particle dispersion at Stokes numbers of order unity. Both the model and full simulation furthermore exhibit oscillatory particle motion, as well as the formation of two bands of high particle concentrations, for larger Stokes numbers. The particle dispersion as a function of time and the Stokes number is quantified by means of two different integral scales. These show that the number of dispersed particles does not reach a maximum for intermediate Stokes number. However, when the distance is weighted, optimum dispersion is observed for Stokes numbers around unity. By tracing the dispersed particles backwards in time, they are found to originate in inclined, narrow bands that initially stretch from the braid region into the seeded free stream. This suggests that particle dispersion can be optimized by phase coupling the injection device with the forcing signal for the continuous phase. In the presence of a subharmonic perturbation, enhanced particle dispersion is observed as a result of the motion of the vortices, whereby a larger part of the flow field is swept out.

The initial evolution of a wave packet in a laminar boundary layer
View Description Hide DescriptionThe initial evolution of a wave packet in a laminar boundary layer is examined. By comparing the experimental streamwise growth of individual modes with the prediction of a linear model, two kinds of nonlinearities are explored. The first of which is associated with oblique waves with a frequency range centered around half the frequency of the most amplified linear mode. This type of nonlinear growth of subharmonic waves has been observed in the past in cases in which artificial excitations of a single two‐dimensional (2‐D) fundamental wave, with or without simultaneous excitations of three‐dimensional (3‐D) subharmonic waves were used, and in the case in which the disturbance was generated by a continuous harmonic point source disturbance. However, unlike all the cases mentioned above, the nonlinear growth of subharmonic waves in the present case begins well upstream of the neutral point of the fundamental wave at branch II, long before the streamwise position, where the amplitudes of the fundamental Tollmien–Schlichting (TS) waves (in the previous cases), attained a saturation level. The second type of nonlinear growth is associated with waves having shorter wavelengths than the wavelengths of the most amplified waves. In this case the nonlinear growth of the waves is first observed at their neutral points (at branch II), characterized by an enhanced streamwise growth rate of their amplitudes. The linear model, which includes the solution of the vertical vorticity equation in addition to the Orr–Sommerfeld equation and accounts for the effect of the mean flow divergence, is found to be very successful in predicting the downstream growth of the most amplified modes and the normalized cross‐stream distributions of the various modes of all three velocity components.

Transient rotational flow of an Oldroyd‐B fluid over a disk
View Description Hide DescriptionThe transient flow of an Oldroyd‐B fluid over an infinite disk set in rotation impulsively is studied under the similarity assumption. The unsteady velocity and stress field is calculated exactly for short times by a power series expansion in time. The order of magnitude of the velocity and stress components is found to depend on the relative magnitude of the Deborah number (De) and the ratio of solvent to polymericviscosities (μ_{ r }). When either one becomes very small, a solution using singular perturbations and Laplace transforms is developed. It is found that the diffusive mechanism for momentum transfer, which exists for about μ_{ r }≳0.1 (depending on De) dramatically changes and turns into a propagating wave for μ_{ r }<0.1. Numerical calculations are used to determine the extent of validity of the present results.

Nonlinear interaction of small‐scale Rossby waves with an intense large‐scale zonal flow
View Description Hide DescriptionThe system of Rossby waves is unstable with respect to modulations. This instability results in the generation of a large‐scale zonal flow with zero mean vorticity. It is shown that in both stable and unstable situations, a finite‐time singularity formation takes place in the system. Such a singularity has the form of peaks on the vorticity profile of the zonal flow. The situation is also considered when some zonal flow with nonzero mean vorticity is initially present. Solitary‐wave solutions appropriate for the description of nonlinear behavior of such systems are found. In the case of weak mean vorticity of the zonal flow, the solitons break and the singularities develop. If the mean vorticity is strong, then the evolution of the system can be considered as the dynamics of a soliton gas. Soliton dynamics possesses some interesting properties, such as formation of soliton pairs and annihilation of solitons during collision.

Spin‐up in a rectangular tank with low angular velocity
View Description Hide DescriptionA comparison is made between numerical and experimental results for spin‐up from rest in a rectangular container. The numerical results were obtained by using a three‐dimensional finite volume method on a supercomputer. The experiments were performed with water, using tracer particles floating at the free surface in order to visualize the flow field. The numerical and experimental results are in good agreement. They show the formation of a stable three‐cell pattern. In contrast to similar experiments performed at higher angular velocities, the center cell of this pattern appears to be anticyclonic. Initially, the relation between vorticity ω and streamfunction ψ of this organized flow is linear, but it is seen to evolve slowly into a relation with ∂^{2}ω/∂ψ^{2}<0.

On the modulational stability of traveling and standing water waves
View Description Hide DescriptionAsymptotically exact evolution equations are derived for trains of small amplitude counterpropagating water waves over finite depth. Surface tension is included. The resulting equations are nonlocal and generalize the equations derived by Davey and Stewartson for unidirectional wave trains. The stability properties of stationary standing and quasiperiodic waves are determined as a function of surface tension and fluid depth for both long wavelength longitudinal and transverse perturbations.

Experiments on single oblique laminar‐instability waves in a boundary layer: Introduction, growth, and transition
View Description Hide DescriptionThe laminar–turbulent transition in an incompressible flat‐plate boundary layer was studied experimentally by using a spanwise array of computer‐controlled surface heating elements to generate small disturbances. Oblique Tollmien–Schlichting waves were successfully introduced, and their downstream development into the intermittent region was studied using flush‐mounted hot‐film wall‐shear sensors and dye flow visualization. Comparative studies of the development of single oblique waves were made for various wave angles, frequencies, and amplitudes. As these single oblique waves grew and began to break down, higher harmonics and subharmonics appeared in the wall shear. The amplitude of the subharmonic component decreased rapidly with increasing oblique‐wave angle, so that a 10° oblique wave had a subharmonic amplitude an order of magnitude below that for a two‐dimensional (2‐D) wave. Thus, the nonlinear mechanism that produces the subharmonic is affected by the symmetry of the primary wave. Intermittency measurements, carried out farther downstream, show that a 2‐D wave is most effective in moving the transition point upstream, for a given power input.

Long nonlinear waves in an unbounded rotating jet or rotating two‐fluid flow
View Description Hide DescriptionThe objective of this paper is to study weakly nonlinear waves in an infinitely long rotating jet and a rotating two‐fluid flow bounded by an infinitely long rigid cylinder with surface tension at the interface. The critical values for Rossby number, a nondimensional wave speed, are found. When the Rossby number is near one of the critical values, nonlinear theory is developed under long‐wave approximation and the well‐known Korteweg–de Vries (KdV) equations for the free surface and free interface are obtained. Then the solitary wave solutions are given as the first‐order approximations of the solutions of the equations governing the motion of the flows. The analogy between the rotating fluid flows and a two‐dimensional flow with density stratification is discussed.

Generation of Tollmien–Schlichting waves by harmonic excitation
View Description Hide DescriptionThe creation of wavy disturbances (Tollmien–Schlichting waves) by a localized two‐dimensional disturbance source vibrating periodically at a fixed frequency in a Blasius boundary layer is discussed. The initial‐boundary value problem mathematically models the vibrating ribbon experiment of Schubauer and Skramstad. By considering the ribbon vibration to start at t=0, in the model one gets the transient and the eventual periodic part of the solution. The results of the full simulation are compared with the time‐asymptotic solution.

Long waves at the interface between two viscous fluids
View Description Hide DescriptionUsing a perturbation method up to the second order, the equation for long waves at the interface between two viscous fluids is derived for plane Couette–Poiseuille flow, and for moderate surface tension. The leading order equation is the Burgers equation. Higher order terms take into account linear dispersive effects and stabilizing effect of surface tension, and involve two more nonlinear terms. The exactness of the coefficients of this equation has been checked by using symmetry properties. For zero gravity, Poiseuille flow is shown to be stable if and only if the velocity profile is convex. Stable Couette flow can become unstable when the Reynolds number of one fluid is decreased, while keeping the other dimensionless parameters unchanged. Introducing characteristic length scales, the interfaceequation is put into ‘‘canonical’’ form.

On long‐lived vortices in 2‐D viscous flows, most probable states of inviscid 2‐D flows and a soliton equation
View Description Hide DescriptionThis paper considers, in contraposition to the most probable states of quasi‐inviscid theories, the status of the vorticity(ω)–streamfunction(ψ) relations satisfied by the long‐lived vortices observed in some numerical simulations of decaying two‐dimensional turbulence and in experiments in stratified fluids. For the case ω=−Δψ=sinh ψ, the circular solutions can be expressed in terms of the IIIrd Painlevé transcendent and dipolar solutions can be constructed by means of a Bäcklund transform.

Vortex structures and microfronts
View Description Hide DescriptionThis study addresses the relationship between thermal microfronts and coherent vortex structures in homogeneous turbulence. The turbulence is created by mean shear in a weakly stratified flow. The data set is generated by direct numerical simulation providing highly resolved instantaneous three‐dimensional fields of fluctuating velocity and temperature (160^{3} data points for each field). Vertically inclined large‐scale horseshoe vortices develop due to stretching and rotation by the mean shear rate, as would also occur in neutrally stratified flow. In a homogeneous shear flow, the structures on the tilted plane are oriented both upward and downward with equal probability, and are referred to as ‘‘head‐up’’ and ‘‘head‐down’’ horseshoe eddies.Vorticity structures are sampled in those regions of the flow where the strongest coherent local temperature gradients occur. The sampled fields are composited. It is found that the microfronts are caused by the local outflow between the legs of the horseshoe eddies. A head‐up eddy always forms a cold microfront (moving toward warmer fluid) and a head‐down eddy forms a warm microfront. In most of the sampled cases, the two vortex structures occur in pairs, such that the head‐down vortex always lies above the head‐up vortex. Therefore, local shear layers with enhanced cross‐stream vorticity form between the outflows of the structures. The strongest temperature gradients also occur at this location. Typical length, width, and thickness of a coherent vortex structure are found to be 1.4l, 1.4l, and 0.72l, respectively, where l is the integral length scale (based on the three‐dimensional energy–density spectrum). The typical distance between two vortices forming a pair is about one integral length.

Instability of strained counter‐rotating vortices
View Description Hide DescriptionThe stability of counter‐rotating vortices subjected to a uniform plane straining flow has been examined in this study. The straining flow field flattens the vortices in a way that the resulting slender shape can be considered as a double shear layer. The linear analysis based on this assumption has indicated two regions of instability: decreasing slenderness ratio and increasing slenderness ratio with time, where the Orr–Sommerfeld equation has been found to describe the phenomenon when the slenderness ratio is time independent. Decreasing slenderness ratio with time, coupled with an extension of the line vortex, is a stabilizing mechanism, where the rate of change of the slenderness ratio plays an important role in this process. On the other hand, in the case of a contracting line vortex with decreasing slenderness ratio, instability will not reach full growth and the vortex will undergo irreversible elongation. Increasing slenderness ratio is found to be an unconditionally destabilizing mechanism. The nonlinear phenomenon, computed by the numerical solution, indicates that in the case of an extending line vortex and decreasing slenderness ratio, the vortices will collapse to form a pair of vortices with finite cores. In some other cases, the vortex pair will undergo irreversible elongation, where increased slenderness ratio is found numerically to be a destabilizing mechanism.

Turbulence transition and internal wave generation in density stratified jets
View Description Hide DescriptionThe nonlinear evolution of an unstable symmetric jet in incompressible, density stratified fluid is simulated numerically. When N ^{2} is constant and near zero, like‐signed vortices pair by way of an instability of the mean flow to a subharmonic disturbance with wavelength twice that of the most unstable mode of linear theory. For small but finite and constant values of N ^{2}, however, the individual vortex cores are strained and vorticity is generated at small scales by the action of baroclinic torques. In this case, the mean flow of the fully evolved jet is stable to subharmonic disturbances. The linear stability of the two‐dimensional nonlinear basic states to three‐dimensional perturbations is examined in detail. From this stability analysis, it is inferred that jet flow with stratification characterized by constant N ^{2} is a poor candidate for IGW generation. However, the existence of an efficient mechanism whereby IGW may be radiated to infinity from the jet core is demonstrated via simulations initialized with a density profile such that N ^{2}=J tanh^{2}(z/R). This mechanism is expected to be an important contributor to the wave field observed in a variety of geophysical circumstances.

Comparison of spectral method and lattice Boltzmann simulations of two‐dimensional hydrodynamics
View Description Hide DescriptionNumerical solutions of the two‐dimensional Navier–Stokes equations are presented by two methods; spectral and the novel lattice Boltzmannequation (LBE) scheme. Very good agreement is found for global quantities as well as energy spectra. The LBE scheme is, indeed, providing reasonably accurate solutions of the Navier–Stokes equations with an isothermal equation of state, in the nearly incompressible limit. Relaxation to a previously reported ‘‘sinh‐Poisson’’ state is also observed for both runs.