Volume 7, Issue 4, April 1995
Index of content:

Hysteresis effect in stationary reflection of shock waves
View Description Hide DescriptionThe problem of transition between regular and Mach reflection of planar shock waves over straight wedges in steady flows was numerically studied by the DSMC method. It is shown that the transition from regular to Mach reflection takes place, in accordance with the detachment criterion, while the opposite transition occurs at smaller angles. The hysteresis effect was observed at increasing and decreasing shock wave angles.

Vorticity transport in modeling three‐dimensional unsteady shear flows
View Description Hide DescriptionThis Letter examines how intrinsic three‐dimensionality influences the flow structure of unsteady shear flows, and models its effects by considering spanwise averages of vorticityfluctuations. In particular, uniform flow past a half‐cylinder is considered: the two‐dimensional wake is very different from its three‐dimensional counterpart and the evolution equation for the spanwise spatially averaged vorticity is analyzed. It is shown that the vector flux which transports spanwise vorticity due to spatial vorticityfluctuations is of considerable magnitude (large in comparison with molecular diffusion) and is far from co‐linear with the gradient of the spanwise averaged vorticity. Modeling this flux using classical eddyviscosity ideas would therefore seem inappropriate.

Reevaluation of the experimental support for the Kolmogorov refined similarity hypothesis
View Description Hide DescriptionExperiments in high Reynolds number turbulence call into question the experimental verification of Kolmogorov’s refined similarity hypothesis. Previous experimental support for the hypothesis is shown to be distorted by the severe simplifications used in estimating the turbulence energy dissipation, which has invariably been based only on the streamwise gradient of the streamwise velocity component. However, when the dissipation is estimated by the streamwise gradient of a transverse velocity component the dependence between ‖Δu _{ r }‖ and (rε_{ r } ^{*})^{1/3} all but disappears. These results suggest that previous experimental support for the hypothesis may have been premature.

On the existence of uniform momentum zones in a turbulent boundary layer
View Description Hide DescriptionInstantaneous velocity fields in the x‐y plane of a zero pressure gradient turbulent boundary layer are measured using particle image velocimetry. It is found that there exist random, time‐varying zones in the u‐ν fields in which the streamwise momentum is remarkably uniform. The largest dimension of a typical zone is proportional to the boundary layer thickness. The zone closest to the wall contains viscous‐inertial inclined structures similar to those found in low Reynolds number wall turbulence. A second zone is located above the wall zone in a region that coincides roughly with the logarithmic layer. The wake region of the boundary layer contains a complicated, time‐varying pattern of several nearly‐constant‐momentum zones. The zones are separated from each other and from the free stream by thin viscous shear layers that contain concentrations of spanwise vorticity.

Electrophoresis of a thin charged disk
View Description Hide DescriptionThe electrophoretic velocity of a charged disk of zero thickness is computed in the limit of small surface potentials, but with arbitrary double layer thickness. The disk represents an idealized clay particle, and has uniform surface charge over its flat surface, together with a uniform line charge around its edge. The contributions of these two charges to the electrophoretic velocity are considered separately. Asymptotic results are obtained for thin and thick double layers, and intermediate results are obtained by numerical integration. The singularities in both the electrical and hydrodynamic fields at the edge of the particle enhance the importance of the edge charge when the double layer thickness κ^{−1} is small compared to the disk radius a.

Equilibrium salt‐fingering convection
View Description Hide DescriptionThe possible existence of equilibrium salt fingering in a deep fingering domain, equivalent to large Rayleigh number fingering convection, is investigated with a numerical model of a continuously heat and salt stratified fluid system. The growth of fingering heat and salt fluxes in this system is shown to be limited by the instability of fingering cells, and an equilibrium state is achieved when the instability has increased the energy dissipation sufficiently to balance the buoyancy forcing generated by double diffusion. The equilibrium fluxes are shown to vary proportional to the mean vertical T and S gradients, in contrast to the inverse relationship that holds for convection across a finite height, or low Rayleigh number, fingering zone. The structure of equilibrium convection is shown to be disorganized and turbulent‐like, characterized by incoherently rising and sinking blobs of anomalous density fluid. A model of equilibrium fingering based on the blob structure is presented to interpret the convective processes. Estimates of the finger blob scale and the heat‐to‐salt flux ratio are also made from the model and are shown to be consistent with those determined from direct numerical simulations. Finally, a critical Rayleigh number is suggested and estimated to delineate the equilibrium fingering regime from the low Rayleigh number fingering regime.

Bifurcation phenomena in confined thermosolutal convection with lateral heating: Commencement of the double‐diffusive region
View Description Hide DescriptionAs has recently been reported by Tsitverblit and Kit [Phys. Fluids A 5, 1062 (1993)], a vertical rectangular enclosure containing stably stratified brine and differentially heated from its side walls is characterized by complex steady bifurcation phenomena. In the present work, the structure of steady solutions in the enclosure has been studied in detail for several values of the salinity Rayleigh number, Ra_{ S }, fixed near the commencement of the double‐diffusive region. It was found that when the thermal Rayleigh number, Ra_{ T }, is either very small or sufficiently large, the steady solution is unique while in an intermediate region of this parameter, there exists a great variety of the multiple steady flows, being the result of nondegenerate hysteresis points and isolas of asymmetric solutions forming as Ra_{ S } is increased. In particular, at the maximal value of Ra_{ S } considered there have been observed symmetric and asymmetric one‐, two‐, three‐, four‐, and five‐cell flows. Despite the multiplicity of the flow patterns, a critical interval of the buoyancy ratio has been distinguished, above and below which the generic characteristics of the steady solutions were found to resemble the respective features of the ‘‘successive’’ and ‘‘simultaneous’’ regimes of layer formation whose existence was established in previous studies. Although the set of the steady solutions has been found to contain no linearly stable multicell flows, the perturbation was so long retained in the close proximity of the unstable steady solutions that such flows could be easily observable in the experiment. In spite of the appreciably different range of the Rayleigh numbers, the physically meaningful parameters suggested in previous studies were found to be represented in the present results.

Vortex rings generated by drops just coalescing with a pool
View Description Hide DescriptionA detailed experimental study has been made of the formation and motion of vortex rings generated when drops of liquid are allowed to come into contact at zero velocity with a quiescent flat surface of the same liquid. In this case the vortexmotion is driven principally by the dropsurface energy. It is shown that in the case considered, the phenomenon depends on two dimensionless parameters alone, the reciprocal of a Bond number B^{−1} and a global Reynolds number Re where the velocity scale is based on the surface energy. Using a video camera,measurements were made, over the whole trajectory, of the ring velocity and diameter for a number of drop sizes and liquids. Vortex rings generated by small drops reach higher peak velocities but decelerate more rapidly than ones generated by large drops. The latter, however, may become turbulent before relaminarizing during deceleration. Although the most interesting finding of the investigation was the discovery of oscillations in the ring translational velocity and diameter during deceleration, the identification of the appropriate scaling laws and a suggestion regarding the early stages of vorticity generation should also be of interest.

A phase field model of capillarity
View Description Hide DescriptionThe phenomenological derivation of a phase field model of capillarity that accounts for the structure of an interfacial layer formed by two immiscible incompressible liquids is addressed. A rheological expression for the reversible component of capillary stresses is obtained in terms of the free energy of a binary fluid, which depends on the absolute temperature, composition, and gradient of composition. This model can be applied to those flows that involve change of topology of a capillary interface, such as coalescence and breakup of drops. As an illustration, an equilibrium of a binary fluid with either a flat or spherical interfacial layer is analyzed, and a thermocapillary flow in an infinite gap is considered.

On the determination of solenoidal or compressible velocity fields from measurements of passive or reactive scalars
View Description Hide DescriptionSeveral techniques have been proposed for determining two‐ or three‐dimensional velocity fields from measurements of one passive scalar. It is shown that measurements of one scalar and knowledge of the equation governing its transport determine a velocity field, only up to an additive vector field locally perpendicular to the gradient of the scalar field but otherwise arbitrary. Three previously proposed procedures for selecting a unique velocity field from among the uncountable infinity consistent with the scalar transport data and equation are then discussed, and it is shown that a recent ‘‘iterative inversion’’ procedure for ‘‘solution’’ of a singular linear equation system (obtained using only measurements of one scalar and the equation governing its transport) cannot converge as claimed. A method for determining the correct n‐dimensional (n=2 or 3) divergence‐free velocity field from measurements of n−1 passive or reactive scalars is then developed. Finally, it is shown how the velocity field in an n‐dimensional compressible flow can be determined from measurements of density and n−1 passive or reactive scalars.

Direct numerical simulation of rotating fluid flow in a closed cylinder
View Description Hide DescriptionPresent numerical simulations of the transition scenario of a rotating fluidflow in a closed cylinder are presented, where the motion is created by a rotating lid. The numerical algorithm, which is based on a finite‐difference discretization of the axisymmetric Navier‐Stokes equations, is validated against experimental visualizations of both transient and stable periodic flows. The complexity of the flow problem is illuminated numerically by injecting flow tracers into the flow domain and following their evolution in time. The vortex dynamics appears as stretching, folding and squeezing of flow structures which wave along the contour of a central vortex core. The main purpose of the study is to clarify the mechanisms of the transition scenario and relate these to experiences known from other dynamical systems and bifurcation theory. The dynamical system was observed to exhibit up to three multiple solutions for the same Reynolds number, and to contain four discernible branches. The transition to strange attractor behavior was identified as a nontrivial Ruelle‐Takens transition through a transient torus. The various solution branches of the rotating flow problem are illustrated by phase portraits and summarized on a frequency diagram.

On the transition of the cylinder wake
View Description Hide DescriptionThe transition of the cylinder wake is investigated experimentally in a water channel and is computed numerically using a finite‐difference scheme. Four physically different instabilities are observed: a local ‘‘vortex‐adhesion mode,’’ and three near‐wake instabilities, which are associated with three different spanwise wavelengths of approximately 1, 2, and 4 diam. All four instability processes can originate in a narrow Reynolds‐number interval between 160 and 230, and may give rise to different transition scenarios. Thus, Williamson’s [Phys. Fluids 31, 3165 (1988)] experimental observation of a hard transition is for the first time numerically reproduced, and is found to be induced by the vortex‐adhesion mode. Without vortexadhesion, a soft onset of three‐dimensionality is numerically and experimentally obtained. A control‐wire technique is proposed, which suppresses transition up to a Reynolds number of 230.

Unsteady evolutions of vortex rings
View Description Hide DescriptionUnsteady evolutions of vortex rings with linearly distributed vorticity and various core parameters are considered in an unbounded, inviscid fluid. The instability of a Norbury vortex with a moderate core thickness parameter α is also investigated. Contour integral expressions based on the Biot–Savart law for the velocity field induced by a vortex ring are derived. Numerical results show that all vortex rings except the Norbury vortices will undergo an unsteady evolution process to reach an asymptotic state. The process may be roughly divided into two major stages: initial large deformation stage and later asymptotic oscillating stage. Vortexfilamentation is often observed during the first stage. In the second stage, the vortex oscillates periodically with nearly constant amplitude; its core closely resembles a Norbury vortex having the same circulation and impulse, but the dynamic properties and kinetic energies are different.

Stabilization of trapped vortices by alternating blowing suction
View Description Hide DescriptionMotion of a free point vortex near a circular cylinder in a uniform oncoming flow is considered. There is a point at which the point vortex velocity is zero. This stationary position is unstable. Introducing two periodic point sources of zero mean flow rate modifies the point vortex velocity field so that instead of the stationary position there is a periodic trajectory. The corresponding periodic motion of the vortex is found to be stable over a certain range of characteristics of periodic blowing/suction modeled with periodic sources. In particular, for stabilization the amplitude of the source flow rate oscillations must not exceed a certain limit. The results are believed to bear on the problem of enhancing lift at high angle of attack with the use of trapped vortices.

Two counter‐rotating diffusing vortices
View Description Hide DescriptionThe problem of the diffusion of two counter‐rotating vortices of equal strength is studied numerically and analytically. Asymptotic expansions are derived for the limiting behavior of the solution for small times, for small Reynolds numbers, and for large times. The results are used to more fully understand the drift and decay of the vortex system. Thus it is shown that different measures for the position of the vortex system used by previous authors may give significantly different values for the drift velocity of the vortices. The expansion for small Reynolds number shows that these differences remain even in the Stokes limit Re→0, in which the vorticity system becomes symmetric about the line connecting the vortex centers. But surprisingly, the large time expansion shows that for large times all drift velocities become identical. Moreover, this universal velocity is different from the average velocity in each half plane although it equals the velocity of the centers of vorticity of those planes. The small time expansion shows that increasing Reynolds number makes the vortices more symmetric. This tends to reduce the differences between the drift velocities. The small time expansion describes the numerical solution well as long as the vortices remain small compared to their spacing. The numerical results show that the Stokes solution describes various flow quantities fairly well for Reynolds numbers up to 600 based on the circulation; however, nonzero Reynolds number reduces the decay of the circulation of the vortices even on a diffusive time scale.

Transition experiments in a boundary layer with embedded streamwise vortices
View Description Hide DescriptionThe stability of a flat plate boundary layer modulated by stationary streamwise vortices was studied experimentally in the T‐324 low speed wind tunnel in Novosibirsk. Vortices were generated inside the boundary layer by means of roughness elements arranged in a regular array along the spanwise (z‐) direction. Transition is not caused directly by these structures, but by the growth of small amplitude traveling waves riding on top of the steady vortices. This situation is analogous to the transition process in Görtler and cross‐flows. The waves were found to amplify up to a stage where higher harmonics are generated, leading to turbulentbreakdown and disintegration of the spanwise boundary layer structure. For strong modulations, the observed instability is quite powerful, and can be excited ‘‘naturally’’ by small uncontrollable background disturbances. Controlled oscillations were then introduced by means of a vibrating ribbon, allowing a detailed investigation of the wave characteristics. The instability seems to be associated with the spanwise gradients of the mean flow, ∂U/∂z, and at all z‐positions, the maximum wave amplitude was found at a wall‐normal position where the mean velocity is equal to the phase velocity of the wave, U(y)=c, i.e., at the local critical layer. Unstable waves were observed at frequencies well above those for which Tollmien–Schlichting (TS) waves amplify in the Blasius boundary layer. Excitation at lower frequencies and milder basic flow modulations showed that TS‐type waves may also develop. The relation between TS‐type waves and the observed high‐frequency instability is discussed in the light of previous authors’ findings.

A mostly linear model of transition to turbulence
View Description Hide DescriptionA simple model in three real dimensions is proposed, illustrating a possible mechanism of transition to turbulence. The linear part of the model is stable but highly non‐normal, so that certain inputs experience a great deal of growth before they eventually decay. The nonlinear terms of the model contribute no energy growth, but recycle some of the linear outputs into inputs, closing a feedback loop and allowing initially small solutions to ‘‘bootstrap’’ to a much larger amplitude. Although different choices of parameters in the nonlinearity lead to a variety of long‐term behaviors, the bootstrapping process is essentially independent of the details of the nonlinearity and varies predictably with the Reynolds number. The bootstrapping scenario demonstrated by this model is the basis of some recent explanations for the failure of classical hydrodynamic stability analysis to predict the onset of turbulence in certain flow configurations.

Large‐eddy simulation of rotating channel flows using a localized dynamic model
View Description Hide DescriptionMost applications of the dynamic subgrid‐scale stress model use volume‐ or planar‐averaging to avoid ill‐conditioning of the model coefficient, which may result in numerical instabilities. Furthermore, a spatially‐varying coefficient is mathematically inconsistent with the original derivation of the model. A localization procedure is proposed here that removes the mathematical inconsistency to any desired order of accuracy in time. This model is applied to the simulation of rotating channel flow, and results in improved prediction of the turbulence statistics. The model coefficient vanishes in regions of quiescent flow, reproducing accurately the intermittent character of the flow on the stable side of the channel. Large‐scale longitudinal vortices can be identified, consistent with the observation from experiments and direct simulations. The effect of the unresolved scales on higher‐order statistics is also discussed.

Pressure spectra for vortex models of fine‐scale homogeneous turbulence
View Description Hide DescriptionPressure spectra at large wave numbers are calculated for Lundgren–Townsend vortexmodels of the fine scales of homogeneous turbulence. Specific results are given for the Burgers vortex and also for the Lundgren‐strained spiral vortex. For the latter case, it is found that the contribution to the shell‐summed spectrum produced by the interaction between the axisymmetric and nonaxisymmetric components of the velocity field is proportional to k ^{−7/3} (k=‖k‖ is the modulus of the wave number) in agreement with Kolmogorov‐type dimensional arguments. Numerical estimates of the dimensionless prefactors for this component are obtained in Kolmogorov scaling variables and comparisons are made with results from the Batchelor–Kolmogorov theory, and with experimental measurement.

Viscous sublayer flow visualizations at R _{θ}≂1 500 000
View Description Hide DescriptionPlan view flow visualization experiments were conducted in the atmospheric surface layer that flows over the Great Salt Lake Desert at the U.S. Army Dugway Proving Ground, Dugway, Utah. Measurements were acquired on a nonconductive, polyethylene platform made flush with the desert floor. Surface conditions upstream of the measurement site were flat, devoid of vegetation, and because of the dried mud/clay/salt composition, essentially dust free. Local surface variations ranged between 1 and 3 mm, which corresponded to three to ten viscous units during the experiments. Flow visualizations were accomplished by continuously injecting theatrical fog through a tangential slit covering a smoke reservoir buried under the platform. During the visualizations, the atmospheric surface layer flow was near neutral thermal stability. Flow velocities at 2.0 m above the surface maintained directional constancy, with a magnitude of about 1.5 m/s. A single element hot‐wire probe positioned near y ^{+}=3.4 was used to measure the wall shear. Visualization results indicate the existence of the pocket and streak motions seen at much lower Reynolds number. The average inner normalized streak spacing was found to be λ^{+}=93.1, with a positively skewed, nearly lognormal distribution. The average maximum inner normalized pocket width was found to be w ^{+}=127.2, with a positively skewed distribution. The average time between pockets was determined to be T ^{+}=36.6. Comparisons are made with existing low Reynolds number results, and a brief discussion is provided regarding the physics underlying the present observations.