Volume 6, Issue 7, July 1994
Index of content:

Capillary–gravity periodic and solitary waves
View Description Hide DescriptionIt is now well‐known that capillary–gravity waves of solitary type exist in deep water. These waves bifurcate from a train of trivial periodic waves when the wave number corresponds to the minimum of the dispersion curve. Such solitary waves are characterized by decaying oscillations in their tail. In this Letter, it is shown how these recently discovered waves can be obtained as the limit of periodic waves as the wavelength is continuously increased. Moreover, the existence of new branches of solitary waves is conjectured.

Near‐singular collapse and local intensification of a ‘‘Lissajous‐elliptic’’ vortex ring: Nonmonotonic behavior and zero‐approaching local energy densities
View Description Hide DescriptionThis Letter observes collapse and intensification of the two parameter (b,c) ‘‘Lissajous‐elliptic’’ vortex ring. Laboratory and direct numerical studies of this ring are proposed to elucidate near‐singular and intermittent fluid phenomena at very high Reynolds number. Quantifications of single filament Biot–Savart numerical simulations with various core ‘‘radii’’ show that collapse may be nonmonotonic in time. In the collapsing region, the largest positive strain‐rate eigenvalue, α, is off the filament and exhibits self‐similar, singular‐like behavior. A signature of collapse is found in the local approach to zero of the filament energy density in the collapsing regions.

Spectral transfer of self‐similar passive scalar fields in isotropic turbulence
View Description Hide DescriptionThe spectral transfer of passive scalars in a self‐similar state is studied by direct numerical simulations of stationary isotropic turbulence. At high wave numbers the dominant process is forward cascading local transfer as a result of nonlocal triadic interactions that couple together two similarly high wave‐number scalar modes by one low wave‐number velocity mode. This indicates that advection by the energy‐containing eddies transfers the scalar fluctuations towards smaller scales.

The Stokes drag due to the sliding of a smooth plate over a finned plate
View Description Hide DescriptionThe relative sliding motion of a smooth plate over a periodically finned plate is studied under the Stokes limit. The resulting harmonic or biharmonic equations with mixed boundary conditions are solved by eigenfunction expansions and collocation. The drag is determined as a function of fin height and fin spacing.

Computer simulations of rapid granular flows of spheres interacting with a flat, frictional boundary
View Description Hide DescriptionThis paper employs computer simulations to test the theory of Jenkins [J. Applied Mech. 59, 120 (1992)] for the interaction between a rapid granular flow of spheres and a flat, frictional wall. This paper examines the boundary conditions that relate the shear stress and energy flux at the wall to the normal stress, slip velocity, and fluctuation energy, and to the parameters that characterize a collision. It is found that while the theory captures the trends of the boundary conditions at low friction, it does not anticipate their behavior at large friction. A critical evaluation of Jenkins’ assumptions suggests where his theory may be improved.

Convection effects on radial segregation and crystal–melt interface in vertical Bridgman growth
View Description Hide DescriptionThe influence of convection caused by horizontal heat transfer through the sides of a vertical Bridgman apparatus is studied analytically. The case when the heat transfer across the side walls is small is considered, so that the resulting interfacial deformation and fluid velocities are also small. This allows one to linearize the Navier–Stokes equations and express the interfacial conditions about a planar interface through a Taylor expansion. Using a no‐tangential stress condition on the side walls, asymptotic expressions for both the interfacial slope and radial segregation at the crystal–melt interface are obtained in closed form in the limit of large thermal Rayleigh number. It is suggested that these can be reduced by appropriately controlling a specific heat transferproperty at the edge of the insulation zone on the solid side.

Gasdynamic agglomeration of aerosols. I. Acoustic waves
View Description Hide DescriptionIn this paper acoustic agglomeration of aerosols is studied in its simplest form, using a time‐averaged form of the coalescence equations. An acoustic coagulation kernel is introduced that neglects all nonlinear effects, as well as particle interactions. Based on this model, it is shown that an acoustic frequency exists that optimizes the coalescence process. This optimum frequency does not have a unique value for all distributions, but for size distributions that are not too wide, it is given by ω_{opt}=1/τ*, where τ* is the relaxation time of a particle having a diameter corresponding to the mode count diameter of the initial size distribution function. The paper also includes the result of a numerical integration of the coalescence equations for typical aerosol distributions, which show their evolution under the influence of a sound wave.

Large‐scale flow and pattern drift in finite amplitude convection
View Description Hide DescriptionThe connection between the large‐scale flows observed in thermal convection and the drift of the roll structures in large aspect ratios is examined. The analysis is based on a perturbative expansion of slightly deformed, finite‐amplitude straight rolls. The derivation assumes that the aspect ratio of the container is large, so that the flow is translation invariant. A scale separation technique is used to smooth out the small‐scale dynamics. If the unperturbed rolls cannot support a mean velocity field, the dynamics of the perturbations is diffusive and the mean flow is driven by roll bending, stretching, or pinching. If the unperturbed state supports a mean velocity field, the perturbations in the roll structure can be advected or even can propagate as a dispersive wave. In the present formulation, pattern defects arise from resonance and multivaluedness, with the implication that they can be advected or even travel.

On the structure of an electrostatic spray of monodisperse droplets
View Description Hide DescriptionAn experimental study has been performed on the structure of an electrostatic spray of monodisperse droplets. Such a spray is established when a liquid with sufficient electric conductivity and moderate surface tension, in the present case heptane doped with an antistatic additive, is fed through a small metal tube maintained at several kilovolts relative to a ground electrode a few centimeters away. The liquidmeniscus at the outlet of the capillary takes a conical shape under the action of the electric field, with a thin jet emerging from the cone tip. This jet breaks up into charged droplets that disperse into a fine spray. Flash shadowgraph of the breakup region showed that the jet initially breaks into droplets of bimodal size distribution by varicose wave instabilities. The spray monodispersity is established farther downstream by a segregation process of electrostatic and inertial nature that confines the bulk of the mass flow rate (97%) and 85% of the total current in a core of nearly monodisperse primary droplets, with the remainder in a shroud of satellites. Droplet size, axial velocity, and concentration were measured throughout the spray by phase Doppler anemometry (PDA). The complementary use of these measurements permitted the determination of the electric field via the spray momentum equation.
It was found that droplets are ejected from the jet at a relatively high velocity in a region characterized by a very intense electric field. They maintain this velocity farther downstream because of inertia, even though the field is precipitously decreasing, and ultimately decelerate under the action of the drag force and a progressively weaker electrostatic force. Velocity and concentration fields were shown to be self‐similar. Comparison between the external field, due to the potential difference applied between the electrodes, and the space charge field shows that the dropletaxial motion is driven primarily by the external field, whereas the dropletradial motion and, consequently, the jet lateral spreading, is controlled primarily by the space charge field. The latter is typically at least one order of magnitude smaller than the external one, except at off‐axis locations near the breakup region of the spray, where the two fields can be comparable. The droplet charge distribution was also determined via the spray momentum equation and the simultaneous measurements of droplet size and velocity in a region where droplets experience negligible acceleration. The charge distribution was found to be narrow, with a ratio of standard deviation over mean of 0.15.

Numerical study on two‐dimensional spin‐up in a rectangle
View Description Hide DescriptionTwo‐dimensional spin‐up in a rectangular geometry is examined by a numerical computation of the Navier–Stokes equations. Cells are, in most cases, created by the vorticity generated on the top and bottom boundaries. Further, the movement and interaction of those vortices play a key role in establishing the cell pattern in the quasisteady state. The critical phenomenon in the cell‐pattern selection (a small perturbation can result in a completely different cell pattern) observed by previous investigators turns out to be caused by the critical movement of vortices; that is, a pair of vortices, for instance, can merge with or part away from each other by a small perturbation. The final quasisteady flow decays slowly at high Reynolds numbers having a linear relationship between the streamfunction and the vorticity in the core inviscid region.

Diffusion in laminar Rayleigh–Bénard convection: Boundary layers versus boundary tubes
View Description Hide DescriptionNew results on the advection–diffusion of a passive tracer in a periodic system of hexagonal Rayleigh–Bénard convection cells at high Péclet number P=Lv/D _{0}≫1 are presented, where L is the characteristic length scale of the flow,v is the velocity amplitude, and D _{0} is the molecular diffusivity. It is shown that the transport properties of this three‐dimensional (3‐D) laminar flow are drastically different from those of the well‐studied two‐dimensional convection rolls. The 3‐D topology of the streamlines in the hexagonal convection leads to the formation of boundary tubes near the axes and the edges of the hexagons, in addition to the standard boundary layers found near the faces and the bases. A scaling theory is given and confirmed by test‐particle simulations that show that the transport enhancement due to the hexagonal cells is controlled by the boundary tubes and scales only logarithmically with P. On the other hand, it is found that the subdiffusive regimes of transport in hexagons are similar to those found in other flows with constrained streamlines. The described effects can be used for the experimental investigation of structures in thermal convection.

On the nonspherical collapse and rebound of a cavitation bubble
View Description Hide DescriptionThe behavior of a cavitation bubble adjacent to a rigid wall is studied numerically with the boundary integral method described in Zhang, Duncan, and Chahine [J. Fluid Mech. 257, 147 (1993)]. In the previous work, the pressure inside the bubble was held constant (this is referred to herein as the empty bubble case). In the present calculations, an internal gas pressure, which is a function of the bubble volume, is included in the model. The present results are qualitatively similar to those in the empty bubble case in several ways: a wall‐directed reentrant jet is formed in the later phase of the collapse; this jet impacts with the side of the bubble closest to the wall creating a toroidal‐shaped bubble; and a shear layer develops along the impact interface. However, unlike the empty bubble, whose volume decreases monotonically to zero at the end of the collapse, the present gas‐filled bubble reaches a minimum volume and then, due to its high internal gas pressure, begins to grow again (rebound). In the empty bubble case, the hydrodynamicpressure on the wall rises rapidly at the end of the calculation making it impossible to compute the maximum value of the pressure. In the present calculations, the pressure on the wall is found to reach a maximum value when the bubble starts to rebound. This timing of the pressure peak is in agreement with the experimental data of Tomita and Shima [J. Fluid Mech. 169, 535 (1986)] and Kimoto [International Symposium on Cavitation Research Facilities and Techniques (American Society of Mechanical Engineers, New York, 1987), Vol. 57, pp. 535–564], as are the orders of magnitude of the maximum pressures. Direct comparison with the numerical results of Best [J. Fluid Mech. 251, 79 (1993)] are also presented. Large differences in bubble shapes and flow fields are found.

Effects of body curvature and nonparallelism on the stability of flow over a swept cylinder
View Description Hide DescriptionThe linear stability of three‐dimensional incompressible flow over an infinite swept cylinder is considered. In particular, the effects of convex body curvature and nonparallelism on both stationary and traveling disturbances are studied. Both an exact approach and a perturbation approach are used to account for the body‐curvature effect and the results from both approaches agree very well. The nonparallel effects are accounted for by using a perturbation method. The influence of convex body curvature and nonparallelism with variation in the sweep angle, the unit Reynolds number, the cylinder radius, the location on the cylinder, and the disturbance spanwise wave number and frequency is studied. Both stationary and traveling disturbances are destabilized by the nonparallel effects and are stabilized by the convex body‐curvature effects. This is in agreement with the results obtained by solving linear parabolized stability equations. The effect of convex body curvature is gradually less stabilizing as the disturbance frequency increases. The variation of the combined effects of convex body curvature and nonparallelism in the parameter space is controlled mainly by the variation (in the parameter space) of the convex body‐curvature effects.

Investigation of the use of Prandtl/Navier–Stokes equation procedures for two‐dimensional incompressible flows
View Description Hide DescriptionThe technique of combining solutions of the Prandtl equations with solutions of the Navier–Stokes equations to compute incompressible flow around two‐dimensional bodies is investigated herein. Computational evidence is presented which shows that if the ‘‘obvious’’ coupling is used to combine the solutions, then the resulting solution is not accurate. An alternate coupling procedure is described which greatly improves the accuracy of the solutions obtained with the combined equation approach. An alternate coupling that can be used to create a more accurate vortex sheet/vortex blob method is then shown.

Feedback control of von Kármán vortex shedding behind a circular cylinder at low Reynolds numbers
View Description Hide DescriptionA computational study of the feedback control of von Kármán vortex shedding behind a circular cylinder at low Reynolds numbers is reported. The two‐dimensional Navier–Stokes equations with feedback are solved numerically. The control actuators are a pair of blowing/suction slots located at ±110° from the leading stagnation point. A single feedback sensor is used and the actuators are 180° out of phase with each other. Complete suppression of vortex shedding is achieved for the simulation at Reynolds number Re=60. The suppression window in the feedback sensor location x _{ s } is narrow. With the feedback sensor location fixed at the optimum location, vortex shedding becomes suppressed with increasing feedback gain α. However, further increase of the feedback gain destabilizes the flow again. At Reynolds number Re=80, and above, the feedback control stabilizes the primary vortex shedding mode, but a secondary mode which may be lower or higher in frequency than the primary depending upon the phase of the feedback (feedback sensor location x _{ s }) and the feedback gain α arises.

On the spanwise correlation of vortex shedding from a circular cylinder at high subcritical Reynolds number
View Description Hide DescriptionThe three‐dimensionality of the vortex shedding from a circular cylinder at high subcritical Reynolds number (4.3×10^{4}) has been studied, focusing on the characteristics of the vortex shedding phase drift and correlation along the cylinder span. Short time integration of the correlation coefficient, based on eight shedding cycles, showed that there were strong oscillations in the degree of correlation of the vortex shedding, and that these oscillations showed strong regularity having periods around 10–20 times the Strouhal period. These oscillations started when exceeding separations of Δz/d≊1 between the measurement points, and remained up to the largest case studied (Δz/d=6). The phase drift angle between two points at different spanwise positions was analyzed, showing that at the transition separation (Δz/d≊1) the probability distribution changed from being narrow banded to become more broad banded and closely Gaussian for an intermediate spacing of 2<Δz/d<4, and almost uniform for Δz/d=6. For all different separations studied (0.25<Δz/d<6), the probability distribution of the phase drift angle had a zero mean, indicating that the vortex shedding was, on the average, parallel to the cylinder.

Inertial organization of a two‐dimensional turbulent vortex street
View Description Hide DescriptionThe formation of organized structures resulting from the evolution of two parallel opposite vortex sheets in a two‐dimensional perfect fluid is studied, employing an equilibrium statistical theory. The system is confined in a channel in which periodic solutions of the statistical equilibrium equations are considered. It is found that the statistical equilibrium state of maximum entropy is a staggered arrangement of alternating vortex patches—the turbulent counterpart to the von‐Kármán vortex street. However, these vortices spread over the whole width of the channel and the pattern with the maximum allowed wavelength is always preferred: the confinement by the boundary conditions is essential. The theoretical predictions are supported by direct numerical simulations of the two‐dimensional Navier–Stokes equation at high Reynolds number.

Characteristics of asymmetric turbulent near wakes
View Description Hide DescriptionThe mean‐velocity and turbulence quantities in a few qualitatively different types of asymmetric two‐dimensional turbulent near wakes in nearly zero‐pressure gradient have been experimentally studied. In all cases, the log‐law similarity of the boundary layers was found to continue into the initial part of the wake, with the same similarity variables as in the boundary layers on the same side, provided the origin of the normal distance is taken at the smooth extension of the respective surfaces. The Reynolds stress profiles show sharp peaks just downstream of the trailing edge, and the magnitudes of the peaks are found to be related to the values of the surface friction at the trailing edge. These peaks grow into the full width of the wake to form the far wake profiles. In this process, the turbulent diffusions of the Reynolds stresses are very important, and when the size asymmetry is severe, is not clearly of the gradient‐transport type.

Supercritical Reynolds number experiments on a freely rotatable cylinder/splitter plate body
View Description Hide DescriptionThe behavior of a circular cylinder with attached splitter plate, under conditions where the entire cylinder/splitter plate body was free to rotate about the cylinder axis, has been studied experimentally as Reynolds number was increased through the critical value (laminar boundary layer separation replaced by turbulent boundary layer separation farther downstream). As in previous experiments (at lower Reynolds numbers), it was found that the plate did not align itself with the free stream, but instead rotated to some off‐axis equilibrium angle, θ. For plates with splitter plate length, L, less than one cylinder diameter, D, θ dropped very quickly to some minimum value in the transitional Reynolds number range, then slowly increased toward some new supercritical value, which in all cases was smaller than that in the subcritical range. For cases with L/D≳1, however, in the transitional Reynolds number range, the splitter plate overshot the centerline and commenced oscillating between extremes on either side. At supercritical Reynolds numbers, for cases with L/D≤2, a new equilibrium angle eventually reestablished itself on one side or the other; but for cases with L/D≥2.5, the cylinder/splitter plate body continued to oscillate between extremes on either side, even at the highest Reynolds numbers tested.

The spacing of streaks in unsteady turbulent wall‐bounded flow
View Description Hide DescriptionThe spacing of streaks of low‐speed fluid has been studied experimentally in a wall‐bounded turbulent flow in which sinusoidal unsteadiness was superposed on an otherwise steady mainstream over a range of frequencies. The modulation of phase‐conditioned streak spacing about its mean value does not follow the steady wall‐flow relation of λu _{τ}/ν equal to a constant. Instead, it is quite accurately described by a local length scale that models the momentary value of the total shear distortion of large eddies of the flow. This single shear‐distortion length scale also correlates well with the streak spacing measured in steady wall‐bounded flows and in unbounded homogeneous turbulent flow at high shear rates. The apparent generality of these results implies that the streak‐spacing selection mechanism depends strongly on the strain history of large‐scale coherent motions and so should be investigated in the context of the coupled straining processes of turbulence production.