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Volume 5, Issue 12, December 1993

Scale invariance and ε expansion in the RNG theory of stirred fluids
View Description Hide DescriptionThe influence of a ‘‘nonscale invariant’’ forcing is investigated in the renormalization group theory of stirred fluids. It is shown that such forcings introduce a free parameter in the theory. However, the difficulties inherent in the use of the ε expansion remain when the results are applied to real turbulence.

Chaotic motion of a solid through ideal fluid
View Description Hide DescriptionNumerical evidence is presented that the motion of a solid body through incompressible, inviscid fluid, moving irrotationally and otherwise at rest, is chaotic.

Two‐ and three‐dimensional models for flow through a junction
View Description Hide DescriptionTwo‐ and three‐dimensional models of the fluid motion through a junction are described for Stokes flow. It is found that for both models separation of the streamlines can occur on the branch of the outer channel wall with smaller flow rate. The relevance of the results for flow at higher Reynolds numbers is also described.

Wave formation in the gravity‐driven low‐Reynolds number flow of two liquid films down an inclined plane
View Description Hide DescriptionWave formation in the gravity‐driven low‐Reynolds number flow of two liquid films down an inclined plane is studied by a linear stability analysis. Wavy motion can appear due to an instability of either the fluid–fluid interface or the fluid‐air free surface. It is shown that the flow is always unstable and wavy motion can occur when the less viscous layer is in the region next to the wall for any Reynolds number and any finite interface and surface tensions. Stability can be achieved for the configuration with the more viscous component adjacent to the wall in the presence of interfacial tension when Reynolds number is small enough.

The transition in the sedimentation pattern of a particle cloud
View Description Hide DescriptionExperiments were carried out to study the sedimentation of a two‐dimensional particle cloud. When a large number of particles (glass beads) of uniform size are released from a two‐dimensional opening into a column of fresh water, the mixture initially descends as a thermal; however, after some time, the particles start settling individually, thus leaving the parent fluid behind. For a given type of particle, the critical depth z _{ c } at which this transition occurs, measured from a virtual origin, was found to change as z _{ cw } _{ s }/ν∼(Q/νw _{ s })^{α}, with α≂0.3, where w _{ s } is the terminal velocity of a single particle, ν is the kinematic viscosity, and Q is the total buoyancy of the released particles per unit length. The descending velocity and the spatial growth of the particle cloud were found to depend on its sedimentation characteristics.

Hydrodynamic modes of a uniform granular medium
View Description Hide DescriptionThis paper uses a fluid‐mechanical model of a granular medium to calculate the hydrodynamic modes of a spatially uniform basic state. These modes are the granular analogs of the heat, sound, and shear modes of the standard fluid. Attention is focused on the possibility of an unstable mode that might result in the spontaneous development of inhomogeneities in density. Two cases are considered: the cooling medium which loses energy without replenishment, and the heated medium which reaches a steady state when an energy source balances the loss of energy through particle collisions. The spatially uniform state of the cooling granular medium is unstable. Two modes, analogous to the shear and heat conduction modes of a standard fluid, are unstable at long wavelengths. The growth of these modes is algebraic, rather than exponential, in time. The shear mode does not involve the formation of density inhomogeneities, but the heat mode does. At long wavelengths the heat mode can be visualized by imagining a converging velocity increasing the density of particles in a certain region. The increased collisional dissipation of granular thermal energy reduces the pressure, and prevents it from reversing the convergent velocities, so the condensation is not checked. The stability of the heated granular medium depends on the energy source. If the energy source selectively deposits energy in hot regions of a disturbance, the diffusion and collisional damping can be overwhelmed, and the disturbance grows exponentially. The standard fluid (completely elastic particles) can be recovered as a special case of the heated granular medium. In all cases, waves analogous to the sound and heat conduction modes are present. In some cases, a second type of sound wave is present at long wavelengths with the peculiar property of being damped more quickly for more elastic particles.

Resonance of longshore currents under topographic forcing
View Description Hide DescriptionThe steady perturbation caused in a longshore flow by a bottom undulation is considered. The bedforms are assumed to be alongshore periodic, with crests in the cross‐shore direction and with a small amplitude in order for linear theory to be applicable. The inviscid shallow‐water equations are considered in order to investigate topographic resonance, that is, the condition under which the perturbation in the flow reaches a maximum. Since upstream edge waves held stationary by the mean flow are solutions to the homogeneous resonance equations, the existence of such flows gives rise to the existence of resonances of infinite amplitude (linear, inviscid theory). For a maximum local Froude number of the basic flow F of less than 1, the flow is found to behave subcritically according to classic channel flow theory. In addition, neither steady edge waves nor infinite amplitude resonances exist in this case. However, by numerical simulation, a finite maximum in the flow perturbation as a function of bedform wavelength is found. This topographic resonance is rather weak and wide banded. For a bedform height of 1% the local water depth, the perturbation on the flow may typically be 4% of the mean current. The resonant wavelength is between two and three times the distance of the peak longshore current to the shoreline, l _{ V }, when the current profile has a maximum at some distance offshore, or nearly four times the cross‐shore length scale of the sandbars, l, for a flow profile monotonically increasing to a constant current far offshore. For F≳1 resonances of infinite amplitude are found. For every F, l _{ V }, and l, there is an infinite set of resonant modes with an increasing cross‐shore complexity when the mode number increases, similarly to edge waves. The resonant wavelength increases with F and with l _{ V }. Some implications on the growth of transverse sandbar families and cuspidal coast are discussed.

A theoretical study of polydisperse liquid sprays in a shear‐layer flow
View Description Hide DescriptionA shear layer formed by two unidirectional gas streams of different velocities with a multisize (polydisperse) spray of evaporating droplets suspended in one of the gas streams is considered here. Similarity solutions are presented for the evolution in droplet size distributions across the shear layer and the effects of various initial droplet size distributions on the profiles of vapor concentrations are examined. A qualitative comparison between the present results for typical computed total mass distributions of the liquid phase and experimental data reported by Lazaro and Lasheras [Phys. Fluids A 1, 1035 (1989); Proceedings of the 22nd Symposium (International) on Combustion (The Combustion Institute, Pittsburgh, 1988), pp. 1991–1998; J. Fluid Mech. 235, 143 (1992)] shows strong similarity between the two sets of profiles. This supports the assumptions and boundary conditions employed in the present theoretical study. The general behavior of the theoretical SMD (Sauter mean diameter) distribution of the spray across the shear layer also compares well with the reported experimental results of Lazaro and Lasheras.

The unsteady structure of two‐dimensional steady laminar separation
View Description Hide DescriptionThe two‐dimensional unsteady incompressible Navier–Stokes equations, solved by a fractional time‐step method, were used to investigate separation due to the application of an adverse pressure gradient to a low‐Reynolds number boundary layer flow. The inviscid pressure distribution of Gaster [AGARD CP 4, 813 (1966)] was applied in the present computations to study the development of a laminar separation bubble. In all cases studied, periodic vortex shedding occurred from the primary separation region. The shed vortices initially lifted from the boundary layer and then returned towards the surface downstream. The shedding frequency nondimensionalized by the momentum thickness was found to be independent of Reynolds number. The value of the nondimensional Strouhal number, however, was found to differ from the results of Pauley et al. [J. Fluid Mech. 220, 397 (1990)], indicating that the shedding frequency varies with the nondimensional pressure distribution, C _{ p }. The computational results were time averaged over several shedding cycles and the results were compared with Gaster. The numerical study accurately reproduced the major characteristics of the separation found in Gaster’s study such as the separation point, the pressure plateau within the upstream portion of the separation bubble, and the reattachment point. The similarity between the experimental results and the time‐averaged two‐dimensional computational results indicates that the low‐frequency velocity fluctuations detected by Gaster are primarily due to the motion of large vortex structures. This suggests that large‐scale two‐dimensional structures control bubble reattachment and small‐scale turbulence contributes a secondary role.

On the spatial stability of tube flows subject to body forces
View Description Hide DescriptionThe spatial stability of multiple solutions for fully developed laminar flow through a curved tube and through a straight tube in the presence of buoyant effects was studied using a three‐dimensional numerical representation of the fully elliptic Navier–Stokes and energy equations. Several recent numerical studies have reported the existence of multiple solutions for fully developed laminar flow in curved tubes; see, for example, Yang and Keller [Appl. Numer. Math. 2, 257 (1986)] or Daskopoulos and Lenhoff [J. Fluid Mech. 203, 125 (1989)]. A similar solution multiplicity for the case of fully developed laminar flow in a heated straight tube has been discussed by Nandakumar et al. [J. Fluid Mech. 152, 145 (1985)]. Each of these studies has reported solutions of a two‐ and four‐cell type, which are symmetric about the midplane of the tube and invariant in the axial (streamwise) flow direction. In addition, the flow visualization study of Cheng and Yuen [J. Heat Trans. 109, 55 (1987)] indicates that two‐ and four‐cell solutions can be obtained experimentally for the flow developing in a curved tube with a 180° bend. The present study investigates the stability of two‐ and four‐cell solutions with respect to perturbations that are asymmetric about the midplane of the tube. Perturbed two‐ and four‐cell solutions are imposed as entrance boundary conditions for three‐dimensional simulations of the full tube cross section and the evolution of the flow is observed as it progresses downstream from the entrance plane. For the conditions explored, present results indicate that the two‐cell solutions are stable, and asymmetric perturbations are observed to decay as the flow travels downstream. Four‐cell solutions are unstable when asymmetric perturbations are introduced, and growth of the perturbations eventually results in a complex three‐dimensional transition from four‐cell to two‐cell flow. Details of the transition flow are described and transition mechanisms are identified.

Nonaxisymmetric instability in slowly swirling jet flows
View Description Hide DescriptionVery small amounts of swirl are shown to strongly destabilize a round jet flow to large n inertial modes, where (k,n) are, respectively, the axial and azimuthal wave numbers of the disturbance. This mechanism is operative if the scale of the small swirl, ε, is O(n ^{−1}) for k=O(1). Further, viscosity is shown to exert a stabilizing influence on these modes if the Reynolds number is no larger than O(n ^{3}), hence, neutral curves have been obtained for a particular example—the slowly‐swirling Squire jet. For this case, there is a very broad range of unstable wave numbers—all k between εn/2 and n. It is further determined that in the context of an initial‐value problem for such a weak vortex on which is imposed a concentrated initial disturbance, the entire vortex is temporally unstable downstream of the initial disturbance.

On the stability of nonisothermal circular Couette flow
View Description Hide DescriptionThe stability of a circular Couette flow with internal heat generation is studied in the region between two coaxial cylinders. The inner cylinder is rotating with constant velocity while the outer one is kept at rest. The investigation is carried out for wide ranges of the Prandtl number and radius ratio R, for the axisymmetric mode (n=0) and the first two asymmetric modes (n=1,2), respectively. For small to large gaps (R=0.95, 0.7, 0.4), axisymmetric perturbations are the single main cause of instability. The growths of the Prandtl and Taylor numbers lead to a decrease of the critical Grashof number. Stabilization of the Couette flow is possible for very large gaps (R=0.1), but the region of stabilization decreases for large Prandtl numbers. The stability boundary is determined by the concurrence of axisymmetric (n=0) and spiral (n=1) modes in the case of very large gaps.

Study of the parametric oscillator driven by narrow‐band noise to model the response of a fluid surface to time‐dependent accelerations
View Description Hide DescriptionA stochastic formulation is introduced to study the large amplitude and high‐frequency components of residual accelerations found in a typical microgravity environment (or g‐jitter). The linear response of a fluid surface to such residual accelerations is discussed in detail. The analysis of the stability of a free fluid surface can be reduced in the underdamped limit to studying the equation of the parametric harmonic oscillator for each of the Fourier components of the surface displacement. A narrow‐band noise is introduced to describe a realistic spectrum of accelerations, that interpolates between white noise and monochromatic noise. Analytic results for the stability of the second moments of the stochastic parametric oscillator are presented in the limits of low‐frequency oscillations, and near the region of subharmonic parametric resonance. Based upon simple physical considerations, an explicit form of the stability boundary valid for arbitrary frequencies is proposed, which interpolates smoothly between the low frequency and the near resonance limits with no adjustable parameter, and extrapolates to higher frequencies. A second‐order numerical algorithm has also been implemented to simulate the parametric stochastic oscillator driven with narrow‐band noise. The simulations are in excellent agreement with our theoretical predictions for a very wide range of noise parameters. The validity of previous approximate theories for the particular case of Ornstein–Uhlenbeck noise is also checked numerically. Finally, the results obtained are applied to typical microgravity conditions to determine the characteristic wavelength for instability of a fluid surface as a function of the intensity of residual acceleration and its spectral width.

Wall reflections of solitary waves in Marangoni–Bénard convection
View Description Hide DescriptionIn Marangoni–Bénard convection with mass transfer from the air to the liquid, an experimental exploration of reflections of solitary waves at walls has been carried out. Mach stems are observed for small enough incident angles. Upon increasing the incident angle the stem disappears thus leading to regular reflections with, however, the angle of reflection being generally larger than the incident angle.

Corewise cross‐flow transport in hairpin vortices—The ‘‘tornado effect’’
View Description Hide DescriptionThere is increasing awareness that even in fully turbulent boundary layers, large‐scale structures in the form of hairpin vortices abound. Although their implications are not all that clear at the present time, they seem to play an important role in turbulent flows. Due to the inherent unpredictability of hairpin vortices in their natural state, in the past effort has been made to generate synthetic hairpin vortices in a laminar boundary layer; from their study considerable insight into the processes underlying various features of turbulent flows has been gained. Contrary to those preceding studies where attention has been directed to the flows external to hairpin vortices, interest here is focused solely upon their interiors: the possible existence of cross‐flow transport inside the cores of the hairpin legs. In synthetic hairpin vortices, the presence of such a corewise transport away from a wall surface, or the ‘‘tornado effect,’’ is substantiated in a water tunnel with flow visualization techniques. The effect is also verified using heated fluid injected near the wall surface by measuring the temperature at various points along the hairpin vortex core.

The effect of slowly varying gap width on Dean vortices
View Description Hide DescriptionTaylor–Dean vortices are considered in channels whose thickness and curvature vary slowly on a scale 1/δ. The effect of divergence of the channel is shown to be destabilizing and of convergence to be stabilizing. Results for neutral curves are presented for a class of particular channels, and a method of computing such curves for more general channels is given.

Self‐similarity and multifractals in turbulence
View Description Hide DescriptionA simple argument based on self‐similarity is used to derive a relationship between pointwise energy‐dissipation‐rate moments, 〈ε^{ q }〉, and inertial‐range volume‐averaged moments, 〈ε^{ q } _{ r }〉, in homogeneous, isotropic and stationary turbulence. These results support the multifractal description of energy dissipation. The moment relationship implies that pointwise and inertial‐range volume‐averaged energy‐dissipation rates cannot both be lognormally distributed. Some pointwise moments may not even exist if the volume‐average counterpart is lognormal. The Schwartz inequalities for moments satisfying the self‐similar relationship are examined and support the realizability of such processes.

A dynamic mixed subgrid‐scale model and its application to turbulent recirculating flows
View Description Hide DescriptionThe dynamic subgrid‐scale eddy viscosity model of Germano et al. [Phys. Fluids A 3, 1760 (1991)] (DSM) is modified by employing the mixed model of Bardina et al. [Ph.D dissertation, Stanford University (1983)] as the base model. The new dynamic mixed model explicitly calculates the modified Leonard term and only models the cross term and the SGS Reynolds stress. It retains the favorable features of DSM and, at the same time, does not require that the principal axes of the stress tensor be aligned with those of the strain rate tensor. The model coefficient is computed using local flow variables. The new model is incorporated in a finite‐volume solution method and large‐eddy simulations of flows in a lid‐driven cavity at Reynolds numbers of 3200, 7500, and 10 000 show excellent agreement with the experimental data. Better agreement is achieved by using the new model compared to the DSM. The magnitude of the dynamically computed model coefficient of the new model is significantly smaller than that from DSM.

The influence of linear mechanisms during the adjustment of sheared turbulence to flow curvature
View Description Hide DescriptionRecent experimental studies on homogeneous curved shear flow have shown that the imposition of strong mean flow curvature can cause a reversal of the turbulent shear stress, giving it the same sign as the gradient of mean velocity. Measurements of the coherence spectrum for these flows has revealed that this reversal is not uniform across all scales, and that eddies of different sizes can have opposite orientations and transport momentum in opposite directions. To evaluate the influence of linear mechanisms in the shear stress reversal a ‘‘rapid distortion’’ type of model was applied to those flows which demonstrated this phenomenon. The model predicts that flow curvature causes a periodic modulation of the structure of sheared turbulence, and that the sign of the shear stress reverses because of these oscillations. The period of the modulation, in terms of the total strain, was found to decrease as the turning rate increases relative to the shearing rate. For those flows which showed a reversal of the shear stress, the range of experimental observation was only a fraction of the predicted period, but interpreting the observed development as a portion of an oscillation the measurements were found to be qualitatively similar to the predictions of the linear theory. In cases of stronger turbulence a self‐preserving structure developed, before the shear stress could reverse, and the measurements deviated significantly from the predictions.

Conditional scalar dissipation rates in turbulent wakes, jets, and boundary layers
View Description Hide DescriptionThe expected value E _{χ}≡E(χ,θ=θ_{0}) of the dissipation rate χ of a passive scalar θ conditioned on the scalar value θ=θ_{0} has been measured in three varieties of turbulent shear flows: heated wakes, dyed liquid jets, and the atmospheric surface layer. The quantity E _{χ} depends fairly strongly on θ_{0} and on the flow. For the wake, E _{χ} exhibits two peaks—one on the low‐temperature end and the other on the high‐temperature end—and the peaks are separated by an approximately flat region. The relative strength of the two peaks varies with the spatial position. Measured in the turbulent part alone, E _{χ} tends to have only one peak on the hot side, but is still nonuniform. The related quantity, E _{θ‘ }≡(∇^{2}θ,θ=θ_{0}), which is the expected value of the Laplacian of the scalar conditioned on the scalar concentration, has also been measured on the wake centerline and shows a simpler dependence on θ_{0} than E _{χ}. For jets, E _{χ} has a single peak on the high‐concentration side. This feature appears to be essentially independent of the use of Taylor’s hypothesis and on whether or not the dissipation rate χ is approximated by only one of its components. It is, however, sensitive to the resolution of measurement. For the temperature fluctuation in the atmospheric boundary layer, the peak in E _{χ} on the cold side is far weaker than that on the hot side. From this combination of experiments, it is argued that the different shapes of E _{χ} in different flows are related to differences in the nature of the scalar pdf itself and, for the high‐Schmidt‐number dyes in water flows, on whether or not the finest scales of the scalar are resolved.