Showing papers in "Journal of Fluid Mechanics in 1992"

Turbulence - The filtering approach

Abstract: The paper describes a new operatorial approach to the study of turbulence, based on the general algebraic properties of the filtered representations of a turbulence field at different levels. The main results of this analysis is the averaging invariance of the filtered Navier-Stokes eaquations in terms of the generalized central moments, and an algebraic identity that relates the turbulent stresses at different levels. The resolved turbulence is defined, the algebraic consistency rules that relate these resolved quantities to the turbulent stresses at different levels are derived, and their possible uses in subgrid modeling is discussed.

Toward the large-eddy simulation of compressible turbulent flows

Abstract: New subgrid-scale models for the large-eddy simulation of compressible turbulent flows are developed and tested based on the Favre-filtered equations of motion for an ideal gas. A compressible generalization of the linear combination of the Smagorinsky model and scale-similarity model, in terms of Favre-filtered fields, is obtained for the subgrid-scale stress tensor. An analogous thermal linear combination model is also developed for the subgrid-scale heat flux vector. The two dimensionless constants associated with these subgrid-scale models are obtained by correlating with the results of direct numerical simulations of compressible isotropic turbulence performed on a 96 (exp 3) grid using Fourier collocation methods. Extensive comparisons between the direct and modeled subgrid-scale fields are provided in order to validate the models. A large-eddy simulation of the decay of compressible isotropic turbulence (conducted on a coarse 32(exp 3) grid) is shown to yield results that are in excellent agreement with the fine-grid direct simulation. Future applications of these compressible subgrid-scale models to the large-eddy simulation of more complex supersonic flows are discussed briefly.

A new modelling of cavitating flows : a numerical study of unsteady cavitation on a hydrofoil section

Abstract: A new cavity model that can explain the interaction between viscous effects including vortices and cavitation bubbles is presented in this study. This model, which is named a bubble two-phase flow (BTF) model, treats the inside and outside of a cavity as one continuum by regarding the cavity as a compressible viscous fluid whose density changes greatly. Navier–Stokes equations including cavitation bubble clusters are solved in finite-difference form by a time-marching scheme, where the growth and collapse of a bubble cluster is given by a modified Rayleigh's equation. Computation was made on a two-dimensional flow field around a hydrofoil NACA0015 at angles of attack of 8° and 20°. The Reynolds number was 3 × 105. The experiments were also performed at the same Reynolds number for comparison. The computed results by the BTF cavity model can express the feature of cloud-type cavitation shed from the trailing edge of the attached cavities when the angle of attack is 8°. It shows the mechanism of cavitation cloud generation and large-scale vortices. The boundary layer separates at the cavity leading edge. Then it rolls up and produces the cavitation cloud. In other words, the instability of the shear layer may produce the cavitation cloud. When the angle of attack is 20°, the flow was fully separated from the leading edge of the hydrofoil and vortex cavitation occurs in the separated region. The BTF cavity model can also express the generation of such vortex cavitation and the effect of cavitation nuclei in the uniform flow.

Stochastic backscatter in large-eddy simulations of boundary layers

Abstract: The ability of a large-eddy simulation to represent the large-scale motions in the interior of a turbulent flow is well established. However, concerns remain for the behaviour close to rigid surfaces where, with the exception of low-Reynolds-number flows, the large-eddy description must be matched to some description of the flow in which all except the larger-scale ‘inactive’ motions are averaged. The performance of large-eddy simulations in this near-surface region is investigated and it is pointed out that in previous simulations the mean velocity profile in the matching region has not had a logarithmic form. A number of new simulations are conducted with the Smagorinsky (1963) subgrid model. These also show departures from the logarithmic profile and suggest that it may not be possible to eliminate the error by adjustments of the subgrid lengthscale. An obvious defect of the Smagorinsky model is its failure to represent stochastic subgrid stress variations. It is shown that inclusion of these variations leads to a marked improvement in the near-wall flow simulation. The constant of proportionality between the magnitude of the fluctuations in stress and the Smagorinsky stresses has been empirically determined to give an accurate logarithmic flow profile. This value provides an energy backscatter rate slightly larger than the dissipation rate and equal to idealized theoretical predictions (Chasnov 1991).

Linear and nonlinear stability of the Blasius boundary layer

Abstract: Two new techniques for the study of the linear and nonlinear instability in growing boundary layers are presented. The first technique employs partial differential equations of parabolic type exploiting the slow change of the mean flow, disturbance velocity profiles, wavelengths, and growth rates in the streamwise direction. The second technique solves the Navier-Stokes equation for spatially evolving disturbances using buffer zones adjacent to the inflow and outflow boundaries. Results of both techniques are in excellent agreement. The linear and nonlinear development of Tollmien-Schlichting (TS) waves in the Blasius boundary layer is investigated with both techniques and with a local procedure based on a system of ordinary differential equations. The results are compared with previous work and the effects of non-parallelism and nonlinearity are clarified. The effect of nonparallelism is confirmed to be weak and, consequently, not responsible for the discrepancies between measurements and theoretical results for parallel flow.

Drop Formation in a One-Dimensional Approximation of the Navier-Stokes Equation

Abstract: We consider the viscous motion of a thin axisymmetric column of fluid with a free surface. A one-dimensional equation of motion for the velocity and the radius is derived from the Navier–Strokes equation. We compare our results with recent experiments on the breakup of a liquid jet and on the bifurcation of a drop suspended from an orifice. The equations form singularities as the fluid neck is pinching off. The nature of the singularities is investigated in detail.

Spectral large-eddy simulation of isotropic and stably stratified turbulence

Abstract: We first recall the concepts of spectral eddy viscosity and diffusivity, derived from the two-point closures of turbulence, in the framework of large-eddy simulations in Fourier space. The case of a spectrum which does not decrease as (Ck is Kolmogorov's constant), with α ≈ 1.32.

The role of streamwise vorticity in the near-field entrainment of round jets

Abstract: The role of streamwise vortex structures in the near-field (x/d < 10) evolution of a round jet is examined. In free shear layers the streamwise vorticity develops into Bernal-Roshko structures which are streamwise vortex pairs. Similar structures are shown to exist in round jets. These structures, which evolve and amplify in the braid region between primary vortical structures, are shown to drastically alter the entrainment process in the near field and to increase the rate at which fluid is entrained into the jet. As the flow evolves downstream, the efficiency of the streamwise vorticity in entraining fluid increases relative to that of the azimuthal vorticity. Beyond the end of the potential core regime, the entrainment process is mainly controlled by streamwise vorticity. These processes are identified via flow visualization and confirmed by detailed global entrainment measurements.

Comparison between rough- and smooth-wall turbulent boundary layers

Abstract: Measurements in a zero-pressure-gradient turbulent boundary layer over a mesh-screen rough wall indicate several differences, in both inner and outer regions, in comparison to a smooth-wall boundary layer. The mean velocity distribution indicates that, apart from the expected k-type roughness function shift in the inner region, the strength of the rough-wall outer region ‘wake’ is larger than on a smooth wall. Normalizing on the wall shear stress, there is a significant increase in the normal turbulence intensity and a moderate increase in the Reynolds shear stress over the rough wall. The longitudinal turbulence intensity distribution is essentially the same for both surfaces. Normalized contributions to the Reynolds shear stress from the second (Q2) and fourth (Q4) quadrants are greater over the rough wall. The data indicate that not only are Q2 and Q4 events stronger on the rough wall but their frequency of occurrence is nearly twice as large for the rough wall as for the smooth wall. Comparison between smooth- and rough-wall spectra of the normal velocity fluctuation suggests that the strength of the active motion may depend on the nature of the surface.

Direct simulation of particle dispersion in a decaying isotropic turbulence

Abstract: Results of a numerical investigation of the dispersion of solid particles in decaying isotropic turbulence are presented. The 3D time-dependent velocity field of a homogeneous nonstationary turbulence is computed using the method of direct numerical simulation (DNS). The dispersion characteristics of three different solid particles (corn, copper, and glass) injected in the flow are obtained by integrating the complete equation of particle motion along the instantaneous trajectories of 22-cubed particles for each particle type, and then by performing ensemble averaging. Good agreement was achieved between the present DNS results and the measured time development of the mean-square displacement of the particles. Questions of how and why the dispersion statistics of a solid particle differ from those of its corresponding fluid point and surrounding fluid and what influences inertia and gravity have on these statistics are also discussed.

Numerical simulation of low-Reynolds-number turbulent flow through a straight square duct

Abstract: The mean flow and turbulent statistics obtained from the numerical simulation of the fully developed turbulent flow through a straight duct of square cross-section are reported. The Reynolds number based on the bulk velocity and hydraulic diameter is 4410. Spatial and temporal approximations of the equations of motion were derived from standard finite-difference techniques. To achieve sufficient spatial resolution 16.1 × 106 grid nodes were employed. Turbulent statistics along the wall bisectors show good agreement with plane channel data despite the influence of the sidewalls in the former flow. The mean secondary flow field consists of two counter-rotating cells symmetrically placed about the corner bisectors with their common flow towards each corner with strong evidence for the existence of a smaller and much weaker pair situated about the wall bisectors. The mean streamwise vorticity of each corner cell is found to be associated with a stronger vorticity distribution of the opposite sign having an absolute maximum on the nearest duct wall.

Kinematic simulation of homogeneous turbulence by unsteady random Fourier modes

Abstract: The velocity field of homogeneous isotropic turbulence is simulated by a large number (38–1200) of random Fourier modes varying in space and time over a large number (> 100) of realizations. They are chosen so that the flow field has certain properties, namely (i) it satisfies continuity, (ii) the two-point Eulerian spatial spectra have a known form (e.g. the Kolmogorov inertial subrange), (iii) the time dependence is modelled by dividing the turbulence into large- and small-scales eddies, and by assuming that the large eddies advect the small eddies which also decorrelate as they are advected, (iv) the amplitudes of the large- and small-scale Fourier modes are each statistically independent and each Gaussian. The structure of the velocity field is found to be similar to that computed by direct numerical simulation with the same spectrum, although this simulation underestimates the lengths of tubes of intense vorticity.Some new results and concepts have been obtained using this kinematic simulation: (a) for the inertial subrange (which cannot yet be simulated by other means) the simulation confirms the form of the Eulerian frequency spectrum , where e,U0,ω are the rate of energy dissipation per unit mass, large-scale r.m.s. velocity, and frequency. For isotropic Gaussian large-scale turbulence at very high Reynolds number, CE ≈ 0.78, which is close to the computed value of 0.82; (b) for an observer moving with the large eddies the ‘Eulerian—Lagrangian’ spectrum is ϕEL11 = CELeω−2, where CEL ≈ 0.73; (c) for an observer moving with a fluid particle the Lagrangian spectrum ϕL11 = CLeω−2, where CL ≈ 0.8, a value consistent with the atmospheric turbulence measurements by Hanna (1981) and approximately equal to CEL; (d) the mean-square relative displacement of a pair of particles 〈Δ2〉 tends to the Richardson (1926) and Obukhov (1941) form 〈Δ2〉 = GΔet3, provided that the subrange extends over four decades in energy, and a suitable origin is chosen for the time t. The constant GΔ is computed and is equal to 0.1 (which is close to Tatarski's 1960 estimate of 0.06); (e) difference statistics (i.e. displacement from the initial trajectory) of single particles are also calculated. The exact result that Y2 = GYet3 with GY = 2πCL is approximately confirmed (although it requires an even larger inertial subrange than that for 〈Δ2〉). It is found that the ratio [Rscr ]G = 2〈Y2〉/〈Δ2〉≈ 100, whereas in previous estimates [Rscr ]G≈ 1, because for much of the time pairs of particles move together around vortical regions and only separate for the proportion of the time (of O(fc)) they spend in straining regions where streamlines diverge. It is estimated that [Rscr ]G ≈ O(fc−3). Thus relative diffusion is both a ‘structural’ (or ‘topological’) process as well as an intermittent inverse cascade process determined by increasing eddy scales as the particles separate; (f) statistics of large-scale turbulence are also computed, including the Lagrangian timescale, the pressure spectra and correlations, and these agree with predictions of Batchelor (1951), Hinzc (1975) and George et al. (1984).

Aspect ratio and end plate effects on vortex shedding from a circular cylinder

Abstract: Aspect ratio effects on the vortex shedding flow from a circular cylinder have been studied by using moveable end plates. Experiments were carried out to measure fluctuating forces, shedding frequency and spanwise correlation whilst varying end plate separation and Reynolds number. The aspect ratio (0.25-12) was found to have a most striking effect on the fluctuating lift. Within a certain range of Reynolds number an increase of the sectional fluctuating lift was obtained for reduced aspect ratio, and showed a maximum for an aspect ratio of 1, where the fluctuating lift could be almost twice the value for very large aspect ratios. This increase of the lift amplitude was found to be accompanied by enhanced spanwise correlation of the flow. The measurements were carried out over the Reynolds number range 8 > 103 < Re < 1.4 x 105. The strong increase in fluctuating lift with small aspect ratio did not occur at the lower and upper boundaries of this range. In the lower Reynolds number range (Re < 2 x 104) the trend could be reversed, i.e. the fluctuating lift decreased with decreasing aspect ratio. Also, with small aspect ratio, a shedding breakdown was found in the upper Reynolds number range (Re = 1.3 x 105). The main three dimensional feature observed was a spanwise variation in the phase of vortex shedding, accompanied by amplitude modulation in the lift signal. However, the level of three-dimensionality can be reduced by using a small aspect ratio. Three-dimensional vortex shedding features are discussed and comparison of the results with those from both two-dimensional numerical simulations and other experiments using large aspect ratios are presented.

The natural and forced formation of spot-like vortex dislocations in the transition of a wake

Abstract: The three-dimensional transition of the flow behind a bluff body is studied, with an emphasis placed on the evolution of large-scale structures in the wake. It has previously been found that there are two fundamental modes of three-dimensional vortex shedding in the wake of a circular cylinder (each mode being dependent on the range of Reynolds number), with a spanwise lengthscale of the same order as the primary streamwise wavelength of the vortex street. However. it is shown in the present study that the wake transition also involves the appearance of large-scale spot-like ‘vortex dislocations’, that grow downstream to a size of the order of 10–20 primary wavelengths. Vortex dislocations are generated between spanwise vortex-shedding cells of different frequency. The presence of these dislocations explains the large intermittent velocity irregularities that were originally found by Roshko (1954) and later by Bloor (1964) to characterize transition. The presence of these vortex dislocations in wake transition is largely responsible for the break-up to turbulence of the wake as it travels downstream.In order to study their evolution in detail, dislocations have been (passively) forced to occur at a local spanwise position with the use of a small ring disturbance. It is found that ‘two-sided’ dislocations are stable in a symmetric in-phase configuration, and that they induce quasi-periodic velocity spectra and (beat) dislocation-frequency oscillations in the near wake. Intrinsic to these dislocations is a mechanism by which they spread rapidly in the spanwise direction, involving helical twisting of the vortices and axial core flows. This is felt to be a fundamental mechanism by which vortices develop large-scale distortions in natural transition. As the wake travels downstream, the energy at the low dislocation frequency decays slowly (in contrast to the rapid decay of other frequencies), leaving the downstream wake dominated by the large dislocation structures. Distinct similarities are found between the periodic forced dislocations and the intermittent dislocations that occur in natural transition. Further similarities of dislocations in different types of flow suggest that vortex or phase dislocations could conceivably be a generic feature of transition in all shear flows.

An experimental investigation on the collision behaviour of hydrocarbon droplets

Abstract: The collisional dynamics of equal-sized water and normal-alkane droplets, in the 150 μm radius range, have been experimentally studied for situations involving 0(1) droplet Weber numbers and head-on to grazing impact parameters. Results show that in the parametric range investigated the behaviour of hydrocarbon droplets is significantly more complex than that of water droplets. For head-on collisions, while permanent coalescence always results for water droplets, the outcome is quite nonmonotonic for the hydrocarbon droplets in that, with increasing droplet Weber number, the collision can result in permanent coalescence, bouncing, permanent coalescence again, and coalescence followed by separation with or without production of satellite droplets. Similar complexities exist for off-centre collisions. Phenomenological explanations are offered for these observations based on the material properties of the fluids, the relative influences of the normal and shearing aspects of the collision, and the nature and extent of energy dissipation due to droplet deformation during collision.

Experimental investigation of the field of velocity gradients in turbulent flows

Abstract: We present results of experiments on a turbulent grid flow and a few results on measurements in the outer region of a boundary layer over a smooth plate. The air flow measurements included three velocity components and their nine gradients. This was done by a twelve-wire hot-wire probe (3 arrays × 4 wires), produced for this purpose using specially made equipment (micromanipulators and some other auxiliary special equipment), calibration unit and calibration procedure. The probe had no common prongs and the calibration procedure was based on constructing a calibration function for each combination of three wires in each array (total 12) as a three-dimensional Chebishev polynomial of fourth order. A variety of checks were made in order to estimate the reliability of the results.Among the results the most prominent are the experimental confirmation of the strong tendency for alignment between vorticity and the intermediate eigenvector of the rate-of-strain tensor, the positiveness of the total enstrophy-generating term ωiωjsij (sij = ½(∂ui/∂xj+∂uj/∂xi), ωi = eijk∂uj/∂xk) even for rather short records and the tendency for alignment in the strict sense between vorticity and the vortex stretching vector Wi = ωjsij. An emphasis is put on the necessity to measure invariant quantities, i.e. independent of the choice of the system of reference (e.g. sijsij and ωiωjsij) as the most appropriate to describe physical processes. From the methodological point of view the main result is that the multi-hot-wire technique can be successfully used for measurements of all the nine velocity derivatives in turbulent flows, at least at moderate Reynolds numbers.

Structure of a particle-laden round jet

Abstract: The interaction of solid particles with the temporal features of a turbulent flow has direct relevance to problems in particle and spray combustion and the processing of particulate solids. The object of the present study was to examine the behaviour of particles in a jet dominated by vortex ring structures. An axisymmetric air jet laden with 55 μm glass particles was forced axially with an acoustic speaker to organize the vortex ring structures rolling up in the free shear layer downstream of the nozzle exit. Visualization studies of forced and unforced flow with Reynolds number of the order of 20000 were completed using a pulsed copper vapour laser. Instantaneous photographs and videotapes of strobed forced flow show that particles become clustered in the saddle regions downstream of the vortex rings and are propelled away from the jet axis by the outwardly moving flow in these regions. Phase-averaged spatial distribution of particle number density computed from digitized photographs and phase-averaged particle velocity measurements yield further evidence that local particle dispersion and concentration are governed by convection due to large-scale turbulence structures. The large-scale structures and convection mechanisms were shown to persist for particle-to-air mass loading ratios up to 0.65.

The three-dimensional evolution of a plane mixing layer - The Kelvin-Helmholtz rollup

Abstract: The Kelvin Helmholtz roll up of three dimensional, temporally evolving, plane mixing layers were simulated numerically. All simulations were begun from a few low wavenumber disturbances, usually derived from linear stability theory, in addition to the mean velocity profile. The spanwise disturbance wavelength was taken to be less than or equal to the streamwise wavelength associated with the Kelvin Helmholtz roll up. A standard set of clean structures develop in most of the simulations. The spanwise vorticity rolls up into a corrugated spanwise roller, with vortex stretching creating strong spanwise vorticity in a cup shaped region at the vends of the roller. Predominantly streamwise rib vortices develop in the braid region between the rollers. For sufficiently strong initial three dimensional disturbances, these ribs collapse into compact axisymmetric vortices. The rib vortex lines connect to neighboring ribs and are kinked in the opposite direction of the roller vortex lines. Because of this, these two sets of vortex lines remain distinct. For certain initial conditions, persistent ribs do not develop. In such cases the development of significant three dimensionality is delayed. When the initial three dimensional disturbance energy is about equal to, or less than, the two dimensional fundamental disturbance energy, the evolution of the three dimensional disturbance is nearly linear (with respect to the mean and the two dimensional disturbances), at least until the first Kelvin Helmholtz roll up is completed.

Experiments on transition in plane Couette flow

Abstract: The first flow visualization experimental results of transition in plane Couette flow are reported. The Couette flow water channel was of an infinite-belt type with counter-moving walls. The belt and channel walls were transparent making it possible to visualize the flow pattern in the streamwise-spanwise plane by utilizing fluid-suspended reflective flakes. Transition was triggered by a high-amplitude pointwise disturbance. The transitional Reynolds number, i.e. the lowest Reynolds number for which turbulence can be sustained, was determined to be 360 ± 10, based on half-channel height and half the velocity difference between the walls. For Reynolds numbers above this value a large enough amplitude of the initial disturbance gave rise to a growing turbulent spot. Its shape and spreading rate was determined for Reynolds numbers up to 1000.

Three-dimensional dynamics and transition to turbulence in the wake of bluff objects

Abstract: The wakes of bluff objects and in particular of circular cylinders are known to undergo a ‘fast’ transition, from a laminar two-dimensional state at Reynolds number 200 to a turbulent state at Reynolds number 400. The process has been documented in several experimental investigations, but the underlying physical mechanisms have remained largely unknown so far. In this paper, the transition process is investigated numerically, through direct simulation of the Navier—Stokes equations at representative Reynolds numbers, up to 500. A high-order time-accurate, mixed spectral/spectral element technique is used. It is shown that the wake first becomes three-dimensional, as a result of a secondary instability of the two-dimensional vortex street. This secondary instability appears at a Reynolds number close to 200. For slightly supercritical Reynolds numbers, a harmonic state develops, in which the flow oscillates at its fundamental frequency (Strouhal number) around a spanwise modulated time-average flow. In the near wake the modulation wavelength of the time-average flow is half of the spanwise wavelength of the perturbation flow, consistently with linear instability theory. The vortex filaments have a spanwise wavy shape in the near wake, and form rib-like structures further downstream. At higher Reynolds numbers the three-dimensional flow oscillation undergoes a period-doubling bifurcation, in which the flow alternates between two different states. Phase-space analysis of the flow shows that the basic limit cycle has branched into two connected limit cycles. In physical space the period doubling appears as the shedding of two distinct types of vortex filaments.Further increases of the Reynolds number result in a cascade of period-doubling bifurcations, which create a chaotic state in the flow at a Reynolds number of about 500. The flow is characterized by broadband power spectra, and the appearance of intermittent phenomena. It is concluded that the wake undergoes transition to turbulence following the period-doubling route.

Insoluble surfactant spreading on a thin viscous film: Shock evolution and film rupture

Abstract: Lubrication theory and similarity methods are used to determine the spreading rate of a localized monolayer of insoluble surfactant on the surface of a thin viscous film, in the limit of weak capillarity and weak surface diffusion. If the total mass of surfactant increases as t(alpha), then at early times, when spreading is driven predominantly by Marangoni forces, a planar (axisymmetric) region of surfactant is shown to spread as t(1 + alpha)/3 (t(1 + alpha)/4) . A shock exists at the leading edge of the monolayer; asymptotic methods are used to show that a wavetrain due to capillary forces exists ahead of the shock at small times, but that after a finite time it is swamped by diffusive effects. For alpha < 1/2 (alpha < 1), the diffusive lengthscale at the shock grows faster than the length of the monolayer, ultimately destroying the shock; subsequently, spreading is driven by diffusion, and proceeds as t1/2. The asymptotic results are shown to be good approximations of numerical solutions of the governing partial differential equations in the appropriate limits. Additional forces are also considered: weak vertical gravity can also destroy the shock in finite time, while effects usually neglected from lubrication theory are important only early in spreading. Experiments have shown that the severe thinning of the film behind the shock can cause it to rupture: the dryout process is modelled by introducing van der Waals forces.

Numerical simulations of Lewis number effects in turbulent premixed flames

Abstract: The structure of a premixed flame front propagating in a region of two-dimensional turbulence is investigated using full numerical simulation including heat release, variable properties, and one-step Arrhenius chemistry. The influence of reactant Lewis number (Le = ratio of thermal to species diffusivity) is reported for Le = 0.8, Le = 1.0, and Le = 1.2 flames. Local flame behaviour is described by comparing the local instantaneous turbulent flame structure (local consumption rate of reactants) to the steady one-dimensional laminar flame structure for the same thermochemical parameters. Statistics of flame front strain rates and curvature are calculated and global quantities of interest in modelling (flame surface area, mean reactant consumption rate per unit area of flame, and turbulent flame speed) are reported. Principal findings are: that probability density functions (p.d.f.s) of flame curvature are nearly symmetric about a near-zero mean; that the flame tends to align preferentially with extensive tangential strain rates; that the local flame structure of the non-unity Lewis number flames correlates more strongly with local flame curvature than with tangential strain rate; that the mean consumption rate per unit area is relatively insensitive to curvature and is controlled by the mean tangential strain rate; and, that more flame area is generated for Le 1. Implications of the results for flamelet models of turbulent premixed combustion are discussed.

Satellite and subsatellite formation in capillary breakup

Abstract: An investigation of the interfacial-tension-driven fragmentation of a very long fluid filament in a quiescent viscous fluid is presented. Experiments covering almost three orders of magnitude in viscosity ratio reveal as many as 19 satellite droplets in between the largest droplets; complementary boundary-integral calculations are used to study numerically the evolution of the filament as a function of the viscosity ratio of the fluids and the initial wavenumber of the interface perturbation. Satellite drops are generated owing to multiple breakup sequences around the neck region of a highly deformed filament. In low-viscosity ratio systems, p < O(0.1), the breakup mechanism is self-repeating in the sense that every pinch-off is always associated with the formation of a neck, the neck undergoes pinch-off, and the process repeats. In general the agreement between computations and experiments is excellent; both indicate that the initial wavenumber of the disturbance is important in the quantitative details of the generated drop size distributions. However, these details are insignificant when compared with the large variations produced in the drop size distributions owing to variation in the viscosity ratio.

Instabilities of the liquid and mushy regions during solidification of alloys

Abstract: The solidification of melts can be profoundly influenced by convection. In alloys, compositional convection can be driven by solute gradients generated as one component of the alloy is preferentially incorporated within the solid, even when the thermal field is stabilizing. In this paper, two modes of compositional convection during solidification from below are uncovered using a linear-stability analysis : one, which we shall call the ‘mushy-layer mode’, is driven by buoyant residual fluid within a mushy layer, or porous medium, of dendritic crystals; the other, which we shall call the ‘ boundary-layer mode ’ is associated with a narrow compositional boundary layer in the melt just above the mush-liquid interface. Either mode can be the first to become unstable depending on the thermodynamical and physical properties of the alloy. The marginally stable eigenfunctions suggest that the boundary-layer mode results in fine-scale convection in the melt above the mushy layer and leaves the interstitial fluid of the mushy layer virtually stagnant. In contrast, the mushy-layer mode causes perturbations to the solid fraction of the mushy layer that are indicative of a tendency to form chimneys, which are vertical channels of reduced or zero solid fraction that have been observed experimentally. Particular attention is focused on the mushy-layer mode and its dependence upon the thermodynamical properties of the alloy. The results of this analysis are used to make a number of interpretations of earlier experimental studies such as the observations that some systems are less prone to form chimneys and that the regions of melt in these systems evolve to supersaturated conditions, while the melt evolves to unsaturated conditions once chimneys have formed. In addition, good quantitative agreement is found between the results of the linear-stability analysis and the experimental results of Tait & Jaupart (1992) for the onset of the mushy-layer mode of convection.

Spiralling columnar convection in rapidly rotating spherical fluid shells

Abstract: It is shown that the fundamental features of both thermal instabilities and the corresponding nonlinear convection in rapidly rotating spherical systems (in the range of the Taylor number 109 < T < 1012) are determined by the fluid properties characterized by the size of the Prandtl number. Coefficients of the asymptotic power law for the onset of convection at large Taylor number are estimated in the range of the Prandtl number 0.1 ≤ Pr ≤ 100. For fluids of moderately small Prandtl number, a new type of convective instability in the form of prograde spiralling drifting columnar rolls is discovered. The linear columnar rolls extend spirally from near latitude 60° to the equatorial region, and each spans azimuthally approximately five wavelengths with the inclination angle between a spirally elongated roll and the radial direction exceeding 45°. As a consequence, the radial lengthscale of the linear roll becomes comparable with the azimuthal lengthscale. A particularly significant finding is the connection between the new instability and the predominantly axisymmetric convection. Though non-axisymmetric motions are preferred at the onset of convection, the nonlinear convection (at the Rayleigh number of the order of (R—Rc)/Rc = O(0.1)) bifurcating supercritically from the spiralling mode is primarily dominated by the component of the axisymmetric zonal flow, which contains nearly 90% of the total kinetic energy. For fluids of moderately large Prandtl numbers, thermal instabilities at the onset of convection are concentrated in a cylindrical annulus coaxial with the axis of rotation; the position of the convection cylinder is strongly dependent on the size of the Prandtl number. The associated nonlinear convection consists of predominantly non-axisymmetric columnar rolls together with a superimposed weak mean flow that contains less than 10% of the total kinetic energy at (R—Rc)/Rc = O(0.1). A double-layer structure of the temperature field (with respect to the basic state) forms as a result of strong nonlinear interactions between the nonlinear flow and the temperature field. It is also demonstrated that the aspect ratio of the spherical shell does not substantially influence the fundamental properties of convection.

On the applicability of various scaling laws to the turbulent wall jet

Abstract: The spatial distribution of the mean velocity in a two-dimensional turbulent wall jet was measured for a variety of nozzle Reynolds numbers. It was determined that the bulk of the flow is self-similar and it depends on the momentum flux at the nozzle and on the viscosity and density of the fluid. The width of the nozzle which was commonly used to reduce these data has no part in the similarity considerations as has already been suggested by Narasimha et al. (1973). This type of self-similarity can be easily applied to determine the skin friction, which can otherwise only be determined with considerable difficulty. It was also shown that the ‘law of the wall’ applies only to the viscous sublayer. The Reynolds stress in the inviscid, inner portion of the flow is not constant thus the assumption of a ‘constant stress layer’ is not applicable. The applicability and universality of the ‘outer scaling law’ (i.e. Coles’ law of the wake) has been verified throughout the inviscid inner portion of the wall jet. The logarithmic velocity distribution cannot be derived by making the usual assumptions based on the constancy of the Reynolds stresses or on the thinness of the logarithmic region relative to the thickness of the inner layer.

Internal solitary wave breaking and run-up on a uniform slope

Abstract: Laboratory experiments have been conducted to study the shoaling of internal solitary waves of depression in a two-layer system on a uniform slope. The shoaling of a single solitary wave results in wave breaking and the production of multiple turbulent surges, or boluses, which propagate up the slope. Significant vertical mixing occurs everywhere inshore of the breaking location. The kinematics of the breaking and bolus runup are described and a breaking criterion is found. The energetics of the breaking are investigated. Over the range of parameters examined, 15 (±5) % of the energy lost from first-mode wave motion inshore of the break point goes into vertical mixing.

Low-Reynolds-number effects in a fully developed turbulent channel flow

Abstract: Low-Reynolds-number effects are observed in the inner region of a fully developed turbulent channel flow, using data obtained either from experiments or by direct numerical simulations. The Reynolds-number influence is observed on the turbulence intensities and to a lesser degree on the average production and dissipation of the turbulent energy. In the near-wall region, the data confirm Wei and Willmarth's (1989) conclusion that the Reynolds stresses do not scale on wall variables. One of the reasons proposed to account for this behavior, namely, the 'geometry' effect or direct interaction between inner regions on opposite walls, was investigated in some detail by introducing temperature at one of the walls, both in experiment and simulation. Although the extent of penetration of thermal excursions into the opposite side of the channel can be significant at low Reynolds numbers, the contribution these excursions make to the Reynolds shear stress and the spanwise vorticity in the opposite wall region is negligible. In the inner region, spectra and cospectra of the velocity fluctuations u and v change rapidly with the Reynolds number, the variations being mainly confined to low wavenumbers in the u spectrum.

Capillary rollers and bores

Abstract: A very intriguing phenomenon, which throws much light on the small-scale structure of the sea surface, is the occurrence of “parasitic capillaries” on the forward face of moderately short gravity waves, especially those with wavelengths 5 to 50 cm; see Figure la. These capillary waves were first studied experimentally by Cox (1958). Evidently their existence depends on the fact that a gravity wave and a much shorter capillary wave may have the same phase speed. The dynamical theory of the generation of parasitic capillaries has been developed by Longuet-Higgins (1963), Crapper (1970) and Ruvinsky et al. (1981, 1985, 1991). This so far takes into account only the first-order effects of viscous damping.

A numerical study of the evolution and structure of homogeneous stably stratified sheared turbulence

Abstract: The structure and evolution of homogeneous stably stratified sheared turbulence have been investigated through direct numerical simulation. In these simulations the primary dimensionless parameter is the Richardson number which measures the relative importance of stratification and mean shear.For Richardson numbers less than the transition value the Reynolds stress and vertical density flux are down-gradient. Some of the vertical kinetic energy gained indirectly through production is expended in creating potential energy. Included in this shear-dominated regime is the stationary Richardson number at which the turbulent kinetic energy is constant in time although the spectra are evolving. At low dimensionless shear rate the stationary Richardson number increases with increasing Reynolds number.At the transition Richardson number the maximum anisotropy and energy partition are achieved. For larger Richardson numbers potential energy is released into vertical kinetic energy and the vertical density flux becomes counter-gradient. The associated production reversal enhances the decay rate of the turbulent kinetic energy.The effects of other dimensionless parameters have been investigated. After initial transients the developed flow is rather insensitive to the presence of significant initial potential energy. An increase in the Schmidt number increases the effect of stable stratification, e.g. the counter-gradient vertical density flux occurs earlier.In the shear dominated case the down-gradient fluxes are produced by the pumping of fluid through coherent hairpin-shaped vorticity. In the buoyancy dominated flow the counter-gradient fluid parcels induce helical vorticity structures as they move toward a position of neutral buoyancy.