Showing papers in "Journal of Fluid Mechanics in 1998"

Oscillating foils of high propulsive efficiency

Abstract: Thrust-producing harmonically oscillating foils are studied through force and power measurements, as well as visualization data, to classify the principal characteristics of the flow around and in the wake of the foil. Visualization data are obtained using digital particle image velocimetry at Reynolds number 1100, and force and power data are measured at Reynolds number 40 000. The experimental results are compared with theoretical predictions of linear and nonlinear inviscid theory and it is found that agreement between theory and experiment is good over a certain parametric range, when the wake consists of an array of alternating vortices and either very weak or no leading-edge vortices form. High propulsive efficiency, as high as 87%, is measured experimentally under conditions of optimal wake formation. Visualization results elucidate the basic mechanisms involved and show that conditions of high efficiency are associated with the formation on alternating sides of the foil of a moderately strong leading-edge vortex per half-cycle, which is convected downstream and interacts with trailing-edge vorticity, resulting eventually in the formation of a reverse Karman street. The phase angle between transverse oscillation and angular motion is the critical parameter affecting the interaction of leading-edge and trailing-edge vorticity, as well as the efficiency of propulsion.

A universal time scale for vortex ring formation

Abstract: The formation of vortex rings generated through impulsively started jets is studied experimentally. Utilizing a piston/cylinder arrangement in a water tank, the velocity and vorticity fields of vortex rings are obtained using digital particle image velocimetry (DPIV) for a wide range of piston stroke to diameter (L/D) ratios. The results indicate that the flow field generated by large L/D consists of a leading vortex ring followed by a trailing jet. The vorticity field of the leading vortex ring formed is disconnected from that of the trailing jet. On the other hand, flow fields generated by small stroke ratios show only a single vortex ring. The transition between these two distinct states is observed to occur at a stroke ratio of approximately 4, which, in this paper, is referred to as the ‘formation number’. In all cases, the maximum circulation that a vortex ring can attain during its formation is reached at this non-dimensional time or formation number. The universality of this number was tested by generating vortex rings with different jet exit diameters and boundaries, as well as with various non-impulsive piston velocities. It is shown that the ‘formation number’ lies in the range of 3.6–4.5 for a broad range of flow conditions. An explanation is provided for the existence of the formation number based on the Kelvin–Benjamin variational principle for steady axis-touching vortex rings. It is shown that based on the measured impulse, circulation and energy of the observed vortex rings, the Kelvin–Benjamin principle correctly predicts the range of observed formation numbers.

A numerical study of breaking waves in the surf zone

Abstract: This paper describes the development of a numerical model for studying the evolution of a wave train, shoaling and breaking in the surf zone. The model solves the Reynolds equations for the mean (ensemble average) flow field and the k–e equations for the turbulent kinetic energy, k, and the turbulence dissipation rate, e. A nonlinear Reynolds stress model (Shih, Zhu & Lumley 1996) is employed to relate the Reynolds stresses and the strain rates of the mean flow. To track free-surface movements, the volume of fluid (VOF) method is employed. To ensure the accuracy of each component of the numerical model, several steps have been taken to verify numerical solutions with either analytical solutions or experimental data. For non-breaking waves, very accurate results are obtained for a solitary wave propagating over a long distance in a constant depth. Good agreement between numerical results and experimental data has also been observed for shoaling and breaking cnoidal waves on a sloping beach in terms of free-surface profiles, mean velocities, and turbulent kinetic energy. Based on the numerical results, turbulence transport mechanisms under breaking waves are discussed.

Mean-flow scaling of turbulent pipe flow

Abstract: Measurements of the mean velocity profile and pressure drop were performed in a fully developed, smooth pipe flow for Reynolds numbers from 31×103 to 35×106. Analysis of the mean velocity profiles indicates two overlap regions: a power law for 60 9×103). Von Karman's constant was shown to be 0.436 which is consistent with the friction factor data and the mean velocity profiles for 600 5%) than those predicted by Prandtl's relation. A new friction factor relation is proposed which is within ±1.2% of the data for Reynolds numbers between 10×103 and 35×106, and includes a term to account for the near-wall velocity profile.

Cavitation erosion by single laser-produced bubbles

Abstract: In order to elucidate the mechanism of cavitation erosion, the dynamics of a single laser-generated cavitation bubble in water and the resulting surface damage on a flat metal specimen are investigated in detail. The characteristic effects of bubble dynamics, in particular the formation of a high-speed liquid jet and the emission of shock waves at the moment of collapse are recorded with high-speed photography with framing rates of up to one million frames/s. Damage is observed when the bubble is generated at a distance less than twice its maximum radius from a solid boundary (γ=2, where γ=s/Rmax, s is the distance between the boundary and the bubble centre at the moment of formation and Rmax is the maximum bubble radius). The impact of the jet contributes to the damage only at small initial distances (γ[les ]0.7). In this region, the impact velocity rises to 83 m s−1, corresponding to a water hammer pressure of about 0.1 GPa, whereas at γ>1, the impact velocity is smaller than 25 m s−1. The largest erosive force is caused by the collapse of a bubble in direct contact with the boundary, where pressures of up to several GPa act on the material surface. Therefore, it is essential for the damaging effect that bubbles are accelerated towards the boundary during the collapse phases due to Bjerknes forces. The bubble touches the boundary at the moment of second collapse when γ<2 and at the moment of first collapse when γ<1. Indentations on an aluminium specimen are found at the contact locations of the collapsing bubble. In the range γ=1.7 to 2, where the bubble collapses mainly down to a single point, one pit below the bubble centre is observed. At γ[les ]1.7, the bubble shape has become toroidal, induced by the jet flow through the bubble centre. Corresponding to the decay of this bubble torus into multiple tiny bubbles each collapsing separately along the circumference of the torus, the observed damage is circular as well. Bubbles in the ranges γ[les ]0.3 and γ=1.2 to 1.4 caused the greatest damage. The overall diameter of the damaged area is found to scale with the maximum bubble radius. Owing to the possibility of generating thousands of nearly identical bubbles, the cavitation resistance of even hard steel specimens can be tested.

Mixing, structure and scaling of the jet in crossflow

Abstract: The mixing of the round jet normal to a uniform crossflow is studied for a range of jet-to-crossflow velocity ratios, r, from 5 to 25. Planar laser-induced fluorescence (PLIF) of acetone vapour seeded into the jet is used to acquire quantitative two-dimensional images of the scalar concentration field. Emphasis is placed on r=10 and r=20 and a few select images are acquired up to r=200. The Reynolds number based on the jet exit diameter, d, and the exit velocity varies from 8400 to 41 500. Images are acquired for conditions in which the product rd is held constant, requiring decreasing d for increasing r.Results from this experimental study concern structural events of the vortex interaction region, and mixing and mean centreline concentration decay in the near and far fields. The results cover all three regions of the transverse jet, and suggest that the jet scales with three length scales: d, rd and r2d.Events within the vortex interaction region display d-scaling, including the crossflow boundary layer separation and roll-up. Over the range of velocity ratios studied, the vortex interaction region shows r-dependent variations in the flow field, including the emergence of jet fluid in the wake structures for r>10 and a slower development of the counter-rotating vortex pair (CVP) in higher-r jets.The trajectory and physical dimension of the jet in both the near and far field display rd-scaling. The near field is characterized by a centreline concentration decay along the centreline coordinate s of s−1.3, different from the decay rate (s−1) of the free jet. When normalized by rd, the decay of each velocity-ratio jet branches away from the s−1.3 decay, approaching a decay of s−2/3, a rate predicted by modelling efforts. The branch points represent a transition in the flow field from enhanced mixing to reduced mixing compared to the free jet. When normalized by r2d, the branch points occur at a uniform jet position, s/r2d=0.3, which is viewed to be the division between the near and far fields. Self-similarity is not seen in the near field, but may be present in the far field.The view of the branch points as a place of transition in the flow is supported by the probability density function (p.d.f.) of concentration along the upper edge of the jet. Before the branch points, the p.d.f.s are non-marching in character, and after the branch points, they are tilted in character.Instantaneously, the CVP is asymmetric in shape and concentration. End views reveal extensive motion of the CVP and plan views show this motion can occur in both axisymmetric and sinusoidal motion. Ensemble-averaged images show the jet concentration is asymmetric about the centreline plane.

Direct analysis of particulate suspensions with inertia using the discrete Boltzmann equation

Abstract: An efficient and robust computational method, based on the lattice-Boltzmann method, is presented for analysis of impermeable solid particle(s) suspended in fluid with inertia. In contrast to previous lattice-Boltzmann approaches, the present method can be used for any solid-to-fluid density ratio. The details of the numerical technique and implementation of the boundary conditions are presented. The accuracy and robustness of the method is demonstrated by simulating the flow over a circular cylinder in a two-dimensional channel, a circular cylinder in simple shear flow, sedimentation of a circular cylinder in a two-dimensional channel, and sedimentation of a sphere in a three-dimensional channel. With a solid-to-fluid density ratio close to one, new results from two-dimensional and three-dimensional computational analysis of dynamics of an ellipse and an ellipsoid in a simple shear flow, as well as two-dimensional and three-dimensional results for sedimenting ellipses and prolate spheroids, are presented.

The lift force on a spherical bubble in a viscous linear shear flow

Abstract: The three-dimensional flow around a spherical bubble moving steadily in a viscous linear shear flow is studied numerically by solving the full Navier–Stokes equations. The bubble surface is assumed to be clean so that the outer flow obeys a zero-shear-stress condition and does not induce any rotation of the bubble. The main goal of the present study is to provide a complete description of the lift force experienced by the bubble and of the mechanisms responsible for this force over a wide range of Reynolds number (0.1[les ]Re[les ]500, Re being based on the bubble diameter) and shear rate (0[les ]Sr[les ]1, Sr being the ratio between the velocity difference across the bubble and the relative velocity). For that purpose the structure of the flow field, the influence of the Reynolds number on the streamwise vorticity field and the distribution of the tangential velocities at the surface of the bubble are first studied in detail. It is shown that the latter distribution which plays a central role in the production of the lift force is dramatically dependent on viscous effects. The numerical results concerning the lift coefficient reveal very different behaviours at low and high Reynolds numbers. These two asymptotic regimes shed light on the respective roles played by the vorticity produced at the bubble surface and by that contained in the undisturbed flow. At low Reynolds number it is found that the lift coefficient depends strongly on both the Reynolds number and the shear rate. In contrast, for moderate to high Reynolds numbers these dependences are found to be very weak. The numerical values obtained for the lift coefficient agree very well with available asymptotic results in the low- and high-Reynolds-number limits. The range of validity of these asymptotic solutions is specified by varying the characteristic parameters of the problem and examining the corresponding evolution of the lift coefficient. The numerical results are also used for obtaining empirical correlations useful for practical calculations at finite Reynolds number. The transient behaviour of the lift force is then examined. It is found that, starting from the undisturbed flow, the value of the lift force at short time differs from its steady value, even when the Reynolds number is high, because the vorticity field needs a finite time to reach its steady distribution. This finding is confirmed by an analytical derivation of the initial value of the lift coefficient in an inviscid shear flow. Finally, a specific investigation of the evolution of the lift and drag coefficients with the shear rate at high Reynolds number is carried out. It is found that when the shear rate becomes large, i.e. Sr=O(1), a small but consistent decrease of the lift coefficient occurs while a very significant increase of the drag coefficient, essentially produced by the modifications of the pressure distribution, is observed. Some of the foregoing results are used to show that the well-known equality between the added mass coefficient and the lift coefficient holds only in the limit of weak shears and nearly steady flows.

Deformation of liquid capsules enclosed by elastic membranes in simple shear flow: large deformations and the effect of fluid viscosities

Abstract: The deformation of a liquid capsule enclosed by an elastic membrane in an infinite simple shear flow is studied numerically at vanishing Reynolds numbers using a boundary-element method. The surface of the capsule is discretized into quadratic triangular elements that form an evolving unstructured grid. The elastic membrane tensions are expressed in terms of the surface deformation gradient, which is evaluated from the position of the grid points. Compared to an earlier formulation that uses global curvilinear coordinates, the triangular-element formulation suppresses numerical instabilities due to uneven discretization and thus enables the study of large deformations and the investigation of the effect of fluid viscosities. Computations are performed for capsules with spherical, spheroidal, and discoidal unstressed shapes over an extended range of the dimensionless shear rate and for a broad range of the ratio of the internal to surrounding fluid viscosities. Results for small deformations of spherical capsules are in quantitative agreement with the predictions of perturbation theories. Results for large deformations of spherical capsules and deformations of non-spherical capsules are in qualitative agreement with experimental observations of synthetic capsules and red blood cells. We find that initially spherical capsules deform into steady elongated shapes whose aspect ratios increase with the magnitude of the shear rate. A critical shear rate above which capsules exhibit continuous elongation is not observed for any value of the viscosity ratio. This behaviour contrasts with that of liquid drops with uniform surface tension and with that of axisymmetric capsules subject to a stagnation-point flow. When the shear rate is sufficiently high and the viscosity ratio is sufficiently low, liquid drops exhibit continuous elongation leading to breakup. Axisymmetric capsules deform into thinning needles at sufficiently high rates of elongation, independent of the fluid viscosities. In the case of capsules in shear flow, large elastic tensions develop at large deformations and prevent continued elongation, stressing the importance of the vorticity of the incident flow. The long-time behaviour of deformed capsules depends strongly on the unstressed shape. Oblate capsules exhibit unsteady motions including oscillation about a mean configuration at low viscosity ratios and continuous rotation accompanied by periodic deformation at high viscosity ratios. The viscosity ratio at which the transition from oscillations to tumbling occurs decreases with the sphericity of the unstressed shape. Results on the effective rheological properties of dilute suspensions confirm a non-Newtonian shear-thinning behaviour.

Direct numerical simulation of turbulence modulation by particles in isotropic turbulence

Abstract: The modulation of isotropic turbulence by particles has been investigated using direct numerical simulation (DNS). The particular focus of the present work is on the class of dilute flows in which particle volume fractions and inter-particle collisions are negligible. Gravitational settling is also neglected and particle motion is assumed to be governed by drag with particle relaxation times ranging from the Kolmogorov scale to the Eulerian time scale of the turbulence and particle mass loadings up to 1. The velocity field was made statistically stationary by forcing the low wavenumbers of the flow. The calculations were performed using 963 collocation points and the Taylor-scale Reynolds number for the stationary flow was 62. The effect of particles on the turbulence was included in the Navier–Stokes equations using the point-force approximation in which 963 particles were used in the calculations. DNS results show that particles increasingly dissipate fluid kinetic energy with increased loading, with the reduction in kinetic energy being relatively independent of the particle relaxation time. Viscous dissipation in the fluid decreases with increased loading and is larger for particles with smaller relaxation times. Fluid energy spectra show that there is a non-uniform distortion of the turbulence with a relative increase in small-scale energy. The non-uniform distortion significantly affects the transport of the dissipation rate, with the production and destruction of dissipation exhibiting completely different behaviours. The spectrum of the fluid–particle energy exchange rate shows that the fluid drags particles at low wavenumbers while the converse is true at high wavenumbers for small particles. A spectral analysis shows that the increase of the high-wavenumber portion of the fluid energy spectrum can be attributed to transfer of the fluid–particle covariance by the fluid turbulence. This in turn explains the relative increase of small-scale energy caused by small particles observed in the present simulations as well as those of Squires & Eaton (1990) and Elghobashi & Truesdell (1993).

Experimental study of vortex breakdown in swirling jets

Abstract: The goal of this study is to characterize the various breakdown states taking place in a swirling water jet as the swirl ratio S and Reynolds number Re are varied. A pressure-driven water jet discharges into a large tank, swirl being imparted by means of a motor which sets into rotation a honeycomb within a settling chamber. The experiments are conducted for two distinct jet diameters by varying the swirl ratio S while maintaining the Reynolds number Re fixed in the range 300

Break-up and atomization of a round water jet by a high-speed annular air jet

Abstract: The near- and far-field break-up and atomization of a water jet by a high-speed annular air jet are examined by means of high-speed flow visualizations and phase Doppler particle sizing techniques. Visualization of the jet's near field and measurements of the frequencies associated with the gas–liquid interfacial instabilities are used to study the underlying physical mechanisms involved in the primary break-up of the water jet. This process is shown to consist of the stripping of water sheets, or ligaments, which subsequently break into smaller lumps or drops. An entrainment model of the near-field stripping of the liquid is proposed, and shown to describe the measured liquid shedding frequencies. This simplified model explains qualitatively the dependence of the shedding frequency on the air/water momentum ratio in both initially laminar and turbulent water jets. The role of the secondary liquid break-up in the far-field atomization of the water jet is also investigated, and an attempt is made to apply the classical concepts of local isotropy to explain qualitatively the measurement of the far-field droplet size distribution and its dependence on the water to air mass and momentum ratios. Models accounting for the effect of the local turbulent dissipation rate in the gas on both the break-up and coalescence of the droplets are developed and compared with the measurements of the variation of the droplet size along the jet's centreline. The total flux of kinetic energy supplied by the gas per unit total mass of the spray jet was found to be the primary parameter determining the secondary break-up and coalescence of the droplets in the far field.

Low-Reynolds-number flow around an oscillating circular cylinder at low Keulegan–Carpenter numbers

Abstract: Time-averaged LDA measurements and time-resolved numerical flow predictions were performed to investigate the laminar flow induced by the harmonic in-line oscillation of a circular cylinder in water at rest. The key parameters, Reynolds number Re and Keulegan–Carpenter number KC, were varied to study three parameter combinations in detail. Good agreement was observed for Re=100 and KC=5 between measurements and predictions comparing phase-averaged velocity vectors. For Re=200 and KC=10 weakly stable and non-periodic flow patterns occurred, which made repeatable time-averaged measurements impossible. Nevertheless, the experimentally visualized vortex dynamics was reproduced by the two-dimensional computations. For the third combination, Re=210 and KC=6, which refers to a totally different flow regime, the computations again resulted in the correct fluid behaviour. Applying the widely used model of Morison et al. (1950) to the computed in-line force history, the drag and the added-mass coefficients were calculated and compared for different grid levels and time steps. Using these to reproduce the force functions revealed deviations from those originally computed as already noted in previous studies. They were found to be much higher than the deviations for the coarsest computational grid or the largest time step. The comparison of several in-line force coefficients with results obtained experimentally by Kuhtz (1996) for β=35 confirmed that force predictions could also be reliably obtained by the computations.

Optimal and robust control and estimation of linear paths to transition

Abstract: Optimal and robust control theories are used to determine effective, estimator-based feedback control rules for laminar plane channel flows that effectively stabilize linearly unstable flow perturbations at Re=10 000 and linearly stable flow perturbations, characterized by mechanisms for very large disturbance amplification, at Re=5000. Wall transpiration (unsteady blowing/suction) with zero net mass flux is used as the control, and the flow measurement is derived from the wall skin friction. The control objective, beyond simply stabilizing any unstable eigenvalues (which is relatively easy to accomplish), is to minimize the energy of the flow perturbations created by external disturbance forcing. This is important because, when mechanisms for large disturbance amplification are present, small-amplitude external disturbance forcing may excite flow perturbations with sufficiently large amplitude to induce nonlinear flow instability.The control algorithms used in the present work account for system disturbances and measurement noise in a rigorous fashion by application of modern linear control techniques to the discretized linear stability problem. The disturbances are accounted for both as uncorrelated white Gaussian processes ([Hscr ]2 or ‘optimal’ control) and as finite ‘worst case’ inputs which are maximally detrimental to the control objective ([Hscr ]∞ or ‘robust’ control). Root loci and transient energy growth analyses are shown to be inadequate measures to characterize overall system performance. Instead, appropriately defined transfer function norms are used to characterize all systems considered in a consistent and relevant manner. In order to make a parametric study tractable in this high-dimensional system, a convenient new scaling to the estimation problem is introduced such that three scalar parameters {γ, α, [lscr ]} may be individually adjusted to achieve desired closed-loop characteristics of the resulting systems. These scalar parameters may be intuitively explained, and are defined such that the resulting control equations retain the natural dual structure between the control parameter, [lscr ], and the estimation parameter, α. The performance of the present systems with respect to these parameters is thoroughly investigated, and comparisons are made to simple proportional schemes where appropriate.

Post-stall flow control on an airfoil by local unsteady forcing

Abstract: By using a Reynolds-averaged two-dimensional computation of a turbulent flow over an airfoil at post-stall angles of attack, we show that the massively separated and disordered unsteady flow can be effectively controlled by periodic blowing–suction near the leading edge with low-level power input. This unsteady forcing can modulate the evolution of the separated shear layer to promote the formation of concentrated lifting vortices, which in turn interact with trailing-edge vortices in a favourable manner and thereby alter the global deep-stall flow field. In a certain range of post-stall angles of attack and forcing frequencies, the unforced random separated flow can become periodic or quasi-periodic, associated with a significant lift enhancement. This opens a promising possibility for flight beyond the static stall to a much higher angle of attack. The same local control also leads, in some situations, to a reduction of drag. On a part of the airfoil the pressure fluctuation is suppressed as well, which would be beneficial for high-α buffet control. The computations are in qualitative agreement with several recent post-stall flow control experiments. The physical mechanisms responsible for post-stall flow control, as observed from the numerical data, are explored in terms of nonlinear mode competition and resonance, as well as vortex dynamics. The leading-edge shear layer and vortex shedding from the trailing edge are two basic constituents of unsteady post-stall flow and its control. Since the former has a rich spectrum of response to various disturbances, in a quite wide range the natural frequency of both constituents can shift and lock-in to the forcing frequency or its harmonics. Thus, most of the separated flow becomes resonant, associated with much more organized flow patterns. During this nonlinear process the coalescence of small vortices from the disturbed leading-edge shear layer is enhanced, causing a stronger entrainment and hence a stronger lifting vortex. Meanwhile, the unfavourable trailing-edge vortex is pushed downstream. The wake pattern also has a corresponding change: the shed vortices of alternate signs tend to be aligned, forming a train of close vortex couples with stronger downwash, rather than a Karman street.

On stability of streamwise streaks and transition thresholds in plane channel flows

Abstract: Streak breakdown caused by a spanwise inflectional instability is one phase of the following transition scenarios, which occur in plane Poiseuille and Couette flow. The streamwise vortex scenario is described byformula hereThe oblique wave scenario is described byformula hereThe purpose of this paper is to investigate the streak breakdown phase of the above scenarios by a linear stability analysis and compare threshold energies for transition for the above scenarios with those for transition initiated by Tollmien-Schlichting waves (TS), two-dimensional optimals (2DOPT), and random noise (N) at subcritical Reynolds numbers.We find that if instability occurs, it is confined to disturbances with streamwise wavenumbers α0 satisfying 0<αmin< [mid ]α0[mid ]<αmax. In these unstable cases, the least stable mode is located near the centre of the channel with a phase velocity approximately equal to the centreline velocity of the mean flow. Growth rates for instability increase with streak amplitude. For Couette flow streak breakdown is inhibited by mean shear. Using the linear stability analysis, we determine lower bounds on threshold amplitude for transition for scenario (SV) that are consistent with thresholds determined by direct numerical simulations.In the final part of the paper we show that the threshold energies for transition in Poiseuille flow at subcritical Reynolds numbers for scenarios (SV) and (OW) are two orders of magnitude lower than the threshold for transition initiated by Tollmien–Schlichting waves (TS) and an order of magnitude lower than that for (2DOPT). Scenarios (SV) and (OW) occur on a viscous time scale. However, even when transition times are taken into account, the threshold energy required for transition at a given time for (SV) and (OW) is lower than that for the (TS) and (2DOPT) scenarios at Reynolds number 1500.

Analyses of slow high-concentration flows of granular materials

Abstract: A theory intended for slow, dense flows of cohesionless granular materials is developed for the case of planar deformations. By considering granular flows on very fine scales, one can conveniently split the individual particle velocities into fluctuating and mean transport components, and employ the notion of granular temperature that plays a central role in rapid granular flows. On somewhat larger scales, one can think of analogous fluctuations in strain rates. Both kinds of fluctuations are utilized in the present paper. Following the standard continuum approach, the conservation equations for mass, momentum and particle translational fluctuation energy are presented. The latter two equations involve constitutive coefficients, whose determination is one of the main concerns of the present paper. We begin with an associated flow rule for the case of a compressible, frictional, plastic continuum. The functional dependence of the flow rule is chosen so that the limiting behaviours of the resulting constitutive relations are consistent with the results of the kinetic theories developed for rapid flow regimes. Following Hibler (1977) and assuming that there are fluctuations in the strain rates that have, for example, a Gaussian distribution function, it is possible to obtain a relationship between the mean stress and the mean strain rate. It turns out, perhaps surprisingly, that this relationship has a viscous-like character. For low shear rates, the constitutive behaviour is similar to that of a liquid in the sense that the effective viscosity decreases with increasing granular temperature, whereas for rapid granular flows, the viscosity increases with increasing granular temperature as in a gas. The rate of energy dissipation can be determined in a manner similar to that used to derive the viscosity coefficients. After assuming that the magnitude of the strain-rate fluctuations can be related to the granular temperature, we obtain a closed system of equations that can be used to solve boundary value problems. The theory is used to consider the case of a simple shear flow. The resulting expressions for the stress components are similar to models previously proposed on a more ad hoc basis in which quasi-static stress contributions were directly patched to rate-dependent stresses. The problem of slow granular flow in rough-walled vertical chutes is then considered and the velocity, concentration and granular temperature profiles are determined. Thin boundary layers next to the vertical sidewalls arise with the concentration boundary layer being thicker than the velocity boundary layer. This kind of behaviour is observed in both laboratory experiments and in granular dynamics simulations of vertical chute flows.

Passive scalar statistics in high-Péclet-number grid turbulence

Abstract: The statistics of a turbulent passive scalar (temperature) and their Reynolds number dependence are studied in decaying grid turbulence for the Taylor-microscale Reynolds number, Rλ, varying from 30 to 731 (21[les ]Peλ[les ]512). A principal objective is, using a single (and simple) flow, to bridge the gap between the existing passive grid-generated low-Peclet-number laboratory experiments and those done at high Peclet number in the atmosphere and oceans. The turbulence is generated by means of an active grid and the passive temperature fluctuations are generated by a mean transverse temperature gradient, formed at the entrance to the wind tunnel plenum chamber by an array of differentially heated elements. A well-defined inertial–convective scaling range for the scalar with a slope, nθ, close to the Obukhov–Corrsin value of 5/3, is observed for all Reynolds numbers. This is in sharp contrast with the velocity field, in which a 5/3 slope is only approached at high Rλ. The Obukhov–Corrsin constant, Cθ, is estimated to be 0.45–0.55. Unlike the velocity spectrum, a bump occurs in the spectrum of the scalar at the dissipation scales, with increasing prominence as the Reynolds number is increased. A scaling range for the heat flux cospectrum was also observed, but with a slope around 2, less than the 7/3 expected from scaling theory. Transverse structure functions of temperature exist at the third and fifth orders, and, as for even-order structure functions, the width of their inertial subranges dilates with Reynolds number in a systematic way. As previously shown for shear flows, the existence of these odd-order structure functions is a violation of local isotropy for the scalar differences, as is the existence of non-zero values of the transverse temperature derivative skewness (of order unity) and hyperskewness (of order 100). The ratio of the temperature derivative standard deviation along and normal to the gradient is 1.2±0.1, and is independent of Reynolds number. The refined similarity hypothesis for the passive scalar was found to hold for all Rλ, which was not the case for the velocity field. The intermittency exponent for the scalar, μθ, was found to be 0.25±0.05 with a possible weak Rλ dependence, unlike the velocity field, where μ was a strong function of Reynolds number. New, higher-Reynolds-number results for the velocity field, which smoothly follow the trends of Mydlarski & Warhaft (1996), are also presented.

A Herschel–Bulkley model for mud flow down a slope

Abstract: The spreading and sediment deposit of a two-dimensional, unsteady, laminar mud flow from a constant-volume source on a relatively steep slope is studied theoretically and experimentally. The mud under consideration has the rheological properties of a Herschel–Bulkley fluid. The flow is of low-Reynolds-number type and has a well-formed wave front moving a substantial distance downslope. Due to the nonlinear rheological characteristics, a set of nonlinear partial differential equations is needed for this transient problem. Depth-integrated continuity and momentum equations are derived by applying von Karman's momentum integral method. A matched-asymptotic perturbation method is implemented analytically to get asymptotic solutions for both the outer region away from, and the inner region near, the wave front. The outer solution gives accurate results for spreading characteristics, while the inner solution, which is shown to agree well with experimental results of Liu & Mei (1989) for a Bingham fluid, predicts fairly well the free-surface profile near the wave front. A composite solution uniformly valid over the whole spreading length is then achieved through a matching of the inner and outer solutions in an overlapping region. The range of accuracy of the solution and the size of the inner and overlapping regions are quantified by physical scaling analyses. Rheological and dynamic measurements are obtained through laboratory experiments. Theoretical predictions are compared with experimental results, showing reasonable agreement. The impact of shear thinning on the runout characteristics, free-surface profiles and final deposit of the mud flow is examined. A mud flow with shear thinning spreads beyond the runout distance estimated by a Bingham model, and has a long and thin deposit.

Cooperative elliptic instability of a vortex pair.

Abstract: In this paper, we investigate the three-dimensional instability of a counter-rotating vortex pair to short waves, which are of the order of the vortex core size, and less than the inter-vortex spacing. Our experiments involve detailed visualizations and velocimetry to reveal the spatial structure of the instability for a vortex pair, which is generated underwater by two rotating plates. We discover, in this work, a symmetry-breaking phase relationship between the two vortices, which we show to be consistent with a kinematic matching condition for the disturbances evolving on each vortex. In this sense, the instabilities in each vortex evolve in a coupled, or ‘cooperative’, manner. Further results demonstrate that this instability is a manifestation of an elliptic instability of the vortex cores, which is here identified clearly for the first time in a real open flow. We establish a relationship between elliptic instability and other theoretical instability studies involving Kelvin modes. In particular, we note that the perturbation shape near the vortex centres is unaffected by the finite size of the cores. We find that the long-term evolution of the flow involves the inception of secondary transverse vortex pairs, which develop near the leading stagnation point of the pair. The interaction of these short-wavelength structures with the long-wavelength Crow instability is studied, and we observe significant modifications in the longevity of large vortical structures.

Direct numerical simulations of bubbly flows. Part 1. Low Reynolds number arrays

Abstract: Direct numerical simulations of the motion of two- and three-dimensional finite Reynolds number buoyant bubbles in a periodic domain are presented. The full Navier–Stokes equations are solved by a finite difference/front tracking method that allows a fully deformable interface between the bubbles and the ambient fluid and the inclusion of surface tension. The rise Reynolds numbers are around 20–30 for the lowest volume fraction, but decrease as the volume fraction is increased. The rise of a regular array of bubbles, where the relative positions of the bubbles are fixed, is compared with the evolution of a freely evolving array. Generally, the freely evolving array rises slower than the regular one, in contrast to what has been found earlier for low Reynolds number arrays. The structure of the bubble distribution is examined and it is found that while the three-dimensional bubbles show a tendency to line up horizontally, the two-dimensional bubbles are nearly randomly distributed. The effect of the number of bubbles in each period is examined for the two-dimensional system and it is found that although the rise Reynolds number is nearly independent of the number of bubbles, the velocity fluctuations in the liquid (the Reynolds stresses) increase with the size of the system. While some aspects of the fully three-dimensional flows, such as the reduction in the rise velocity, are predicted by results for two-dimensional bubbles, the structure of the bubble distribution is not. The magnitude of the Reynolds stresses is also greatly over-predicted by the two-dimensional results.

Direct numerical simulation of a separated turbulent boundary layer

Abstract: A separated turbulent boundary layer over a flat plate was investigated by direct numerical simulation of the incompressible Navier–Stokes equations. A suction-blowing velocity distribution was prescribed along the upper boundary of the computational domain to create an adverse-to-favourable pressure gradient that produces a closed separation bubble. The Reynolds number based on inlet free-stream velocity and momentum thickness is 300. Neither instantaneous detachment nor reattachment points are fixed in space but fluctuate significantly. The mean detachment and reattachment locations determined by three different definitions, i.e. (i) location of 50% forward flow fraction, (ii) mean dividing streamline (ψ=0), (iii) location of zero wall-shear stress (τw=0), are in good agreement. Instantaneous vorticity contours show that the turbulent structures emanating upstream of separation move upwards into the shear layer in the detachment region and then turn around the bubble. The locations of the maximum turbulence intensities as well as Reynolds shear stress occur in the middle of the shear layer. In the detached flow region, Reynolds shear stresses and their gradients are large away from the wall and thus the largest pressure fluctuations are in the middle of the shear layer. Iso-surfaces of negative pressure fluctuations which correspond to the core region of the vortices show that large-scale structures grow in the shear layer and agglomerate. They then impinge on the wall and subsequently convect downstream. The characteristic Strouhal number St=fδ*in/U0 associated with this motion ranges from 0.0025 to 0.01. The kinetic energy budget in the detachment region is very similar to that of a plane mixing layer.

Hydrodynamic equations for rapid flows of smooth inelastic spheres, to Burnett order

Abstract: The Chapman–Enskog expansion is generalized in order to derive constitutive relations for flows of inelastically colliding spheres in three dimensions – to Burnett order. To this end, the pertinent (nonlinear) Boltzmann equation is perturbatively solved by performing a (double) expansion in the Knudsen number and the degree of inelasticity. One of the results is that the normal stress differences and the ‘temperature anisotropy’, characterizing granular fluids, are Burnett effects. The constitutive relations derived in this work differ, both qualitatively and quantitatively, from those obtained in previous studies. In particular, the Navier–Stokes (order) terms have a different dependence on the degree of inelasticity and the number density than in previously derived constitutive relations; for instance, the expression for the heat flux contains a term which is proportional to e∇ log n, where e is a measure of the degree of inelasticity and n denotes the number density. This contribution to the heat flux is of zeroth order in the density; a similar term, i.e. one that is proportional to e∇n, has been previously obtained by using the Enskog correction but this term is O(n) and it vanishes in the Boltzmann limit. These discrepancies are resolved by an analysis of the Chapman–Enskog and Grad expansions, pertaining to granular flows, which reveals that the quasi-microscopic rate of decay of the temperature, which has not been taken into account heretofore, provides an important scale that affects the constitutive relations. Some (minor) quantitative differences between our results and previous ones exist as well. These are due to the fact that we take into account an isotropic correction to the leading Maxwellian distribution, which has not been considered before, and also because we consider the full dependence of the corrections to the Maxwellian distribution on the (fluctuating) speed.

Physical analysis of the transition to turbulence in the wake of a circular cylinder by three-dimensional Navier–Stokes simulation

Abstract: The transition to turbulence of the flow around a circular cylinder is studied by a three-dimensional numerical simulation of the Navier–Stokes equations system in the Reynolds number range 100–300. The numerical method is second-order accurate in space and time and Neumann boundary conditions are used at the two boundaries in the spanwise direction; non-reflecting boundary conditions are specified for the outlet downstream boundary. This study predicts the frequency modulation and the formation of a discontinuity region delimited by two frequency steps within the present Reynolds number range. These features are related to the birth of streamwise vorticity and to the kinetic energy distribution in the near wake. The development of the mean dynamic quantities, the Reynolds stress correlations and the variation of their maximum values are provided in this region, where the similarity laws do not hold. The spatial evolution of the von Karman mode and of its spectral amplitude are quantified and the variation laws of the maximum spectral amplitude and of its location as a function of Reynolds number are established. The critical Reynolds number for the appearance of the first discontinuity in the present flow system is evaluated by the fully nonlinear approach.

The initial stages of dam-break flow

Abstract: Experiments have been undertaken to investigate dam-break flows where a thin plate separating water at different levels is withdrawn impulsively in a vertically upwards direction. Depth ratios of 0, 0.1 and 0.45 were investigated for two larger depth values of 10 cm and 36 cm. The resulting sequence of surface profiles is shown to satisfy approximately Froude scaling. For the dry-bed case a horizontal jet forms at small times and for the other cases a vertical, mushroom-like jet occurs, none of which have been observed previously. We analyse the initial-release problem in which the plate is instantaneously removed or dissolved. Although this shows singular behaviour, jet-like formations are predicted. Artificially smoothing out the singularity enables a fully nonlinear, potential-flow computation to follow the jet formation for small times. There is qualitative agreement between theory and experiment.In the experiments, after a bore has developed downstream as a result of highly complex flow interactions, the surface profiles agree remarkably well with exact solutions of the shallow-water equations which assume hydrostatic pressure and uniform velocity over depth.

Observations of shock waves in cloud cavitation

Abstract: This paper describes an investigation of the dynamics and acoustics of cloud cavitation, the structures which are often formed by the periodic breakup and collapse of a sheet or vortex cavity. This form of cavitation frequently causes severe noise and damage, though the precise mechanism responsible for the enhancement of these adverse effects is not fully understood. In this paper, we investigate the large impulsive surface pressures generated by this type of cavitation and correlate these with the images from high-speed motion pictures. This reveals that several types of propagating structures (shock waves) are formed in a collapsing cloud and dictate the dynamics and acoustics of collapse. One type of shock wave structure is associated with the coherent collapse of a well-defined and separate cloud when it is convected into a region of higher pressure. This type of global structure causes the largest impulsive pressures and radiated noise. But two other types of structure, termed 'crescent-shaped regions' and 'leading-edge structures' occur during the less-coherent collapse of clouds. These local events are smaller and therefore produce less radiated noise but the interior pressure pulse magnitudes are almost as large as those produced by the global events. The ubiquity and severity of these propagating shock wave structures provides a new perspective on the mechanisms reponsible for noise and damage in cavitating flows involving clouds of bubbles. It would appear that shock wave dynamics rather than the collapse dynamics of single bubbles determine the damage and noise in many cavitating flows.

Local energy flux and subgrid-scale statistics in three-dimensional turbulence

Abstract: Statistical properties of the subgrid-scale stress tensor, the local energy flux and filtered velocity gradients are analysed in numerical simulations of forced three-dimensional homogeneous turbulence. High Reynolds numbers are achieved by using hyperviscous dissipation. It is found that in the inertial range the subgrid-scale stress tensor and the local energy flux allow simple parametrization based on a tensor eddy viscosity. This parametrization underlines the role that negative skewness of filtered velocity gradients plays in the local energy transfer. It is found that the local energy flux only weakly correlates with the locally averaged energy dissipation rate. This fact reflects basic difficulties of large-eddy simulations of turbulence, namely the possibility of predicting the locally averaged energy dissipation rate through inertial-range quantities such as the local energy flux is limited. Statistical properties of subgrid-scale velocity gradients are systematically studied in an attempt to reveal the mechanism of local energy transfer.

An experimental study of the motion of concentrated suspensions in two-dimensional channel flow. Part 2. Bidisperse systems

Abstract: In this paper we report experimental velocity and concentration profiles for suspensions possessing a bidisperse distribution of particle size undergoing pressure-driven flow through a parallel-wall channel. In addition to the overall concentration distributions determined by implementing the modified laser Doppler velocimetry method described in Part 1 (Lyon & Leal 1998), concentration profiles for the particles of each size were measured by sampling the position of marked tracer particles across 60% of the channel gap. Non-uniform overall particle concentration distributions and blunted velocity profiles were found at bulk particle volume fractions of 0.30 and 0.40, which were equal to the monodisperse data of Part 1, within experimental uncertainty. The large-particle concentration profiles were non-uniform down to a large-particle bulk volume fraction of 0.075, while non-uniform distributions of the small particles were only found when the volume fraction of small particles in the bulk was greater than or equal to 0.20. Experiments in which at least half the suspended particulate volume was occupied by large particles revealed enrichment of the large particles in the centreline region of the channel. This size segregation was found to increase as the total number of suspended particles decreased. Finally, the data from experiments in which a uniform small-particle concentration profile was measured were compared with suspension balance model (McTigue & Jenkins 1992; Nott & Brady 1994) predictions for parameter values that corresponded only to the large particles. While close agreement with the large-particle concentration profiles was found, this comparison also reflected the fact that the small particles bring the suspension viscosity to a regime that is more sensitive to the particle concentration, rather than simply providing an increment in background viscosity to the suspending liquid.

Experimental study of the fine-scale structure of conserved scalar mixing in turbulent shear flows. Part 2. Sc 1

Abstract: We present results from an experimental investigation into the fine-scale structure associated with the mixing of a dynamically passive conserved scalar quantity on the inner scales of turbulent shear flows. The present study was based on highly resolved two- and three-dimensional spatio-temporal imaging measurements. For the conditions studied, the Schmidt number (Sc ≡ v/D) was approximately 2000 and the local outerscale Reynolds number (Reσ ≡ uσ/v) ranged from 2000 to 10000. The resolution and signal quality allow direct differentiation of the measured scalar field ζ(x, t) to give the instantaneous scalar energy dissipation rate field (Re Sc)−1 ∇ζċ∇ζ(x, t). The results show that the fine-scale structure of the scalar dissipation field, when viewed on the inner-flow scales for Sc ≡ 1, consists entirely of thin strained laminar sheet-like diffusion layers. The internal structure of these scalar dissipation sheets agrees with the one-dimensional self-similar solution for the local strain–diffusion competition in the presence of a spatially uniform but time-varying strain rate field. This similarity solution also shows that line-like structures in the scalar dissipation field decay exponentially in time, while in the vorticity field both line-like and sheet-like structures can be sustained. This sheet-like structure produces a high level of intermittency in the scalar dissipation field – at these conditions approximately 4% of the flow volume accounts for nearly 25% of the total mixing achieved. The scalar gradient vector field ∇ζ(x, t) for large Sc is found to be nearly isotropic, with a weak tendency for the dissipation sheets to align with the principal axes of the mean flow strain rate tensor. Joint probability densities of the conserved scalar and scalar dissipation rate have a shape consistent with this canonical layer-like fine-scale structure. Statistics of the conserved scalar and scalar dissipation rate fields are found to demonstrate similarity on inner-scale variables even at the relatively low Reynolds numbers investigated.