Reliable prediction of the erosion rate of sediment beds is important for many applications in coastal and river engineering. Theoretical understanding of empirically derived scaling relations is still lacking. This applies in particular for the scaling anomaly between low and high Shields number conditions. In this work, the erosion process is studied from the perspective of the phase-averaged turbulent kinetic energy (TKE) equations. The multi-phase TKE equations are written in a form that allows for a direct comparison with the TKE equation that appears for a stratified single-phase flow under the Boussinesq approximation. This reveals that next to buoyancy destruction, several other TKE modulation mechanisms become important at high Shields numbers and concentrations. Two scaling laws are derived for both moderate and high Shields numbers, and are tested against a wide range of experimental data.@en

Particle-resolved Direct Numerical Simulations have been performed on the gravitational settling of mono-disperse solid spheres in a viscous fluid and triply periodic domain. In a comprehensive study, the bulk solid volume concentration was varied from ϕ=0.5 to 30%. To study the effect of inertia, three different Galileo numbers were considered in the inertial regime, Ga=144, 178 and 210, for which a single settling sphere exhibits distinctly different wake and path characteristics. The particle/fluid mass density ratio was fixed at 1.5. We find that for ϕ=2−30% the suspension microstructure and dynamics depend predominantly on the bulk concentration. In qualitative agreement with previous studies in literature, three different sedimentation regimes can be distinguished: (1) the dilute concentration regime for ϕ≲2% with preferential settling of particles in vertical trains, (2) the moderate concentration regime for 2%≲ϕ≲10% with preferential settling of particles in horizontal pairs with an interparticle distance of ∼ 1.5 particle diameters, and (3) the dense concentration regime for ϕ≳10% with a nearly random (“hard-sphere”) distribution of the particles in space. The clustering of particles is dictated by, respectively, trapping of particles in the wake of other particles, a drafting–kissing–tumbling (DKT) instability by which two vertically aligned particles quickly reorient themselves into a horizontally aligned particle pair, and short-range multiparticle interactions through viscous lubrication and to a lesser extent collisions between particles. In all cases, hindered settling at a reduced speed is observed as compared to a single settling sphere. The well-known Richardson–Zaki relation for the mean sedimentation velocity appears valid only for the dense concentration regime. We provide ample evidence that in the dense regime the characteristic velocity and time scales of particle motion are proportional to gD_p and D_p/g, respectively, with g the gravitational acceleration and D_p the particle diameter. We also observe an ω⁻³ scaling of the particle velocity spectra for ωD_p/g≳0.4 and we propose a model to explain this scaling behavior, based on the inertial response of the particles to small-scale flow perturbations. Kinematic waves, i.e., vertically propagating plane waves in the local concentration field, are observed in all cases, though unrelated particle motions are responsible for significant loss of the spatio-temporal coherence of the waves. The wave speed was determined from repeated space–time autocorrelations of the local concentration field and appears in reasonable agreement with Kynch sedimentation theory using the Richardson–Zaki relation. The passage of kinematic waves causes perturbations in the particle velocity at a frequency that matches well with peak frequencies in the particle velocity spectra for concentrations up to ϕ≈10%. The time-lagged cross-correlation of the vertical and horizontal particle velocity suggests that kinematic waves may trigger DKT instabilities, while conversely DKT instabilities may be responsible for the onset of kinematic waves. Finally, we suggest that obstruction and perturbation of the particle wake by neighboring particles could offer an explanation for the small influence of the Galileo number on the suspension behavior for ϕ=2−30%.

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We experimentally and numerically study the dynamics of a liquid jet issued from a rotating orifice, whose breakup is regulated by a vibrating piezo element. The helical trajectory of the spiralling jet yields fictitious forces varying along the jet whose longitudinal projections stretch and thin the jet, affecting the growth of perturbations. We show that by quantifying these fictitious forces, one can estimate the jet intact length and size distribution of drops formed at jet breakup. The presence of the locally varying fictitious forces may render high-frequency perturbations, that would otherwise be stable in the abscence of stretching, unstable, as observed similarly in the case of straight jets stretching under gravity. The perturbation amplitude then dictates how strong the perturbation is coupled to the jet compared with random noise that is inherently present in any experimental set-up. In the present study we exploit the slenderness of the jet to separate the calculation of the base flow and the growth of perturbations. The fictitious forces calculated from the base flow trajectory are then used in a nonlinear slender-jet model, which treats the spiralling jet as a quasi-straight jet with locally varying body forces. We show both experimentally and numerically that jet breakup characteristics (e.g. intact length and drop size distribution) can be controlled by finite-amplitude perturbations created by mechanically induced pressure modulations. Finally, we revisit the integrated net gain approach developed for straight jets under gravity and we provide simple analogous relations for spiralling jets.@en

Magnetic resonance imaging (MRI) experiments have been performed in conjunction with direct numerical simulations (DNS) to study neutrally buoyant particle-laden pipe flows. The flows are characterized by the suspension liquid Reynolds number (Res), based on the bulk liquid velocity and suspension viscosity obtained from Eilers' correlation, the bulk solid volume fraction (φb), and the particle-to-pipe diameter ratio (d/D). Six different cases have been studied, each with a unique combination of Res and φ, while d/D is kept constant at 0.058. The selected cases ensure that the comparison is performed across different flow regimes, each exhibiting characteristic behavior. In general, an excellent agreement is found between experiment and simulation for the average liquid velocity and solid volume fraction profiles. Root-mean-square errors as low as 1.7% and 5.3% are found for the velocity and volume fraction profiles, respectively. This study presents accurate and quantitative velocity and volume fraction profiles of semidilute up to dense suspension flows using both experimental and numerical methods. Three different flow regimes are identified, based on the experimental and numerical solid volume fraction profiles. These profiles explain observations in the drag change. For low bulk solid volume fractions a drag increase (with respect to an equal Res single-phase case) is observed. For moderate volume fraction distributions the drag is found to decrease, due to particle accumulation at the pipe center. For high volume fractions the drag is found to decrease further. For solid volume fractions of 0.4 a drag reduction higher than 25% is found. This drag reduction is linked to the strong viscosity gradient in the radial direction, where the relatively low viscosity near the pipe wall acts as a lubrication layer between the pipe wall and the dense core.

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Path instabilities of a sphere rising or falling in a quiescent Newtonian fluid have been studied experimentally. The rich palette of possible instabilities is dependent upon two dimensionless quantities, namely the Galileo number (Ga) and the particle/fluid mass density ratio (ρ¯). In recent literature, several (Ga,ρ¯) regime maps have been proposed to characterize path instabilities, based on both numerical and experimental studies, with substantial disagreements among them. The present study attempts to shed light on path instabilities for which previous studies disagree. A detailed experimental investigation has been conducted for 219 different combinations of Ga and ρ¯, grouped around four values of ρ¯ (∼ 0.87, 1.12, 3.19 and 3.9) and Ga in the range of ∼ 100 to 700. Our results agree well with literature for the low Ga range in which a particle takes a steady vertical or steady oblique path and for which all previous studies agree with each other. For the higher and more controversial Ga range, we discuss consensus and disagreements with previous studies. Some regimes, which were only recently observed in numerical simulations, have been observed experimentally for the first time. Also, intriguing bi-stable regimes (i.e., coexistence of two stable asymptotic states) have been observed. For all four investigated density ratios, an update of the regime map is proposed. Finally, for both the rising and falling spheres, the drag coefficient as function of terminal settling Reynolds number has been determined, which for the investigated density ratios does not differ significantly from that of flow past a fixed sphere.

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When two spherical particles submerged in a viscous fluid are subjected to an oscillatory flow, they align themselves perpendicular to the direction of the flow leaving a small gap between them. The formation of this compact structure is attributed to a nonzero residual flow known as steady streaming. We have performed direct numerical simulations of a fully resolved, oscillating flow in which the pair of particles is modeled using an immersed boundary method. Our simulations show that the particles oscillate both parallel and perpendicular to the oscillating flow in elongated figure-8 trajectories. In absence of bottom friction, the mean gap between the particles depends only on the normalized Stokes boundary layer thickness δ∗, and on the normalized, streamwise excursion length of the particles relative to the fluid Ar∗ (equivalent to the Keulegan-Carpenter number). For Ar∗≲1, viscous effects dominate and the mean particle separation only depends on δ∗. For larger Ar∗ values, advection becomes important and the gap widens. Overall, the normalized mean gap between the particles scales as L∗≈3.0δ∗1.5+0.03Ar∗3, which also agrees well with previous experimental results. The two regimes are also observed in the magnitude of the oscillations of the gap perpendicular to the flow, which increases in the viscous regime and decreases in the advective regime. When bottom friction is considered, particle rotation increases and the gap widens. Our results stress the importance of simulating the particle motion with all its degrees of freedom to accurately model the system and reproduce experimental results. The insights of the particle pairs provide an important step towards understanding denser and more complex systems.

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The alignment of a pair of spherical particles perpendicular to a horizontally oscillating flow is attributed to a non-zero residual flow, known as steady streaming. This phenomenon is the basis of complex patterns in denser systems, such as particle chains and the initial stages of rolling-grain ripples. Previous studies on such self-organization processes used two distinct systems: an oscillating box filled with viscous fluid and an oscillating channel flow, where the fluid oscillates relative to the bottom boundary. In this paper, we show that particle pair dynamics in these two systems are fundamentally different, due to the presence of a Stokes boundary layer above the bottom in the oscillating channel flow. The results are obtained from direct numerical simulations in which the dynamics of a pair of particles are simulated using an immersed boundary method. The oscillating box and the oscillating channel flow are only equivalent in a limited region of the parameter space, where both the normalized Stokes boundary layer thickness and the normalized relative particle excursion length are small. Overall, the particle dynamics in the oscillating channel flow, compared to the oscillating box, are governed by an additional dimensionless parameter, that is, the particle–fluid density ratio.@en

Using a slender-jet approach, we numerically investigate the control of jet breakup using temperature modulation at the nozzle with a specified frequency and amplitude. Our results show that temperature modulation does lead to instability through capillary and Marangoni stresses, providing control of the droplet formation in terms of intact length and resultant drop size distribution, which is otherwise irregular due to inevitable presence of background noise. For understanding the mechanisms underpinning the breakup of a thermally modulated jet in the presence of noise, it is useful to decompose the surface tension forces into a contribution from curvature-gradient forces and a contribution from surface tension-gradient forces, associated with axial variations in the jet curvature and the temperature-dependent surface tension coefficient, respectively. We show that in the limit of slow axial heat diffusion and slow cooling to the ambient, as considered here, the breakup of a thermally modulated jet is governed by the ratio of the surface tension-gradient force to the imposed random perturbation force at nozzle exit. This so-called “thermal modulation strength number” depends on the amplitude and frequency of the thermal modulation, the sensitivity of the surface tension coefficient to variations in temperature, the Weber number, and the strength of the Gaussian white noise added to the nozzle exit velocity. We show that the thermal modulation strength number governs the shift in breakup characteristics from forward to rear pinchoff for increasing modulation strength as well as the nature of the instability. When thermal modulation is weak, the surface tension-gradient forces act only as a trigger, and curvature-gradient forces soon take over and grow exponentially downstream from the jet due to inertio-capillary growth. When thermal modulation is strong, the surface tension-gradient forces not only act as a trigger, but remain significant until breakup. The thermal modulation strength number is thus useful to the design of thermal modulation in practical applications as a possible alternative to often-used mechanical excitation mechanisms to control jet breakup.

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Natural particles are frequently applied in drinking water treatment processes in fixed bed reactors, fluidised bed reactors, and sedimentation processes to clarify water and to concentrate solids. When particles settle, it has been found that, in terms of hydraulics, natural particles behave differently when compared to perfectly round spheres. To estimate the terminal settling velocity of single solid particles in a liquid system, a comprehensive collection of equations is available. For perfectly round spheres, the settling velocity can be calculated quite accurately. However, for naturally polydisperse non-spherical particles, experimentally measured settling velocities of individual particles show considerable spread from the calculated average values.

This work aims to analyse and explain the different causes of this spread. To this end, terminal settling experiments were conducted in a quiescent fluid with particles varying in density, size, and shape. For the settling experiments, opaque and transparent spherical polydisperse and monodisperse glass beads were selected. In this study, we also examined drinking-water-related particles, like calcite pellets and crushed calcite seeding material grains, which are both applied in drinking water softening. Polydisperse calcite pellets were sieved and separated to acquire more uniformly dispersed samples. In addition, a wide variety of grains with different densities, sizes, and shapes were investigated for their terminal settling velocity and behaviour. The derived drag coefficient was compared with well-known models such as the one of Brown and Lawler (2003).

A sensitivity analysis showed that the spread is caused, to a lesser extent, by variations in fluid properties, measurement errors, and wall effects. Natural variations in specific particle density, path trajectory instabilities, and distinctive multi-particle settling behaviour caused a slightly larger degree of the spread. In contrast, a greater spread is caused by variations in particle size, shape, and orientation.

In terms of robust process designs and adequate process optimisation for fluidisation and sedimentation of natural granules, it is therefore crucial to take into consideration the influence of the natural variations in the settling velocity when using predictive models of round spheres.@en

The effect of drag reducing riblets on the flow structure was examined experimentally for a turbulent boundary layer at Re_θ = 9890 and riblet spacing s⁺ = 13.4. Trapezoidal riblets were used, which were attached to the water tunnel wall as a coating. Force measurements were performed to quantify the amount of drag reduction. Then, the mechanism underlying this reduction was investigated by stereo-PIV measurements in the cross-stream plane. To determine the effect of the drag reducing riblets, the results were compared with the smooth flat plate. Time-averaged turbulent statistics such as turbulent kinetic energy and Reynolds shear stress were found to be lower over the riblets compared to the flat surface. Two-point correlations of the fluctuating velocity components were calculated to elucidate the average flow structure size and strength, where riblets significantly suppressed the turbulent structures. Quadrant analysis of the Reynolds shear stress was performed to assess the change in ejection and sweep events and the results were found to be in correspondence with previous works.

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Liquid-solid fluidisation is frequently encountered in drinking water treatment processes, often to obtain a large liquid-solid interfacial surface area. A large surface area is crucial for optimal seeded crystallisation in full-scale softening reactors. Due to crystallisation, particles grow and migrate to a lower zone in the reactor which leads to a stratified bed. Larger particles adversely affect the surface area. To maintain optimal process conditions in the fluidised beds, information is needed about the distribution of particle size, local voidage and available surface area, over the reactor height.

In this work, a sensor is developed to obtain the hydraulic state gradient, based on Archimedes’ principle. A cylindrical heavy object is submerged in the fluidised bed and lowered gradually while its weight is measured at various heights using a sensitive force measuring device.

Based on accurate fluidisation experiments with calcite grains, the voidage is determined and a straightforward empirical model is developed to estimate the particle size as a function of superficial fluid velocity, kinematic viscosity, suspension density, voidage and particle density. The surface area and specific space velocity can be estimated accordingly, which represent key performance indicators regarding the hydraulic state of the fluidised bed reactor. The prediction error for voidage is 5 ± 2 % and for particle size 9 ± 4 %.

The newly developed soft sensor is a more time-effective method for obtaining the hydraulic state in full-scale liquid-solid fluidised bed reactors.@en

We use direct numerical simulations (DNS) to investigate the turbulent modulation due to the presence of bubbles in vertical channels flowing downward. The Reynolds number for single-phase flow based on half channel height h^* and friction velocity is Re_τ= 180. A density and viscosity ratio of ρ_d^*/ρ_c^*=0.01 and μ_d^*/μ_c^*=0.018 is chosen for two void fractions of ϵ=1.2% and ϵ=2.4%. For each void fraction three different bubble sizes are simulated: D/h=0.2130, 0.2684 and 0.3382, where D denotes the diameter of the bubbles. Numerical simulations are based on multiple markers Coupled Level-Set/Volume-of-Fluid (CLSVOF) method. To improve the efficiency of this method, a fast pressure-correction method is used in order to enable the simulation to exploit a constant coefficient Poisson equation which can be solved with FFT-based technique. Extensive verification and validation were performed and perfect accuracy and agreement are obtained. In all the simulations performed in this work, the new Poisson solver showed a minimum speedup of 22 times. Accumulation of bubbles in the core region of the channel for all cases is observed, which forms a bubble-free layer in the near-wall region. The presence of bubbles resulted in considerable modification in the mean velocity profile compared to single-phase flow. Another common observation is that all the components of velocity fluctuations in the near-wall region decrease with increasing void fraction and decreasing wall layer thickness. The opposite happens in the core region, where the presence of bubbles favours turbulence. With respect to the bubble size, the wall-normal and spanwise velocity fluctuations decrease in the near-wall region for smaller bubbles, however, the streamwise velocity fluctuations remained almost unaffected. The investigation of turbulent kinetic budgets shows that, unlike single-phase flow, the dissipation terms rises to large values in the core region of the channel. This behaviour is referred to the presence of bubbles and hence enhancement of turbulent kinetic energy in the core region.

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The purpose of this work is to experimentally establish the combined influence on the flow and thermal resistance of an exhaust pipe wall formed by a porous, compliant layer with overlying discrete roughness elements exposed to the pulsating exhaust gas flow of a combustion engine. Through measuring the streamwise pressure drop over and radial temperature differences in different pipe samples for a range of flow states with different Reynolds numbers and non-dimensional pulsation frequencies, the effects were discerned. The configurations of the sample walls covered a range of mesh pitches, compliant-layer densities, and compliant-layer compression ratios. The (non-sinusoidally) pulsating exhaust gas flow spanned the following range: Re_b (= u_bD/ν_b) = 1⋅ 10⁴ - 3⋅ 10⁴, T_b = 500 - 800 ^∘C, ω⁺(= ωνb/uτ2) = 0.003 - 0.040. The friction factors were found to be effectively constant with Reynolds number and non-dimensional pulsation frequency while the variation with insulation density/compression was not significant. Additionally, for both mesh pitches, the measured friction factors were in line with those reported in literature for similar geometries with steady flow and solid walls. Together this indicates that neither compliance nor the pulsations in the exhaust gas flow significantly affect the friction for this configuration. Comparison of the samples based on the derived thermal resistance showed a similar influence of the fluid-wall interface as for the friction. Additionally a distinct influence of compression, independent of the insulation density, was observed that increases with increasing temperature. It was concluded that the increased resistance was due to additional radiation resistance because of fibre reorientation due to compression.

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Macroscopic properties of sedimenting suspensions have been studied extensively and can be characterized using the Galileo number (Ga), solid-to-fluid density ratio (π_p) and mean solid volume concentration (ϕ¯). However, the particle–particle and particle–fluid interactions that dictate these macroscopic trends have been challenging to study. We examine the effect of concentration on the structure and dynamics of sedimenting suspensions by performing direct numerical simulation based on an Immersed Boundary Method of monodisperse sedimenting suspensions of spherical particles at fixed Ga= 144 , π_p= 1.5 , and concentrations ranging from ϕ¯ = 0.5 to ϕ¯ = 30 %. The corresponding particle terminal Reynolds number for a single settling particle is Re_T= 186. Our simulations reproduce the macroscopic trends observed in experiments and are in good agreement with semi-empirical correlations in literature. From our studies, we observe, first, a change in trend in the mean settling velocities, the dispersive time scales and the structural arrangement of particles in the sedimenting suspension at different concentrations, indicating a gradual transition from a dilute regime (ϕ¯ ≲ 2 %) to a dense regime (ϕ¯ ≳ 10 %). Second, we observe the vertical propagation of kinematic waves as fluctuations in the local horizontally-averaged concentration of the sedimenting suspension in the dense regime.

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During the last decade, many approaches for resolved-particle simulation (RPS) have been developed for numerical studies of finite-size particle-laden turbulent flows. In this paper, three RPS approaches are compared for a particle-laden decaying turbulence case. These methods are, the Volume-of-Fluid Lagrangian method, based on the viscosity penalty method (VoF-Lag); a direct forcing Immersed Boundary Method, based on a regularized delta function approach for the fluid/solid coupling (IBM); and the Bounce Back scheme developed for Lattice Boltzmann method (LBM-BB). The physics and the numerical performances of the methods are analyzed. Modulation of turbulence is observed for all the methods, with a faster decay of turbulent kinetic energy compared to the single-phase case. Lagrangian particle statistics, such as the velocity probability density function and the velocity autocorrelation function, show minor differences among the three methods. However, major differences between the codes are observed in the evolution of the particle kinetic energy. These differences are related to the treatment of the initial condition when the particles are inserted in an initially single-phase turbulence. The averaged particle/fluid slip velocity is also analyzed, showing similar behavior as compared to the results referred in the literature. The computational performances of the different methods differ significantly. The VoF-Lag method appears to be computationally most expensive. Indeed, this method is not adapted to turbulent cases. The IBM and LBM-BB implementations show very good scaling.

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The present paper is focused on the development of an accurate 1D numerical model for pig motion in two-phase flow. The focus will be on the liquid slug that is accumulated in front of the pig, the so-called pig-generated slug. Under the assumption of a stratified flow, we first discuss the academic case of liquid slug accumulation where we neglect the viscosity of the fluids. The size of the liquid slug will then effectively be determined by the speed of the hydrostatic wave which runs ahead of the pig. We also consider the more realistic case which includes the viscosity of the fluid. Finally, we discuss the effect of the presence of a by-pass in the pig on the accumulated liquid slug.@en

We investigate the deformation of a linear viscoelastic compliant coating in a turbulent flow for a wide range of coating parameters. A one-way coupling model is proposed in which the turbulent surface stresses are expressed as a sum of streamwise-travelling waves with amplitudes determined from the stress spectra of the corresponding flow over a rigid wall. The analytically calculated coating deformation is analysed in terms of the root-mean-square (r.m.s.) surface displacement and the corresponding point frequency spectra. The present study systematically investigates the influence of five coating properties namely density, stiffness, thickness, viscoelasticity and compressibility. The surface displacements increase linearly with the fluid/solid density ratio. They are linearly proportional to the coating thickness for thin coatings, while they become independent of the thickness for thick coatings. Very soft coatings show resonant behaviour, but the displacement for stiffer coatings is proportional to the inverse of the shear modulus. The viscoelastic loss angle has only a significant influence when resonances occur in the coating response, while Poisson's ratio has a minor effect for most cases. The modelled surface displacement is qualitatively compared with recent measurements on the deformation of three different coatings in a turbulent boundary-layer flow. The model predicts the order of magnitude of the surface displacement, and it captures the increase of the coating displacement with the Reynolds number and the coating softness. Finally, we propose a scaling that collapses all the experimental data for the r.m.s. of the vertical surface displacement onto a single curve.

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