Turbulent transport of particles in a straight square duct

Turbulent flow through a duct of square cross-section gives rise to off-axis secondary flows, which are known to transfer momentum between fluid layers thereby flattening the velocity profile. The aim of this study is to investigate the role of the secondary flows in the transport and dispersion of particles suspended in a turbulent square duct flow. We have numerically simulated a flow through a square duct having a Reynolds number of Re_τ = 300 through discretization of the Navier–Stokes equations, and followed the trajectories of a large number of passive tracers and finite-inertia particles under a one-way coupling assumption. Snapshots of particle locations and statistics of single-particle and particle pair dispersion were analyzed. It was found that lateral mixing is enhanced for passive tracers and low-inertia particles due to the lateral advective transport that is absent in straight pipe and channels flows. Higher inertia particles accumulate close to the wall, and thus tend to mix more efficiently in the streamwise direction since a large number of the particles spend more time in a region where the mean fluid velocity is small compared to the bulk. Passive tracers tend to remain within the secondary swirling flows, circulating between the core and boundary of the duct.     

Kinetic modeling of particles in stratified flow – Evaluation of dispersion tensors in inhomogeneous turbulence

The continuum equations for a dilute particle distribution in inhomogeneous turbulence are tested against results from a Langevin particle tracking simulation. Reeks’ version of the kinetic theory is used to generate the mass, momentum and kinetic stress equations for the particle distribution. The particle tracking data are used to directly evaluate the dispersion tensors λ and μ which serve as closure relations for the continuum equations. These exact forms are compared to approximate, local forms. Even for low Stokes numbers (corresponding to low particle inertia defined by τ/τ_p ? 1), the tensor λ is strongly affected by the inhomogeneity and depends on turbulence parameters in the volume corresponding to the particle path dispersion over the particle Lagrangian integral timescale τ. In contrast, the locally homogeneous form of the velocity dispersion tensor μ is a sufficient approximation, since it depends on the dispersion volume over the much smaller particle relaxation time τ_p. It is demonstrated that the body force due to the dispersion vector γ cannot be neglected. In the limit of passive tracers (zero stopping distance), γ is equal to the gradient of λ, if the physical setting is such that we can invoke constant tracer density in this limit.     

Tensile-shear transition in mixed mode I/III fracture

The propensity of the transition of fracture type in either brittle or ductile cracked solid under mixed-mode I and III loading conditions is investigated. A fracture criterion based on the competition of the maximum normal stress and maximum shear stress is utilized. The prediction of the fracture type is determined by comparing τ_max/σ_max at a critical distance from the crack tip to the material strength ratio τ_C/σ_C, i.e., (τ_max/σ_max)<(τ_C/σ_C) for tensile fracture and (τ_max/σ_max)>(τ_C/σ_C) for shear fracture, where σ_C (τ_C) is the fracture strength of materials in tension (shear). Mixed mode I/III fracture tests were performed using circumferentially notched cylindrical bars made of PMMA and 7050 aluminum alloy. Fracture surface morphology of the specimens reveals that: (1) for the brittle material, PMMA, only tensile type of fracture occurs, and (2) for the ductile material, 7050 aluminum alloy, either tensile or shear type of fracture occurs depending on the mode mixity. The transition (in ductile material) or non-transition (in brittle material) of the fracture type and the fracture path observed in experiments were properly predicted by the theory. Additional test data from open literature are also included to validate the proposed theory.     

A model for tracking inertial particles in a lattice Boltzmann turbulent flow simulation

A computationally inexpensive model for tracking inertial particles through a turbulent flow is presented and applied to the turbulent flow through a square duct having a friction Reynolds number of Re_τ = 300. Prior to introducing particles into the model, the flow is simulated using a lattice Boltzmann computation, which is allowed to evolve until a steady state turbulent flow is achieved. A snapshot of the flow is then stored, and the trajectories of particles are computed through the flow domain under the influence of this static probability field. Although the flow is not computationally evolving during the particle tracking simulation, the local velocity is obtained stochastically from the local probability function, thus allowing the dynamics of the turbulent flow to be resolved from the point of view of the suspended particles. Particle inertia is modeled by using a relaxation parameter based on the particle Stokes number that allows for a particle velocity history to be incorporated during each time step. Wall deposition rates and deposition patterns are obtained and exhibit a high level of agreement with previously obtained DNS computational results and experimental results for a wide range of particle inertia. These results suggest that accurate particle tracking through complex turbulent flows may be feasible given a suitable probability field, such as one obtained from a lattice Boltzmann simulation. This in turn presents a new paradigm for the rapid acquisition of particle transport statistics without the need for concurrent computations of fluid flow evolution.     

Lagrangian velocity and acceleration statistics of fluid and inertial particles measured in pipe flow with 3D particle tracking velocimetry

Three-dimensional particle tracking velocimetry (3D-PTV) has been applied to particle-laden pipe flow at Reynolds number 10,300, based on the bulk velocity and the pipe diameter. The volume fraction of the inertial particles was equal to 1.4 × 10⁻⁵. Lagrangian velocity and acceleration statistics were determined both for tracers and for inertial particles with Stokes number equal to 2.3, based on the particle relaxation time and the viscous time scale. The decay of Lagrangian velocity and acceleration correlation functions was measured both for the fluid and for the dispersed phase at various radial positions. The decay of Lagrangian velocity correlations is faster for inertial particles than for flow tracers, whereas the decay of Lagrangian acceleration correlations is about 25% slower for inertial particles than for flow tracers. Further differences between inertial and tracer particles are found in velocity fluctuations evaluated for both positive and negative time lags. The asymmetry in time of velocity cross-correlations is more pronounced for inertial particles. Quadrant analysis revealed another difference still near the wall: ejection and sweep events are less frequent for inertial particles than for tracers.     

Numerical study of particle deposition in a turbulent channel flow with transverse roughness elements on one wall

A numerical study is presented for the effect of wall roughness on the deposition of solid spherical particles in a fully developed turbulent channel flow based on large eddy simulation combined with a Lagrangian particle-tracking scheme. The interest is focused on particles with response times in wall units in the range of 2.5 ≤ τ_p⁺ ≤ 600 depositing onto a vertical rough surface consisting of two-dimensional transverse square bars separated by a rectangular cavity. Predictions of particle deposition rates are obtained for several values of the cavity width to roughness element height ratio and particle response time. It is shown that the accumulation of particles in the near wall region and their preferential concentration in flow areas of low streamwise fluid velocity that occur in turbulent flows at flat channels are significantly affected by the roughness elements. Particle deposition onto the rough wall is considerably increased, exhibiting a subtle dependence on the particle inertia and the spacing between the bars. The observed augmentation of deposition coefficient can be attributed to the flow modifications induced by the roughness elements and to the inertial impaction of particles onto the frontal deposition area of the protruding square bars.     

Columnar heat transport via advection induced by inertial waves

Inertial waves are oscillations in a rotating fluid that arise due to the restoring action of the Coriolis force. Low-frequency inertial waves are known to create columnar flow structures inrapidly rotating systems. Columnar heat transport away from the equator has been observed in some strongly forced, rapidly-rotating geodynamo simulations of the Earth’s core. In this study, we investigate the mechanism governing this heat transport by performing direct numerical simulations of model problems comprising buoyant blobs under rapid rotation in a periodic box. We consider a wide range of Rossby numbers (Ro), the ratio of advection to Coriolis force, and Peclet numbers (Pe), the ratio of thermal advection to thermal diffusion. Columnar flow structures, that comprise inertial wave packets, are observed to emerge from the buoyant regions and travel towards the box boundary. We find that the columnar heat transport occurs by advection governed by the local Pe (for instance, a larger vertical elongation in the blob is observed for larger Pe at the same Ro). The magnitude of the advection velocity is determined by the balance between the buoyancy and Coriolis forces. Moreover, the direction of advection is determined by the direction of the wave-induced flow in the columns above and below the blob. Our results suggest that the local Pe could be important for the columnar heat transport in strongly forced dynamo simulations.     

Turbulent Couette–Poiseuille flow with zero wall shear

A particular pressure-driven flow in a plane channel is considered, in which one of the walls moves with a constant speed that makes the mean shear rate and the friction at the moving wall vanish. The Reynolds number considered based on the friction velocity at the stationary wall (u_τ,S) and half the channel height (h) is Re_τ,S = 180. The resulting mean velocity increases monotonically from the stationary to the moving wall and exhibits a substantial logarithmic region. Conventional near-wall streaks are observed only near the stationary wall, whereas the turbulence in the vicinity of the shear-free moving wall is qualitatively different from typical near-wall turbulence. Large-scale-structures (LSS) dominate in the center region and their spanwise spacing increases almost linearly from about 2.3 to 4.2 channel half-heights at this Re_τ,S. The presence of LSS adds to the transport of turbulent kinetic energy from the core region towards the moving wall where the energy production is negligible. Energy is supplied to this particular flow only by the driving pressure gradient and the wall motion enhances this energy input from the mean flow. About half of the supplied mechanical energy is directly lost by viscous dissipation whereas the other half is first converted from mean-flow energy to turbulent kinetic energy and thereafter dissipated.     

A stochastic Langevin model of turbulent particle dispersion in the presence of thermophoresis

Direct numerical simulation (DNS) and experimental data have shown that inertial particles exhibit concentration peaks in isothermal turbulent boundary layers, whereas tracer-like particles remain well mixed in the domain. It is therefore expected that the interactions between turbulence and thermophoresis will be strong in particle-laden flows where walls and carrier fluid are at significantly different temperatures. To capture turbulent particle dispersion with active thermophoresis, a coupled CFD-Lagrangian continuous random walk (CRW) model is developed. The model uses 3D mean flow velocities obtained from the Fluent 6.3 CFD code, to which are added turbulent fluid velocities derived from the normalized Langevin equation which accounts for turbulence inhomogeneities. The mean thermophoretic force is included as a body force on the particle following the Talbot formulation. Validation of the model is performed against recent integral thermophoretic deposition data in long pipes as well as the TUBA TT28 test with its detailed local deposition measurements. In all cases, the agreement with the data is very good. In separate parametric studies in a hypothetical cooled channel flow, it is found that turbulence strongly enhances thermophoretic deposition of particles with dimensionless relaxation times τ⁺ of order 1 or more. On the other hand, the thermophoretic deposition of very small inertia particles (τ⁺ < 0.2) in the asymptotic region far from the injection point tends to that which characterizes stagnant flow conditions, in agreement with the DNS results of Thakurta et al.     

NUMERICAL STUDIES OF TRANSITIONAL TURBULENT PULSATILE FLOW IN PIPES WITH RING–TYPE CONSTRICTIONS

Pulsatile flows in the vicinity of mechanical ring-type constrictions in pipes were studied for transitional turbulent flow with a Reynolds number (Re) of the order of 10⁴. The Womersley number (Nw) is in the range 30–50, with a corresponding Strouhal number (St) range of 0·0143–0·0398. The pulsatile flows considered are a pure sinusoidal flow, a physiological flow and an experimental pulsatile flow profile for mechanical aortic valve flow simulations. Transitional laminar and turbulent flow characteristics in an alternating manner within the pulsatile flow fields were studied numerically. It was observed that fluid accelerations tend to suppress the development of flow disturbances. All the instantaneous maximum values of turbulent kinetic energy, turbulent viscosity and turbulent shear stress are smaller during the acceleration phase than during the deceleration period. Various parametric equations have been formulated through numerical experimentation to better describe the relationships between the instantaneous flow rate (Q), the pressure loss (ΔP), the maximum velocity (V_max), the maximum vorticity (ζ_max), the maximum wall vorticity (ζ_w,max), the maximum shear stress (τ_max) and the maximum wall shear stress (τ_w,max) for turbulent pulsatile flow in the vicinity of constrictions in the vascular tube. An elliptic relationship has been found to exist between the instantaneous flow rate and the instantaneous pressure gradient. Other linear and quadratic relations between various flow parameters were also obtained.     

Liquid/liquid dispersion in a chaotic advection flow

Mixing by chaotic advection in a twisted-pipe flow is used here to investigate the efficiency of this flow in the liquid/liquid dispersion process. This study focuses on water/oil dispersions produced by continuous water injection into a main oil flow, for small Dean numbers. The drop sizes obtained with the chaotic-advection twisted-pipe flow are compared with those in a straight pipe and a helically coiled flow for the same conditions. It is found that the resulting dispersions are finer and more mono-dispersed in the chaotic advection flow. These results are compared with the theoretical maximum diameter _dmax$d_{\text{max}}$ determined by the Grace theory in which the viscous stress controls the breakup phenomena. For this purpose, the kinematic field is computed from the theoretical formulae for Dean flow. The strain rate fields in the pipe cross-section are then analytically computed and used to predict the maximum drop diameter. The theoretical values are identical for the three configurations (straight, helically coiled, and twisted pipe) up to a critical Dean number, where the secondary flow becomes significant. Beyond this value, the shear stress is enhanced in the twisted-pipe flow compared with the straight-pipe flow, and the predicted drop diameters are smaller. An interpretation of the higher dispersive performance of the chaotic flow is provided by the Lagrangian trajectories of the particles.     

Direct numerical simulation of supersonic pipe flow at moderate Reynolds number

We study compressible turbulent flow in a circular pipe at computationally high Reynolds number. Classical related issues are addressed and discussed in light of the DNS data, including validity of compressibility transformations, velocity/temperature relations, passive scalar statistics, and size of turbulent eddies. Regarding velocity statistics, we find that Huang’s transformation yields excellent universality of the scaled Reynolds stresses distributions, whereas the transformation proposed by Trettel and Larsson (2016) yields better representation of the effects of strong variation of density and viscosity occurring in the buffer layer on the mean velocity distribution. A clear logarithmic layer is recovered in terms of transformed velocity and wall distance coordinates at the higher Reynolds number under scrutiny (Re_τ ≈ 1000), whereas the core part of the flow is found to be characterized by a universal parabolic velocity profile. Based on formal similarity between the streamwise velocity and the passive scalar transport equations, we further propose an extension of the above compressibility transformations to also achieve universality of passive scalar statistics. Analysis of the velocity/temperature relationship provides evidence for quadratic dependence which is very well approximated by the thermal analogy proposed by Zhang et al. (2014). The azimuthal velocity and scalar spectra show an organization very similar to canonical incompressible flow, with a bump-shaped distribution across the flow scales, whose peak increases with the wall distance. We find that the size growth effect is well accounted for through an effective length scale accounting for the local friction velocity and for the local mean shear.     

Delay in response of turbulent heat transfer against acceleration or deceleration of flow in a pipe

To predict the heat transfer enhancements that result from the application of a pulsating flow in a pipe, we experimentally investigated the turbulent heat transfer variations produced in response to sudden accelerations or decelerations to flows within a pipe. To accomplish this, the Reynolds numbers with the valve open (Re₁) and close (Re₀) were systematically varied in the range of 8,000 ≤ Re₁ ≤ 34,000 and 700 ≤ Re₀ ≤ 23,000, respectively, and in-pipe spatiotemporal heat transfer variations were measured using infrared thermography simultaneously with temporal variations to the in-pipe flow properties. Based on the experimental results, it was found that the heat transfer delays that occur in response to accelerations or decelerations can be characterized using the corresponding time lag Δt and first-order time constant τ. The values of Δt and τ can be expressed as non-dimensional forms of Δt/(ν/u_τ²) and τ/(R/u_τ), respectively, where u_τ is the pipe wall friction velocity, ν is the kinematic viscosity of the fluid, and R is the pipe radius.     

Effect of particles on turbulent thermal field of channel flow with different Prandtl numbers

The direct numerical simulation(DNS) of heat transfer in a fully developed non-isothermal particle-laden turbulent channel flow is performed.The focus of this paper is on the modulation of the particles on turbulent thermal statistics in the particle-laden flow with three Prandtl numbers(P r = 0.71,1.5,and 3.0) and a shear Reynolds number(Reτ = 180).Some typical thermal statistics,including normalized mean temperature and their fluctuations,turbulent heat fluxes,Nusselt number and so on,are analyzed.The results show that the particles have less effects on turbulent thermal fields with the increase of Prandtl number.Two reasons can explain this.First,the correlation between fluid thermal field and velocity field decreases as the Prandtl number increases,and the modulation of turbulent velocity field induced by the particles has less influence on the turbulent thermal field.Second,the heat exchange between turbulence and particles decreases for the particle-laden flow with the larger Prandtl number,and the thermal feedback of the particles to turbulence becomes weak.     

Extensional rheology of concentrated emulsions as probed by capillary breakup elongational rheometry (CaBER)

Elongational flow behavior of w/o emulsions has been investigated using a capillary breakup elongational rheometer (CaBER) equipped with an advanced image processing system allowing for precise assessment of the full filament shape. The transient neck diameter D(t), time evolution of the neck curvature κ(t), the region of deformation l _def and the filament lifetime t _c are extracted in order to characterize non-uniform filament thinning. Effects of disperse volume fraction ?, droplet size d _sv , and continuous phase viscosity η _c on the flow properties have been investigated. At a critical volume fraction ? _c , strong shear thinning, and an apparent shear yield stress τ _y,s occur and shear flow curves are well described by a Herschel–Bulkley model. In CaBER filaments exhibit sharp necking and t _c as well as κ _max ?=?κ (t?=?t _c ) increase, whereas l _def decreases drastically with increasing ?. For ? <?? _c , D(t) data can be described by a power-law model based on a cylindrical filament approximation using the exponent n and consistency index k from shear experiments. For ??≥?? _c , D(t) data are fitted using a one-dimensional Herschel–Bulkley approach, but k and τ _y,s progressively deviate from shear results as ? increases. We attribute this to the failure of the cylindrical filament assumption. Filament lifetime is proportional to η _c at all ?. Above ? _c, κ _max as well as t _c /η _c scale linearly with τ _y,s . The Laplace pressure at the critical stretch ratio ε _c which is needed to induce capillary thinning can be identified as the elongational yield stress τ _y,e , if the experimental parameters are chosen such that the axial curvature of the filament profile can be neglected. This is a unique and robust method to determine this quantity for soft matter with τ _y ?< 1,000 Pa. For the emulsion series investigated here a ratio τ _y,e /τ _y,s = 2.8 ± 0.4 is found independent of ?. This result is captured by a generalized Herschel–Bulkley model including the third invariant of the strain-rate tensor proposed here for the first time, which implies that τ _y,e and τ _y,s are independent material parameters.     

Particle response in a planar sudden expansion flow

Particle velocity and concentration statistics were measured in a vertically downward planar sudden expansion flow for large-eddy particle Stokes numbers (τ_pU_o/5H) ranging from 0.5 to 7.4. Particles with Stokes numbers greater than 3 did not enter the recirculation zone, exhibited substantial attenuation of cross-stream velocity fluctuations, and had large streamwise velocity fluctuations in regions of strong velocity gradient. The smallest particles filled the recirculation zone and showed strong response to the large eddies in the flow. Phase-locked particle concentration measurements showed that these particles were centrifuged away from vortex cores and concentrated between vortices. Intermediate-size particles with Stokes numbers of 1.4 were injected intermittently into the recirculation zone as tongues of particles moving down between vortices. Particle Reynolds number was found to have negligible effect on the particle velocity statistics.     

Numerical study of effects of pulsatile amplitude for transitional turbulent pulsatile flow in pipes with ring‐type constrictions

The effects of pulsatile amplitude on sinusoidal transitional turbulent flows through a rigid pipe in the vicinity of a sharp‐edged mechanical ring‐type constriction have been studied numerically. Pulsatile flows were studied for transitional turbulent flow with Reynolds number (Re) of the order of 10⁴, Womersley number (Nw) of the order of 50 with a corresponding Strouhal number (St) of the order of 0.04. The pulsatile flow considered is a sinusoidal flow with dimensionless amplitudes varying from 0.0 to 1.0. Transitional laminar and turbulent flow characteristics in an alternative manner within the pulsatile flow fields were observed and studied numerically. The flow characteristics were studied through the pulsatile contours of streamlines, vorticity, shear stress and isobars. It was observed that fluid accelerations tend to suppress the development of flow disturbances. All the instantaneous maximum values of turbulent kinetic energy, turbulent viscosity, turbulent shear stress are smaller during the acceleration phase when compared with those during deceleration period. Various parametric equations within a pulsatile cycle have also been formulated through numerical experimentations with different pulsatile amplitudes. In the vicinity of constrictions, the empirical relationships were obtained for the instantaneous flow rate (Q), the pressure gradient (dp/dz), the pressure loss (P_loss), the maximum velocity (V_max), the maximum vorticity (ζ_max), the maximum wall vorticity (ζ_w,max), the maximum shear stress (τ_max) and the maximum wall shear stress (τ_w,max). Elliptic relation was observed between flow rate and pressure gradient. Quadratic relations were observed between flow rate and the pressure loss, the maximum values of shear stress, wall shear stress, turbulent kinematic energy and the turbulent viscosity. Linear relationships exist between the instantaneous flow rate and the maximum values of vorticity, wall vorticity and velocity. The time‐average axial pressure gradient and the time average pressure loss across the constriction were observed to increase linearly with the pulsatile amplitude. Copyright © 1999 John Wiley & Sons, Ltd.     

Space-time correlations in turbulent Rayleigh–Bnard convection

The recent development of the elliptic model(He,et al. Phy. Rev. E, 2006), which predicts that the space-time correlation function Cu(r, τ) in a turbulent flow has a scaling form Cu(rE, 0) with rEbeing a combined space-time separation involving spatial separation r and time delay τ, has stimulated considerable experimental efforts aimed at testing the model in various turbulent flows. In this paper, we review some recent experimental investigations of the space-time correlation function in turbulent Rayleigh–B′enard convection. The experiments conducted at different representative locations in the convection cell confirmed the predictions of the elliptic model for the velocity field and passive scalar field, such as local temperature and shadowgraph images.The understanding of the functional form of Cu(r, τ) has a wide variety of applications in the analysis of experimental and numerical data and in the study of the statistical properties of small-scale turbulence. A few examples are discussed in the review.     

Passive Scalar Transport in a Turbulent Cylinder Wake in the Presence of a Downstream Cylinder

This work aims to investigate how the presence of a downstream cylinder affects the passive scalar transport in a cylinder wake. The wake was generated by two tandem brass circular cylinders of the same diameter (d). The cylinder centre-to-centre spacing L/d was 1.3, 2.5 and 4.0, respectively, covering the three typical flow regimes of this flow. The upstream cylinder was slightly heated. Measurements were conducted at x/d= 10 and Re (≡ dU _∞/ν, where U _∞ is the free-stream velocity and ν is the kinematic viscosity of fluid) = 7000. A three-wire probe consisting of an X-wire and a cold wire was used to measure the velocity and temperature fluctuations, while an X-wire provided a phase reference. The phase-averaged velocity vectors and vorticity display single vortex street behind the downstream cylinder, irrespective of the flow regimes. However, the detailed flow structure exhibits strong dependence on L/d in terms of the Strouhal number, the vortex strength and its downstream evolution. This naturally affects passive scalar transport. The coherent and incoherent heat flux vectors show significant variation for different L/d.     

Effects of anisotropy in permeability on the two-phase flow and heat transfer in a porous cavity

This paper reports on the results of a numerical study of convection flow and heat transfer in a rectangular porous cavity filled with a phase change material under steady state conditions. The two vertical walls of the cavity are subject respectively to temperatures below and above the melting point of the PCM while adiabatic conditions are imposed on the horizontal walls. The porous medium is characterized by an anisotropic permeability tensor with the principal axes arbitrarily oriented with respect to the gravity vector. The problem is governed by the aspect ratioA, the Rayleigh numberRa, the anisotropy ratioR and the orientation angle θ of the permeability tensor. Attention is focused on these two latter parameters in order to investigate the effects of the anisotropic permeability on the fluid flow and heat transfer of the liquid/solid phase change process. The method of solution is based on the control volume approach in conjunction with the Landau-transformation to map the irregular flow domain into a rectangular one. The results are obtained for the flow field, temperature distribution, interface position and heat transfer rate forA=2.5,Ra=40, 0≤θ≤π, 0.25≤R≤4. It was found that the equilibrium state of the solid/liquid phase change process may be strongly influenced by the anisotropy ratioR as well as by the orientation angle θ of the permeability tensor. First, for a given set of parametersA,Ra andR, there exists an optimum orientation θ_max for which the flow strength, the liquid volume and the heat transfer rate are maximum. There also exists an orientation θ_min=θ_max+π/2 for which these quantities are minimum. Second, when an anisotropic medium is oriented along the optimum direction θ_max, an increase of the permeability component along that direction will increase the flow and heat transfer rate in a same order while an increase of the other permeability component only has a negligible effect. For the parameter ranges considered in the present study, it was found that the optimum direction is lying between the gravity vector and the dominant flow direction.