Drag law for monodisperse gas–solid systems using particle-resolved direct numerical simulation of flow past fixed assemblies of spheres

Gas–solid momentum transfer is a fundamental problem that is characterized by the dependence of normalized average fluid–particle force F on solid volume fraction ? and the Reynolds number based on the mean slip velocity Re_m. In this work we report particle-resolved direct numerical simulation (DNS) results of interphase momentum transfer in flow past fixed random assemblies of monodisperse spheres with finite fluid inertia using a continuum Navier–Stokes solver. This solver is based on a new formulation we refer to as the Particle-resolved Uncontaminated-fluid Reconcilable Immersed Boundary Method (PUReIBM). The principal advantage of this formulation is that the fluid stress at the particle surface is calculated directly from the flow solution (velocity and pressure fields), which when integrated over the surfaces of all particles yields the average fluid–particle force. We demonstrate that PUReIBM is a consistent numerical method to study gas–solid flow because it results in a force density on particle surfaces that is reconcilable with the averaged two-fluid theory. The numerical convergence and accuracy of PUReIBM are established through a comprehensive suite of validation tests. The normalized average fluid–particle force F is obtained as a function of solid volume fraction ? (0.1 ? ? ? 0.5) and mean flow Reynolds number Re_m (0.01 ? Re_m ? 300) for random assemblies of monodisperse spheres. These results extend previously reported results of  and  to a wider range of ?, Re_m, and are more accurate than those reported by Beetstra et al. (2007). Differences between the drag values obtained from PUReIBM and the drag correlation of Beetstra et al. (2007) are as high as 30% for Re_m in the range 100–300. We take advantage of PUReIBM’s ability to directly calculate the relative contributions of pressure and viscous stress to the total fluid–particle force, which is useful in developing drag correlations. Using a scaling argument, Hill et al. (2001b) proposed that the viscous contribution is independent of Re_m but the pressure contribution is linear in Re_m (for Re_m > 50). However, from PUReIBM simulations we find that the viscous contribution is not independent of the mean flow Reynolds number, although the pressure contribution does indeed vary linearly with Re_m in accord with the analysis of Hill et al. (2001b). An improved correlation for F in terms of ? and Re_m is proposed that corrects the existing correlations in Re_m range 100–300. Since this drag correlation has been inferred from simulations of fixed particle assemblies, it does not include the effect of mobility of the particles. However, the fixed-bed simulation approach is a good approximation for high Stokes number particles, which are encountered in most gas–solid flows. This improved drag correlation can be used in CFD simulations of fluidized beds that solve the average two-fluid equations where the accuracy of the drag law affects the prediction of overall flow behavior.     

Rolling/sliding of a particle on a flat wall in a linear shear flow at finite Re

Recently Lee and Balachandar proposed analytically-based expressions for drag and lift coefficients for a spherical particle moving on a flat wall in a linear shear flow at finite Reynolds number. In order to evaluate the accuracy of these expressions, we have conducted direct numerical simulations of a rolling particle for shear Reynolds number up to 100. We assume that the particle rolls on a horizontal flat wall with a small gap separating the particle from the wall (L = 0.505) and thus avoiding the logarithmic singularity. The influence of the shear Reynolds number and the translational velocity of the particle on the hydrodynamic forces of the particle was investigated under both transient and the final drag-free and torque-free steady state. It is observed that the quasi-steady drag and lift expressions of Lee and Balachandar provide good approximation for the terminal state of the particle motion ranging from perfect sliding to perfect rolling. With regards to transient particle motion in a wall-bounded shear flow it is observed that the above validated quasi-steady drag and lift forces must be supplemented with appropriate wall-corrected added-mass and history forces in order to accurately predict the time-dependent approach to the terminal steady state. Quantitative comparison with the actual particle motion computed in the numerical simulations shows that the theoretical models quite effective in predicting rolling/sliding motion of a particle in a wall-bounded shear flow at moderate Re.     

Fluid forces on a very low Reynolds number airfoil and their prediction

This paper presents the measurements of mean and fluctuating forces on an NACA0012 airfoil over a large range of angle (α) of attack (0-90°) and low to small chord Reynolds numbers (Re_c), 5.3 × 10³-5.1 × 10⁴, which is of both fundamental and practical importance. The forces, measured using a load cell, display good agreement with the estimate from the LDA-measured cross-flow distributions of velocities in the wake based on the momentum conservation. The dependence of the forces on both α and Re_c is determined and discussed in detail. It has been found that the stall of an airfoil, characterized by a drop in the lift force and a jump in the drag force, occurs at Re_c ? 1.05 × 10⁴ but is absent at Re_c = 5.3 × 10³. A theoretical analysis is developed to predict and explain the observed dependence of the mean lift and drag on α.     

Experimental study on a non-dilute two-phase coflowing jet: Dynamics of particles in the near flow field

The dynamics of particles in multi-phase jets has been widely studied due to its importance for a broad range of practical applications. The present work describes an experimental investigation on an initially non-dilute two-phase jet, aimed at improving the understanding in this field. A two-color PDPA has been employed to measure simultaneously the velocity and size of particles. The measurements are post-processed to check the reliability of the results and to derive information on particle volume flux as an indication of their concentration. Acoustic forcing is applied in order to control coherent structures, which are responsible for mixing and transport phenomena, and also to get phase-locked measurements. Phase-averaged statistics enabled to freeze the jet structure, not visible in the time-averaged data. The results along the jet centerline confirm that drag forces and the spread angle of the jet initially control particle dispersion, very near the nozzle exit (x/D < 4). However, as the vortical structures evolve forming tongue-shaped structures, the total particle volume flux is augmented when these structures connect with the main stream (x/D > 5). This is due to an increase of the number of smaller size particles, even when a decrease of the number of larger size particle is observed. Further analysis at five cross-stream sections across two consecutive vortices confirm that small particles are convected around the coherent structure and then incorporated to the main stream, increasing the particle concentration at the jet core. On the other hand, the number of larger particles (as well as their contribution to axial volume flux) starts to decay in regions of high azymuthal vorticity. This behaviour is partly ascribed to the transversal lift force, associated to the large spatial gradients observed in these regions. Saffman and Magnus forces have been estimated to be comparable or even greater than radial drag forces. The results suggest that the Saffman force might accelerate particles in radial direction, inducing a high radial volumetric flow rate from high to low axial velocity regions.     

Drag forces of interacting spheres in power-law fluids

Drag forces of interacting particles suspended in power-law fluid flows were investigated in this study. The drag forces of interacting spheres were directly measured by using a micro-force measuring system. The tested particles include a pair of interacting spheres in tandem and individual spheres in a cubic matrix of multi-sphere in flows with the particle Reynolds number from 0.7 to 23. Aqueous carboxymethycellulose (CMC) solutions and glycerin solutions were used as the fluid media in which the interacting spheres were suspended. The range of power-law index varied from 0.6 to 1.0. In conjunction to the drag force measurements, the flow patterns and velocity fields of power-law flows over a pair of interacting spheres were also obtained from the laser assisted flow visualization and numerical simulation.

Both experimental and computational results suggest that, while the drag force of an isolated sphere depends on the power-index, the drag coefficient ratio of an interacting sphere is independent from the power-law index but strongly depends on the separation distance and the particle Reynolds number. Our study also shows that the drag force of a particle in an assemblage is strongly positions dependent, with a maximum difference up to 38%.     

Flow-induced force fluctuations on a sphere at high Strouhal number

An empirical model is developed to estimate the broadband unsteady force spectrum induced on a rigid sphere in a nominally steady, uniform flow. The Reynolds number is sub-critical, and the frequency range considered is above the low-mode Strouhal shedding frequency of the sphere (0.5⩽fd/U₀⩽100, where f is the frequency, d is the diameter, and U₀ is the mean flow speed). The model uses the separation of variables assumption for the cross-power spectral densities of the surface pressure fluctuations. The assumption is shown to be a proper engineering approximation except in the lower part of the considered frequency range. In addition, the flow-induced unsteady lift and drag forces are measured independently of each other using towed spheres in a basin of water. Both estimations, from the empirical model and the data measured in the tow tank, show that the dimensionless power spectral densities of broadband unsteady lift and drag forces are constant for fd/U₀<1, and (fd/U₀)⁻³ dependent for 1⩽fd/U₀⩽100. The model predicts that the broadband spectral density of the unsteady lift force is about 5 dB higher than that of the unsteady drag force, while the measured data show the level difference between 3 and 7 dB. The empirical model presented here has application in predicting the flow-induced noise of underwater hydrophones that sense acoustic particle velocity or acceleration.     

The aerodynamics of a cylinder submerged in the wake of another

Flow-induced fluctuating lift (C_Lf) and drag (C_Df) forces and Strouhal numbers (St) of a cylinder submerged in the wake of another cylinder are investigated experimentally for Reynolds number (Re)=9.7×10³–6.5×10⁴. The spacing ratio L^⁎ (=L/D) between the cylinders is varied from 1.1 to 4.5, where L is the spacing between the cylinders and D is the cylinder diameter. The results show that C_Lf, C_Df and St are highly sensitive to Re due to change in the inherent nature of the flow structure. How the flow structure is dependent on Re and L^⁎ is presented in a flow structure map. Zdravkovich and Pridden (1977) observed a ‘kink’ in time-mean drag distribution at L^⁎≈2.5 for Re>3.1×10⁴, but not for Re≤3.1×10⁴. The physics is provided here behind the presence and absence of the ‘kink’ that was left unexplained since then.     

History force on a sphere in a weak linear shear flow

A numerical study of history forces acting on a spherical particle in a linear shear flow, over a range of finite Re, is presented. In each of the cases considered, the particle undergoes rapid acceleration from Re₁ to Re₂ over a short-time period. After acceleration, the particle is maintained at Re₂ in order to allow for clean extraction of drag and lift kernels. Good agreement is observed between current drag kernel results and previous investigations. Furthermore, ambient shear is found to have little influence on the drag kernel. The lift kernel is observed to be oscillatory, which translates to a non-monotonic change in lift force to the final steady state. In addition, strong dependence on the start and end conditions of acceleration is observed. Unlike drag, the lift history kernel scales linearly with Reynolds number and shear rate. This behavior is consistent with a short-time inviscid evolution. A simple expression for the lift history kernel is presented.     

The effect of trip wire roughness on the performance of the Wortmann FX 63-137 airfoil at low Reynolds numbers

An experimental investigation was conducted on the performance and boundary layer characteristics of the Wortmann FX 63-137 airfoil with and without trip wire roughness. Data were obtained through use of a three-component strain gage force balance and static pressure measurement equipment at a test Reynolds number of R _c = 100,000. Emphasis was placed on determining the effect of trip wire placement and size on such performance parameters as (C _l /C _d )_max and (C _l ^3/2 /C _d )_max. Prediction of transition location by the criterion due to Tani and Gibbings was found to have limited application. Most trip wire locations resulted in degraded performance, but for some locations, minimum drag was reduced, maximum lift to drag ratio increased, and hysteresis averted.     

Effect of laden solid particles on the turbulent flow structure of a round free jet

The effect of solid particles on the flow structure of a round air jet in a stagnant surrounding was investigated experimentally. Information on the averaged two-component velocities, the kinetic energies, and the u′ v′-properties were obtained for both phases by means of a monochromatic three beam laser Doppler anemometer. The particle number density was also measured by this system. Glass beads of 64 μm and 132 μm diameter were used for a constant mass loading ratio of 0.3 in a jet with a Reynolds number of 20 000. The lateral mean velocity and number density profiles were expressed by best fitting functions and several invariable coefficients were found. The standard drag force coefficient C _D for a single particle was applicable for a dilute particle cloud even in a non-uniform air velocity field.     

Experimental study of sedimentation characteristics of spheroidal particles

The drag of non-spherical particles is a basic, important parameter for multi-phase flow. As the first step in research in this area, the terminal velocities, Ut, of hemispherical and spherical segment particles with maximal diameters of 6-21 mm were measured in static fluids by using a high-speed video camera. The drag coefficient, CD, measured for Reynolds number, Re of 10^1-10^5, has been obtained and compared with those for a sphere. The Re based on the terminal velocity has a logarithmic linear relationship with Ar number for both the facet facing upwards or downwards for the two experimental spheroidal particles, and their Co values are greater than those of spheres. A shape function that depends on the initial orientation of the particle facet is presented to correct for the shape effects.     

Lattice–Boltzmann simulations for analysing the detachment of micron-sized spherical particles from surfaces with large-scale roughness structures

Fully resolved numerical simulations of a micron-sized spherical particle residing on a surface with large-scale roughness are performed by using the Lattice–Boltzmann method. The aim is to investigate the influence of surface roughness on the detachment of fine drug particles from larger carrier particles for transporting fine drug particles in a DPI (dry powder inhaler). Often the carrier surface is modified by mechanical treatments for modifying the surface roughness in order to reduce the adhesion force of drug particles. Therefore, drug particle removal from the carrier surface is equivalent to the detachment of a sphere from a rough plane surface. Here a sphere with a diameter of 5 μm at a particle Reynolds number of 1.0, 3.5 and 10 are considered. The surface roughness is described as regularly spaced semi-cylindrical asperities (with the axes oriented normal to the flow direction) on a smooth surface. The influence of asperity distance and size ratio (i.e. the radius of the semi-cylinder to the particle radius, R_c/R_d) on particle adhesion and detachment are studied. The asperity distance is varied in the range 1.2 < L/R_d < 2 and the semi-cylinder radius between 0.5 < R_c/R_d < 0.75. The required particle resolution and domain size are appropriately selected based on numerical studies, and a parametric analysis is performed to investigate the relationship between the contact distance (i.e. half the distance between the particle contact points on two neighbouring semi-cylinders), the asperity distance, the size ratio, and the height of the particle centroid from the plane wall. The drag, lift and torque acting on the spherical particle are measured for different particle Reynolds numbers, asperity distances and sizes or diameters. The detachment of particles from rough surfaces can occur through lift-off, sliding and rolling, and the corresponding detachment models are constructed for the case of rough surfaces. These studies will be the basis for developing Lagrangian detachment models that eventually should allow the optimisation of dry powder inhaler performance through computational fluid dynamics.     

Agglomeration of inertial particles in a random rotating symmetric straining flow

This paper is concerned with the development and validation of a simple Lagrangian model for particle agglomeration in a turbulent flow involving the collision of particles in a sequence of correlated straining and vortical structures which simulate the Kolmogorov small scales of motion of the turbulence responsible for particle pair dispersion and collision. In this particular study we consider the collision rate of monodisperse spherical particles in a symmetric (pure) straining flow which is randomly rotated to create an isotropic flow. The model is similar to the classical model of Saffman and Turner (S&T) (1956) for the collision (agglomeration) of tracer particles suspended in a turbulent flow. However unlike S&T, the straining flow is not frozen in time persisting only for timescales ∼Kolmogorov timescale. Furthermore, we consider the collision of inertial particles as well as tracer particles, and study their behavior not only at the collision boundary but also in its vicinity. In the simulation, particles are injected continuously at the boundaries of the straining flow, the size of the straining region being typical of the Kolmogorov length scale η_K of the turbulence. For steady state conditions, we calculate the flux of particles colliding with a test particle at the centre of the straining flow and consider its dependence on the inertia of the colliding particles (characterized by the particle Stokes number, St). The model replicates the segregation and accumulation observed in DNS and in particular the maximum segregation for St ∼ 1 (where St is the ratio of the particle response time to the Kolmogorov timescale). We also calculate the contributions of the various turbulent forces in the momentum balance equation for satellite particles and show for instance that for small Stokes number, there is a balance between turbulent diffusion and turbophoresis (gradient of kinetic stresses) which in turn is responsible for the build-up of concentration at the collision boundary. As found in previous studies, for the case of inertialess tracer particles, the collision rate turns out to be significantly smaller than the S&T prediction due to a lowering of the concentration at the collision boundary compared to the fully mixed value. The increase in collision rate for St ∼ 0.5 is shown to be a combination of particle segregation (build-up of concentration near the collision boundary) and the decorrelation of the relative velocity between the local fluid and a colliding particle. The difference from the S&T value for the agglomeration kernel is shown to be a consequence of the choice of perfectly absorbing boundary conditions at collision and the influence of the time scale of the turbulence (eddy lifetime). We draw the analogy between turbulent agglomeration and particle deposition in a fully developed turbulent boundary layer.     

Influence of heat transfer on the aerodynamic performance of a plunging and pitching NACA0012 airfoil at low Reynolds numbers

The unsteady low Reynolds number aerodynamics phenomena around flapping wings are addressed in several investigations. Elsewhere, airfoils at higher Mach numbers and Reynolds numbers have been treated quite comprehensively in the literature. It is duly noted that the influence of heat transfer phenomena on the aerodynamic performance of flapping wings configurations is not well studied. The objective of the present study is to investigate the effect of heat transfer upon the aerodynamic performance of a pitching and plunging NACA0012 airfoil in the low Reynolds number flow regime with particular emphasis upon the airfoil's lift and drag coefficients. The compressible Navier–Stokes equations are solved using a finite volume method. To consider the variation of fluid properties with temperature, the values of dynamic viscosity and thermal diffusivity are evaluated with Sutherland's formula and the Eucken model, respectively. Instantaneous and mean lift and drag coefficients are calculated for several temperature differences between the airfoil surface and freestream within the range 0–100 K. Simulations are performed for a prescribed airfoil motion schedule and flow parameters. It is learnt that the aerodynamic performance in terms of the lift C_L and drag C_D behavior is strongly dependent upon the heat transfer rate from the airfoil to the flow field. In the plunging case, the mean value of C_D tends to increase, whereas the amplitude of C_L tends to decrease with increasing temperature difference. In the pitching case, on the other hand, the mean value and the amplitude of both C_D and C_L decrease. A spectral analysis of C_D and C_L in the pitching case shows that the amplitudes of both C_D and C_L decrease with increasing surface temperature, whereas the harmonic frequencies are not affected.     

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.     

Two-phase flow laden with spherical particles in a microcapillary

Solid–liquid two-phase flow in a finite Reynolds number range (2 < Re < 12), transporting neutrally-buoyant microspheres with diameters of 6, 10, and 16 μm through a 260-μm microcapillary, is investigated. A standard microparticle-tracking velocimetry (μ-PTV) that consists of a double-pulsed Nd:YAG laser, an epi-fluorescent microscope, and a cooled-CCD camera is used to examine the flow. The solid particles are visualized in view of their spatial distributions. We observe a strong radial migration of the particles across the flow streamlines at substantially small Re. The degree of particle migration is presented in terms of probability density function. Some applications based on this radial migration phenomena are discussed in conjunction with particle separation/concentration in microfluidic devices, where the spatial distribution of particles is of great importance. In doing so, we propose a particle-trajectory function to empirically construct the spatial distribution of solid particles, which is well correlated with our experimental data. It is believed that this function provides a simple method for estimating the spatial distribution of particles undergoing radial migration in solid–liquid two-phase flows.     

DRAG FORCE IN DENSE GAS—PARTICLE TWO—PHASE FLOW

Numerical simulations of flow over a stationary particle in a dense gas-particle two-phase flow have been carried out for small Reynolds numbers (less than 100). In order to study the influence of the particles interaction on the drag force, three particle arrangements have been tested: a single particle, two particles placed in the flow direction and many particles located regularly in the flow field. The Navier-Stokes equations are discretized in the three-dimensional space using finite volume method. For the first and second cases, the numerical results agree reasonably well with the data in literature. For the third case, i.e., the multiparticle case, the influence of the particle volume fraction and Reynolds numbers on the drag force has been investigated. The results show that the computational values of the drag ratio agree approximately with the published results at higher Reynolds numbers (from 34.2 to 68.4), but there is a large difference between them at small Reynolds numbers. The project supported by the Special Funds for Major State Basis Research Projects in China (G19990222).     

Numerical simulation of sedimentation of rectangular particle in Newtonian fluid

The sedimentation of a rectangular particle falling in a two-dimensional channel filled with Newtonian fluid was simulated with finite element arbitrary Lagrangian–Eulerian domain method. The numerical procedure was validated by comparison of the simulation results with existing numerical work. Moreover, good agreement was obtained between the simulation results and experimental measurements performed in the current study. The equilibrium position, stable orientation and drag coefficient of a rectangular particle for different particle Reynolds numbers (Re_p) were studied. The results show that there is a critical particle Reynolds number for the preferred orientation of a rectangular particle falling in a Newtonian fluid. When Re_p is smaller than the critical value, the particle falls with its long side parallel to gravity; otherwise the particle falls with its long side perpendicular to gravity. The critical particle Reynolds number is a decreasing function of the blockage ratio and aspect ratio. The distributions of pressure and shear stress on rectangular particle surface were analyzed. Moreover, the drag coefficient of the rectangular particle decreases as Re_p or the blockage ratio increases; however, it appears to be independent of aspect ratio.     

An experimental study of a bio-inspired corrugated airfoil for micro air vehicle applications

An experimental study was conducted to investigate the aerodynamic characteristics of a bio-inspired corrugated airfoil compared with a smooth-surfaced airfoil and a flat plate at the chord Reynolds number of Re _C  = 58,000–125,000 to explore the potential applications of such bio-inspired corrugated airfoils for micro air vehicle designs. In addition to measuring the aerodynamic lift and drag forces acting on the tested airfoils, a digital particle image velocimetry system was used to conduct detailed flowfield measurements to quantify the transient behavior of vortex and turbulent flow structures around the airfoils. The measurement result revealed clearly that the corrugated airfoil has better performance over the smooth-surfaced airfoil and the flat plate in providing higher lift and preventing large-scale flow separation and airfoil stall at low Reynolds numbers (Re _C  < 100,000). While aerodynamic performance of the smooth-surfaced airfoil and the flat plate would vary considerably with the changing of the chord Reynolds numbers, the aerodynamic performance of the corrugated airfoil was found to be almost insensitive to the Reynolds numbers. The detailed flow field measurements were correlated with the aerodynamic force measurement data to elucidate underlying physics to improve our understanding about how and why the corrugation feature found in dragonfly wings holds aerodynamic advantages for low Reynolds number flight applications.     

Effect of suspended CuO nanoparticles on mass transfer to a rotating disc electrode

The effect of suspended CuO nanoparticles on the mass transfer to a rotating disc electrode was investigated experimentally, using the electrochemical limiting diffusion current technique. The particle volume fraction was from 0.39% to 1.94%. The rotating speed ranged from 100 to 1000 rpm, which yielded the Reynolds number between 10 and 110, based on the electrochemically active disc radius. The results showed that the addition of the suspended particles increased the limiting current and the plot of log I vs. log ω resulted in linear lines, of which slopes decreased with increasing particle volume fraction. The ratio of Sh/Sh_o ranged from 1 to 1.5. The Sherwood number correlation as function of the Reynolds number and the particle volume fraction was also given.