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.     

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.     

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.     

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.     

Low Reynolds number airfoil aerodynamic loads determination via line integral of velocity obtained with particle image velocimetry

The small magnitude lift forces generated by both a NACA 0012 airfoil and a thin flat plate at Re?=?29,000 and 54,000 were determined through the line integral of velocity, obtained with particle image velocimetry, via the application of the Kutta–Joukowsky theorem. Surface pressure measurements of the NACA0012 airfoil were also obtained to validate the lift coefficient C _l. The bound circulation was found to be insensitive to the size and aspect ratio of the rectangular integration loop for pre-stall angles. The present C _l data were also found to agree very well with the surface pressure-determined lift coefficient for pre-stall conditions. A large variation in C _l with the loop size and aspect ratio for post-stall conditions was, however, observed. Nevertheless, the present flat-plate C _l data were also found to collectively agree with the published force-balance measurements at small angles of attack, despite the large disparity exhibited among the various published data at high angles. Finally, the ensemble-averaged wake velocity profiles were also used to compute the drag coefficient and, subsequently, the lift-to-drag ratio.     

On the interaction between two fixed spherical particles

The variation of the drag (C_D) and lift coefficients (C_L) of two fixed solid spherical particles placed at different positions relative each other is studied. Simulations are carried out for particle Reynolds numbers of 50, 100 and 200 and the particle position is defined by the angle between the line connecting the centers of the particles and the free-stream direction (α) and the separation distance (d₀) between the particles. The flow around the particles is simulated using two different methods; the Lattice Boltzmann Method (LBM), using two different computational codes, and a conventional finite difference approach, where the Volume of Solid Method (VOS) is used to represent the particles. Comparisons with available numerical and experimental data show that both methods can be used to accurately resolve the flow field around particles and calculate the forces the particles are subjected to. Independent of the Reynolds number, the largest change in drag, as compared to the single particle case, occurs for particles placed in tandem formation. Compared to a single particle, the drag reduction for the secondary particle in tandem arrangement is as high as 60%, 70% and 80% for Re = 50, 100 and 200, respectively. The development of the recirculation zone is found to have a significant influence on the drag force. Depending on the flow situation in-between the particles for various particle arrangements, attraction and repulsion forces are detected due to low and high pressure regions, respectively. The results show that the inter-particle forces are not negligible even under very dilute conditions.     

Numerical simulation of flow around a square cylinder in uniform-shear flow

This paper presents results obtained from a numerical simulation of a two-dimensional (2-D) incompressible linear shear flow over a square cylinder. Numerical simulations are performed, using the lattice Boltzmann method, in the ranges of 50⩽Re⩽200 and 0⩽K⩽0.5, where Re and K are the Reynolds number and the shear rate, respectively. The effect of the shear rate on the frequency of vortex shedding from the cylinder, and the lift and drag forces exerted on the cylinder are quantified together with the flow patterns around the cylinder. The present results show that vortex structure behind the cylinder is strongly dependant on both the shear rate and Reynolds number. When Re=50, a small K can disturb the steady state and cause an alternative vortex shedding with uneven intensity. In contrast, a large value of K will suppress the vortex shedding from the cylinder. When Re>50, the differences in the strength and size of vortices shed from the upper and lower sides of the cylinder become more pronounced as K increases. Vortex shedding disappears when K is larger than a critical value, which depends on Re. The flow patterns around the cylinder for different Re tend towards self-similarity with increasing K. The lift and drag forces exerted on the cylinder, in general, decrease with increasing K. Unlike a shear flow past a circular cylinder, the vortex shedding frequency past a square cylinder decreases with increasing the shear rate. A significant reduction of the drag force occurs in the range 0.15<K<0.3.     

Three dimensional flow motions in the viscous sublayer

We employ novel digital Fresnel reflection holography to capture the 3 D flows within the viscous sublayer of a smooth-wall turbulent channel flow at Re τ= 400. The measurements reveal unsteady and diverse flow patterns in the sublayer including nearly uniform high and low speed flows and strong small-scale(on the order of viscous wall units) spanwise meandering motions. The probability density functions(PDFs) of wall shear stresses show a clear discrepancy in high stress range with those from direct numerical simulation(DNS), which is attributed to the unresolved streamwise and spanwise motions by DNS. Moreover,the PDF of Lagrangian particle accelerations yields a stretched exponential shape like that in homogenous isotropic turbulence, indicating strong intermittency in the sublayer. We find a significant fraction of high accelerations is associated with the small-scale meandering motions. Our study helps explain the effect of sublayer-scale roughness on reducing drag and flow separation reported in the literature.     

Efficient calculation of the history force at finite Reynolds numbers

Well-acknowledged problems associated with modeling the history force in large, many-particle simulations are related to the need to store and integrate over the entire lifetime of the particle. To address this concern, a computationally efficient method for calculating the history force (the “window model”) was developed based on the assumption of weak changes in acceleration in the recent relevant history of the particle. This assumption leads to the design of a model with a truncated integration interval which requires storage of and integration over a much shorter period of the particle’s history compared to other history force models. The truncation of the integration window can yield more than an order of magnitude savings in CPU time. In a related study, the two empirical coefficients of the Mei & Adrian history force kernel have been optimized (based on comparison with experimental data for falling particles) to give improved predictions of the data. Both the new history force kernel and the window model have been investigated for a large range of experimental data yielding, to the authors’ knowledge, the most extensive comparison yet conducted. For falling particles, the new history force kernel shows good predictions for particle Reynolds numbers ranging from 9 to 853 and density ratios from 1.17 to 9.32. Good predictions were also obtained using the window model when changes in particle relative acceleration over the window period were modest. For particles under forced oscillating in a quiescent fluid, the history kernel was generally reasonable but did not predict the peak forces well in all cases. This may be explained by noting that the assumption of a t⁻² long-time dependence for the finite Reynolds number history force kernel may become invalid during rapid deceleration and wake ingestion (which can lead to exponential or t⁻¹ behavior). However, the finite Reynolds number kernel gives better predictions in all cases than those made using the Basset history force. The window model was only reasonable for the oscillating particle cases when the changes in the relative particle acceleration over the integration window were small.     

Wall proximity effects on the effectiveness of upstream control rod

Two dimensional flow over a circular cylinder with an upstream control rod of same diameter is simulated in unbound condition and in wall bounded conditions. The cylinders are placed at various heights from the wall and the inter-distance between cylinders is also varied. The control rod is subjected to different rotation rates. It is found that, in unbound condition, rotating the control rod decreases the critical pitch length (S/D_cr) and increases the drag and Strouhal number of the main cylinder. In presence of plane wall, the shielding provided by the separated shear layers from the control rod in cavity regime is deteriorated due to deflection of shear layers which results in higher drag and large fluctuation of lift coefficient. However, in wake impingement regime, the binary vortices from the control rod are weakened due to diffusion of vorticity and hence, the main cylinder experiences a lower drag and small lift fluctuations than that of unbound condition. The critical height of vortex suppression (H/D_cr) is higher in cavity regime than that of wake impingement regime due to the single extended-bluff body like configuration. The rotation of control rod energizes the wall boundary layer and increases the critical height of vortex suppression. Increasing the rotational rate of control rod decreases the drag force and reduces the amplitude of lift fluctuation. Analysis of the wall shear stress distribution reveals that it suffers a sudden drop at moderate height where the normal Karman vortex shedding changes to irregular shedding consisting of single row of negative vortices. Modal structures obtained from dynamic mode decomposition (DMD) reveal that the flow structures behind the main cylinder are suppressed due to wall and the flow is dominated by the wake of control rod.     

Effects of a trapped vortex cell on a thick wing airfoil

The effects of a trapped vortex cell (TVC) on the aerodynamic performance of a NACA0024 wing model were investigated experimentally at Re = 10⁶ and 6.67×10⁵6.67\times 10^{5}. The static pressure distributions around the model and the wake velocity profiles were measured to obtain lift and drag coefficients, for both the clean airfoil and the controlled configurations. Suction was applied in the cavity region to stabilize the trapped vortex. For comparison, a classical boundary layer suction configuration was also tested. The drag coefficient curve of the TVC-controlled airfoil showed sharp discontinuities and bifurcative behavior, generating two drag modes. A strong influence of the angle of attack, the suction rate and the Reynolds number on the drag coefficient was observed. With respect to the clean airfoil, the control led to a drag reduction only if the suction was high enough. Compared to the classical boundary layer suction configuration, the drag reduction was higher for the same amount of suction only in a specific range of incidence, i.e., α = −2° to α = 6° and only for the higher Reynolds number. For all the other conditions, the classical boundary layer suction configuration gave better drag performances. Moderate increments of lift were observed for the TVC-controlled airfoil at low incidence, while a 20% lift enhancement was observed in the stall region with respect to the baseline. However, the same lift increments were also observed for the classical boundary layer suction configuration. Pressure fluctuation measurements in the cavity region suggested a very complex interaction of several flow features. The two drag modes were characterized by typical unsteady phenomena observed in rectangular cavity flows, namely the shear layer mode and the wake mode.     

Measurement of the aerodynamic forces on a small particle attached to a wall

Using operating principles similar to that applied in atomic force microscopes, we have developed a novel measuring method to study the aerodynamic forces, in particular the lift and drag force, acting on a small particle attached to a wall and immersed in a linear shear flow. Results thus far have shown that the system is capable of measuring both the minute aerodynamic lift and drag forces that a particle experiences as a result of the flow.C. Muthanna has also published under the name C. M. Kolera

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Influence of inlet shear on the 3-D flow past a square cylinder at moderate Reynolds number

Unsteady three-dimensional (3-D) numerical simulations of linear shear flow past a square cylinder at moderate Reynolds number (Re=200) are performed. The shear parameter (K) considered in this study is varied as 0.0, 0.1, and 0.2. For the uniform flow (K=0.0) case, the chosen Re falls in the transition Reynolds number range. The low frequency force pulsations of square cylinder transition phenomena are observed to decrease with increasing shear parameter. The evolution of streamwise vortical structures indicates a mode A spanwise instability in the uniform flow. Unlike in uniform flow, mixed mode A and mode B spanwise instability is observed in the case of a shear flow. The autocorrelation function of the lift and the drag coefficients is improved for any particular separation distance with increasing K.     

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.     

On the development of lift and drag in a rotating and translating cylinder

The two-dimensional flow around a rotating cylinder is investigated numerically using a vorticity forces formulation with the aim of analyzing quantitatively the flow structures, and their evolutions, that contribute to the lift and drag forces on the cylinder. The Reynolds number considered, based on the cylinder diameter and steady free stream speed, is Re=200, while the non-dimensional rotation rate (ratio of the surface speed and free stream speed) selected was α=1 and 3. For α=1 the wake behind the cylinder for the fully developed flow is oscillatory due to vortex shedding, and so are the lift and drag forces. For α=3 the fully developed flow is steady with constant (high) lift and (low) drag. Each of these cases is considered in two different transient problems, one with angular acceleration of the cylinder and constant speed, and the other one with translating acceleration of the cylinder and constant rotation. We characterize quantitatively the contributions of individual fluid elements (vortices) to aerodynamic forces, explaining and quantifying the mechanisms by which the lift is generated in each case. In particular, for high rotation (when α=3), we explain the relation between the mechanisms of vortex shedding suppression and those by which the lift is enhanced and the drag is almost suppressed when the fully developed flow is reached.     

Optimal airfoil shapes for low Reynolds number flows

Flow over NACA 0012 airfoil is studied at α = 4^° and 12^° for Re?500. It is seen that the flow is very sensitive to Re. A continuous adjoint based method is formulated and implemented for the design of airfoils at low Reynolds numbers. The airfoil shape is parametrized with a non‐uniform rational B‐splines (NURBS). Optimization studies are carried out using different objective functions namely: (1) minimize drag, (2) maximize lift, (3) maximize lift to drag ratio, (4) minimize drag and maximize lift and (5) minimize drag at constant lift. The effect of Reynolds number and definition of the objective function on the optimization process is investigated. Very interesting shapes are discovered at low Re. It is found that, for the range of Re studied, none of the objective functions considered show a clear preference with respect to the maximum lift that can be achieved. The five objective functions result in fairly diverse geometries. With the addition of an inverse constraint on the volume of the airfoil the range of optimal shapes, produced by different objective functions, is smaller. The non‐monotonic behavior of the objective functions with respect to the design variables is demonstrated. The effect of the number of design parameters on the optimal shapes is studied. As expected, richer design space leads to geometries with better aerodynamic properties. This study demonstrates the need to consider several objective functions to achieve an optimal design when an algorithm that seeks local optima is used. Copyright © 2008 John Wiley & Sons, Ltd.     

Passive control of wake flow by two small control cylinders at Reynolds number 80

Passive control of the wake behind a circular cylinder in uniform flow is studied by numerical simulation at Re_D=80. Two small control cylinders are placed symmetrically along the separating shear layers at various stream locations. In the present study, the detailed flow mechanisms that lead to a significant reduction in the fluctuating lift but maintain the shedding vortex street are clearly revealed. When the stream locations lie within 0.8≤X_C/D≤3.0, the alternate shedding vortex street remains behind the control cylinders. In this case, the symmetric standing eddies immediately behind the main cylinder and the downstream delay of the shedding vortex street are the two primary mechanisms that lead to a 70–80% reduction of the fluctuating lift on the main cylinder. Furthermore, the total drag of all the cylinders still has a maximum 5% reduction. This benefit is primarily attributed to the significant reduction of the pressure drag on the main cylinder. Within X_C/D>3.0, the symmetry of the standing eddy breaks down and the staggered vortex street is similar to that behind a single cylinder at the same Reynolds number. In the latter case, the mean pressure drag and the fluctuating lift coefficients on the main cylinder will recover to the values of a single cylinder.     

Flow past a square cylinder at low Reynolds numbers

Results are presented for the flow past a stationary square cylinder at zero incidence for Reynolds number, Re ? 150. A stabilized finite‐element formulation is employed to discretize the equations of incompressible fluid flow in two‐dimensions. For the first time, values of the laminar separation Reynolds number, Re_s, and separation angle, θ_s, at Re_s are predicted. Also, the variation of θ_s with Re is presented. It is found that the steady separation initiates at Re = 1.15. Contrary to the popular belief that separation originates at the rear sharp corners, it is found to originate from the base point, i.e. θ_s=180^° at Re = Re_s. For Re > 5, θ_s approaches the limit of 135 ^°. The length of the separation bubble increases approximately linearly with increasing Re. The drag coefficient varies as Re^?0.66. Flow characteristics at Re ? 40 are also presented for elliptical cylinders of aspect ratios 0.2, 0.5, 0.8 and 1 (circle) having the same characteristic dimension as the square and major axis oriented normal to the free‐stream. Compared with a circular cylinder, the flow separates at a much lower Re from a square cylinder leading to the formation of a bigger wake (larger bubble length and width). Consequently, at a given Re, the drag on a square cylinder is more than the drag of a circular cylinder. This suggests that a cylinder with square section is more bluff than the one with circular section. Among all the cylinder shapes studied, the square cylinder with sharp corners generates the largest amount of drag. Copyright © 2010 John Wiley & Sons, Ltd.