Viscosity of a Suspension with a Cubic Array of Spheres in a Shear Flow

Viscous fluid flow past an infinite periodic array of rigid spheres of the same radius is considered. A solution of the Stokes equations periodic in three variables is obtained for viscous incompressible flow with a linear velocity profile. The solution takes into account the hydrodynamic interaction of an infinite number of particles in the array. An expression for the effective viscosity of a suspension with a cubic array of particles is obtained.     

Viscous Flow Past a Periodic Array of Spheres

Viscous flow past an infinite periodic array of rigid spheres is considered. The hydrodynamic interaction of all the particles in the array is taken into account. An analytical solution of the problem is proposed. The forces exerted by the fluid on the array particles are calculated and an expression for the velocity of fluid filtration through the array is obtained. The results are compared with the previous theoretical and experimental results.     

Bulk stress of a periodic suspension of solid particles of arbitrary shape

An analytical study is made of the bulk stress of a periodic array of solid identical and force-free particles of arbitrary shape in an incompressible Newtonian fluid. Asymptotic expressions are derived for the bulk stress of dilute periodic suspensions. Asymptotic expressions are also derived for a concentrated suspension of almost-touching ellipsoids at an instant when the suspension has orthotropic symmetry.     

Analysis of periodic 3D viscous flows using a quadratic discrete Galerkin boundary element method

A discrete Galerkin boundary element technique with a quadratic approximatión of the variables was developed to simulate the three-dimensional (3D) viscous flow established in periodic assemblages of particles in suspensions and within a periodic porous medium. The Batchelor's unit-cell approach is used. The Galerkin formulation effectively handles the discontinuity in the traction arising in flow boundaries with edges or corners, such as the unit cell in this case. For an ellipsoidal dilute suspension over the range of aspect ratio studied (1 to 54), the numerical solutions of the rotational velocity of the particles and the viscosity correction were found to agree with the analytic values within 0.2% and 2% respectively, even with coarse meshes. In a suspension of cylindrical particles the calculated period of rotation agreed with the experimental data. However, Burgers' predictions for the correction to the suspension viscosity were found to be 30% too low and therefore the concept of the equivalent ellipsoidal ratio is judged to be inadequate. For pressure-driven flow through a fixed bed of fibres, the prediction on the permeability was shown to deviate by as much as 10% from the value calculated based on approximate permeability additivity rules using the corresponding values for planar flow past a periodic array of parallel cylinders. These applications show the versatility of the technique for studying viscous flows in complicated 3D geometries.     

PIV measurements of slow flow of a viscoelastic fluid within a porous medium

Particle image velocimetry (PIV) and pressure loss measurements were used to investigate slow flow through a square array of cylinders having a solid fraction of 10%. The test fluids were a Newtonian fluid and a Boger fluid, both of high viscosity such that the Reynolds number did not exceed 0.1. The pressure loss data reveal that the onset of elastic effects occurred at a Deborah number around 0.5 and that flow resistance was up to several times Newtonian values at Deborah numbers up to 3. PIV showed that the transverse velocity profiles for the Newtonian and non-Newtonian fluid were the same at Deborah numbers below onset. Above onset, the profiles became skewed, increasingly so as the Deborah number increased. In the wake regions between cylinders in a column, periodic flow structures formed in the spanwise direction. The structures were staggered from column to column, consistent with the skewing and were offset. These flow patterns are the result of an apparent elastic instability.     

Numerical predictions for unsteady viscous flow past an array of cylinders

The unsteady two-dimensional flow around an array of circular cylinders submerged in a uniform onset flow is analysed. The fluid is taken to be viscous and incompressible. The array of cylinders consists of two horizontal rows extending to infinity in the upstream and downstream directions. The centre-to-centre distance between adjacent cylinders is fixed at three diameters, and the rows are staggered. Advantage is taken of spatially periodic boundary conditions in the flow direction. This reduces the computational domain to a rectangular region surrounding a single circular cylinder. Two cases, for Reynolds numbers of 1000 and 10,000, are presented.     

Phase-resolved PIV measurements of the flow between a pair of corotating disks in a cylindrical enclosure

This paper presents the measurements of the flow in the space between an enclosed corotating disk pair using particle image velocimetry (PIV) and laser doppler velocimetry (LDV). LDV gives the time history of velocity for time-domain analysis, while PIV provides the spatial distribution of the instantaneous velocity. A flow visualization technique displaying the concentration distribution of seeding particles was also employed to visualize the flow patterns. Experiments were conducted on the interdisk midplane with a Reynolds number of 5.25×10⁵. Based on the LDV measured rotating frequency of the vortices around the hub, the phase-resolved PIV measurements were achieved, and a rotating reference coordinate system was employed to represent the flow patterns. The phase-resolved measurements reveal that the circumferential flow velocity oscillates periodically in both the inner and outer regions but in opposite trends. Based on the phase averaged data, the contributions of the periodic and random motions to the Reynolds stresses were evaluated, and the spatial distributions of the periodic Reynolds stresses were displayed. It is found that, the local rotation of the fluid induced by the deformation of the inner region contribute to a significant portion of the momentum transport.     

Effect of a periodic action on average flow in an inhomogeneous liquid-saturated porous medium

The problem of the average flow of a viscous incompressible fluid saturating a stationary porous incompressible matrix under a periodic action is considered. It is shown that a spatial inhomogeneity of the medium porosity leads to an average fluid flow, quadratically dependent on the action amplitude, in the direction of increase in porosity. In particular, this means that wave action on an oil reservoir could lead to fluid flow on the interfaces from low-porosity,weakly permeable collector regions into high-porosity regions, for example, to flow from blocks to fractures in fractured porous reservoirs, which makes it possible to enhance oil production. It is shown that in the presence of a constant pressure gradient the flow component generated by a periodic action can provide a substantial proportion of the total flow, especially on the boundaries with low-porosity strata or blocks. With increase in amplitude this may significantly exceed the component associated with the constant pressure gradient.     

The theoretical study of the efficiency of diffusion deposition of nanoaerosols in the extended range of the Peclet numbers

The efficiencies of the diffusion deposition of nanoaerosols for a single fiber for the models of aerosol filter and wire mesh screen are studied numerically in the extended range of the Peclet number Pe. The rectangular periodic cell model for fluid flow and convective-diffusive transport of small aerosol particles is used. Most of the previous theoretical and experimental studies of single fiber diffusion deposition efficiency were for the case of Pe > 1. The array with uniform square or chess grid of fibers and of a row of circular cylindrical fibers are considered as the filter and wire mesh screen models. The flow and particles transport equations are solved numerically using the Boundary Element Method.The obtained numerical data are used to derive the approximate formulas for the deposition efficiency in the entire range of the Peclet number for the various porosities of the filter medium or distances between fibers in a wire mesh screen. The derived dependencies take into account nonlinearity of the deposition efficiency at the low Peclet numbers. The obtained analytical dependencies compare well with the numerical and experimental data.     

Flow of wormlike micelle solutions past a confined circular cylinder

In this paper, we present the results of an investigation into the flow of a series of viscoelastic wormlike micelle solutions past a confined circular cylinder. Although this benchmark flow has been studied in great detail for polymer solutions, this paper reports the first experiments to use a viscoelastic wormlike micelle solution as the test fluid. The flow kinematics, stability and pressure drop were examined for two different wormlike micelle solutions over a wide range of Deborah numbers and cylinder to channel aspect ratios. A combination of particle image velocimetry and pressure drop measurements were used to characterize the flow kinematics, while flow-induced birefringence measurements were used to measure the micelle deformation and alignment in the flow. The pressure drop was found to decrease initially due to the shear thinning of the test fluid before increasing at higher flow rates as elastic effects begin to dominate the flow. Above a critical Deborah number, an elastic instability was observed for just one of the test fluids studied, the other remained stable for all Deborah number tested. Flow-induced birefringence and velocimetry measurements showed that observed instability originates in the extensional flow in the wake of the cylinder and appears not as periodic counter-rotating vortices as has been observed in the flow of polymer solutions past circular cylinders, but as a chaotic rupture event in the wake of the cylinder that propagates axially along the cylinder. Reducing the cylinder to channel aspect ratio and the degree of shearing introduced by the channel walls had a weak impact on the stability of the flow. These measurements, when taken in conjunction with previous work on flow of wormlike micelle solutions through a periodic array of cylinders, definitively show that the instability can be attributed to a breakdown of the wormlike micelle solutions in the extensional flow in the wake of the cylinder.     

Complex transition to chaotic flow in a periodic array of cylinders

This paper presents a numerical study of the transition to chaos of the flow of a Newtonian fluid in a periodic array of cylinders between two parallel walls. Using tools from dynamical system theory, we identify and characterize the different solutions to the Navier-Stokes equations at different values of the Reynolds number. We show that a very complex transition to chaos occurs for this problem where we first observe two incommensurate frequencies and then a frequency locking followed by a few period doublings following Feigenbaum's route to turbulence.     

Flow of a viscous fluid past an array of spheres

The Stokes flow of a viscous incompressible fluid through a periodic array of impenetrable spheres with linear friction on the boundary is considered. A solution and an expression for the drag are obtained to terms of order c^5/3 compared with unity (c is the volume concentration of the spheres). The proposed algorithm permits solution with any required degree of accuracy. The solution contains as limits the cases of perfect slip and no-slip on the surfaces of the spheres. In the problem with the no-slip condition, an asymptotically exact lower bound for the drag, which is valid for all values of the concentration c, is constructed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 37–44, July–August, 1981.     

On solute dispersion in pulsatile flow through a channel with absorbing walls

The paper presents the longitudinal dispersion of passive tracer molecules released in an incompressible viscous fluid flowing through a channel with reactive walls under the action of a periodic pressure gradient. A finite-difference implicit scheme is adopted to solve the unsteady advection-diffusion equation based on the Aris-Barton method of moments for all time period. Here it is shown how the spreading of tracers is influenced by the shear flow, lateral diffusion about its mean position due to the action of absorption at both the walls. The analysis has been performed for three different cases: steady, periodic and the combined effect of steady and periodic currents, separately. The results show that for all cases the dispersion coefficient asymptotically reaches a stationary state after a certain critical time and it achieves a stationary state at earlier instant of time, when absorption at the walls increases. The axial distributions of mean concentration are determined from the first four central moments by using Hermite polynomial representation for all three different flow velocities.     

Slow flow through a periodic array of spheres

Slow flow through a periodic array of spheres is studied theoretically, and the drag force by the fluid on a sphere forming the periodic array is calculated using a modification of the method developed by Hashimoto (1959). Results for the complete range of volume fraction $\text{c}$ of spheres are given for simple cubic, body-centered cubic, and face-centered cubic arrays and these agree well with the corresponding values reported by previous investigators. Also, series expansions for the drag force to 0($\text{c}{}^{10}$) are derived for each of these cubic arrays. The method is also applied to determine the drag force to 0($\text{c}{}^3$) on infinitely long cylinders in square and hexagonal arrays.     

The response of a turbulent boundary layer to artificial disturbances

With periodic fluid injection through small slots, a turbulent boundary layer is artificially disturbed on scales that are of the order of those of the natural quasi-periodic events. The periodic phase-average of the streamwise fluid velocity is determined from hot-film measurements, and used to find the coherent velocity component as defined by the triple decomposition. It appears that, when a disturbance is active, the generated flow pattern is very similar to the one caused by the interaction of a crossflow and a jet. However, when it is terminated, the turbulent boundary layer returns to its undisturbed state.     

Measurements of a stator-induced circumferentially varying flow

The flow physics associated with the generation of both axisymmetric and non-axisymmetric swirl by various deflection patterns of a stator array was investigated experimentally through surface pressure and Stereoscopic Particle Image Velocimetry measurements. A three-dimensional rendering technique was developed to reconstruct the flow field around the model and in its wake. The three-dimensional fluid volume was reconstructed from multiple two-dimensional measurement planes. A cyclic distribution of the stators’ deflections resulted in non-axisymmetric distributions of the surface pressure and the flow field downstream of the stator array. The addition of a shroud had an amplifying effect: accelerating the flow through the stator array while reducing the non-uniform tangential velocity component generated by the stators. In the model near wake the flow field is associated with secondary flow patterns in the form of coherent streamwise vortical structures that can be described by potential flow mechanisms. The collective pitch distribution of the stators produces a flow field that resembles a potential Rankine vortex, whereas the cyclic pitch distribution generates a flow pattern that can be described by a potential vortex pair in a cross-flow.     

A model of two-velocity particles transport in a porous medium

A one-dimensional flow of suspension with two types of solid particles moving with different velocities in a porous medium is considered. A mathematical model of deep bed filtration which generalizes the known equations of mass balance and particle capture kinetics for a flow of fluid with identical particles is developed. The exact solution is evaluated at the filter inlet and on the concentration front of fast suspended and retained particles, asymptotic solutions are provided in certain vicinities of these lines. A global asymptotic solution to the problem with a small limit deposit is constructed. The asymptotics rapidly converges to the numerical solution.     

Transient growth mechanisms of low Reynolds number flow over a low-pressure turbine blade

A direct transient growth analysis for three-dimensional perturbations to flow past a periodic array of T-106/300 low-pressure turbine fan blades is presented. The methodology is based on a singular value decomposition of the flow evolution operator, linearised about a steady or periodic base flow. This analysis yields the optimal growth modes. Previous work on global mode stability analysis of this flow geometry showed the flow is asymptotically stable, indicating a non-modal explanation of transition may be more appropriate. The present work extends previous investigations into the transient growth around a steady base flow, to higher Reynolds numbers and periodic base flows. It is found that the notable transient growth of the optimal modes suggests a plausible route to transition in comparison to modal growth for this configuration. The spatial extent and localisation of the optimal modes is examined and possible physical triggering mechanisms are discussed. It is found that for longer times and longer spanwise wavelengths, a separation in the shear layer excites the wake mode. For shorter times and spanwise wavelengths, smaller growth associated with excitation of the near wake are observed.     

Chaotic advection and nonlinear resonances in an oceanic flow above submerged obstacle

The effect of an isolated submarine obstacle on the motion of fluid particles in a periodic external flow is studied within the framework of the barotropic, quasi-geostrophic approximation on f-plane. The concept of background currents advanced by Kozlov [1995. Background currents in geophysical hydrodynamics. Izvestia, Atmos. Oceanic Phys. 31 (2), 245-250] is used to construct a dynamically consistent stream function satisfying the potential vorticity conservation law. It is shown that a system of two topographic vortices revolving about a rotation center can form in a circular external flow. Unsteady periodic perturbations, associated with either variations in the background current or deviations of the external flow from circulation, are analyzed. Unsteadiness in the external flow essentially complicates the pattern of the motion of fluid particles. Vortex-type quasi-periodic structures, identified with nonlinear resonances that form in Lagrangian equations of fluid particle advection, are examined. They either surround the stationary configuration by a vortex chain—a ringlet-like structure [Kennelly, M.A., Evans, R.H., Joyce, T.M., 1985. Small-scale cyclones on the periphery of Gulf Stream warm-core rings. J. Geophys. Res. 90(5), 8845-8857], or they form a complex-structure multivortex domain. Asymptotic estimates and numerical modeling are used to study the distribution and widths of the nonlinear resonance domains that appear under unsteady perturbations of different types. The onset of chaotic regimes owing to the overlapping of nonlinear resonance domains is analyzed. Transport fluxes determined by chaotic advection and barriers for transport (KAM-tori) and the conditions of their existence are studied. The relation of the rotation frequency of fluid particles on their initial position (when the dependence is calculated in the undisturbed system) is shown to completely determine the main features of the pattern of Lagrangian trajectories and chaotization effects. Because of nonlinear effects, the domain involved in quasi-periodic and chaotic motions can be much larger than the domain occupied by steady topographic vortices. The results of study by Sokolovskiy et al. [1988. On the influence of an isolated submerged obstacle on a barotropic tidal flow. Geophys. Astrophys. Fluid Dyn. 88(1), 1-30] concerning the due regard on the irrotational background component as the necessary factor for the transportation of fluid particles from the vortex domain to infinity are confirmed.     

Dissipation and inter-scale transfer in fully coupled particle and fluid motions in homogeneous isotropic forced turbulence

This work examines in detail the coupling mechanism between a stationary, homogeneous and isotropic turbulent (HIT) flow and particles, including the effect of particle-particle collisions. In order to illustrate how the physics can be elucidated of four-way interactions, a series of coupled Direct Numerical Simulations (DNS) of forced HIT are performed on a 128³ periodic box at two Taylor Reynolds numbers, 35.4 and 58.0, with interacting particles of different global Stokes numbers and volume fractions. The results show that fluid dissipation decreases up to 32% with increasing global Stokes numbers and particle volume fractions. Moreover, the corresponding dissipation when ignoring particle-particle collisions is over-estimated by up to 7% compared to the fully coupled simulations. A spectral analysis of the coupling mechanism reveals that the particles transfer energy from the large to the small scales, thereby explaining the difference in dissipation. Finally, a model spectrum for the coupling between the turbulent fluid and the particles is proposed.