Numerical simulation of flow kinematics for suspensions containing continuously dispersed aspect ratio fibers

Fiber suspension flow and fiber orientation through a parallel-plate channel were numerically simulated for fiber suspensions including continuously dispersed aspect ratios from 10 to 50. In the simulations, both the fiber–fiber and fiber–wall interactions were not taken into account. A statistical scheme that proceeds by evaluating the orientation evolution of a large number of fibers from the solution of the Jeffery equation along the streamlines was confirmed to be a very useful and feasible method to accurately analyze the orientation distribution of fibers with continuously dispersed aspect ratios. For monodisperse suspensions with small-aspect-ratio fibers, flip-over or oscillation phenomenon of the orientation ellipsoid caused the wavy patterns of the velocity profile and the streamlines as well as the abrupt and complex variation of the shear stress and the normal stress difference near the channel wall as proven in one of our former works. On the other hand, continuous dispersions containing from small- to large-aspect-ratio fibers were able to induce smoother evolutions of the fiber orientation and the flow kinematics. In the processing of fiber composites, the length of suspended fibers is always continuously distributed because of fiber breakage during processing; thus, the smooth evolutions of the flow kinematics and the stress distribution can be attained.This paper was presented at the Annual Meeting of the European Society of Rheology, Grenoble, April 2005.     

Flow-driven orientation dynamics of semiflexible fiber systems

We study the flow-induced orientation dynamics of semiflexible fibers in dilute fiber suspensions. Starting from the equations of motion for a two-rod model of flexible fibers in Stokes flow, the Smoluchowski equation for a connected monomer orientation distribution function is derived. We then obtain a set of equations for the time dependence of the first and second moments of the orientation distribution function, thus extending the Folgar Tucker equations for short rigid fiber suspensions to flexible fiber suspensions. The resulting generalized equations for the orientation dynamics of a suspension of flexible fibers are solved for simple channel flow. It is shown that all qualitative effects of bending and straightening of fibers and their influence on the orientation of flexible fibers are captured within our model. A scalar measure for the distribution of bending in a flow is introduced, which allows to detect the degree of bending of fibers. Paper was presented at the 3rd Annual Rheology Conference, AERC 2006, April 27–29, 2006, Crete, Greece.     

Fiber suspension flow in a tapered channel: The effect of flow/fiber coupling

A numerical model for predicting the flow and orientation state of semi-dilute, rigid fiber suspensions in a tapered channel is presented. The effect of the two-way flow/fiber coupling is investigated for low Reynolds number flow using the constitutive model of Shaqfeh and Fredrickson. An orientation distribution function is used to describe the local orientation state of the suspension and evolves according to a Fokker–Plank type equation. The planar orientation distribution function is determined along streamlines of the flow and is coupled with the fluid momentum equations through a fourth-order orientation tensor. The coupling term accounts for the two-way interaction and momentum exchange between the fluid and fiber phases. The fibers are free to interact through long range hydrodynamic fiber–fiber interactions which are modeled using a rotary diffusion coefficient, an approach outlined by Folgar and Tucker. Numerical predictions are made for two different orientation states at the inlet to the contraction, namely a fully random and a partially aligned fiber orientation state. Results from these numerical predictions show that the streamlines of the flow are altered and that velocity profiles change from Jeffery–Hamel, to something resembling a plug flow when the fiber phase is considered in the fluid momentum equations. This phenomenon was found when the suspension enters the channel in either a pre-aligned, or in a fully random orientation state. When the suspension enters the channel in an aligned orientation state, fiber orientation is shown to be only marginally changed when the two-way coupling is included. However, significant differences between coupled and uncoupled predictions of fiber orientation were found when the suspension enters the channel in a random orientation state. In this case, the suspension was shown to align much more quickly when the mutual coupling was accounted for and profiles of the orientation anisotropy were considerably different both qualitatively and quantitatively.     

Sedimentation of particles in shear flows of viscoelastic fluids with fibers

We study the effect of fiber additives on rheology and sedimentation of particle suspensions in a base viscoelastic suspending fluid in the case when the suspension is subjected to shear flow. We found experimentally that fiber additives (3–6 mm in length and 8–12 μm in diameter at a mass fraction of 0–0.4%) increase the suspension viscosity and retard the particle sedimentation significantly. At the same mass concentration, long and thin fibers reduce the sedimentation velocity and increase the viscosity to a much greater extent than short and thick fibers. We revealed that both rheology and sedimentation are controlled by a single conformational parameter (overlap parameter) defined as the number of fibers per unit volume multiplied by fiber length cubed.     

Orientation distribution of fibers and rheological property in fiber suspensions flowing in a turbulent boundary layer

A model relating the translational and rotational transport of orientation distribution function （ODF） of fibers to the gradient of mean ODF and the dispersion coefficients is proposed to derive the mean equation for the ODE Then the ODF of fibers is predicted by numerically solving the mean equation for the ODF together with the equations of turbulent boundary layer flow. Finally the shear stress and first normal stress difference of fiber suspensions are obtained. The results, some of which agree with the available relevant experimental data, show that the most fibers tend to orient to the flow direction. The fiber aspect ratio and Reynolds number have significant and negligible effects on the orientation dis- tribution of fibers, respectively. The additional normal stress due to the presence of fibers is anisotropic. The shear stress of fiber suspension is larger than that of Newtonian solvent, and the first normal stress difference is much less than the shear stress. Both the additional shear stress and the first normal stress difference increase with increasing the fiber concentration and decreasing fiber aspect ratio.     

EFFECTS OF TENSOR CLOSURE MODELS AND 3-D ORIENTATION ON THE STABILITY OF FIBER SUSPENSIONS IN A CHANNEL FLOW

IntroductionFlowoffibresuspensionshasbeenveryfamiliarinmanyindustrialfields.Fibreadditivesplayanimportantroleindragreductioninmanytypesofflow[1- 3].Inthesuspensions,somebehavioroftheflowmaybealteredbythefibres.Oneoftheimportantexamplesisthehydrodynamicsta…     

Stability analysis in spatial mode for channel flow of fiber suspensions

Different from previous temporal evolution assumption, the spatially growing mode was employed to analyze the linear stability for the channel flow of fiber suspensions. The stability equation applicable to fiber suspensions was established and solutions for a wide range of Reynolds number and angular frequency were given numerically . The results show that, the flow instability is governed by a parameter H which represents a ratio between the axial stretching resistance of fiber and the inertial force of the fluid. An increase of H leads to a raise of the critical Reynolds number, a decrease of corresponding wave number, a slowdown of the decreasing of phase velocity , a growth of the spatial attenuation rate and a diminishment of the peak value of disturbance velocity. Although the unstable region is reduced on the whole, long wave disturbances are susceptible to fibers.     

An experimental study of flow-induced fiber orientation and concentration distributions in a concentrated suspension flow through a slit channel containing a cylinder

Flow-induced fiber orientation and concentration distributions were measured in a concentrated fiber suspension (CFS) and a dilute one (DFS). The channel has a thin slit geometry containing a circular cylinder. In the previous work, many researchers have qualitatively studied fiber orientation and concentration distributions in injection-molded products of fiber-reinforced plastics. In the present work, however, they are quantitatively estimated by direct observation of fibers in the concentrated suspension flow. For the CFS, some fibers rotate in an expansion part between the channel wall and the circular cylinder, and the fiber orientation becomes almost random state. On the other hand, fibers are perfectly aligned along the flow direction owing to the elongational flow near the centerline downstream of the cylinder. The fiber concentration has a flat distribution except near the channel wall and the centerline. For the DFS a minimum in the fiber concentration distribution was clearly observed on the centerline, and two peaks beside the centerline and near the channel wall. This characteristic distribution is caused by the fiber-wall and fiber-cylinder interactions. It is found that the obstacle such as the circular cylinder in the channel significantly affects the fiber orientation downstream of the obstacle for the CFD, while it affects the fiber concentration distribution for the DFS.     

Comparative numerical study of two concentrated fiber suspension models

We consider two rheological models for concentrated fiber suspensions. In both models the equations for orientation and flow are fully coupled, i.e., the orientation influences the flow via a constitutive relation for the viscosity and the orientation of the fibers is determined by the flow field. The orientation state of the fibers is characterized by the Advani–Tucker orientation tensor. We are investigating suspensions of fibers in which the kinetic energies of the fibers are large compared to the thermal energies, i.e., the influence of Brownian motion may be neglected. The first model is the Folgar–Tucker model with backcoupling to the flow (FT model). The second model is an extension of Folgar–Tucker, which models phenomenologically the topological exclusion interaction in dense suspensions (FTMS model). As test cases for the simulation are considered channel flow, 8:1 contraction flow and flow around a cylinder.     

Structure and properties of sheared fiber suspensions with mechanical contacts

The properties of fiber suspensions are highly sensitive to the suspension microstructure. In dilute or semi-dilute suspensions, nL²d≪1, the fibers' orientation distribution is controlled by hydrodynamic interactions among the fibers. However, direct mechanical contacts among the fibers play an important role in semi-concentrated suspensions, nL²d=O(1). Here, n is the number of fibers per unit volume, L is the fiber length and d is the fiber diameter. We have performed dynamic simulations of fiber suspensions including contact forces that prevent any two fibers from passing through one another. Collisions between the fibers cause them to flip more frequently in the shear flow, leading to a spread of the orientation distribution away from the flow direction. Both this increased orientational dispersion and the direct stress transmitted through the contacts enhance the shear viscosity of the suspension significantly. The contacts also give rise to normal stress differences. The results of the simulation are compared with experiments and the relative importance of contacts and hydrodynamic interactions is discussed.     

Direct simulation of flexible fibers

A numerical simulation of multiple flexible fibers in suspension in Newtonian simple shear flow is presented. The method used is similar to those of previous recent simulation works by Fan et al. [J. Non-Newtonian Fluid Mech. 74 (1998) 113] and Yamane et al. [J. Non-Newtonian Fluid Mech. 54 (1994) 405], however, the method has been modified to allow a small amount of bending and torsion in the fibers. A restoring moment acts to straighten the fibers as they interact in the flow.It is demonstrated that this simulation can be used to extract basic rheological information about the suspension including fiber orientations and suspension viscosity. The viscosity of semi-concentrated to concentrated flexible fiber suspensions are shown to increase by a magnitude of the order 7–10% greater than the equivalent rigid fiber suspension tested. This is in qualitative agreement with previous experimental work by Goto et al. [Rheologica Acta 25 (1986) 119] and Blakeney [J. Colloid Interface Sci. 22 (1966) 324]. The implication is that any constitutive relation involving particulate suspensions described by orientation vectors may quantitatively underestimate suspension viscosity, particularly for fibers of large aspect ratio, or low Young’s modulus, whereby the tendency to flex is greater [Rheologica Acta 25 (1986) 119]. If particulate deformation were accounted for (by whatever means) in the existing constitutive relationship, predictions of bulk suspension parameters such as viscosity should be noticeably improved. A method is developed to modify an existing rigid-fiber viscosity to an equivalent flexible fiber viscosity, hence improving viscosity prediction ability.     

Finite-length effects on dynamical behavior of rod-like particles in wall-bounded turbulent flow

Combined Particle Image Velocimetry (PIV) and Particle Tracking Velocimetry (PTV) measurements have been performed in dilute suspensions of rod-like particles in wall turbulence. PIV results for the turbulence field in the water table flow apparatus compared favorably with data from Direct Numerical Simulations (DNS) of channel flow turbulence and the universality of near-wall turbulence justified comparisons with DNS of fiber-laden channel flow. In order to examine any shape effects on the dynamical behavior of elongated particles in wall-bounded turbulent flow, fibers with three different lengths but the same diameter were used. In the logarithmic part of the wall-layer, the translational fiber velocity was practically unaffected by the fiber length l. In the buffer layer, however, the fiber dynamics turned out to be severely constrained by the distance z to the wall. The short fibers accumulated preferentially in low-speed areas and adhered to the local fluid speed. The longer fibers (l/z > 1) exhibited a bi-modal probability distribution for the fiber velocity, which reflected an almost equal likelihood for a long fiber to reside in an ejection or in a sweep. It was also observed that in the buffer region, high-speed long fibers were almost randomly oriented whereas for all size cases the slowly moving fibers preferentially oriented in the streamwise direction. These phenomena have not been observed in DNS studies of fiber suspension flows and suggested l/z to be an essential parameter in a new generation of wall-collision models to be used in numerical studies.     

Orientation behaviour of fibers in suspension flow through a branching channel

Fiber orientation and dispersion in the fiber suspension that flows through a T-shaped branching channel have been experimentally studied. The fiber dispersion is not uniform in a channel flow. In particular, in the suspension containing long fibers, there are no fibers near a wall at a low Reynolds number. However, the content ratio of fibers near the wall increases with an increasing Reynolds number. Fiber orientation angles of short fibers are widely distributed. Furthermore, a variance of orientation angle in the middle region of the channel has a minimum value at a Reynolds number from 10 to 20, and a size of secondary vortex in the branching corner also has a minimum value at a similar Reynolds number.     

Characteristics of fiber suspension flow in a rectangular channel

Velocity profile of fiber suspension flow in a rectangular channel is measured by pulsed ultrasonic Doppler velocimetry (PUDV), and the effect of fiber concentration and Reynolds number on the shape of the velocity profile is investigated. Five types of flow behavior are observed when fiber concentration increases or flow rate decreases progressively. The turbulent velocity profiles of fiber suspension can be described by a correlation with fiber concentration, nl³, and Reynolds number, Re as the main parameters. The presence of fiber in the suspension will reduce the turbulence intensity and thus reduce the turbulent momentum transfer. On the other hand, fibers in the suspension have the tendency to form fiber networks, which will increase the momentum transfer. The relative contribution of these two types of momentum flux will determine the final shape of the velocity profile.     

Numerical prediction of fiber motion in a branching channel flow of fiber suspensions

Fiber orientation and dispersion in the dilute fiber suspension that flows through a T-shaped branching channel are simulated numerically based on the slender-body theory. The simulated results are consistent qualitatively with the experimental data available in the literature. The results show that the spatial distribution of fibers is dependent on the fiber aspect ratio, but has no relation with the volume fraction of fiber. The content ratio of fibers near the upper wall increases monotonically with an increasing Re number, and the situation is reverse for the region near the bottom wall. The orientation of fibers depends on Re number, however, the function of fiber volume fraction and aspect ratio is negligible. The fibers near the wall and in the central region of the channel align along the flow direction at all times, but the fibers in the other parts of the channel tend to align along the flow direction only in the downstream region.The project supported by the National Natural Science Foundation of China (10372090) and Doctoral Program of Higher Education in China (20030335001)The English text was polished by Ron Marshall     

Research on the motion of particles in the turbulent pipe flow of fiber suspensions

The motion of fibers in turbulent pipe flow was simulated by 3-D integral method based on the slender body theory and simplified model of turbulence. The orientation distribution of fibers in the computational area for different Re numbers was computed. The results which were consistent with the experimental ones show that the fluctuation velocity of turbulence cause fibers to orient randomly. The orientation distributions become broader as the Re number increases. Then the fluctuation velocity and angular velocity of fibers were obtained. Both are affected by the fluctuation velocity of turbulence. The fluctuation velocity intensity of fiber is stronger at longitudinal than at lateral, while it was opposite for the fluctuation angular velocity intensity of fibers. Finally, the spatial distribution of fiber was given. It is obvious that the fiber dispersion is strenghened with the increase of Re numbers.     

On translational and rotational relative velocities of fibers and fluid in a turbulent channel flow with a backward-facing step

A dilute fiber suspension in a turbulent channel with a backward-facing step is investigated by means of Feature Tracking. Its combination with a phase-discrimination methodology, which is described in detail, allows simultaneous and separate measurement of carrier and dispersed phases velocity fields, the orientation and rotation rate of fibers as well as the fiber–fluid translational and rotational slip velocities. The patterns of fibers concentration, angular velocity and the probability distribution of fibers velocity appear to be dominated by the mechanical interactions with the wall and the local high shear rather than by near-wall turbulent structures. The translational slip velocity obtained from instantaneous data shows that fibers move faster than the surrounding fluid inside the buffer layer, the velocity gap reducing gradually when approaching the channel centerline. On the other hand, the rotational slip profile suggests a gradual decoupling of the translational and rotational dynamics. Downstream of the step, the excess of streamwise velocity displayed by fibers is still observed and extends in the free-shear region, whereas the rotation rate slip decreases at a relatively short distance from the step, as the effect of the wall presence fades away.     

Evaluation of steerable filter for detection of fibers in flowing suspensions

Steerable filters are concluded to be useful in order to determine the orientation of fibers captured in digital images. The fiber orientation is a key variable in the study of flowing fiber suspensions. Here, digital image analysis based on a filter within the class of steerable filters is evaluated for suitability of finding the position and orientation of fibers suspended in flowing suspensions. In sharp images with small noise levels, the steerable filter succeeds in determining the orientation of artificially generated fibers with well-defined angles. The influence of reduced image quality on the orientation has been quantified. The effect of unsharpness and noise is studied and the results show that the error in orientation is less than 1° for moderate levels. Images from two flow cases, one laminar shear flow and one turbulent, are also analyzed. The fiber orientation distribution is determined in the flow-vorticity plane. For the laminar case a comparison is made to a robust, but computationally more expensive, method involving convolutions with an oriented elliptic filter. A good agreement is found when comparing the resulting fiber orientation distributions obtained with the two methods. For the turbulent case, it is demonstrated that correct results are obtained and that the method can handle overlapping fibers.     

A numerical study of the influence of the Basset force on the statistics of LDV velocity data sampled in a flow region with a large spatial velocity gradient

 The motion of small particles, such as those typically used as seeding particles for tracer particle flow velocity measurement techniques, is studied numerically for a flow region with a large spatial velocity gradient. The influence of the Basset history integral on the statistics of results of particle motion calculations which are based on multi-disperse particle size distributions is investigated. The biasing of the measured velocity data, with regard to the actual flow velocity, which results as a consequence of such particle size distributions is discussed. It is found that the net effect of the Basset integral on the calculations is indeed to reduce the maximum RMS deviation associated with the multi-disperse distribution and that the relative reduction increases with a decreasing particle density. The main result of this study is, thus, that it is desirable to use light tracer particles not only because they more readily adjust to a changed flow velocity but in particular also because they tend to contribute less to the overall RMS deviation of velocity data sampled in a region with a large spatial gradient of the flow velocity. Received: 9 August 1996/Accepted: 25 January 1997