Inertial migration regimes of a neutrally buoyant sphere in pipe Poiseuille flow

This study investigates the structure and dynamic behavior of a linear system which describes the perturbation of a laminar pipe flow. In the preceding papers a numerical method for calculating the eigenvalues with sufficient accuracy was proposed, and the characteristic distribution pattern of eigenvalues for Poiseuille pipe flow and the classification of the modes of the perturbations were presented. This paper discusses the effects of the Reynolds number on the distribution pattern of eigenvalues. In the first place, for the Poiseuille pipe flow, it is shown how the distribution of eigenvalues in a complex phase velocity plane takes a tree like shape as the Reynolds number increases. Then the distributions of eigenvalues are calculated for the laminar developing pipe flow and the Poiseuille pipe flow with rigid rotation, which are known to be unstable for high Reynolds number. It is found that these distribution patterns take the same shape as that of Poiseuille pipe flow, and that the unstable eigenvalues belong to the specified modes of perturbations, as predicted in the former report.

The radial migration of a single neutrally buoyant particle in Poiseuille flow is numerically investigated by direct numerical simulations. The simulation results show that the Segre and Silberberg equilibrium position moves towards the wall as the Reynolds number increases and as the particle size decreases. At high Reynolds numbers, inner equilibrium positions are found at positions closer to the centerline and move towards the centerline as the Reynolds number increases. At higher Reynolds numbers, the Segre and Silberberg equilibrium position disappears and only the inner equilibrium position exists. We prove that the inner annuluses in the measurements of Matas, Morris & Guazzelli (J. Fluid Mech. 515, 171-195, 2004) are not transient radial positions, but are real equilibrium positions. The results on the inner equilibrium positions and unstable equilibrium positions are new and convince us of the existence of multiple equilibrium radial positions for neutrally buoyant particles.

The finite element method based on fluid-structure interaction is used to systematically study the inertial migration of polymer vesicles in microtubule flow with a two-dimensional model, and the mechanism of the vesicles deformed by the fluid and the inertial migration phenomena are analyzed. The studies show that with the increase Reynolds number, the equilibrium position of vesicle inertial migration is farther and farther from its initial position; with the increase of blocking ratio, the equilibrium position of vesicle inertial migration is closer to the wall surface. For the modulus and viscosity of the vesicle membrane and for the membrane thickness, the results show that the modulus and viscosity determine the degree of deformation of the vesicle, and the modulus has little effect on the equilibrium position of the vesicle, but increases the viscosity, and the membrane thickness will promote the equilibrium position of the vesicle to be biased toward the center of the tube. This study helps to further clarify the deformation and equilibrium position of vesicles during inertial migration, and provides a reliable computational basis for the application of vesicles in drug transport, chemical reactions and physiological processes.

Inertial migration of spherical elastic phytoplankton in a microscale pipe was investigated in Reynolds number range of 10–100. Three-dimensional position of migrating cells was obtained using digital in-line holographic microscopy. Characteristics of inertial migration were studied by analyzing the spatial distribution of cells. As Reynolds number increases, Segre–Silberberg annulus clearly appeared at the same measurement location. The effect of elastic shell compliance was experimentally investigated by comparing the probability density function of normal and hardened cells. As Re increases, the equilibrium positions of both normal and hardened cells drift toward the tube wall as the result of balanced lifts. The degree of inertial-migration development of elastic cells is slower than that of hardened cells. These results will be useful for better understanding of dynamic behaviors of phytoplankton, especially inertial migration in pipe flow.

By combining the lattice Boltzmann model of fluid flow with the molecular dynamics model of copolymers, we investigate the inertial migration of cylindrical micelles, which is obtained by controlling the length ratios of hydrophilic and hydrophobic segments in a comb-like copolymer. Our results demonstrate that cylindrical micelles gradually deviate from the center of the nanochannel with increasing Reynolds number (Re). For the same Re, the larger the cylindrical micelle is, the closer it is to the center of the nanochannel. Importantly, we find that the change in the equilibrium position is particularly pronounced at Re less than 0.1, while the trend becomes smoother at Re greater than 0.1, which is because of the transition of micelles from cylindrical to disk-like shapes when Re is smaller than 0.1, and does not change as Re further increases. This work provides an understanding of cylindrical micelles' inertial migration, particularly in identifying the effect of morphological changes on the equilibrium position, which could lead to significant advancements in the inertial migration of polymer micelles.

Inertial migration phenomena of phytoplankton in pipe flows were investigated using a digital holography technique. As the Reynolds number increases, the microorganisms suspended in a pipe flow are focused at a certain radial position which is called equilibrium position or pinch point. In this study, the effects of the size of microorganism and Reynolds number in the range of 1 < Re < 78 on the inertial migration were investigated and the results are compared with those for solid particles under similar experimental conditions. As a result, the equilibrium position for the elastic microorganisms is not so distinct, compared to the solid particles. This results from deformation of elastic body shape caused by shear-gradient of surrounding flow.

Abstract

Numerical Investigation of Flow and Heat Transfer Characteristics of Two Tandem Circular Cylinders of Different Diameters

A theoretical study is made of initial algebraic growth for small angular-dependent disturbances in pipe Poiseuille flow. The analysis is based on the homogeneous equation for the pressure for which the eigenvalue problem is solved numerically. In the limit of small streamwise wave numbers asymptotic results for the eigenvalues are derived. On the basis of the modes of the system, which are all damped, the initial value problem is considered and in particular the largest possible growth of the disturbance energy density is determined following the ideas of Butler and Farrell [Phys. Fluids A 4, 1637 (1992)]. The results show that a large amplification of the disturbance energy is possible. The largest amplification is obtained for disturbances with a small streamwise wave number and with an azimuthal wave number of one. The energy growth is then only due to the growth of the streamwise disturbance component. However, for disturbances of shorter wavelength, the energy growth is also substantial and not only concentrated to the streamwise velocity component. The wall shear corresponding to disturbances with the largest energy growth also shows a large amplification and the dependence of wave numbers and the Reynolds number is the same as for the energy. However, the wall pressure of a long wavelength disturbance of the largest growth just decays from its initial value, but for disturbances of shorter wavelength, it is also amplified.

An analysis is given of a secondary instability that obtains in a wide class of wall-bounded parallel shear flows, including plane Poiseuille flow, plane Couette flow, flat-plate boundary layers, and pipe Poiseuille flow. In these flows it is shown that two-dimensional finite-amplitude waves are (exponentially) unstable to infinitesimal three-dimensional disturbances. This secondary instability seems to be the prototype of transitional instability in these flows in that it has the characteristic (convective) timescales observed in the typical transitions. In the case of plane Poiseuille flow, two-dimensional nonlinear equilibria and quasi-equilibria exist, and the stability of the secondary flow is determined by a three-dimensional linear eigenvalue calculation. In flows without equilibria (e.g. pipe flow), a time-dependent stability analysis is performed by direct spectral numerical calculation of the incompressible three-dimensional Navier–Stokes equations. The energetics and vorticity dynamics of the instability are discussed. It is shown that the two-dimensional wave mediates the transfer of energy from the mean flow to the three-dimensional perturbation but does not directly provide energy to the disturbance. The instability is of an inviscid character as it persists to high Reynolds numbers and grows on convective timescales. Maximum vorticity (inflexion-point) arguments predict some features of the instability like phase-locking of the two-dimensional and three-dimensional waves, but they do not explain its essential three-dimensionality. The inviscid vorticity dynamics of the instability shows that vortext-stretching and tilting effects are both required to explain the persistent exponential growth. The instability is not centrifugal in nature. The three-dimensional instability requires that a threshold two-dimensional amplitude be achieved (about 1% of the centreline velocity in plane Poiseuille flow): the growth rates are relatively insensitive to amplitude for moderate two-dimensional amplitudes. With moderate two-dimensional amplitudes, the critical Reynolds numbers for substantial three-dimensional growth are about 1000 in plane Poiseuille and Couette flows and several thousand in pipe Poiseuille flow. The asymptotic (as R approaches infinity) growth rate in plane Poiseuille flow is approximately 0·15h/U0, where h is the half-channel width and U0 is the centreline velocity. It is possible to make some progress identifying experimental features of transitional spot structure with aspects of the nonlinear two-dimensional/linear three-dimensional instability. The principal excitation of the eigenfunction of the three-dimensional (growing) disturbance is localized within a given periodicity length (in both the stream and cross-stream directions) near the vorticity maxima of the two-dimensional flow; its planwise structure corresponds to that of observed streaks in early transitional spots; its vortical structure resembles that of a streamwise vortex lifting off the wall. As the three-dimensional disturbance grows to finite amplitude, the flows become chaotic with statistical structure similar to that observed experimentally in moderate-Reynolds-number turbulent shear flows.

A computational investigation, supported by a theoretical analysis, is performed to investigate a pressure-driven flow around a line of equispaced spheres moving at a prescribed velocity along the axis of a circular tube. This fundamental study underpins a range of applications including physiological circulation research. A spectral-element formulation in cylindrical coordinates is employed to solve for the incompressible fluid flow past the spheres, and the flows are computed in the reference frame of the translating spheres.Both the volume flow rate relative to the spheres and the forces acting on each sphere are computed for specific sphere-to-tube diameter ratios and sphere spacing ratios. Conditions at which zero axial force on the spheres are identified, and a region of unsteady flow is detected at higher Reynolds numbers (based on tube diameter and sphere velocity). A regular perturbation analysis and the reciprocal theorem are employed to predict flow rate and drag coefficient trends at low Reynolds numbers. Importantly, the zero drag condition is well-described by theory, and states that at this condition, the sphere velocity is proportional to the applied pressure gradient. This result was verified for a range of spacing and diameter ratios. Theoretical approximations agree with computational results for Reynolds numbers up toO(100).The geometry dependence of the zero axial force condition is examined, and for a particular choice of the applied dimensionless pressure gradient, it is found that this condition occurs at increasing Reynolds numbers with increasing diameter ratio, and decreasing Reynolds number with increasing sphere spacing.Three-dimensional simulations and predictions of a Floquet linear stability analysis independently elucidate the bifurcation scenario with increasing Reynolds number for a specific diameter ratio and sphere spacing. The steady axisymmetric flow first experiences a small region of time-dependent non-axisymmetric instability, before undergoing a regular bifurcation to a steady non-axisymmetric state with azimuthal symmetrym= 1. Landau modelling verifies that both the regular non-axisymmetric mode and the axisymmetric Hopf transition occur through a supercritical (non-hysteretic) bifurcation.

Great benefits of biomimetic textured surfaces have been recognised through recent research. Such studies highly impact transportation-related fuel consumption and emission of carbon-dioxide. With 10% reduction of friction drag, 4 billion GBP/year savings are expected in the ship industry alone. The shape, spacing and alignment of the textured geometries together with the Reynolds number are the main parameters to characterise turbulent flow and drag performance associated with the use of these surfaces. The present research aims at investigating the effect of a novel biomimetic texture surface on turbulent flow behaviour and drag reduction. The texture is inspired by an aerodynamically-efficient insect called the Backswimmer also known as Notonecta Glauca. Direct Numerical Simulations (DNS) has been used to examine turbulence and drag reduction for the texture geometry. An in-house code for channel and pipe flows, CHAPSim, is used for the simulations. The code is adopted for (i) treating the novel textured surface using an Immersed Boundary Method (IBM) and (ii) improving the solver scalability by implementing a hybrid parallelisation approach. Verification and validation (where possible) for smooth and textured simulations have been undertaken. The effect of using the backswimmer geometry is studied at different geometry conditions with a constant bulk velocity at Reynolds number Re = 2800. Additionally, the effect of changing the Reynolds number was studied for one geometry condition at Re = 2800, 3500, 5500 and 7400. Compared to smooth channel, the backswimmer cases showed an increase in the wall-normal and spanwise fluctuating velocities, near wall vorticity and Reynolds shear stress. Among all the backswimmer cases, the increase in the wall normal and spanwise vorticity profiles with the textured element was almost unaffected by either the changes in the geometry or the increase in Reynolds numbers. Compared against smooth channel, the backswimmer geometry case showed a decrease in the ejection events as Reynolds number increases but a significant increase in the sweep events. The flow visualisations and vortex identification were simulated by calculating low- and high- speed streaks and second-largest Eigen value of the symmetric tensor (λ₂). It was shown that the streaky structures and vortex strength within the roughness elements were dependent on the change in the geometry conditions and Reynolds number. As the Reynolds number increased, the intensity of λ₂ increased within the roughness for the smaller Reynolds numbers and above the roughness crest for the higher Reynolds numbers.

Numerical Investigation of the Effect of Dimpled Surface on Convective Heat Transfer and Friction Factor in a Rectangular Channel

The phenomenon of vortex reconnection is analysed numerically and the results are compared qualitatively with the predictions of a model of reconnection recently proposed by Saffman. Using spectral methods over both uniform and strained meshes, numerical simulations are performed of two nearly parallel, counter-rotating vortex tubes, over the range of Reynolds numbers Re = 1000–3500. The calculations utilizing a uniform mesh are performed for Re ≤ 1500 with a resolution of 128 points in each direction. The calculations utilizing a stretched mesh are performed for 1500 &lt; Re ≤ 3500 with a resolution of up to 160 points in each direction and with a fourfold stretching about the region of reconnection. We present results for the variation of the maximum of vorticity, the time to reconnection, and other diagnostics of this flow as functions of the Reynolds number. From numerical simulation of the model equations, we infer and demonstrate the existence of exact solutions to the model to which its solutions arising from more general initial conditions are attracted at late times. In the limit of infinite Reynolds number, the model predicts eventual saturation of the axial strain, a feature observed in the recent work of Pumir &amp; Siggia and also observed in our full numerical simulations. In this respect the model captures the observed local dynamics of vortex stretching. However, because the global effects of external flows are not included in the model, the model predicts that the axial strain eventually decays and the maximum vorticity grows linearly at late times. In contrast, from the full simulations, we see the possible emergence of the behaviour of the axial strain at infinite Reynolds number. As our simulations are affected by non-local effects, we do observe saturation of the strain but no subsequent decay. It is also shown analytically that the model predicts a reconnection time which varies logarithmically with increasing Reynolds number. Comparison with the full numerical simulations shows a much slower variation of the reconnection time with increasing Reynolds number than predicted by the model. Other points of agreement and disagreement between the Saffman model and the simulations are discussed, Reconnection is also discussed from the point of view of its relation to the possible onset of nearly singular behaviour of the Euler equation. In agreement with the recent numerical results of Pumir &amp; Siggia, our results suggest that no singularity in the vorticity will form in a finite time for this initial condition.

Using a hybrid simulation approach that combines a lattice-Boltzmann method for fluid flow and a molecular dynamics model for polymers, we investigate the inertial migration of star-like and crew-cut polymer micelles in a square microchannel. It is found that they exhibit two types of equilibrium positions, which shift further away from the center of the microchannel when the Reynolds number (Re) increases, as can be observed for soft particles. What differs from the behaviors of soft particles is that here, the blockage ratio is no longer the decisive factor. When the sizes are the same, the star-like micelles are always relatively closer to the microchannel wall as they gradually transition from spherical to disc-like with the increase of Re. In comparison, the crew-cut micelles are only transformed into an ellipsoid. Conversely, when the hydrophobic core sizes are the same, the equilibrium position of the star-like micelles becomes closer to that of the crew-cut micelles. Our results demonstrate that for polymer micelles with a core-shell structure, the equilibrium position is no longer solely determined by their overall dimensions but depends on the core and shell's specific dimensions, especially the hydrophobic core size. This finding opens up a new approach for achieving the separation of micelles in inertial migration.