Ring Of Functions Research Articles

The effect of Reynolds number on the dynamics and wakes of freely rising and falling spheres

In this paper, we study the effect of the Reynolds number (Re) on the dynamics and vortex formation modes of spheres rising or falling freely through a fluid (whereRe= 100–15000). Since the oscillation of freely falling spheres was first reported by Newton (University of California Press, 3rd edn, 1726, translated in 1999), the fundamental question of whether a sphere will vibrate, as it rises or falls, has been the subject of a number of investigations, and it is clear that the mass ratiom* (defined as the relative density of the sphere compared to the fluid) is an important parameter to define when vibration occurs. Although all rising spheres (m* < 1) were previously found to oscillate, either chaotically or in a periodic zigzag motion or even to follow helical trajectories, there is no consensus regarding precise values of the mass ratio (m*crit) separating vibrating and rectilinear regimes. There is also a large scatter in measurements of sphere drag in both the vibrating and rectilinear regimes.In our experiments, we employ spheres with 133 combinations ofm* andRe, to provide a comprehensive study of the sphere dynamics and vortex wakes occurring over a wide range of Reynolds numbers. We find that falling spheres (m* > 1) always move without vibration. However, in contrast with previous studies, we discover that a whole regime of buoyant spheres rise through a fluid without vibration. It is only when one passes below a critical value of the mass ratio, that the sphere suddenly begins to vibrate periodically and vigorously in a zigzag trajectory within a vertical plane. The critical mass is nearly constant over two ranges of Reynolds number (m*crit≈ 0.4 forRe= 260–1550 andm*crit≈ 0.6 forRe> 1550). We do not observe helical or spiral trajectories, or indeed chaotic types of trajectory, unless the experiments are conducted in disturbed background fluid. The wakes for spheres moving rectilinearly are comparable with wakes of non-vibrating spheres. We find that these wakes comprise single-sided and double-sided periodic sequences of vortex rings, which we define as the ‘R’ and ‘2R’ modes. However, in the zigzag regime, we discover a new ‘4R’ mode, in which four vortex rings are created per cycle of oscillation. We find a number of changes to occur at a Reynolds number of 1550, and we suggest the possibility of a resonance between the shear layer instability and the vortex shedding (loop) instability. From this study, ensuring minimal background disturbances, we have been able to present a new regime map of dynamics and vortex wake modes as a function of the mass ratio and Reynolds number {m*,Re}, as well as a reasonable collapse of the drag measurements, as a function ofRe, onto principally two curves, one for the vibrating regime and one for the rectilinear trajectories.

An Experimental Investigation for Bubble Rising in Non-Newtonian Fluids and Empirical Correlation of Drag Coefficient

The velocity, shape, and trajectory of the rising bubble in polyacrylamide (PAM) and carboxymethylcellulose (CMC) aqueous solutions were experimentally investigated using a set of homemade velocimeters and a video camera. The effects of gas the flowrate and solution concentration on the bubble terminal velocity were examined respectively. Results show that the terminal velocity of the bubble increases with the increase in the gas flowrate and the decrease in the solution concentration. The shape of the bubble is gradually flattened horizontally to an ellipsoid with the increase in the Reynolds number (Re), Eötvös number (Eo), and Morton number (Mo). With the increase in the Re and Eo, the rising bubble in PAM aqueous solutions begin to oscillate, but there is no oscillation phenomena for CMC aqueous solutions. By dimensional analysis, the drag coefficient of a single bubble in non-Newtonian fluids in a moderate Reynolds number was correlated as a function of Re, Eo, and Archimedes number (Ar) based on the equivalent bubble diameter. The predicted results by the present correlation agree well with the experimental data.

Simulations of lateral mixing in cross-channel flow

Cross flow phenomena between connected sub-channels are studied by means of numerical simulations based on lattice-Boltzmann discretization. The cross (that is lateral) transfer is largely due to macroscopic instabilities developing at two shear layers. The characteristic size and advection velocity of the instabilities favorably compare with experimental results from the literature on a geometrically similar system. The strength of the cross flow strongly depends on the Reynolds number, with cross flow developing only for Reynolds numbers (based on macroscopic flow quantities) larger than 1360. Mass transfer between the sub-channels has been assessed by adding a passive scalar to the flow and solving its transport equation. As a result of the intimate connection of cross flow and lateral mass transfer, also the mass transfer coefficient is a pronounced function of Re.

Roughness Length of Water Vapor over Land Surfaces and Its Influence on Latent Heat Flux

Latent heat flux at the surface is largely dependent on the roughness length for water vapor (z(subscript 0q)). The determination of z(subscript 0q) is still uncertain because of its multifaceted characteristics of surface properties, atmospheric conditions and insufficient observations. In this study, observed values from the Fluxes Over Snow Surface Ⅱ field experiment (FLOSS-Ⅱ) from November 2002 to March 2003 were utilized to estimate z(subscript 0q) over various land surfaces: bare soil, snow, and senescent grass. The present results indicate that the estimated z(subscript 0q) over bare soil is much smaller than the roughness length of momentum (z(subscript 0m)); thus, the ratio z(subscript 0m)/z(subscript 0q) is larger than those of previous studies by a factor of 20-150 for the available flow regime of the roughness Reynolds number, Re(subscript *)＞0.1. On the snow surface, the ratio is comparable to a previous estimation for the rough flow (Re(subscript *)＞1), but smaller by a factor of 10-50 as the flow became smooth (Re(subscript *)＜1). Using the estimated ratio, an optimal regression equation of z(subscript 0m)/z(subscript 0q) is determined as a function of Re(subscript *) for each surface type. The present parameterization of the ratio is found to greatly reduce biases of latent heat flux estimation compared with that estimated by the conventional method, suggesting the usefulness of current parameterization for numerical modeling.

Numerical Simulations of Misalignment Effects in Microfluidic Interconnects

AbstractNumerical simulations were performed to see the effect of geometrical misalignment in pressure driven flows. Geometric misalignment effects on flow characteristics arising in three types of interconnection methods a) end-to-end interconnection, b) channel overlap when chips are stacked on top of each other, and c) the misalignment occurring due to the offset between the external tubing and the reservoir were investigated. For the case of end-to-end interconnection, the effect of misalignment was investigated for 0, 13, 50, 58, and 75% reduction in the available flow area at the location of geometrical misalignment. In the interconnection through channel overlap, various possible misalignment configurations were simulated by maintaining the same amount of misalignment (75% flow area reduction) for all the configurations. The effect of misalignment in a Tube-in-Reservoir interconnection was investigated by positioning the tube at an offset of 164μm from the reservoir center. All the results were evaluated in terms of the equivalent length of a straight pipe. The effect of reynolds number (Re) was also taken into account by performing additional simulations of aforementioned cases at reynolds numbers ranging from 0.075 to 75. The results are interpreted in terms of equivalent length (Le) as a function of Re and misalignment area ratio (A1:A2), where A1 is the original cross-sectional area of the channel and A2 is the available flow area at mismatch location. Equivalent length calculations revealed that the effect of misalignment in tube-in-reservoir interconnection method was the most insignificant when compared to the other two methods of interconnection

Effect of some geometric parameters on performance of PF 2 heat exchangers in periodic frosting

The thermal performance under conditions of initial frost growth and subsequent refrosting after a defrost is experimentally studied for the parallel flow parallel fin (PF 2) heat exchanger, a new style of heat exchanger that uses louvered bent fins on flat tubes to enhance water drainage when the flat tubes are horizontal (typically outdoor heat exchanger in heat pump operation). This paper focuses on quantification of the effects of geometry (i.e. fin pitch 12–22 fpi and louver pitch 1.4–2.8 mm) on defrost and refrost times. Eight heat exchangers differing in louver pitch and fin spacing are studied. A series of tests are conducted in search for the best geometry. The effects of geometry on heat transfer (thermal performance) and pressure drop for air face velocities of 0.9, 2, and 3 m/s are determined and used for comparison. Characteristics of initial heat transfer coefficient and pressure drop during the first frosting cycle are reported in terms of Colburn j 0 factor and Fanning friction f 0 factor, as a function of Re L p and geometry. The newly developed air-side correlations for PF 2 heat exchangers (Colburn j 0 factor and Fanning friction f 0 factor) predict the test data within rms error of ±10%, and with mean deviation of 2.95% and 4.98%, respectively. The correlations are base on a very low Reynolds numbers in the range of 100–620, and 8 PF 2 heat exchangers using 48 experimental data.

Segal's conjecture and the Burnside rings of fusion systems

For a given saturated fusion system ℱ we define the ring A(S)ℱ of the ℱ-invariants of the Burnside ring functor. We show how this ring is related to the Burnside ring of the fusion system ℱ and how it appears naturally in the analogue of Segal's conjecture for the classifying spectrum 𝔹ℱ. We give an explicit description of A(S)ℱ and we prove it is a local ring.

Tambara Functors on Profinite Groups and Generalized Burnside Functors

The Tambara functor was defined by Tambara in the name of TNR-functor, to treat certain ring-valued Mackey functors on a finite group. Recently Brun revealed the importance of Tambara functors in the Witt–Burnside construction. In this article, we define the Tambara functor on the Mackey system of Bley and Boltje. Yoshida's generalized Burnside ring functor is the first example. Consequently, we can consider a Tambara functor on any profinite group. In relation with the Witt–Burnside construction, we can give a Tambara-functor structure on Elliott's functor V M , which generalizes the completed Burnside ring functor of Dress and Siebeneicher.

Friction factor of two-dimensional rough-boundary turbulent soap film flows

We use momentum-transfer arguments to predict the friction factor f in two-dimensional turbulent soap film flows with rough boundaries (an analog of three-dimensional pipe flow) as a function of Reynolds number Re and roughness r , considering separately the inverse energy cascade and the forward enstrophy cascade. At intermediate Re, we predict a Blasius-like friction factor scaling of f proportional, variant Re{-1/2} in flows dominated by the enstrophy cascade, distinct from the energy cascade scaling of Re{-1/4} . For large Re, f approximately r in the enstrophy-dominated case. We use conformal map techniques to perform direct numerical simulations that are in satisfactory agreement with theory and exhibit data collapse scaling of roughness-induced criticality, previously shown to arise in the three-dimensional pipe data of Nikuradse.

Anomalous scaling of a passive vector advected by the Navier–Stokes velocity field

Using the field theoretic renormalization group and the operator-product expansion, the model of a passive vector field (a weak magnetic field in the framework of the kinematic MHD) advected by the velocity field which is governed by the stochastic Navier–Stokes equation with the Gaussian random stirring force δ-correlated in time and with the correlator proportional to k4−d−2ε is investigated to the first order in ε (one-loop approximation). It is shown that the single-time correlation functions of the advected vector field have anomalous scaling behavior and the corresponding exponents are calculated in the isotropic case, as well as in the case with the presence of large-scale anisotropy. The hierarchy of the anisotropic critical dimensions is briefly discussed and the persistence of the anisotropy inside the inertial range is demonstrated on the behavior of the skewness and hyperskewness (dimensionless ratios of correlation functions) as functions of the Reynolds number Re. It is shown that even though the present model of a passive vector field advected by the realistic velocity field is mathematically more complicated than, on one hand, the corresponding models of a passive vector field advected by ‘synthetic’ Gaussian velocity fields and, on the other hand, than the corresponding model of a passive scalar quantity advected by the velocity field driven by the stochastic Navier–Stokes equation, the final one-loop approximate asymptotic scaling behavior of the single-time correlation or structure functions of the advected fields of all models are defined by the same anomalous dimensions (up to normalization).

Reynolds number dependence of gas-phase turbulence in particle-laden flows: Effects of particle inertia and particle loading

A gas-particle flow experiment at a low particle loading ( m = 0.4) in a vertical downward pipe is conducted at three different Reynolds numbers (Re = 6000, 10,000, and 13,000) to investigate the Re influence on the gas-phase turbulence modulation. The mean and fluctuating velocity data of both phases are acquired using a two-component LDV/PDA system. Two particles of varying degrees of inertia (i.e. high-density 70 µm glass beads and low-density 60 µm cenospheres) are used as the model particles to examine the effect of particle inertia on the trend in the turbulence modulation as a function of Re. An experiment at a higher particle loading ( m = 4.0) using the glass beads is also conducted to examine the effect of particle concentration. In the presence of high inertia particles (St T > 500) at a low particle loading, the gas-phase turbulence intensity in the pipe core is increased with increasing Re resulting in turbulence enhancement relative to the unladen flow. The turbulence enhancement is attributed to 1) a modification of the turbulence production by the Reynolds stress due to interparticle collision and/or 2) a reduction in the fluctuating drag force due to a change in the radial profile of the particle concentration. In contrast, the gas-phase turbulence intensity in the presence of low inertia particles (St T < 500) is found to decrease with increasing Re similar to the trend in the unladen flow. Lastly, the turbulence enhancement at high Re is not observed at a high particle loading where the turbulent energy dissipation by the fluctuating drag force is dominant.

Decomposition of the Witt–Burnside ring and Burnside ring of an abelian profinite group

Let G, H be abelian profinite groups whose orders are coprime and assume that q ranges over the set of integers. The aim of this paper is to establish an isomorphism of functors W G q ○ W H q ≅ W G × H q , where W G q denotes the q-deformed Witt–Burnside ring functor of G introduced in [Y.-T. Oh, q-Deformation of Witt–Burnside rings, Math. Z. 207 (1) (2007) 151–191]. To do this, we first establish an isomorphism of functors B G q ○ B H q ≅ B G × H q , where B G q denotes the q-deformed Burnside ring functor of G which was also introduced in [Y.-T. Oh, q-Deformation of Witt–Burnside rings, Math. Z. 207 (1) (2007) 151–191]. As an application, we derive a pseudo-multiplicative property of the q-Möbius function associated to the lattice of open subgroups of the direct sum of G and H.

Clusters of sedimenting high-Reynolds-number particles

We report experiments wherein groups of particles were allowed to sediment in an otherwise quiescent fluid contained in a large tank. The Reynolds number of the particles, defined as Re = aU/ν, ranged from 93 to 425; here, a is the radius of the spherical particle, U its settling velocity and ν the kinematic viscosity of the fluid. The characteristic size of a cluster, in a plane transverse to gravity, was measured by a ‘cluster variance’(〈r2t〉); the latter is defined as the mean square of the transverse coordinates of all constituent particles, averaged over a series of runs. The cluster variance, when plotted as a function of time, exhibited two regimes. There was a quadratic growth in the variance at short times(〈r2t〉 ∝ t2), while for long times, the cluster variance exhibited a slower sublinear growth with 〈r2t〉 ∝ t0.67. A theory, based on isotropic repulsive hydrodynamic interactions between particles, predicts the cluster variance to grow as t2/3 in the limit of long times. The theoretical framework was originally proposed to describe the long-time self-similar evolution of dilute clusters in the limit Re ≪ 1 Subramanian &amp; Koch (J. Fluid Mech., vol. 603, 2008, p. 63), when the probability of wake-mediated interactions between particles remains asymptotically small; the latter requirement is satisfied for homogeneous spherical clusters larger than a critical radius, and is evidently satisfied for planar clusters oriented transversely to gravity. The isotropy of the interactions therefore stems from the isotropy, at large distances, of the disturbance velocity field produced by a single sedimenting particle outside its wake(which contains the compensating inflow to satisfy mass conservation). Herein, the theory is extended to large Re using an empirical correlation for the drag on a sedimenting particle. This allows one to predict, as a function of Re, the numerical prefactors in the expressions for the cluster variance of both spherical and planar clusters; the predictions for the growth exponent remain unchanged. The agreement between the theoretical and experimental growth exponents supports the hypothesis of a self-similar expansion at long times. The prefactor determined from the experimental observations is found to lie between the theoretical predictions for planar and spherical clusters.

A long gas bubble moving in a tube with flowing liquid

The steady axisymmetric behavior of a long gas bubble moving with a flowing liquid in a straight round tube is studied by computationally solving the nonlinear Navier–Stokes equations using a Galerkin finite-element method with a boundary-fitted mesh for wide ranges of capillary number Ca and Reynolds number Re. As illustrated with a series of computed results, the hydrodynamic stresses due to liquid flow around the bubble tend to shape the middle section of long bubbles into a cylinder of constant radius, with a uniform annular liquid flow adjacent to the tube wall. But the surface tension effect tends to cause nonuniformities in the annular liquid film thickness. The annular liquid film thickness generally increases with increasing Ca, but decreases with increasing Re. In units of the bubble velocity relative to the tube wall, the average liquid flow velocity U ¯ is always less than unity, indicating that the bubble always moves faster than the average liquid flow. For convenient practical applications, a least-square fitted empirical formula is obtained in a form of U ¯ = 1 - A ∼ / ( 1 + B ∼ Ca - C ∼ ) with A ∼ , B ∼ , and C ∼ being functions of Re. The fact that the behavior of long bubbles moving in a tube appears independent of the bubble length is consistent with the inconsequential influence of the uniform annular film flow in bubble’s middle section to the bubble dynamics. Whereas all the long bubbles exhibit similar nose profile, various tail shapes can be obtained by adjusting the values of Re and Ca.

Density functional study of collective electron localization: detection by persistentcurrent

We apply the optimized effective potential (OEP) implementation of density functionaltheory (DFT) to the model system of interacting spinless electrons on a quantum ring.The ring encircles a magnetic flux that induces a persistent current. In a perfectrotationally invariant system the current does not depend on the electron–electroninteraction (the latter is characterized by a standard dimensionless parameterrS) and hence is not sensitive to the microscopic structure of the electron correlated state.This changes, however, if a symmetry-breaking external potential is introduced or, in arealistic system, due to the crystal lattice potential (Hamer et al 1987 J. Phys. A: Math.Gen. 20 5677–93). In our model, we calculate the persistent current as a function ofrS in thepresence of a weak Gaussian-shaped ‘impurity’ potential. We find that while below a threshold valuerS<rSc≈2.05 the current isindependent of rS, itdecays exponentially for rS>rSc. This signals the formation of an electron Wigner crystal pinnedby the impurity potential. The electron density, homogeneous belowrSc, indeed shows a periodic modulation atrS>rSc. The modulationamplitude follows a (rS−rSc)1/2 behaviour which is characteristic for a second-order phase transition, as expected in themean-field-type DFT-OEP approach. Our calculation shows that the macroscopic current,which is a quantity directly accessible in DFT, can serve as an indicator of the formation ofa correlated electron state.

Viscous oscillations in a circular cylinder with an elastic cover on the free surface

The effects of an elastic membrane on the viscous oscillations of liquid filling a circular cylindrical container are studied by using the natural viscous complex eigenfunctions of the problem. The free surface of the liquid is assumed to be fully covered by the membrane. By projecting the governing equations onto an appropriate basis, a nonlinear eigenvalue problem for the complex frequencies is obtained. This is then solved to obtain the modal frequencies as a function of the Reynolds number Re, the tension parameter τ, the mass parameter ζ and the liquid depth h. The zero velocity conditions on both the side and bottom walls are satisfied unlike in earlier studies where either only the sidewall or only the bottom wall conditions were met. Results are presented for the four lowest non-axisymmetric modes as a function of Re, h, τ and ζ. The elastic cover increases the slosh frequencies but only in comparison with an uncovered free surface with a contact line that is free to move; the frequencies are lower when compared with those of a free surface with pinned contact line. There are ranges of Re, h, τ and ζ for which the oscillations are overdamped and the sloshing is aperiodic. Though the frequencies and damping rates decrease for an increasing mass of the elastic cover, there exist ranges of Re, h and τ for which the heavier cover produces higher slosh frequencies.

Numerical investigation of the hydrodynamics of anguilliform swimming in the transitional and inertial flow regimes

We employ numerical simulation to investigate the hydrodynamic performance of anguilliform locomotion and compare it with that of carangiform swimming as the Reynolds number (Re) and the tail-beat frequency (Strouhal number, St) are systematically varied. The virtual swimmer is a 3-D lamprey-like flexible body undulating with prescribed experimental kinematics of anguilliform type. Simulations are carried out for three Reynolds numbers spanning the transitional and inertial flow regimes, Re=300, 4000 (viscous flow), and infinity (inviscid flow). The net mean force is found to be mainly dependent on the tail-beat frequency rather than the tail-beat amplitude. The critical Strouhal number, St, at which the net mean force becomes zero (constant-speed self-propulsion) is, similar to carangiform swimming, a decreasing function of Re and approaches the range of St numbers at which most anguilliform swimmers swim in nature (St approximately 0.45) only as Re increases. The anguilliform swimmer's force time series is characterized by significantly smaller fluctuations above the mean than that for carangiform swimmers. In stark contrast with carangiform swimmers, the propulsive efficiency of anguilliform swimmers at St is not an increasing function of Re but instead is maximized in the transitional regime. Furthermore, the power required for anguilliform swimming is less than that for the carangiform swimmer at the same Re. We also show that the form drag decreases while viscous drag increases as St increases. Finally, our simulations reinforce our previous finding for carangiform swimmers that the 3-D wake structure depends primarily on the Strouhal number.

Modelling of laminar flow in the inlet section of rectangular microchannels

This paper is a study of laminar flow in rectangular microchannels. The behaviour of compressible and incompressible fluids in microchannels was simulated using CFD software. Numerical data were compared to experimental measurements to test the validity of CFD models. The velocity profile of flow developing inside the channel is described as a function of the Reynolds number Re, varying from 100 to 2000, and the aspect ratio h/w, ranging from 1 to 0.125. The fundamental importance of the entrance length of microchannels is highlighted. Numerical data were applied to define analytical formulae covering the minimum entrance length for fully developed laminar flow of compressible fluids, viscous stress and incremental pressure drop effect during flow development, and the velocity profiles of flow for compressible and incompressible fluids.

Reynolds number dependence of mixing in a lock-exchange system from direct numerical and large eddy simulations

Turbulent mixing of water masses of different temperatures and salinities is an important process for both coastal and large-scale ocean circulation. It is, however, difficult to capture computationally. One of the reasons is that mixing in the ocean occurs at a wide range of complexity, with the Reynolds number reaching Re = O ( 10 6 ) , or even higher. In this study, we continue to investigate whether large eddy simulation ( LES) can be a reliable computational tool for stratified mixing in turbulent oceanic flows. LES is attractive because it can be O ( 1000 ) times faster than a direct numerical simulation ( DNS) of stratified mixing in turbulent flows. Before using the LES methodology to compute mixing in realistic oceanic flows, however, a careful assessment of the LES sensitivity with respect to Re needs to be performed first. The main objectives of this study are: (i) to investigate the performance of different LES models at high Re, such as those encountered in oceanic flows; and (ii) to study how mixing varies as a function of Re. To this end, as a benchmark we use the lock-exchange problem, which is described by unambigous and simple initial and boundary conditions. The background potential energy, which accurately quantifies irreversible mixing in an enclosed system, is used as the main criterion in a posteriori testing of LES. This study has two main achievements. The first is that we investigate the accuracy of six combinations of two different classes of LES models, namely eddy-viscosity and approximate deconvolution types, for 3 × 10 3 ⩽ Re ⩽ 3 × 10 4 , for which DNS data is computed. We find that all LES models almost always provide significantly more accurate results than cases without LES models. Nevertheless, no single LES model that is persistently superior to others over this Re range could be identified. Then, an ensemble of the four best performing LES models is selected in order to estimate mixing taking place in this system at Re = 10 5 and 10 6 , for which DNS is presently not feasible. Thus the second achievement of this study is to quantify mixing taking place in this system over an Re range that changes by three orders of magnitude. We find that the background potential energy increases by about 67% when Re is increased from Re = 10 3 to Re = 10 6 , within the computation period, with the most significant increase taking place from Re = 3 × 10 3 to Re = 10 5 .

Potential of a Novel Safety Evaluation of Nanomaterials Using a Proteomic Approach

We studied changes in protein expressions using human monoblastic leukemia cells (U937) exposed to carbon black (CB) to explore the possibility of a new safety evaluation method using a proteomic approach. The cells exposed to CB with a particle size of approximately 85 nm did not show cell growth inhibition by 96 hr. However, when we performed two-dimensional electrophoresis with U937 cell lysates four times and analyzed the gel images, we found that the CB-exposed cells had changed significantly in 14 protein spots, with expression amounts being different by more than two-fold those at 96 hr. Only one protein was up-regulated while the others were downregulated. Ten of the 14 spots were identified using peptide mass fingerprinting: transportin 1, thioredoxin domain containing 5, annexin A2, otubain 1, 14-3-3 protein γ, etc .T he functions of these protein sa re associated with metabolism, responses to stress, signal transduction, and cell differentiation. According to our research, CB undoubtedly causes biological responses, but those responses are not detectable by conventional methods. Regarding safety risk, the proteomic approach could possibly detect biological responses more sensitively than conventional in vitro evaluation methods.