Showing papers on "Reynolds number published in 2007"

Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes

Abstract: Advanced Transport Phenomena is ideal as a graduate textbook. It contains a detailed discussion of modern analytic methods for the solution of fluid mechanics and heat and mass transfer problems, focusing on approximations based on scaling and asymptotic methods, beginning with the derivation of basic equations and boundary conditions and concluding with linear stability theory. Also covered are unidirectional flows, lubrication and thin-film theory, creeping flows, boundary layer theory, and convective heat and mass transport at high and low Reynolds numbers. The emphasis is on basic physics, scaling and nondimensionalization, and approximations that can be used to obtain solutions that are due either to geometric simplifications, or large or small values of dimensionless parameters. The author emphasizes setting up problems and extracting as much information as possible short of obtaining detailed solutions of differential equations. The book also focuses on the solutions of representative problems. This reflects the book's goal of teaching readers to think about the solution of transport problems.

Drag force of intermediate Reynolds number flow past mono- and bidisperse arrays of spheres

Abstract: Extensive lattice-Boltzmann simulations were performed to obtain the drag force for random arrays of monodisperse and bidisperse spheres. For the monodisperse systems, 35 different combinations of the Reynolds number Re (up to Re = 1,000) and packing fraction were studied, whereas for the bidisperse systems we also varied the diameter ratio (from 1:1.5 to 1:4) and composition, which brings the total number of different systems that we considered to 150. For monodisperse systems, the data was found to be markedly different from the Ergun equation and consistent with a correlation, based on similar type of simulations up to Re = 120. For bidisperse systems, it was found that the correction of the monodisperse drag force for bidispersity, which was derived for the limit Re = 0, also applies for higher-Reynolds numbers. On the basis of the data, a new drag law is suggested for general polydisperse systems, which is on average within 10% of the simulation data for Reynolds numbers up to 1,000, and diameter ratios up to 1:4

Large-scale influences in near-wall turbulence

Abstract: Hot-wire data acquired in a high Reynolds number facility are used to illustrate the need for adequate scale separation when considering the coherent structure in wall-bounded turbulence. It is found that a large-scale motion in the log region becomes increasingly comparable in energy to the near-wall cycle as the Reynolds number increases. Through decomposition of fluctuating velocity signals, it is shown that this large-scale motion has a distinct modulating influence on the small-scale energy (akin to amplitude modulation). Reassessment of DNS data, in light of these results, shows similar trends, with the rate and intensity of production due to the near-wall cycle subject to a modulating influence from the largest-scale motions.

A turbulent jet in crossflow analysed with proper orthogonal decomposition

Abstract: Detailed instantaneous velocity fields of a jet in crossflow have been measured with stereoscopic particle image velocimetry (PIV). The jet originated from a fully developed turbulent pipe flow and entered a crossflow with a turbulent boundary layer. The Reynolds number based on crossflow velocity and pipe diameter was 2400 and the jet to crossflow velocity ratios were R=3.3 and R=1.3. The experimental data have been analysed by proper orthogonal decomposition (POD). For R=3.3, the results in several different planes indicate that the wake vortices are the dominant dynamic flow structures and that they interact strongly with the jet core. The analysis identifies jet shear-layer vortices and finds that these vortical structures are more local and thus less dominant. For R=1.3, on the other hand, jet shear-layer vortices are the most dominant, while the wake vortices are much less important. For both cases, the analysis finds that the shear-layer vortices are not coupled to the dynamics of the wake vortices. Finally, the hanging vortices are identified and their contribution to the counter-rotating vortex pair (CVP) and interaction with the newly created wake vortices are described.

Generation of a magnetic field by dynamo action in a turbulent flow of liquid sodium.

Abstract: We report the observation of dynamo action in the von Karman sodium experiment, i.e., the generation of a magnetic field by a strongly turbulent swirling flow of liquid sodium. Both mean and fluctuating parts of the field are studied. The dynamo threshold corresponds to a magnetic Reynolds number R-m similar to 30. A mean magnetic field of the order of 40 G is observed 30\% above threshold at the flow lateral boundary. The rms fluctuations are larger than the corresponding mean value for two of the components. The scaling of the mean square magnetic field is compared to a prediction previously made for high Reynolds number flows.

Global stability of base and mean flows : a general approach and its applications to cylinder and open cavity flows

Abstract: This article deals with the first Hopf bifurcation of a cylinder flow, and more particularly with the properties of the unsteady periodic Karman vortex street regime that sets in for supercritical Reynolds numbers Re > 46. Barkley (Europhys. Lett. vol.75, 2006, p. 750) has recently studied the linear properties of the associated mean flow, i.e. the flow which is obtained by a time average of this unsteady periodic flow. He observed, thanks to a global mode analysis, that the mean flow is marginally stable and that the eigenfrequencies associated with the global modes of the mean flow fit the Strouhal to Reynolds experimental function well in the range 46 < Re < 180. The aim of this article is to give a theoretical proof of this result near the bifurcation. For this, we do a global weakly nonlinear analysis valid in the vicinity of the critical Reynolds number Rec based on the small parameter e = Rec−1 − Re−1 ≪ 1. We compute numerically the complex constants λ and μ′ which appear in the Stuart-Landau amplitude equation: dA/dt = e λA − eμ′ A|A|2. Here A is the scalar complex amplitude of the critical global mode. By analysing carefully the nonlinear interactions yielding the term μ′, we show for the cylinder flow that the mean flow is approximately marginally stable and that the linear dynamics of the mean flow yields the frequency of the saturated Stuart-Landau limit cycle. We will finally show that these results are not general, by studying the case of the bifurcation of an open cavity flow. In particular, we show that the mean flow in this case remains strongly unstable and that the frequencies associated with the eigenmodes do not exactly match those of the nonlinear unsteady periodic cavity flow. It will be demonstrated that two precise conditions must hold for a linear stability analysis of a mean flow to be relevant and useful.

Heavy particle concentration in turbulence at dissipative and inertial scales.

Abstract: Spatial distributions of heavy particles suspended in an incompressible isotropic and homogeneous turbulent flow are investigated by means of high resolution direct numerical simulations. In the dissipative range, it is shown that particles form fractal clusters with properties independent of the Reynolds number. Clustering is there optimal when the particle response time is of the order of the Kolmogorov time scale � � . In the inertial range, the particle distribution is no longer scale invariant. It is, however, shown that deviations from uniformity depend on a rescaled contraction rate, which is different from the local Stokes number given by dimensional analysis. Particle distribution is characterized by voids spanning all scales of the turbulent flow; their signature in the coarse-grained mass probability distribution is an algebraic behavior at small densities.

Active control of high-speed and high-Reynolds-number jets using plasma actuators

Abstract: Localized arc filament plasma actuators are used to control an axisymmetric Mach 1.3 ideally expanded jet of 2.54 cm exit diameter and a Reynolds number based on the nozzle exit diameter of about 1.1×106. Measurements of growth and decay of perturbations seeded in the flow by the actuators, laser-based planar flow visualizations, and particle imaging velocimetry measurements are used to evaluate the effects of control. Eight actuators distributed azimuthally inside the nozzle, approximately 1 mm upstream of the nozzle exit, are used to force various azimuthal modes over a large frequency range (StDF of 0.13 to 1.3). The jet responded to the forcing over the entire range of frequencies, but the response was optimum (in terms of the development of large coherent structures and mixing enhancement) around the jet preferred Strouhal number of 0.33 (f = 5 kHz), in good agreement with the results in the literature for low-speed and low-Reynolds-number jets. The jet (with a thin boundary layer, D/θ ∼ 250) also responded to forcing with various azimuthal modes (m = 0 to 3 and m = ±1, ±2, ±4), again in agreement with instability analysis and experimental results in the literature for low-speed and low-Reynolds-number jets. Forcing the jet with the azimuthal mode m = ±1 at the jet preferred-mode frequency provided the maximum mixing enhancement, with a significant reduction in the jet potential core length and a significant increase in the jet centreline velocity decay rate beyond the end of the potential core.

Scaling analysis and simulation of strongly stratified turbulent flows

Abstract: Direct numerical simulations of stably and strongly stratified turbulent flows with Reynolds number Re >> 1 and horizontal Froude number F-h > 1, viscous forces are unimportant and l(v) scales as l ...

Flapping dynamics of a flag in a uniform stream

Abstract: We consider the flapping stability and response of a thin two-dimensional flag of high extensional rigidity and low bending rigidity. The three relevant non-dimensional parameters governing the problem are the structure-to-fluid mass ratio, μ = ρ s h /(ρ f L); the Reynolds number, Re=VL/ν; and the non-dimensional bending rigidity, K B = EI / (ρfV 2 L 3 ). The soft cloth of a flag is represented by very low bending rigidity and the subsequent dominance of flow-induced tension as the main structural restoring force. We first perform linear analysis to help understand the relevant mechanisms of the problem and guide the computational investigation. To study the nonlinear stability and response, we develop a fluid-structure direct simulation (FSDS) capability, coupling a direct numerical simulation of the Navier-Stokes equations to a solver for thin-membrane dynamics of arbitrarily large motion. With the flow grid fitted to the structural boundary, external forcing to the structure is calculated from the boundary fluid dynamics. Using a systematic series of FSDS runs, we pursue a detailed analysis of the response as a function of mass ratio for the case of very low bending rigidity (K B = to-4) and relatively high Reynolds number (Re=10 3 ). We discover three distinct regimes of response as a function of mass ratio μ: (I) a small μ regime of fixed-point stability; (II) an intermediate μ regime of period-one limit-cycle flapping with amplitude increasing with increasing μ; and (III) a large μ regime of chaotic flapping. Parametric stability dependencies predicted by the linear analysis are confirmed by the nonlinear FSDS, and hysteresis in stability is explained with a nonlinear softening spring model. The chaotic flapping response shows up as a breaking of the limit cycle by inclusion of the 3/2 superharmonic. This occurs as the increased flapping amplitude yields a flapping Strouhal number (St=2Af/V) in the neighbourhood of the natural vortex wake Strouhal number, St ≃ 0.2. The limit-cycle von Karman vortex wake transitions in chaos to a wake with clusters of higher intensity vortices. For the largest mass ratios, strong vortex pairs are distributed away from the wake centreline during intermittent violent snapping events, characterized by rapid changes in tension and dynamic buckling.

Flexible flapping airfoil propulsion at low Reynolds numbers

Abstract: Water tunnel experiments on a flexible airfoil plunging with constant amplitude have been carried out for Reynolds numbers of 0 to 27000. Peaks in thrust coefficient at intermediate values of airfoil stiffness were observed at both zero and non-zero Reynolds numbers, indicating that a degree of flexibility is beneficial at low Reynolds numbers. Time-averaged velocity fields and momentum flux data revealed a broader, higher-velocity jet in cases of optimum airfoil stiffness. Stronger vortices, separated by a larger lateral distance, characterised the corresponding instantaneous velocity fields. The flexibility causes the airfoil to pitch passively; the phase angle of the pitch was found to lead the plunge. Pitch amplitude and trailing-edge amplitude were found to be single-valued functions of pitch phase angle. The shape characteristics of the airfoil could therefore be described by the pitch phase angle only. Thrust coefficient was found to be a function of only two parameters: Strouhal number and pitch phase angle. For each Strouhal number, a peak in thrust coefficient was observed at a particular value of the pitch phase angle. The optimum pitch phase angle was found to tend to a limit of 105±5 degrees at very large Strouhal numbers. A significant thrust benefit was observed over very stiff airfoils when the optimum flexibility is utilized.

Scalings and decay of fractal-generated turbulence

Abstract: A total of 21 planar fractal grids pertaining to three different fractal families have been used in two different wind tunnels to generate turbulence The resulting turbulent flows have been studied using hot wire anemometry Irrespective of fractal family, the fractal-generated turbulent flows and their homogeneity, isotropy, and decay properties are strongly dependent on the fractal dimension Df≤2 of the grid, its effective mesh size Meff (which we introduce and define) and its ratio tr of largest to smallest bar thicknesses, tr=tmax∕tmin With relatively small blockage ratios, as low as σ=25%, the fractal grids generate turbulent flows with higher turbulence intensities and Reynolds numbers than can be achieved with higher blockage ratio classical grids in similar wind tunnels and wind speeds U The scalings and decay of the turbulence intensity u′∕U in the x direction along the tunnel’s center line are as follows (in terms of the normalized pressure drop CΔP and with similar results for v′∕U and w′∕U)

MHD simulations of the magnetorotational instability in a shearing box with zero net flux II. The effect of transport coefficients

Abstract: Aims. We study the influence of the choice of transport coefficients (viscosity and resistivity) on MHD turbulence driven by the magnetorotational instability (MRI) in accretion disks. Methods. We follow the methodology described in Paper I: we adopt an unstratified shearing box model and focus on the case where the net vertical magnetic flux threading the box vanishes. For the most part we use the operator split code ZEUS, including explicit transport coefficients in the calculations. However, we also compare our results with those obtained using other algorithms (NIRVANA, the PENCIL code and a spectral code) to demonstrate both the convergence of our results and their independence of the numerical scheme. Results. We find that small scale dissipation affects the saturated state of MHD turbulence. In agreement with recent similar numerical simulations done in the presence of a net vertical magnetic flux, we find that turbulent activity (measured by the rate of angular momentum transport) is an increasing function of the magnetic Prandtl number Pm for all values of the Reynolds number Re that we investigated. We also found that turbulence disappears when the Prandtl number falls below a critical value Pmc that is apparently a decreasing function of Re. For the limited region of parameter space that can be probed with current computational resources, we always obtained Pmc > 1. Conclusions. We conclude that the magnitudes of the transport coefficients are important in determining the properties of MHD turbulence in numerical simulations in the shearing box with zero net flux, at least for Reynolds numbers and magnetic Prandtl numbers that are such that transport is not dominated by numerical effects and thus can be probed using current computational resources.

Impact of dimensionless numbers on the efficiency of magnetorotational instability induced turbulent transport

Abstract: The magnetorotational instability (MRI) is presently the most promising source of turbulent transport in accretion discs. However, some important issues still need to be addressed to quantify the role of MRI in discs; in particular no systematic investigation of the role of the physical dimensionless parameters of the problem on the dimensionless transport has been undertaken yet. For completeness, we first generalize existing results on the marginal stability limit in the presence of both viscous and resistive dissipation, exhibit simple scalings for all relevant limits, and give them a physical interpretation. We then re-examine the question of transport efficiency through numerical simulations in the simplest setting of a local, unstratified shearing box, with the help of a pseudo-spectral incompressible 3D code; viscosity and resistivity are explicitly accounted for. We focus on the effect of the dimensionless magnetic field strength, the Reynolds number and the magnetic Prandtl number. First, we complete existing investigations on the field strength dependence by showing that the transport in high magnetic pressure discs close to marginal stability is highly time dependent and surprisingly efficient. Secondly, we bring to light a significant dependence of the global transport on the magnetic Prandtl number, with α ∝ Pm δ for the explored range: 0.12 < Pm < 8 and 200 < Re < 6400 (δ being in the range 0.25‐0.5). We show that the dimensionless transport is not correlated to the dimensionless linear growth rate, contrary to a largely held expectation. For large enough Reynolds numbers, one would expect that the reported Prandtl number scaling of the transport should saturate, but such a saturation is out of reach of the present generation of supercomputers. Understanding this saturation process is nevertheless quite critical to accretion disc transport theory, as the magnetic Prandtl number Pm is expected to vary by many orders of magnitude between the various classes of discs, from Pm � 1 in young stellar object discs to Pm or � 1 in active galactic nucleus discs. More generally, these results stress the need to control dissipation processes in astrophysical simulations.

Fluid dynamics of self-propelled microorganisms, from individuals to concentrated populations

Abstract: Nearly close-packed populations of the swimming bacterium Bacillus subtilis form a collective phase, the “Zooming BioNematic” (ZBN). This state exhibits large-scale orientational coherence, analogous to the molecular alignment of nematic liquid crystals, coupled with remarkable spatial and temporal correlations of velocity and vorticity, as measured by both novel and standard applications of particle imaging velocimetry. The appearance of turbulent dynamics in a system which is nominally in the regime of Stokes flow can be understood by accounting for the local energy input by the swimmers, with a new dimensionless ratio analogous to the Reynolds number. The interaction between organisms and boundaries, and with one another, is modeled by application of the methods of regularized Stokeslets.

A passive planar micromixer with obstructions for mixing at low Reynolds numbers

Abstract: Passive mixers rely on the channel geometry to mix fluids. However, many previously reported designs either work efficiently only at moderate to high Reynolds numbers (Re), or require a complex 3D channel geometry that is often difficult to fabricate. In this paper, we report design, simulation, fabrication and characterization of a planar passive microfluidic mixer capable of mixing at low Reynolds numbers. The design incorporates diamond-shaped obstructions within the microchannel to break-up and recombine the flow. Simulation and experimental results of the developed micromixer show excellent mixing performance over a wide range of flow conditions (numerically: 0.01 < Re < 100, experimentally: 0.02 < Re < 10). The micromixer is also characterized by low pressure drop, an important characteristic for integration into complex, cascading microfluidic systems. Due to the simple planar structure of the micromixer, it can be easily realized and integrated with on-chip microfluidic systems, such as micro total analysis systems (μTAS) or lab on a chip (LOC).

Stability of a Jet in Confined Pressure-Driven Biphasic Flows at Low Reynolds Numbers

Abstract: Motivated by its importance for microfluidic applications, we study the stability of jets formed by pressure-driven concentric biphasic flows in cylindrical capillaries. The specificity of this variant of the classical Rayleigh-Plateau instability is the role of the geometry which imposes confinement and Poiseuille flow profiles. We experimentally evidence a transition between situations where the flow takes the form of a jet and regimes where drops are produced. We describe this as the transition from convective to absolute instability, within a simple linear analysis using lubrication theory for flows at low Reynolds number, and reach remarkable agreement with the data.

Visualizing the geometry of state space in plane Couette flow

Abstract: Motivated by recent experimental and numerical studies of coherent structures in wall-bounded shear flows, we initiate a systematic exploration of the hierarchy of unstable invariant solutions of the Navier-Stokes equations. We construct a dynamical, 10^5-dimensional state-space representation of plane Couette flow at Re = 400 in a small, periodic cell and offer a new method of visualizing invariant manifolds embedded in such high dimensions. We compute a new equilibrium solution of plane Couette flow and the leading eigenvalues and eigenfunctions of known equilibria at this Reynolds number and cell size. What emerges from global continuations of their unstable manifolds is a surprisingly elegant dynamical-systems visualization of moderate-Reynolds turbulence. The invariant manifolds tessellate the region of state space explored by transiently turbulent dynamics with a rigid web of continuous and discrete symmetry-induced heteroclinic connections.

Role of the elasticity number in the entry flow of dilute polymer solutions in micro-fabricated contraction geometries

Abstract: We explore the interplay of fluid inertia and fluid elasticity in planar entry flows by studying the flow of weakly elastic solutions through micro-fabricated planar contraction geometries. The small characteristic lengthscales make it possible to achieve a wide range of Weissenberg numbers (0.4 < Wi < 42) and Reynolds numbers (0.03 < Re < 12), allowing access to a large region of Wi-Re space that is typically unattainable in conventional macroscale entry flow experiments. Experiments are carried out using a series of dilute solutions (0.78 < clc* < 1.09) of a high molecular weight polyethylene oxide, in which the solvent viscosity is varied in order to achieve a range of elasticity numbers, 2.8 < El = WilRe < 68. Fluorescent streak imaging and micro-particle image velocimetry ([L-PIV) are used to characterize the kinematics, which are classified into a number of flow regimes including Newtonian-like flow at low Wi, steady viscoelastic flow, unsteady diverging flow and vortex growth regimes. Progressive changes in the centreline velocity profilt are used to identify each of the flow regimes and to map the resulting stability boundaries in Wi-Re space. The same flow transitions can also be detected through measurements of the enhanced pressure drop across the contraction/expansion which arise from fluid viscoelasticity. The results of this work have significant design implications for lab-on-a-chip devices, which commonly contain complex geometric features and transport complex fluids, such as those containing DNA or proteins. The results also illustrate the potential for using micro-fabricated devices as rheometric tools for measuring the extensional properties of weakly elastic fluids. (C) 2007 Elsevier B.V. All rights reserved.

Small-scale statistics in high-resolution direct numerical simulation of turbulence: Reynolds number dependence of one-point velocity gradient statistics

Abstract: One-point statistics of velocity gradients and Eulerian and Lagrangian accelerations are studied by analysing the data from high-resolution direct numerical simulations (DNS) of turbulence in a periodic box, with up to 4096 3 grid points. The DNS consist of two series of runs; one is with k max η∼ 1 (Series 1) and the other is with k max η∼2 (Series 2), where k max is the maximum wavenumber and η the Kolmogorov length scale. The maximum Taylor-microscale Reynolds number R λ in Series 1 is about 1130, and it is about 675 in Series 2. Particular attention is paid to the possible Reynolds number (Re) dependence of the statistics. The visualization of the intense vorticity regions shows that the turbulence field at high Re consists of clusters of small intense vorticity regions, and their structure is to be distinguished from those of small eddies. The possible dependence on Re of the probability distribution functions of velocity gradients is analysed through the dependence on R λ of the skewness and flatness factors (S and F). The DNS data suggest that the R λ dependence of S and F of the longitudinal velocity gradients fit well with a simple power law: S∼-0.32R λ 0.11 and F∼1.14R λ 0.34 , in fairly good agreement with previous experimental data. They also suggest that all the fourth-order moments of velocity gradients scale with R λ similarly to each other at R λ >00, in contrast to R λ < 100. Regarding the statistics of time derivatives, the second-order time derivatives of turbulent velocities are more intermittent than the first-order ones for both the Eulerian and Lagrangian velocities, and the Lagrangian time derivatives of turbulent velocities are more intermittent than the Eulerian time derivatives, as would be expected. The flatness factor of the Lagrangian acceleration is as large as 90 at R λ ≈430. The flatness factors of the Eulerian and Lagrangian accelerations increase with R λ approximately proportional to R λ αE and R λ αL , respectively, where α E ≈0.5 and α L ≈1.0, while those of the second-order time derivatives of the Eulerian and Lagrangian velocities increases approximately proportional to R λ βE and R λ βL , respectively, where β E ≈1.5 and β L ≈3.0.

Scalar mixing in direct numerical simulations of temporally evolving plane jet flames with skeletal CO/H2 kinetics ☆

Abstract: Direct numerical simulations of three-dimensional turbulent temporally evolving plane CO/H2 jet flames are performed with detailed chemistry at Reynolds numbers of up to 9000 and with up to 500 million grid points. The effect of Reynolds number on turbulent mixing properties and flame structure is quantified for low Damkohler number flames. These flames exhibit strong flame–turbulence interactions resulting in local extinction followed by re-ignition. The probability density of the stoichiometric scalar dissipation rate is found to be nearly log-normal with some negative skewness. Conditional statistics of the hydroxyl radical reveal increasing levels of extinction and longer re-ignition times with increasing Reynolds number. The mechanical-to-scalar mixing time scale ratio, a key quantity in transported probability density function (pdf) modeling, is investigated for both conserved and reacting scalars. The conserved scalar timescale ratio is found to be consistent with prior studies. For reacting scalars, the effects of molecular diffusivity and chemical reaction on the timescale ratio are quantified.