Intermediate Reynolds Numbers Research Articles

A flow-reconstruction based approach for the computation of hydrodynamic stresses on immersed body surface

This study focuses on obtaining reliable hydrodynamic stresses on the surface of rigid bodies in the simulation of fluid-solid interactions using the diffuse-interface immersed boundary method (IBM). In order to obtain the pressure and velocity gradient information on the immersed boundary, we employ a model to reconstruct the near-boundary fluid flow along the surface normal direction. Different from most existing flow reconstruction models in diffuse-interface IBMs based on the boundary layer approximation, we propose a new reconstruction model relying only on the non-slip velocity and local flow information. With this improvement, the present reconstruction model is theoretically applicable to fluid flows with intermediate Reynolds numbers (with the existence of a laminar boundary layer) and low Reynolds numbers (in the absence of a boundary layer). To examine the validity of the present reconstruction model, we conduct a series of simulations, including the uniform flow past a fixed body and the settling of a rigid body in a quiescent fluid. It is demonstrated that the present reconstruction model has a better performance than existing models in predicting hydrodynamic stresses on the immersed boundary within a wide range of Reynolds numbers from O(1) to O(100). The capability and robustness of the proposed model demonstrate its potential to address fluid-solid interaction problems with a-priori unknown or time-varying Reynolds numbers, such as fully resolved simulations of particle laden flows.

Wake Formation Flow Physics and Boundary Layer Analysis on the Sides of the Isosceles Triangular Cylinder with Apex Pointing Downstream

Boundary layer interaction with downstream flow structures was numerically studied to find the region of inactivity behind an 75° isosceles triangular cylinder with apex pointing downstream at intermediate Reynolds numbers (Re = 520, 640, 840 and 1040). The Standard k-ε model in OpenFOAM was used in the study. Numerical results were validated against Particle Image Velocimetry data. Results revealed the stable region of inactivity characterized by low turbulent kinetic energy and vorticity. The onset of secondary vortex and separation point, independent of Reynolds number, was identified. The onset of the secondary vortex was located at (x = 2 mm) from the base and (y = 1.5 mm) from the apex on either side of the cylinder. The ratio of modulus of absolute primary z-component of vorticity, |ω_z^1 |, to the modulus of secondary z-component of vorticity, |ω_z^2 |, was found to be approximately equal to 1.2. This ratio is invariant of the Reynolds number of the study. These findings have practical implications. The unique properties of the inactivity region forms an ideal location that can be used for injecting fluid, placing measurement probe, active flow control and drag reduction. The research problem is formulated in the introduction. Literature is reviewed next providing the background. Details about the range of parameters, governing equations, numerical study details and software used are given in the methodology section. The results section gives the numerical results, verified by mesh refinement test and validated against experimental results. The results are finally discussed in the next section.

Experimental investigation on inter-particle settling dynamics of multiple spherical particles released side by side at intermediate Reynolds numbers

The settling of solid particles in fluid constitutes a fundamental and crucial aspect with applications spanning various natural phenomena and engineering processes, including sediment transport and wastewater treatments. This paper delves into an experimental investigation aimed at comprehending the settling dynamics and self-organization of multiple spherical particles settling side by side at intermediate Reynolds numbers. The study employs an electromagnetic release device, previously developed for controlled settling of particles under gravity, ensuring simultaneous release with zero initial rotation and velocity. This research captures settling trajectories and provides insight into the flow fields surrounding particles by utilizing particle tracking and particle image velocimetry. The experiments systematically investigate the influence of the settling patterns, the flow fields, the velocities of particles, and their dependence on Reynolds number Re (Re = 52–258), the number of particles n (n = 3–8), as well as the initial spacing between particles l0* (l0* = 0–2). The results consistently reveal a left–right symmetry about the centerline in settling patterns, flow fields, and particle rotations across all values of n, l0*, and Re. The final settling pattern exhibits distinct shapes dependent on l0*: a “V” or “M” shape for l0* < 0.2, a “concave-downward” shape for 0.2 < l0* < 2, and a “straight-line” shape for l0* ≥ 2. The lateral spread of particles increases with time, particularly pronounced with smaller l0* and larger Re, attributed to strong repulsive forces between neighboring particles. Correspondingly, the maximum of horizontal velocities reduces from outside to inside and increases with decreasing l0* and increasing Re. The inner vortices are smaller than the outer vortices, which causes the lateral spread. The vertical spread increases with n but remains insensitive to Re. The average terminal settling velocities for all particles in the array are consistently smaller than those for single particles, as a portion of kinetic energy contributes to horizontal motions.

Simple tracking of occluded self-propelled organisms

Deepening our understanding of animals’ collective motions represents a multidisciplinary goal. Yet, quantifying the motions of hundreds of animals in the laboratory and nature posits a fundamental challenge for digital image processing: How do we track each object out of the crowd while allowing them to move freely in a three-dimensional (3D) domain? Here, we present a simple tracking strategy to reconstruct 3D trajectories with the aid of a mirror, even if moving objects experience occlusion. We explain the method using synthetically generated datasets and apply it to measure collective motions of phototactic zooplankton, Daphnia magna, swimming in a lab-scale aquarium at intermediate Reynolds numbers, 1<Re<13 . The method enables measuring statistics of characteristic features of D. magna swarm, including sinking velocities and flapping frequencies. Beyond the lab-scale animal tracking, we foresee further implementations of the method to study wild animals freely behaving in 3D environments irrespective of their species.

The flow around a stepped cylinder with turbulent wake and stable shear layer

The turbulent external flow around a three-dimensional stepped cylinder is studied by means of direct numerical simulations with the adaptive mesh refinement technique. We give a broad perspective of the flow regimes from laminar to turbulent wake at $Re_D=5000$ , which is the highest ever considered for this flow case. In particular, we focus on the intermediate Reynolds number $Re_D=1000$ that reveals a turbulent wake coupled with a stable cylinder shear layer (subcritical regime). This flow shows a junction dynamics similar to the laminar $Re_D=150$ , where no hairpin vortex appears around the edges, and just two horseshoe vortices are visible. A new stable vortex in the form of a ring, which coils around the rear area, is also identified. In the turbulent wake, the presence of three wake cells is pointed out: the large and small cylinder cells together with the modulation region. However, the modulation dynamics varies between the subcritical and turbulent regimes. A time-averaged, three-dimensional set of statistics is computed, and spatially coherent structures are extracted via proper orthogonal decomposition (POD). The POD identifies the (long-debated) connection between the N-cell and the downwash behind the junction. Furthermore, as the Reynolds number increases, the downwash phenomenon becomes less prominent. Eventually, a reduced-order reconstruction with the most energetically relevant modes is defined to explain the wake vortex interactions. This also serves as a valuable starting point for simulating the stepped cylinder wake behaviour within complex frameworks, e.g. fluid–structure interaction.

A new propulsion enhancement mechanism in metachronal rowing at intermediate Reynolds numbers

Metachronal rowing is a biological propulsion mechanism employed by many swimming invertebrates (e.g. copepods, ctenophores, krill and shrimp). Animals that swim using this mechanism feature rows of appendages that oscillate in a coordinated wave. In this study, we used observations of a swimming ctenophore (comb jelly) to examine the hydrodynamic performance and vortex dynamics associated with metachronal rowing. We first reconstructed the beating kinematics of ctenophore appendages based on a high-speed video of a metachronally coordinated row. Following the reconstruction, two numerical models were developed and simulated using an in-house immersed-boundary-method-based computational fluid dynamics solver. The two models included the original geometry (16 appendages in a row) and a sparse geometry (8 appendages, formed by removing every other appendage along the row). We found that appendage tip vortex interactions contribute to hydrodynamic performance via a vortex-weakening mechanism. Through this mechanism, appendage tip vortices are significantly weakened during the drag-producing recovery stroke. As a result, the swimming ctenophore produces less overall drag, and its thrust-to-power ratio is significantly improved (up to 55.0 % compared with the sparse model). Our parametric study indicated that such a propulsion enhancement mechanism is less effective at higher Reynolds numbers. Simulations were also used to investigate the effects of substrate curvature on the unsteady hydrodynamics. Our results illustrated that, compared with a flat substrate, arranging appendages on a curved substrate can boost the overall thrust generation by up to 29.5 %. These findings provide new insights into the fluid dynamic principles of propulsion enhancement underlying metachronal rowing.

Sphericity based correlations for flow resistance coefficients of non-spherical particles of irregular shape beyond the Stokes regime

In this brief paper, a novel set of correlations for the drag, lift and torque coefficients for non-spherical particles of irregular shape is proposed. Such correlations are developed for a sphericity range between 0.7 and 0.95 and for intermediate Reynolds numbers in the range 1,200, common in industrial and environmental processes. The proposed expressions are derived by fitting the results obtained by means of Particle-Resolved Direct Numerical Simulations (PR-DNS) for a uniform flow around different sets of irregular particles and considering a large number of random orientations. The resulting flow resistance coefficients can be approximated by normal distributions whose first and second order statistical moments (i.e., mean and standard deviation) are described in this work by fitting functions depending of particle Reynolds number and sphericity. The derived correlations are simple, can be easily implemented and allow the tracking of irregular particles in a Lagrangian stochastic framework.

Outer-layer similarity and energy transfer in a rough-wall turbulent channel flow

Direct numerical simulations (DNSs) are performed to investigate the roughness effects on the statistical properties and the large-scale coherent structures in the turbulent channel flow over three-dimensional sinusoidal rough walls. The outer-layer similarities of mean streamwise velocity and Reynolds stresses are examined by systematically varying the roughness Reynolds number $k^{+}$ and the ratio of the roughness height to the half-channel height $k / \delta$ . The energy transfer mechanism of turbulent motions in the presence of roughness elements with different sizes is explored through spectral analysis of the transport equation of the two-point velocity correlation and the scale-energy path display of the generalized Kolmogorov equation. The results show that, with increasing $k^+$ , the downward shift of the mean streamwise velocity profile in the logarithmic region increases and the peak intensities of turbulent Reynolds stresses decrease. At an intermediate Reynolds number ( $Re_{\tau }= 1080$ ), the length scale and intensity of the large-scale coherent structures increase for a small roughness ( $k^{+}=10$ ), which leads to failure of the outer-layer similarity in rough-wall turbulence, and decrease for a large roughness ( $k^{+}=60$ ), as compared with the smooth-wall case. The existence of the small roughness ( $k^{+}=10$ ) enhances the mechanism of inverse energy cascade from the inner-layer small-scale structures to the outer-layer large-scale structures. Correspondingly, the self-sustaining processes of the outer-layer large-scale coherent structures, including turbulent production, interscale transport, pressure transport and spatial turbulent transport, are all enhanced, whereas the large roughness ( $k^{+}=60$ ) weakens the energy transfer between the inner and outer regions.

An analytic solution of Navier–Stokes flow past a sphere in the region of intermediate Reynolds number

We study an analytical solution of steady, laminar, and incompressible flow past a sphere in the region of intermediate Reynolds number. The flow is governed by the Navier–Stokes (N–S) equation and the continuity equation. By applying a simple perturbation method to solve the equations, a second-order approximation cannot be obtained, as well-known (Whitehead’s paradox). Many analytical studies, such as Oseen approximation, matching technique, the homotopy analysis method, etc, have been conducted to resolve the paradox. The drag coefficients of these solutions are valid in the region of Reynolds number (R d is the diameter-based Reynolds number) However, the solution cannot express the flow separation behind a sphere observed in experiments. We also develop a perturbation technique to construct a solution of the N–S equation asymptotically to solve the paradox. The solution consists of power series of , where h is an arbitrary constant . By setting the value of h to avoid divergence of the solution in the all-region where steady flow exists, the solution expresses the flow separation behind a sphere, coinciding with experiments in the region of intermediate Reynolds number (), although the existing analytical solutions could not express. Also, the present solution gives the drag coefficient which agrees with experimental and numerical values in the region of .

Three-dimensional simulations of fluid flows in oscillating lid-driven cavities using lattice Boltzmann method

Abstract We utilize the lattice Boltzmann method to conduct three-dimensional simulations of incompressible flows in oscillating cubic lid-driven cavities. Our investigation focuses on examining the impact of Reynolds number and oscillating frequency on the flow field. Notably, we observe that the flow field can be adequately approximated as two-dimensional within the low and intermediate Reynolds number range, but this approximation is no longer valid for high Reynolds numbers. Additionally, we find that high Reynolds numbers correspond to transient flow fields, while low and moderate Reynolds numbers exhibit quasi-steady periodic flow fields. Our study holds significant relevance for industrial processing applications, where the Reynolds numbers and oscillating frequencies can be optimized to achieve a desired flow field.

Pleobot: a modular robotic solution for metachronal swimming

Metachronal propulsion is widespread in aquatic swarming organisms to achieve performance and maneuverability at intermediate Reynolds numbers. Studying only live organisms limits our understanding of the mechanisms driving these abilities. Thus, we present the design, manufacture, and validation of the Pleobot—a unique krill-inspired robotic swimming appendage constituting the first platform to study metachronal propulsion comprehensively. We combine a multi-link 3D printed mechanism with active and passive actuation of the joints to generate natural kinematics. Using force and fluid flow measurements in parallel with biological data, we show the link between the flow around the appendage and thrust. Further, we provide the first account of a leading-edge suction effect contributing to lift during the power stroke. The repeatability and modularity of the Pleobot enable the independent manipulation of particular motions and traits to test hypotheses central to understanding the relationship between form and function. Lastly, we outline future directions for the Pleobot, including adapting morphological features. We foresee a broad appeal to a wide array of scientific disciplines, from fundamental studies in ecology, biology, and engineering, to developing new bio-inspired platforms for studying oceans across the solar system.

Hydrodynamics of the fast-start caridoid escape response in Antarctic krill, Euphausia superba

Krill are shrimp-like crustaceans with a high degree of mobility and variety of documented swimming behaviors. The caridoid escape response, a fast-start mechanism unique to crustaceans, occurs when the animal performs a series of rapid abdominal flexions and tail flipping that results in powerful backward strokes. The current results quantify the animal kinematics and three-dimensional flow field around a free-swimming Euphausia superba as it performs the caridoid escape maneuver. The specimen performs a single abdominal flexion-tail flip combination that leads to an acceleration over a 42 ms interval allowing it to reach a maximum speed of 57.0 cm/s (17.3 body lengths/s). The krill’s tail flipping during the abdominal closure is a significant contributor to the thrust generation during the maneuver. The krill sheds a complex chain of vortex rings in its wake due to the viscous flow effects while the organism accelerates. The vortex ring structure reveals a strong suction flow in the wake, which suggests that the pressure distribution and form drag play a role in the force balance for this maneuver. Antarctic krill typically swim in a low to intermediate Reynolds number (Re) regime where viscous forces are significant, but as shown by this analysis, its high maneuverability allows it to quickly change its body angle and swimming speed.

Hydrodynamics of finite-length pipes at intermediate Reynolds numbers

Extensive studies of the hydraulics of pipes have focused on limiting cases, such as fully-developed laminar or turbulent flow through long conduits and the accelerating flow through an orifice, for which there exist laws relating pressure drop and flow rate. We carry out experiments on smooth, circular pipes for dimensions and flow rates that interrogate intermediate conditions between the well-studied limits. Organizing this information in terms of dimensionless friction factor, Reynolds number and pipe aspect ratio yields a surface$f_D(Re,\alpha )$that is shown to match the three laws associated with developed laminar, developed turbulent, and orifice flows. While each law fails outside its applicable range of$(Re,\alpha )$, we present a hybrid theoretical–empirical model that includes inlet, development and transition effects, and that proves accurate to approximately 10 % over wide ranges of$Re$and$\alpha$. We also present simple formulas for the boundaries between the three hydraulic regimes, which intersect at a triple point. Measurements show that sipping through a straw is an everyday example of such intermediate conditions not accounted for by existing laws but described accurately by our model. More generally, our findings provide formulas for predicting frictional resistance for intermediate-$Re$flows through finite-length pipes.

Swimming of the midge larva: Principles and tricks of locomotion at intermediate Reynolds number

At the millimeter scale and in the intermediate Reynolds number (Re) regime, the midge and mosquito larvae can reach swimming speeds of more than one body length per cycle performing a “figure eight” gait, in which their elongated bodies periodically bend nearly into circles and then fully unfold. To elucidate the propulsion mechanism of this cycle of motion, we conducted a three-dimensional (3D) numerical study, which investigates the hydrodynamics of undergoing the prescribed kinematics. We found novel propulsion mechanisms, such as modulating the body deformation rate to dynamically increase the maximum net propulsion force, using asymmetric kinematics to generate torque and the appropriate rotation, and controlling the radius of the curled body to manipulate the moment of inertia. The figure eight gait is found to achieve propulsion at a wide range of Re but is most effective at intermediate Re. The results were further validated experimentally, via the development of a soft millimeter-sized robot that can reach comparable speeds using the figure eight gait.

Hydrodynamics of inhomogeneous agglomerates for a fractal dimension of 2.5 at intermediate Reynolds numbers: Effect of agglomerate sizes and Reynolds numbers

AbstractIn this study, the effect of agglomerate sizes for a fractal dimension (Df) of 2.5 on the hydrodynamics at intermediate Reynolds numbers (Re) of 1–120 was assessed. The results show that a core behaves like a solid sphere that exists in the central region inside the agglomerate. In addition, increasing the agglomerate diameter represents adding an extra permeable layer outside the agglomerate. For a larger Re or a smaller agglomerate diameter, the fluid can enter and penetrate through the agglomerate more easily, and the hydrodynamic characteristics of agglomerates deviate more from those of solid spheres. The effect of diameters on the velocity and pressure profiles becomes less significant with the increase in the diameter. Based on the simulated results, the drag ratio has an approximately linear relationship with Re, and its intercept has an exponential relationship with the dimensionless agglomerate diameter. Compared with homogeneous porous spheres, the drag ratio of the agglomerate is different. The effect of diameters on the drag ratio decreases as the diameter increases. It should be noted that the effect of radially varying permeability on inhomogeneous agglomerates should not be ignored and that the effect weakens as Re increases.

Computational aerodynamics of insect flight using volume penalization

The state-of-the-art of insect flight research using advanced computational fluid dynamics techniques on supercomputers is reviewed, focusing mostly on the work of the present authors. We present a brief historical overview, discuss numerical challenges and introduce the governing model equations. Two open source codes, one based on Fourier, the other based on wavelet representation, are succinctly presented and a mass-spring flexible wing model is described. Various illustrations of numerical simulations of flapping insects at low, intermediate and high Reynolds numbers are presented. The role of flexible wings, data-driven modeling and fluid–structure interaction issues are likewise discussed.

Reciprocal swimming at intermediate Reynolds number

In Stokes flow, Purcell's scallop theorem forbids objects with time-reversible (reciprocal) swimming strokes from moving. In the presence of inertia, this restriction is eased and reciprocally deforming bodies can swim. A number of recent works have investigated dimer models that swim reciprocally at intermediate Reynolds numbers${\textit Re} \approx 1$–1000. These show interesting results (e.g. switches of the swim direction as a function of inertia) but the results vary and seem to be case specific. Here, we introduce a general model and investigate the behaviour of an asymmetric spherical dimer of oscillating length for small-amplitude motion at intermediate${\textit {Re}}$. In our analysis we make the important distinction between particle and fluid inertia, both of which need to be considered separately. We asymptotically expand the Navier–Stokes equations in the small-amplitude limit to obtain a system of linear partial differential equations. Using a combination of numerical (finite element) and analytical (reciprocal theorem, method of reflections) methods we solve the system to obtain the dimer's swim speed and show that there are two mechanisms that give rise to motion: boundary conditions (an effective slip velocity) and Reynolds stresses. Each mechanism is driven by two classes of sphere–sphere interactions, between one sphere's motion and (1) the oscillating background flow induced by the other's motion, and (2) a geometric asymmetry induced by the other's presence. We can thus unify and explain behaviours observed in other works. Our results show how sensitive, counterintuitive and rich motility is in the parameter space of finite inertia of particles and fluid.

A traveling wave bifurcation analysis of turbulent pipe flow

Using various techniques from dynamical systems theory, we rigorously study an experimentally validated model by [Barkley et al 2015 Nature 526 550–3], which describes the rise of turbulent pipe flow via a PDE system of reduced complexity. The fast evolution of turbulence is governed by reaction-diffusion dynamics coupled to the centerline velocity, which evolves with advection of Burgers’ type and a slow relaminarization term. Applying to this model a spatial dynamics ansatz and geometric singular perturbation theory, we prove the existence of a heteroclinic loop between a turbulent and a laminar steady state and establish a cascade of bifurcations of various traveling waves mediating the transition to turbulence. The most complicated behaviour can be found in an intermediate Reynolds number regime, where the traveling waves exhibit arbitrarily long periodic-like dynamics indicating the onset of chaos. Our analysis provides a systematic mathematical approach to identifying the transition to spatio–temporal turbulent structures that may also be applicable to other models arising in fluid dynamics.

Theoretical and numerical analysis of drag force at the interface between the dilute and dense phases

Resolving the sharp interface between the dilute and dense phases in gas–solid flows is a bottleneck in fine-grid simulations. This study addresses this issue through the theoretical analysis of an infinite gas–structure interface with arbitrary flow directions. The drag force at the interface is decomposed into three parts: the homogeneous drag forces of the dilute and dense regions and a stress divergence difference term. All the three parts are expressed as the functions of the solid volume fractions, particle Reynolds numbers, and stress divergences of the interface grid and its adjacent grids. The developed theoretical drag models at the interface are verified and improved based on particle-resolved direct numerical simulations (PR-DNSs) of flows past plug-like structures. The models are then tested against PR-DNSs of flows past bubble-containing, spherical, ellipsoidal structures. They yield significantly better performance than the traditional Beetstra et al.'s model [Beetstra et al., “Drag force of intermediate Reynolds number flow past mono- and bidisperse arrays of spheres,” AIChE J. 53(2), 489–501 (2007)].

Flow Over a Square Cylinder at Intermediate Reynolds Numbers

Abstract This study experimentally investigates the flow over a stationary square cylinder in the test section of a subsonic wind tunnel. Hotwire anemometry (CTA type), particle image velocimetry (2C-PIV), and flow visualization techniques are used in the study to characterize the flow properties. The study is carried out at four intermediate Reynolds numbers (blocking width 'B' as the characteristic length) 600, 800, 1000, and 2000. The flow over the circular cylinder and square cylinder differs, particularly at intermediate Reynolds numbers, but the literature lacks detailed flow behavior over a square cylinder beyond the wake transition regime. Re = 600 is the critical Reynolds number as the flow changes from a disordered fine-scale three-dimensional regime to a shear layer transition regime. The effects of the regime alterations on the flow properties, vortex shedding frequency, and mean drag coefficient are reported in the study for all Reynolds numbers. The flow visualization supplements the CTA and PIV measurements. The turbulent kinetic energy budget has also been compared for all Reynolds numbers.