The low-Reynolds-number Stokes flow driven by rotation of two parallel cylinders of equal unit radius is investigated by both analytical and numerical techniques. In Part I, the case of counter-rotating cylinders is considered. A numerical (finite-element) solution is obtained by enclosing the system in an outer cylinder of radius $R_{0}\!\gg\!1$, on which the no-slip condition is imposed. A model problem with the same symmetries is first solved exactly, and the limit of validity of the Stokes approximation is determined; this model has some relevance for ciliary propulsion. For the two-cylinder problem, attention is focused on the small-gap situation $\varepsilon \ll 1$. An exact analytic solution is obtained in the contact limit $\varepsilon=0$, and a net force $F_{c}$ acting on the pair of cylinders in this contact limit is identified; this contributes to the torque that each cylinder experiences about its axis. The far-field torque doublet (`torquelet') is also identified.
Part II treats the case of co-rotating cylinders, for which again a finite-element numerical solution is obtained for $R_{0}\!\gg \!1$. The theory of Watson (1995) is elucidated and shown to agree well with the numerical solution. In contrast to the counter-rotating case, inertia effects are negligible throughout the fluid domain, however large, provided Re $\ll 1$.
In the concluding section, the main results for both cases are summarised, and the situation when the fluid is unbounded ($R_{0}=\infty$) is discussed. (...)
We study the hydrodynamic interaction between two closely-spaced waving elastic cylinders immersed within a viscous liquid, at the creeping flow regime. The cylinders are actuated by a forced oscillation of the slope at their clamped end and are free at the opposite end. We obtain an expression for the interaction force and apply an asymptotic expansion based on a small parameter representing the ratio between the elastic deflections and the distance between the cylinders. The leading-order solution is an asymmetric oscillation pattern at the two frequencies ($ω_1,ω_2$) in which the cylinders …
This paper deals with the small oscillations of two circular cylinders immersed in a viscous stagnant fluid. A new theoretical approach based on an Helmholtz expansion and a bipolar coordinate system is presented to estimate the fluid forces acting on the two bodies. We show that these forces are linear combinations of the {\textcolor{black}{cylinder accelerations}} and velocities, through viscous fluid added coefficients. {\textcolor{black}{To assess the validity of this theory, we consider the case of two equal size cylinders, one of them being stationary while the other one is forced sinuso…
The effects of cavitation on the flow around a circular-cylinder array are studied by using a molecular dynamics simulation. Cavitation significantly affects on vortex shedding characteristics. As the cavitation develops,the vibration acting on the cylinders decreases and eventually disappears. The further cavitation development generates a longer vapor region next to the cylinders, and the vortex streets are formed at further positions from the cylinders. The neighboring Karman vortexes are synchronized in the antiphase in the absence of the cavitation. This synchronization is weakened by the…
The study of vortex flows in the vicinity of multiple solid obstacles is of considerable theoretical interest and practical importance. In particular, the case of flows past a circular cylinder placed above a plane wall has attracted a lot of attention recently. In this case, a stationary vortex is formed in front of the cylinder, in contradistinction to the usual case without the plane where a vortex pair is observed behind the cylinder. In the present work, we apply modern complex analysis techniques to obtain the complex potential for the problem of one point-vortex placed in a uniform stre…
The different types of instabilities of free cylinders (diameter $D$, length $L$) have been studied in a viscous flow (velocity $U$) between parallel vertical walls of horizontal width $W$ at a distance $H$: the influence of the confinement parameters $D/H$ and $L/W$ has been investigated. As $D/H$ increases, there is a transition from stable flow to oscillations transverse to the walls and then to a fluttering motion with oscillations of the angle of the axis with respect to the horizontal. The two types of oscillations may be superimposed in the transition domain. The frequency $f$ of the tr…
In 1982, in his classical work, L. Rayleigh considered the instability of a cylinder of viscous liquid under capillary force, the so-called Plateau-Rayleigh instability. In this work, in linear approximation, he obtained a dispersion equation describing the increment of this instability as a function of wavelength, the radius of cylinder, the mass density, surface tension, and viscosity of the liquid. Hundreds of authors referred to this work, but none of them used his dispersion equation in its complete form; they used only the asymptotic solutions of his equation for zero and infinitely larg…
We investigate the dynamics of flows past a stationary circular cylinder embedded on spherical and cylindrical surfaces at a fixed Reynolds number of 100. For flows on surfaces, it is convenient to express the Navier-Stokes equations in exterior calculus notation. We employ the discrete exterior calculus (DEC) method (Mohamed et al. (2016)) to numerically solve the equations. We explore the role played by curvature on dynamical quantities of interest: the drag coefficient (Cd), the lift coefficient (Cl), and the Strouhal number (St). Our simulations indicate that the effect of surface curvatur…
Direct numerical simulation is performed to study compressible, viscous flow around a circular cylinder. The present study considers two-dimensional, shock-free continuum flow by varying the Reynolds number between 20 and 100 and the freestream Mach number between 0 and 0.5. The results indicate that compressibility effects elongate the near wake for cases above and below the critical Reynolds number for two-dimensional flow under shedding. The wake elongation becomes more pronounced as the Reynolds number approaches this critical value. Moreover, we determine the growth rate and frequency of …
In a wide range of applications, microfluidic channels are implemented in soft substrates. In such configurations, where fluidic inertia and compressibility are negligible, the propagation of fluids in channels is governed by a balance between fluid viscosity and elasticity of the surrounding solid. The viscous-elastic interactions between elastic substrates and non-Newtonian fluids are particularly of interest due to the dependence of viscosity on the state of the system. In this work, we study the fluid-structure interaction dynamics between an incompressible non-Newtonian fluid and a slende…
Direct numerical simulations of closely interacting infinite circular cylinders in a Bingham fluid are presented, and results compared to asymptotic solutions based on lubrication theory in the gap. Unlike for a Newtonian fluid, the macroscopic flow outside of the gap between the cylinders is shown to have a large effect on the pressure profile within the gap and the resulting lubrication force on the cylinders. The presented results indicate that the asymptotic lubrication solution can be used to predict the lubrication pressure only if the surrounding viscoplastic matrix is yielded by a macr…
This paper proposes an Arbitrary Lagrangian-Eulerian incompressible finite volume scheme based on Consistent Flux Reconstruction method (ALE-CFR) on deforming two-dimensional unstructured triangular grids. The scheme is further applied to solve for a problem of two-dimensional incompressible viscous flow past a rotationally oscillating circular cylinder when subjected to forced transverse oscillations at Re = 100. The paper first focuses on the mathematical development and validation of the ALE-CFR scheme and then its application on the problem. The rotational oscillation frequencies were take…
Numerical simulations using the Lattice Boltzmann Method are presented of the following two-dimensional incompressible flow problem. Starting from configurations corresponding to translating inviscid equilibria, namely, the translating Föppl equilibria (counter-rotating point vortex pair, either fore or aft of a translating circular cylinder) and the translating Hill equilibria (counter-rotating point vortex pair, either fore or aft of an elliptic cylinder), viscosity is turned on for $t >0$ and the subsequent viscous interaction is simulated. The interaction is in a dynamically coupled settin…
A novel mechanism for the transport of microscale particles in viscous fluids is demonstrated. The mechanism exploits the trapping of such particles by rotational streaming cells established in the vicinity of an oscillating cylinder, recently analyzed in previous work. The present work explores a strategy of transporting particles between the trapping points established by multiple cylinders undergoing oscillations in sequential intervals. It is demonstrated that, by controlling the sequence of oscillation intervals, an inertial particle is effectively and predictably transported between the …
We study the steady streaming between two infinitely long circular cylinders produced by small amplitude transverse vibrations of the inner cylinder about the axis of the outer cylinder. The Vishik-Lyusternik method is employed to construct an asymptotic expansion of the solution of the Navier-Stokes equations in the limit of high-frequency vibrations for Reynolds numbers of order of unity. The effect of the Stokes drift of fluid particles is also studied. It is shown that it is nonzero not only within the boundary layers but also in higher order terms of the expansion of the averaged outer fl…
In this paper we have investigated a circular band formation of fluid-rigid particle mixtures in a fully filled cylinder horizontally rotating about its cylinder axis by direct numerical simulation. These phenomena are modeled by the Navier-Stokes equations coupled to the Euler-Newton equations describing the rigid solid motion of the non-neutrally particles. The formation of circular bands studied in this paper is not resulted by mutual interaction between the particles and the periodic inertial waves in the cylinder axis direction (as suggested in Phys. Rev. E, 72, 021407 (2005)), but due to…
Flows over two side-by-side circular cylinders exhibit fascinating flow physics due to complex interactions between the coupled wakes. However, their mutual interference effects have not been elucidated in a quantitative manner thus far. In this paper, we create a mismatch between the two wakes by introducing a size difference in the cylinder pair, such that the effects of one wake on the other can be distinguished. Depending on the size and gap ratios between the two cylinders, the coupled wake exhibits distinct dynamical features including the quasi-periodic flow, synchronized flow, and chao…
The scale factors of an arbitrary orthogonal space are a measure of its content of homogeneous orthogonal space. In the present study, it is shown, that their spatial and temporal rates of variation do not contribute to the differential calculus of arbitrary functions in orthogonal space. Based on this, the Navier-Stokes equations are formulated accordingly, to provide a method of studying the kinematics of fluid motion in orthogonal plane space. Employing this formulation and regular perturbation theory, the kinematic physical measures of the flow of an incompressible, viscous fluid around tw…
The classical problem of suppressing vortex shedding in the wake of a circular cylinder by using body rotation is revisited in an adjoint-based optimal control framework. The cylinder's unsteady and fully unconstrained rotation rate is optimized at Reynolds numbers between 75 and 200 and over horizons that are longer than in previous studies, where they are typically of the order of a vortex shedding period or shorter. In the best configuration, the drag is reduced by 19%, the vortex shedding is effectively suppressed, and this low drag state is maintained with minimal cylinder rotation after …
A new eigenvalue analysis is developed and applied to the circular cylinder laminar flow configuration to investigate the various mechanisms at play in the nonlinear saturation of perturbations yielding to limit cycles for supercritical values of the Reynolds number. Unlike the mean-flow analysis, which only accounts for the interaction of the first-harmonic of the time-periodic flow with its mean-flow, the so-called extended mean-flow analysis also accounts for its interaction with the second-harmonic. The results reveal the existence of two eigenvalues both having a growth rate exactly equal…
In this paper, we formulated the non-steady flow due to the uniformly accelerated and rotating circular cylinder from rest in a stationary, viscous, incompressible and micropolar fluid. This flow problem is examined numerically by adopting a special scheme comprising the Adams-Bashforth Temporal Fourier Series method and the Runge-Kutta Temporal Special Finite-Difference method. This numerical scheme transforms the governing equation for micropolar fluids for this problem into system of finite-difference equations. This system was further solved numerically by point SOR-method. These results w…
A number of new closed-form fundamental solutions for the generalized unsteady Oseen and Stokes flows associated with arbitrary time-dependent translational and rotational motions have been developed. These solutions are decomposed into two parts corresponding to a longitudinal wave and a transversal wave. As examples of application, the hydrodynamic force acting on a sphere or on a circular cylinder translating in an unsteady flow field at low Reynolds numbers are calculated using the new generalized fundamental solutions. The results for a rotating viscous flow past an impulsively moving sph…
The motion of a pair of counter-rotating point vortices placed in a uniform flow around a circular cylinder forms a rich nonlinear system that is often used to model vortex shedding. The phase portrait of the Hamiltonian governing the dynamics of a vortex pair that moves symmetrically with respect to the centerline---a case that can be realized experimentally by placing a splitter plate in the center plane---is presented. The analysis provides new insights and reveals novel dynamical features of the system, such as a nilpotent saddle point at infinity whose homoclinic orbits define the region …
The stability of a two-dimensional viscous flow between two rotating porous cylinders is studied. The basic steady flow is the most general rotationally-invariant solution of the Navier-Stokes equations in which the velocity has both radial and azimuthal components, and the azimuthal velocity profile depends on the Reynolds number. It is shown that for a wide range of the parameters of the problem, the basic flow is unstable to small two-dimensional perturbations. Neutral curves in the space of parameters of the problem are computed. Calculations show that the stability properties of this flow…
The slow viscous flow through a doubly-periodic array of cylinders does not have an analytical solution. However, as a reduced model for the flow within fibrous porous media, this solution is important for many real-world systems. We asymptotically determine the flow around a doubly-periodic array of infinite slender cylinders, by placing doubly-periodic two-dimensional singularity solutions within the cylinder and expanding the no-slip condition on the cylinder's surface in powers of the cylinder radius. The asymptotic solution provides a closed-form estimate for the flow and forces as a func…
The motion of two pairs of counter-rotating point vortices placed in a uniform flow past a circular cylinder is studied analytically and numerically. When the dynamics is restricted to the symmetric subspace---a case that can be realized experimentally by placing a splitter plate in the center plane---, it is found that there is a family of linearly stable equilibria for same-signed vortex pairs. The nonlinear dynamics in the symmetric subspace is investigated and several types of orbits are presented. The analysis reported here provides new insights and reveals novel features of this four-vor…
Vortex-Induced Vibrations (VIV) offer a safe, renewable, and environmentally friendly energy source for energy harvesting. To enhance the energy harvesting capability of the circular cylinder-based devices, the authors explore the placement of the normal slit by determining the most effective slit offset location from the cylinder's center. Using the open-source Computational Fluid Dynamics (CFD) program OpenFOAM, a series of numerical simulations are conducted to determine the utility of the slit placement on the circular cylinder and how it influences (positively/negatively) the harvester's …
A simple method to increase the flow power extraction efficiency of a circular cylinder, undergoing vortex-induced vibration (VIV), by confining it between two parallel plates is proposed. A two-dimensional numerical study was performed on VIV of a circular cylinder inside a parallel plate channel of height H at Reynolds number 150 to quantify the improvement. The cylinder is elastically mounted with a spring such that it is only free to vibrate in the direction transverse to the channel flow and has a fixed mass ratio (m*) of 10. The energy extraction process is modelled as a damper, with spa…
Phase synchronization between the vortex shedding behind a two-dimensional circular cylinder and its vibrations is investigated using the phase-reduction analysis. Leveraging this approach enables the development of a one-dimensional, linear model with respect to the limit-cycle attractor of the laminar wake, which accurately describes the phase dynamics of the high-dimensional, nonlinear fluid flow and its response to rotational, transverse and longitudinal vibrations of the cylinder. This phase-based model is derived by assessing the phase-response and sensitivity of the wake dynamics to imp…
A numerical study of two-dimensional flow past a confined circular cylinder with slip wall is performed. A dimensionless number, Knudsen number ($Kn$) is used to describe the slip length of cylinder wall. The Reynolds number ($Re$) and Knudsen number ($Kn$) ranges considered are $Re = [1, 180]$ and $Kn = [0, \infty)$, respectively. Time-averaged flow separation angle ($\bar{θ_s}$), dimensionless recirculation length ($\bar{L_s}$) and the tangential velocity ($\bar{u_τ}$) distributed on the cylinder's wall, drag coefficient ($\bar{C_d}$) and drag reduction ($DR$) are investigated. The time-aver…
Many microorganisms swim in a highly heterogeneous environment with obstacles such as fibers or polymers. To better understand how this environment affects microorganism swimming, we study propulsion of a cylinder or filament in a fluid with a sparse, stationary network of obstructions modeled by the Brinkman equation. The mathematical analysis of swimming speeds is investigated by studying an infinite-length cylinder propagating lateral or spiral displacement waves. For fixed bending kinematics, we find that swimming speeds are enhanced due to the added resistance from the fibers. In addition…
In this paper, we discuss whether the instability of viscoelastic flow around a circular cylinder is subcritical or supercritical by numerical simulation. The Oldroyd-B model is selected to describe the viscoelastic constitutive relationship. The Log-conformation reformulation is employed to stabilize numerical simulation. The parameter ranges investigated are the Reynolds numbers ($Re$) spanning from 5 to 100 and the Weissenberg number ($Wi$) spanning from 0 to 10, with a fixed viscosity ratio of $β= 0.9$. Simulations are performed under two paths, i.e., 1) increasing $Re$ (or $Wi$) and 2) de…
This work applies a combined approach a reduced-order model (ROM) together with experiments and direct numerical simulations to investigate the optimal efficiency of fluid-flow energy harvesting from transverse vortex-induced vibration (VIV) of a circular cylinder. High resolution efficiency maps were predicted over wide ranges of flow reduced velocities and structural damping ratios, and the maximum efficiency and optimal settings of damping ratio and reduced velocity were then examined for different mass ratios and Reynolds numbers. Efficiencies predicted by the ROM were also validated again…
Unsteady flow fields over a circular cylinder are trained and predicted using four different deep learning networks: convolutional neural networks with and without consideration of conservation laws, generative adversarial networks with and without consideration of conservation laws. Flow fields at future occasions are predicted based on information of flow fields at previous occasions. Deep learning networks are trained first using flow fields at Reynolds numbers of 100, 200, 300, and 400, while flow fields at Reynolds numbers of 500 and 3000 are predicted using the trained deep learning netw…
We perform direct numerical simulations to characterize the three-dimensional wake dynamics of long inclined circular cylinders with inhomogeneous end boundary conditions. Three Reynolds numbers, $\Rey=100$, 200 and 300 are considered to reveal the roles of the intrinsic secondary instabilities and the extrinsic end boundary effects in shaping the three-dimensional flows. At $\Rey=100$, the end boundary effects are felt over the entire cylinder span by inducing oblique vortex shedding, which is associated with stronger spanwise flow in the wake than a parallel shedding. The Strouhal number of …
The study numerically investigates the three-dimensionality in the flow and two-degree-of-freedom (2 DOF) vortex-induced-vibrations (VIV) characteristics of an elastically mounted circular cylinder. The cylinder is allowed to vibrate in both streamwise and transverse directions. A low value of mass-ratio with the zero damping coefficient is taken for the simulations. The primary aim is to understand the vortex shedding behind the cylinder and the transition characteristics of the wake-flow from two-dimensional (2D) to three-dimensional (3D). The Reynolds number (Re) is varied from 150 (fully 2…
Turbulent flow evolution and energy cascades are significantly different in two-dimensional (2D) and three-dimensional (3D) flows. Studies have investigated these differences in obstacle-free turbulent flows, but solid boundaries have an important impact on the cross-over between 3D to 2D turbulence dynamics. In this work, we investigate the span effect on the turbulence nature of flow past a circular cylinder at Re=10000. It is found that even for highly anisotropic geometries, 3D small-scale structures detach from the walls. Additionally, the natural large-scale rotation of the Kármán vortic…
The present article aims to study the suppression of vortex shedding using a passive flow control technique (slit through a circular cylinder) in the laminar regime (Re=100-500). The slit width ratio S/D (slit width/diameter) on the modified cylinder plays an essential role to control the vortex shedding. The additional flow through the slit leads to the suppression of the global instability and vortex shedding, whereas a large amount of flow through the slit drastically alters the behavior of vortex shedding. The nature of vortex shedding remains periodic for all S/D, and the root mean square…
The present study investigates the passive flow control phenomena over a two-dimensional circular cylinder using numerical simulations in the laminar regime. The aim is to explore one of the passive control techniques, which involves the introduction of a slit to the geometry of the cylinder. The two parameters, slit width ratio S/D (slit width/diameter) and slit angle (measured with respect to the incoming flow direction), play an essential role in determining the trend of critical Reynolds number (Rec). Most of the analysis invokes flow visualization and saturation amplitude methods to obtai…
It is known that an object translating parallel to a soft wall in a viscous fluid produces hydro- dynamic stresses that deform the wall, which, in turn, results in a lift force on the object. Recent experiments with cylinders sliding under gravity near a soft incline, which confirmed theoretical arguments for the lift force, also reported an unexplained steady-state rotation of the cylinders [Saintyves et al. PNAS 113(21), 2016]. Motivated by these observations, we show, in the lubrication limit, that an infinite cylinder that translates in a viscous fluid parallel to a soft wall at constant s…
The combined effect of fluid elasticity and yield-stress on the flow past a circular cylinder is studied by two-dimensional direct numerical simulation. We analyze the effects of yield-stress, elasticity, shear-thinning, and shear-thickening on the wake characteristics using the Saramito constitutive model. The elastoviscoplastic (EVP) wake flow is studied at a moderate Reynolds number (Re = 100) where two-dimensional vortex shedding occurs in the Newtonian case. We find that in the shear-thinning elastoviscoplastic flow, when yield stress increases, the drag coefficient and root mean square o…
In this paper I consider the self-excited rotation of an elliptical cylinder towed in a viscous fluid as a canonical model of nonlinear fluid structure interactions with possible applications in the design of sensors and energy extraction devices. First, the self-excited ellipse system is shown to be analogous to the forced bistable oscillators studied in classic chaos theory. A single variable, the distance between the pivot and the centroid, governs the system bifurcation into bi-stability. Next, fully coupled computational fluid dynamics simulations of the motion of the cylinder demonstrate…
Viscous streaming flows generated by objects of constant curvature (circular cylinders, infinite plates) have been well understood. Yet, characterization and understanding of such flows when multiple body length-scales are involved has not been looked into, in rigorous detail. We propose a simplified setting to understand and explore the effect of multiple body curvatures on streaming flows, analyzing the system through the lens of bifurcation theory. Our setup consists of periodic, regular lattices of cylinders characterized by two distinct radii, so as to inject discrete curvatures into the …
The hydrodynamic and thermal characteristics of a freely-vibrating circular cylinder in mixed convection flow are numerically investigated at low Reynolds numbers. The numerical investigations are conducted for a range of parameters, Ur = [2.0, 10], Pr = [0.7, 10] and Ri = [0.5, 2.0]. Whereas the Reynolds number, the mass ratio and the damping ratio are fixed. A secondary VIV lock-in region is found in the cases of high Richardson number Ri=2.0 for high reduced velocity values, in which the buoyancy-driven flow is non-trivial. A wide VIV lock-in region is formed with tremendous energy transfer…
In this paper we found, by means of numerical simulations, a transition in the oscillatory character of the flow field for a particular combination of buoyancy and spacing in an array of six circular cylinders at a Reynolds number of 100 and Prandtl number of 0.7. The cylinders are iso-thermal and they are aligned with the Earth acceleration (g). According to the array orientation, an aiding or an opposing buoyancy is considered. The effect of natural convection with respect to the forced convection is modulated with the Richardson number, Ri, ranging between -1 and 1. Two values of center to …
Deep reinforcement learning-based active flow control of an elliptical cylinder: transitioning from an elliptical cylinder to a circular cylinder and a flat plate
