Barotropic elliptical dipoles in a rotating fluid

Barotropic f-plane dipolar vortices were generated in a rotating fluid and a comparison was made with the so-called supersmooth f-plane solution which—in contrast to the classical Lamb–Chaplygin solution—is marked by an elliptical separatrix and a doubly continuously differentiable vorticity field. Dye-visualization and high-resolution particle-tracking techniques revealed that the observed dipole characteristics (separatrix aspect ratio, cross-sectional vorticity distribution and vorticity versus streamfunction relationship) are in close agreement with those of the supersmooth f-plane solution for the entire lifespan of the dipolar vortex.     

The evolution of an initially circular vortex near an escarpment. Part I: analytical results

The motion of a quasigeostrophic, equivalent-barotropic, initially circular vortex patch near an infinitely long topographic escarpment is studied using f-plane dynamics. There are two time scales in the problem: the advective time scale associated with the vortex, and the time scale for topographic vortex stretching. Analytical progress is possible when these two time scales are well-separated and results are presented here.If topographic vortex stretching dominates advection by the vortex the vortex is said to be ‘weak’. The vortex patch remains circular on the topographic time scale, and dispersive topographic waves rapidly propagate the initial disturbance away from the vicinity of the vortex. Subsequently cross-isobath motion is inhibited, and the vortex moves as though the escarpment were a plane wall. The same behaviour was observed for the motion of a weak singular vortex near an escarpment by Dunn, McDonald and Johnson [7], who named the phenomenon the ‘pseudoimage’ of the vortex.If advection dominates over topographic effects, the vortex is said to be ‘intense’. The vortex also remains circular to leading order, but the relative vorticity produced by the swirl of the vortex is less able to escape the vicinity of the vortex. The vortex follows a similar curved trajectory to those observed for intense vortices on the β-plane. The dipolar mechanism for this behaviour is described. Large time solutions are inhibited by the form of the escarpment topography, but examination of the equations leads to the conclusion that the leading order solution may be predict the motion for times beyond its formal range of validity.     

On the nonlinear theory of the wing in a plane unsteady flow

The main aspects of the nonlinear theory of the wing in a plane unsteady fluid flow are generalized on the basis of the author’s previous results. An initial-boundary problem for complex velocity is formulated. A system of differential equations with conditions at points of vortex wake shedding is presented, which allows a large class of problems to be solved correctly. The Cauchy problem is solved by using a standard discretization procedure. The boundary-value problem is reduced at each time step to singular integral equations of the first and second kind. The accuracy of solving these equations by the method of discrete vortices and by the method of panels is compared. Specific features of pressure calculations in the case of a separated flow around the airfoil contour are discussed     

Dean instability and secondary flow structure in curved rectangular ducts

The pressure driven, fully developed turbulent flow of incompressible viscous fluid (water) in 120° curved ducts of rectangular cross-section is investigated experimentally and numerically. Three different types of curved duct (A-CL, B-SL and C-IL) with continuously varying curvature conform to blade profile as the inner and outer curvature walls to simplify and guide the impeller design of pumps. After validating the numerical method against Particle Image Velocimetry (PIV) measurements, the flow development in the ducts is analyzed in detail by Computational Fluid Dynamics (CFD) for a wide range of Reynolds numbers (Re = 2.4 × 10⁴–1.4 × 10⁵) and aspect ratios (Ar > 1.0, =1.0 and <1.0). The results clearly depict the existence of multiple Dean vortices along the duct: while the axial velocity profile is more related to an inner Dean vortex (called split base vortex), the wall pressure is more influenced by the Dean vortex attached to the inner curvature wall (called ICW Dean vortex). The induced multiple Dean vortices and the secondary flow patterns in the duct cannot be faithfully predicted by using traditional techniques. Therefore, a new criterion based on the vortex core velocities is proposed. With this approach, the effects of Re, Cr and Ar on the Dean instabilities in curved ducts are carefully studied. Decreasing Re promotes the generation of Dean vortices closer to the duct inlet, a trend that is as opposed to laminar flow. In addition, a new pair of vortices called entrainment Dean vortex occurs near the outlet of the curved duct with Ar = 1.0, which has not been previously reported in the literature.     

Structure of a bathtub vortex: importance of the bottom boundary layer

A bathtub vortex in a cylindrical tank rotating at a constant angular velocity Ω is studied by means of a laboratory experiment, a numerical experiment and a boundary layer theory. The laboratory and numerical experiments show that two regimes of vortices in the steady-state can occur depending on Ω and the volume flux Q through the drain hole: when Q is large and Ω is small, a potential vortex is formed in which angular momentum outside the vortex core is constant in the non-rotating frame. However, when Q is small or Ω is large, a vortex is generated in which the angular momentum decreases with decreasing radius. Boundary layer theory shows that the vortex regimes strongly depend on the theoretical radial volume flux through the bottom boundary layer under a potential vortex : when the ratio of Q to the theoretical boundary-layer radial volume flux Q _b (scaled by ${2\pi R^2 ( \Omega \nu )^\frac{1}{2}}$ ) at the outer rim of the vortex core is larger than a critical value (of order 1), the radial flow in the interior exists at all radii and Regime I is realized, where R is the inner radius of the tank and ν the kinematic viscosity. When the ratio is less than the critical value, the radial flow in the interior nearly vanishes inside a critical radius and almost all of the radial volume flux occurs only in the boundary layer, resulting in Regime II in which the angular momentum is not constant with radius. This criterion is found to explain the results of the laboratory and numerical experiments very well.     

FLUID–STRUCTURE RESONANCE PRODUCED BY ONCOMING ALTERNATING VORTICES

The present investigation examines a simple fluid–structure interaction problem, which is represented by the unsteady response of an airfoil/blade to a Karman vortex street in an inviscid uniform flow. Two different cases were examined; one with a rigid airfoil/blade, where the structural stiffness is infinite, another with an elastic blade. In both cases, the flow remains attached to the airfoil/blade surface. A time-marching technique solving the Euler equations and a two-degree-of-freedom structural dynamic model is used to examine the interactions between the fluid and the structure. The interactions between the convected vortices and the structure modify the shed wake whose energy, in turn, feeds into the forces and moments acting on the structure. For a rigid airfoil/blade, it is found that the amplitude of the aerodynamic response is not proportional to the density of the oncoming vortex street, but depends on c/d , the ratio of the chord length (c) to the axial spacing (d) of the convected vortices. When the number of vortices per unit length is increased, the amplitudes of the aerodynamic response increase and then decrease even though the density of the vorticity keeps increasing and so is the energy of the excitation wake. Maxima are observed at c/d=0·5, 1·5 and 2·5. This behaviour is analogous to the structural resonance phenomenon and is labeled “aerodynamic resonance”. The existence of such an “aerodynamic resonance” is important to turbomachinery applications where the blade is elastic, the flow is unsteady and the shed vortices from the previous row are convected downstream by the mean flow. Thus, “aerodynamic resonance” alone or in conjunction with structural resonance could impact negatively on the fatigue life of turbine blades and their combined effects should be accounted for in blade design. A preliminary attempt to assess this impact has been carried out. It is found that the relative fatigue life of a blade could be reduced by four orders of magnitude as a result.     

Asymmetric vortex shedding flow past an inclined flat plate at high incidence

This paper reports an experimental investigation of the vortex shedding wake behind a long flat plate inclined at a small angle of attack to a main flow stream. Detailed velocity fields are obtained with particle-image velocimetry (PIV) at successive phases in a vortex shedding cycle at three angles of attack, α=20°, 25° and 30°, at a Reynolds number Re≈5,300. Coherent patterns and dynamics of the vortices in the wake are revealed by the phase-averaged PIV vectors and derived turbulent properties. A vortex street pattern comprising a train of leading edge vortices alternating with a train of trailing edge vortices is found in the wake. The trailing edge vortex is shed directly from the sharp trailing edge while there are evidences that the formation and shedding of the leading edge vortex involve a more complicated mechanism. The leading edge vortex seems to be shed into the wake from an axial location near the trailing edge. After shedding, the vortices are convected downstream in the wake with a convection speed roughly equal to 0.8 the free-stream velocity. On reaching the same axial location, the trailing edge vortex, as compared to the leading edge vortex, is found to possess a higher peak vorticity level at its centre and induce more intense fluid circulation and Reynolds stresses production around it. It is found that the results at the three angles of attack can be collapsed into similar trends by using the projected plate width as the characteristic length of the flow.     

Multisensor Hot Wire Vorticity Probe Measurements of the Formation Field of Two Corotating Vortices

Experimental evidence is reported, regarding the formation of a pair of co-rotating tip vortices by a split wing configuration, consisting of two half wings at equal and opposite angles of attack. Simultaneous measurements of the three-dimensional vector fields of velocity and vorticity were conducted on a cross plane at a downstream distance corresponding to 0.3 cord lengths (near wake), using an in-house constructed 12-sensor hot wire anemometry vorticity probe. The probe consists of three closely separated orthogonal 4-wire velocity sensor arrays, measuring simultaneously the three-dimensional velocity vector at three closely spaced locations on a cross plane of the flow filed. This configuration makes possible the estimation of spatial velocity derivatives by means of a forward difference scheme of first order accuracy. Velocity measurements obtained with an X-wire are also presented for comparison. In this near wake location, the flow field is dictated by the pressure distribution established by the flow around the wings, mobilizing large masses of air and leading to the roll up of fluid sheets. Fluid streams penetrating between the wings collide, creating on the cross plane flow a stagnation point and an “impermeable” line joining the two vortex centres. Along this line fluid is directed towards the two vortices, expanding their cores and increasing their separation distance. This feeding process generates a dipole of opposite sign streamwise mean vorticity within each vortex. The rotational flow within the vortices obligates an adverse streamwise pressure gradient leading to a significant streamwise velocity deficit characterizing the vortices. The turbulent flow field is the result of temporal changes in the intensity of the vortex formation and changes in the position of the cores (wandering).     

Coherent structures in unsteady swirling jet flow

An LDA technique and phase-averaging analysis were used to study unsteady precessing flow in a model vortex burner. Detailed measurements were made for Re=15,000 and S=1.01. On the basis of the analysis of phase-averaged data and vortex detection by the λ₂-technique of Joeng and Hussain (1995), three precessing spiral vortex structures were identified: primary vortex (PV), inner secondary vortex (ISV), and outer secondary vortex (OSV). The PV is the primary and most powerful structure as it includes primary vorticity generated by the swirler; the ISV and OSV are considered here as secondary vortical structures. The jet breakdown zone is the conjunction of a pair of co-rotating co-winding spiral vortices, PV and ISV. The interesting new feature described is that the secondary vortices form a three-dimensional vortex dipole with a helical geometry. The effect of coupling of secondary vortices was suggested as a mechanism of enhanced stability reflected in their increased axial extent.     

Chaotic advection and nonlinear resonances in an oceanic flow above submerged obstacle

The effect of an isolated submarine obstacle on the motion of fluid particles in a periodic external flow is studied within the framework of the barotropic, quasi-geostrophic approximation on f-plane. The concept of background currents advanced by Kozlov [1995. Background currents in geophysical hydrodynamics. Izvestia, Atmos. Oceanic Phys. 31 (2), 245-250] is used to construct a dynamically consistent stream function satisfying the potential vorticity conservation law. It is shown that a system of two topographic vortices revolving about a rotation center can form in a circular external flow. Unsteady periodic perturbations, associated with either variations in the background current or deviations of the external flow from circulation, are analyzed. Unsteadiness in the external flow essentially complicates the pattern of the motion of fluid particles. Vortex-type quasi-periodic structures, identified with nonlinear resonances that form in Lagrangian equations of fluid particle advection, are examined. They either surround the stationary configuration by a vortex chain—a ringlet-like structure [Kennelly, M.A., Evans, R.H., Joyce, T.M., 1985. Small-scale cyclones on the periphery of Gulf Stream warm-core rings. J. Geophys. Res. 90(5), 8845-8857], or they form a complex-structure multivortex domain. Asymptotic estimates and numerical modeling are used to study the distribution and widths of the nonlinear resonance domains that appear under unsteady perturbations of different types. The onset of chaotic regimes owing to the overlapping of nonlinear resonance domains is analyzed. Transport fluxes determined by chaotic advection and barriers for transport (KAM-tori) and the conditions of their existence are studied. The relation of the rotation frequency of fluid particles on their initial position (when the dependence is calculated in the undisturbed system) is shown to completely determine the main features of the pattern of Lagrangian trajectories and chaotization effects. Because of nonlinear effects, the domain involved in quasi-periodic and chaotic motions can be much larger than the domain occupied by steady topographic vortices. The results of study by Sokolovskiy et al. [1988. On the influence of an isolated submerged obstacle on a barotropic tidal flow. Geophys. Astrophys. Fluid Dyn. 88(1), 1-30] concerning the due regard on the irrotational background component as the necessary factor for the transportation of fluid particles from the vortex domain to infinity are confirmed.     

Vortex shedding flow behind a slowly rotating circular cylinder

This paper investigates flow past a rotating circular cylinder at 3600?Re?5000 and α?2.5. The flow parameter α is the circumferential speed at the cylinder surface normalized by the free-stream velocity of the uniform cross-flow. With particle image velocimetry (PIV), vortex shedding from the cylinder is clearly observed at α<1.9. The vortex pattern is very similar to the vortex street behind a stationary circular cylinder; but with increasing cylinder rotation speed, the wake is observed to become increasing narrower and deflected sideways. Properties of large-scale vortices developed from the shear layers and shed into the wake are investigated with the vorticity field derived from the PIV data. The vortex formation length is found to decrease with increasing α. This leads to a slow increase in vortex shedding frequency with α. At α=0.65, vortex shedding is found to synchronize with cylinder rotation, with one vortex being shed every rotation cycle of the cylinder. Vortex dynamics are studied at this value of α with the phase-locked eduction technique. It is found that although the shear layers at two different sides of the cylinder possess unequal vorticity levels, alternating vortices subsequently shed from the cylinder to join the two trains of vortices in the vortex street pattern exhibit very little difference in vortex strength.     

A circular cylinder undergoing large-amplitude transverse oscillations in a slow uniform cross flow

This study explores the vortex patterns formed by a circular cylinder undergoing lateral cylinder oscillations with large amplitudes and in the presence of a slow uniform cross flow. It is an extension of our previous study (Lam et al., 2010b) in which formation of the 2S, 2P and P+S vortex modes were discussed from the viewpoint of interaction of a uniform cross-flow with the vortex street patterns of a cylinder oscillating in an otherwise quiescent fluid at Keulegan–Carpenter numbers up to KC=8.9. The present paper reports three additional experimental sets in which the amplitudes of cylinder oscillations have even larger values, at A/D>2.5, and lie beyond the vortex mode map usually quoted from Williamson and Roshko (1988). It is found that the slow uniform cross-flow at λ/D≈3 and Reynolds number based on cross-flow velocity at 232 acts to convect the corresponding vortex patterns in the absence of cross-flow downstream across the line of cylinder oscillation. Vortex–vortex interaction and vortex–cylinder interaction are observed to affect the subsequent development of vortices. The P+S vortex mode is found to occur up to KC=16. At KC between 16 and 24, a new vortex mode is observed in which only one vortex pair can be convected downstream every cylinder oscillation cycle. Another new vortex mode with two vortex pairs and two stationary vortices are found at KC>24.     

Localized dipoles: from 2D to rotating shallow water

The progress made in the theory of localized dipoles over the course of the past century is overviewed. The dependence between the dipole shape, on the one hand, and the vorticity–streamfunction relation in the frame of reference co-moving with the dipole, on the other hand, is discussed. We show that, in 2D non-divergent and quasi-geostrophic dipoles, circularity of the trapped-fluid region and linearity of the vorticity–streamfunction relation in this region are equivalent. The existence of elliptical dipoles of high smoothness is demonstrated. A generalization of the dipole theory to the rotating shallow water model is offered. This includes the construction of localized f-plane dipole solutions (modons) and demonstration of their soliton nature, and derivation of a necessary condition for an eastward-traveling β-plane modon to exist. General properties of such ageostrophic modons are discussed, and the fundamental dissimilarity of fast and/or large dipoles in the rotating shallow water model from quasi-geostrophic dipoles is demonstrated and explained.     

Influence of rounding corners on the wake of a finite-length cylinder: An experimental study

This study aims to investigate experimentally the influence of rounding corners (r) as well as aspect ratio (AR) on the flow structures of a surface-mounted finite cylinder. The cylinders with sharp (r* = r/D = 0) and rounded corners (r*=0.167, 0.25 and 0.5) and aspect ratio or height-to-width/diameter ratio (AR = H/D) between 2 and 7 are utilized. The experiments are based on the five-hole probe and hot-wire measurements as well as the oil flow visualization. Wake measurements are made in an open return wind tunnel at the Reynolds number, Re = 1.6 × 10⁴, where Re is defined based on the side width/diameter (D) of the cylinder cross-section and the freestream velocity. It is found that r* and AR have significant effects on the flow structure from the perspective of wake topology, strength of streamwise vortices, and vortex shedding frequency. For all r* considered, the wake is characterized by a quadrupole type (both the tip and base vortices are present) at AR = 7, while a dipole type occurs for AR = 2 and 4 (the base vortices are absent). The strength (circulation) of the streamwise vortex structures is affected by r*. For all AR examined in the present study, the strengths of tip and base vortex structures decrease with increasing r*. The oil flow visualization demonstrates that the features of the horseshoe vortex are sensitive to r* and AR. With increasing r*, the location of the separation line moves downstream and the distance between horseshoe vortex legs decreases. Velocity measurements reveal that the downwash flow enhances with increasing r*. It is also found that the Strouhal number increases progressively by 60% as r* increases from 0 to 0.5, regardless of AR.     

Collapse and concentration of vortex sheets in two-dimensional flow

Numerical evidence is presented for the existence of collapse configurations of vortex sheets (one-dimensional singular distributions of vorticity) in a two-dimensional ideal fluid. Point vortices are used to approximate the vortex sheets. These and related motions cause a significant concentration of vorticity, with possible relevance to the concentration seen in the evolution of turbulent flows.     

Order in chaotic pseudoplastic flow between coaxial cylinders

Order is found within the chaotic nonlinear flow between rotating coaxial cylinders. The flow stability analysis is carried out for a pseudoplastic fluid through bifurcation diagram and Lyapunov exponent histogram. The fluid is assumed to follow the Carreau–Bird model, and mixed boundary conditions are imposed. The low-order dynamical system, resulted from Galerkin projection of the conservation of mass and momentum equations, includes additional nonlinear terms in the velocity components originated from the shear-dependent viscosity. It is observed that the base flow loses its radial flow stability to the vortex structure at a lower critical Taylor number, as the shear-thinning effects increase. The emergence of the vortices corresponds to the onset of a supercritical bifurcation, which is also seen in the flow of a linear fluid. However, unlike the Newtonian case, shear-thinning Taylor vortices lose their stability as the Taylor number reaches a second critical number corresponding to the onset of a Hopf bifurcation. Complete flow field together with viscosity maps are given for different scenarios in the bifurcation diagram.     

Unsteady flow calculation in a radial flow centrifugal pump with spiral casing

The prediction of the two-dimensional unsteady flow established in a radial flow centrifugal pump is considered. Assuming the fluid incompressible and inviscid, the velocity field is represented by means of source and vorticity surface distributions as well as a set of point vortices. Using this representation, a grid-free (Lagrangian) numerical method is derived based on the coupling of the boundary element and vortex particle methods. In this context the source and vorticity surface distributions are determined through the non-entry boundary condition together with the unsteady Kutta condition. In order to satisfy Kelvin's theorem, vorticity is shed at the trailing edges of the impeller blades. Then the vortex particle method is used to approximate the convection of the free vorticity distribution. Results are given for a pump configuration experimentally tested by Centre Technique des Industries Mécaniques (CETIM). Comparisons between predictions and experimental data show the capability of the proposed method to reproduce the main features of the flow considered.     

Explicit formulations for evaluation of velocity gradients using boundary‐domain integral equations in 2D and 3D viscous flows

In this paper, explicit boundary‐domain integral equations for evaluating velocity gradients are derived from the basic velocity integral equations. A free term is produced in the new strongly singular integral equation, which is not included in recent formulations using the complex variable differentiation method (CVDM) to compute velocity gradients (Int. J. Numer. Meth. Fluids 2004; 45 :463–484; Int. J. Numer. Meth. Fluids 2005; 47 :19–43). The strongly singular domain integrals involved in the new integral equations are accurately evaluated using the radial integration method (RIM). Considerable computational time for evaluating integrals of velocity gradients can be saved by using present formulation than using CVDM. The formulation derived in this paper together with those presented in reference (Int. J. Numer. Meth. Fluids 2004; 45 :463–484) for 2D and in (Int. J. Numer. Meth. Fluids 2005; 47 :19–43) for 3D problems constitutes a complete boundary‐domain integral equation system for solving full Navier–Stokes equations using primitive variables. Three numerical examples for steady incompressible viscous flow are given to validate the derived formulations. Copyright © 2007 John Wiley & Sons, Ltd.     

The axisymmetric contraction–expansion: the role of extensional rheology on vortex growth dynamics and the enhanced pressure drop

The flow of a polystyrene Boger fluid through axisymmetric contraction–expansions having various contraction ratios (2≤β≤8) and varying degrees of re-entrant corner curvatures are studied experimentally over a large range of Deborah numbers. The ideal elastic fluid is dilute, monodisperse and well characterized in both shear and transient uniaxial extension. A large enhanced pressure drop above that of a Newtonian fluid is observed independent of contraction ratio and re-entrant corner curvature. Streak images, laser Doppler velocimetry (LDV) and digital particle image velocimetry (DPIV) are used to investigate the flow kinematics upstream of the contraction plane. LDV is used to measure velocity fluctuation in the mean flow field and to characterize a global elastic flow instability which occurs at large Deborah numbers. For a contraction ratio of β=2, a steady elastic lip vortex is observed while for contraction ratios of 4≤β≤8, no lip vortex is observed and a corner vortex is seen. Rounding the re-entrant corner leads to shifts in the onset of the flow transitions at larger Deborah numbers, but does not qualitatively change the overall structure of the flow field. We describe a simple rescaling of the deformation rate which incorporates the effects of lip curvature and allows measurements of vortex size, enhanced pressure drop and critical Deborah number for the onset of elastic instability to be collapsed onto master curves. Transient extensional rheology measurements are utilized to explain the significant differences in vortex growth pathways (i.e. elastic corner vortex versus lip vortex growth) observed between the polystyrene Boger fluids used in this research and polyisobutylene and polyacrylamide Boger fluids used in previous contraction flow experiments. We show that the role of contraction ratio on vortex growth dynamics can be rationalized by considering the dimensionless ratio of the elastic normal stress difference in steady shear flow to those in transient uniaxial extension. It appears that the differences in this normal stress ratio for different fluids at a given Deborah number arise from variations in solvent quality or excluded volume effects.