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A new mathematical approach to kinematics and dynamics of planar uniform vortices in an incompressible inviscid fluid is presented. It is based on an integral relation between Schwarz function of the vortex boundary and induced velocity. This relation is firstly used for investigating the kinematics of a vortex having its Schwarz function with two simple poles in a transformed plane. The vortex boundary is the image of the unit circle through the conformal map obtained by conjugating its Schwarz function. The resulting analysis is based on geometric and algebraic properties of that map. Moreover, it is shown that the steady configurations of a uniform vortex, possibly in presence of point vortices, can be also investigated by means of the integral relation. The vortex equilibria are divided in two classes, depending on the behavior of the velocity on the boundary, measured in a reference system rotating with this curve. If it vanishes, the analysis is rather simple. However, vortices having nonvanishing relative velocity are also investigated, in presence of a polygonal symmetry. In order to study the vortex dynamics, the definition of Schwarz function is then extended to a Lagrangian framework. This Lagrangian Schwarz function solves a nonlinear integrodifferential Cauchy problem, that is transformed in a singular integral equation. Its analytical solution is here approached in terms of successive approximations. The self-induced dynamics, as well as the interactions with a point vortex, or between two uniform vortices are analyzed.

We present an extensive study of vortex dynamics in a high-quality single crystal of HgBa 2 CuO 4+ δ , a highly anisotropic superconductor that is a model system for studying the effects of anisotropy. From magnetization M measurements over a wide range of temperatures T and fields H , we construct a detailed vortex phase diagram. We find that the temperature-dependent vortex penetration field H p ( T ), second magnetization peak H smp ( T ), and irreversibility field H irr ( T ) all decay exponentially at low temperatures and exhibit an abrupt change in behavior at high temperatures T / T c  >~ 0.5. By measuring the rates of thermally activated vortex motion (creep) S ( T , H ) = | d ln M ( T , H )/ d ln t |, we reveal glassy behavior involving collective creep of bundles of 2D pancake vortices as well as temperature- and time-tuned crossovers from elastic (collective) dynamics to plastic flow. Based on the creep results, we show that the second magnetization peak coincides with the elastic-to-plastic crossover at low T , yet the mechanism changes at higher temperatures.

Galilean non-invariance of the shallow-water equations describing the motion of a rotating fluid implies that a homogeneous background flow modifies the dynamics of localized vortices even without the$\\beta $-effect. In particular, in a divergent quasi-geostrophic model on a$\\beta $-plane, which originates from the shallow-water model, the equation of motion in the reference frame attached to a uniform zonal background flow has the same form as in the absence of this flow, but with a modified$\\beta $-parameter depending linearly on the flow velocity$\\bar{U}$. The evolution of a singular vortex (SV) embedded in such a flow consists of two stages. In the first, quasi-linear stage, the SV motion is induced by the secondary dipole ($\\beta $-gyres) generated in the neighbourhood of the SV. During the next, nonlinear stage, the SV merges with the$\\beta $-gyre of opposite sign to form a compact vortex pair interacting with far-field Rossby waves radiated previously by the SV, while the other$\\beta $-gyre loses connection with the SV and disappears. In the absolute reference frame and with$\\beta = 0$, the SV drifts downstream and at an angle to the background flow. The SV always lags behind the background flow, with the strongest resistance during the quasi-linear stage and weakening resistance at the nonlinear stage of SV evolution. In the general case where$\\beta \\gt 0$, the SV can move both upstream (for small-to-moderate$\\bar{U} \\gt 0$) and downstream (for$\\bar{U} \\lt 0$or sufficiently large$\\bar{U} \\gt 0$). Under weak-to-moderate westward and all eastward flows the SV cyclone (anticyclone) also moves northward (southward), its meridional drift increasing with$\\bar{U}$.

This paper presents experimental and numerical investigations on the modification of local spanwise skinfriction over triangular riblets under the total drag reduction condition. Specifically, the mean and fluctuating vortical flow fields were measured using 2-components X-wires and computed using LES, respectively. Besides, the relationship between local skin-friction along the riblet spanwise and associated vortex evolution was also built using the vortex dynamic method. Based on these results, it was found that, compared with the smooth case, the impaired and enhanced vortex strength, and resultant viscous diffusion/energy dissipation, determined the reduction and augment of the viscous drag force over the local spanwise riblet groove, i.e., decreasing and increasing cases of local drag, respectively. Furthermore, the mean normal diffusion fluxes of normal and spanwise vorticities contributed more to the viscous drag under these two cases. Correspondingly, the relevant flow physics related to these phenomena was discussed in detail.

Fascination with glassy states has persisted since Fisher introduced the vortex-glass as a new thermodynamic phase that is a true superconductor that lacks conventional long-range order. Though Fisher’s original model considered point disorder, it was later predicted that columnar defects (CDs) could also induce glassiness — specifically, a Bose-glass phase. In YBa 2 Cu 3 O 7−x (YBCO), glassy states can cause distinct behavior in the temperature ( T  ) dependent rate of thermally activated vortex motion ( S ). The vortex-glass state produces a plateau in S ( T  ) whereas a Bose-glass can transition into a state hosting vortex excitations called double-kinks that can expand, creating a large peak in S ( T  ). Although glass phases have been well-studied in YBCO, few studies exist of other materials containing CDs that could contribute to distinguishing universal behavior. Here, we report on the effectiveness of CDs tilted ~30° from the c -axis in reducing S in a NbSe 2 crystal. The magnetization is 5 times higher and S is minimized when the field is parallel to the defects versus aligned with the c -axis. We see signatures of glassiness in both field orientations, but do not observe a peak in S ( T  ) nor a plateau at values observed in YBCO. Finally, we discuss the possibility that competing disorder induces a field-orientation-driven transition from a Bose-glass to an anisotropic glass involving both point and columnar disorder.

Flow past three identical circular cylinders is numerically investigated using the immersed boundary method. The cylinders are arranged in an equilateral-triangle configuration with one cylinder placed upstream and the other two side-by-side downstream. The focus is on the effect of the spacing ratio$L/D(=1.0{-}6.0)$, Reynolds number$Re(=50{-}300)$and three-dimensionality on the flow structures, hydrodynamic forces and Strouhal numbers, where$L$is the cylinder centre-to-centre spacing and$D$is the cylinder diameter. The fluid dynamics involved is highly sensitive to both$Re$and$L/D$, leading to nine distinct flow structures, namely single bluff-body flow, deflected flow, flip-flopping flow, steady symmetric flow, steady asymmetric flow, hybrid flow, anti-phase flow, in-phase flow and fully developed in-phase co-shedding flow. The time-mean drag and lift of each cylinder are more sensitive to$L/D$than$Re$while fluctuating forces are less sensitive to$L/D$than$Re$. The three-dimensionality of the flow affects the development of the wake patterns, changing the$L/D$ranges of different flow structures. A diagram of flow regimes, together with the contours of hydrodynamic forces, in the$Re-L/D$space, is given, providing physical insights into the complex interactions of the three cylinders.

Numerical simulations are used to investigate the hydrodynamic benefits of body–fin and fin–fin interactions in a fish model in carangiform swimming. The geometry and kinematics of the model are reconstructed in three-dimensions from high-speed videos of a live fish, Crevalle Jack (Caranx hippos), during steady swimming. The simulations employ an immersed-boundary-method-based incompressible Navier–Stokes flow solver that allows us to quantitatively characterize the propulsive performance of the fish median fins (the dorsal and the anal fins) and the caudal fin using three-dimensional full body simulations. This includes a detailed analysis of associated performance enhancement mechanisms and their connection to the vortex dynamics. Comparisons are made using three different models containing different combinations of the fish body and fins to provide insights into the force production. The results indicate that the fish produces high performance propulsion by utilizing complex interactions among the fins and the body. By connecting the vortex dynamics and surface force distribution, it is found that the leading-edge vortices produced by the caudal fin are associated with most of the thrust production in this fish model. These vortices could be strengthened by the vorticity capture from the vortices generated by the posterior body during undulatory motion. Meanwhile, the pressure difference between the two sides of posterior body resulting from the posterior body vortices (PBVs) helps with the alleviation of the body drag. The appearance of the median fins in the posterior region further strengthens the PBVs and caudal-fin wake capture mechanism. This work provides new physical insights into how body–fin and fin–fin interactions enhance thrust production in swimming fishes, and emphasizes that movements of both the body and fins contribute to overall swimming performance in fish locomotion.

Rotationally coherent Lagrangian vortices are formed by tubes of deforming fluid elements that complete equal bulk material rotation relative to the mean rotation of the deforming fluid volume. We show that the initial positions of such tubes coincide with tubular level surfaces of the Lagrangian-averaged vorticity deviation (LAVD), the trajectory integral of the normed difference of the vorticity from its spatial mean. The LAVD-based vortices are objective, i.e. remain unchanged under time-dependent rotations and translations of the coordinate frame. In the limit of vanishing Rossby numbers in geostrophic flows, cyclonic LAVD vortex centres are precisely the observed attractors for light particles. A similar result holds for heavy particles in anticyclonic LAVD vortices. We also establish a relationship between rotationally coherent Lagrangian vortices and their instantaneous Eulerian counterparts. The latter are formed by tubular surfaces of equal material rotation rate, objectively measured by the instantaneous vorticity deviation (IVD). We illustrate the use of the LAVD and the IVD to detect rotationally coherent Lagrangian and Eulerian vortices objectively in several two- and three-dimensional flows.

This paper presents the response and the wake modes of a freely vibrating D-section prism with varying angles of attack ($\\alpha = 0^\\circ \\text {--}180^\\circ$) and reduced velocity ($U^* = 2\\text {--}20$) by a numerical investigation. The Reynolds number, based on the effective diameter, is fixed at 100. The vibration of the prism is allowed only in the transverse direction. We found six types of response with increasing angle of attack: typical vortex-induced vibration (VIV) at $\\alpha = 0^\\circ \\text {--}35^\\circ$; extended VIV at $\\alpha = 40^\\circ \\text {--}65^\\circ$; combined VIV and galloping at $\\alpha = 70^\\circ \\text {--}80^\\circ$; narrowed VIV at $\\alpha = 85^\\circ \\text {--}150^\\circ$; transition response, from narrowed VIV to pure galloping, at $\\alpha = 155^\\circ \\text {--}160^\\circ$; and pure galloping at $\\alpha = 165^\\circ \\text {--}180^\\circ$. The typical and narrowed VIVs are characterized by linearly increasing normalized vibration frequency with increasing $U^*$, which is attributed to the stationary separation points of the boundary layer. On the other hand, in the extended VIV, the vortex shedding frequency matches the natural frequency in a large $U^*$ range with increasing $\\alpha$ generally. The galloping is characterized by monotonically increasing amplitude with enlarging $U^*$, with the largest amplitude being $A^* = 3.2$. For the combined VIV and galloping, the vibration amplitude is marginal in the VIV branch while it significantly increases with $U^*$ in the galloping branch. In the transition from narrowed VIV to pure galloping, the vibration frequency shows a galloping-like feature, but the amplitude does not monotonically increase with increasing $U^*$. Moreover, a partition of the wake modes in the $U^*$–$\\alpha$ parametric plane is presented, and the flow physics is elucidated through time variations of the displacement, drag and lift coefficients and vortex dynamics. The angle-of-attack range of galloping is largely predicted by performing a quasi-steady analysis of the galloping instability. Finally, the effects of $m^*$ and ${\\textit {Re}}$, the roles of afterbody and the roles of separation point in determining vibration responses and vortex shedding frequency are further discussed.

A numerical investigation was conducted on $Re_{\\varGamma _{0}}=3000$ vortex rings colliding with wall-mounted hemispheres to study how their relative sizes affect the resulting vortex dynamics and structures. The hemisphere to vortex ring diameter ratio ranges from $D/d=0.5$ to $D/d=2$. Secondary/tertiary vortex rings are observed to result from hemispheric surface boundary layer separations rather than wall boundary layer separations as the diameter ratio increases. While those for $D/d\\leq 1$ hemispheres can be attributed to sequential hemispheric and wall boundary layer separations, the primary vortex ring produces a series of secondary/tertiary vortex rings only along the $D/d=2$ hemispheric surface. This indicates that the presence of the wall makes little difference when the hemisphere is sufficiently large. On top of comparing vortex ring circulations and translational velocities between hemisphere and flat-wall based collisions, present collision outcomes have also been compared with those predicted by specific discharge velocity models. Additionally, comparisons of vortex core trajectories and vortex ring formation locations with earlier cylindrical convex surface based collisions provide more clarity on differences between two- and three-dimensional convex surfaces. Finally, vortex flow models are presented to account for the significantly different flow behaviour as the hemisphere size varies. Specifically, the vortex flow model for the $D/d=2$ hemisphere hypothesizes that the recurring tertiary vortex ring formations cease only when the primary vortex ring slows down sufficiently for the last tertiary vortex ring to entangle with it and render it incoherent. Until that happens, the primary vortex ring will continue to induce more tertiary vortex rings to form, with potential implications for heat/mass transfer optimizations.