Analytic solution of in-plane vortex–rotor interactions with arbitrary orientation and its impact on rotor trim

The general aerodynamic problem of arbitrarily oriented in-plane vortex-rotor interaction was investigated in the past only by numerical simulation. Just one special case of in-plane vortex-rotor interaction with the vortex axis in flight direction was recently solved analytically. In this article, the analytical solution for arbitrary in-plane vortex orientation and position relative to the rotor is given for the first time. The solution of the integrals involved as derived here encompasses and simplifies the previous derivation of the special case significantly. Results provide the vortex impact on rotor trim (thrust, aerodynamic rolling and pitching moments about the hub) and the rotor controls required to mitigate these disturbances. For the special case with the vortex axis in flight direction, the results are identical to the former solution and results for the other in-plane vortex orientations and positions agree with the numerical results obtained so far.

The response of the copepod Acartia tonsa to the hydrodynamic cues of small-scale, dissipative eddies in turbulence

This study quantifies the behavioral response of a marine copepod (Acartia tonsa) to individual, small-scale, dissipative vortices that are ubiquitous in turbulence. Vortex structures were created in the laboratory using a physical model of a Burgers vortex with characteristics corresponding to typical dissipative vortices that copepods are likely to encounter in the turbulent cascade. To examine the directional response of copepods, vortices were generated with the vortex axis aligned in either horizontal or vertical directions. Tomographic particle image velocimetry was used to measure the volumetric velocity field of the vortex. Three-dimensional copepod trajectories were digitally reconstructed and overlaid on the vortex flow field to quantify A. tonsa’s swimming kinematics relative to the velocity field and to provide insight to the copepod behavioral response to hydrodynamic cues. The data show significant changes in swimming kinematics and an increase in relative swimming velocity and hop frequency with increasing vortex strength. Furthermore, in moderate-to-strong vortices, A. tonsa moved at elevated speed in the same direction as the swirling flow and followed spiral trajectories around the vortex, which would retain the copepod within the feature and increase encounter rates with other similarly behaving Acartia. While changes in swimming kinematics depended on vortex intensity, orientation of the vortex axis showed minimal significant effect. Hop and escape jump densities were largest in the vortex core, which is spatially coincident with the peak in vorticity suggesting that vorticity is the hydrodynamic cue that evokes these behaviors.

Vortex axis tracking by iterative propagation (VATIP): a method for analysing three-dimensional turbulent structures

Vortex is a central concept in the understanding of turbulent dynamics. Objective algorithms for the detection and extraction of vortex structures can facilitate the physical understanding of turbulence regeneration dynamics by enabling automated and quantitative analyses of these structures. Despite the wide availability of vortex identification criteria, they only label spatial regions belonging to vortices, without any information on the identity, topology and shape of individual vortices. This latter information is stored in the axis lines lining the contours of vortex tubes. In this study, a new tracking algorithm is proposed which propagates along the vortex axis lines and iteratively searches for new directions for growth. The method is validated in flow fields from transient simulations where vortices of different shapes are controllably generated. It is then applied to statistical turbulence for the analysis of vortex configurations and distributions. It is shown to reliably extract axis lines for complex three-dimensional vortices generated from the walls. A new procedure is also proposed that classifies vortices into commonly observed shapes, including quasi-streamwise vortices, hairpins, hooks and branches, based on their axis-line topology. Clustering analysis is performed on the extracted axis lines to reveal vortex organization patterns and their potential connection to large-scale motions in turbulence.

Experimental investigation of the scaling of vortex wandering in turbulent surroundings

The wandering of a wing-tip vortex in free-stream turbulence was documented by analysis of multi-probe hot-wire measurements in a wind tunnel and flow visualisation and particle image velocimetry measurements in a water tunnel. An error-minimisation approach was applied to the hot-wire measurements to estimate the time history of the location of the vortex axis, whereas flow visualisation from two orthogonal views permitted the reconstruction of relatively long sections of the vortex axis. The amplitude of the wandering motion was found to scale with the turbulence intensity, the core radius and the vortex turnover time; this amplitude was insensitive to changes in the integral length and time scales of the turbulence. The period of the vortex wandering was distributed in the range between 1 and 10 vortex turnover times. The wavelength of wandering was distributed at a relatively long value, which scaled with the vortex turnover time. The velocity of vortex wandering depended on the vortex turnover time, but also contained an additional contribution that was consistent with motion induced by bending waves. The prevalence of the vortex turnover time as the scale for vortex wandering was interpreted as evidence that vortex-induced straining of the free-stream eddies bounds the interaction time between the two, thus limiting the time available for linear and angular momentum transfer.

Effects of System Rotation on Vortical Structure in Wall Turbulence

Direct numerical simulations (DNSs) of rotating turbulent Poiseuille flows are performed to study the effects of both cyclonic and anticyclonic system rotation on the kinematics of the quasi-streamwise vortices. By using the second invariant of the deformation tensor, a number of streamwise vortices are detected and averaged in the wall vicinity where the intense sweep motion, i.e., the inrush motion of high-speed fluid toward the wall, is related to the quasi-streamwise vortices. The effects of the system rotation on the angle of vortex axis are clearly observed as studied in longitudinal vortices of the homogeneous shear flow. Moreover, by calculating the probability of the emergence of the counterclockwise vortices (CCVs) around a clockwise vortex (CV), we find that with increase in the anticyclonic system rotation, the probability increases and decreases in the ejection and sweep sides of a CV, respectively. In contrast, cyclonic system rotation attenuates CCVs in both sides of a CV, though it increases at the top of the CV. This distribution of CCVs is found to affect sweep motion related to the quasi-streamwise vortices.

Topological structure of shock induced vortex breakdown

Using a combination of critical point theory of ordinary differential equations and numerical simulation for the three-dimensional unsteady Navier–Stokes equations, we study possible flow structures of the vortical flow, especially the unsteady vortex breakdown in the interaction between a normal shock wave and a longitudinal vortex. The topological structure contains two parts. One is the sectional streamline pattern in the cross-section perpendicular to the vortex axis. The other is the sectional streamline pattern in the symmetrical plane. In the cross-section perpendicular to the vortex axis, the sectional streamlines have spiral or centre patterns depending on a function λ (x, t) = 1/ρ(∂ρ/∂t+∂ρu/∂x), where x is the coordinate corresponding to the vortex axis. If λ > 0, the sectional streamlines spiral inwards in the near region of the centre. If λ < 0, the sectional streamlines spiral outwards in the same region. If λ = 0, the sectional streamlines form a nonlinear centre. If λ changes its sign along the vortex axis, one or more limit cycles appear in the sectional streamlines in the cross-section perpendicular to the vortex axis. Numerical simulation for two typical cases of shock induced vortex breakdown (Erlebacher, Hussaini & Shu, J. Fluid Mech., vol. 337, 1997, p. 129) is performed. The onset and time evolution of the vortex breakdown are studied. It is found that there are more limit cycles for the sectional streamlines in the cross-section perpendicular to the vortex axis. In addition, we find that there are quadru-helix structures in the tail of the vortex breakdown.

Tornado funnel-shaped cloud as convection in a cloudy layer

Analytical model of convection in a thick horizontal cloud layer with free upper and lower boundaries is constructed. The cloud layer is supposed to be subjected to the Coriolis force due to the cloud rotation, which is a typical condition for tornado formation. It is obtained that convection in such system can look as just one rotating cell in contrast to the usual many-cells Benard convection. The tornado-type vortex is different from spatially periodic convective cells because their amplitudes vanish with distance from the vortex axis. The lower boundary at this convection can substantially move out of the initially horizontal cloud layer forming a single vertical vortex with intense upward and downward flows. The results are also applicable to convection in water layer with negative temperature gradient.

A generalized vortex ring model

A conventional laminar vortex ring model is generalized by assuming that the time dependence of the vortex ring thickness ℓ is given by the relation ℓ = atb, where a is a positive number and 1/4 ≤ b ≤ 1/2. In the case in which $a=\sqrt{2\nu}$, where ν is the laminar kinematic viscosity, and b = 1/2, the predictions of the generalized model are identical with the predictions of the conventional laminar model. In the case of b = 1/4 some of its predictions are similar to the turbulent vortex ring models, assuming that the time-dependent effective turbulent viscosity ν∗ is equal to ℓℓ′. This generalization is performed both in the case of a fixed vortex ring radius R0 and increasing vortex ring radius. In the latter case, the so-called second Saffman's formula is modified. In the case of fixed R0, the predicted vorticity distribution for short times shows a close agreement with a Gaussian form for all b and compares favourably with available experimental data. The time evolution of the location of the region of maximal vorticity and the region in which the velocity of the fluid in the frame of reference moving with the vortex ring centroid is equal to zero is analysed. It is noted that the locations of both regions depend upon b, the latter region being always further away from the vortex axis than the first one. It is shown that the axial velocities of the fluid in the first region are always greater than the axial velocities in the second region. Both velocities depend strongly upon b. Although the radial component of velocity in both of these regions is equal to zero, the location of both of these regions changes with time. This leads to the introduction of an effective radial velocity component; the latter case depends upon b. The predictions of the model are compared with the results of experimental measurements of vortex ring parameters reported in the literature.