Identification of vortex cores of three-dimensional large-vortical structures

In this paper, we propose a nonlocal method to identify vortex cores in three-dimensional flows as a complement to the existing list of local and nonlocal methods of the bibliography. The method is based on the vector field of the instantaneous rotation of a particle around a center. This center is defined using the Darboux vector field along the path-particle lines; the vortex core is detected using their Frenet–Serret frame. We illustrate the application of the method to identify the core of large-vortical structures in analytical and numerically simulated laminar and turbulent natural convection flows.     

Finite-span rotating wings: three-dimensional vortex formation and variations with aspect ratio

We investigate experimentally the effect of aspect ratio ( ) on the time-varying, three-dimensional flow structure of flat-plate wings rotating from rest at 45° angle of attack. Plates of = 2 and 4 are tested in a 50 % by mass glycerin–water mixture, with a total rotation of ? = 120° and a matched tip Reynolds number of 5,000. The time-varying, three-component volumetric velocity field is reconstructed using phase-locked, phase-averaged stereoscopic digital particle image velocimetry in multiple, closely-spaced chordwise planes. The vortex structure is analyzed using the $\mathcal{Q}$ -criterion, helicity density, and spanwise quantities. For both s, the flow initially consists of a connected and coherent leading-edge vortex (LEV), tip vortex (TV), and trailing-edge vortex (TEV) loop; the LEV increases in size with span and tilts aft. Smaller, discrete vortices are present in the separated shear layers at the trailing and tip edges, which wrap around the primary TEV and TV. After about ? = 20°, the outboard-span LEV lifts off the plate and becomes arch-like. A second, smaller LEV and the formation of corner vortex structures follow. For = 4, the outboard LEV moves farther aft, multiple LEVs form ahead of it, and after about ? = 50° a breakdown of the lifted-off LEV and the TV occurs. However, for = 2, the outboard LEV lift-off is not progressive, and the overall LEV-TV flow remains more coherent and closer to the plate, with evidence of breakdown late in the motion. Inboard of about 50 % span, the = 4 LEV is stable for the motion duration. Up to approximately 60 % span, the = 2 LEV is distinct from the TV and is similarly stable. The = 2 LEV exhibits substantially higher spanwise vorticity and velocity. The latter possesses a “four-lobed” distribution at the periphery of the LEV core having adjacent positive (outboard) and negative (inboard) components, corresponding to a helical streamline structure. Both s show substantial root-to-tip velocity aft of the stable LEV, which drives outboard spanwise vorticity flux; flux toward the root is also present in the front portion of the LEV. For = 2, there is a strong flux of spanwise vorticity from the outboard LEV to the tip, which may mitigate LEV lift-off and is not found for = 4. The TV circulation for each is similar in magnitude and growth when plotted versus the chord lengths travelled by the tip, prior to breakdown. Streamwise vorticity due to the TV induces high spanwise velocity, and for = 2, the tilted LEV creates further streamwise vorticity which corresponds well to spanwise-elongated regions of spanwise velocity. For = 2, the TV influences a relatively greater portion of the span and is more coherent at later times, which coupled with the tilted LEV strongly contributes to the higher overall spanwise velocity and vorticity flux.     

Generation of large-scale structures and vortex systems in numerical experiments for rotating annular channels

Methods for solving shallow-water equations that describe flows in rotating annular channels are considered and the results of numerical calculations are analyzed for the possible generation of global large-scale flows, narrow jets, and numerous small-scale vortices in laboratory experiments. External effects in fluids are induced using a mass source–sink and the MHD-method of interaction of radial electric current with the magnetic field generated by the field of permanent magnets. A central–upwind scheme modified to suit the specific aspects of geophysical hydrodynamics. Initially, this method was used to solve shallow-water equations only in hydraulic problems, such as for flows in dam breaks, channels, rivers, and lakes. Geophysical hydrodynamics (in addition to free surface and topography) requires a rotation of the system as a whole, which is accompanied by the appearance of a complex system of vortices, jets, and turbulence (these should be taken into account in the formulation of the problem). Accordingly, the basic features of the central–upwind method should be changed. The modifications should ensure that the scheme is well-balanced and choose interpolation methods for desired variables. The main result of this modification is the control over numerical viscosity affecting the fluid motion variety. The active dynamics of a large number of vortices transformed into jets or generating large-scale streams is the general result of modifications suitable for geophysical hydrodynamics. Because there are technical difficulties in the creation of an appropriate laboratory setup for modeling of geophysical flows with the help of numerous source–sinks, it will be appropriate to use numerical experiments for studying the motions generated by this method. Unlike this method, the MHD-method can be rather easily used in laboratory conditions to generate a large variety of flows and vortex currents in the channel by a relatively small number of permanent magnets. Specifically, this method made it possible to obtain large-scale circular flows over the entire channel area, jets, and systems of interacting vortices. For the purpose of experiments, the distributions of source–sinks and systems of permanent magnets over the bottom of annular channels are determined.     

Experimental study of vortex structures in a hypersonic shock layer on a flat plate

The characteristics of travelling perturbations of density in a hypersonic shock layer on a flat plate for the Mach number M_∞=21 and unit Reynolds numberRe ₁=6·10⁵ m⁻¹ were experimentally studied by the method of electron-beam fluorescence. The perturbations were generated by interaction of the shock layer behind an oblique gas-dynamic whistle and the leading edge of the plate. The cases of unsteady and quasi-steady interaction were considered. In both cases, vortex disturbances of finite amplitude were generated. The measurements were performed at the fundamental frequency F=0.6·10⁻⁴ and at the harmonic; the streamwise phase velocities, the growth rates of the disturbances, and the angles of wave propagation were obtained. The measurement results are compared with some experimental data for subsonic flows, some particular results of the linear stability theory for compressible flows, and the results obtained on the basis of a simple model of the nonlinear stage of disturbance evolution in a hypersonic boundary layer. Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 6, pp. 41–47, November–December, 1999.     

Numerical modeling of three-dimensional vortex structures in hypersonic transverse flow past a cylinder

A new mechanism of the formation of spatially periodic structures on the nose surfaces of cylindrically blunted bodies in a hypersonic transverse flow is investigated. According to this mechanism, a curved shock wave produces a vortex flow, while the vortex, which is conserved in the presence of weak dissipation, acts on the shock and maintains its curved shape. The realizability of this vortex formation mechanism is verified by direct numerical simulation using the FLUENT software package. It is confirmed that in the case of uniform hypersonic freestream both plane and three-dimensional modes of the steady flow past the cylinder nose can exist. The three-dimensional mode is characterized by periodic-in-span vortex structures and considerable heat flux peaks on the nose surface. The calculated results are compared with the experimental data.     

Experimental investigation of three-dimensional flow instabilities in a rotating lid-driven cavity

The swirling flow between a rotating lid and a stationary cylinder is studied experimentally. The flow is governed by two parameters: the ratio of container height to disk radius, h, and the Reynolds number, Re, based on the disk angular velocity, cylinder radius and kinematic viscosity of the working liquid. For the first time, the onset of three-dimensional flow behavior is measured by combining the high spatial resolution of particle image velocimetry and the temporal accuracy of laser Doppler anemometry. A detailed mapping of the transition scenario from steady and axisymmetric flow to unsteady and three-dimensional flow is investigated for 1 ≥ h ≥ 3.5. The flow is characterized by the development of azimuthal modes of different wave numbers. A range of different modes is detected and critical Reynolds numbers and associated frequencies are identified. The results are compared to the numerical stability analysis of Gelfgat et al. (J Fluid Mech 438:363–377, 2001). In most cases, the measured onset of three-dimensionality is in good agreement with the numerical results and disagreements can be explained by bifurcations not accounted for by the numerical stability analysis.     

Non-linear viscoelastodynamic equations of three-dimensional rotating structures in finite displacement and finite element discretization

This paper deals with the non-linear viscoelastodynamics of three-dimensional rotating structure undergoing finite displacement. In addition, the non-linear dynamics is studied with respect to geometrical and mechanical perturbations. On part of the boundary of the structure, a rigid body displacement field is applied which moves the structure in a rotation motion. A time-dependent Dirichlet condition is applied to another part of the boundary. For instance, this corresponds to the cycle step of a helicopter rotor blade. A surface force field is applied to the third part of the boundary and depends on the time history of the structural displacement field. For example, this might corresponds to general unsteady aerodynamics forces applied to the structure. The objective of this paper is to model the non-linear dynamic behavior of such a rotating viscoelastic structure undergoing finite displacements, and to allow small geometrical and mechanical (mass, constitutive equations) perturbations analysis to be performed. The model is constructed by the introduction of a reference configuration which is deduced from the non-linear steady boundary value problem. A constitutive equation deduced from the Coleman and Noll theory concerning the viscoelasticity in finite displacement is used. Thereafter, the weak formulation of the boundary value problem is constructed and discretized using the finite element method. In order to simplify the mathematical study of the equations, multilinear forms are introduced in the algebraic calculation and their mathematical properties are presented.     

On the development of lift and drag in a rotating and translating cylinder

The two-dimensional flow around a rotating cylinder is investigated numerically using a vorticity forces formulation with the aim of analyzing quantitatively the flow structures, and their evolutions, that contribute to the lift and drag forces on the cylinder. The Reynolds number considered, based on the cylinder diameter and steady free stream speed, is Re=200, while the non-dimensional rotation rate (ratio of the surface speed and free stream speed) selected was α=1 and 3. For α=1 the wake behind the cylinder for the fully developed flow is oscillatory due to vortex shedding, and so are the lift and drag forces. For α=3 the fully developed flow is steady with constant (high) lift and (low) drag. Each of these cases is considered in two different transient problems, one with angular acceleration of the cylinder and constant speed, and the other one with translating acceleration of the cylinder and constant rotation. We characterize quantitatively the contributions of individual fluid elements (vortices) to aerodynamic forces, explaining and quantifying the mechanisms by which the lift is generated in each case. In particular, for high rotation (when α=3), we explain the relation between the mechanisms of vortex shedding suppression and those by which the lift is enhanced and the drag is almost suppressed when the fully developed flow is reached.     

Three-dimensional vortex structures in cavities

There are two main approaches in theoretical studies on the flow of liquids in planar cavities.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 54–59, March–April, 1981.     

Experimental study of supersonic three-dimensional jets

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 75–80, May–June, 1988.     

Experimental study of oblique circular cylindrical apertures in plates

Ten steel plates weakened by the penetration of an oblique circular cylindrical aperture have been tested. The generator of the aperture makes 0-, 15-, 30- and 45-deg angles with the normal to the plate surface. In the case of the first seven models, the tractions were applied in the direction normal and parallel to the plane of symmetry. The strain distributions around the aperture are presented in nondimensional forms along three generators 45 deg apart. The last three models, with large aperture diameter, were tested to investigate the effect of thickness-to-diameter ratio for the three angles of skewness. The experimental results are compared with the theoretical predictions. It is noted that, for most of the plates, there was a fairly good agreement for the stress distribution throughout the thickness. The results of the experiments for uniaxial state of stress have been superimposed in order to obtain the response for various biaxial conditions.     

Resistance of rough plates in a rotating fluid

Generalizing Navier’s partial slip condition, the flow due to a rough or striated plate moving in a rotating fluid is studied. It is found that the motion of the plate, the fluid surface velocity, and the shear stress are in general not in the same direction. The solution is extended to the case of finite depth, or Couette slip flow in a rotating system. In this case an optimum depth for minimum drag is found. The solutions are also closed form exact solutions of the Navier–Stokes equations. The results are fundamental to flows with Coriolis effects.     

Numerical study of vortex shedding from a rotating cylinder immersed in a uniform flow field

A numerical study is made of the unsteady two‐dimensional, incompressible flow past an impulsively started translating and rotating circular cylinder. The Reynolds number (Re) and the rotating‐to‐translating speed ratio (α) are two controlled parameters, and the influence of their different combinations on vortex shedding from the cylinder is investigated by the numerical scheme sketched below. Associated with the streamfunction (ψ)–vorticity (ω) formulation of the Navier–Stokes equations, the Poisson equation for ψ is solved by a Fourier/finite‐analytic, separation of variable approach. This approach allows one to attenuate the artificial far‐field boundary, and also yields a global conditioning on the wall vorticity in response to the no‐slip condition. As for the vorticity transport equation, spatial discretization is done by means of finite difference in which the convection terms are handled with the aid of an ENO (essentially non‐oscillatory)‐like data reconstruction process. Finally, the interior vorticity is updated by an explicit, second‐order Runge–Kutta method. Present computations fall into two categories. One with Re=10³ and α≤3; the other with Re=10⁴ and α≤2. Comparisons with other numerical or physical experiments are included. Copyright © 2000 John Wiley & Sons, Ltd.