Regions Of Concentrated Vorticity Research Articles

Onset of nonlinearity in a two-dimensional thin shear layer

This study numerically investigates an early stage of nonlinear interaction for the better understanding of the onset of nonlinear behaviors. Two-dimensional shear flow is chosen as a canonical flow. When two disturbances of different wavenumbers satisfying no resonance condition are initially given, new components appear one after another while the original disturbances grow. The vorticity budget analysis shows that the beat of the two exciting modes plays an essential role in forming the sum and difference nonlinear components, namely the secondary modes. For the nonlinear interaction mechanism, the high vorticity around the center of the shear layer is locally transported in the transverse direction at specific streamwise sections where the amplitude of the vertical velocity fluctuation becomes relatively larger compared to other sections. The distance between these specific sections corresponds to the wavelength of the beat. The vertically dispersed vorticity will then be convected in the horizontal directions by the mean flow. As a result, several regions of concentrated vorticity appear which eventually develop into vortices. The amplification mechanism is found to be the same for both the primary and secondary modes, though the secondary modes arise from the additional perturbation deriving from the initial perturbation.

Exit-Slot Orientation on Pumping Blades for a Rotor in Ground Effect

High-resolution particle image velocimetry measurements were taken to document the evolution of the tip vortices trailed from a rotor system operating in ground effect, with centrifugal-driven (i.e., pumping) spanwise blowing blade designs. All pumping blade designs used the same intake and internal slot, with exit slots oriented at angles of 0, 30, 45, and 60 deg above the horizontal axis. Each blade set was tested in a hovering state in ground effect at a blade loading coefficient of 0.080. Additional measurements were also performed to examine the effect of mass flow rate on tip vortex diffusion. Each pumping blade designs incurred a power penalty, resulting in part from the increased profile losses associated with the exit slots. The 30, 45, and 60 deg pumping blade designs had a degraded performance when compared to that of the 0 deg pumping blade, which was hypothesized to result from negative thrust effects. At the higher thrust condition, the tip vortices trailed from the pumping blade designs persisted in the wake as concentrated regions of vorticity, and at older wake ages, produced relatively similar flowfields. However, at the lower thrust condition, the tip vortices were found to be significantly more diffused.

Chaos, patterns, coherent structures, and turbulence: Reflections on nonlinear science.

The paradigms of nonlinear science were succinctly articulated over 25 years ago as deterministic chaos, pattern formation, coherent structures, and adaptation/evolution/learning. For chaos, the main unifying concept was universal routes to chaos in general nonlinear dynamical systems, built upon a framework of bifurcation theory. Pattern formation focused on spatially extended nonlinear systems, taking advantage of symmetry properties to develop highly quantitative amplitude equations of the Ginzburg-Landau type to describe early nonlinear phenomena in the vicinity of critical points. Solitons, mathematically precise localized nonlinear wave states, were generalized to a larger and less precise class of coherent structures such as, for example, concentrated regions of vorticity from laboratory wake flows to the Jovian Great Red Spot. The combination of these three ideas was hoped to provide the tools and concepts for the understanding and characterization of the strongly nonlinear problem of fluid turbulence. Although this early promise has been largely unfulfilled, steady progress has been made using the approaches of nonlinear science. I provide a series of examples of bifurcations and chaos, of one-dimensional and two-dimensional pattern formation, and of turbulence to illustrate both the progress and limitations of the nonlinear science approach. As experimental and computational methods continue to improve, the promise of nonlinear science to elucidate fluid turbulence continues to advance in a steady manner, indicative of the grand challenge nature of strongly nonlinear multi-scale dynamical systems.

A Numerical Study of Hurricane Erin (2001). Part II: Shear and the Organization of Eyewall Vertical Motion

Abstract A high-resolution numerical simulation of Hurricane Erin (2001) is used to examine the organization of vertical motion in the eyewall and how that organization responds to a large and rapid increase in the environmental vertical wind shear and subsequent decrease in shear. During the early intensification period, prior to the onset of significant shear, the upward motion in the eyewall was concentrated in small-scale convective updrafts that formed in association with regions of concentrated vorticity (herein termed mesovortices) with no preferred formation region around the eyewall. Asymmetric flow within the eye was weak. As the shear increased, an azimuthal wavenumber-1 asymmetry in storm structure developed with updrafts tending to occur on the downshear to downshear-left side of the eyewall. Continued intensification of the shear led to increasing wavenumber-1 asymmetry, large vortex tilt, and a change in eyewall structure and vertical motion organization. During this time, the eyewall structure was dominated by a vortex couplet with a cyclonic (anticyclonic) vortex on the downtilt-left (downtilt-right) side of the eyewall and strong asymmetric flow across the eye that led to strong mixing of eyewall vorticity into the eye. Upward motion was concentrated over an azimuthally broader region on the downtilt side of the eyewall, upstream of the cyclonic vortex, where low-level environmental inflow converged with the asymmetric outflow from the eye. As the shear diminished, the vortex tilt and wavenumber-1 asymmetry decreased, while the organization of updrafts trended back toward that seen during the weak shear period. Based upon the results for the Erin case, as well as that for a similar simulation of Hurricane Bonnie (1998), a conceptual model is developed for the organization of vertical motion in the eyewall as a function of the strength of the vertical wind shear. In weak to moderate shear, higher wavenumber asymmetries associated with eyewall mesovortices dominate the wavenumber-1 asymmetry associated with the shear so that convective-scale updrafts form when the mesovortices move into the downtilt side of the eyewall and dissipate on the uptilt side. Under strong shear conditions, the wavenumber-1 asymmetry, characterized by a prominent vortex couplet in the eyewall, dominates the vertical motion organization so that mesoscale ascent (with embedded convection) occurs over an azimuthally broader region on the downtilt side of the eyewall. Further research is needed to determine if these results apply more generally.

Chemical fronts in Hele-Shaw cells: Linear stability analysis based on the three-dimensional Stokes equations

We present linear stability results based on the three-dimensional Stokes equations for chemically reacting, propagating fronts giving rise to an unstable density stratification in a Hele-Shaw cell. The results are compared with the experiments in M. Böckmann and S. C. Müller [Phys. Rev. Lett. 85, 2506 (2000)], as well as with a corresponding linear stability analysis based on the Darcy equations that was performed in A. De Wit [Phys. Rev. Lett. 87, 054502 (2001)]. The reason for the good agreement between these earlier Darcy data and the experimentally observed growth rates is found in the relatively low experimental value of the Rayleigh number, Ra=79, for which the flow is approximately of Poiseuille type. Already for Ra values as low as 300, we observe a discrepancy between the stability results based on the Darcy and Stokes equations, respectively, with the Darcy results overpredicting both the most amplified wavenumber, as well as the corresponding growth rate, by about a factor of two. This indicates that three-dimensional effects quickly gain importance as Ra increases, so that the stability analysis needs to be based on the full, three-dimensional Stokes equations. The stability results based on the Stokes equations furthermore demonstrate the stabilizing influences of both an increasing interfacial thickness, as well as increasing frontal propagation velocities, confirming the earlier Darcy-based findings by De Wit. An argument in terms of vorticity is forwarded to explain the latter effect. A more rapidly advancing front deposits vorticity over a wider layer of fluid particles, so that the concentrated regions of vorticity needed for rapid instability growth cannot form. Somewhat surprisingly, however, slowly propagating fronts are seen to be more unstable than nonreacting fronts of equivalent thickness, as the chemical reaction leads to the formation of more compact perturbations in the interfacial region.

A model of flapping motion in a plane jet

Flow visualization near the exit of a plane jet shows a small-amplitude disturbance whose wavelength is large relative to the jet shear layer thickness. Further downstream, in the transition region, concentrated regions of vorticity are observed which drive the flapping motion of the jet. These observations motivate an inviscid, two-dimensional model for the transitional region of the jet. Linear stability analysis of a piecewise-uniform shear layer model indicates that small-amplitude, long wavelength disturbances are unstable. Long wave theory shows that regions of high circulation convect downstream faster than regions of low circulation resulting in nonlinear steepening and that the rate of the steepening is directly proportional to the strength of the local shear. The long wave theory also shows that finite-amplitude sinuous disturbances at the jet centerline will grow linearly as they convect downstream. The results predict the steepening and growth of the jet centerline observed in the flow visualization.

Is concentrated vorticity that important?

The main purpose of this paper is to bring about a better balance between views of the exaggerated importance of concentrated vorticity, on one hand, and the underestimated role of three kinds of regions other than concentrated vorticity: i — ‘structureless’ background, ii — regions of strong vorticity/strain (self) interaction and strong enstrophy generation, and iii — regions with negative enstrophy production, on the other hand, with the emphasis on the latter. Results of experiments on turbulent grid flow and DNS of decaying turbulent flow in a periodic ‘box’ at the same Reynolds number (Re λ ≈ 75) are used in order to demonstrate that all these regions are strongly non-Gaussian, dynamically significant and possess structure. It is argued that due to the strong nonlocality of turbulence in physical space all the four regions are in continuous interaction and are strongly correlated. Thus the answer to the question posed in the title is that — though important — regions of concentrated vorticity are not as important as is commonly believed.

The interaction of spatially modulated vortex pairs with free surfaces

Spatially modulated vortex pairs were generated below a free surface by two counter-rotating flaps whose edges approximate a sinusoid. The surface interactions of the vertically approaching vortex pairs were visualized by the shadowgraph technique. Two limiting cases were investigated in detail: the interaction with a surfactant-rich (contaminated) surface and with a surfactant-poor (‘clean’) surface. In the latter case shadowgraph images showed that the underlying vortex core formed a line of circular surface depressions. Subsequent measurements of the temporally evolving velocity fields using digital particle image velocimetry (DPIV) of the vortex pair cross-sections and the subsurface plane confirmed the connection process of the main vortex core with the surface. As a result of the connection the initially modulated vortex tube was broken into a line of U-vortices. In the presence of surfactants this connection could not be observed; rather a Reynolds ridge (or stagnation line) was formed and a very weak connection of the secondary separation vortex could be seen in the shadowgraphs as well as measured with the time-resolved DPIV technique.A prerequisite for connection of the vortex with the surface is that the flow's kinematics force the vortex core, that is, regions of concentrated vorticity, toward the surface. The ensuing locally concentrated viscous flux of surface-parallel vorticity through the surface is balanced by a local surface deceleration. Surface-normal vorticity appears on each side of the decelerated region whose gradually increasing circulation is directly balanced by the loss of circulation of the surface-parallel vortex. However, the shear forces caused by small amounts of surface contamination and its associated subsurface boundary layer inhibit the connection process by preventing the essential viscous flux of parallel vorticity through the surface. Instead, the subsurface boundary layer is associated with a flux of parallel vorticity into the surface which then concentrates into the observable secondary separation vortex.

Design considerations for numerical filters used in Vortex-in-cell algorithms

Vortex methods constitute a class of numerical algorithms for the simulation of flow fields characterized by regions of concentrated vorticity. Vortex-in-cell (VIC) algorithms call for some choices in the distribution functions (numerical filters) implicit in the distribution of vorticity or circulation onto a fixed mesh and interpolation of velocities from the mesh back onto the Lagrangian points. This choice is dictated by several considerations, including accuracy, grid independence, Fourier transformability and computational efficiency. In this paper, we examine and compare five numerical filters found in the literature and one developed by one of us (Bartholomew) in terms of local conservation of circulation, subgrid spatial resolution, reduction of undesirable grid effects or aliassing effects and computer work load (i.e. execution time). Based upon these comparisons, an optimum filter with respect to accuracy considerations is established.

Perspective: Numerical Simulations of Wakes and Blade-Vortex Interaction

A method for simulating incompressible flows past airfoils and their wakes is described. Vorticity panels are used to represent the body, and vortex blobs (vortex points with their singularities removed) are used to represent the wake. The procedure can be applied to the simulation of completely attached flow past an oscillating airfoil. The rate at which vorticity is shed from the trailing edge of the airfoil into the wake is determined by simultaneously requiring the pressure along the upper and lower surface streamlines to approach the same value at the trailing edge and the circulation around both the airfoil and its wake to remain constant. The motion of the airfoil is discretized, and a vortex is shed from the trailing edge at each time step. The vortices are convected at the local velocity of fluid particles, a procedure that renders the pressure continuous in an inviscid fluid. When the vortices in the wake begin to separate they are split into more vortices, and when they begin to collect they are combined. The numerical simulation reveals that the wake, which is originally smooth, eventually coils, or wraps, around itself, primarily under the influence of the velocity it induces on itself, and forms regions of relatively concentrated vorticity. Although discrete vortices are used to represent the wake, the spatial density of the vortices is so high that the computed velocity profiles across a typical region of concentrated vorticity are quite smooth. Although the computed wake evolves in an entirely inviscid model of the flowfield, these profiles appear to have a viscous core. The computed spacing between the regions of concentrated vorticity in the wake and the circulations around them are in good agreement with the experimental results. As an application, a simulation of the interaction between vorticity in the oncoming stream and a stationary airfoil is also discussed.

Collision of two vortex rings

The interaction of two identical circular viscous vortex rings starting in a side-by-side configuration is investigated by solving the Navier–Stokes equation using a spectral method with 643grid points. This study covers initial Reynolds numbers (ratio of circulation to viscosity) up to 1153. The vortices undergo two successive reconnections,fusionandfission, as has been visualized experimentally, but the simulation shows topological details not observed in experiments. The shapes of the evolving vortex rings are different for different initial conditions, but the mechanism of the reconnection is explained bybridging(Melander & Hussain 1988) except that the bridges are created on the front of the dipole close to the position of the maximum strain rate. Spatial structures of various field quantities are compared. It is found that domains of high energy dissipation and high enstrophy production overlap, and that they are highly localized in space compared with the regions of concentrated vorticity. The kinetic energy decays according to the same power laws as found in fully developed turbulence, consistent with concentrated regions of energy dissipation. The main vortex cores survive for a relatively long time. On the other hand, the helicity density which is higher in roots of bridges and threads (or legs) changes rapidly in time. The high-helicity-density and high-energy-dissipation regions overlap significantly although their peaks do not always do so. Thus a long-lived structure may carry high-vorticity rather than necessarily high-helicity density. It is shown that the time evolution of concentration of a passive scalar is quite different from that of the vorticity field, confirming our longstanding warning against relying too heavily on flow visualization in laboratory experiments for studying vortex dynamics and coherent structures.

Cyclic variations and turbulence structure in spark-ignition engines

Cyclic variations in pressure and burning time in a single cylinder spark-ignition engine have been determined as a function of equivalence ratio and engine speed. For the same operating conditions, measurements of turbulence intensity are available and these have been used with an estimate of the integral scale to estimate the magnitude of the Taylor microscale. It has been shown that the standard deviation in the burning time associated with the early stages of burning is predictable from knowledge of the Taylor microscale and the laminar burning velocity. This result is an implication of the Tennekes model of small scale turbulence and the Chomiak explanation of high flame propagation rate in regions of concentrated vorticity.

Combination of Kinematics with Flow Visualization to Compute Total Circulation

A method is described in which kinematics is exploited to compute the total circulation of a vortex from relatively simple flow visualization experiments. There are several advantages in the technique, including the newly acquired ability to calculate the changes in strength of a single vortex as it evolves. The main concepts and methodology are discussed in a general way for application to vortices which carry along with them definable regions of essentially irrotational fluid; however, the approach might be generalized to other flows which contain regions of concentrated vorticity. As an illustrative example, an application to the study of the transient changes in total circulation of individual vortex rings as they travel up a tube is described, taking into account the effect of the tube boundary. The accuracy of the method, assessed in part by a direct comparison with a laser Doppler measurement is felt to be well within experimental precision for vortex rings over a wide range of Reynolds numbers.

Structure of turbulence

Studies of turbulent flow structure at Southampton over the past ten years have been primarily concerned with the quantitative description of those components concerned with aerodynamic noise generation and those concerned with the unsteady loading of structures. The aims of, and experimental approach to, individual experiments has changed over this period with the growth of knowledge and development of facilities. For example, excellent analogue facilities for the measurement of the statistical properties of turbulent flow structure have existed throughout the ten years while fast digital data processing equipment has recently provided what amounts to a new environment for our research. Initially the studies concerned statistical descriptions of the structure of turbulent jets and boundary layers based on time delayed cross-correlation measurements. These provide useful descriptions of practical existing flows but offer no obvious hints how these observed flows may be controlled or modified. More recently the aim has been to describe in more detail the physical structure of the individual eddies that constitute the turbulence and to clearly identify those features of the flow that led to the observed state. Three lines of active development are described, with some promising initial results. The first concerns time domain analysis of the velocity, displacement and pressure records from turbulent flows to identify the details of the characteristic eddy patterns in the flow. The second concerns potential flow modelling of impulsively started flows where arrays of elementary vortices represent vortex sheets and regions of concentrated vorticity. The third concerns modified Orr-Sommerfeld stability analysis of shear flow turbulence. These approaches are clearly complementary and so provide the basis of a comprehensive attack on the problem.