Peter S. Bernard

Discretization of a vortex sheet with an example of roll-up

Abstract The point vortex approximation of a vortex sheet in two space dimensions is examined and a remedy for some of its shortcomings is suggested. The approximation is then applied to the study of the roll-up of a vortex sheet induced by an elliptically loaded wing.

Vortex dynamics and the production of Reynolds stress

The physical mechanisms by which the Reynolds shear stress is produced from dynamically evolving vortical structures in the wall region of a direct numerical simulation of turbulent channel flow are explored. The complete set of quasistreamwise vortices are systematically located and tracked through the flow by the locus of the points of intersection of their centres of rotation with the (y, z) numerical grid planes. This approach assures positive identification of vortices of widely differing strengths, including those whose amplitude changes significantly in time. The process of vortex regeneration, and the means by which vortices grow, distort and interact over time are noted. Ensembles of particle paths arriving on fixed planes in the flow are used to represent the physical processes of displacement and acceleration transport (Bernard & Handler 1990a) from which the Reynolds stress is produced. By interweaving the most dynamically significant of the particle paths with the evolving vortical structures, the dynamical role of the vortices in producing Reynolds stress is exposed. This is found to include ejections of low-speed fluid particles by convecting structures and the acceleration and deceleration of fluid particles in the cores of vortices. Sweep dominated Reynolds stress close to the wall appears to be a manifestation of the regeneration process by which new vortices are created in the flow.

Discretization of a Vortex Sheet, with an Example of Roll-Up*

Abstract : The point vortex approximation of a vortex sheet in two space dimensions is examined and a remedy for some of its shortcomings is suggested. The approximation is then applied to the study of the roll-up of a vortex sheet induced by an elliptically loaded wing. (Author)

Reynolds stress and the physics of turbulent momentum transport

Abstract : The nature of the momentum transport processes responsible for the Reynolds shear stress is investigated using several ensembles of fluid particle paths obtained from a direct numerical simulation of turbulent channel flow. It is found that the Reynolds stress can be viewed as arising from two fundamentally different mechanisms. The more significant entails transport in the manner described by Prandtl in which momentum is carried unchanged from one point to another by the random displacement of fluid particles. One point models, such as the gradient law are found to be inherently unsuitable for representing this process. However, a potentially useful non-local approximation to displacement transport, depending on the global distribution of the mean velocity gradient, may be developed as a natural consequence of its definition. A second important transport mechanism involves fluid particles experiencing systematic acceleration and decelerations. Close to the wall this results in a reduction in Reynolds stress due to the slowing of sweep type motions. Further away Reynolds stress is produced in spiraling motions for which particles accelerate or decelerate while changing direction. Both transport mechanisms appear to be closely associated with the dynamics of vortical structures in the wall region.

An analysis of particle trajectories in computer-simulated turbulent channel flow

The origin of Reynolds stress in turbulent channel flow is analyzed using several ensembles of particle paths computed in a direct numerical simulation. The time interval over which the paths are calculated is shown by several criteria to be sufficiently long so that complete mixing of the particle momenta with the surroundings has occurred. The corresponding mixing length is determined and found to be within the range required by a gradient transport model. However, a small fraction of the particles, which tend to be associated with highly vortical sweep and ejection events, travel well beyond the mixing length and collectively make a major nongradient contribution to Reynolds stress. It is suggested that further analysis of these motions may lead to useful formulas for predicting the nongradient component of momentum transport.

Grid-Free Simulation of the Spatially Growing Turbulent Mixing Layer

The spatially developing, unforced, turbulent mixing layer is simulated via a grid-free vortex method. Vortex filaments composed of straight tubes are used as the computational element with new vortex tubes produced as the filaments stretch. A loop removal algorithm serves as a de facto subgrid model limiting growth in the number of elements to practical levels. The computations are high resolution and well resolve the mixing layer from its unforced inception as a laminar flow through transition to a self-similar turbulent state. Mean velocity statistics including growth rate and Reynolds stresses agree well with experimental values. The vortical composition of the transition region is found to develop in one or another of the modes that have been documented in previous experiments and computations: roller/rib vortices, the chain-link fence structure in a diamond shaped pattern, and somewhat oblique roller/rib configuration with partial pairing. Evidently, small perturbative effects that are intrinsic to the numerical scheme influence which transitional mode appears locally in the simulations. The computations offer a clear view of the downstream dissolution of the identifiable structure into turbulence in the late transition and the salient aspects of the process are noted.

On the physical accuracy of scalar transport modeling in inhomogeneous turbulence

Direct numerical simulations of scalar fields produced by uniform and line sources in channel flow are used as the basis for examining the accuracy of random flight and closure models in predicting turbulent scalar transport rates. Closure models of gradient form with an anisotropic eddy diffusivity tensor perform well for the uniform source flow and the far field of plumes. In the near field, the plumes are seriously distorted due to the inappropriateness of gradient transport in modeling the streamwise flux rate. Random flight models are most successful in producing a qualitative rendering of the near field of plumes, but are subject to significant quantitative inaccuracies for the low Reynolds and Schmidt number flows considered here. Ensembles of particle paths having common end points are used to explore the physics of the scalar transport correlation. For plume flows, transport in the near field is found to be primarily due to the average effect of the meandering of the turbulent fluid over the sour...

Vortex filament simulation of the turbulent coflowing jet

A high resolution grid-free vortex filament scheme is applied to the prediction and simulation of coflowing round jets with a view to acquiring a new perspective on their physics and establishing the validity of the numerical technique. Vortex loop removal at inertial range scales provides a nondiffusive model of local dissipation that remains compatible with the presence of backscatter. Consistency with the Kolmogorov −5/3 inertial range spectrum is used to estimate the effective dissipation rate and subsequently the local and global Reynolds numbers associated with the turbulentvortex field. New insights into the accuracy of the scheme are presented including dependencies on numerical parameters. Comparisons of the computed statistics and structural features of the coflowing jet versus experiments show the method to provide an accurate rendering of the flow. Some consideration is also given to the dispersion of a scalar contaminant in the jet and its comparison to data.

Turbulent flow properties of large-scale vortex systems.

Large-scale computations of dynamically interacting vortex tubes forming filaments are performed with a view toward investigating their relationship to turbulent fluid flow. It is shown that the statistical properties of the tubes are consistent with commonly accepted observations about turbulence such as the Kolmogorov inertial range spectrum and lognormality of the vorticity distribution. A loop-removal algorithm is demonstrated to reduce the nominally exponential growth rate in the number of tubes to linear growth without apparent harm to the underlying physics. In this form, a vortex tube method may become a practical means for simulating high Reynolds number turbulent flows.

The hairpin vortex illusion

It has long been customary to assume that vortical structures in turbulent flows are synonymous with regions of rotational motion. Mathematical implementations of this idea using numerical and experimental velocity data from turbulent boundary layers reveal the presence of hairpin vortices, both singly and in groups called packets. However, vorticity may be present that does not directly cause rotation, and by failing to take this into account it is possible to be misled as to the true nature of the vortical structures. In this work a vortex filament scheme is applied to boundary layer flow that allows for a view of structures unrestrained by the requirement that they be regions of rotational motion. It is found that furrow-like streamwise aligned eruptions of the nominally spanwise near-wall vorticity overlying low-speed streaks are the primary structural feature of the transitioning boundary layer. These progress from an arch-like form at their upstream end to either one or two-lobed mushroom-shapes at their downstream end. The rotational motion associated with the furrows has the appearance of hairpins. Mushroom-shaped structures rapidly breakdown into complex forms in the post-transitional region that may have rotational signatures similar to that of packets.