A.W. Leonard

Vortex methods for flow simulation

Recent progress in the development of vortex methods and their applications to the numerical simulation of incompressible fluid flows are reviewed. Emphasis is on recent results concerning the accuracy of these methods, improvements in computational efficiency, and the development of three-dimensional methods. Simulations of several example flows which display some of the strengths and weaknesses of vortex methods are presented.

An analytical study of transport, mixing and chaos in an unsteady vortical flow

We examine the transport properties of a particular two-dimensional, inviscid incompressible flow using dynamical systems techniques. The velocity field is time periodic and consists of the field induced by a vortex pair plus an oscillating strainrate field. In the absence of the strain-rate field the vortex pair moves with a constant velocity and carries with it a constant body of fluid. When the strain-rate field is added the picture changes dramatically; fluid is entrained and detrained from the neighbourhood of the vortices and chaotic particle motion occurs. We investigate the mechanism for this phenomenon and study the transport and mixing of fluid in this flow. Our work consists of both numerical and analytical studies. The analytical studies include the interpretation of the invariant manifolds as the underlying structure which govern the transport. For small values of strain-rate amplitude we use Melnikovs technique to investigate the behaviour of the manifolds as the parameters of the problem change and to prove the existence of a horseshoe map and thus the existence of chaotic particle paths in the flow. Using the Melnikov technique once more we develop an analytical estimate of the flux rate into and out of the vortex neighbourhood. We then develop a technique for determining the residence time distribution for fluid particles near the vortices that is valid for arbitrary strainrate amplitudes. The technique involves an understanding of the geometry of the tangling of the stable and unstable manifolds and results in a dramatic reduction in computational effort required for the determination of the residence time distributions. Additionally, we investigate the total stretch of material elements while they are in the vicinity of the vortex pair, using this quantity as a measure of the effect of the horseshoes on trajectories passing through this region. The numerical work verifies the analytical predictions regarding the structure of the invariant manifolds, the mechanism for entrainment and detrainment and the flux rate.

High-resolution simulations of the flow around an impulsively started cylinder using vortex methods

The development of a two-dimensional viscous incompressible flow generated from a circular cylinder impulsively started into rectilinear motion is studied computationally. An adaptative numerical scheme, based on vortex methods, is used to integrate the vorticity/velocity formulation of the Navier–Stokes equations for a wide range of Reynolds numbers (Re = 40 to 9500). A novel technique is implemented to resolve diffusion effects and enforce the no-slip boundary condition. The Biot–Savart law is employed to compute the velocities, thus eliminating the need for imposing the far-field boundary conditions. An efficient fast summation algorithm was implemented that allows a large number of computational elements, thus producing unprecedented high-resolution simulations. Results are compared to those from other theoretical, experimental and computational works and the relation between the unsteady vorticity field and the forces experienced by the body is discussed.

A spectral numerical method for the Navier-Stokes equations with applications to Taylor-Couette flow

A new spectral method for solving the incompressible Navier-Stokes equations in a plane channel and between concentric cylinders is presented. The method uses spectral expansions which inherently satisfy the boundary conditions and the continuity equation and yield banded matrices which are efficiently solved at each time step. In addition, the number of dependent variables is reduced, resulting in a reduction in computer memory requirements. Several test problems have been computed for the channel flow and for flow between concentric cylinders, including Taylor-Couette flow with axisymmetric Taylor vortices and wavy vortices. In all cases, agreement with available experimental and theoretical results is very good.

Evolution of a curved vortex filament into a vortex ring

The deformation of a hairpin-shaped vortex filament under self-induction and in the presence of shear is studied numerically using the Biot–Savart law. It is shown that the tip region of an elongated hairpin vortex evolves into a vortex ring and that the presence of mean shear impedes the process. In addition, evolution of a finite-thickness vortex sheet under self-induction is investigated using the Navier–Stokes equations. The layer evolves into a hairpin vortex, which in turn produces a vortex ring of high Reynolds stress content. These results indicate a mechanism for the generation of ring vortices in turbulent shear flows, and a link between the experimental and numerical observation of hairpin vortices and the observation of Falco [Phys. Fluids 20, S124 (1977)] of ring vortices in the outer regions of turbulent boundary layers.

Invariant manifold templates for chaotic advection

Abstract Invariant manifolds can serve as a geometric template for the study of chaotic advection. In particular, global stable and unstable manifolds of invariant hyperbolic or normally hyperbolic sets form the boundaries of chaotic tangles in physical space, and these manifolds criss-cross one another to form an intricate network of lobes in the tangles. Manifold invariance implies that these lobes of fluid evolve from one to another in a well-defined manner, and it is in the context of lobe dynamics in the tangles that one exploits the invariant manifold template as a skeletal backbone for the study of transport, stretching, and mixing of fluid under chaotic advection, all of which are intimately connected. More concretely, in the study of fluid transport, one can use segments of the invariant manifolds to partition the irregular flow into regions of qualitatively different fluid motion, such as open flow versus closed flow, or different rolls, and then perform a lobe-dynamic study of fluid transport into, and out of, these regions (e.g. entrainment and detrainment) in a geometrically exact context, specified in terms of images and pre-images of a set of turnstile lobes. Mixing can be thought of as a consequence of barrier destruction, and transport across partially destroyed barriers can be studied in a lobe-dynamic context, providing a basic measure of mixing. A practical example of a transport calculation is the use of Melnikov theory to obtain analytical expressions for lobe areas, and hence a measure of flux in the mixing region, which allows for efficient searches through parameter space in order to enhance or diminish a mixing process. The stretching of fluid elements can be studied in the context of the lobes repeatedly stretching, folding, and wrapping around one another in an approximately self-similar manner. Such a picture can be viewed as a generalization of the horseshoe map paradigm for stretching in 2D chaotic tangles, which restricts its interest to a Cantor set near hyperbolic fixed points. A symbolic description of the evolution of lobe boundaries provides a framework for studying the global topology of, and mechanisms for, enhanced stretching in chaotic tangles. In particular, it is found that, though stretching under chaotic flows can be viewed in terms of products of weakly correlated events, there is a range of stretch, spatial, and temporal scales associated with these products, leading to a range of stretch processes. Invariant manifold geometry plays a central role in these stretch processes. For example, the invariant hyperbolic or normally hyperbolic sets have special relevance as engines of good stretching, the turnstile lobes play a fundamental role of re-orienting line elements between each successive pass by the hyperbolic sets, and intersections of stable and unstable manifolds act as partitions between segments of lobe boundaries that experience qualitatively different stretch histories. A notable consequence of the range of scales in the stretch processes is the range of statistics, and hence multifractal properties, associated with the high-stretch tails of finite-time Lyapunov exponent distributions, which has significant impact on interfacial evolution and striation width in the small-scale limit. Regions in physical space of vanishingly small initial measure associated with these tails are thus shown to be able to play a significant role under the flow. The mixing of passive scalars or vectors as a result of such stochastic effects as molecular diffusion or chemical reactions tends to wash out the structure associated with invariant manifolds. However, one can study this interplay between advection and mixing in the context of lobe evolution. In particular, mixing efficiency is seen to depend not only on the stretch experienced by lobe boundaries, but also on the thickness of the lobes, and their separation from other lobes. Though essentially all chaotic advection studies have been in the context of 2D time-periodic velocity fields, we show how an invariant manifold template can apply to quite general circumstances, such as quasiperiodic and indeed aperiodic time dependences, and 3D velocity fields. In all cases, the invariant manifold templates are seen to have physical reality, and the unstable manifolds act as a dominant structure in the regions of irregular flow.

Direct Numerical Simulation of Equilibrium Turbulent Boundary Layers

This paper describes the simulation of turbulent boundary layers by direct numerical solution of the three-dimensional time-dependent Navier-Stokes equations, using a spectral method. The flow is incompressible, and Re δ * = 1000 for most cases. The equations are written in the similarity coordinate system and normalized by the local friction velocity. Periodic streamwise and spanwise boundary conditions are then imposed. A family of nine “equilibrium” boundary layers, from the strongly accelerated “sink” flow to Stratford’s separating flow, is treated. Good general agreement with experiments is observed. The effects of the pressure gradient and of the Reynolds number are discussed.

Investigation of a drag reduction on a circular cylinder in rotary oscillation

Drag reduction in two-dimensional flow over a circular cylinder, achieved using rotary oscillation, was investigated with computational simulations. In the experiments of Tokumaru & Dimotakis (1991), this mechanism was observed to yield up to 80% drag reduction at Re = 15 000 for certain ranges of frequency and amplitude of sinusoidal rotary oscillation. Simulations with a high-resolution viscous vortex method were carried out over a range of Reynolds numbers (150–15 000) to explore the effects of oscillatory rotational forcing. Significant drag reduction was observed for a rotational forcing which had been very effective in the experiments. The impact of the forcing is strongly Reynolds number dependent. The cylinder oscillation appears to trigger a distinctive shedding pattern which is related to the Reynolds number dependence of the drag reduction. It appears that the source of this unusual shedding pattern and associated drag reduction is vortex dynamics in the boundary layer initiated by the oscillatory cylinder rotation. The practical efficiency of the drag reduction procedure is also discussed.

Chaotic transport in the homoclinic and heteroclinic tangle regions of quasiperiodically forced two-dimensional dynamical systems

The authors generalize notions of transport in phase space associated with the classical Poincare map reduction of a periodically forced two-dimensional system to apply to a sequence of nonautonomous maps derived from a quasiperiodically forced two-dimensional system. They obtain a global picture of the dynamics in homoclinic and heteroclinic tangles using a sequence of time-dependent two-dimensional lobe structures derived from the invariant global stable and unstable manifolds of one or more normally hyperbolic invariant sets in a Poincare section of an associated autonomous system phase space. The invariant manifold geometry is studied via a generalized Melnikov function. Transport in phase space is specified in terms of two-dimensional lobes mapping from one to another within the sequence of lobe structures, which provides the framework for studying several features of the dynamics associated with chaotic tangles.

Lévy stable distributions for velocity and velocity difference in systems of vortex elements

The probability density functions (PDFs) of the velocity and the velocity difference field induced by a distribution of a large number of discrete vortex elements are investigated numerically and analytically. Tails of PDFs of the velocity and velocity difference induced by a single vortex element are found. Treating velocities induced by different vortex elements as independent random variables, PDFs of the velocity and velocity difference induced by all vortex elements are found using limit distribution theorems for stable distributions. Our results generalize and extend the analysis by Takayasu [Prog. Theor. Phys. 72, 471 (1984)]. In particular, we are able to treat general distributions of vorticity, and obtain results for velocity differences and velocity derivatives of arbitrary order. The PDF for velocity differences of a system of singular vortex elements is shown to be Cauchy in the case of small separation r, both in 2 and 3 dimensions. A similar type of analysis is also applied to non-singular vortex blobs. We perform numerical simulations of the system of vortex elements in two dimensions, and find that the results compare favorably with the theory based on the independence assumption. These results are related to the experimental and numerical measurements of velocity and velocity difference statistics in the literature. In particular, the appearance of the Cauchy distribution for the velocity difference can be used to explain the experimental observations of Tong and Goldburg [Phys. Lett. A 127, 147 (1988); Phys. Rev. A 37, 2125, (1988); Phys. Fluids 31, 2841 (1988)] for turbulent flows. In addition, for intermediate values of the separation distance, near exponential tails are found.