Mogens V. Melander

Cross‐linking of two antiparallel vortex tubes

The detailed mechanisms in vortex cross‐linking are unveiled by adequately resolved, direct numerical simulation of two viscous vortex tubes. There are three characteristic phases: (i) inviscid induction followed by core flattening and stretching; (ii) bridging of the two vortices by accumulation of annihilated and then cross‐linked vortex lines; and (iii) threading of the remnants of the initial vortex pair in between the two bridges as they pull apart. These phases and the role of threading—along with bridging—in the mixing and the enstrophy cascade are explained, and it is shown that the mechanism is insensitive to asymmetries.

Polarized vorticity dynamics on a vortex column

It is shown that vortex core dynamics results from the interaction of two slowly deforming, but overlapping, helical vortex structures. These are the left‐ and right‐handed components of the vortex, and are obtained by a generalized Helmholtz decomposition (the complex helical wave decomposition) of the vorticity field. This decomposition is based on a Fourier expansion in eigenfunctions of the curl operator, which has only real eigenvalues λ. The sum of eigenmodes with λ≳0 (λ<0) constitutes the right (left) polarized component, and the vector lines of the field (e.g., vortex lines) are locally right (left) handed helixes. It is found that for a localized vortex the polarized structures are also localized, a crucial result for physical space applications. The polarized vortex structures deform slowly (compared to unpolarized structures) and behave almost like solitary waves when isolated. It is shown that this is because the nonlinearity in the Navier–Stokes equations is largely suppressed between eigenmodes of the same polarity. Moreover, the helicity of polarized structures is very high. The interaction between overlapping polarized structures however gives each structure a different propagation velocity and also results in some additional deformation. The latter is shown to occur mainly in two places: at the front of the structure where a low enstrophy bubble forms (which is a permanent feature in each of the polarized packets), and at the back where a tail develops. Otherwise, the deformation occurs on a much slower time scale compared to that for unpolarized vortices. Thus the rapid changes in the total vorticity field result from the superposition of two slowly deforming structures moving in opposite directions, as is the case for the one‐dimensional (1‐D) wave equation. The decomposition can also be applied to turbulent flows. In fact, it offers new insight into the structure of turbulent shear flows. The organization of small‐scale vortical threads forming in the neighborhood of a coherent structure and their polarization serve as a prime example.

Core dynamics on a vortex column

Through a detailed study of vortex core dynamics we show that variations in core size play a crucial role in the dynamics in 3D-vortex flows and that they in general cannot be ignored as has previously been assumed. To arrive at this conclusion we examine the core dynamics of an isolated axisymmetric vortex column with a nonuniform core. In this simple and idealized configuration, the effects of the core dynamics are seen most clearly because they are decoupled from other effects, such as interactions with other vortices and selfinduced displacement of the axis. We show analytically that the core dynamics results from a distinct physical mechanism, namely differential rotation along axisymmetric vortex surfaces. Furthermore, we show that the core dynamics is neither pure wavemotion nor pure mass transport, but a combination of both. We show and explain why core dynamics is highly Re-sensitive. Moreover, we find that the core variations—which are likely in practical flow situations—do not disappear by inviscid effects. By examining viscous dissipation we find that core dynamics results in a significantly higher dissipation rate and that dissipation is the only effect that reduces core size variations. In fact, we find that the frequency of the core size oscillations increases with Re, but to a finite limit as Re → ∞. A striking, newly observed feature resulting from core dynamics is the appearance and disappearance of low enstrophy pockets inside the vortex.

Understanding Turbulence Via Vortex Dynamics

Turbulence research in the past half century has seen notable progress, even bursts of hectic activity, on three seemingly independent fronts: Statistical approaches starting in the forties (E.G. Kolmogorov 1941), structural approaches starting in the sixties (E.G. Kline et al 1967; Crow & Champagne 1971; Brown & Roshko 1974) and dynamical approaches starting in the eighties (E.G. Berge et al 1984). While researchers typically tend to show their partiality for one approach or another depending on their familiarity or ignorance, even sometimes displaying their disapproval or disdain for approaches outside their own preferred one, there should be little doubt that all the three approaches are helpful, perhaps even essential, to understand turbulence. There are few whose repertoire of expertise covers statistical, structural and dynamical turbulence. John Lumley is one such exception, having had made profound contributions in all three areas (E.G. Lumley 1970; 1980; Aubry etal 1988)

Transients in the decay of isotropic turbulence

Using a shell model, we examine how initial conditions affect the decay of isotropic 3D turbulence when no confining spatial boundaries exist, Re λ is high and the spectrum has a single peak initially. We base our investigation on large ensembles of realizations with the same initial energy spectrum. The initial shell phases, however, consist of a coherent part, common to all realizations within an ensemble, and a random part. The effects of the initial conditions are transient as a decay law ⟨ E ⟩ = C0(t − t0)α0 with a common exponent α0 emerges from all initial conditions. In addition, the same energy spectrum emerges, and the normalized helicity disappears from all ensembles whatever their initial helicity. Thus, we argue that asymptotically a single state of decay exists. By considering the compensated decay, we find the transient is itself a power law to leading order. What is more, our data suggest a universal transient exponent. Our analysis focuses on the decay signature F (⟨ E ⟩), which we define...

Self-similar enstrophy divergence in a shell model of isotropic turbulence

We focus on the early evolution of energy E and enstrophy Z when the dissipation grows in significance from negligible to important. By considering a sequence of viscous shell model solutions we find that both energy and dissipation are continuous functions of time in the inviscid limit. Inviscidly, Z takes only a finite time t* to diverge, where t* depends on initial conditions

The Complementary Roles of Experiments and Simulation in Coherent Structure Studies

The past two decades’ vigorous studies of coherent structures (CS) have failed to produce a consensus on what CS are, let alone a CS-based turbulence theory or even an objective, mathematical definition of CS. What started out as a promise for a mechanistic explanation for fluid turbulence—as researchers found or reinvented CS in their ‘search for order in disorder’ and presumed to have discovered a deterministic, tractable route to turbulence phenomena—has unfolded itself as a Pandora’s box. Successive studies of CS continue to raise more questions than they answer. Thus, even though we understand more about turbulence via CS concepts, we have become painfully even more aware of how complex turbulence is. CS are not the panacea they were initially presumed to be, nor are they as simple as we all had hoped. Despite some progress through CS research, the secrets of turbulence remain ever impenetrable.

New studies in vortex dynamics: Incompressible and compressible vortex reconnection, core dynamics, and coupling between large and small scales

Coherent structure dynamics in turbulent flows are explored by direct numerical simulations of the Navier-Stokes equations for idealized vortex configurations. For this purpose, two dynamically significant coherent structure interactions are examined: (i) incompressible and compressible vortex reconnection and (ii) core dynamics (with and without superimposed small-scale turbulence). Reconnection is studied for two antiparallel vortex tubes at a Reynolds number (Re) of 103. Incompressible reconnection consists of three distinct phases: inviscid advection, bridging and threading. The key mechanism, bridging, involves the ‘cutting’ of vortex lines by viscous cross diffusion and their subsequent reconnection in front of the advancing vortex dipole. We conjecture that reconnection occurs in successive bursts and is a physical mechanism of cascade to smaller scales.Compressible reconnection is seen to be significantly affected by the choice of pressure and density initial conditions. We propose a polytropic initial condition which is consistent with experimental results and low-Mach number asymptotic theories. We also explain how compressibility initiates an early reconnection due to shocklet formation, but slows down the circulation transfer at late times. Thus, the reconnection timescale increases with increasing Mach number.Motivated by the important role of helical vortex lines in the reconnected vortices (bridges), we focus our attention on the dynamics of an axisymmetric vortex column with axial variation of core size. The resulting core dynamics is first explained via coupling between swirl and meridional flows. We then show that core dynamics can be better understood by applying a powerful analytical tool —helical wave decomposition — which extracts vorticity wave packets, thereby providing a simple explanation of the dynamics. The increase in core size variation with increasing Re in such a vortex demonstrates the limitation of the prevalent vortex filament models which assume constant core size. By studying the columnar vortex with superimposed small-scale, homogeneous, isotropic turbulence, we address the mutual interactions between large and small scales in turbulent flows. At its boundary the columnar vortex organizes the small scales, which, if Re is sufficiently high, induce bending waves on the vortex which further organize the small scales. Such backscatter from small scales cannot be modelled by an eddy viscosity. Based on the observation of such close coupling between large and small scales, we question the local isotropy assumption and conjecture a fractal vortex model for high Re turbulent flows.

Intermittency via Self-Similarity in New Variables

We consider the classical subject of inertial range scaling in high Reynolds number isotropic turbulence driven into statistical equilibrium by steady large scale forcing. Using DNS data from a periodic box calculation, we look at the problem two ways. First, we use longitudinal velocity increments. Their statistics show intermittency, i.e., the pdf varies from Gaussian at the largest scale to highly non-Gaussian at the smallest scale. However, when we employ a new set of variables to look at the same data, we find self-similarity. The new variables are constructed in wave number space and the self-similarity expresses a particular symmetry. We discuss how the traditional variables (velocity increments) hide this symmetry.

Model Coherent Structure Dynamics: Vortex Reconnection, Core Dynamics and Interaction with Turbulence

Direct numerical simulations of the Navier-Stokes equations for three idealized vortex interactions are analyzed to obtain insights into the dynamics of coherent structures in turbulent flows. We first study the vortex reconnection mechanism which presumably is important in cascade and mixing in turbulent flows. The dynamical significance of axial flow in coherent structures inferred from this interaction is explored further by studying the evolution of an axisymmetric laminar vortex tube with nonuniform core. Evolution of vorticity wavepackets in such a vortex tube is first explained as a coupling between swirl and meridional flow. We show how this evolution is better understood by the motion of polarized vorticity components, obtained via complex helical wave decomposition of the vorticity and velocity fields. Finally, by treating such a vortex tube as a prototypical segment of a coherent structure we study its interaction with background, fine-scale, isotropic turbulence. This study shows polarization and organization of small scales by the coherent structure, and subsequent feedback on the structure from the small scales, thus providing the mechanism for a direct coupling between large and fine scales in turbulent flows.