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2 - Structure and Dynamics of Vorticity in Turbulence
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- By Jörg Schumacher, Ilmenau University of Technology, Robert M. Kerr, University of Warwick, Kiyosi Horiuti, Tokyo Institute of Technology
- Edited by Peter A. Davidson, University of Cambridge, Yukio Kaneda, Katepalli R. Sreenivasan, New York University
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- Book:
- Ten Chapters in Turbulence
- Published online:
- 05 February 2013
- Print publication:
- 06 December 2012, pp 43-86
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Summary
Introduction
Ancient depictions of fluids, going back to the Minoans, envisaged waves and moving streams. They missed what we would call vortices and turbulence. The first artist to depict the rotational properties of fluids, vortical motion and turbulent flows was da Vinci (1506 to 1510). He would recognize the term vortical motion as it comes from the Latin vortere or vertere: to turn, meaning that vorticity is where a gas or liquid is rapidly turning or spiraling. Mathematically, one represents this effect as twists in the velocity derivative, that is the curl or the anti-symmetric component of the velocity gradient tensor. If the velocity field is u, then for the vorticity is ω = ∇ × u.
The aspect of turbulence which this chapter will focus upon is the structure, dynamics and evolution of vorticity in idealized turbulence – either the products of homogeneous, isotropic, statistically stationary states in forced, periodic simulations, or flows using idealized initial conditions designed to let us understand those states. The isotropic state is often viewed as a tangle of vorticity (at least when the amplitudes are large), an example of which is given in Fig. 2.1. This visualization shows isosurfaces of the magnitude of the vorticity, and similar techniques have been discussed before (see e.g. Pullin and Saffman, 1998; Ishihara et al., 2009; Tsinober, 2009). The goal of this chapter is to relate these graphics to basic relations between the vorticity and strain, to how this subject has evolved to using vorticity as a measure of regularity, then focus on the structure and dynamics of vorticity in turbulence, in experiments and numerical investigations, before considering theoretical explanations. Our discussions will focus upon three-dimensional turbulence.
Geometry of enstrophy and dissipation, grid resolution effects and proximity issues in turbulence
- IVÁN BERMEJO-MORENO, D. I. PULLIN, KIYOSI HORIUTI
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- Journal:
- Journal of Fluid Mechanics / Volume 620 / 10 February 2009
- Published online by Cambridge University Press:
- 10 February 2009, pp. 121-166
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We perform a multi-scale non-local geometrical analysis of the structures extracted from the enstrophy and kinetic energy dissipation-rate, instantaneous fields of a numerical database of incompressible homogeneous isotropic turbulence decaying in time obtained by DNS in a periodic box. Three different resolutions are considered: 2563, 5123 and 10243 grid points, with kmax
approximately 1, 2 and 4, respectively, the same initial conditions and Reλ ≈ 77. This allows a comparison of the geometry of the structures obtained for different resolutions. For the highest resolution, structures of enstrophy and dissipation evolve in a continuous distribution from blob-like and moderately stretched tube-like shapes at the large scales to highly stretched sheet-like structures at the small scales. The intermediate scales show a predominance of tube-like structures for both fields, much more pronounced for the enstrophy field. The dissipation field shows a tendency towards structures with lower curvedness than those of the enstrophy, for intermediate and small scales. The 2563 grid resolution case (kmax ≈ 1) was unable to detect the predominance of highly stretched sheet-like structures at the smaller scales in both fields. The same non-local methodology for the study of the geometry of structures, but without the multi-scale decomposition, is applied to two scalar fields used by existing local criteria for the eduction of tube- and sheet-like structures in turbulence, Q and [Aij]+, respectively, obtained from invariants of the velocity-gradient tensor and alike in the 10243 case. This adds the non-local geometrical characterization and classification to those local criteria, assessing their validity in educing particular geometries. Finally, we introduce a new methodology for the study of proximity issues among structures of different fields, based on geometrical considerations and non-local analysis, by taking into account the spatial extent of the structures. We apply it to the four fields previously studied. Tube-like structures of Q are predominantly surrounded by sheet-like structures of [Aij]+, which appear at closer distances. For the enstrophy, tube-like structures at an intermediate scale are primarily surrounded by sheets of smaller scales of the enstrophy and structures of dissipation at the same and smaller scales. A secondary contribution results from tubes of enstrophy at smaller scales appearing at farther distances. Different configurations of composite structures are presented.
The multi-mode stretched spiral vortex in homogeneous isotropic turbulence
- KIYOSI HORIUTI, TAKEHARU FUJISAWA
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- Journal:
- Journal of Fluid Mechanics / Volume 595 / 25 January 2008
- Published online by Cambridge University Press:
- 08 January 2008, pp. 341-366
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The stretched spiral vortex is identified using direct numerical simulation (DNS) data for homogeneous isotropic turbulence and its properties are studied. Its genesis, growth and annihilation are elucidated, and its role in the generation of turbulence is shown. Aside from the two symmetric modes of configurations with regard to the vorticity alignment along two spiral sheets and the vortex tube in the core region studied in previous works, a third asymmetric mode is found. One of the two symmetric modes and the asymmetric mode are created not by a conventional rolling-up of a single vortex sheet but through the interaction among several sheets. The stagnation flow caused by the two sheets converges to form recirculating flow through its interaction with the vortex along the third sheet. This recirculating flow strains and stretches the sheets. The vortex tube is formed by axial straining, lowering of pressure and the intensification of the swirling motion in the recirculating region. As a result of the differential rotation induced by the tube and that self-induced by the sheet, the vortex sheets are entrained by the tube and form spiral turns. The transition between the three modes is examined. The initial configuration is in one of two symmetric modes, but it is transformed into another set of two modes due to the occurrence of reorientation in the vorticity direction along the stretched sheets. The symmetric mode tends to be more persistent than the asymmetric mode, among the two transformed modes. The tightening of the spiral turns of the spiral sheets produces a cascade of velocity fluctuations to smaller scales and generates a strongly intermittent dissipation field. To precisely capture the spiral turns, a grid resolution with at least (kmax is the largest wavenumber, is the averaged Kolmogorov scale) is required. At a higher Reynolds number, self-similar spiral vortices are successively produced by the instability cascade along the stretched vortex sheets. A cluster consisting of spiral vortices with an extensive range of length scales is formed and this cluster induces an energy cascade.
Assessment of two-equation models of turbulent passive-scalar diffusion in channel flow
- Kiyosi Horiuti
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- Journal:
- Journal of Fluid Mechanics / Volume 238 / May 1992
- Published online by Cambridge University Press:
- 26 April 2006, pp. 405-433
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Models for the transport of passive scalar in turbulent flow were investigated using databases derived from numerical solutions of the Navier—Stokes equations for fully developed plane channel flow, these databases being generated using large-eddy and direct numerical simulation techniques. Their reliability has been established by comparison with the experimental measurements of Hishida. Nagano & Tagawa (1986). The present paper compares these simulations and calculations using the Nagano & Kim (1988) ‘two-equation’ model for the scalar variance (kθ) and scalar variance dissipation (εθ). This model accounts for the dependence of flow quantities on the Prandtl number by expressing eddy diffusivity in terms of the ratio of the timescales of velocity and scalar fluctuations. However, the statistical analysis by Yoshizawa (1988) showed that there was an inconsistency in the definition of the isotropic eddy diffusivity in the Nagano—Kim model, the implications of which are clearly demonstrated by the results of this paper where large-eddy simulation and direct numerical simulation (LES/DNS) databases are used to compute the quantities contained in both models. An extension of the Nagano-Kim model is proposed which resolves these inconsistencies, and a further development of this model is given in which the anisotropic scalar fluxes are calculated. Near a rigid surface, a third-order ‘anisotropic representation’ of scalar fluxes may be used as an alternative model for reducing the eddy diffusivity, instead of the conventional ‘damping functions’. This model is similar but distinct from the algebraic scalar flux model of Rogers, Mansour & Reynolds (1989). A third aspect of this paper is the use of the LES/DNS databases to evaluate certain coefficients (those for modelling the pressure-scalar gradient terms) of another model of a similar type, namely the algebraic scalar flux model of Launder (1975).
Roles of non-aligned eigenvectors of strain-rate and subgrid-scale stress tensors in turbulence generation
- KIYOSI HORIUTI
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- Journal:
- Journal of Fluid Mechanics / Volume 491 / 25 September 2003
- Published online by Cambridge University Press:
- 27 August 2003, pp. 65-100
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Alignment of the eigenvectors for strain-rate tensors and subgrid-scale (SGS) stress tensors in large-eddy simulation (LES) is studied in homogeneous isotropic turbulence. Non-alignment of these two eigenvectors was shown in Tao, Katz & Meneveau (2002). In the present study, the specific term in the decomposition of the SGS stress tensor, which is primarily responsible for causing this non-alignment, is identified using the nonlinear model. The bimodal behaviour of the alignment configuration reported in Tao et al. (2002) was eliminated by reordering the eigenvalues according to the degree of alignment of the corresponding eigenvectors with the vorticity vector. The preferred relative orientation of the eigenvectors was ${\approx}\,42^\circ$. The alignment trends were conditionally sampled based on the relative dominance of strain and vorticity. The effect of the identified term on the alignment was the largest in the region in which the magnitudes of strain and vorticity were comparable and large (flat sheet). The most probable alignment configuration in the flat-sheet region was different from those in the strain-dominated and vorticity-dominated regions. The relative orientation of the eigenvectors was dependent on the degree of resolution for the flat sheet region yielded on the LES mesh. When the alignment was conditionally sampled on the events with the backward scatter of the SGS energy into the grid scale, the interchange of the alignment of the eigenvectors took place. Relevance of the identified term for the generation of turbulence is investigated. It is shown that the identified term makes no contribution to the production of the total SGS energy, but contributes significantly to the generation of the SGS enstrophy. The identified term causes a time-lag in the evolution of the turbulent energy and enstrophy. It is shown that generation of vorticity is markedly attenuated when the magnitude of the identified term is modified, and the original nonlinear model yielded the results which are in the closest agreement with the direct numerical simulation data.