Hostname: page-component-848d4c4894-pjpqr Total loading time: 0 Render date: 2024-06-22T23:59:13.775Z Has data issue: false hasContentIssue false
  • English
  • Français

Eulerian and Lagrangian scaling properties of randomly advected vortex tubes

Published online by Cambridge University Press:  26 April 2006

N. A. Malik  and
J. C. Vassilicos
N. A. Malik
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
J. C. Vassilicos
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the Eulerian and Lagrangian spectral scaling properties of vortex tubes, and the consistency of these properties with Tennekes’ (1975) statistical advection analysis and universal equilibrium arguments. We consider three different vortex tubes with power-law wavenumber spectra: a Burgers vortex tube, an inviscid Lundgren single spiral vortex sheet, and a vortex tube solution of the Euler equation. While the Burgers vortex is a steady solution of the Navier–Stokes equation, the other two are unsteady solutions of, respectively, the Navier–Stokes and the Euler equations. In our numerical experiments we study the vortex tubes by subjecting each of them to external ‘large-scale’ sinusoidal advection of characteristic frequency f and length scale ρ.

Not only do we find that the Eulerian frequency spectrum ϕE(ω) can be derived from the wavenumber spectrum E(k) using the simple Tennekes advection relation ω ∼ k for all finite advection frequencies f when the vortex is steady, but also when the vortex is unsteady, and in the Lundgren case even when f = 0 owing to the self-advection of the Lundgren vortex by its own differential rotation.

An analytical calculation using the method of stationary phases for f = 0 shows that for large enough Reynolds numbers the combination of strain with differential rotation implies that ϕL(ω) ∼ ω−2+Const for large values of ω. We verify numerically that ϕL(ω) does not change when f ≠ 0. With the Burgers vortex tube we are in a position to investigate the spectral broadening of the Eulerian frequency spectrum with respect to the Lagrangian frequency spectrum. A spectral broadening does exist but is different from the spectral broadening predicted by Tennekes (1975).

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 326 , 10 November 1996 , pp. 417 - 436
Copyright
© 1996 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Gilbert, A. D. 1988 Spiral structures and spectra in two-dimensional turbulence. J. Fluid Mech. 193, 475.Google Scholar
Hunt, J. C. R. & Vassilicos, J. C. 1991 Kolmogorov's contributions to the physical and geometrical understanding of small-scale turbulence and recent developments. Proc. R. Soc. Lond. A 434, 183.Google Scholar
Inoue, E. 1951 On turbulent diffusion in the atmosphere. J. Met. Soc. Japan 29, 246.Google Scholar
Jiménez, J., Wray, A. A., Saffman, P. G. & Rogallo, R. S. 1993 The structure of intense vorticity in homogeneous isotropic turbulence. J. Fluid Mech. 255, 65.Google Scholar
Lundgren, T. S. 1982 Strained spiral vortex model for turbulent fine structure. Phys. Fluids 25, 2193.Google Scholar
Lundgren, T. S. 1993 A small-scale turbulence model. Phys. Fluids A 5, 1472.Google Scholar
Moffatt, H. K. 1984 Simple topological aspects of turbulent vorticity dynamics. In Turbulence and Chaotic Phenomena in Fluids (ed. T. Tatsumi). Elsevier.
Moffatt, H. K. 1993 Spiral structures in turbulent flow. In Fractals, Wavelets and Fourier Transforms (ed. M. Farge, J. C. R. Hunt & J. C. Vassilicos). Oxford University Press.
Moffatt, H. K., Kida, S. & Okhitani, K. 1994 Stretched vortices — the sinews of turbulence; large Reynolds-number asymptotics. J. Fluid Mech. 259, 241.Google Scholar
Pullin, D. & Saffman, P. G. 1993 On the Lundgren-Townsend model of turbulent fine scales. Phys. Fluids A 5, 126.Google Scholar
Tennekes, H. 1975 Eulerian and Lagrangian time microscales in isotropic turbulence. J. Fluid Mech. 67, 561.Google Scholar
Townsend, A. A. 1951 On the fine scale structure of turbulence. Proc. R. Soc. Lond. A 208, 534.Google Scholar
Vassilicos, J. C. & Brasseur, J. G. 1996 Self-similar flow structure in low Reynolds number isotropic and decaying turbulence. Phys. Rev. E 54, No 1, 467.Google Scholar
Vassilicos, J. C. & Hunt, J. C. R. 1991 Fractal dimensions and spectra of interfaces with application to turbulence. Proc. R. Soc. Lond. A 435, 505.Google Scholar
Villermaux, E., Sixou, B. & Gagne, Y. 1995 Intense vortical structures in grid-generated turbulence. Phys. Fluids 7, 2008.Google Scholar
Yeung, P. K., Brasseur, J. G. & Wang, Q. 1995 Dynamics of direct large-small scale couplings in coherently forced turbulence: concurrent physical- and Fourier-space views. J. Fluid Mech. 283, 43.CrossRefGoogle Scholar