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Universal constants and equations of turbulent motion

Profile image of Helmut Z. BaumertHelmut Z. Baumert

2011

Abstract

In the spirit of Prandtl's conjecture of 1926, for turbulence at high Reynolds number we present an analogy with the kinetic theory of gases, with dipoles made of quasi-rigid and 'dressed' vortex tubes as frictionless, incompressible but deformable quasi-particles. Their movements are governed by Helmholtz' elementary vortex rules applied locally. A contact interaction or 'collision' leads either to random scatter of a trajectory or to the formation of two likewise rotating, fundamentally unstable whirls forming a dissipative patch slowly rotating around its center of mass which is almost at rest. This approach predicts von Karman's constant as 1/sqrt(2 pi) = 0.399 and the spatio-temporal dynamics of energy-containing time and length scales controlling turbulent mixing [Baumert 2009]. A link to turbulence spectra was missing so far. In the present paper it is shown that the above image of random vortex-dipole movements is compatible with Kolmogorov's turbulence spectra if dissipative patches, beginning as two likewise rotating eddies, evolve locally into a space-filling bearing in the sense of Herrmann [1990], i.e. into an "Apollonian gear". Its parts and pieces are incompressible and flexibly deformable vortex tubes which are frictionless, excepting the dissipative scale of size zero. For steady and locally homogeneous conditions our approach predicts the dimensionless pre-factor in the 3D Eulerian wavenumber spectrum as [(4 pi)^2/3]/3 = 1.8, and in the Lagrangian frequency spectrum as 2. Our derivations rest on geometry, methods from many-particle physics, and on elementary conservation laws.

Figures (7)
Baumert: Universal constants and equations Figure 1. Overturning and overturning scales in the sense of |Imberger and Boashash| 1986} Fig. 6 there]. The cartoon depicts a 180° overturning of a 20 m thick layer that brings relatively heavy water up and water down. Note the characteristic Z-pattern vertical displacement ¢. The original stable densi ighter of the ty pro- file is the blue line, the unstable density in the overturn is red and dashed. By courtesy of Dr. Hartmut Earth and Space Research, Seattle, USA. Peters,
Baumert: Universal constants and equations Figure 1. Overturning and overturning scales in the sense of |Imberger and Boashash| 1986} Fig. 6 there]. The cartoon depicts a 180° overturning of a 20 m thick layer that brings relatively heavy water up and water down. Note the characteristic Z-pattern vertical displacement ¢. The original stable densi ighter of the ty pro- file is the blue line, the unstable density in the overturn is red and dashed. By courtesy of Dr. Hartmut Earth and Space Research, Seattle, USA. Peters,
Baumert: Universal constants and equations Figure 2. A big overturn in the Pacific Equatorial Undercurrent at 0°,140°W during the Tropic Heat II cruise [Peters et _al.| /1995, adapted from ]}: (a) poten- tial temperature (0, red) and potential density (ce, black), Thorpe-sorted ae (blue), and Thorpe scale Lrp, (shaded); (b) turbulent vertical displacement ¢. By courtesy of Dr. Hartmut Peters.
Baumert: Universal constants and equations Figure 2. A big overturn in the Pacific Equatorial Undercurrent at 0°,140°W during the Tropic Heat II cruise [Peters et _al.| /1995, adapted from ]}: (a) poten- tial temperature (0, red) and potential density (ce, black), Thorpe-sorted ae (blue), and Thorpe scale Lrp, (shaded); (b) turbulent vertical displacement ¢. By courtesy of Dr. Hartmut Peters.
Baumert: Universal constants and equations Figure 3. Turbulent velocity g.=(el)!/? derived from measured € via Taylor scaling, (4), versus measured tur- bulent velocity fluctuation q. Adapted from|Peters et al! where !=1.6 Ly; and where q is the spectral ve- locity variance at vertical wavenumbers > 1/1. Shown are data for individual overturns at least 1 m thick from depths 60-350 m in the Equatorial Undercurrent at 0°, 140°W. The gray line indicates the median ratio of qe over g. By courtesy of Dr. Hartmut Peters.
Baumert: Universal constants and equations Figure 3. Turbulent velocity g.=(el)!/? derived from measured € via Taylor scaling, (4), versus measured tur- bulent velocity fluctuation q. Adapted from|Peters et al! where !=1.6 Ly; and where q is the spectral ve- locity variance at vertical wavenumbers > 1/1. Shown are data for individual overturns at least 1 m thick from depths 60-350 m in the Equatorial Undercurrent at 0°, 140°W. The gray line indicates the median ratio of qe over g. By courtesy of Dr. Hartmut Peters.
Baumert: Universal constants and equations Figure 4. Two possible collision results of two dipoles: the left branch is “diffusive” because it leads to a recom- bination of the dipole elements and scatter of trajecto- ries which is known as turbulent diffusion. The right branch is “dissipative” because it leads to an unsta- ble vortex configuration which decays “somehow” into heat!4] For symmetry reasons both branches have iden- tical probabilities of 0.5. Note that the circular form of the vortex cross sections presented here are chosen for reasons of clarity. They hold as an statistical average picture only. Real vortices have elliptic or even strongly deformatted cross sections.
Baumert: Universal constants and equations Figure 4. Two possible collision results of two dipoles: the left branch is “diffusive” because it leads to a recom- bination of the dipole elements and scatter of trajecto- ries which is known as turbulent diffusion. The right branch is “dissipative” because it leads to an unsta- ble vortex configuration which decays “somehow” into heat!4] For symmetry reasons both branches have iden- tical probabilities of 0.5. Note that the circular form of the vortex cross sections presented here are chosen for reasons of clarity. They hold as an statistical average picture only. Real vortices have elliptic or even strongly deformatted cross sections.
Baumert: Universal constants and equations Figure 5. Local cross section through the first de- velopmental stage of an unstable pair of likewise ro- tating vortices resulting from a dipole-dipole collision right branch in Fig. |4). The green circles represent the primary energy-containing vortices with identical radii r. Note that the green circles do not touch each other! They are separated by the red circles who sym- bolize secondary vortices in the beginning phase of a whole vortex cascade. The dipole cloud surrounding the above structure not only generates the necessary pressure head to keep individual vortices stable but it also acts as a source of perturbations initiating roll-up instabilities and thus tertiary and higher-order vortices and eventually a fully developed dissipative patch, see text. The broken blue line and the arrows symbolize the slow rotation of the whole patch around the com- mon center of mass.
Baumert: Universal constants and equations Figure 5. Local cross section through the first de- velopmental stage of an unstable pair of likewise ro- tating vortices resulting from a dipole-dipole collision right branch in Fig. |4). The green circles represent the primary energy-containing vortices with identical radii r. Note that the green circles do not touch each other! They are separated by the red circles who sym- bolize secondary vortices in the beginning phase of a whole vortex cascade. The dipole cloud surrounding the above structure not only generates the necessary pressure head to keep individual vortices stable but it also acts as a source of perturbations initiating roll-up instabilities and thus tertiary and higher-order vortices and eventually a fully developed dissipative patch, see text. The broken blue line and the arrows symbolize the slow rotation of the whole patch around the com- mon center of mass.
We finally notice that the transformation of the un- stable configuration of two likewise rotating vortices deserves time to set the greater masses of the smaller scales into motion. This inertia effect might play a role in highly dynamic scenarios. Figure 6. Outer limits of a dissipative patch (c.f. Fig. 5). The maximum wavelength is obviously equal to Xo =2r.
We finally notice that the transformation of the un- stable configuration of two likewise rotating vortices deserves time to set the greater masses of the smaller scales into motion. This inertia effect might play a role in highly dynamic scenarios. Figure 6. Outer limits of a dissipative patch (c.f. Fig. 5). The maximum wavelength is obviously equal to Xo =2r.
Later works have been analysed in an important study by |Sreenivasan}|1995) who possibly gave the most comprehensive literature review of experimental and ob- servational values for the number a;/3 until now. Later |Yeung and Zhou| [1997] reported a value of a, = 1.62 based on high-resolution DNS studies with up to 512° erid points. Figure 7. Experimental and observational results for a1/3 measured, collected from the literature, and analysed by |Sreenivasan [1995 . The solid green line follows our somewhat arbitrary approximation 0.6 x V Rey /(VRex + VRe)) wherein a,/3 = 0.6 is the the- oretically derived asymptotic value. Here we took Re, = 10. A similar presentation has been chosen by Sreenivasan] (1995) for a1/3 in his Fig. 3, and by [Yeung et al.| [2006] for 7, in their Fig. 2.
Later works have been analysed in an important study by |Sreenivasan}|1995) who possibly gave the most comprehensive literature review of experimental and ob- servational values for the number a;/3 until now. Later |Yeung and Zhou| [1997] reported a value of a, = 1.62 based on high-resolution DNS studies with up to 512° erid points. Figure 7. Experimental and observational results for a1/3 measured, collected from the literature, and analysed by |Sreenivasan [1995 . The solid green line follows our somewhat arbitrary approximation 0.6 x V Rey /(VRex + VRe)) wherein a,/3 = 0.6 is the the- oretically derived asymptotic value. Here we took Re, = 10. A similar presentation has been chosen by Sreenivasan] (1995) for a1/3 in his Fig. 3, and by [Yeung et al.| [2006] for 7, in their Fig. 2.

Key takeaways

  • Thus we have the challenging theory situation that a mean flow exhibits turbulent friction and circulation while the turbulent vortices responsible for fluctuations and fricition have no circulation and are essentially frictionless, excepting the singular dissipation scale of size zero, see further below.
  • These assumptions describe our image of vortices in a turbulent vortex tangle without violating the conservation laws of kinetic energy, momentum, angular momentum and circulation.
  • In stable stratification turbulent vortices are directly recognizable in vertical profiles of density, ρ, and other scalar variables, such as temperature and salinity, as "overturning" of density profiles.
  • Geophysical fluid dynamics is a domain of pioneering turbulence stud-6 The idea of a rigid vortex is compatible with fluid mechanics because Helmholtz' laws also apply here.
  • The turbulence properties (iii') and (v') of section 4 above establish an analogy of turbulent dipole movements with the Brownian motion of particles suspended in a fluid at rest [in the sense of Einstein, 1905].

Related papers

Universal equations and constants of turbulent motion
Helmut Z. Baumert

Physica Scripta, 2013

This paper presents a parameter-free theory of shear-generated turbulence at asymptotically high Reynolds numbers in incompressible fluids. It is based on a two-fluids concept. Both components are materially identical and inviscid. The first component is an ensemble of quasi-rigid dipole-vortex tubes (vortex filaments, excitations) as quasiparticles in chaotic motion. The second is a superfluid performing evasive motions between the tubes. The local dipole motions follow Helmholtz' law. The vortex radii scale with the energy-containing length scale. Collisions between quasiparticles lead either to annihilation (likewise rotation, turbulent dissipation) or to scattering (counterrotation, turbulent diffusion). There are analogies with birth and death processes of population dynamics and their master equations and with Landau's two-flluid theory of liequid Helium. For free homogeneous decay the theory predicts the TKE to follow t −1 . With an adiabatic wall condition it predicts the logarithmic law with von Kármán's constant as 1/ √ 2 π = 0.399. Likewise rotating couples form dissipative patches almost at rest (→ intermittency) wherein under local quasi-steady conditions the spectrum evolves into an "Apollonian gear" as discussed first by . Dissipation happens exclusively at scale zero and at finite scales this system is frictionless and reminds of Prigogine's (1947) law of minimum (here: zero) entropy production. The theory predicts further the prefactor of the 3D-wavenumber spectrum (a Kolmogorov constant) as 1 3 (4 π) 2/3 = 1.802, well within the scatter range of observational, experimental and DNS results.

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Universal constants and
Helmut Z. Baumert

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In the spirit of Prandtl's [1926] conjecture, for turbulence at Re → ∞ we present an analogy with the kinetic theory of gases, with dipoles made of quasi-rigid and "dressed" vortex tubes as frictionless, incompressible but deformable quasi-particles. Their movements are governed by Helmholtz' elementary vortex rules applied locally. A contact interaction or "collision" leads either to random scatter of a trajectory or to the formation of two likewise rotating, fundamentally unstable whirls forming a dissipative patch slowly rotating around its center of mass which is almost at rest. This approach predicts von Karman's constant as κ = 1/ √ 2π 0.399 and the spatio-temporal dynamics of energy-containing time and length scales controlling turbulent mixing [Baumert, 2009] in agreement with observations. A link to turbulence spectra was missing so far. In the present paper it is shown that the above image of random vortex-dipole movements is compatible with Kolmogorov's turbulence spectra if dissipative patches, beginning as two likewise rotating eddies, evolve locally into a space-filling bearing in the sense of Herrmann [1990], i.e. into an "Apollonian gear" consisting of incompressible and flexibly deformable vortex tubes which are frictionless, excepting the dissipative scale of size zero. For steady and locally homogeneous conditions our approach predicts the pre-factor in the three-dimensional Eulerian wavenumber spectrum, E(k) = α 1 ε 2/3 k −5/3 , as α 1 = 1 3 (4 π) 2/3 1.802, and in the Lagrangian frequency spectrum, E(ω) = β 1 ε ω −2 , as β 1 = 2. The unique values for α 1 , β 1 and κ are situated well within the broad scatter range of observational, experimental and approximative results. Our derivations rest on geometry, methods from many-particle physics, and on elementary conservation laws.

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