  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.
