Kolmogorov hypotheses

11.3 Kolmogorov hypotheses

The basis of the Kolmogorov theory are three hypotheses, which are supposed

p to be valid for high Reynolds numbers Re t D , where v D TKE D k

vL

is the characteristic fluctuation velocity. The Kolmogorov hypothesis of local isotropy reads:

At sufficiently high Reynolds number Re t , the small-scale turbulent motions EI ) are statistically isotropic

Here l EI is the lengthscale as the demarcation between the anisotropic large eddies and the isotropic small eddies. Kolmogorov argued that all informa- tion about the geometry of the large eddies - determined by the mean flow field and boundary conditions - is also lost. Directional information at small scales is lost. With the other words, the direction of the vorticity vector ! of small turbulent vortices is uniformly distributed over the sphere. As a consequence, the statistics of the small-scale motions are in a sense universal - similar in every high-Reynolds-number turbulent flow (see [14]).

The Kolmogorov first similarity hypothesis reads:

In every turbulent flow at sufficiently high Reynolds number Re t , the statistics of the small-scale motions (l < l EI ) have a universal form that is uniquely

(11.5) The analysis of dimension allows one to derive the following dependences:

u is the characteristic velocity of turning of Kolmogorov vortices, is characteristic turn over time of Kolmogorov vortices. Using expressions

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Very remarkable is the first formula defining the ratio between the smallest and largest vortices in the flow. If L is, say one meter, and the fluctua- tion 1m=s, the turbulent Reynolds number in water is Re 6

D 10 . The Kolmogorov scale is in this case 32000 as less as the flow macroscale L. Es-

timations of the Kolmogorov scale in the jet mixer with nozzle diameter of

d D 1cm and closing pipe of D D 5cm diameter is shown in Fig. 11.4. The Kolmogorov second similarity hypothesis reads:

In every turbulent flow at sufficiently high Reynolds number, the statistics of the motions of scale uniquely determined by

This range is called as the inertial subrange. Since the vortices of this range are much larger than Kolmogorov vortices, we can assume that their Reynolds numbers lu l

are large and their motion is little affected by the viscosity. The energy density depends on the wave number k and the dissipation rate "

˛ E.k/ D " ˇ k

Figure 1 Figure 11.4: Distribution of the Kolmogrov scale along the centerline of the jet mixer 1.4: Distribution of the Kolmogorov scale along the centerline of and free jet [47]. The dissipation rate the jet mixer and free jet. The dissipation rate " is calculated from the ε is calculated from the k – ε model and the

experimental estimation of the Miller and Dimotakis (1991) k " model and the experimental estimatin of Miller and Dimotakis (1991) 3 ε = 48(U d /d )((x – x 0 )/d ) –4 .

" D 48.U 3 d =d /..x x /=d / 4 0 .

The analysis of dimension leads to the Kolmogorov law

E.k/ D ˛" 5=3 k

11 .4 T hree different scale ranges of turbulent flow

Three different ranges can be distinguished in the spectrum of scales in the full developed turbulence at high Reynolds numbers Re t (Fig.11.5):

Energy containing range at l > l 1 EI (according to Pope [14], l EI 6 L ). Within this range the kinetic energy of turbulence is generated and big turbulent eddies are created.

Inertial subrange at l DI <l<l EI (according to Pope [14], l DI ). Within this subrange the energy is transferred along the scales towards

density obeys the Kolmogorov law (11.10). Dissipation range l < l DI . The dissipation of the energy of big vortices

occurs within the dissipation range. The inertial subrange and dissipation range belong to the universal equilib-

rium range. Three corresponding ranges can be distinguished in the distribu- tion of the energy density over the wave numbers k Fig. 11.6. The presence of the inertial and dissipation subranges was confirmed in numerous exper- imental measurements performed after development of the K41 theory (see Fig. 11.7, 11.8).

Figure 11.5: Three typical scale ranges in the turbulent flow at high Reynolds numbers.

Figure 11.6: Three typical ranges of the energy density spectrum in the turbulent flow at high Reynolds number. 1- energy containing range, 2- inertial subrange, 3- dissipation range.

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