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Propagation of Flames in Dust Clouds 327

large scale, intermediate scale, and small scale. Large-scale turbulence is closely
linked to the geometry of the structure in which the flow exists. It is characterized by
strong coherence and high degree of organization of the turbulence structures, reflect-
ing the geometry of the structure. For plane flow, the coherent large-scale structures
are essentially two-dimensional vortices with their axes parallel to the boundary
walls. For flow in axisymmetric systems, concentric large-scale vortex rings are
formed. The theoretical description of the three-dimensional, large-scale vortex struc-
tures encountered in practice presents a real challenge. Also, experimental investi-
gation of such structures is very difficult. According to Beer et al., the lack of research
in this area is the most serious obstacle to further advances in turbulent combustion
theory.
On all scale levels, turbulence has to be considered a collection of long-lasting vortex
structures, tangled and folded in the fluid. This picture is quite different from the ideal-
ized hypothetical stochastic fluctuation model of isotropic turbulence. Beer et al. argue
againsit the common idea that the small-scale structures are randomly distributed “little
whirls.” According to these authors, it is known that the fine-scale structures of high
Reynolds number turbulence become less and less space filling as the scale size decreases
and the Reynolds number increases.
According to Hinze (1975), Kolmogoroff postulated that, if the Reynolds number is
infinitely large, the energy spectrum of the small-scale turbulence is independent of the
viscosity and dependent on only the rate of dissipation of kinetic energy into heat, per
mass unit of fluid, E. For this range, Kolmogoroff arrived at his well-known energy
spectrum law for high Reynolds numbers:

213 -513
E(a, t)=AE a (4.82)
E(a, t) is called the three-dimensional energy spectrum function of turbulence; a is the
wave number 2nn/ v,where n is the frequency of the turbulent fluctuation of the veloc-
ity, and V is the mean global flow velocity; A is a constant; and E is the rate of dissipa-
tion of turbulent kinetic energy into heat per unit mass of fluid.
Figure 4.37 illustrates the entire three-dimensional energy spectrum of turbulence, from
the largest, primary eddies via those containing most of the kinetic energy to the low-
energy range of very high wave numbers (or very high frequencies). Figure 4.37 includes
the Kolmogoroff law for the universal equilibrium range.
In the range of low Reynolds numbers, other theoretical descriptions than
Kolrnogoroff’s law are required. In principle, the kinetic energy of turbulence is iden-
tical to the integral of the energy spectrum curve E(a, t) in Figure 4.37 over all wave
numbers.
A formally exact equation for E may be derived from the Navier-Stokes equations.
However, the unknown statistical turbulence correlations must be approximated by
known or calculable quantities. Fully comprehensive calculation requires extensive com-
putational capacity, and it is not yet a realistic approach for solving practical problems.
Therefore, simpler and more approximate approaches are needed. One widely used
approximate theoiy, assuming isotropic turbulence, is the k-Emodel by Jones and Launder
(1972, 1973), where k is the kinetic energy of turbulence, and E the rate of dissipation
of the Wunetic energy of turbulence into heat. The k-E model contains Equation (4.82) as