Inertial subrange

Note that the Kolmogorov power spectrum is unphysical at low frequencies— the variance is infinite at k = 0. In fact the turbulence is only homogeneous within a finite range—the inertial subrange. The modified von Karman spectral model includes effects of finite inner and outer scales. 

Tennekes, H. (1979). The exponential Lagrangian correlation function and turbulent diffusion in the inertial subrange. Atmos. Environ. 13, 1565-1567. 

According to Kolmogorov s (1941) inertial subrange theory of turbulence, the exponent in Eq. 1 is m = 3. Inserting into Eq. 4 ... 

Most correlations show that di2 is proportional to the Weber number raised to the power of -0.6, which is consistent with the theory of drop breakup by turbulent shear forces. Strictly, these correlations should be applied only where the drop size is in the inertial subrange of turbulence, i.e.,... 

However, checks should be made that flow is fully turbulent at both scales and that the drop size remains within the inertial subrange of turbulence [Eq. (46)]. As a minimum, residence time should be maintained, i.e.,... 

Kolmogoroff (KIO) developed the following expression for the energy spectrum function E k) applicable to the inertial subrange ... 

The kinetic energy responsible for breakup comes from eddies smaller than the drop diameter, since larger eddies would presumably only carry the drop along with them without breaking it. The kinetic turbulent energy is proportional to (7r/6)a p , where for drops of size within the inertial subrange of an isotropic turbulent flow ... 

Furthermore, the assumption is made that the motion of the centers of mass of daughter droplets to be formed (binary breakage) is similar to the relative motion of two lumps of fluid in a turbulent flow field as described by Batchelor (B6). Thus, for the inertial subrange eddies... 

Experiments (G6, V4) show that the probability density distribution in the inertial subrange can be written as Gaussian with variance u and cutoff velocity Uc- Thus,... 

The h rpotheses of Kolmogorov allow a number of additional deductions to be formulated on the statistical characteristics of the small-scale components of turbulence. The most important of them is the two-third-law deduced by Kolmogorov [84]. This law states that the mean square of the difference between the velocities at two points of a turbulent flow, being a distance x apart, equals C(ex) / when x lies in the inertial subrange. (7 is a universal model constant. Another form of this assertion (apparently first put forward by Obukhov [116] [117]) is the five-third law. This law states that the spectral density of the kinetic energy of turbulence over the spectrum of wave numbers, k, has the form Cke / k / in the inertial subrange. Cj, is a new model constant (see e.g., [8], sect. 6.4). 

Closure of the mean scalar field equation requires a model for the scalar flux term. This term represents the scalar transport due to velocity fluctuations in the inertial subrange of the energy spectrum and is normally independent of the molecular diffusivity. The gradient diffusion model is often successfully employed (e.g., [15, 78, 2]) ... 

To estimate the average turbulent fluid particle velocity, the second-order structure function (9.14) is interpreted as an absolute particle velocity squared and defined for two points in the fluid separated by a distance equal to the bubble diameter d. The structure function is then given as 5v [d) = uz z + d) — Vz(z)]. If the magnitude of the diameter d lies within the inertial subrange of turbulence, the structure function can be calculated as 5v d) = C(ed) / . The discrete absolute mean velocity of bubbles with diameter di is thus approximated as ... 

The length of the relative velocity between a pair of unlike bubbles is approximated by a mean velocity representative for bubbles with size corresponding to the inertial subrange of isotropic turbulence (9.f3) ... 

The mean turbulent velocity of eddies with size A in the inertial subrange of isotropic turbulence was assumed equal to the velocity of the neutrally buoyant droplets measured by Kuboi et al [53, 54]. Kuboi et al [53, 54] found that the turbulent velocity of droplets could be expressed by the Maxwell distribution function (9.32), thus the mean eddy velocity was approximated by ... 

The functional form of the energy spectrum in the inertial subrange of turbulence is defined as (e.g., [89]) ... 

The size of these bubbles is assumed to be within the inertial subrange of turbulence, thus the average deformation stress, which results from velocity fluctuations existing in the liquid between two points separated by a distance d, was estimated as ... 

This model further assumes that the size of the parent particles is in the inertial subrange of turbulence. Therefore, it implies that dmin < d < dmax provided that dmin > Ad, where Ad is the Kolmogorov length scale of the underlying turbulence. Otherwise, dmin is taken to be equal to Ad. However, no assumption needs to be made about the minimum and maximum eddy size that can cause particle breakage. All eddies with sizes between the Kolmogorov scale and the integral scale are taken into account. 

Where e is the rate of kinetic energy dissipation per unit mass and Cx is of order 1. Equation (36) is valid for drops whose diameter falls within the inertial subrange of turbulence, < d < L, where L is the integral scale of turbulence and q = is the Kolmogorov microscale of turbulence. 

The method proposed by Kolmogorov (53) and Hinze (54) was extended by Ba+dyga and Podgorska (56) and Ba+dyga and Bourne (57) to the case of more realistic intermittent turbulence, which was described by means of multifractal formalism. In this model drop size in the inertial subrange also depends on the integral scale of turbulence, which is related to the scale of the system. This new formalism predicts... 

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