Inertial range

The validity of Eqs. (3-5) are bond on the condition of fully developed turbulent flow which only exists if the macro turbulence is not influenced by the viscosity. This is the case if the macro turbulence is clearly separated from the dissipation range by the inertial range. This is given if the macro scale A is large in comparison to Kolmogorov s micro scale qp Liepe [1] and Mockel [24] found out by measurement of turbulence spectra s the following condition ... 

The research on the flow regimes in packed tubes suggests that laminar flow CFD simulations should be reasonable for Re <100 approximately, and turbulent simulations for Re >600, also approximately. Just as RANS models provide steady solutions that are regarded as time averages of the real time-dependent turbulent flow, it may be suggested that CFD simulations in the unsteady laminar inertial range 100 time-averaged picture of the flow field. As with wall functions, comparisons with experimental data and an improved assessment of what information is really needed from the simulations will inform us as to how to proceed in these areas. 

Furthermore, the universal equilibrium range is composed of the inertial range and the dissipation range. As its name indicates, at high Reynolds numbers the universal equilibrium range should have approximately the same form in all turbulent flows. 

From this definition, it can be observed that T,(k. t) is the net rate at which turbulent kinetic energy is transferred from wavenumbers less than k to wavenumbers greater than k. In fully developed turbulent flow, the net flux of turbulent kinetic energy is from large to small scales. Thus, the stationary spectral energy transfer rate Tu(k) will be positive at spectral equilibrium. Moreover, by definition of the inertial range, the net rate of transfer through wavenumbers /cei and kdi will be identical in a fully developed turbulent flow, and thus... 

In the inertial range of fully developed turbulence, the spectral transfer time scale becomes... 

As discussed in Section 2.1, in high-Reynolds-number turbulent flows the scalar dissipation rate is equal to the rate of energy transfer through the inertial range of the turbulence energy spectrum. The usual modeling approach is thus to use a transport equation for the transfer rate instead of the detailed balance equation for the dissipation rate derived from (1.27). Nevertheless, in order to understand better the small-scale physical phenomena that determine e, we will derive its transport equation starting from (2.99). 

For example, the vortex-stretching term is a triple-correlation term that corresponds to the rate at which dissipation is created by spectral energy passing from the inertial range to the dissipative range of the energy spectrum (see (2.75), p. 43). Letting /cdi 0.1 denote... 

The right-hand side of (2.142) scales as Re 2 /re. In the usual situation where the spectral flux is controlled by the rate at which energy enters the inertial range from the large scales, 7bl(0 will vary on time scales of order re. Thus, at large Reynolds numbers, 7di(0 will be in a quasi-steady state with respect to e(f). 

In the discussion that follows, we will assume that 1 < Sc so that a high Reynolds number suffices to imply the existence of an inertial range for the turbulence and a convective range for the scalar. 

The diffusive exponent is 3 /4 in the inertial range and 1 /2 in the viscous range. The values of cD and the exponent in (3.67) have been chosen to reproduce roughly the spectral bump observed in compensated spectra for 0.1 < Sc < 1 (Yeung et al. 2002). 

For Sc < 1, a similar expression can be derived by taking /cd = kci- However, because /cci < /cdi lies in the inertial range, the characteristic spectral transport time for wavenumbers greater than /cci cannot be taken as constant. 

Strictly speaking, this scaling is valid for 1 < Sc. For smaller Schmidt numbers, an inertial-range time scale can be used. 

A detailed description of LES filtering is beyond the scope of this book (see, for example, Meneveau and Katz (2000) or Pope (2000)). However, the basic idea can be understood by considering a so-called sharp-spectral filter in wavenumber space. For this filter, a cut-off frequency kc in the inertial range of the turbulent energy spectrum is chosen (see Fig. 4.1), and a low-pass filter is applied to the Navier-Stokes equation to separate the... 

These relations originally appeared in Fox (1999) with a different definition for b and for three inertial-range stages. By comparing with the earlier definitions, the reader will note a pattern that can be used to find yn for any arbitrary number of stages. 

For example, a well defined inertial range exists only for R, > 240. Thus the proportionality constant in (5.251) should depend on the Reynolds number in most laboratory-scale experiments. 

As discussed in Section 4.3, the linear-eddy model solves a one-dimensional reaction-diffusion equation for all length scales. Inertial-range fluid-particle interactions are accounted for by a random rearrangement process. This leads to significant computational inefficiency since step (3) is not the rate-controlling step. Simplifications have thus been introduced to avoid this problem (Baldyga and Bourne 1989). 

The mixing parameter Q must be chosen to yield the correct mixture-fraction-variance dissipation rate. However, inertial-range scaling arguments suggest that its value should be near unity.165... 

Some workers reserve the term LES for the case where the filter scale A corresponds to a wavenumber in the k A inertial range, and the term VLES when the residual field begins before the inertial range [128]. The reason for... 

Unlike diffusion, which is a stochastic process, particle motion in the inertial range is deterministic, except for the very important case of turbulent transport. The calculation of inertial deposition rates Is usually based either on a force balance on a particle or on a direct analysis of the equations of fluid motion in the case of colli Jing spheres. Few simple, exact solutions of the fundamental equations are available, and it is usually necessary to resort to dimensional analysis and/or numerical compulations. For a detailed review of earlier experimental and theoretical studies of the behavior of particles in the inertial range, the reader is referred to Fuchs (1964). 

Fuchs, N. A. (1964) Mechanics of Aero.sots, Pergamon, New York. This reference contains a thorough review of the literature on the behavior of aerosols in the inertial range through the 1950s. It lisLs 886 references. Both theory and experiment arc covered. 

Different behaviour at different spatio-temporal scales, ranging from small-scale concentration fluctuations in the turbulence inertial range to large-scale air movements in synoptic events affecting the entire troposphere. 

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