Inertial-convective sub-range

Note that as Re/, goes to infinity with Sc constant, both the turbulent energy spectrum and the scalar energy spectrum will be dominated by the energy-containing and inertial/inertial-convective sub-ranges. Thus, in this limit, the characteristic time scale for scalar variance dissipation defined by (3.55) becomes... 

In a fully developed turbulent flow,22 the scalar spectral transfer rate in the inertial-convective sub-range is equal to the scalar dissipation rate, i.e., T k) = for /cei < < Kn. Likewise, when Sc 1, so that a viscous-convective sub-range exists, the scalar trans-... 

In addition to the Reynolds number, local isotropy for the scalar field will depend on the Schmidt number Sc must be large enough to allow for a inertial-convective sub-range in the scalar energy spectmm. 

Figure 4.8. Sketch of wavenumber bands in the spectral relaxation (SR) model. The scalar-dissipation wavenumber kd lies one decade below the Batchelor-scale wavenumber kb. All scalar dissipation is assumed to occur in wavenumber band [/cd, oo). Wavenumber band [0, k ) denotes the energy-containing scales. The inertial-convective sub-range falls in wavenumber bands [k, k3 ), while wavenumber bands [/c3, /cD) contain the viscous-convective sub-range.

Reynolds numbers, the inertial-convective sub-range contains two stages. The wavenumber bands are defined by the cut-off wavenumbers 26... 

Note that the right-hand sides of these expressions can be extracted from DNS data for homogeneous turbulence in order to explore the dependence of the rate constants on Rei and Sc. Results from a preliminary investigation (Fox and Yeung 1999) for Rx = 90 have revealed that the backscatter rate constant from the dissipative range has a Schmidt-number dependence like/Son Sc1/2 for Schmidt numbers in the range [1/8, 1], On the other hand, for cut-off wavenumbers in the inertial-convective sub-range, one would expect a 1) and... 

Note that at spectral equilibrium the integral in (A.33) will be constant and proportional to ea (i.e., the scalar spectral energy transfer rate in the inertial-convective sub-range will be constant). The forward rate constants a j will thus depend on the chosen cut-off wavenumbers through their effect on (computationally efficient spectral model possible, the total number of wavenumber bands is minimized subject to the condition that... 

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