The inertial-convective range

The inertial-convective range of three-dimensional turbulence covers the range of length scales where inertial forces dominate over viscous forces in the dynamics of the velocity field and advection is the dominant transport process with respect to diffusion. This is valid below the integral scale and limited at small scales by the larger of the Kolmogorov scale (rf) or the diffusive scale (Id)- For 

Decay-type and Stable Reaction Dynamics in Flows 

The concentration fluctuations produced by the source at large scales, which is assumed to have variations on lengthscales comparable to the integral scale of the flow, are distorted by advection generating smaller scales in the concentration field. Therefore the characteristic wavenumber associated to a wavepacket representing concentration fluctuations in a narrow spectral range increases in time as 

This is the same as for a non-reactive passive scalar since the linear decay affects all scales equally, so it does not change the shape of the concentration iso-contours but only reduces the contrast between them. 

The source term can be included as a constant flux boundary condition at the lowest wavenumber, ks = 2n/Ls The terms on the right represent the decay of fluctuations due to (i) the reaction acting uniformly on all scales, and (ii) due to diffusion, that becomes stronger at large wavenumbers. For large Pe the diffusive decay is relatively weak in the inertial convective range and can be neglected for simplicity. 

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Thus analogously to the inertial scale of turbulence, the statistical properties of the scalar fluctuations in the inertial-convective range, i.e. in a range of scales below the forcing scale where both diffusion and viscosity are negligible, can only depend on the dissipation rate ee, the energy dissipation rate e, and on the length scale. Thus the only dimensionally correct form of the second order scalar structure function of the concentration fluctuations is... 

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-... 

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... 

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. 

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 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. 

In the previous sections we considered flows with a smooth spatial structure in which the relative dispersion of fluid trajectories is exponential in time and can be characterized by a single timescale, the inverse of the Lyapunov exponent. This is also valid for two-dimensional turbulent flows that have a smooth velocity field in the small-scale enstrophy cascade range (Bennett, 1984). A similar behavior occurs in any dimension at scales below the Kolmogorov scale (the so-called Batchelor or viscous-convective range, see below). In the inertial range of fully developed three-dimensional turbulence, however, the velocity field has a broad range of timescales and they all contribute to the relative dispersion of particle trajectories and affect the transport properties of the flow. 

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