Mechanical-to-scalar time-scale ratio

In the literature on turbulent mixing, the mechanical-to-scalar time-scale ratio is defined by... 

Fig. 4. Mechanical-to-scalar time-scale ratio found from the normalized model scalar energy spectra.

Having defined the model scalar energy spectrum, it can now be used to compute the scalar mixing time as a function of Sc and Rk. In the turbulent mixing literature, the scalar mixing time is usually reported in a dimensionless form referred to as the mechanical-to-scalar time-scale ratio R defined by... 

Figure 3.14. The mechanical-to-scalar time-scale ratio for a fully developed scalar energy spectrum as afunction of the Schmidt number at various Reynolds numbers R> = 50,100, 200,400, and 800. The arrow indicates the direction of increasing Reynolds number.

In Fig. 3.14, the mechanical-to-scalar time-scale ratio computed from the model scalar energy spectrum is plotted as a function of the Schmidt number at various Reynolds numbers. Consistent with (3.15), p. 61, for 1 Sc the mechanical-to-scalar time-scale ratio decreases with increasing Schmidt number as ln(Sc). Likewise, the scalar integral scale can be computed from the model spectrum. The ratio L Lu is plotted in Fig. 3.15, where it can be seen that it approaches unity at high Reynolds numbers. 

Chasnov (1994) has carried out detailed studies of inert-scalar mixing at moderate Reynolds numbers using direct numerical simulations. He found that for decaying scalar fields the scalar spectrum at low wavenumbers is dependent on the initial scalar spectrum, and that this sensitivity is reflected in the mechanical-to-scalar time-scale ratio. Likewise, R is found to depend on both the Reynolds number and the Schmidt number in a non-trivial manner for decaying velocity and/or scalar fields (Chasnov 1991 Chasnov 1998). 

The model scalar energy spectrum was derived for the limiting case of a fully developed scalar spectrum. As mentioned at the end of Section 3.1, in many applications the scalar energy spectrum cannot be assumed to be in spectral equilibrium. This implies that the mechanical-to-scalar time-scale ratio will depend on how the scalar spectrum was initialized, i.e., on E (k. 0). In order to compute R for non-equilibrium scalar mixing, we can make use of models based on the scalar spectral transport equation described below. 

Despite the explicit dependence on Reynolds number, in its present form the model does not describe low-Reynolds-number effects on the steady-state mechanical-to-scalar time-scale ratio (R defined by (3.72), p. 76). In order to include such effects, they would need to be incorporated in the scalar spectral energy transfer rates. In the original model, the spectral energy transfer rates were chosen such that R(t) —> AV, = 2 for Sc = 1 and V

model parameter. DNS data for 90 < R-,. suggest that Re, is nearly constant. However, for lower... 

In the definition of b, Ro is the equilibrium mechanical-to-scalar time-scale ratio found with Sc = 1 and = 0.34 The parameters Cd, Cb, and Cd appear in the SR model for the scalar dissipation rate discussed below. Note that, by definition, xi + Yi + K3 + Kd = 1. ... 

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