Decaying scalars

In the absence of mean scalar gradients, the scalar covariances and joint dissipation rates will decay towards zero. For this case, it is convenient to work with the governing equations for g p and p p directly. These expressions can be derived from (3.179) and (3.180)  

In addition, we will need the governing equations for the scalar time scales  

Note that the first term in the series on the right-hand side of this expression is independent of Schmidt number. 

Using (3.189) in (3.185) and (3.186), and keeping only the terms of leading order in Rei, yields 

at high Reynolds numbers, the correlation functions both decay at the same rate, which is proportional to the turbulence integral time scale ru = k/e. 

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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). 

We will next consider the case of decaying scalars where V p = 0. For this case, it is convenient to assume that the scalars are initially perfectly correlated so that pap(0) = gap(0) = pd(0) = 1. The multi-variate SR model can then be used to describe how the scalars de-correlate with time. We will again consider a case with constant Rc>, = 90 and Schmidt number pair Sc = (1, 1/8). 

Typical model predictions without and with backscatter are shown in Figs. 4.16 and 4.17, respectively. It can be noted that for decaying scalars the effect of backscatter on de-correlation is dramatic. For the case without backscatter (Fig. 4.16), after a short transient period the correlation coefficients all approach steady-state values. In contrast, when backscatter is included (Fig. 4.17), the correlation coefficients slowly approach zero. The rate of long-time de-correlation in the multi-variate SR model is thus proportional to the backscatter constant Cb-... 

Figure 4.16. Predictions of the multi-variate SR model for Re = 90 and Sc = (1, 1/8) for decaying scalars with no backscatter (cb = 0). For these initial conditions, the scalars are perfectly correlated paj9(0) = gap(0) = 1. The correlation coefficient for the dissipation range, pD, is included for comparison with pap. 

The distribution of a decaying scalar field advected by a turbulent flow was studied by Corrsin (1961) who generalized the Obukhov-Corrsin theory of passive scalar turbulence for the linear decay problem F(C) = S(x) — bC. As in the case of the passive non-decaying scalar field, depending on the length scales considered, one can identify inertial-convective and viscous-convective regimes with qualitatively different characteristics. 

Nam et al. (2000) have shown that the theoretical framework introduced in the previous sections for the description of passively ad-vected decaying scalar field also applies to the description of the small scale structure of the vorticity field in a two-dimensional turbulent flow with linear damping. The vorticity dynamics in this case is described by the Navier-Stokes equation... 

Figure 6.9 Comparison of power spectra of decaying scalar (x) and vorticity (+) from numerical simulation of a two-dimensional turbulent flow with Ekman friction. Inset shows the ratio Z(k)/Eg(k), which is roughly constant for large k. (From Boffetta et al. (2002))...

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