Time scales scalar

In the discussion above, we have considered only the velocity field in a turbulent flow. What about the length and time scales for turbulent mixing of a scalar field The general answer to this question is discussed in detail in Fox (2003). Here, we will only consider the simplest case where the scalar field 4> is inert and initially nonpremixed with a scalar integral length scale that is approximately equal to Lu. If we denote the molecular diffusivity of the scalar by T, we can use the kinematic viscosity to define a dimensionless number in the following way ... 

The degree of local mixing in a RANS simulation is measured by the scalar variance (complete mixing (i.e., (j> — (j>) is uniform at the SGS) up to (4>max — (4>))((4>) — 4>min) where () is the mean concentration and max and r/>min are the maximum and minimum values, respectively. The rate of local mixing is controlled by the scalar dissipation rate (Fox, 2003). The scalar time scale analogous to the turbulence integral time scale is (Fox, 2003) as follows ... 

To determine how the scalar time scale defined in Eq. (15) is related to the turbulence integral time scale given in Table I, we can introduce a normalized model scalar energy spectrum (Fox, 2003) as follows ... 

In Fig. 2, the normalized model scalar energy spectrum is plotted for a fixed Reynolds number (ReL = 104) and a range of Schmidt numbers. In Fig. 3, it is shown for Sc = 1000 and a range of Reynolds numbers. The reader interested in the meaning of the different slopes observed in the scalar spectrum can consult Fox (2003). By definition, the ratio of the time scales is equal to the area under the normalized scalar energy spectrum as follows ... 

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.

In a fully developed turbulent flow, the rate at which the size of a scalar eddy of length l,P decreases depends on its size relative to the turbulence integral scale L and the Kolmogorov scale ij. For scalar eddies in the inertial sub-range (ij < Ip, < Lu), the scalar mixing rate can be approximated by the inverse of the spectral transfer time scale defined in (2.68), p. 42 8... 

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... 

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. 

For a passive scalar, the turbulent flow will be unaffected by the presence of the scalar. This implies that for wavenumbers above the scalar dissipation range, the characteristic time scale for scalar spectral transport should be equal to that for velocity spectral transport tst defined by (2.67), p. 42. Thus, by equating the scalar and velocity spectral transport time scales, we have23 t)... 

Krt < k scalar spectral transport time scale defined in terms of the velocity spectrum (e.g., rst). 

Equation (3.82) illustrates the importance of the scalar spectral energy transfer rate in determining the scalar dissipation rate in high-Reynolds-number turbulent flows. Indeed, near spectral equilibrium, 7 (/cd, 0 (like Tu(kDi, 0) will vary on time scales of the order of the eddy turnover time re, while the characteristic time scale of (3.82) is xn <[Pg.99]

The scalar spectral energy transfer rate XA m)- t) will vary on time scales proportional to the eddy turnover time re. At high Reynolds numbers, (3.131) and (3.132) quickly attain a quasi-steady state wherein... 

Note that in (3.163) the characteristic time scale for scalar-covariance dissipation is... 

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

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... 

As described in Fox (1995), the wavenumber bands are chosen to be as large as possible, subject to die condition that the characteristic time scales decrease as the band numbers increase. This condition is needed to ensure that scalar energy does not pile up at intermediate wavenumber bands. The rate-controlling step in equilibrium spectral decay is then die scalar spectral energy transfer rate (T ) from die lowest wavenumber band. 

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. ... 

As discussed in Section 3.4, differential-diffusion effects will decrease with increasing Reynolds number. A single molecular mixing time scale T[Pg.168]

Due to the conservation of elements, the rank of J will lie less than or equal to K — E 1 In general, rank(J) = Ny < K - E, which implies that V = K — T eigenvalues of J are null. Moreover, since M is a similarity transformation, (5.51) implies that the eigenvalues of J and those of J are identical. We can thus limit the definition of the chemical time scales to include only the Nr finite ra found from (5.50). The other N components of the transformed composition vector correspond to conserved scalars for which no chemical-source-term closure is required. The same comments would apply if the Nr non-zero singular values of J were used to define the chemical time scales. 

The one-point PDF contains no length-scale information so that scalar dissipation must be modeled. In particular, the mixing time scale must be related to turbulence time scales through a model for the scalar dissipation rate. 

Note that the dimensions of the fast and slow manifolds will depend upon the time step. In the limit where At is much larger than all chemical time scales, the slow manifold will be zero-dimensional. Note also that the fast and slow manifolds are defined locally in composition space. Hence, depending on the location of 0q], the dimensions of the slow manifold can vary greatly. In contrast to the ILDM method, wherein the dimension of the slow manifold must be globally constant (and less than two or three ), ISAT is applicable to slow manifolds of any dimension. Naturally this flexibility comes with a cost ISAT does not reduce the number (Ns) of scalars that are needed to describe a reacting flow.168... 

The final term in (A.41) (Daa) is modeled by the product of the inverse of a characteristic time scale for the scalar dissipation range and eau- The characteristic time scale is taken to be proportional to ,2)i)/ u so that the final term has the form Cd 2D/(2)D. The proportionality constant Cd is thus defined by... 

Beguier, C., I. Dekeyser, and B. E. Launder (1978). Ratio of scalar and velocity dissipation time scales in shear flow turbulence. The Physics of Fluids 21, 307-310. 

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