Scalar energy spectrum

In a RANS simulation of scalar mixing, a model for i ,/, must be supplied to compute (4>a). In fully developed turbulence, t,p can be related to tu by considering the scalar energy spectrum, as first done by Corrsin (1964). 

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 ... 

Since we have a compact region (all the momenta are now discrete) it is more convenient to employ the energy spectrum formulation to obtain the Casimir energy. The wave function for the massless scalar field in the cavity is... 

As discussed in Section 2.1, in high-Reynolds-number turbulent flows the scalar dissipation rate is equal to the rate of energy transfer through the inertial range of the turbulence energy spectrum. The usual modeling approach is thus to use a transport equation for the transfer rate instead of the detailed balance equation for the dissipation rate derived from (1.27). Nevertheless, in order to understand better the small-scale physical phenomena that determine e, we will derive its transport equation starting from (2.99). 

As seen in Chapter 2 for turbulent flow, the length-scale information needed to describe a homogeneous scalar field is contained in the scalar energy spectrum E k, t), which we will look at in some detail in Section 3.2. However, in order to gain valuable intuition into the essential physics of scalar mixing, we will look first at the relevant length scales of a turbulent scalar field, and we develop a simple phenomenological model valid for fully developed, statistically stationary turbulent flow. Readers interested in the detailed structure of the scalar fields in turbulent flow should have a look at the remarkable experimental data reported in Dahm et al. (1991), Buch and Dahm (1996) and Buch and Dahm (1998). 

Because the integral scale is defined in terms of the energy spectrum, an appropriate starting point would be the scalar spectral transport equation given in Section 3.2. 

Or, equivalently, the turbulence field will change the scalar energy spectrum. 

As shown below, the rate of change of / (t) is determined from the length-scale distribution of the scalar field as characterized by the scalar energy spectrum. 

The need to add new random variables defined in terms of derivatives of the random fields is simply a manifestation of the lack of two-point information. While it is possible to develop a two-point PDF approach, inevitably it will suffer from the lack of three-point information. Moreover, the two-point PDF approach will be computationally intractable for practical applications. A less ambitious approach that will still provide the length-scale information missing in the one-point PDF can be formulated in terms of the scalar spatial correlation function and scalar energy spectrum described next. 

In general, the scalar Taylor microscale will be a function of the Schmidt number. However, for fully developed turbulent flows,18 l.,p L and /, Sc 1/2Xg. Thus, a model for non-equilibrium scalar mixing could be formulated in terms of a dynamic model for Xassociated with working in terms of the scalar spatial correlation function, a simpler approach is to work with the scalar energy spectrum defined next. 

The scalar energy spectrum E k, t) is defined in terms of the scalar spectrum by integrating out all directional information ... 

For an isotropic scalar field, the scalar spectrum is related to the scalar energy spectrum by... 

By definition, the scalar variance can be found directly from the scalar energy spectrum by integrating over wavenumber space ... 

Thus, E k, t) Ak represents the amount of scalar variance located at wavenumber k. For isotropic turbulence, the scalar integral length scale is related to the scalar energy spectrum by... 

Similar relationships exist for the scalar flux energy spectrum and the scalar covariance energy spectrum. 

As in Section 2.1 for the turbulent energy spectrum, a model scalar energy spectrum can be developed to describe lop(n). However, one must account for the effect of the Schmidt number. For Sc < 1, the scalar-dissipation wavenumbers, defined by19... 

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... 

The form of the scalar energy spectrum for larger wavenumbers will depend on the Schmidt number. Considering first the case where Sc 1, the range of wavenumbers between kc2 and kdi is referred to as the inertial-diffusive sub-range (Batchelor et al. 1959). Note that this range can exist only for Schmidt numbers less than Scid, where... 

The form of the scalar energy spectrum in the inertial-diffusive sub-range can be found starting from the Navier-Stokes equation (see McComb (1990) for details) to be... 

A model scalar energy spectrum can be developed by combining the various theoretical spectra introduced above with appropriately defined cut-off functions and exponents ... 

Figure 3.12. Model scalar energy spectra at Rk = 500 normalized by the integral scales. The velocity energy spectrum is shown as a dotted line for comparison. The Schmidt numbers range from Sc = 10 4 to Sc = 104 in powers of 102.

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. 

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. 

Like the turbulent energy spectrum discussed in Section 2.1, a transport equation can be derived for the scalar energy spectrum lipjn. t) starting from (1.27) and (1.28) for an inert scalar (see McComb (1990) or Lesieur (1997) for details). The resulting equation is21... 

This expression applies to the case where there is no mean scalar gradient. Adding a uniform mean scalar gradient generates an additional source term on the right-hand side involving the scalar-flux energy spectrum. 

In many reacting flows, the reactants are introduced into the reactor with an integral scale L that is significantly different from the turbulence integral scale Lu. For example, in a CSTR, Lu is determined primarily by the actions of the impeller. However, is fixed by the feed tube diameter and feed flow rate. Thus, near the feed point the scalar energy spectrum will not be in equilibrium with the velocity spectrum. A relaxation period of duration on the order of xu is required before equilibrium is attained. In a reacting flow, because the relaxation period is relatively long, most of the fast chemical reactions can occur before the equilibrium model, (4.93), is applicable. 

Figure 4.9. Sketch of CSTR representation of the SR model for 1 < Sc. Each wavenumber band is assumed to be well mixed in the sense that it can be represented by a single variable

In principle, the forward and backward transfer rates can be computed directly from DNS (see Appendix A). However, they are more easily computed by assuming idealized forms for the scalar energy spectrum (Fox 1995). In the general formulation (Fox 1999), they include both a forward cascade (a) and backscatter (/() ... 

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