Energy spectrum dissipation range

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). 

For example, the vortex-stretching term is a triple-correlation term that corresponds to the rate at which dissipation is created by spectral energy passing from the inertial range to the dissipative range of the energy spectrum (see (2.75), p. 43). Letting /cdi 0.1 denote... 

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 scalar-dissipation wavenumber /cd is defined in terms of /cdi by /cd = Sc1/2kdi-Like the fraction of the turbulent kinetic energy in the dissipation range kn ((2.139), p. 54), for a fully developed scalar spectrum the fraction of scalar variance in the scalar dissipation range scales with Reynolds number as... 

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

The standard k-e model employs a single time scale r =kle, called dissipation range timescale, in the e equation to characterize the dynamic processes occurring in the energy spectrum. Thus, Eq. (6) can be rewritten as ... 

Sirovich, L., Smith, L., Yakhot, V. Energy spectrum of homogeneous and isotropic turbulence in far dissipation range. Physical Review Letters 72(3), 344-347 (1994)... 

The absorption and emission spectra of a fluorophore are bands spread over a range of wavelengths with at least one peak of maximal absorbance and emission that corresponds to the So-Si and Si—S0 transitions, respectively. There are several vibrational levels within an electronic state and transitions from one electronic to several vibrational states are potentially possible. This determines that the spectra are not sharp but consist of broad bands. The emission spectrum is independent of the excitation wavelength. The energy used to excite the fluorophore to higher electronic and vibrational levels is very rapidly dissipated, sending the fluorophore to the lowest vibrational level of the first electronic excited state (Si) from where the main fluorescent transition occurs [3] (see Fig. 6.1). 

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