Spectral transport

The model turbulent energy spectrum given in (2.53) was introduced to describe fully developed turbulence, i.e., the case where Eu(k, t) does not depend explicitly on t. The case where the turbulent energy spectrum depends explicitly on time can be handled by deriving a transport equation for the velocity spectrum tensor 4 ,(k, t) starting from the Navier-Stokes equation for homogeneous velocity fields with zero or constant mean velocity (McComb 1990 Lesieur 1997). The resultant expression can be simplified for isotropic turbulence to a transport equation for / ,(/ . /) of the form14 

The second term on the right-hand side of (2.61) can be rewritten in terms of the turbulent energy dissipation spectrum as 

The first term on the right-hand side of (2.61) is the spectral transfer function, and involves two-point correlations between three components of the velocity vector (see McComb (1990) for the exact form). The spectral transfer function is thus unclosed, and models must be formulated in order to proceed in finding solutions to (2.61). However, some useful properties of Tu(k, t) can be deduced from the spectral transport equation. For example, integrating (2.61) over all wavenumbers yields the transport equation for the turbulent kinetic energy  

In words, (2.64) implies that T (k, t) is responsible for transferring energy between different wavenumbers without changing the total turbulent kinetic energy. 

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The material covered in the appendices is provided as a supplement for readers interested in more detail than could be provided in the main text. Appendix A discusses the derivation of the spectral relaxation (SR) model starting from the scalar spectral transport equation. The SR model is introduced in Chapter 4 as a non-equilibrium model for the scalar dissipation rate. The material in Appendix A is an attempt to connect the model to a more fundamental description based on two-point spectral transport. This connection can be exploited to extract model parameters from direct-numerical simulation data of homogeneous turbulent scalar mixing (Fox and Yeung 1999). 

The spectral transport equation can also be used to generate a spectral model for the dissipation rate e. Multiplying (2.61) by 2vk2 yields the spectral transport equation for... 

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. 

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. 

The scalar spectral transport equation can be easily extended to the cospectrum of two inert scalars Eap(K, t). The resulting equation is... 

In order to understand better the physics of scalar spectral transport, it will again be useful to introduce the scalar spectral energy transfer rates T,f, and T p defined by... 

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). 

For Sc < 1, a similar expression can be derived by taking /cd = kci- However, because /cci < /cdi lies in the inertial range, the characteristic spectral transport time for wavenumbers greater than /cci cannot be taken as constant. 

A spectral model similar to (3.82) can be derived from (3.75) for the joint scalar dissipation rate eap defined by (3.139), p. 90. We will use these models in Section 3.4 to understand the importance of spectral transport in determining differential-diffusion effects. As we shall see in the next section, the spectral interpretation of scalar energy transport has important ramifications on the transport equations for one-point scalar statistics for inhomogeneous turbulent mixing. 

In homogeneous turbulence, spectral transport can be quantified by the scalar cospectral energy transfer rate Tap(ic, t). We can also define the wavenumber that separates the viscous-convective and the viscous-diffusive sub-ranges nf by introducing the arithmetic-mean molecular diffusivity Tap defined by... 

A logical extension of the spectral equilibrium model for e would be to consider nonequilibrium spectral transport from large to small scales. Such models are an active area of current research (e.g., Schiestel 1987 and Hanjalic et al. 1997). Low-Reynolds-number models for e typically add new terms to correct for the viscous sub-layer near walls, and adjust the model coefficients to include a dependency on Re/,. These models are still ad hoc in the sense that there is little physical justification - instead models are validated and tuned for particular flows. 

The spectral transport term for the first wavenumber band is defined by... 

In the SR model, forward and backscatter rate constants are employed to model the spectral transport terms ... 

Besnard, D. C., F. H. Harlow, R. M. Rauenzahn, and C. Zemach (1990). Spectral transport model of turbulence. Report LA-11821-MS, Los Alamos National Laboratory. 

Spectral transport model for turbulence. Report LA-UR92-1666, Los Alamos National Laboratory. 

Yeung, P. K. (1994). Spectral transport of self-similar passive scalar fields in isotropic turbulence. Physics of Fluids 6, 2245-2247. 

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