Turbulent Flow, Reynolds Number

Chapter 2 reviews the statistical theory of turbulent flows. The emphasis, however, is on collecting in one place all of the necessary concepts and formulae needed in subsequent chapters. The discussion of these concepts is necessarily brief, and the reader is referred to Pope (2000) for further details. It is, nonetheless, essential that the reader become familiar with the basic scaling arguments and length/time scales needed to describe high-Reynolds-number turbulent flows. Likewise, the transport equations for important one-point statistics in inhomogeneous turbulent flows are derived in Chapter 2 for future reference. 

The molecular transport term vV2(m m ) is closed, but negligible (order ReL 1) in high-Reynolds-number turbulent flows. The production term... 

Thus, only the normal Reynolds stresses (i = j) are directly dissipated in a high-Reynolds-number turbulent flow. The shear stresses (i / j), on the other hand, are dissipated indirectly, i.e., the pressure-rate-of-strain tensor first transfers their energy to the normal stresses, where it can be dissipated directly. Without this redistribution of energy, the shear stresses would grow unbounded in a simple shear flow due to the unbalanced production term Vu given by (2.108). This fact is just one illustration of the key role played by 7 ., -in the Reynolds stress balance equation. 

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). 

Both of these terms are large in high-Reynolds-number turbulent flows. 

For convenience, the turbulence statistics used in engineering calculations of inhomogeneous, high-Reynolds-number turbulent flows are summarized in Table 2.4 along with the unclosed terms that appear in their transport equations. Models for the unclosed terms are discussed in Chapter 4. 

We are essentially assuming that the small scales are in dynamic equilibrium with the large scales. This may also hold in low-Reynolds-number turbulent flows. However, for low-Reynolds-number flows, one may need to account also for dissipation rate anisotropy by modeling all components in the dissipation-rate tensor s j. 

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]

Thus, the closure problem reduces to finding an appropriate expression for the scalar flux (Ujtp). In high-Reynolds-number turbulent flows, the molecular transport term is again negligible. Thus, the scalar-flux term is responsible for the rapid mixing observed in turbulent flows. 

The firsttwo terms on the right-hand side of this expression are responsible for spatial transport of scalar dissipation. In high-Reynolds-number turbulent flows, the scalar-dissipation flux (iijC ) is the dominant term. The other terms on the right-hand side are similar to the corresponding terms in the dissipation transport equation ((2.125), p. 52), and are defined as follows. 

The remaining two terms in (3.114) will be large in high-Reynolds-number turbulent flows. The vortex-stretching term Vf is defined by... 

Equation (5.2a) is valid for any DG/ i value, Reynolds number, turbulent flow zone, or laminar flow zone. First calculate a Reynolds number from DG/[i. Then use Kern s Fig. 24, which appears in App. A as Fig. A.l. You may also derive this value by using Eq. (5.2a) for Jh. This equation is simply a curve-fit to Kern s figure. 

In high-Reynolds-number turbulent flows, the molecular diffusion term Fp( ) will be negligible compared with Ft (Fox, 2003). Except for the last term, Eq. (2.49) is closed. 

The theory of mixing of a passive scalar concentration field subject to advection and diffusion in a high Reynolds number turbulent flow is based on the works of Obukhov (1949) and Corrsin (1951). Consider a statistically stationary state with a large-scale source of scalar fluctuations in the case when both Pe and Re are large. The... 

Shih, T.H. Uou, W.W. Shabbir, A Yang, Z G et al. 1995. A new k-e eddy viscosity model for high Reynolds number turbulent flows. Comput Fluid, 24 3) 227-238. 

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