Reynolds number turbulence

Clemens, N.T., Paul, P.H., and Mungal, M.G., The structure of OH fields in high Reynolds number turbulent jet diffusion flames. Combust. Sci, Technol., 129,165,1997. 

LES is suitable for the simulation of turbulence at moderate Reynolds numbers. Turbulence generation, transport, and dissipation are described very accurately... 

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. 

Notwithstanding these difficulties, it is sometimes possible to manipulate a non-linear expression using the usual rules of calculus into a form that can be further simplified using high-Reynolds-number turbulence theory. For example, using... 

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... 

As discussed in Chapter 3, at very high Reynolds numbers, turbulent mixing theory predicts that the scalar dissipation rate will be independent of Re and Sc. Thus, most molecular models ignore all dependencies on these parameters, even at moderate Reynolds numbers. In general, the inclusion of dependencies on Re, Sc, or Da is difficult and, most likely, will have to be done on a case-by-case basis. 

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. 

Hanks, R. W. 1978. Low Reynolds number turbulent pipeline flow of pseudohomogeneous slurries, in Proceedings of the Fifth International Conference on the Hydraulic Transport ofSolids in Pipes Hydrotransport. May 8-11. Paper C2, p. C2-23 to C2-34, Hanover, West Germany, cited in Garcia and Steffe 1987. 

There are two main approaches to modeling the near-wall region. In one approach, the so-called wall function approach, the viscosity-affected inner regions (viscous and buffer layers) are not modeled. Instead, semi-empirical formulae (wall functions) are used to bridge the viscosity-affected region between the wall and the fully turbulent region. In another approach, special, low Reynolds number turbulence models are developed to simulate the near-wall region flow. These two approaches are shown schematically in Fig. 3.5(b) and 3.5(c). 

The phrases similarity hypothesis and universal form refer to a mathematical consequence of the Kolmogorov h3q)othesis denoting that on the small scales all high-Reynolds-number turbulent velocity fields are statistically similar. That is, they are statistically identical when they are scaled by the Kolmogorov velocity scale ([121], p. 186). 

Shih T-H, Liou WW, Shabbir A, Zhu J (1995) A New k-e Eddy Viscosity Model for High Reynolds Number Turbulent Elows. Comp Eluids 24(3) 227-238 Smith LM, Reynolds WC (1992) On the Yakhot-Orszag renormalization group method for deriving turbulence statistics and models. Phys Fluids A 4(2) 364-390... 

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. 

---