Turbulence scaling theory

The nonlinear interaction between turbulence and the hydrodynamic instability is an important, difficult, and unresolved problem of turbulent-flame theory for large-scale turbulence in the reaction-sheet regime. 

The subject of Chapter 4 (originally the third lecture) is the problem of the reduction of turbulent losses by polymer chains in a solvent. The topic appears to be especially intricate in view of the coupling between the hydrodynamic aspects of turbulent flow and the viscoelastic behavior of chains in strongly perturbed conformations. The lecture followed closely a paper Towards a scaling theory of drag reduction published in 1986 by Professor de Gennes in Physica which is here reprinted by permission of the publisher. Because this lecture is the most tentative (and difficult) part in the series we have put it in the last chapter. 

Boundary layer flows are a special class of flows in which the flow far from the surface of an object is inviscid, and the effects of viscosity are manifest only in a thin region near the surface where steep velocity gradients occur to satisfy the no-slip condition at the solid surface. The thin layer where the velocity decreases from the inviscid, potential flow velocity to zero (relative velocity) at the sohd surface is called the boundary layer The thickness of the boundary layer is indefinite because the velocity asymptotically approaches the free-stream velocity at the outer edge. The boundaiy layer thickness is conventionally t en to be the distance for which the velocity equals 0.99 times the free-stream velocity. The boundary layer may be either laminar or turbulent. Particularly in the former case, the equations of motion may be simphfied by scaling arguments. Schhchting Boundary Layer Theory, 8th ed., McGraw-HiU, New York, 1987) is the most comprehensive source for information on boundary layer flows. 

In this approach, it is assumed that turbulence dies out at the interface and that a laminar layer exists in each of the two fluids. Outside the laminar layer, turbulent eddies supplement the action caused by the random movement of the molecules, and the resistance to transfer becomes progressively smaller. For equimolecular counterdiffusion the concentration gradient is therefore linear close to the interface, and gradually becomes less at greater distances as shown in Figure 10.5 by the full lines ABC and DEF. The basis of the theory is the assumption that the zones in which the resistance to transfer lies can be replaced by two hypothetical layers, one on each side of the interface, in which the transfer is entirely by molecular diffusion. The concentration gradient is therefore linear in each of these layers and zero outside. The broken lines AGC and DHF indicate the hypothetical concentration distributions, and the thicknesses of the two films arc L and L2. Equilibrium is assumed to exist at the interface and therefore the relative positions of the points C and D are determined by the equilibrium relation between the phases. In Figure 10.5, the scales are not necessarily the same on the two sides of the interface. 

P. Clavin and F.A. Williams. Theory of premixed-flame propagation in large-scale turbulence. Journal of Fluid Mechanics, 90(pt 3) 589-604, 1979. 

Stresses acting on micro-organisms in (a) to (c) are derived on the premise that the flow forces originate from the turbulent motion of the carrier medium. In almost all cases, turbulence is assumed to be locally isotropic and homogeneous which greatly simplifies the analysis and allows the application of the Kolmogoroff s theory of turbulence to the problem [81]. The Kolomogoroff micro-scale of turbulence,... 

The RNG theory as applied to turbulence reduces the Reynolds number to an effective Reynolds number (Reeff) by increasing an effective viscosity (/small-scale eddies are eliminated, which reduces computational demand considerably. The new equation for the variation of the effective viscosity is as follows ... 

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. 

For small-scale, high-intensity turbulence, Damkohler reasoned that the transport properties of the flame are altered from laminar kinetic theory viscosity y0 to the turbulent exchange coefficient e so that... 

The definition of X incorporates the fact that the Lagrangian integral time scale is of the order of Zilw,. The statistical theory of turbulent diffusion outlined in the beginning of Section VIII,B can be used to estimate the functional dependence of g as... 

McCready et al., 1986). The surface renewal theory can be made to fit the transfer data at fluid-fluid interfaces. The exception to this is bubbles with a diameter less than approximately 0.5 mm. Even though there is a fluid on both sides, surface tension causes these small bubbles to behave as though they have a solid-fluid interface. There is also some debate about this 1 /2 power relationship at free surfaces exposed to low shear, such as wind-wave flumes at low wind velocity (Jahne et al., 1987) and tanks with surfactants and low turbulence generation (Asher et al., 1996). The difficulty is that these results are influenced by the small facilities used to measure Kl, where surfactants wiU be more able to restrict free-surface turbulence and the impact on field scale gas transfer has not been demonstrated. 

In many of these experiments, interfacial turbulence was the obvious visible cause of the unusual features of the rate of mass transfer. There are, however, experimental results in which no interfacial activity was observed. Brian et al. [108] have drawn attention to the severe disagreement existing between the penetration theory and data for the absorption of carbon dioxide in monoethanolamine. They have performed experiments on the absorption of C02 with simultaneous desorption of propylene in a short, wetted wall column. The desorption of propylene without absorption of C02 agrees closely with the predictions of the penetration theory. If, however, both processes take place simultaneously, the rate of desorption is greatly increased. This enhancement must be linked to a hydrodynamic effect induced by the absorption of C02 and the only one which can occur appears to be the interfacial turbulence caused by the Marangoni effect. No interfacial activity was observed because of the small scale and small intensity of the induced turbulence. 

Recall our short discussion in Section 18.5 where we learned that turbulence is kind of an analytical trick introduced into the theory of fluid flow to separate the large-scale motion called advection from the small-scale fluctuations called turbulence. Since the turbulent velocities are deviations from the mean, their average size is zero, but not their kinetic energy. The kinetic energy is proportional to the mean value of the squared turbulent velocities, Mt2urb, that is, of the variance of the turbulent velocity (see Box 18.2). The square root of this quantity (the standard deviation of the turbulent velocities) has the dimension of a velocity. Thus, we can express the turbulent kinetic energy content of a fluid by a quantity with the dimension of a velocity. In the boundary layer theory, which is used to describe wind-induced turbulence, this quantity is called friction velocity and denoted by u. In contrast, in river hydraulics turbulence is mainly caused by the friction at the... 

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