Turbulent flow analogy solutions

At present analytical solutions of the equations describing the microscopic aspects of material transport in turbulent flow are not available. Nearly all the equations representing component balances are nonlinear in character even after many simplifications as to the form of the equation of state and the effect of the momentum transport upon the eddy diffusivity are made. For this reason it is not to be expected that, except by assumption of the Reynolds analogy or some simple consequence of this relationship, it will be possible to obtain analytical expressions to describe the spatial variation in concentration of a component under conditions of nonuniform material transport. 

ANALOGY SOLUTIONS FOR HEAT TRANSFER IN TURBULENT FLOW... 

In outlining the main steps in obtaining an analogy solution, attention will here be given to two-dimensional flow. The total shear stress and total heat transfer rate are made up of the molecular and turbulent contributions, i.e. ... 

As discussed in the previous chapter, most early efforts at trying to theoretically predict heat transfer rates in turbulent flow concentrated on trying to relate the wall heat transfer rate to the wall shear stress [1],[2],[3],[41. The reason for this is that a considerable body of experimental and semi-theoretical knowledge concerning the shear stress in various flow situations is available and that the mechanism of heat transfer in turbulent flow is obviously similar to the mechanism of momentum transfer. In the present section an attempt will be made to outline some of the simpler such analogy solutions for boundary layer flows, attention mainly being restricted to flow over a flat plate. 

The ideas used in the previous chapter to derive analogy solutions for boundary layer flows can easily be extended to obtain such analogy solutions for turbulent 304... 

Some simple methods of determining heat transfer rates to turbulent flows in a duct have been considered in this chapter. Fully developed flow in a pipe was first considered. Analogy solutions for this situation were discussed. In such solutions, the heat transfer rate is predicted from a knowledge of the wall shear stress. In fully developed pipe flow, the wall shear stress is conventionally expressed in terms of the friction factor and methods of finding the friction factor were discussed. The Reynolds analogy was first discussed. This solution really only applies to fluids with a Prandtl number of 1. A three-layer analogy solution which applies for all Prandtl numbers was then discussed. 

The so-called Taylor-Prandtl analogy was applied to boundary layer flow in Chapter 6. Use this analogy solution to derive an expression for the Nusselt number in fully developed turbulent pipe flow. 

Analogy Solutions for Heat Transfer in Turbulent Flow 244... 

Qfis may ask the reason for the functional form of Eq. (6-4). Physical reasoning, based on the experience gained with the analyses of Chap. 5, would certainly indicate a dependence of the heat-transfer process on the flow field, and hence on the Reynolds number. The relative rates of diffusion of heat and momentum are related by the Prandtl number, so that the Prandtl nunfber is expected to be a significant parameter in the final solution. We can be rather confident of the dependence of the heat transfer on the Reynolds and Prandtl numbers. But the question arises as to the correct functional form of the relation i.e., would one necessarily expect a product of two exponential functions of the Reynolds and Prandtl numbers The answer is that one might expect this functional form since it appears in the flat-plate analytical solutions of Chap. 5, as well as the Reynolds analogy for turbulent flow. In addition, this type of functional relation is convenient to use in correlating experimental data, as described below. 

Analogies Between Mass, Heat and Momentum Transfer Fluxes A comparison of the partial differential equations for the conservation of heat, mass and momentum in a turbulent flow field (5.240), (5.241) and (5.242) shows that the equations are mathematically similar provided that the pressure term in the momentum equation is negligible [135]. If the corresponding boundary contitions are similar too, the normalized solution of these equations will have the same form. 

Metzner and Friend [73] measured turbulent heat transfer rates with aqueous solutions of Carbopol, corn syrup, and slurries of Attagel in circular-tube flow. They developed a semi-theoretical correlation to predict the Stanton number for purely viscous fluids as a function of the friction factor and Prandtl number, applying Reichardt s general formulation for the analogy between heat and momentum transfer in turbulent flow ... 

The possibility of carrying out not only fast polymerisation processes in turbulent flows, but also polymer-analogous reactions of polymer modification [18,20-22] was exemplified by butyl rubber (BR) chlorination in solution with molecular chlorine. 

Ideal Gases at High Temperature. Three fundamentally different approaches have been applied to the treatment of the turbulent boundary layer with variable fluid properties all are restricted to air behaving as an ideal, calorically perfect gas. First, the Couette flow solutions have been extended to permit variations in viscosity and density. Second, mathematical transformations, analogous to Eq. 6.36 for a laminar boundary layer, have been used to transform the variable-property turbulent boundary layer differential equations into constant-property equations in order to provide a direct link between the low-speed boundary layer and its high-speed counterpart. Third, empirical correlations have been found that directly relate the variable-property results to incompressible skin friction and Stanton number relationships. Examples of the latter are reference temperature or enthalpy methods analogous to those used for the laminar boundary layer, and the method of Spalding and Chi [104]. 

At high enough qualities and mass fluxes, however, it would be expected that the nucleate boiling would be suppressed and the heat transfer would be by forced convection, analogous to that for the evaporation for pure fluids. Shock [282] considered heat and mass transfer in annular flow evaporation of ethanol water mixtures in a vertical tube. He obtained numerical solutions of the turbulent transport equations and carried out calculations with mass transfer resistance calculated in both phases and with mass transfer resistance omitted in one or both phases. The results for interfacial concentration as a function of distance are illustrated in Fig. 15.112. These results show that the liquid phase mass transfer resistance is likely to be small and that the main resistance is in the vapor phase. A similar conclusion was reached in recent work by Zhang et al. [283] these latter authors show that mass transfer effects would not have a large effect on forced convective evaporation, particularly if account is taken of the enhancement of the gas mass transfer coefficient as a result of interfacial waves. 

Another analogous relationship is that of mass transfer, represented by Pick s law of diffusion for mass flux, J, of a dilute component, I, into a second fluid, 2, which is proportional to the gradient of its mass concentration, mi. Thus we have, J = p Du Vmt, where the constant Z)/2 is the binary diffusion coefficient and p is density. By using similar solutions we can find generalized descriptions of diffusion of electrons, homogeneous illumination, laminar flow of a liquid along a spherical body (assuming a low-viscosity, non-compressible and turbulent-lree fluid) or even viscous flow applied to the surface tension of a plane membrane. 

---