Since there is no good way of predicting the Reynolds stress terms, we must rely on theoretical models. A number of these have been postulated, and we will review the major ones. Note, though, that these tend to include very big, very shaky assumptions -- the theory really isn't all that good. Fortunately, the major results do work well enough to allow decent fitting of the data.
Boussineq (ca. 1877) adopted the straightforward approach of setting up a "Newton-like" relationship between the turbulent shear stress and the shear strain. This meant defining an eddy viscosity or eddy diffusivity such that
It is also possible to define eddy diffusivities for heat (EH) and mass (EM) and set up turbulent versions of Fourier's and Fick's Laws.The next stage in "classical" turbulence analysis is owed to Prandtl (cir 1925). He proposed defining a mixing length: the distance a particle of fluid can travel before becoming indistinguishable from the bulk. This idea is analogous to the mean free path in the molecular theory of gases. The mixing length is characteristic of the degree of turbulence, but small enough that it can be used to define the velocity perturbation
With some extremely broad assumptions, the mixing length formulas can be integrated to obtain a velocity profile. These assumptions (some of which we now know to be wrong) include constant shear stress, 2-D geometry, and most importantly the "Prandtl Hypothesis" that the mixing length is proportional to the distance from the wall. Integration under these circumstances yields a logarithmic expression of the form
Other researchers have since improved on the assumptions, but the same general form recurs in much of their work. Thus, some version of the logarithmic relationship appears in most turbulent velocity profiles. The most common of these is the universal velocity profile. This applies to smooth tubes, channels, and boundary layers and is based on Prandtl's theory. The constants, however, are based strictly on empirical fits.
The universal velocity distribution divides a conduit into three regions: (1) a viscous (or "laminar") sublayer near the wall, where turbulent momentum transfer is negligible, (2) a turbulent core region surrounding the centerline, and (3) a single intermediate region, the "eddy generation zone" between the two. (A more accurate profile can be obtained by including more than one intermediate region, but this is usually more work than people want to do ...)
Universal velocity profiles are generally expressed in terms of dimensionless positions and velocities, given by:
Experimental velocity measurements are then fitted to obtain appropriate constants, and the universal profile results:
Need figure showing regions
The universal distribution is empirical, and it does show some inconsistencies. Take a look at the centerline value of the derivative -- it isn't zero as it should be.
When calculating Q and uavg, the logarithmic form should be used. The laminar sublayer is very thin and shouldn't make a difference. The sublayer also shrinks as flow increases, since larger eddies tend to approach the wall more closely.
The distribution given is that for smooth pipe -- wall roughness will change the fit of the core region, although not that of the sublayer. Rough pipe (most commercial applications) changes the boundary conditions for integration. The form which results is:
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