A weakness of both the power-law and logarithmic velocity profiles is that they give

at y = R. Because symmetry requires that the shear stress vanish at the centerline, this implies that ε → 0 as y → R. The idea that turbulent eddies somehow disappear at the center of a tube is unrealistic and can create errors in heat and mass transfer calculations. A way around this problem was proposed by Reichardt (1951), who suggested that the eddy diffusivity outside the wall region be approximated as

where η = r⁄R, R+ is as defined in Eq. (10.2-6), and κ is an empirical constant.

(a) Show that the Reichardt eddy diffusivity leads to

where C is a constant.

(c) Where the law of the wall applies, η → 1. By choosing κ and C for consistency with Eq. (10.5-16), show that

Note that the right-hand side could be rewritten in terms of η alone, if desired, by using y⁺ = R⁺(1 − η).

Transcribed Image Text:

dv₂/dy # 0