Re From this we can expect that the boundary layer is relatively thick for

Re from this we can expect that the boundary layer is

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Re ). From this we can expect that the boundary layer is relatively thick for laminar flows, corresponding to low Re , that the entire boundary layer acts as the viscous sublayer of a turbulent flow would and in particular, no matter how rough the pipe surface is, the mean “roughness height” will always lie within the highly viscous (molecular) shear stress dominated boundary layer. As the Reynolds number is increased and transition to turbulence occurs, the boundary layer thins and so also does the thickness of the viscous sublayer. To some extent this has been quanti- fied experimentally through the following formula for complete (not just boundary layer) velocity profiles: u U c = parenleftBig 1 r R parenrightBig 1 /n , (4.53) where U c is pipe centerline velocity. Velocities obtained from this formula are provided in Fig. 4.16 for several values of the exponent n . The value of n used in Eq. (4.53) depends on the Reynolds 0 0.2 0.4 0.6 0.8 1 u / U c 0 0.2 0.4 0.6 0.8 1 1.0 0.0 0.2 0.4 0.6 0.8 ( R - r 29 R 1.0 0.0 0.2 0.4 0.6 0.8 laminar n = 6 n = 8 n = 10 Figure 4.16: Empirical turbulent pipe flow velocity profiles for different exponents in Eq. (4.53). number and increases from n = 6 at Re 2 × 10 4 to n = 10 at Re 3 × 10 6 in a nearly linear (on a semi-log plot) fashion. For moderate Re , n = 7 is widely used, almost independent of the actual value of Re . In turn, this implies that as Re increases more of the rough edges of the surface are extending beyond the viscous sublayer and into the buffer and inertial layers (see Fig. 4.17) where the turbulent fluctuations are dominant. Because these are inertially driven the effect of their interactions with protrusions of the rough surface is to create a drag, thus significantly increasing the effective internal friction. In particular, one can envision the rough, irregular protrusions as locations where the flow actually stagnates creating locally-high “stagnation pressures,” and thus slowing the flow. Effectively, this leads to far more internal friction than does viscosity, so the result is considerable pressure loss and increased friction factor in comparison with laminar flow.
138 CHAPTER 4. APPLICATIONS OF THE NAVIER–STOKES EQUATIONS This interaction between the viscous sublayer and surface roughness suggests, simply on physical grounds, that the friction factor should be a function of Reynolds number and a dimensionless surface roughness, defined for pipe flow as ε/D , with ε being a mean roughness height. There are several ways in which this might be defined. Figure 4.17 displays one of these: the difference between mean values of protrusion maxima and protrusion minima. (b) edge of viscous sublayer ε ε edge of viscous sublayer (a) viscous sublayer viscous sublayer Figure 4.17: Comparison of surface roughness height with viscous sublayer thickness for (a) low Re , and (b) high Re .

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