136 CHAPTER 4 APPLICATIONS OF THE NAVIERSTOKES EQUATIONS time averaged velocity

# 136 chapter 4 applications of the navierstokes

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136 CHAPTER 4. APPLICATIONS OF THE NAVIER–STOKES EQUATIONS time-averaged velocity fluctuating velocity complete velocity u, u t Figure 4.15: Graphical depiction of components of Reynolds decomposition. It can be shown using the definition of time average, Eq. (4.50), that u = v = 0 (an exercise we leave to the reader), so application of the averaging operator to both sides of Eq. (4.51), along with commuting averaging and differentiation, results in the simplification ( uv ) y = ( u v ) y + ( u v ) y . (4.52) The second quantity on the right-hand side arises strictly from turbulent fluctuations and is of inertial origin, as are all the terms in Eq. (4.52). But if one recalls that in the derivation of the N.–S. equations viscous stresses, τ , appear in terms of the form, e.g. , ∂τ/∂y , it is natural to call quantities such as ρ u v turbulent stresses , or because they often arise as the result of a Reynolds decomposition, Reynolds stresses . Note that ρ must be inserted in the preceding expression for dimensional consistency with units of shear stress, but u v is often loosely termed a turbulent stress. We can now return to Fig. 4.14 and describe the physics of each of the regions indicated there. The first of these is the viscous sublayer . This lies immediately adjacent to the solid wall, and in this region we expect velocities to be quite small due to the no-slip condition. Moreover, they increase roughly linearly coming away from the wall, similar to what occurs in Couette flow. Because of the low speed, turbulent fluctuations are small (sometimes, this layer is incorrectly termed the “laminar sublayer”), and they are readily damped by viscous forces. Thus, the dominant physics in this layer is the result of molecular viscosity—nonlinear inertial effects arising from u v are relatively small. As we move farther from the wall, into the buffer layer (sometimes called the “overlap region”) the magnitude of turbulent fluctuations increases, and both molecular diffusion and turbulent stress terms are important aspects of the physics in this region, with the former decreasing in importance as we move farther from the wall. As we move yet further from the wall, into the inertial sublayer , effects of viscous diffusion become negligible, and only the turbulent stresses are important. Finally, an “outer layer” can be identified in which most of the flow energy is contained in motion on scales of the order of the pipe radius. The motivation for the preceding discussions is to provide a rational framework for under- standing the effects of surface roughness on turbulent friction factors. We have already indicated that even for laminar flow the friction factor is never zero, but in addition it does not depend on
4.5. PIPE FLOW 137 roughness of the pipe wall. In a turbulent flow wall roughness is a major factor, and the degree to which this is the case also depends on the Reynolds number. To see how this occurs we need to first recall one of the basic properties of boundary layers, viz. , their thickness decreases with increasing Re (recall that the boundary-layer thickness is δ 1 /

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