8. Flow in Pipes The flow in a pipe is a viscous flow, and there exist 2 distinct regions: laminar and turbulent region, connected by a transition region. • Laminar flow : dye remains a fine filament y x • Turbulent flow : y x dye is rapidly dispersed randomly The shear stress, , is a function of the velocity gradient. The greater the change in U with respect to y , the greater is the shear stress. Therefore, the shear stress and the frictional losses are higher in turbulent flow. Whether the flow is laminar or turbulent depends on the Reynolds number. Transition from laminar to turbulent flow occurs at Re =2300. However, if conditions such as pipe smoothness and velocity are well controlled, transition can be delayed up to Re =10 5 for incompressible flow. 74
As the flow enters a pipe uniformly, the velocity at the walls must decrease to zero (no-slip Boundary Condition). Continuity dictates that the velocity at the center must increase. Thus, the velocity profile is changing continuously from the pipe entrance until it reaches fully developed conditions. This distance, L , is called the entrance length. fully-developed profile L 8.1. Poiseuille Flow Consider the steady, fully developed laminar flow in a straight pipe of circular cross section having a constant diameter, D . τ τ y x b The condition of equilibrium in x -direction requires that the pressure force ( p 1 - p 2 )π y 2 acting on the faces of the cylinder be equal to the shear stress 2π yb acting on the circumferential area: = ( p 1 - p 2 ) y /2 b 75
In accordance with the law of friction (Newtonian fluid), we have: = d u /d y . Since u decreases with y , therefore: d u d y = – p 1 – p 2 μ b y 2 and upon integration: 4 d d 1 4 ) ( 2 2 2 1 y C x p y C b p
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- Fall '08
- Fluid Dynamics, Incompressible Flow, Turbulent Flow