57:020 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall 2013 1 Chapter 8 Flow in Conduits Entrance and developed flows Le = f(D, V, , ) i theorem Le/D = f(Re) Laminar flow: Recrit 2000, i.e., for Re < Recrit laminar Re > Recrit turbulent Le/D = .06Re from experiments Lemax = .06RecritD 138D maximum Le for laminar flow57:020 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall 2013 2 Turbulent flow: DLe 6/1Re4.4 from experiment Laminar vs. Turbulent Flow laminar turbulent spark photo Reynolds 1883 showed difference depends on Re = VD Re Le/D 4000 18 104 20 105 30 106 44 107 65 108 95 i.e., relatively shorter than for laminar flow57:020 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall 2013 3 Shear-Stress Distribution Across a Pipe Section Continuity: Q1 = Q2 = constant, i.e., V1 = V2 since A1 = A2 Momentum: AVuFs = 222111AVVAVV = 0VVQ12 0dsr2sinWAdsdsdpppA AdsW dsdzsin 0dsr2dsdzAdsdsAdsdp Ads zpdsd2r 02Wrdpzds varies linearly from 0.0 at r = 0 (centerline) to max (= w) at r = r0 (wall), which is valid for laminar and turbulent flow.57:020 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall 2013 4 no slip condition Laminar Flow in Pipes zpdsd2rdrdVdydV y = wall coordinate = ro r dydVdrdydydVdrdV zpdsd2rdrdV Czpdsd4rV2 zpdsd4rC0rV2oo 222014oCrrdrV r p z Vds r where AdVQ = or0rdr2rV 2rdrrdrddA Exact solution to Navier-Stokes equations for laminar flow in circular pipe 24oCrdV p zds 57:020 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall 2013 5 zpdsd8rQ4o 282oCrVQdV p zA ds For a horizontal pipe, 2244ooCrrdpV p zds L where L = length of pipe = ds 222200144orp r pV r r rL r L 0422002 128rp D pQ r r rdrLL Energy equation: L22221211hzg2Vpzg2Vp 1212pph z z 57:020 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall 2013 6 friction coefficient for pipe flow boundary layer flow 121202( ) [ ] [ ]WLppdh L d Lh z z h L p zds ds r Define friction factor 2wV8f 2wfV21C 2200282[ ] [ ]2WLffVL L L Vh h fr r D g Darcy – Weisbach Equation, which is valid for both laminar and turbulent flow. Friction factor definition based on turbulent flow analysis where ( , , , , )w w or V k thus n=6, m=3 and r=3 such that i=1,2,3 = 2wV8f, ReVD VD, k/D; or f=f(Re, k/D) where k=roughness height. For turbulent flow f determined from turbulence modeling since exact solutions not known, as will be discussed next. For laminar flow f not affected k and f(Re) determined from exact analytic solution to Navier-Stokes equations. Exact solution: 284= = 22oowoorrd V Vpzds r r 57:020 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall 2013 7 For laminar flow( , , )w w orV thus n=4, m=3 and r=1 such that 1WorV=constant. The constant depends on duct shape (circular, rectangular, etc.) and is referred to as Poiseuille number=Po. Po=4 for circular duct. Re64DV64Vr32fo or 2LfDVL32hh hf = head loss due to friction for z=0: pVas per Hagen!57:020 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall 2013 8 Stability and Transition Stability: can a physical state withstand a disturbance and still return to its original state. In fluid mechanics, there are two problems of particular interest: change in flow conditions resulting in (1) transition from one to another laminar flow; and (2) transition from laminar to turbulent flow. (1) Example of transition from one to another laminar flow: Centrifugal instability for Couette flow between two rotating cylinders when centrifugal force > viscous force 3 2 2cr 02Ta =1708 ( )i i oiircTa c r r r , which is predicted by small-disturbance/linear stability theory.57:020 Mechanics of Fluids and Transport Processes Chapter 8 Professor Fred Stern Fall 2013 9 (2) Transition from laminar to turbulent flow Not all laminar flows have different equilibrium states, but all laminar flows for sufficiently large Re become unstable and undergo transition to turbulence. Transition: change over space and time and Re range of laminar flow into a turbulent flow. UcrRe ~ 1000, δ = transverse viscous thickness Retrans > Recr with xtrans ~ 10-20 xcr Small-disturbance/linear stability theory also predicts Recr with some success for parallel viscous flow such as plane Couette flow, plane or pipe Poiseuille flow, boundary layers without or with pressure gradient, and free shear flows (jets, wakes, and mixing layers). No theory for transition, but recent Direct Numerical Simualtions is helpful. In general: Retrans=Retrans(geometry, Re, pressure gradient/velocity profile shape, free stream turbulence, roughness, etc.)57:020 Mechanics of Fluids and Transport Processes Chapter 8
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