058 0160 Professor Fred Stern Chapters 6 1 Fall 2007 Chapter 6 Viscous Flow in Ducts Entrance developing and fully developed flow Le f D V L theorem f Re f Re from AFD and EFD D e i Laminar Flow Recrit 2000 L D 06 Re Re Recrit Re Recrit laminar turbulent e L e max 06 Re D 138 D crit Max Le for laminar flow 058 0160 Professor Fred Stern Chapters 6 2 Fall 2007 Turbulent flow Re 4000 104 105 106 107 108 Le D 18 20 30 44 65 95 L D 4 4 Re 1 6 e Relatively shorter than for laminar flow Laminar vs Turbulent Flow Hagen 1839 noted difference in p p u but could not explain two regimes Laminar Turbulent Spark photo Reynolds 1883 showed that the difference depends on Re VD 058 0160 Professor Fred Stern Chapters 6 3 Fall 2007 Laminar pipe flow 1 CV Analysis Continuity 0 V dA Q1 Q2 const CS i e V1 V2 sin ce A1 A2 const and V Vave Momentum Fx p1 p2 R 2 w 2 RL R 2 L sin m 2V2 1V1 W z L 0 p p R 2 RL R z 0 2 L p z R 2 2 w w 058 0160 Professor Fred Stern Fall 2007 h h1 h2 p z or w Chapters 6 4 2 w L R R h R dh 2 L 2 dx R d p z 2 dx r d p z 2 dx i e shear stress varies linearly in r across pipe for either laminar or turbulent flow Energy p1 1 2g V1 z1 p2 h h L 2 2g V2 z 2 hL 2 L R w once w is known we can determine pressure drop In general roughness V D w i Theorem w 058 0160 Professor Fred Stern Fall 2007 8 w V Chapters 6 5 2 f friction factor f Re D D where Re D VD LV2 h hL f D 2g Darcy Weisbach Equation f ReD D still needs to be determined For laminar flow there is an exact solution for f since laminar pipe flow has an exact solution For turbulent flow approximate solution for f using log law as per Moody diagram and discussed late 2 Differential Analysis Continuity V 0 Use cylindrical coordinates r z where z replaces x in previous CV analysis 1 1 r r z 0 z r r r 058 0160 Professor Fred Stern Chapters 6 6 Fall 2007 where V r er e z e z Assume 0 i e no swirl and fully developed z 0 which shows r constant 0 since flow z r R 0 V z e z u r e z Momentum z equation DV p z 2 V Dt u V u p z 2 u z t 0 1 u p z r z r r r f z f r both terms must be constant r dp u A ln r B 4 dz 2 u r 0 finite p p z A 0 058 0160 Professor Fred Stern Chapters 6 7 Fall 2007 u r R 0 r R d p 4 dz 2 u r B 2 R dp 4 dz 2 u max u 0 u u r r z r As per CV analysis r p 2 z w y R r u u R p y r R r r R 2 z R 4 d p 1 Q u r 2 r dr umax R 2 dz 8 2 0 Q R2 d p 1 Vave u R 2 2 max 8 dz R Substituting V Vave f 8 w V 2 R dp 4 dz 2 058 0160 Professor Fred Stern Chapters 6 8 Fall 2007 R 8 Vave 4 Vave 8 V 2 R D 2 R w f 64 64 DV Re D L V 2 64 L V 2 32 LV h hL f D 2 g DV D 2 g gD 2 for z 0 or Cf w 1 V 2 2 V p V 16 Re D Both f and Cf based on V2 normalization which is appropriate for turbulent but not laminar flow The more appropriate case for laminar flow is Poiseuille P C Re 16 0 f for pipe flow Compare with previous solution for flow between parallel plates with p x 058 0160 Professor Fred Stern Fall 2007 y u u 1 h 2 max 4 q hu 3 2h 3 3 max Chapters 6 9 h p 2 2 u max x p x q h 2 v p u 2 h 3 3 2 x max w 3 V h f C 12 f Re 24 48 Vh Re 2 h Po 12 2h Same as pipe other than constants Exact laminar solutions are available for any arbitrary cross section for laminar steady fully developed duct flow 058 0160 Professor Fred Stern Chapters 6 10 Fall 2007 BVP u 0 x 0 p u u x YY ZZ u h 0 Re only enters through stability and transition y y h z z h u u U U h 2 p x Related umax u 1 Poisson equation u 1 0 Dirichlet boundary condition 2 Can be solved by many methods such as complex variables and conformed mapping transformation into Laplace equation by redefinition of dependent variables and numerical methods 058 0160 Professor Fred Stern Fall 2007 Chapters 6 11 058 0160 Professor Fred Stern Fall 2007 Chapters 6 12 058 0160 Professor Fred Stern Fall 2007 Chapters 6 13 058 0160 Professor Fred Stern Fall 2007 Chapters 6 14 058 0160 Professor Fred Stern Chapters 6 15 Fall 2007 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 Transition from one to another laminar flow a Thermal instability Bernard Problem A layer of fluid heated from below is top heavy but only undergoes convective cellular motion for bouyancy force viscous force 1 coefficient of thermal expansion T 1 T T d dT dz Raleigh g d g d 4 Ra Racr k w d 2 P 0 d depth of layer k thermal viscous diffusivities w velocity scale convection w diffusion k d from energy equation i e w k d 058 0160 Professor Fred Stern Fall 2007 Chapters 6 16 Solution for two rigid plates Racr 1708 for progressive wave disturbance i x ct ct cr d 3 12 w we e i cos x ct sin x ct i x ct T T e cr 2 2d r c cr ici For temporal stability r 2 wavenumber cr wave speed ci 0 0 0 unstable neutral stable Ra 5 x 104 transition to turbulent flow Thumb curve stable for low Ra 1708 and very long or short 058 0160 Professor Fred Stern Chapters 6 17 Fall 2007 b finger oscillatory instability hot salty …
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