058 0160 Professor Fred Stern Chapters 6 1 Fall 2006 Chapter 6 Viscous Flow in Ducts Entrance developing and fully developed flow Le f D V theorem i Laminar Flow Recrit 2000 L D 06 Re L e D f 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 2006 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 2006 Laminar pipe flow 1 CV Analysis Continuity 0 V dA Q1 Q2 const CS i e V1 V2 due to A1 A2 const and V Vave Momentum 2 2 m B V B2V2 Fx p1 p2 R w 2 RL R L sin 123 14242244 1424 3 3 p W z L 0 058 0160 Professor Fred Stern Chapters 6 4 Fall 2006 p R 2 RL R z 0 2 L p z R 2 2 w w h h h p z 1 Or w 2 2 L 8 R w 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 h L 2 L R w once w is known we can determine pressure drop 058 0160 Professor Fred Stern Chapters 6 5 Fall 2006 In general roughness V D w w i Theorem 8 w V 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 the solution for f using log log as per Moody diagrams that is discussed later 2 Differential Analysis Continuity V 0 058 0160 Professor Fred Stern Chapters 6 6 Fall 2006 Use cylindrical coordinates r z where z replaces x in previous CV analysis z 1 1 r r 0 r r r z where V r er e z e z Assume 0 i e no swirl and fully developed z flow 0 which shows r constant 0 since z r R 0 V z e z u r e z Momentum DV p z 2 V Dt only z equation u V u p z 2 u z t 1 u 0 p z r z 43 1r42 r 43 4 r 1 42 4 f z f r both terms must be constant 058 0160 Professor Fred Stern Chapters 6 7 Fall 2006 r dp u A ln r B 4 dz 2 u r 0 finite u r 0 0 u r p p z A 0 B r R d p 4 dz 2 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 R u u R p y r R r r R 2 z Q u r 2 r dr 0 4 R dp 4 dz 2 R d p 1 Vmax R 2 8 dz 2 058 0160 Professor Fred Stern Vave Chapters 6 8 Fall 2006 2 Q 1 R d p V max 2 2 8 dz R Substituting V Vave f 8 w V 2 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 058 0160 Professor Fred Stern Chapters 6 9 Fall 2006 Poiseuille P C Re 16 0 for pipe flow f Compare with previous solution for flow between parallel plates with p x y u u 1 h 2 4 q hu 3 3 max max 2h 3 h p 2 2 u max x p x q h 2 v p u 2h 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 058 0160 Professor Fred Stern Chapters 6 10 Fall 2006 Exact laminar solutions are available for any arbitrary cross section for laminar steady fully developed duct flow 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 x Related umax u 1 Poisson equation u y 0 Dirichlet boundary condition 2 p 058 0160 Professor Fred Stern Fall 2006 Chapters 6 11 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 2006 Chapters 6 12 058 0160 Professor Fred Stern Fall 2006 Chapters 6 13 058 0160 Professor Fred Stern Fall 2006 Chapters 6 14 058 0160 Professor Fred Stern Fall 2006 Chapters 6 15 Stability and Transition Outline 1 Definition 2 Basic steps for stability analysis 3 Examples of instability a Transition from laminar to laminar flow express the disturbance as a superposition basic possible modes b Transition from laminar to turbulent flow express disturbance as Tollmien Schlichting wave Stability Can a physical state withstand a disturbance and still return to its original state If a system is disturbed will the disturbance 1 Gradually die down stable system Stability implies that there exists no mode of disturbance for which it is unstable 2 Grow in amplitude and depart from the initial state unstable system even if there is only one special mode of disturbance with respect to which it is unstable 3 The locus which separates the two classes of system is called the marginal state or neutral state 058 0160 Professor Fred Stern Fall 2006 Chapters 6 16 Basic steps for linear stability analysis 1 Select a basic solution of the flow 2 Add a disturbance 3 Find the disturbance equation 4 Linearize assumption perturbation is infinitesimally small 5 Simplify 6 Solve for the eigenvalues 7 Interpret the stability conditions and draw a chart showing the neutral curves and the growth and decay rates Examples of instability 1 Laminar to laminar transition i Thermal Instability Bernard problem Finger or oscillatory instability ii Centrifugal instability between coaxial flow cylinders …
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