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UI ME 5160 - Chapter 6 - Viscous Flow in Ducts

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058:0160 Chapters 6 Professor Fred Stern Fall 2006 1 Chapter 6: Viscous Flow in Ducts Entrance, developing, and fully developed flow Le = f (D, V, ρ, µ) (Re)fDLtheoremei=→Π Laminar Flow: Recrit ~ 2000 Re < Recrit laminar Re > Recrit turbulent Re06./ ≅DLe DDLcrite138~Re06.max= Max Le for laminar flow058:0160 Chapters 6 Professor Fred Stern Fall 2006 2 Turbulent flow: 6/1Re4.4~/DLe (Relatively shorter than for laminar flow) Laminar vs. Turbulent Flow Re Le/D 4000 18 10420 10530 10644 10765 10895 Hagen 1839 noted difference in ∆p=∆p(u) but could not explain two regimes Laminar Turbulent Reynolds 1883 showed that the difference depends on Re = VD/ν Spark photo058:0160 Chapters 6 Professor Fred Stern Fall 2006 3 Laminar pipe flow: 1. CV Analysis Continuity: .021constQQdAVCS==→⋅=∫ρρρ aveVVandconstAAtodueVVei ==== .,,..2121ρ Momentum: {4434421321434210)(/sin2)(22222221=−=∆+−∆−=•∑VBVBmLzWLRRLRpppFwxϑγππτπ058:0160 Chapters 6 Professor Fred Stern Fall 2006 4 0222=∆+−∆zRRLRpwγππτπ RLzpwτγ2=∆+∆ RLzphhhw82)/(21τγ=+∆=−=∆ Or )(222zpdxdRdxdhRLhRwγγγτ+−=−=∆= )(2zpdxdrγτ+−= i.e. shear stress varies linearly in r across pipe for either laminar or turbulent flow Energy: LhzVgpzVgp+++=++2222111122αγαγ RLhhwLγτ2==∆ ∴ once τw is known, we can determine pressure drop058:0160 Chapters 6 Professor Fred Stern Fall 2006 5 In general, ),,,,(εµρττDVww= roughness Πi Theorem )/,(Re82DffactorfrictionfVDwερτ=== where υVDD=Re gVDLfhhL22==∆ Darcy-Weisbach Equation f (ReD, ε/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: 0=⋅∇V058:0160 Chapters 6 Professor Fred Stern Fall 2006 6 Use cylindrical coordinates (r, θ, z) where z replaces x in previous CV analysis 0)(1)(1=∂∂+∂∂+∂∂zrrrrzrϑϑθϑθ where ^^^zzrreeeVϑϑϑθθ++= Assume θϑ = 0 i.e. no swirl and fully developed flow 0=∂∂zzϑ, which shows rϑ = constant = 0 since )(Rrϑ =0 ^^)(z zzerueV ==∴ϑ Momentum: VzpDtVD2)( ∇++−∇=µγρ only z equation: uzpzuVtu2)( ∇++∂∂−=⎥⎦⎤⎢⎣⎡∇⋅+∂∂µγρ 443442143421)()(1)(0rfzfrurrrzpz⎟⎠⎞⎜⎝⎛∂∂∂∂++∂∂−=µγ ∴both terms must be constant058:0160 Chapters 6 Professor Fred Stern Fall 2006 7 BrAdzpdru ++= ln4^2µ zppγ+=^ finite Æ A = 0 )0( =ru u (r = 0) = 0 Æ dzpdRB^42µ−= dzpdRrru^4)(22µ−= dzpdRuu^4)0(2maxµ−== zpRruyuzprruruzRrRrwr∂∂−=∂∂−=∂∂−=∂∂=∂∂=⎥⎦⎤⎢⎣⎡∂∂+∂∂===^^22µµτµϑµτ As per CV analysis y=R-r, 2max402182)(^RVdzpdRdrrruQRπµππ=−==∫058:0160 Chapters 6 Professor Fred Stern Fall 2006 8 dzpdRVRQVave^8212max2µπ−=== Substituting V = Vave 28Vfwρτ= DVRVRVRaveavewµµµτ84822==−×−= DDVfRe6464==ρµ VgDLVgVDLDVgVDLfhhL∝=××===∆222322642ρµρµ for Vpz∝∆→=∆ 0 or DwfVCRe16212==ρτ 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 Chapters 6 Professor Fred Stern Fall 2006 9 16Re#0===fCPPoiseuille for pipe flow Compare with previous solution for flow between parallel plates with . ^xp ⎟⎟⎠⎞⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛−=2max1hyuu xphu^22maxµ−= ⎟⎠⎞⎜⎝⎛−==xphhuq^32343maxµ max23232^uphhqvx=⎟⎠⎞⎜⎝⎛−==µ hVwµτ3= hVhf2Re4824==ρµ hfC2Re12= Po =12 Same as pipe other than constants!058:0160 Chapters 6 Professor Fred Stern Fall 2006 10 Exact laminar solutions are available for any “arbitrary” cross section for laminar steady fully developed duct flow BVP 0)()(00^=++−==huuupuZZYYxxµ hyy/*= hzz /*= Uuu /*= )(^2xphU −=µ Re only enters through stability and transition Related umax12−=∇ u Poisson equation u(y) = 0 Dirichlet boundary condition058:0160 Chapters 6 Professor Fred Stern Fall 2006 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 Chapters 6 Professor Fred Stern Fall 2006 12 058:0160 Chapters 6 Professor Fred Stern Fall 2006 12058:0160 Chapters 6 Professor Fred Stern Fall 2006 13 058:0160 Chapters 6 Professor Fred Stern Fall 2006 13058:0160 Chapters 6 Professor Fred Stern Fall 2006 14 058:0160 Chapters 6 Professor Fred Stern Fall 2006 14058:0160 Chapters 6 Professor Fred Stern Fall 2006 15 Stability and TransitionOutline: 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 Chapters 6 Professor Fred Stern Fall 2006 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 (Couette flow, Gortler Instability, Couette with


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UI ME 5160 - Chapter 6 - Viscous Flow in Ducts

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