External Flow

Outline • Review head loss in internal flows • Definition of lift and drag forces Ex ternal Flow – Result of pressure and shear stress forces • • • • • External flows over a flat plate Laminar and turbulent regions Flows over cylinders and spheres Pressure distributions and lift Lift coefficients for an airfoil 2 Review Energy Equation Review Head Loss • Energy equation between inlet (1) and outlet (2) p V2 p V2 z2 + 2 + 2 = z1 + 1 + 1 + hs − hL γ 2g γ 2g • During the last two weeks we learned to compute the head loss • Previous applications allowed us to compute the head loss from other data in this equation • Call this the design head loss ⎛ l hL = ⎜ f + ⎝ D ⎞V 2 ∑ K L ⎟⎠ 2 g – Allows us to determine the head loss from empirical relations among Re, f, and ε/D • Also have various KL for minor losses • Once we know the head loss, we can find other terms in the energy equation – Call this the measured head loss • We can compute it, but we have no way of knowing its cause 3 4 Review Energy/Head Loss Review Energy/Head Loss II • Combine energy equation with f and KL p V2 p V2 z2 + 2 + 2 = z1 + 1 + 1 + hs − hL γ 2g γ 2g 2 2 2 p V p V ⎛ l ⎞V z2 + 2 + 2 = z1 + 1 + 1 + hs − ⎜ f + KL ⎟ γ 2g γ 2g ⎝ D ⎠ 2g ∑ • Top equation: compute hL if you know all other terms in the equation • Bottom equation: find flow properties accounting for calculated head loss 5 z2 + p2 V22 p V2 ⎛ l + = z1 + 1 + 1 + hs − ⎜ f + γ 2g γ 2g ⎝ D ∑ 2 ⎞V KL ⎟ ⎠ 2g • Simplest case: z1 – z2 =V1 – V2 = hs = 0 2 2 ⎛ l ⎞ ρV ⎛ l ⎞V p1 − p2 = γ⎜ f + KL ⎟ KL ⎟ =⎜ f + ⎠ 2 ⎝ D ⎠ 2g ⎝ D • Often have V2 = V1 so head loss is ∑ ∑ ⎛ l p1 + γz1 − ( p2 + γz2 ) = ⎜ f + ⎝ D ⎞ ρV KL ⎟ ⎠ 2 ∑ 2 6 1 Review Moody Diagram Review Moody Equations • Colebrook equation 1 (turbulent) f • Haaland equation (turbulent) ⎛ε D 2.51 = −2.0 log10 ⎜ + ⎜ 3.7 Re f ⎝ ⎞ ⎟ ⎟ ⎠ ⎛ 6.9 ⎛ ε D ⎞1.11 ⎞ 1 ≈ −1.8 log10 ⎜ +⎜ ⎟ ⎟ ⎜ Re ⎝ 3.7 ⎠ ⎟ f ⎠ ⎝ • Laminar 128μlQ 256μ π 2 V D 4 3 64 64 Δp 4 π D = = πD = = f = 2 ρVD Re 1 l 1 l 2 2 ρ V ρV ρV μ 2D 2D Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and Theodore Okiishi. Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. 7 Review Laminar vs. Turbulent 8 Other Review Items • Not for partially filled pipes • Find minor loss coefficients from various tables and charts • Parabolic laminar velocity profile – Adjustment coefficients for momentum and kinetic energy flows terms • Laminar flows have smooth layers of fluid • Flat turbulent velocity profile – No adjustment terms required • Turbulent flows have fluctuations • Non-circular pipes use Dh = 4A/P 9 Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and Theodore Okiishi. Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Other Review Items II 10 Review Head Loss Problems • Find the pressure drop given fluid data, pipe dimensions, ε, and flow (volume flow, mass flow, or velocity) • Wall shear stress related to pressure drop: Δp = 2τwℓ/R = 4τwℓ/D (for horizontal pipe) • Entry lengths for developing flows le l = 4.4 Re1 6 – laminar e = 0.06 Re turbulent D D – ℓe ≈ 10D for turbulent flows • hL varies as D-4 for laminar flows and D-5 for fully turbulent flows with fixed Q 11 – Get A = πD2/4 & /ρA if not given V – Get V = Q/A or V = m – Find ρ and μ for fluid at given T and P – Compute Re = ρVD/μ and ε/D – Find f from diagram or equation • Laminar f = 64/Re; Colebrook for turbulent – Compute hL = f (ℓ/D) V2/2g 12 2 Review Lead Loss Problems II • Forces due to pressure distribution and shear stress • Net result is a lift force, L, and a drag force, D • Find the diameter for a given head loss given fluid data, ε, and flow (volume flow, mass flow, or velocity) – Find ρ and μ for fluid at given T and P – Guess D; get A = πD2/4 & /ρA if not given V – Get V = Q/A or V = m – Compute Re = ρVD/μ and ε/D – Find f from diagram or equation • Laminar f = 64/Re; Colebrook for turbulent – Compute hL,calculated = f (ℓ/D) V2/2g – Iterate on D until hL,calculated = hL,required External Flow Forces 13 Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and 14 Theodore Okiishi. Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. External Flow Characteristics • Flow over a flat plate is the simplest example – Re = U∞ℓ/ν • U∞ = velocity upstream and far from plate • ℓ = length of plate in direction of flow • Low Reynolds number flow has viscous effects important over wide region • As Re increases region of significant viscous effects narrows 15 Small Reynolds Number Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and 17 Theodore Okiishi. Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. • Shaded area is where viscous effects are important for (a) small, (b) moderate and (c) large Re Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and 16 Theodore Okiishi. Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Moderate Reynolds Number Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and 18 Theodore Okiishi. Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. 3 Large Reynolds Number Flow Over a Sphere • Similar effects of different flow regimes as Reynolds number increases • Drag force due to both pressure drag and friction drag – Flat plate only has friction drag • No simple relation between drag coefficient and Reynolds number – Nature of flow changes as Reynolds number increases Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and 19 Theodore Okiishi. Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Low Reynolds Number Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and 21 Theodore Okiishi. Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Large ReynoldsNumber Number Low Reynolds 20 Moderate Reynolds Number Low Reynolds Number Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and 22 Theodore Okiishi. Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Flat Plate Boundary Layer • Transition to turbulent flow at Rex = U∞x/ν = 2x105 to 5x105 Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and 23 Theodore Okiishi. Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and 24 Theodore Okiishi. Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. 4 Friction and Drag Coefficients Laminar : c f ( x) = D = lb l τ w ( x) ρU ∞2 2 = 0.664 Re x C Df = D ρU ∞2 bl 2 = Flat Plate Drag Coefficients 1.328 Re l 1 U z τ w ( x)dx l = length b = width Re z = ∞ ν l ∫ 0 Length of plate is small enough that entire flow is laminar Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and Theodore Okiishi. Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Re l < Re crit = 5 x105 Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, 25 Donald Young, and Theodore Okiishi. Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Cylinder and Sphere Drag Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and Theodore Okiishi. Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Cylinder and Sphere Drag 27 Cylinder and Sphere Drag Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and Theodore Okiishi. Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. 26 Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and Theodore Okiishi. Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. 28 Surface Roughness Effect 29 Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and Theodore Okiishi. Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. 30 5 Automobile Drag Coefficients Figures 9.28 to 9.30 (pp 536538 give drag coefficients for “objects of interest” Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and Theodore Okiishi. Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. PD = D V = C D ρV 3 AF 2 Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and 31 Theodore Okiishi. Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. 32 Lift • Example: pressure distribution on a car Lift Coefficient, CL • Defined similarly to drag coefficient CL = L ρU ∞2 A 2 b is width c is chord length α is angle of attack Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and 33 Theodore Okiishi. Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Area A ≈ bc = (width)(cord length) Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and 34 Theodore Okiishi. Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. Lift and Drag Coefficients CL and CD vary with angle of attack, a, and aspect ratio A = b2/A Fundamentals of Fluid Mechanics, 5/E by Bruce Munson, Donald Young, and 35 Theodore Okiishi. Copyright © 2005 by John Wiley & Sons, Inc. All rights reserved. 6