57:020 Fluids Mechanics Fall2013 1 Review for Exam3 12. 11. 2013 Hyunse Yoon, Ph.D. Assistant Research Scientist IIHR-Hydroscience & Engineering University of Iowa
57:020 Fluids Mechanics Fall2013 2 Chapter 8 Flow in Conduits
57:020 Fluids Mechanics Fall2013 3 Pipe Flow: Laminar vs. Turbulent Reynolds number RRRR = ρρρρρρ μμ RRRR < RRee cccccccc 2,000 RRee cccccccc < RRRR < RRee tttttttttt RRRR > RRee tttttttttt 4,000
57:020 Fluids Mechanics Fall2013 4 Entrance Region and Fully Developed Entrance Length, LL ee : o Laminar flow: LLLL DD = 0.06RRRR (LL ee,mmmmmm = 0.06RRee cccccccc 138DD) o Turbulent flow: LLLL DD = 4.4RRee 1 6 (20DD < LL ee < 30DD for 10 4 < RRRR < 10 5 )
57:020 Fluids Mechanics Fall2013 5 Pressure Drop and Shear Stress Pressure drop, Δpp = pp 1 pp 2, is needed to overcome viscous shear stress. The nature of shear stress is strongly dependent of whether the flow is laminar or turbulent. Friction factor (or Darcy friction factor) ff = 8ττ ww ρρvv 2
57:020 Fluids Mechanics Fall2013 6 Fully-developed Laminar Flow Exact solution, uu rr = VV cc 1 rr RR Wall sear stress ττ ww = μμ dddd ddrr rr=rr Where, VV = QQ AA Friction factor, ff = 8ττ ww ρρvv 2 = 64 2 = 8μμμμ DD ρρρρρρ μμ = 64 RRRR
57:020 Fluids Mechanics Fall2013 7 Fully-developed Turbulent Flow ττ ww = ff DD, VV, μμ, ρρ, εε kk rr = 6 3 = 3 Π ss ff = 8ττ ww ρρρρρρ ; RRRR = ρρvv2 μμ ; Roughness = εε DD ff = φφ RRRR, εε DD
Moody Chart 57:020 Fluids Mechanics Fall2013 8
57:020 Fluids Mechanics Fall2013 9 Moody Chart Contd. Colebrook equation εε DD = 2 log 1ff 3.7 + 2.51 RRRR ff Haaland equation 1 εε = 1.8 log ff DD 3.7 1.1 + 6.9 RRRR
57:020 Fluids Mechanics Fall2013 10 Major Loss and Minor Losses Energy equation pp 1 γγ + αα 1 VV 1 2 2gg + zz 1 + h pp = pp 2 γγ + αα 2 VV 2 2 2gg + zz 2 + h tt + h LL Head loss: h LL = h ff + h mm Major loss due to friction: VV 2 h ff = ff LL DD 2gg (Darcy Weisbach equation) Minor loss due to pipe system components VV 2 h mm = KK LL (KK 2gg LL : Loss coefficient)
57:020 Fluids Mechanics Fall2013 11 Pipe Flow Examples Type I: Determine head loss h LL (or pressure drop) Type II: Determine flow rate QQ (or the average velocity VV) Type III: Determine pipe diameter DD For types II and III, iteration process is needed
57:020 Fluids Mechanics Fall2013 12 Type I Problem Typically, VV and DD are given RRRR and εε DD ff = φφ ρρρρρρ μμ, εε DD h LL = ff LL DD VV 2 2gg
57:020 Fluids Mechanics Fall2013 13 Type II Problem QQ (thus VV) is unknown RRRR? Solve energy equation for VV = function ff, for example Or h pp = VV2 2gg + ff LL DD VV 2 2gg + KK LL VV 2 2gg VV = 2ggh pp 1 + ff LL DD + KK LL Guess ff VV RRRR ff nnnnnn ; Repeat until ff is converged
57:020 Fluids Mechanics Fall2013 14 Type III Problem DD is unknown RRRR and εε DD? Solve energy equation for DD = function(ff), for example, 1 DD = 8LLQQ2 5 1 ff ππ 2 5 ggh ff Guess ff DD RRRR and εε DD ff nnnnnn ; Repeat until ff is converged
Chapter 9 Flow over Immersed Bodies 57:020 Fluids Mechanics Fall2013 15
57:020 Fluids Mechanics Fall2013 16 Fluid Flow Categories Internal flow: Bounded by walls or fluid interfaces Ex) Duct/pipe (Ch. 8), turbo machinery, open channel/river External flow: Unbounded or partially bounded. Viscous and inviscid flow regions Ex) Flow around vehicles and structures Boundary layer flow: High Reynolds number flow around streamlined bodies without flow separation Bluff body flow: Flow around bluff bodies with flow separation Free shear flow: Absence of walls Ex) Jets, wakes, mixing layers
57:020 Fluids Mechanics Fall2013 17 Basic Considerations Drag, DD: Resultant force in the direction of the upstream velocity = DD 1 = 1 2 ρρvv2 1 AA 2 ρρvv2 AA CC DD Drag coefficient pp pp nn ii dddd SS CC DDDD = Pressure drag (or Form drag) tt l 1 CC ff CC DDDD Streamlined body tt l ~1 CC DDDD CC ff Bluff body where, tt is the thickness and l the length of the body Lift, LL: Resultant force normal to the upstream velocity = LL 1 = 1 2 ρρvv2 1 AA 2 ρρvv2 AA CC LL Lift coefficient pp pp nn jj dddd SS + ττ ww tt ii dddd SS CC ff =Friction drag
57:020 Fluids Mechanics Fall2013 18 Boundary Layer Boundary layer theory assumes that viscous effects are confined to a thin layer, δδ There is a dominant flow direction (e.g., xx) such that uu~uu and vv uu Gradients across δδ are very large in order to satisfy the no-slip condition; thus, + = 0 uu + vv = + νν 2 uu yy 2
57:020 Fluids Mechanics Fall2013 19 Laminar boundary layer Prandtl/Blasius solution uu = UU f ηη vv = 1 2 ννuu xx ηηf f ττ ww = 0.332UU 3 2 ρρρρ xx δδ xx = 5 ; cc ff = 0.664 ; CC ff = 1.328 RRee xx RRee xx RRee LL
57:020 Fluids Mechanics Fall2013 20 Turbulent boundary layer uu UU yy δδ 1 7 one seventh power law cc ff 0.02RRee δδ 1 6 power law fit δδ = 0.16 1 xx RRee7 xx ; cc ff = 0.027 1 RRee7 xx ; CC ff = 0.031 1 RRee7 LL Valid for a fully turbulent flow over a smooth flat plate from the leading edge. Better results for sufficiently large RRee LL
57:020 Fluids Mechanics Fall2013 21 Turbulent boundary layer Contd. Alternate forms by using an experimentally determined shear stress formula: ττ ww = 0.0225ρρUU 2 νν UUUU 1 4 δδ = 0.37RRee 1 5 xx xx ; cc ff = 0.058 1 RRee5 xx ; CC ff = 0.074 1 RRee5 LL Valid only in the range of the experimental data; RRee LL = 5 10 5 ~10 7 for smooth flat plate
57:020 Fluids Mechanics Fall2013 22 Turbulent boundary layer Contd. Other empirical formulas for smooth flat plates ( tripped by some roughness or leading edge disturbance to make the flow turbulent from the leading edge): δδ LL = cc ff 0.98 log RRee LL 0.732 cc ff = 2 log RRee xx 0.65 2.3 CC ff = 0.455 log 10 RRee LL 2.58
57:020 Fluids Mechanics Fall2013 23 Turbulent boundary layer Contd. Composite formulas (for flows initially laminar and subsequently turbulent with RRee tt = 5 10 5 ): CC ff = 0.031 1 7 RRee LL CC ff = 0.074 1 5 RRee LL 1440 RRee LL 1700 RRee LL CC ff = 0.455 log 10 RRee 2.58 1700 LL RRee LL
57:020 Fluids Mechanics Fall2013 24 Turbulent boundary layer Contd.
57:020 Fluids Mechanics Fall2013 25 Bluff Body Drag In general, DD = ff VV, LL, ρρ, μμ, cc, tt, εε, Drag coefficient: CC DD = DD 1 2 ρρvv2 AA = φφ AAAA, tt LL, RRRR, cc VV, εε LL, For bluff bodies experimental data are used to determine CC DD
57:020 Fluids Mechanics Fall2013 26 Shape dependence The blunter the body, the larger the drag coefficient The amount of streamlining can have a considerable effect
57:020 Fluids Mechanics Fall2013 27 Reynolds number dependence Very low RRRR flow (RRRR < 1) Inertia effects are negligible (creeping flow) CC DD ~ RRee 1 Streamlining can actually increase the drag (an increase in the area and shear force) Moderate RRRR flow (10 3 < RRRR < 10 5 ) For streamlined bodies, CC DD ~ RRee 1 2 For blunt bodies, CC DD ~ constant Very large RRRR flow (turbulent boundary layer) For streamlined bodies, CC DD increases For relatively blunt bodies, CC DD decreases when the flow becomes turbulent (10 5 < RRRR < 10 6 ) For extremely blunt bodies, CC DD ~ constant
57:020 Fluids Mechanics Fall2013 28 Separation Fluid stream detaches from a surface of a body at sufficiently high velocities. Only appears in viscous flows. Inside a separation region: low-pressure, existence of recirculating /backflows; viscous and rotational effects are the most significant
57:020 Fluids Mechanics Fall2013 29 Surface roughness For streamlined bodies, the drag increases with increasing surface roughness For extremely blunt bodies, the drag is independent of the surface roughness For blunt bodies, an increase in surface roughness can actually cause a decrease in the drag.
57:020 Fluids Mechanics Fall2013 30 Lift CC LL = LL 1 2 ρρuu2 AA LL = CC LL 1 2 ρρuu2 AA
57:020 Fluids Mechanics Fall2013 31 Magnus Effect Lift generation by spinning Breaking the symmetry causes a lift
57:020 Fluids Mechanics Fall2013 32 Minimum Flight Velocity Total weight of an aircraft should be equal to the lift Thus, WW = FF LL = 1 2 CC 2 LL,mmmmmmρρVV mmmmmm AA VV mmmmmm = 2WW ρρcc LL,mmmmmm AA