Viscous Flow in Pipes.pdf

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8.3.4 Turbulence ModelingTime-averaging approach: need to solve the closureproblem for the Reynolds stresses, such as −ρuυ′ ′, etc.Different methods for Reynolds stress closure:• Algebraic (zero-equation) models—by modeling the eddyviscosity or mixing length• One-equation models—solve one additional transport equation• Two-equation models—solve two more additional transportequations, such as k-ε model, k-ω model• Reynolds stress transport models—solve the transport equationfor Reynolds stressesLarge eddy simulations (LES)—explicitly solve for thelarge eddies in a calculation and implicitly account for the smalleddies by using a subgrid-scale model (SGS model).Direct numerical simulations (DNS)—directly solve thetransient 3-D Navier-Stokes equations with very fine grids andtime steps
8.3.5 Chaos and TurbulenceChaos theory involves the behavior of nonlineardynamical systems and their response to initial andboundary conditions. The flow of viscous fluid,governed by the complex nonlinear Navier-Stokesequations, is such a system.The chaotic behaviorassociated with thelogistic equation withincrease of r:Xnext = rX(1 − X)r=3(from Chaos, by James Gleick)
Page 1 and 2: Chapter 8Viscous Flow in Pipes
Page 3 and 4: 8.1.1 Laminar or Turbulent Flow•
Page 5 and 6: 8.1.2 Entrance Region and FullyDeve
Page 7 and 8: 8.1.3 Pressure and Shear Stress•
Page 9 and 10: 2. Increased Wall Shear Stress:⎡
Page 11 and 12: Nature of flow• The nature of the
Page 13 and 14: 8.2.1 From F=ma Applied Directly to
Page 15 and 16: • The above analysis is valid for
Page 17 and 18: • Average velocityVQ2 2π RVcVcΔ
Page 19 and 20: 8.2.2 From the Navier-Stokes Equati
Page 21 and 22: • It is advantageous to describe
Page 23 and 24: 8.3 Fully developed turbulent flow
Page 25 and 26: 8.3.2 Turbulent shear stress• Tur
Page 27 and 28: Time-averaged Navier-Stokes Equatio
Page 29 and 30: Total stress• Total stressduτ =
Page 31 and 32: 8.3.3 Turbulent Velocity Profile•
Page 33: Turbulent Velocity Profile• Empir
Page 37 and 38: About the difficulty to understand
Page 39 and 40: Pressure drop• For turbulent flow
Page 41 and 42: Friction coefficient• For laminar
Page 43 and 44: • Energy Equation2 2p1 V1 p2 V2+
Page 45 and 46: 8.4.2 Minor Losses• Major losses
Page 47 and 48: Loss coefficients for minor lossFor
Page 49 and 50: Pressure loss• Flow patter and pr
Page 51 and 52: Loss coefficient• Loss coefficien
Page 53 and 54: Loss coefficient• Loss coefficien
Page 55 and 56: Loss coefficient
Page 57 and 58: 8.4.3 Noncircular Conduits• For n

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