Viscous Flow in Pipes.pdf

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Turbulence intensity• Fluctuating partu = u + u′ or u′= u−u1 1′ = ( − ) = ⎡TT ⎣−1= ( Tu − Tu ) = 0T′1 t + T=T∫′ > 0t + T t + T t + T∫ ∫ ∫0 0 0u u u dt udt u dtt0 ⎢ t0 t02 0 2u u dtt0• Turbulence intensity⎤⎥⎦Iu= =u⎡1 t0+ T22u′⎤dt′ ⎢T∫t⎣⎥0⎦u12Well designed wind tunnel: I ≈ 0.01 (may be down to 0.0002)
Time-averaged Navier-Stokes EquationsLet u = u + u′ , υ = υ + υ′ , w= w+w′and then take time average2 2 2⎛∂u ∂u ∂u′ ∂u ∂u′ ∂u ∂u′ ⎞ ∂p ⎛∂ u ∂ u ∂ u ⎞x-dir:ρ⎜ + u + u′ + υ + υ′ + w + w′⎟=− + ρgx+ μ⎜ + +2 2 2 ⎟⎝ ∂t ∂x ∂x ∂y ∂y ∂z ∂z ⎠ ∂x ⎝ ∂x ∂y ∂z⎠2∂u′ ∂u′ ∂u′ ∂u′ ∂u′ υ′ ∂υ′ ∂u′ ∂uw ′ ′ ∂w′But u′ = − u′ ; υ′ = − u′ ; w′ = − u′∂x ∂x ∂x ∂y ∂y ∂y ∂z ∂z ∂z∂u′ ∂υ′ ∂w′and + + = 0 (from continuity equation)∂x ∂y ∂z⎛2 2 2 2∂u ∂u ∂u′ ∂u ∂u′ υ′ ∂u ∂uw ′ ′ ⎞ ∂p ⎛∂ u ∂ u ∂ u ⎞→ ρ + u + + υ + + w + =− + ρgx+ μ + +⎜2 2 2t x x y y z z ⎟⎜ ⎟⎝∂ ∂ ∂ ∂ ∂ ∂ ∂⎠∂x ⎝ ∂x ∂y ∂z⎠Terms with fluctuations moved and merged to the r.h.s.Finally,⎛∂u ∂u ∂u ∂u ⎞ ∂p ∂ ⎛ ∂u 2 ⎞ ∂ ⎛ ∂u ⎞ ∂ ⎛ ∂u⎞ρ⎜ + u + υ + w ⎟=− + ρgx+ ⎜μ − ρu′ ⎟+ ⎜μ − ρu′ υ′ ⎟+ ⎜μ −ρuw′ ′ ⎟⎝ ∂t ∂x ∂y ∂z ⎠ ∂x ∂x⎝ ∂x ⎠ ∂y⎝ ∂y ⎠ ∂z⎝ ∂z⎠⎛∂υ ∂υ ∂υ ∂υ ⎞ ∂p∂ ⎛ ∂υ ⎞ ∂ ⎛ ∂υ 2 ⎞ ∂ ⎛ ∂υ⎞ρ⎜ + u + υ + w ⎟=− + ρgy+ ⎜μ − ρu′ υ′ ⎟+ ⎜μ − ρυ′ ⎟+ ⎜μ −ρυ′ w′⎟⎝ ∂t ∂x ∂y ∂z ⎠ ∂y ∂x⎝ ∂x ⎠ ∂y⎝ ∂y ⎠ ∂z⎝∂z⎠⎛∂w ∂w ∂w ∂w⎞ ∂p ∂ ⎛ ∂w ⎞ ∂ ⎛ ∂w ⎞ ∂ ⎛ ∂w2 ⎞ρ⎜ + u + υ + w ⎟=− + ρgz+ ⎜μ − ρuw ′ ′ ⎟+ ⎜μ − ρυ′ w′ ⎟+ ⎜μ −ρw′⎟⎝ ∂t ∂x ∂y ∂z ⎠ ∂z ∂x⎝ ∂x ⎠ ∂y⎝ ∂y ⎠ ∂z⎝ ∂z⎠
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: 8.3.2 Turbulent shear stress• Tur
Page 29 and 30: Total stress• Total stressduτ =
Page 31 and 32: 8.3.3 Turbulent Velocity Profile•
Page 33 and 34: Turbulent Velocity Profile• Empir
Page 35 and 36: 8.3.5 Chaos and TurbulenceChaos the
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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