Chapter 5
Velocity Distributions in Turbulent Flow
• Comparisons of laminar and turbulent flows
• Time-smoothed equations of change for incompressible fluids
• The time-smoothed velocity profile near a wall
• Turbulent flow in ducts
• Turbulent flow in jets
Comparisons of laminar and turbulent flows
Laminar Re < 2100
Turbulent 1e4<Re <1e5
𝑣𝑧
𝑣𝑧,𝑚𝑎𝑥 = 1 − 𝑟 𝑅
2 𝑣𝑧
𝑣𝑧,𝑚𝑎𝑥 = 1 − 𝑟 𝑅
1/7
𝑣𝑧
𝑣𝑧,𝑚𝑎𝑥 = 1 2
𝑣𝑧
𝑣𝑧,𝑚𝑎𝑥 = 4 5
Velocity profiles
"Transport Phenomena" 2nd ed., R.B. Bird, W.E. Stewart, E.N. Lightfoot
Time-smoothed equations of change for incompressible fluids
• The actual velocity in a point
𝑣𝑧 = 𝑣𝑧 + 𝑣𝑧′
𝑣𝑧 = 1 𝑡𝑜 න
𝑡−1 2𝑡𝑜 𝑡+1
2𝑡𝑜
𝑣𝑧 𝑠 𝑑𝑠
• Local velocity as a function of time.
• a) Time independent
• b) Time dependent
"Transport Phenomena" 2nd ed., R.B. Bird, W.E. Stewart, E.N. Lightfoot
Some relationships
𝑣𝑥′ = 0 𝑣𝑦′ = 0 𝑣𝑧′ = 0
𝑣𝑥𝑣𝑥′ = 0 𝑣𝑦𝑣𝑦′ = 0 𝑣𝑧𝑣𝑧′ = 0
𝜕
𝜕𝑥 𝑣𝑧 = 𝜕
𝜕𝑥𝑣𝑧 𝜕
𝜕𝑡𝑣𝑧 = 𝜕
𝜕𝑡 𝑣𝑧
Time smoothing of the equations of change
• Introducing
• into equations of continuity
𝑣𝑥 = 𝑣𝑥 + 𝑣𝑥′
𝑣𝑧 = 𝑣𝑧 + 𝑣𝑧′ 𝑝 = 𝑝 + 𝑝′ 𝑣𝑦 = 𝑣𝑦 + 𝑣𝑦′
𝜕
𝜕𝑥 𝑣𝑥 + 𝑣𝑥′ + 𝜕
𝜕𝑥 𝑣𝑦 + 𝑣𝑦′ + 𝜕
𝜕𝑥 𝑣𝑧 + 𝑣𝑧′ = 0
• Time-smoothed equation of continuity
𝜕
𝜕𝑥 𝑣𝑥 + 𝜕
𝜕𝑥 𝑣𝑦 + 𝜕
𝜕𝑥 𝑣𝑧 = 0
Time smoothing of the equations of change
• into equation of motion (x-component)
𝜕
𝜕𝑡 𝑣𝑥 + 𝑣𝑥′
= − 𝜕
𝜕𝑥 𝑝 + 𝑝′
− ൭
൱
𝜕
𝜕𝑥𝜌 𝑣𝑥 + 𝑣𝑥′ 𝑣𝑥 + 𝑣𝑥′ + 𝜕
𝜕𝑦𝜌 𝑣𝑦 + 𝑣𝑦′ 𝑣𝑥 + 𝑣𝑥′ + 𝜕
𝜕𝑧𝜌 𝑣𝑧 + 𝑣𝑧′ 𝑣𝑥 + 𝑣𝑥′ + 𝜇𝛻2 𝑣𝑥 + 𝑣𝑥′ + 𝜌𝑔𝑥
Time smoothing of the equations of change
𝜕
𝜕𝑡 𝑣𝑥
= − 𝜕
𝜕𝑥 𝑝 − 𝜕
𝜕𝑥 𝜌 𝑣𝑥 𝑣𝑥 + 𝜕
𝜕𝑦𝜌 𝑣𝑦 𝑣𝑥 + 𝜕
𝜕𝑧𝜌 𝑣𝑧 𝑣𝑥
− 𝜕
𝜕𝑥 𝜌 𝑣𝑥′𝑣𝑥′ + 𝜕
𝜕𝑦𝜌 𝑣𝑦′𝑣𝑥′ + 𝜕
𝜕𝑧𝜌 𝑣𝑧′𝑣𝑥′ + 𝜇𝛻2 𝑣𝑥 + 𝜌𝑔𝑥
Momentum transport associated with the turbulent fluctuations
Remarks
• The equation of continuity is the same, except that:
• In the equation of motion v and p are replaced by:
• In addition, it appears the boxed term associated with the turbulent fluctuations.
𝑣 𝑖𝑠 𝑟𝑒𝑝𝑙𝑎𝑐𝑒𝑑 𝑏𝑦 𝑣
𝑣 𝑏𝑦 𝑣 𝑎𝑛𝑑 𝑝 𝑏𝑦 𝑝
Turbulent momentum flux tensor
• Definition
𝜏𝑥𝑥(𝑡) = 𝜌𝑣𝑥′𝑣𝑥′ 𝜏𝑥𝑦(𝑡) = 𝜌𝑣𝑥′𝑣𝑦′ 𝜏𝑥𝑧(𝑡) = 𝜌𝑣𝑥′𝑣𝑧′
• Then
𝛻 ∙ 𝑣 = 0 𝑎𝑛𝑑 𝛻 ∙ 𝑣′ = 0
𝜕
𝜕𝑡𝜌𝑣 = −𝛻𝑝 − 𝛻 ∙ 𝜌𝑣𝑣 − 𝛻 ∙ 𝜏(𝑣) + 𝜏(𝑡) + 𝜌𝑔
The time-smoothed velocity profile near a wall
(1) Viscous sub-layer (2) Buffer layer
(3) Inertial sub-layer (4) Main turbulent stream
The time-smoothed velocity profile near a wall
• Viscous sub-layer, viscosity plays a key role
• Buffer layer, transition between viscous and inertial sub-layers
• Inertial sub-layer, viscosity plays a minor role
• Main turbulent stream, time-smoothed velocity
distribution is nearly flat and viscosity is unimportant
The logarithmic velocity profiles in the inertial sub-layer
• Velocity gradient (does not depend on viscosity)
where 𝜏𝑜 is the shear stress acting on the wall y is the distance from the wall
• Definition of the friction velocity 𝑣∗
𝜕𝑣𝑥
𝜕𝑦 = 1 𝜅
𝜏𝑜 𝜌
1 𝑦
𝑣∗ = 𝜏𝑜 𝜌
The logarithmic velocity profiles in the inertial sub-layer
• Integrating
• 𝜆′ is a integration constant. Constant are determined by experiments
𝑣𝑥 = 𝑣∗
𝜅 𝑙𝑛 𝑦 + 𝜆′
𝑣𝑥
𝑣∗ = 2.5𝑙𝑛 𝑦𝑣∗
𝜈 + 5.5 𝑓𝑜𝑟 𝑦𝑣∗
𝜈 > 30
Taylor-series development in the viscous sub-layer
• Taylor-series development
• Shear stress for steady flow in a slit of thickness 2B
𝑣𝑥 𝑦 = 𝑣𝑥 0 + 𝜕𝑣𝑥ቤ
𝜕𝑦 𝑦=0
𝑦 + 1
2!𝜕2𝑣𝑥อ
𝜕𝑦2
𝑦=0
𝑦2 + 1
3!𝜕3𝑣𝑥อ
𝜕𝑦3
𝑦=0
𝑦3 + ⋯
Coefficient
by no slip condition
Taylor-series development in the viscous sub-layer
• Assuming the next term and
determining the coefficient experimentally
Empirical expressions for the turbulent momentum flux
• The Eddy viscosity of Boussinesq. In analogy with Newton’s law of viscosity
𝜏𝑦𝑥(𝑡) = −𝜇(𝑡) 𝑑𝑣𝑥 𝑑𝑦
• 𝜇(𝑡) is the turbulent viscosity (eddy viscosity: property of flow)
Empirical expressions for the turbulent momentum flux
• Expressions of eddy viscosity for some special cases:
• For wall turbulence, valid only very near the wall
• For free turbulence
where b is width of mixing zone
κo is empirical parameter, dimensionless
Empirical expressions for the turbulent momentum flux
• The mixing length of Prandtl
• For
• Wall turbulence
(y=distance from the wall)
• Free turbulence
(b=width of the mixing zone)
Turbulent flow in ducts
• Experimental measurements
"Transport Phenomena" 2nd ed., R.B. Bird, W.E. Stewart, E.N. Lightfoot
Average velocity in a circular tube
"Transport Phenomena" 2nd ed., R.B. Bird, W.E. Stewart, E.N. Lightfoot