Velocity Distributions in Turbulent Flow

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Chapter 5

• 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

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Comparisons of laminar and turbulent flows

Laminar Re < 2100

Turbulent 1e4<Re <1e5

𝑣_𝑧

𝑣_{𝑧,𝑚𝑎𝑥} = 1 − 𝑟 𝑅

2 𝑣_𝑧

𝑣_{𝑧,𝑚𝑎𝑥} = 1 − 𝑟 𝑅

1/7

𝑣_𝑧

𝑣_{𝑧,𝑚𝑎𝑥} = 1 2

𝑣_𝑧

𝑣_{𝑧,𝑚𝑎𝑥} = 4 5

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Velocity profiles

"Transport Phenomena" 2nd ed., R.B. Bird, W.E. Stewart, E.N. Lightfoot

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Time-smoothed equations of change for incompressible fluids

• The actual velocity in a point

𝑣_𝑧 = 𝑣_𝑧 + 𝑣_𝑧^′

𝑣_𝑧 = 1 𝑡_𝑜 න

𝑡−1 2𝑡^𝑜 𝑡+1

2𝑡_𝑜

𝑣_𝑧 𝑠 𝑑𝑠

• Local velocity as a function of time.

• a) Time independent

• b) Time dependent

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Some relationships

𝑣_𝑥^′ = 0 𝑣_𝑦^′ = 0 𝑣_𝑧^′ = 0

𝑣_𝑥𝑣_𝑥^′ = 0 𝑣_𝑦𝑣_𝑦^′ = 0 𝑣_𝑧𝑣_𝑧^′ = 0

𝜕

𝜕𝑥 𝑣_𝑧 = 𝜕

𝜕𝑥𝑣_𝑧 𝜕

𝜕𝑡𝑣_𝑧 = 𝜕

𝜕𝑡 𝑣_𝑧

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Time smoothing of the equations of change

• Introducing

• into equations of continuity

𝑣_𝑥 = 𝑣_𝑥 + 𝑣_𝑥^′

𝑣_𝑧 = 𝑣_𝑧 + 𝑣_𝑧^′ 𝑝 = 𝑝 + 𝑝^′ 𝑣_𝑦 = 𝑣_𝑦 + 𝑣_𝑦^′

𝜕

𝜕𝑥 𝑣_𝑥 + 𝑣_𝑥^′ + 𝜕

𝜕𝑥 𝑣_𝑦 + 𝑣_𝑦^′ + 𝜕

𝜕𝑥 𝑣_𝑧 + 𝑣_𝑧^′ = 0

• Time-smoothed equation of continuity

𝜕

𝜕𝑥 𝑣_𝑥 + 𝜕

𝜕𝑥 𝑣_𝑦 + 𝜕

𝜕𝑥 𝑣_𝑧 = 0

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• into equation of motion (x-component)

𝜕

𝜕𝑡 𝑣_𝑥 + 𝑣_𝑥^′

= − 𝜕

𝜕𝑥 𝑝 + 𝑝^′

− ൭

൱

𝜕

𝜕𝑥𝜌 𝑣_𝑥 + 𝑣_𝑥^′ 𝑣_𝑥 + 𝑣_𝑥^′ + 𝜕

𝜕𝑦𝜌 𝑣_𝑦 + 𝑣_𝑦^′ 𝑣_𝑥 + 𝑣_𝑥^′ + 𝜕

𝜕𝑧𝜌 𝑣_𝑧 + 𝑣_𝑧^′ 𝑣_𝑥 + 𝑣_𝑥^′ + 𝜇𝛻² 𝑣_𝑥 + 𝑣_𝑥^′ + 𝜌𝑔_𝑥

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𝜕

𝜕𝑡 𝑣_𝑥

= − 𝜕

𝜕𝑥 𝑝 − 𝜕

𝜕𝑥 𝜌 𝑣_𝑥 𝑣_𝑥 + 𝜕

𝜕𝑦𝜌 𝑣_𝑦 𝑣_𝑥 + 𝜕

𝜕𝑧𝜌 𝑣_𝑧 𝑣_𝑥

− 𝜕

𝜕𝑥 𝜌 𝑣_𝑥^′𝑣_𝑥^′ + 𝜕

𝜕𝑦𝜌 𝑣_𝑦^′𝑣_𝑥^′ + 𝜕

𝜕𝑧𝜌 𝑣_𝑧^′𝑣_𝑥^′ + 𝜇𝛻² 𝑣_𝑥 + 𝜌𝑔_𝑥

Momentum transport associated with the turbulent fluctuations

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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

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Turbulent momentum flux tensor

• Definition

𝜏_𝑥𝑥^(𝑡) = 𝜌𝑣_𝑥^′𝑣_𝑥^′ 𝜏_𝑥𝑦^(𝑡) = 𝜌𝑣_𝑥^′𝑣_𝑦^′ 𝜏_𝑥𝑧^(𝑡) = 𝜌𝑣_𝑥^′𝑣_𝑧^′

• Then

𝛻 ∙ 𝑣 = 0 𝑎𝑛𝑑 𝛻 ∙ 𝑣^′ = 0

𝜕

𝜕𝑡𝜌𝑣 = −𝛻𝑝 − 𝛻 ∙ 𝜌𝑣𝑣 − 𝛻 ∙ 𝜏^(𝑣) + 𝜏^(𝑡) + 𝜌𝑔

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The time-smoothed velocity profile near a wall

(1) Viscous sub-layer (2) Buffer layer

(3) Inertial sub-layer (4) Main turbulent stream

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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

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The logarithmic velocity profiles in the inertial sub-layer

• Velocity gradient (does not depend on viscosity)

where 𝜏_𝑜 is the shear stress acting on the wall y is the distance from the wall

• Definition of the friction velocity 𝑣_∗

𝜕𝑣_𝑥

𝜕𝑦 = 1 𝜅

𝜏_𝑜 𝜌

1 𝑦

𝑣_∗ = 𝜏_𝑜 𝜌

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The logarithmic velocity profiles in the inertial sub-layer

• Integrating

• 𝜆^′ is a integration constant. Constant are determined by experiments

𝑣_𝑥 = 𝑣_∗

𝜅 𝑙𝑛 𝑦 + 𝜆^′

𝑣_𝑥

𝑣_∗ = 2.5𝑙𝑛 𝑦𝑣_∗

𝜈 + 5.5 𝑓𝑜𝑟 𝑦𝑣_∗

𝜈 > 30

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Taylor-series development in the viscous sub-layer

• Taylor-series development

• Shear stress for steady flow in a slit of thickness 2B

𝑣_𝑥 𝑦 = 𝑣_𝑥 0 + 𝜕𝑣_𝑥ቤ

𝜕𝑦 𝑦=0

𝑦 + 1

2!𝜕²𝑣_𝑥อ

𝜕𝑦²

𝑦=0

𝑦² + 1

3!𝜕³𝑣_𝑥อ

𝜕𝑦³

𝑦=0

𝑦³ + ⋯

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Coefficient

by no slip condition

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Taylor-series development in the viscous sub-layer

• Assuming the next term and

determining the coefficient experimentally

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Empirical expressions for the turbulent momentum flux

• The Eddy viscosity of Boussinesq. In analogy with Newton’s law of viscosity

𝜏_𝑦𝑥^(𝑡) = −𝜇^(𝑡) 𝑑𝑣_𝑥 𝑑𝑦

• 𝜇^(𝑡) is the turbulent viscosity (eddy viscosity: property of flow)

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• 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

κ_o is empirical parameter, dimensionless

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• The mixing length of Prandtl

• For

• Wall turbulence

(y=distance from the wall)

• Free turbulence

(b=width of the mixing zone)

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Turbulent flow in ducts

• Experimental measurements

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Average velocity in a circular tube