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Velocity Distributions in Turbulent Flow

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(1)

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

(2)

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

(3)

Velocity profiles

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

(4)

Time-smoothed equations of change for incompressible fluids

• The actual velocity in a point

𝑣𝑧 = 𝑣𝑧 + 𝑣𝑧

𝑣𝑧 = 1 𝑡𝑜

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

(5)

Some relationships

𝑣𝑥 = 0 𝑣𝑦 = 0 𝑣𝑧 = 0

𝑣𝑥𝑣𝑥 = 0 𝑣𝑦𝑣𝑦 = 0 𝑣𝑧𝑣𝑧 = 0

𝜕

𝜕𝑥 𝑣𝑧 = 𝜕

𝜕𝑥𝑣𝑧 𝜕

𝜕𝑡𝑣𝑧 = 𝜕

𝜕𝑡 𝑣𝑧

(6)

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

(7)

Time smoothing of the equations of change

• into equation of motion (x-component)

𝜕

𝜕𝑡 𝑣𝑥 + 𝑣𝑥

= − 𝜕

𝜕𝑥 𝑝 + 𝑝

− ൭

𝜕

𝜕𝑥𝜌 𝑣𝑥 + 𝑣𝑥 𝑣𝑥 + 𝑣𝑥 + 𝜕

𝜕𝑦𝜌 𝑣𝑦 + 𝑣𝑦 𝑣𝑥 + 𝑣𝑥 + 𝜕

𝜕𝑧𝜌 𝑣𝑧 + 𝑣𝑧 𝑣𝑥 + 𝑣𝑥 + 𝜇𝛻2 𝑣𝑥 + 𝑣𝑥 + 𝜌𝑔𝑥

(8)

Time smoothing of the equations of change

𝜕

𝜕𝑡 𝑣𝑥

= − 𝜕

𝜕𝑥 𝑝 − 𝜕

𝜕𝑥 𝜌 𝑣𝑥 𝑣𝑥 + 𝜕

𝜕𝑦𝜌 𝑣𝑦 𝑣𝑥 + 𝜕

𝜕𝑧𝜌 𝑣𝑧 𝑣𝑥

𝜕

𝜕𝑥 𝜌 𝑣𝑥𝑣𝑥 + 𝜕

𝜕𝑦𝜌 𝑣𝑦𝑣𝑥 + 𝜕

𝜕𝑧𝜌 𝑣𝑧𝑣𝑥 + 𝜇𝛻2 𝑣𝑥 + 𝜌𝑔𝑥

Momentum transport associated with the turbulent fluctuations

(9)

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

(10)

Turbulent momentum flux tensor

• Definition

𝜏𝑥𝑥(𝑡) = 𝜌𝑣𝑥𝑣𝑥 𝜏𝑥𝑦(𝑡) = 𝜌𝑣𝑥𝑣𝑦 𝜏𝑥𝑧(𝑡) = 𝜌𝑣𝑥𝑣𝑧

• Then

𝛻 ∙ 𝑣 = 0 𝑎𝑛𝑑 𝛻 ∙ 𝑣 = 0

𝜕

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

(11)

The time-smoothed velocity profile near a wall

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

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

(12)

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

(13)

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

(14)

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

(15)

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 + 𝜕𝑣𝑥

𝜕𝑦 𝑦=0

𝑦 + 1

2!𝜕2𝑣𝑥

𝜕𝑦2

𝑦=0

𝑦2 + 1

3!𝜕3𝑣𝑥

𝜕𝑦3

𝑦=0

𝑦3 + ⋯

(16)

Coefficient

by no slip condition

(17)

Taylor-series development in the viscous sub-layer

• Assuming the next term and

determining the coefficient experimentally

(18)

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)

(19)

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

(20)

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)

(21)

Turbulent flow in ducts

• Experimental measurements

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

(22)

Average velocity in a circular tube

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

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