Theoretical Fluid Mechanics Turbulent Flow Velocity Profile

By James C.Y. Guo, Professor and P.E. Civil Engineering, U. of Colorado at Denver

1. Concept of Mixing Process in Turbulent Flow Far away from the solid wall, the flow is free, u=U, from the friction. Near the wall, the flow slows down. On the wall, the water particle moves at the same velocity as the wall or u=0. When the solid wall exists, the flow velocity profile is depicted as:

The velocity profile indicates that the horizontal velocity decays towards the wall. Between

the two adjacent layers, flow particles rotate with the eddy up and down. The vertical fluctuation creates momentum exchange or mixing.

2 Turbulent Flow Velocity Distribution Viscosity Model

dydu)( ηµτ += =laminar viscosity + eddy viscosity (15)

There is no generalized guidance as to how to quantify the value of eddy viscosity. Mixing Length Model

A mixing length is exemplified by the size of eddies in the turbulent flow. This phenomenon can be evidenced by the growth of smoke rollers.

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Flow rotates because the solid surface creates the uneven friction and velocity distribution. Therefore, the solid surface is the source of eddies. A smokestack is also considered as the source of eddies because it releases rollers into the wind flow. There are numerous studies on the mixing length for many different turbulent flow cases. It has been concluded that the mixing length is 40% of the distance from the wall or the source of eddies. The value of 0.4 is termed the universal mixing length constant.

k =0.4 a universal constant. kyl =

The turbulent flow friction is no longer directly related to the flow viscosity. Therefore, it is suggested that the momentum principle be considered to model the flow particles rotating up and down between two adjacent layers.

uvAVQAF ∆∆−=∆−== )(ρρτ

22 )())((dydull

dydul

dyduvu ρρρτ −==−=∆∆−= (16)

The sign, “-“, means the direction of shear stress is against the flow.

Let ρτ

=*u and in which k=0.4 (an universal constant) kyl =

Eq 16 becomes

y

dyk

uy

dyk

du *1==

ρτ

To integrate the above equation yields

21 lnln*)5.2(ln* ConstyConstCyuCyk

uu +×=+×=+×= (17)

It takes two data points, (u,y), to determine the values of Const1 and Const2, noted that y is measured from the wall.

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Example: The centerline of a channel flow has a depth of 10 ft. The velocities of flow were measured to be 5.67 fps at 2 feet below the surface and 5.22 fps at 6-ft above the bed. Estimate the velocity at one ft above the bed. Solution: Const1 + Const2 Log (10-2) = 5.67 Const1 + Const2 Log (6) = 5.22 Const1= 3.5 and Const2= 2.5

Discussion At y =0 on the wall, the non-slip condition requires u =0. Obviously, the logarithmic function fairs the non-slip condition. How to explain it? 3. Logarithmic Velocity Profile in Circular Pipe

The relationship between the two coordinate systems is: y = R- r ( 18) Substituting Eq 18 into Eq 17 yields

CrRkuu +−×= )ln(* (19)

At r = 0 or y=R (the centerline in a circular pipe), Eq 19 depicts u = Um or

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CRkuUm +−×= )0ln(* (20)

Taking the difference between Eq’s 19 and 20 yields the Law of Velocity Defect

rR

Ru

uUm

−=

− ln5.2*

(21)

Re-arranging Eq 21 yields

rRRuU

rRRuUu mm −

−=−

−= log76.5ln5.2 ** (22)

Eq 22 has two unknown: Um and u*. From the laboratory data, the semi-theoretical analyses resulted in two empirical formulas for Um and u*. Both can be related to the friction factor in the pipe as:

)326.11( fVUm += (23)

8*fVu ==

ρτ (24)

4. Discussion on Velocity Profiles

For a laminar flow in a circular pipe: 2)(1Rr

Uu

m

−= ---- parabolic

21

=mU

V and 8*fVu ==

ρτ

For a turbulent flow in a circular pipe:rR

RUu

Uu

mm −−= ln5.21 * ---- logarithmic

fU

V

m 326.111

+= and

8*fVu ==

ρτ

Far away from a solid wall (the free-stream zone), the flow is free from the friction. Near the wall (the boundary layer zone), the flow is subject to the friction from the wall. Therefore, the logarithmic profile was derived for the boundary layer flow. Very close to the wall, the viscous

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force dominates the flow movement in this sub-layer flow zone. As a result, the sub-layer flow is laminar. A linear velocity within the sub-layer thickness can be derived using the shear velocity as:

yu

yuu υµ

ρρτ

===12

*

υ

yuuu *

*

= the Law of the Wall which is a linear velocity distribution

In practice, we found that the sub-layer thickness can have

70*

*

≤=υ

yuuu

The thickness of sub-layer, *

70u

y υδ ==

When the water flow enters a pipe, the velocity profile in front of the entrance is nearly uniform (free-stream flow). Immediately downstream of the entrance, the centerline flow can still move at a uniform velocity, but near the wall, the boundary layer flow begins to be developed. The thickness of the boundary layer increases downstream. After a distance, the boundary layers from both walls meet at the centerline and the flow becomes established and unchanged. Such a distance is termed Entrance Length as:

DRL ee 058.0= when Re<2000 laminar flow.

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4.1 A 10-ft pipe carries a water flow of 1000 cfs. The friction factor for this case is 0.03. Analyze this flow.

(1) Cross section area A = 78.5 sq ft (2) Flow velocity V = 12.73 fps (3) Centerline velocity 66.1581.0)33.11( ==+= VVfUm fps (V/Um = 0.81)

(4) Shear velocity 78.08

* ===fVu

ρτ fps

(5) Shear street 17.1=τ lb/ft2

(6) )5

5ln(95.166.15r

u−

−=

(7) For a pipe length of 100 feet, the energy loss is

74.02.32*2

73.121010003.0

2

==fH ft and 46176==∆ fQHE γ lb-ft/sec

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Footnote: 8.36731001014.317.1 =×××=×== DLAF πτττ lb’s and Power = 46767=VFτ lb-ft/sec (loss calculated by friction force) (8) Sub layer thickness

70101

78.05 =

××

−

δ δ=0.0009 ft = 0.011 inch

(9) The Law of the Wall -- linear velocity distribution within the sub-layer thickness

510178.0

78.0 −××

=yu for y<δ , or (u, y) has a linear relationship.

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