Theoretical Fluid Mechanics Turbulent Flow Velocity Profile By James C.Y. Guo, Professor and P.E. Civil Engineering, U. of Colorado at Denver

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1 Theoretical Flid Mechanics Trblent Flow Velocit Proile B Jaes C.Y. Go, Proessor and P.E. Civil Engineering, U. o Colorado at Denver 1. Concept o Mixing Process in Trblent Flow Far awa ro the solid wall, the low is ree, U, ro the riction. Near the wall, the low slows down. On the wall, the water particle oves at the sae velocit as the wall or 0. When the solid wall exists, the low velocit proile is depicted as: The velocit proile indicates that the horizontal velocit decas towards the wall. Between the two adjacent laers, low particles rotate with the ed p and down. The vertical lctation creates oent exchange or ixing. Trblent Flow Velocit Distribtion Viscosit Model d ( µ + η) lainar viscosit + ed viscosit (15) There is no generalized gidance as to how to qanti the vale o ed viscosit. Mixing Length Model A ixing length is exepliied b the size o eddies in the trblent low. This phenoenon can be evidenced b the growth o soe rollers. 1

2 Flow rotates becase the solid srace creates the neven riction and velocit distribtion. Thereore, the solid srace is the sorce o eddies. A soestac is also considered as the sorce o eddies becase it releases rollers into the wind low. There are neros stdies on the ixing length or an dierent trblent low cases. It has been conclded that the ixing length is 40% o the distance ro the wall or the sorce o eddies. The vale o 0.4 is tered the niversal ixing length constant. l 0.4 a niversal constant. The trblent low riction is no longer directl related to the low viscosit. Thereore, it is sggested that the oent principle be considered to odel the low particles rotating p and down between two adjacent laers. F A ρq V ρ( A v) d d d ρ v ρ( l)( l) ρl ( ) (16) The sign, -, eans the direction o shear stress is against the low. Let and l in which 0.4 (an niversal constant) ρ Eq 16 becoes d 1 ρ To integrate the above eqation ields ln + C (.5) ln + C Const1 ln + Const (17) It taes two data points, (,), to deterine the vales o Const1 and Const, noted that is easred ro the wall.

3 Exaple: The centerline o a channel low has a depth o 10 t. The velocities o low were easred to be 5.67 ps at eet below the srace and 5. ps at 6-t above the bed. Estiate the velocit at one t above the bed. Soltion: Const1 + Const Log (10-) 5.67 Const1 + Const Log (6) 5. Const1 3.5 and Const.5 Discssion At 0 on the wall, the non-slip condition reqires 0. Obviosl, the logarithic nction airs the non-slip condition. How to explain it? 3. Logarithic Velocit Proile in Circlar Pipe The relationship between the two coordinate sstes is: - r ( 18) Sbstitting Eq 18 into Eq 17 ields ln( r) + C (19) At r 0 or (the centerline in a circlar pipe), Eq 19 depicts U or 3

4 U ln( 0) + C (0) Taing the dierence between Eq s 19 and 0 ields the Law o Velocit Deect U.5ln r (1) e-arranging Eq 1 ields U.5 ln U 5.76 log () r r Eq has two nnown: U and. Fro the laborator data, the sei-theoretical analses reslted in two epirical orlas or U and. Both can be related to the riction actor in the pipe as: U V ( ) (3) V ρ 8 (4) 4. Discssion on Velocit Proiles For a lainar low in a circlar pipe: U r 1 ( ) ---- parabolic V U 1 and V ρ 8 For a trblent low in a circlar pipe: U 1.5 ln ---- logarithic U r V 1 and U V ρ 8 Far awa ro a solid wall (the ree-strea zone), the low is ree ro the riction. Near the wall (the bondar laer zone), the low is sbject to the riction ro the wall. Thereore, the logarithic proile was derived or the bondar laer low. Ver close to the wall, the viscos 4

5 orce doinates the low oveent in this sb-laer low zone. As a reslt, the sb-laer low is lainar. A linear velocit within the sb-laer thicness can be derived sing the shear velocit as: ρ 1 µ ρ υ the Law o the Wall which is a linear velocit distribtion υ In practice, we ond that the sb-laer thicness can have υ 70 The thicness o sb-laer, δ 70υ When the water low enters a pipe, the velocit proile in ront o the entrance is nearl nior (ree-strea low). Iediatel downstrea o the entrance, the centerline low can still ove at a nior velocit, bt near the wall, the bondar laer low begins to be developed. The thicness o the bondar laer increases downstrea. Ater a distance, the bondar laers ro both walls eet at the centerline and the low becoes established and nchanged. Sch a distance is tered Entrance Length as: Le eD when e <000 lainar low. 5

6 4.1 A 10-t pipe carries a water low o 1000 cs. The riction actor or this case is Analze this low. (1) Cross section area A 78.5 sq t () Flow velocit V 1.73 ps (3) Centerline velocit U ( ) V 0.81V ps (V/U 0.81) V (4) Shear velocit ρ 8 ps (5) Shear street lb/t 5 (6) ln( ) 5 r (7) For a pipe length o 100 eet, the energ loss is H t and E γ QH lb-t/sec

7 Footnote: F A πdl lb s and Power V lb-t/sec (loss calclated b riction orce) F (8) Sb laer thicness δ δ t inch (9) The Law o the Wall -- linear velocit distribtion within the sb-laer thicness or <δ, or (, ) has a linear relationship. 7

Boundary layer develops in the flow direction, δ = δ (x) τ

Boundary layer develops in the flow direction, δ = δ (x) τ 58:68 Trblent Flos Handot: Bondar Laers Differences to Trblent Channel Flo Bondar laer develops in the flo direction, not knon a priori Oter part of the flo consists of interittent trblent/non-trblent