UNIFORM FLOW. U x. U t

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UNIFORM FLOW coditios are established U x U t 0 Give the cross sectio characteristics the

1 UNIFORM FLOW if : 1) there are o appreciable variatios i the chael geometry (width, slope, roughess/grai size), for a certai legth of a river reach ) flow discharge does ot vary the, UNIFORM FLOW coditios are established U x U t 0 Give the cross sectio characteristics the locatio of the free surface / flow depth will result from the balace betwee gravity ad the flow resistace/frictio

2 Dimesioal aalysis: (,,, g, V, R hydraulicradius, k rough, C geometry ) V VR kr f,, R V gr Reyolds relative roughess Froude I a ope chael flow the flow is certaily turbulet ad the regime certaily rough ; thus at a scale of a river, we - eglect viscous effect ad - lose the depedecy o the Reyolds umber (Re high eough! ) BLACKBOARD Mometum equatio 1+,+ Hydraulic radius R=A/P

3 because the flow is uiform: the bed topography // free surface water slope // eergy grade lie Thus the eergy losses h f i a L reach ca be estimated by : assumig: V fuctio k R f 8 with r f frictio factor Note that L=4R is the key legth scale to cosider whe extedig the Moody diagram from pipe to ope hael flow (D pipe 4R hydraulic radious )

4 solvig for the shear stres VR k V f r,,,... V R gr from the uiform coditio : if γ we fv 8g V R C R 4Rg f Chezy coefficiet (coductace, ot resistace) for C R e solve slope of the eergy grade lie for the velocity icreasig, the cross -sect velocity is icreasig 1 f 8 1 V c f V c f = roughess coeff. we eed a empirical closure providig f or C or h f for a give cross sectioal geometry

5 The mea velocity profile i the smooth wall turbulet boudary layer : 1) viscous sublayer u = τ 0 y μ τ = μ du dy the velocity varies liearly, as a Couette flow (movig upper wall). Thus, the shear stress is costat: τ 0

6 scalig ear wall turbulece We ca defie a velocity scale u* = τ ρ [m/s] characteristic of ear wall turbulece u* = shear velocity or frictio velocity we ca rewrite the liear profile i the viscous sublayer as υ u u = yu υ where is a legth scale (very small, remember υ u =O( ) m /s, while u* is a fractio (~5-10%) of the udisturbed velocity U 0 δ boudary layer height we already have velocity scales: 1) u* ) U 0 How may legth scale? 1) υ u ) δ

7 viscous sublayer cotiued How thick is the viscous sublayer? it depeds o the boudary layer... as u* ad υ defie the viscous legth scale, we ca represet the extesio of the viscous sublayer i terms of multiples of the viscous scale (viscous wall uits) δ υ = 5 υ u Note that as u* δ υ : the viscous sublayer becomes thier Note: roughess protrusio (fixed physical scale) may emerge from the viscous sublayer ad chage the ear wall structure of the flow δ υ

8 above the viscous/roughess sublayer Logarithmic regio u u = 1 k l yu υ +C with u* = τ ρ where u* depeds o the flow ad the surface k is the vo Karma costat(?)= (k=0.41 is a good umber) C is the smooth wall costat(?) of itegratio (C=5.5 is a good umber) ote that is a rough wall boudary layer = 1 l u k where y 0 is the aerodyamic roughess legth: it is a measure of aerodyamic roughess, ot geometrical (surface) roughess u y y0 relatig to y 0 is complicate

9 The mea velocity profile: where is it valid? from about 60 viscous wall uits to about 15% of he boudary layer height it makes sese that the extesio of the log layer has to be determied by both ier scalig ad outer scalig

10 urface roughess 1) affect the mea velocity profile (through the aerodyamic roughess legth y 0 ) ) is related o-trivially with aerodyamic/hydraulic roughess coefficiets 3) affect the mea cross sectioal velocity ( through C, c f, f) 4) requires a empirical closures (Moody diagram, Maig, Gauckler trickler coeffs) VR kr V f,, V R gr uiform flow coditios : γ R e,... with a empirical closure estimatig (f, C, c f ) from cross sectioal geometry ad surface roughess obtai R Y ormal flow depth BLACKBOARD geeric formulatio: bis+tris+ BB 4BI7+

frictio factor i turbulet flows Resistace coefficiet from experimets (Nikuradse) k s /D relative roughess k s is the sad roughess height Keulega trapezoidal

11 frictio factor i turbulet flows Resistace coefficiet from experimets (Nikuradse) k s /D relative roughess k s is the sad roughess height Keulega trapezoidal chael fully rough 1 f =.03 log R/k s +.1 ote that f is a measure of aerodyamic roughess while k s is a measure of geometric roughess 1 f =.0 log 10 Re f 0.8

12 Moody diagram costat Re f 1/ curves (give D, L, K s fid losses h f ) k s / 4R explicit formula f log ks 3.7D Re figure_09_1 Re= 4R V/ν

13 tabulated pipe roughess

14 Empirical closures: roughess ad frictio factors Maig proposed a dimesioal coeff.: V C R 1 origial (old) Maig V K R Gauckler-Maig is the dimesioless Maig coefficiet: while [K ]= m 1/3 /s is dimesioal (K dimesioal is the price we pay to have a umber) K =1 m 1/3 /s (I m, m/s) K =1.49 ft 1/3 /s (Eglish Uits ft, ft/s) large : low velocity, large roughess : e.g. =0.05 for high grass or cobbles small : high velocity, low roughess : e.g. =0.01 clea straight cocrete

15 Note that by itegratig the mea velocity profile o rough walls we should obtai a relatioship betwee the frictio factor ad the maig coeff (for the mea cross sectioal velocity) U=u * /k log (y/y 0 ) Also, we should be able to relate all the roughess coefficiets, e.g. i uiform flow γ R solvig for the velocity K (8g) f R 1/ 6 fv 4Rg V 8g f R Importat lik betwee rough wall boudary layer turbulece (f) ad hydraulics () V K R Ks strickler = 1/ maig. The coefficiet Ks strickler varies from 0 (rough stoe ad rough surface) to 80 (smooth cocrete ad cast iro).

16 ome cosideratios based o the roughess iduced by specific bed material the chael frictio factor is proportioal to the size of the larger grais, as compared to the hydraulic radius equivalet to the term z 0 / i rough wall TBL FLOW

17 Aother attampt to relate k s (geometry) with (fluid mechaics drag)

18 UNIFORM FLOW COMPUTATION Give the geometric characteristics of the cross sectio, we ca determie the roughess coefficiet ad calculate the uiform flow depth (or ormal depth) K V Q K A AR R A P Q K A R P A - - P 5/3 Q K Q VA the wetted perimeter P is a fuctio of y, kow oly if the cross sectio is give slope, discharge are boudary coditios defies the roughess Rectagular cross sectio: V Q K K R by( by) ( b y) P b y A by ( y / b) - - y 1 b 5/3 Q Vby Q K b 8/3 solve for y

19 for a trapezoidal chael Oce we compute the uiform flow depth, we ca determie if the flow is sub- or super- critical b y m 1 Note: P b y(1 m ) 1/ A y( b my) What happes if roughess icreases?

20 Complex cross sectios f frictio factor fo rthe cross sectio c f c ks (Re, R P B,, ) B B supposed to accout for: large scale rouhess, supercritical istabilities fairly costat with depth b ) m ( meaderig amplifyig term b surf roughess, 1 bak roughess,, slope variability i x (pools ad riffles) 3, chael obstructio ad 4 vegetatio p 1 1 Compoud chaels U 1 U U 3 p p 3 3 Eistei ad Chie Horto method (p i wetted perimeters) P i c P P i P 3/ i i Imposig U=U 1 =U = U 3 c Imposig R=R 1 =R = R 3

22 lope classificatio V AR c K c c Q K A R R Q K c 4/3 c R A P F>1, supercritical coditios F<1, subcritical coditios Q VA Rectagular sectio BB

Channel design. riprap-lined channels

Channel design. riprap-lined channels Chael desig Vegetatio ad cobbles are used to cotrol erosio, dissipate eergy ad provide a evirometally sustaiable way to desig differet size chaels riprap-lied chaels Smooth chaels i circular coduits: PVC