Resistance in Open Channel Hydraulics

Transcription

1 Resistance in Open Channel Hdraulics I Manning and Chez equations are compared e = R S CR S n R R C= = n n 1 6 R C= n For laminar low: K R R e1 VR = VR K= 2 2 8gSR 8gVR S 8gR S 8g But K= = = R V V V C 2 8g C = Re1 K VR 8g I R e1 = C 2 14 R e1 For triangular Smooth Channel (Reer: Chow) 24 R e1 For Rectangular Smooth Channel (Reer: Chow) e1 Sand Roughness Fixed to Flume Bed (Photograph - Thandaeswara)

2 Laminar Flow with Roughness 60 or a 90 V shape channel. Roughness mm R e1 33 R e Reerence: "Chow Ven Te- Open Channel Hdraulics", Mc Graw Hill Compan, International student edition, 1959, page R e1 24 R e Laminar Transitional Turbulent R e1 Variation o riction coeicient with Renolds number Re1( = R ) in smooth channels

3 Reerence: "Chow Ven Te- Open Channel Hdraulics", Mc Graw Hill Compan, International student edition, 1959, page cm Varwick 25 cm R e1 33 Re1 60 Re Varwick cm Laminar Transitional Turbulent Re1 Variation o riction coeicient with Renolds number Re1 in rough channels Rectangular Channel - Rough low (Roughness = ) ( = R ) Bazin conducted experiment using (500 measurements were made at greatest care) (1) Grael embedded in cement. (2) Unpolished wood roughened b transerse wooden strip (i) 27 mm long 10 mm high 10 mm spacing. (ii) 27 mm 10 mm at 50 mm spacing. 3) Cement lining 4) Unpolished wood I the behaior o n and C is to be inestigated then a number o basic deinitions regarding the tpes o hdrodnamic low must be recalled. Flow can be diided into

4 (i) Hdro dnamicall smooth turbulent low (ii) Hdro dnamicall Rough turbulent low (iii) Hdro dnamicall transition turbulent low. The boundar laer δ or low past a lat plate is gien b 1/ 2 δ Vx o = 5 Laminar x 1/ 5 δ Vx o 7 = turbulent Re 2 10 logarthmic elocit law holds x > V 99% V δ δ δ0 Pseudo boundar Turbulent Transitional region Viscous sub laer Velocit Velocit distribution

5 Smooth kc = 5 kc is critical roughness height k is roughness height k c = 100 or aerage condition δ k δ0 δ0 δ0 k k kc (a) Smooth kc (b) wa kc Dierent surace roughness (c) rough

6 Viscous sublaer ks (i) Hdrodnamicall smooth turbulent low (R e ) ks Viscous sublaer (ii) Hdrodnamicall transition low (R e, k s /) ks Viscous sublaer (iii) Hdrodnamicall rough turbulent low (k s /) For hdro dnamicall smooth condition, iscous sub laer submerges the roughness elements. For hdro dnamicall transitional case the roughness element are partl exposed with reerence to iscous sub laer. For hdro dnamicall rough turbulent low the roughness elements are completel exposed aboe the iscous sub laer. For hdro dnamicall rough turbulent low resistance is a unction o Renolds number and the roughness height. K s I we deine R e = shear Renolds number. ; and τo = grs =. ρ

7 The low is classiied as ollows: K s < 4 Hdrodnamicall smooth K s 4 < < 100 Hdrodnamicall transition K s > 100 Hdrodnamicall ull deeloped turbulent low Summar o Velocit-Proile Equations or Boundar laers with dp 0 dx = Zone Smooth Walls Rough Walls Law o the wall Uniersal equations Laminar sub laer 4 < = ( δ ) - Buer zone 4< < 30 to Logarithmi c zone (also called turb ulent laer) > 30 to 70 < 0.15 δ Velocit-deect law Inner region 0.15 δ < (oerlaps with logarithmi c wall law) Outer region 0.15 δ < (approxim ate ormula) (3000 < R e < 70,000) - outer region = A log + B 5.6 log = Power Law = 8.74 k = A log + B k = 5.6 log + B B= (roughness size, shape and distribution) V = A log + B δ V = 5.6 log δ 1 7 V = A log δ V = 8.6 log δ - A and B are constants.

8 Table shows elocit distributions or dierent conditions Blasius equation or smooth low Smooth pipe low Nikurads e Rough pipe Nikurads e White and Colebroo k ormula VR VR Pipe low equation Re = Open channel low Re = upto R 0.25 e <10 C= R mks units or g = m/sec e Re Re 0.25 =2log R e Re 8g Re > 10 C = 4 2g log 251. C ( ) 1 = 0.86 ln Re = ln d 1 /d o 2.51 = 0.86 ln Re o 1/8 2 C = log Re 8g 251. C R C = log C e R 8g C = 2 log e C g C = 2 log 8g C = -2 log s 14.83R 12R ks k g + 8g Re Suggested modiication to equation is C ks 2.5 = -2 log + 8g 12R Re [ASCE Task Force Committee 1963]. R is hdraulic mean radius, 4R = Diameter o pipe. In open channel low ollowing aspects come into picture R (, K, C,N, F,U) e (1) (2) (3) In which R e is the Renolds number, K is the Relatie Roughness, C Shape actor o the cross-section, N is the Non- uniormit o the channel both in proile and in plan, F is the Froude number, U is the degree o unsteadiness. In the aboe equation, the irst term corresponds to, Surace Resistance (Friction), the second term corresponds to wae resistance and the third term corresponds to Non uniormit due to acceleration/ deceleration in low.

9 Surace Resistance: To be accounted based on Karman - Prandtl - elocit distribution. The constant in resistance equation is due to the numerical integration, and is a unction o shape o the cross-section. C 1 R = =A log +B 2g ' For circular section A = 2.0, B = For rectangular section: A = 2, B = (or large ratio o width/depth) It has remained customar to delineate roughness in terms o the equialent sand grain dimensions k s. For its proper description, howeer, a statistical characteristic such as surace texture requires a series o lengths or length deriaties, though the signiicance o successie terms in the series rapidl approach a minimum. Morris classiied the low into three categories namel (1) isolated roughness low, (2) Wake intererence low, and (3) Quasi smooth low. The igure proides the necessar details. k s s Isolated - roughness low (k/s) - Form drag dominates The wake and the ortex are dissipated beore the next element is reached. The ratio o (k/s) is a signiicant parameter or this tpe o low

10 k s Wake intererence low (/s) s s When the roughness elements are placed closer, the wake and the ortex at each element will interere with those deeloped b the ollowing element and results in complex orticit and turbulent mixing. The height o the roughness is not important, but the spacing becomes an important parameter. The depth '' controls the ertical extent o the surace region o high leel turbulence. (/s) is an important correlating parameter. k j j j j s s s k is surace roughness height s is the spacing o the elements j is the grooe width is the depth o low Quasi smooth low - k/s or j/s becomes signiicant acts as Pseudo wall Quasi smooth low is also known as skimming low. The roughness elements are so closed placed. The luid that ills in the grooe acts as a pseudo wall and hence low essentiall skims the surace o roughness elements. In such a low (k/s) or (j/s) pla a signiicant role. Concept o three basic tpes o rough surace low k, j, s should describe the characteristics o roughness in one dimensional situations is Areal concentration o or densit distribution o roughness elements. (ater Moris).

11 Areal concentration or Densit Distribution Roughness Elements Spheres Schlichting, Spatial distribution o roughness O'Loughlin and Mcdonald (1964) Cubes arranged as in (1) abd (2) also sand grains (2.5 mm dia)cemented to the bed. Koloseus (1958) and Koloseus and Daidian (1965) conducted experiments using Cubical Roughness Smmetrical diamond shaped pattern.

12 4 Spheres 3 Cubes Sand ks Eectie roughness as a unction o orm pattern, and concentration o roughness elements. (Assuming high Renolds number) 10 Nikuradse λ Areal concentration Schlichting (1936) - Sphere spacing Koloseus (1958) Koloseus and Daidian (1965) Cubical Roughness Smmetrical diamond shaped pattern O'Loughlin and Mcdonald (1964) Cubes arranged as in 1 and in 2. Also sand grains centered to the sand grains (2.5 m diameter) ks 1 Sand λ Areal concentration Logarithmic plot o data rom igure at low concentration Open channel resistance (ater H. Rouse, 1965)

13 b 1.5 F = V d = 3b 3b Froude number, F Resistance o a bridge pier in a wide channel, ater Kobus and Newsham 1.5 S = 5 D CD 0.5 D V d D d = 30 S Froude number, F Variation o pier resistance with lateral spacing "S"

14 /b = 1/16 ζ 0.2 /b = 1/8 2b 0.1 /b = 1/ b b Froude number, F Loss at one o a series o channel bends ater Haet Some o the important Reerences: (i) Task orce on riction actors in open channels Proc. ASCE JI. o Hd. Dn. Vol. 89., No. H2, March 1963, pp (ii) Rouse Hunter, "Critical analsis o open channel resistance", Proceedings o ASCE Journal o Hdraulic diision, Vol.91, Hd 4, pp 1-25, Jul 1965 and discussion pp , No. 1965, March 1966, pp 387 to 409. Schlichting, "Boundar laer theor", Mc Graw Hill Publication Open Channel Resistance There is an optimal area concentration 15% to 25% which produces greater relatie resistance. 1 R A log B DhS + h is the roughness height, S is the areal concentration (<15%), D is the constant which depends on shape and arrangement o the roughness elements. For sanded surace: D = 21 and B = 2.17 The existence o ree surace makes it diicult to assume logarthmic elocit distribution and to integrate oer the entire area o low or dierent cross-sectional shapes. The

15 lograthmic elocit distribution can be integrated onl or the wide rectangular and circular sections. Eect o boundar non-uniormit is normall ignored and particularl so or graduall aried low proile computation. The dependence on Froude number is clearl seen in case o pier. In case o unstead lows such as loods, it is assumed that the inertial eects are small in comparison with resistance. Hence, the resistance o stead uniorm low at the same depths and elocit is taken to be alid. Where the Froude number exceeds unit, the surace has instabilit in the orm o roll waes. Earlier ormulae or determining C (or details reer to Historical deelopment o Empirical relationships) 1. G.K. Formula (MKS) 2. Bazin s Formula 1897 (MKS) 3. Powell Formula (1950) FPS while using Powell ormula C must be multiplied b to get C in m 1/2 s Paloskii Formula (1925) Manning equation is applicable to ull deeloped turbulent rough low. Slope o the straight line is 1:3

16 n 1/3 ks g g C R 2 C ks k s 1/3 ks R I we replace k b diameter o the grain size (d) d R s 1/3 8g 8g R C = = d or MKS units g = m/s R R C = = d d R or C = d R n = C 1 n = d = d A number o empirical methods to relate n diameter o the particle are adanced. 1 Strickler (1923) 2 Henderson's interpretation o Strickler's ormula 3a Raudkii (1976) 3b Raudkii (1976) 3c Raudkii (1976) 4 Garde and Ranga Raju n = d [ d in m ] This is not applicable to mobile n = d [ d in eet ] 50 n = d [ d in m ] bed n = d65 [ d in mm ] d 65 = 65 % o the material b n = d [ d in eet ] 65 weight smaller. 1/ 6 n = d50 [ d in eet ] ( ) ( ) 5 Subramana n = d [ d in m ] 6 Meer and Peter and Muller 50 = = ( 50 ) n = d, d is in 'm' [ ] 90 n = d d in m (Signiicant proportion o coarse grained material)

17 7 Simons and Sentrrk (1976) 8 Lane and Carbon (1953) n = d [ d in mm ] n= d (d in inches and d 75 = 75% o the 75 material b weight is smaller) 8) Consider = g R S ks 4 < < 100 Transition low R R n = but C = C d R d 1 n = = d = d (d in m) R (. ) Conditon or ull deeloped rough low k s n = 100 d = = n n 1 g R S Assuming 6 6 ( ) 6 ( ) = m /s g = m/s n R S n RS

18 Blasius equation ( R e <10 5 ) 1 = 2.0 log ( Re ) 2.51 R e C = 4 2g log ( 8g ) 2.51C Laminar low Smooth suraces R e 1 8 (C = Re, mks) ks = 100 do 2ks = 2R ks Transition zone Full rough zone 1 = C = 2.0 log ( 12R ) or Manning 8g ks Commercial suraces Sand coated surace (Nikuradse) Renolds number R e = 4 V R/ Modiied Mood Diagram showing the Behaior o the Chez C ater Henderson