Hydraulics Prof. B.S.Thandaveswara Indian Institute of Technology Madras 16.3.1 Resistance in Open Channel Hydraulics If Manning and Chezy equations are compared 2 3 1 1 1 1 1 2 2 2 0 0 2 1 1 - 3 2 6 1 6 e e 2 2 e1 2 2 2 2 e1 e1 1 R S CR S n R R C= n n R C= n For laminar flow: K f = R VR R υ VR K= f υ 8gSR 8gVR S 8gR S 8g But f = K= = R υV υV C V 8g C R K VR If R υ = = = ∴ = ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ = 2 8g f = C 14 f = For triangular Smooth Channel (Refer: Chow) R e1 24 f = For Rectangular Smooth Channel (Refer: Chow) R e1 Sand Roughness Fixed to Flume Bed (Photograph - Thandaveswara) Hydraulics Prof. B.S. Thandaveswara Indian Institute of Technology Madras 16.3.2 Laminar Flow with Roughness e1 e1 60 f = for a 90 V shape channel. Roughness 0.3023 mm R 33 f = R ← 10 Laminar Transitional Turbulent 10 2 10 3 10 4 10 5 10 6 2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8 0.004 0.006 0.008 0.01 0.02 0.04 0.06 0.08 0.1 0.2 0.4 0.6 0.8 1.0 R e1 f f = 14 ___ R e1 f = 24 ___ R e1 Reference: "Chow Ven Te- Open Channel Hydraulics", Mc Graw Hill Company, International student edition, 1959, page - 10 Variation of friction coefficient f with Reynolds number Re1 in smooth channels = vR __ υ ( ) Hydraulics Prof. B.S. Thandaveswara Indian Institute of Technology Madras f = 14 ___ Re1 f = 33 ___ Re1 f = 60 Re1 ___ 10 10 2 10 3 10 4 10 5 10 6 2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8 Re1 Variation of friction coefficient f with Reynolds number Re1 in rough channels = vR __ υ ( ) 10 7 2 4 6 8 0.006 0.008 0.01 0.02 0.04 0.06 0.08 0.1 0.2 0.4 0.6 0.8 1.0 f 37.5 cm 25 cm Varwick Varwick 1 1 20 cm Laminar Transitional Turbulent 10 3 10 4 10 5 10 6 2 4 6 8 2 4 6 8 2 4 6 8 10 7 2 4 6 8 0.02 0.04 0.06 0.08 0.1 Reference: "Chow Ven Te- Open Channel Hydraulics", Mc Graw Hill Company, International student edition, 1959, page - 11 Rectangular Channel - Rough flow (Roughness =0.7188) Bazin conducted experiment using (500 measurements were made at greatest care) (1) Gravel embedded in cement. (2) Unpolished wood roughened by transverse 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 If the behavior of n and C is to be investigated then a number of basic definitions regarding the types of hydrodynamic flow must be recalled. Flow can be divided into Hydraulics Prof. B.S. Thandaveswara Indian Institute of Technology Madras (i) Hydro dynamically smooth turbulent flow (ii) Hydro dynamically Rough turbulent flow (iii) Hydro dynamically transition turbulent flow. 1 2 o 1 5 7 o e The boundary layer δ for flow past a flat plate is given by V x δ 5 Laminar x υ V x δ 0 38 turbulent R 2 10 logarthmic velocity law holds x υ / / . * − − ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ = > ⎜ ⎟ ⎝ ⎠ Velocity V 99% V y δ Velocity distribution δ ∗ δ 0 Viscous sub layer Transitional region Turbulent Pseudo boundary y Hydraulics Prof. B.S. Thandaveswara Indian Institute of Technology Madras δ δ0 δ0 δ0 k kc kc k kc Different surface roughness (c) rough k k c = υ v* __ 100 kc = 5υ v * __ Smooth for average condition kc is critical roughness height k is roughness height (b) wavy (a) Smooth Hydraulics Prof. B.S. Thandaveswara Indian Institute of Technology Madras Viscous sublayer k s (i) Hydrodynamically smooth turbulent flow f = f(R e ) Viscous sublayer k s Viscous sublayer k s (ii) Hydrodynamically transition flow f = f (R e , k s /y) (iii) Hydrodynamically rough turbulent flow f = f (k s /y) For hydro dynamically smooth condition, viscous sub layer submerges the roughness elements. For hydro dynamically transitional case the roughness element are partly exposed with reference to viscous sub layer. For hydro dynamically rough turbulent flow the roughness elements are completely exposed above the viscous sub layer. For hydro dynamically rough turbulent flow resistance is a function of Reynolds number and the roughness height. If we define e* R =shear Reynolds number * s v K υ . ; and o * f τ v gRS ρ = = . Hydraulics Prof. B.S. Thandaveswara Indian Institute of Technology Madras The flow is classified as follows: * s * s * s v K 4 Hydrodynamically smooth υ v K 4 100 Hydrodynamically transition υ v K 100 Hydrodynamically fully developed turbulent flow υ < < < > Summary of Velocity-Profile Equations for Boundary layers with dp 0 dx = Zone Smooth Walls Rough Walls Law of the wall Universal equations Laminar sub layer ( y δ ≤ ) * v y 4 υ < * * v y v v υ = - Buffer zone * v y 4 30 to 70 υ < < - - Logarithmi c zone (also called turb ulent layer) * v y 30 to 70 y 0.15 υ δ > < * * * * v y v A log B v v y v 5.6 log 4.9 v υ υ = + = + * * v k A log B v y v k 5.6 log B v y B f = + = − + = (roughness size, shape and distribution) Velocity-defect law Inner region (overlaps with logarithmi c wall law) y 0.15 δ < Outer region (approxim ate formula) y 0.15 δ < * * V v y A log B v V v y 5.6 log 2.5 v δ δ − = − + − = − + * * V v y A log v V v y 8.6 log v δ δ − = − − = − (3000 < e R < 70,000) outer region - Power Law 1 7 * * v v y 8.74 v υ ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ - A and B are constants. Hydraulics Prof. B.S. Thandaveswara Indian Institute of Technology Madras Table shows velocity distributions for different conditions Pipe flow equation e VR R υ = Open channel flow e VR R 4 υ = Blasius equation for smooth flow 5 e 0.25 e e 5 e 0.3164 f = upto R <10 R R f 1 =2log 2 51 f R 10 . > e e 1/8 2 0.25 e e e C=18.755 R mks units for g = 9.806 m/sec 0 223 f R R 8 C = 4 2g log 2 51 C R 8g C = 17.72 log 2 51 C 3.5294R C = 17.72 log C . g . . ⎡ ⎤ ⎣ ⎦ = ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ Smooth pipe flow Nikurads e Rough pipe Nikurads e ( ) o 1 = 0.86 ln Re f - 0.8 f 1 = 1.14 - 0.86 ln d f ∈ e s R 8g C = 2 log C 2 51 8g C 12R = 2 log k 8g * . ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ White and Colebroo k formula o /d 1 2.51 = 0.86 ln 3 7 f Re f . ∈ ⎡ ⎤ + ⎢ ⎥ ⎣ ⎦ s e 2.52 8g k C = -2 log 14.83R 8g R f ⎡ ⎤ + ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ Suggested modification to equation is s e k C 2.5 = -2 log 12R 8g R f ⎡ ⎤ + ⎢ ⎥ ⎣ ⎦ [ASCE Task Force Committee 1963]. R is hydraulic mean radius, 4R =Diameter of pipe. In open channel flow following aspects come into picture ( ) e f = f R K, C,N, F,U (1) (2) (3) , In which R e is the Reynolds number, K is the Relative Roughness, C Shape factor of the cross-section, N is the Non- uniformity of the channel both in profile and in plan, F is the Froude number, U is the degree of unsteadiness. In the above equation, the first term corresponds to, Surface Resistance (Friction), the second term corresponds to wave resistance and the third term corresponds to Non uniformity due to acceleration/ deceleration in flow. Hydraulics Prof. B.S. Thandaveswara Indian Institute of Technology Madras Surface Resistance: To be accounted based on Karman - Prandtl - velocity distribution. The constant in resistance equation is due to the numerical integration, and is a function of shape of the cross-section. C 1 R =A log +B y' 2g f For circular section A = 2.0, B = -0.62 For rectangular section: A = 2, B = -0.79 (for large ratio of width/depth) = It has remained customary to delineate roughness in terms of the equivalent sand grain dimensions k s . For its proper description, however, a statistical characteristic such as surface texture requires a series of lengths or length derivatives, though the significance of successive terms in the series rapidly approach a minimum. Morris classified the flow into three categories namely (1) isolated roughness flow, (2) Wake interference flow, and (3) Quasi smooth flow. The figure provides the necessary details. s y k Isolated - roughness flow (k/s) - Form drag dominates s The wake and the vortex are dissipated before the next element is reached. The ratio of (k/s) is a significant parameter for this type of flow Hydraulics Prof. B.S. Thandaveswara Indian Institute of Technology Madras s s Wake interference flow (y/s) j j j Quasi smooth flow - k/s or j/s becomes significant acts as Pseudo wall s y k y k s s s j k is surface roughness height s is the spacing of the elements j is the groove width y is the depth of flow Concept of three basic types of rough surface flow When the roughness elements are placed closer, the wake and the vortex at each element will interfere with those developed by the following element and results in complex vorticity and turbulent mixing. The height of the roughness is not important, but the spacing becomes an important parameter. The depth 'y' controls the vertical extent of the surface region of high level turbulence. (y/s) is an important correlating parameter. Quasi smooth flow is also known as skimming flow. The roughness elements are so closed placed. The fluid that fills in the groove acts as a pseudo wall and hence flow essentially skims the surface of roughness elements. In such a flow (k/s) or (j/s) play a significant role. k, j, s should describe the characteristics of roughness in one dimensional situations is Areal concentration of or density distribution of roughness elements. (after Moris). Hydraulics Prof. B.S. Thandaveswara Indian Institute of Technology Madras 16.3.3 Areal concentration or Density Distribution Roughness Elements Spheres Spatial distribution of roughness Schlichting, 1936 Koloseus (1958) and Koloseus and Davidian (1965) conducted experiments using Cubical Roughness Symmetrical diamond shaped pattern. O'Loughlin and Mcdonald (1964) Cubes arranged as in (1) abd (2) also sand grains (2.5 mm dia)cemented to the bed . 1 2 Hydraulics Prof. B.S. Thandaveswara Indian Institute of Technology Madras Logarithmic plot of data from figure at low concentration Effective roughness as a function of form pattern, and concentration of roughness elements. (Assuming high Reynolds number) Open channel resistance (after H. Rouse, 1965) 1.0 0.1 0.01 0.001 0.1 1 10 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1.0 Spheres Sand Cubes Nikuradse Sand Schlichting (1936) - Sphere spacing Koloseus (1958) Koloseus and Davidian (1965) Cubical Roughness Symmetrical diamond shaped pattern O'Loughlin and Mcdonald (1964) k s ___ y λ − Areal concentration Cubes arranged as in 1 and in 2. Also sand grains centered to the sand grains (2.5 m diameter) λ − Areal concentration k s ___ y Hydraulics Prof. B.S. Thandaveswara Indian Institute of Technology Madras Resistance of a bridge pier in a wide channel, after Kobus and Newsham F = 1.5 1.0 0.5 b 3b d = 3b V 0 0.5 1.0 1.5 2.0 2.5 0 0.5 1.0 1.5 Froude number, F Variation of pier resistance with lateral spacing "S" 0 0.5 1.0 1.5 2.0 Froude number, F 0 0.5 1.0 1.5 C D S D __ = 5 7.5 30 D S D d = 30 V d Hydraulics Prof. B.S. Thandaveswara Indian Institute of Technology Madras Loss at one of a series of channel bends after Hayet 0.2 0.4 0.6 0.8 1.0 2.0 4.0 0.1 0.2 0.4 0.6 y/b = 1/16 y/b = 1/8 y/b = 1/4 2b 4b 90 0 y b ζ Froude number, F Some of the important References: (i) Task force on friction factors in open channels Proc. ASCE J I. of Hyd. Dn. Vol. 89., No. Hy2, March 1963, pp 97 - 143. (ii) Rouse Hunter, "Critical analysis of open channel resistance" , Proceedings of ASCE J ournal of Hydraulic division, Vol.91, Hyd 4, pp 1 - 25, J uly 1965 and discussion pp 247 - 248, Nov. 1965, March 1966, pp 387 to 409. Schlichting, "Boundary layer theory", Mc Graw Hill Publication. 16.3.4 Open Channel Resistance There is an optimal area concentration 15% to 25% which produces greater relative resistance. 1 R A log B DhS f = + h is the roughness height , S is the areal concentration (<15%), D is the constant which depends on shape and arrangement of the roughness elements. For sanded surface: D =21 and B =2.17 The existence of free surface makes it difficult to assume logarthmic velocity distribution and to integrate over the entire area of flow for different cross-sectional shapes. The Hydraulics Prof. B.S. Thandaveswara Indian Institute of Technology Madras lograthmic velocity distribution can be integrated only for the wide rectangular and circular sections. Effect of boundary non-uniformity is normally ignored and particularly so for gradually varied flow profile computation. The dependence on Froude number is clearly seen in case of pier. In case of unsteady flows such as floods, it is assumed that the inertial effects are small in comparison with resistance. Hence, the resistance of steady uniform flow at the same depths and velocity is taken to be valid. Where the Froude number exceeds unity, the surface has instability in the form of roll waves. Earlier formulae for determining C (for details refer to Historical development of Empirical relationships) 1. G.K. Formula (MKS) 2. Bazin’s Formula 1897 (MKS) 3. Powell Formula (1950) FPS while using Powell formula C must be multiplied by 0.5521 to get C in m 1/2 s -1 4. Pavlovskii Formula (1925) Manning equation is applicable to fully developed turbulent rough flow. Slope of the straight line is 1:3 Hydraulics Prof. B.S. Thandaveswara Indian Institute of Technology Madras 1/ 6 1/ 3 s 2 s 1/ 6 s 1/ 3 s s 1/ 3 k g g f C R k C n k k f = 0.113 R If we replace k by diameter of the grain size (d) d f = 0.113 R 8g 8g R C = f 0.113 d ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ ⎛ = ⎜ ⎝ 1/ 6 2 1/ 6 1/ 6 1/ 6 1/ 6 1/ 6 1/ 6 for MKS units g = 9.806 m/s 8 * 9.806 R R C = 26.3482 0.113 d d R or C = 26.34 d R n C 1 n = *d 0.0379d 26.34 ⎞ ⎟ ⎠ ⎛ ⎞ ⎛ ⎞ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ = A number of empirical methods to relate n diameter of the particle are advanced. 1 Strickler (1923) [ ] 1/6 n = 0.02789 d d in m This is not applicable to mobile bed 2 Henderson's interpretation of Strickler's formula [ ] 1/6 50 n = 0.034 d d in feet 3a Raudkivi (1976) [ ] 1/6 n = 0.047 d d in m 3b Raudkivi (1976) [ ] 1/6 65 n = 0.013 d d in mm d 65 =65 % of the material by weight smaller. 3c Raudkivi (1976) [ ] 1/6 65 n = 0.034 d d in feet 4 Garde and Ranga Raju [ ] 1/6 50 n = 0.039 d d in feet ( ) ( ) ( ) 1/6 1 6 50 0 039 0 3048 0 039 0 82036 0 03199 n = 0.03199 d , d is in 'm' / . * . . . . = = 5 Subramanya [ ] 1/6 50 n = 0.0475 d d in m 6 Meyer and Peter and Muller [ ] 1/6 90 n = 0.038 d d in m (Significant proportion of coarse grained material) Hydraulics Prof. B.S. Thandaveswara Indian Institute of Technology Madras 7 Simons and Sentrvrk (1976) [ ] 1/6 n = 0.047 d d in mm 8 Lane and Carbon (1953) 1/6 75 n= 0.026 d (d in inches and d 75 =75% of the material by weight is smaller) ( ) * f * s 1/6 1/6 1/6 1/6 1/6 1/6 1/6 8) Consider v g R S υ k 4 < 100 Transition flow v R R n = but C = 26.35 C d R d 1 n = d 0 03795 d (d in m) 26 35 R 26 35 Conditon for fully develop . . . = < ∴ = = ( ) ( ) 6 8 6 * s 6 6 f -6 2 2 6 f 6 6 6 ed rough flow v k n 100 d = 3 3458 10 n υ 0.03795 n 1 g R S 0.03795 Assuming = 1.01 * 10 m /s g = 9.806 m/s 9 806 1 n R S 100 1 01 10 0.03795 n . * . . * υ υ − = = ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ = 14 f RS 9 635 10 . * − ≥ Hydraulics Prof. B.S. Thandaveswara Indian Institute of Technology Madras Laminar flow Smooth surfaces Fully rough zone Transition zone Commercial surfaces Sand coated surface (Nikuradse) Reynolds number Re = 4 V R/v Modified Moody Diagram showing the Behavior of the Chezy C after Henderson 10 3 10 4 f = 0.316 _____ Re 0.25 (C = , mks) 15.746 Re 1 __ 8 1 __ f = 2.0 log ( ) Re f _____ 2.51 C = 4 2g log ( ) Re 8g 2.51C _____ Blasius equation ( ) Re <10 5 ______ υ = 100 v* ks 1 __ f = C __ 8g = 2.0 log ( ) 12R ___ ks 10 5 10 6 10 7 10 8 2 4 6 30 40 50 60 70 80 90 100 110 120 130 140 150 180 or Manning 507 252 126 60 30.6 15 10 do _____ 2ks 2R _____ ks =