Prof. B.S. Thandaveswara. Superelevation is defined as the difference in elevation of water surface between inside (1)

Transcription

1 36.4 Superelevation Superelevation is defined as the difference in elevation of water surface between inside and outside wall of the bend at the same section. y=y y (1) 1 This is similar to the road banking in curves. The centrifugal force acting on the fluid particles, will throw the particle away from the centre in radial direction, creating centripetal lift. Superelevation in other words means the greater depth near the concave bank than near convex bank of a bend. This phenomenon was first observed by Ripley in 187, while he was surveying Red River in Loussiana for the removal of the great raft obstructing the stream Transverse water surface slope in bends Gockinga was first to derive the following formula for determining the difference in elevation of the water surface on opposite sides of channel bends. x y=. 35V log 1- r in which V is the velocity in -1 ms, x is the distance at which y is to be determined, r is the radius of the river bend. But above equation is found to fit a particular stream to which it was designed. Also he found that the transverse slope was twice greater than the longitudinal slope. He showed that the increased depth in bend is caused by the helicoidal flow induced by centrifugal force. Fargue while conducting studies on scour in meandering devised the formula and called " Fargue's law of greatest depth " in 198. But unfortunately it was found to be applicable to the stream Garonne at Barsac only. 3 C =.3H -.3H +.78H in which C 1 is the reciprocal of radius of curvature in kilometer and H is the lowest water depth at the deepest point of the channel in meter. Mitchell also derived another equation applicable to Delawave river, Philadelphia.

2 11 x x y = 33 + x 11 r 11 Ripley in 196 arrived at the formulae based on the field observations, having their own limitations. 4x x 1 1 y = D 1 + D 1 x T r T 4x 6.8P 1 4x y = P x r T T In the above equations, parabolic sections are assumed whose focal distances are T D 1 and T P respectively and with the origin at a point on the axis at a distance of D 1 and P 1 1 from apex respectively. O r CENTRIFUGAL FORCE V max r i 9 W 1 = WEIGHT OF FLUID dr dy y v O r MECHANICAL MODEL OF THE HELICOIDAL FLOW AS PROPOSED BY GRASHOF The above two equations when combined yield a simplified form in FPS units. Figure represents the general profile for equation given below. x 17. 5x y = 6. 35D T r

3 r T/ T/ origin 1.445D Y AXIS PROFILE FOR EQUATION Grashof was the first to try an analytical solution for superelevation. He obtained equation by applying Newton's second law of motion to every streamline and integrating the equation of motion. Equation gives a logarithmic profile. Referring to figure given below one may write Centrifugal force = W g 1 max V r c max c max 1 grc W1 V dy g r V dr = W = Assuming the boundary conditions near inside wall of the bend and integrating above equation reduces to the form V y=.3 max r log g ri in which r i and r are the inner and outer radii of the bend respectively. Woodward in 19 assumed the velocity to be zero at banks and to have a maximum value at the centre of the bend and the velocity distribution varying in between

4 according to parabolic curve. Using Newton's second law of motion he obtained the following equation for superelevation. 3 rc r c r c r c+ b y = V max ln 3 b b b rc - b Shukry obtained the following equation for maximum superelevation based on freevortex flow and principle of specific energy. The Euler equation of motion. Since s ( ) C y = r - r max i g rri ( p γ z) s + + ρa = ( p + γz ) = γh h ( p + γz ) = γ s n v H = y+ + z g v = h + h = y + z g ( ) r r i r V g X X n r i isovels y r datum z H Section xx dh dn v = gr

5 differentiating above equation dh v v + = dn g n v v v + = gr g n v v + = r n Thus v decreases and h increases from inner boundary to outer boundary. v r The equation can be rewritten as + =. v r Therefore it can be shown as vr = constant which is in the free vortex condition. Consider a rectangular channel bend, Discharge per unit width q= vy. c q = y or y = q r r c a constant. r r i Total Energy Line V g H Outer Inner Wall y y Wall Subcritical flow F < 1 z Supercritical flow F > 1 Modification to the bed profile to obtain the horizontal water surface in a bend dy y = dr r dy dz 1 = - dr dr 1-F z decreases from inner wall to outer wall for subcritical flow (as shown in the above figure). dz dy y = - ( 1-F ) = - ( 1-F ) dr dr r and dh dz dy dy dy = + = - ( 1-F ) + dr dr dr dr dr dy y v -1+F + 1 = F = dr r gr

6 Transverse bed profile. v v H =z+y+ = y + g g rq C H =z+ + C gr The above equation gives resonably good result as long as the angle of the circular bend in plan is greater than 9 a correction factor was suggested by Shukry for circulation constant C, assuming it to vary linearly from to 9. θ θ r V vr = CU f =C C v θ θ V =w 1 U X =w r 9 9 rw However, by applying Newton's nd law of motion based on one dimensional analysis i.e., all the filamental velocities in the bend are equal to the mean velocity ma ma 1 V mb and that of all the streamlines having the same radii of curvature r c, an equation can be obtained for a rectangular channel namely Vmb y max = g r c b For the channels other than the rectangular channel, the bed width (b) can be replaced by the water surface width (T) then Vmb y max = g r c T The above equation is only a first approximation and gives transverse profile as the straightline. This assumes that the rise and the drop of the water surface level from the normal level is equal on either side of center line of bend. As a better approximation, Ippen and Drinker obtained an equation for superelevation. The derivation of equation is based on the assumptions of free vortex or irrotational flow with the uniform specific head over the cross section, and the mean depth in bend being equal to the mean approaching flow depth.

7 V T 1 y= g r c T 1 4r c The bends in nature will not have the symmetry due to entrance conditions, length of curvature and boundary resistance. Hence above equation will not give accurate result. If the forced vortex condition exists with constant stream cross section and constant average specific energy, then equation for superelevation assumes the form V T 1 y= g r c T 1+ 1r c The above equation is applicable to a smooth rectangular boundary with circular bend with the flowing fluid being ideal. Better results can be obtained by combining the effects of the free and forced vortex conditions simultaneously. The minimum angle of bend is to be 9 for applying the above equation in combination. For the smaller angles the difference in computed values from the above equation becomes larger than the actual ones. However, for a rectangular channel, circular bend with, applying the free vortex formula, velocity at inside wall of the bend becomes as thus a depth of should exist at the boundary (i.e. at ), which is physically impossible. Muramoto obtained an equation for superelevation based on equations of motion. Then vr = C 1 C 1 r g S r U = + 3C ( C 1 + C) 4 g S r SCr y = + 36 C1 gr 3C1 in which C 1 and C are circulation constants obtained, after integration. The special feature of above equation lies in including the effect of bed slope on superelevation. r r i

8 Thus it can be observed from the above discussion that superelevation in a bend is a function of shape of the cross section, Reynolds number, approach flow, slope of the bed, Froude number, θ r c, and boundary resistance. The superelevation is also 18 b affected by the presence of secondary currents and separation. However, before applying to field situations, the validity of the transverse flow profile equation has to be justified. The observations in the field have shown the occurrence of troughs near the concave profile in the bend. Further, these troughs have been observed during rising and as well as falling stages of the channels. The channels of smaller width have exhibited accumulation of debris instead of troughs Superelevation and transverse profile Normalised super elevation y/ ymax was correlated with normalised bend angle θ/θ for all the three bends, for two different Reynolds numbers. From Figure, it may be observed that two peaks of superelevation occur at θ/θ = 17. and θ/θ = 67. for all the cases except for bend B1 for R e = 4, 8. The influence of Reynolds number on the trend of the variation of superelevation at various section of the bend is insignificant in all the three bends.

9 y y max SECTION B1 y y max SECTION B y y max θ/θ SECTION B3 Variation of normalised Super elevation with normalised angle for Reynolds number 48

10 y y max SECTION B1 y y max SECTION B y y max θ/θ SECTION B3 1. Variation of normalised Super elevation with normalised angle for Reynolds number 11,7 The maximum value of superelevation was normalised with the approaching velocity head V/gand correlated with the Froude number. The equations of these lines is in the form ymax =m log F + C V/g in which 'm' is the slope of the line and 'C' a constant. ymax Bend 1: = 1.19 log F +. V/g ymax Bend : = 5.33 log F.49 V/g

11 ymax Bend 3: = 1.96 log F.6 V/g View of SET-UP

12 Flow Condtions D/S of B1 The greatest difference in elevation between the longitudinal profiles at outer and inner walls is maximum at the 3 section of the 18 bend.

13 N A P.G Bend I A P.G Wheel volve Inlet pipe rc = Adjustment valve Entrance chamber Transition Leading Channel Rectangular notch Stilling chamber 87 9 Masonry honey comb 1 Transition 11 Main channel 1 Tail control G.L LAYOUT PLAN Point gauge Bed slope 1 18 rc = 14 Bend II Experimental Set - Up Scale = 1: 1 (all dimensions in cm)

14 All dimensions are in cm m m 75 6 rc 45 r c m Station B 7 r c = 3 A A 15 1 Section AA G.L Inner Outer b) points of measurement Station C 1 m Section and points of measurement 6 3

15 Bend y _ V g / Bend 3. Bend Variation of log F y _ V g / with log F

16 4 3 experimental theoretical bed in cm Observed Theoretical : eqn Experimental : eqn Comparison of Observed and Theoretical Superelevation 1. y/ya 1..8 Bed slope 1:1 STN A B1 STN B B STN C B3 STN D Q = 6.1 lps Re 48 F = y/ya 1..8 Bed slope STN A STN B STN C B1 B B3 STN D Q = 71.9 lps Re 1176 F =.55 Scale x axis 1:1 y axis 5:1 Longitudinal Water Surface Profile

17 C L C L STATION A STATION B C L C L STATION C STATION D ISOVELS in open channel bend [Normalised with V max ] Q = 6.1 lps, F =.18, R e= 365

18 C L C L STATION A STATION B C L C L STATION C STATION D ISOVELS in open channel bend [Normalised with V max ] Q = 71.9 lps, F =.44, R e= 954 C L C L STATION A STATION B C L C L STATION C STATION D ISOVELS in open channel bend [Normalised with V max ] Q = 83.5 lps, F =.41, R e=

19 Reference 1. Ippen A.T and Drinker P.A., "Boundary shear stresses in curved trapezoidal channels", Proc. ASCE. journal Hydraulic Diivision HY5, Part - I, Volume 88, p 373, pp , September 196, and discussion by Shukry A, Proc. ASCE. journal Hydraulic Division, pp 333, May Henderson F.M., "Open Channel Hydraulics", Mac Millan Company Limited Thandaveswara B.S, "Characteristics of flow around a 9 open channel bend", M.Sc Engineering Thesis, Department of Civil and Hydraulic Engineering, Indian Institute of Science, Bangalore - 1, May 1969.