Open-channel hydraulics

Open-channel hydraulics

STEADY FLOW IN OPEN CHANNELS constant discharge, other geometric and flow characteristics depended only on position Uniform flow Non-uniform flow S; y; v const. i i 0 i E y 1 y ; v 1 v i i 0 i E K141 HYAE Open-channel hydraulics 1

Equation of uniform flow pressure forces F 1 F weight of water G ρgsdl dz slope of bottom i 0 tgα sinα dl force in direction of motion G Gsinα ρgsdli against motion friction force Ft τ0odl Equilibrium of forces G F t ρgsdli τ0odl τ τ 0 ρgri (R S/O) 0 gri v - friction velocity ρ * τ0 b friction loss Zt av in quadratic zone b ρg 1 1 0,5 1 v Ri, C m s a a K141 HYAE Open-channel hydraulics v 1 S,O G y F F 1 y 1 G dz C Ri dl F t - Chézy equation

SOLUTION OF CHANNELS 1. Chézy equation (1768) C velocity coefficient, K - conveyance (m s -1 ). Manning equation (1889) n - roughness coefficient comparison of both equations v C R i 0 1 v R i n 1 C R n 1 validity: n > 0,011, 0,m < R < 5m Q C S R i K i 0 0 1 P Pavlovskij (195): C R, n P,5 n 0,1 0,75 R n 0,1 validity: 0,011 < n < 0,04, 0,1m < R < m Bretting (1948): ( ) R C 17,7 log + 1,171 de 1 6 K141 HYAE Open-channel hydraulics

Determination of n: - tables values 0,008 0,150 ( 0,500): Type of channel and description Streams on plain a) clean, straight, full stage, no rifts or deep pools b) same as above but more stones and weeds c) clean winding, some pools and shoals n min. 0,05 0,00 0,0 n nor. 0,00 0,05 0,040 n max. 0,0 0,040 0,045 - photographic method - formulas in dependency on d i 1 1,1 Strickler (19) validity: 4, < R/d e < 76 n 1 d 6 e Grain-size curve - screen analysis (fine-grained) - random sample (course-grained) -... K141 HYAE Open-channel hydraulics 4

different roughness on wetted perimeter equivalent roughness coefficient O n weighted average n i i O O n i i Horton, Einstein, Banks n O. Darcy-Weisbach equation L v Zt λ v 4R g Hey (1979): 1 ar,0 log λ,5 d validity: R/d 84 > 4 84 O,n a 11,1 1,6 coefficient of channel shape O,n O 1,n 1 Relation among C, n and λ: R 1 6 8 C g n g λ K141 HYAE Open-channel hydraulics 5

CHANNEL DESIGN - calculation of velocity and discharge Q basic equations - calculation of bottom slope i 0 basic equations - calculation of depth y 0 semi-graphically y f(q) (rating curve) by numerical approximation y i Q i ; Q y 0 - calculation of channel width b similarly as determination of depth Compound channels! velocities, roughness coefficient, discharge Q Q i S S S 1 S S 1 S O O O 1 K141 HYAE Open-channel hydraulics 6

Part-full circular pipes y v for 0,81 max D y Q 1,087Q for 0,9495 max D D K141 HYAE Open-channel hydraulics 7

y k CRITICAL, SUB-CRITICAL AND SUPERCRITICAL FLOW Critical flow: Q const. E dmin (E d const. Q max ) y Q const. critical flow sub-critical flow supercritical flow α v α Q E y + y + g gs d E d energy head of cross section (specific energy) E d f (y) for Q const. determination of minimum E d f (y) ded α Q ds 1 0 dy g S dy S f (y) ds Bdy E dmin E d K141 HYAE Open-channel hydraulics 8 α Q B 1-0 g S

α Q g S B - general condition of critical flow y k Determination of critical depth y k a) from E d f(y) E dmin y k b) from general condition - analytically α Q g possible only exceptionally: S f (y), B f (y) for rectangle: B b, S k b y k, specific discharge k S b B k yk y α Q k gb α g q q Q b c) from general condition - graph. numer. d) iteratively (approximation) e) empirical formulas K141 HYAE Open-channel hydraulics 9

Transition through critical depth Q y k i k e.g. from Chézy equation Froud number - from general condition of critical flow Fr α Q g application of continuity equation Q B y s v, α 1 : B S 1 Q B gs v ysb gy B s v gy s v gy s Fr y s S B - mean depth Fr 1 - critical flow g y s v k velocity of wave front on water level K141 HYAE Open-channel hydraulics 10

Determination of type of flow (regime of flow) Flow Fr y v i critical Fr 1 y y k v v k i i k sub-critical Fr < 1 y > y k v < v k i < i k supercritical Fr > 1 y < y k v > v k i > i k K141 HYAE Open-channel hydraulics 11

NON-UNIFORM FLOW in direction of flow : depth increases backwater curve depth decreases drawdown curve Profile of free surface - example backwater subcritical flow drawdown subcritical flow i 0 < i k backwater supercritical flow i 0 < i k hydraulic jump subcritical flow i 0 < i k K141 HYAE Open-channel hydraulics 1

Determination of free surface profile Bernoulli equation 1 : i i 0 0 L + y L 1 αv1 + g α v v g + Z ( ) ( 1 ) y y + i L 1 Expression of i E from Chézy equation: v v C R i i y αv + g L Q Z E E Cp Rp Cp Sp Rp v i 1 v y y 1 i 0 L 1 index p values calculated from depth y p 0,5(y 1 +y ) (event. average of values in pf. 1 and ) E i E K141 HYAE Open-channel hydraulics 1

general method step method, both for regular and natural channels principle: utilization of BE Z Z t + Z m : Z Z t m C L p Q S p R p L ( v ) α v d mξ g h z Q Bernoulli equation 1 : α vh α vd h h + h d + + Z g g ( v ) d - vh α hh - h d z + Z g αq 1 1 z g Sd Sh ( 1mξ) K141 HYAE Open-channel hydraulics 14 1 S backwater v d < v h - ; drawdown v d > v h + α g d 1 S h + C + Z p L S R p p

v d < v h sub-critical flow - backwater supercritical flow - drawdown v d > v h sub-critical flow - drawdown supercritical flow - backwater gradual contraction of channel: ξ 0,0 0,1 gradual widening of channel : ξ 0, 1,0 sudden widening, contraction: ξ 0,5 1,0 Calculation for known Q: - sub-critical flow against flow direction; supercritical in flow dir. - known profiles L i,i+1, C i f(h), S i f(h) ( ξ), R i f(h) - known initial level (pf. 1) + estimate in pf. C p, S p, R p - calculation of z improved estimate of level in pf. - when improved estimate previous estimate further reach K141 HYAE Open-channel hydraulics 15

HYDRAULIC JUMP transition from supercritical to sub-critical flow direct (with bottom regime) undular Fr 1 L s y k L s practical significance: kinetic energy dissipation bellow spillways, weirs, dams... stilling pool K141 HYAE Open-channel hydraulics 16