21. Channel flow III (8.10 8.11) 1. Hydraulic jump 2. Non-uniform flow section types 3. Step calculation of water surface 4. Flow measuring in channels 5. Examples E22, E24, and E25
1. Hydraulic jump Occurs at junction between supercritical subcritical flow Froude number F>1 upstream and F<1 downstream In practise a standing wave Implies large energy losses
Equation for corresponding depth Momentum eqn Note! Relation symmetrical
Hydraulic jump energy relations
2. Non-uniform flow section types Problem solving, steps Definitions Principles Type profiles Profile types / mild bottom slope Profile types / steep bottom slope Control sections
Problem solving, steps Problem solving require analysis of possible water surface profiles Analysis based on definitions and principles Control section identification Between control section water surface profile is calculated by step iteration
Q given Definitions Natural depth D N. Equals uniform flow. Critical depth. D C. Occurs: Junction subcritical supercritical flow (abrupt change) Junction supercritical subcritical flow (gradual change) Critical slope S C, slope giving D N = D C Mild slope S o < S C and natural depth > D C Brant lutning: S o > S C and natural depth < D C
Principles Natural depth D N Water surface approaches D N asymptotic Critical depth D C Water surface approaches D C abruptly Supercritical flow - Water surface profile is affected from upstream Subcritical flow - Water surface profile is affected from downstream
Water surface profiles - All possible types in Fig. 8.29 - Classification from bottom slope - Water depth gradient can be determined from Eqn. 8.41: dd/dl = (S o S f ) / (1 F 2 )
Profile types / mild bottom slope
Profile types / steep bottom slope
Profile types / mild bottom slope, steep bottom slope
Control sections Control sections are locations with known relationships flow/depth, Q = Q(D) Example of control sections Increase of bottom slope from mild to steep with critical depth Control structures, e.g., gates with known Q = Q(D) Inflow to channels where D = D N (mild slope) Inflow to channels where D = D C (steep slope)
Control sections examples
3. Step calculation water surface profile Principles Calculation of friction slope S f Step calculation, procedure Standard step method Direct step method
Principles Calculation of water surface is based on energy eqn: [Z + D + V 2 /2g ] 1 = [Z + D + V 2 /2g ] 2 + h f (1) Energy loss h f is calculated as h f = S f L (2) where L = distance between section 1 and section 2 and S f (= energy line slope) from Manning s eqn V = 1/n R 2/3 S f ½ S f = V n / R 4/3 (3) Note that S f is not bottom slope.
Calculation of friction slope S f At step calculation of water surface S f can not be expressed exactly since relations vary between section 1 and 2 Some kind of mean value must be used: Alt. 1 S f = (S f1 + S f2 ) / 2 where S f1 = energy line slope in section 1 S f2 = energy line slope in section 2 Alt. 2 S f = V av n / R av 4/3 where V av = (V 1 + V 2 ) / 2 DR av = (R 1 + R 2 ) / 2
Step calculation, procedure Calculation is done stepwise: - known relationships in section 1 - energy eqn and Manning eqn relation in section 2 - at subcritical flow calculations are done in upstream direction - at supercritical flow calculations are done in downstream direction Two alternative methods for calculations
Standard step method Standard step method: - known relationships in section 1 - assume a distance ΔL to section 2 - guess depth at section 2 - test energy eqn - adjust depth at section 2 - iterate until the error is small
Direct step calculation Direct step method : - known relationships in section 1 - assume a depth in section 2 - use Manning eqn S f -calculateδl with: ΔL= (E 1 E 2 ) / (S f S o ) ΔL Presupposes: prismatic channel, i.e., geometry known at section 2
4. Flow measurements in channels xxx
5. Examples E22, E24, E25 Sharp-crested weirs Wide weirs Bottom outlets, gates
Sharp-crested weirs Energy eqn (ideal flow) + integration Q = (2/3) (2g) 1/2 B H 3/2 Non-ideal flow compensate (multiply) with discharge coefficient
Wide weirs Assume rectangular section Critical conditions Q = g 1/2 B D c 3/2 For practical measurements: Q = C B H 1 3/2
Bottom outlets, gates For practical measurements (fig 9.12 a): Q = C a o (2g H 1 ) 1/2 where a o = opening area