Hdraulic jump (lab. scale) http://www.outube.com/watch?v=c x6-_ejhdxy Hdraulic jump (field scale) http://www.outube.com/watch?v=h 45FauagdXw Downstream elevation (or Tail water) conditions are controlling the H.J. Upstream conditions defines the discharge and the supercritical regime
Momentum equation CV Hdraulic jump supercritical flow F> subcritical flow F< Definition: Momentum function M =h c + (h g c centroid of the area ) In a hdraulic jump the momentum is conserved and the onl external forces acting on the CS are the... pressure forces Momentum equation is applied: )when energ losses are unknown or unpredictable (tpicall for complex flow patterns) ) when CV is clearl identifiable (far from mess BC), 3) and when external forces are well defined at the BC
F ext d dt CV ρvd m v CS CV m v Out Out In In section CS section v F p F where : p Mom. Eq. xdirection with CV defined b sec and ρ( v v ) x F p F p F p ρgh h ρgh c c c ρgh g c h ρ ( c g ) For a rectangular section: h c / b for a rectangular sect. gb b gb M = M assuming hdrostatic pressure derivation 37 BB
RECTNGULR CHNNEL 8F b definition 0 (discard the negative solution) the higher F (the more supercritical the ups. flow) the larger will be / and the more dissipative will be the jump The length of the H.J. is determined experimentall: L ~ 6 (good approx.)
Momentum energ equation: estimate dissipation / / / / E E E E ; L F F g q g q E E E L given,, calculate c and define the supercritical condition F use the momentum equation to obtain use energ equation to estimate dissipation Let us make up an exercise:, b,? find (sequent depth)
M M Definition of the momentum function (rect. section): let us find the minimum Momentum dm/d b / b q g 0 q / g /3 gb gb 0 critical conditions q g q g F= minimum energ and momentum E L increases or decreases with F? E/ c =3/=.5 note that as F>> small, - increases and E loss increases E L F< F> on M/b= const keeping const and reaching the E () keeping const and reaching the E () Energ losses E L = E( )-E( )
Trapezoidal section BB 36, 38-39
TRPEZOIDL/CIRCULR CHNNELS Graphical solutions: enter with Z (given geometrical parameters m,b or d) obtain the sequent depth ratio: / (obviousl one of the two depths must be known) D 3D D vortices, persistent and inducing a larger blockage: is larger for a given F
Generic cross section M h c g let us find the minimum Momentum dm/d 0 d d h c d d ( h c ( h c ) ) d g d d d d ( 0 d) d d d d B g B 0 3 g critical conditions F V (gd) / F Realistic H.J dissipation and roughness 4 BB
Stilling basins SF stilling basin GOL: ) dissipate energ, reduce velocit and erosion in the downstream river reaches ) control the location of the hdraulic jump and its intensit 3) operate correctl for a wide range of discharges
T w T w Tailwater > HJ moves downstream Tailwater > HJ moves upstream Note that: if the HJ moves downstream, we would have extensive erosion on an erodible laer if HJ moves upstream, it would be submerged with limited energ dissipation. So how can we stabilize it? (F>>) (F<) sequent depths Tw = tailwater B.C. 8F The supercritical Froude number F is a ke term in the calculation of the sequent depths. HJ changes with F and thus the roughness characteristics of the stilling basins should depend on the Froude number
max block with w= spacing =.5w height sill=.5 Fr=3.0 TW=4.*3.99 =6.7 OK d=3.8 *3.99=5. OK Tpe II basin.5<fr < 4.5 Tpe III basin Fr > 4.5
Δb Δz design goal : match the sequent depth and the tailwater depth for all discharges what are the design degree of freedom? ) Δz ) widening Δb b increasing width, we lower the depth and increase the Froude number we aim at 4.5<F<9 to have a stable HJ (not wav, more controllable) unrealistic... often Tw depends on downstream conditions and it is independent of the discharge So, we work with MX
example case in which both the sequent depth and the tailwater change with discharge, in a different wa (note that the tailwater is a given b.c., level imposed b valle) case : =Tw at max discharge doubling the width, we reduce the difference between and Tw at all discharges case B: <Tw at max discharge, impling that at MX the HJ will be moving upstream, towards the structure (reduced dissipation, but avoid erosionok) To be safe and keep the HJ in the basin it is often recommended to have Tw=. so 0% larger EXMPLE BB
Tidal bores Surges (unstead hdraulic jumps) note: the B.C. V depends on the gate closure (total shut off in the case of V =0) momentum ) (V continuit g V V V V s s Demonstration BB
BRIDGE PIERS F ext / M M LGEBR q g 4 q g 4 c d av gs