Hydraulic jump (lab. scale) http://www.youtube.com/watch?v=C

x6-_eJHdxY

Hydraulic jump (field scale)

http://www.youtube.com/watch?v=h45FauagdXw

Downstream elevation (or Tail water) conditions are controlling the H.J. Upstream conditions defines the discharge Q and the supercritical regime

y1 y2

Q Q

Hydraulic jump

Momentum equation

CV

supercritical flow F>1 subcritical flow F<1

pressure forces

Momentum equation is applied: 1)when energy losses are unknown or unpredictable (typically for complex flow patterns) 2) when CV is clearly identifiable (far from messy BC), 3) and when external forces are well defined at the BC

Definition: Momentum function

M =𝐴ℎ𝑐 +𝑄2

𝑔𝐴2 (hc centroid of the area A)

In a hydraulic jump the momentum is conserved and the only external forces acting on the CS are the ...

In

CS

InOut

CS

Outext vmvmF

CV

dvρdt

d

section 2 section 1 CV

2

2

22c2

1

2

11c1

12

22c11c

c1

1221

hh

)( ρQρghρgh

ρgh

:

)ρQ(

gA

QA

gA

QA

A

Q

A

QAA

AF

where

vvFF

p

pp

Fp2 Fp1

M1 = M2

derivation 37 BB

For a rectangular section:

pressure chydrostati assuming sect.r rectangula afor 2/hc y

2

22

2

1

22

1

2

y

2

y

gby

Qb

gby

Qb

v2

x

Mom. Eq. xdirection with CV defined by sec 1 and 2

2

811

y

y

2

1

1

2

F

the higher F1 (the more supercritical the ups. flow) the larger will be y2/y1 and the more dissipative will be the jump

The length of the H.J. is determined experimentally: L ~ 6y2 (good approx.)

RECTANGULAR CHANNEL

0y

y definitionby

1

2

(discard the negative solution)

Momentum – energy equation: estimate dissipation

2/1

2/1

2/y

2/y1

E

E1

E

E

;

y

y

2

1

22

1

2

1

2

1

2

2

2

2

1

2

1

L

21

1

2

F

F

gyq

gyq

EEEL

given Q, y1, calculate yc and define the supercritical condition F1

use the momentum equation to obtain y2 use energy equation to estimate dissipation

Let us make up an exercise: Q, b, y1 ? find y2 (sequent depth)

Definition of the momentum function (rect. section):

conditions critical

0/22y

0dM/dy

Momentum minimum thefind uslet

2

y

2

y/

2

y

3/12

1

2

1

2

12

2

1

1

22

1

1

2

22

11

1

22

11

1

g

qy

gy

qy

gy

q

gy

q

ygb

QbM

gby

QbM

E/yc=3/2=1.5 F=1 minimum energy and momentum

F>1

F<1

1

2

1 2 on M/b= const 11’ keeping y const and reaching the E (y) 22’ keeping y const and reaching the E (y)

1’

2’

Energy losses EL = E(1’)-E(2’)

EL

note that as F>>1 y1 small, 1-1’ increases and E loss increases

EL increases or decreases with F1 ?

Trapezoidal section BB 36, 38-39

Graphical solutions: enter with Z (given geometrical parameters m,b or d) obtain the sequent depth ratio: y2/y1 (obviously one of the two depths must be known)

TRAPEZOIDAL/CIRCULAR CHANNELS

2D vortices, persistent and inducing a larger blockage: y2 is larger for a given F1

2D

3D

Generic cross section

1/2

2

3

2

2

2

2

2

2

1

(gD)

VF conditions critical

110

)1

()(

1

0)(

0dM/dy

Momentum

minimum thefind uslet

FgA

BQ

gA

BQA

AdAydAdy

dydA

AA

dy

dAh

dy

d

ydAA

h

dy

dA

gA

QAh

dy

d

gA

QAhM

c

c

c

c

Realistic H.J dissipation and roughness 42 BB

Stilling basins

GOAL: 1) dissipate energy, reduce velocity and erosion in the downstream river reaches 2) control the location of the hydraulic jump and its intensity 3) operate correctly for a wide range of discharges

SAF stilling basin

y1 (F>>1) y2 (F<1) sequent depths Tw = tailwater B.C.

Note that: if the HJ moves downstream, we would have extensive erosion on an erodible layer if HJ moves upstream, it would be submerged with limited energy dissipation. So how can we stabilize it ?

y1

y2

Tw1

Tw2

Tailwater > y2 HJ moves downstream

Tailwater > y1 HJ moves upstream

The supercritical Froude number F1 is a key term in the calculation of the sequent depths. HJ changes with F1 and thus the roughness characteristics of the stilling basins should depend on the Froude number

2

811

y

y

2

1

1

2

F

Type III basin Fr > 4.5 Type II basin 2.5<Fr < 4.5

Fr=3.01 TW=4.2*3.99 =16.7 OK d2=3.8 *3.99=15.1 OK

max block with w=y1 spacing =2.5w height 2y1 sill=1.25y1

design goal : match the sequent depth and the tailwater depth for all discharges

unrealistic... often Tw depends on downstream conditions and it is independent of the discharge Q So, we work with QMAX

what are the design degree of freedom? 1) Δz

2) widening Δb

Δz

Δb

by increasing width, we lower the depth and increase the Froude number we aim at 4.5<F<9 to have a stable HJ (not wavy, more controllable)

example case in which both the sequent depth and the tailwater change with discharge, in a different way (note that the tailwater is a given b.c., level imposed by valley) case A: y2=Tw at max discharge doubling the width, we reduce the difference between y2 and Tw at all discharges case B: y2<Tw at max discharge, implying that at QMAX 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=1.1 y2 so 10% larger

EXAMPLE BB

Tidal bores

Surges (unsteady hydraulic jumps)

note: the B.C. V2 depends on the gate closure (total shut off in the case of V2 =0)

momentum 12

1)V(

continuity

1

2

1

2

1

2

1

12

2211

y

y

y

y

gy

V

yy

yVyVV

s

s

Demonstration BB

BRIDGE PIERS

gs

Vay

gy

qy

gy

qy

MMFext

2

c

22

/

2

11d

4

22

4

1

22

1

21

ALGEBRA