34 location of hydraulic jump 374s24

7/23/2019 34 Location of Hydraulic Jump 374s24

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Objectives: To have understanding of the occurrence and characteristics of

hydraulic jump

To be able to locate the hydraulic jump

Hydraulic jump

The hydraulic jump is an important example of local non-uniform flow. As can be

seen from the water surface profiles in Fig. 1, there is no possibility of a smooth

transition from shooting to tranquil flow since, theoretically, the slope of the water surface should be ertical as it passes through the critical depth. !n practice this cannot

occur, and the transition ta"es the form of the hydraulic jump with a steep upward-

sloping water surface and iolently turbulent conditions accompanied by a substantialloss of energy. As for steady flow, the mass per unit time flowing upstream and

downstream of the jump will be equal and, since the elocity upstream is greater than the

elocity downstream, there will be a change in the momentum of the stream per unit time

as it passes through the jump. For a channel with a moderate bed slope, the force

slowing down the stream and producing this rate of change of momentum is due to thedifference in the resultant forces caused by the hydrostatic pressure at the downstream

and upstream cross-sections.

Fig. 1 #ydraulic jump

!f $ is the olume rate of flow, % the width of the channel, assumed to berectangular, &1 and &', 1 and ', the depths and mean elocities at the upstream and

downstream sections, respectiely, for continuity of flow,

$ ( %1&1 ( %&'' and s), ' ( 1*&1+&'

ate of change of momentum between section 1 and ' ( ρ$*1-'

( ρ%1'&1*1-&1+&'

#ydrostatic force acting downstream at section 1 ( 1+'ρg&1'%,

#ydrostatic force acting upstream at section ' ( 1+'ρg&''%,



esultant force acting upstream (1+'ρg%*&''-&1

'.

%y ewton/s second law,

ρ%1'&1*1-&1+&' ( 1+'ρg%*&'

'-&1'

&''

-&1'

( *'1&1+g&'*&1-&'&' 0 &1 ( '1&1+g&'

&'' 0 &1&' '1

'&1+g ( ),

2,+31*14'

11

'

11' gDv D D ++−= *1

From equation 1, the conjugate depths &1 and &' before and after the jump can be

determined, since 1+√*g&1 ( Fr, the Froude number at the upstream section,

2,31*14'

1 '

11'

Fr D D ++−=

*'

The loss of energy in the jump will be equal to the difference of specific energies

at the upstream and downstream sections.

5oss of energy ( *&1 0 1'+'g - *&' 0 '

'+'g

Location of an hydraulic jump

!t is often desirable to be able to determine the position at which an hydraulic

jump will occur. For example, in the design of a spillway oer a dam *Fig. ', the energyof the fast flowing stream must be partially dissipated to preent erosion of the bed

downstream. This can be done by arranging for the formation of an hydraulic jump, but

to preent damage this must occur on the apron. 6hooting flow down the face of the damis retarded by the flatter slope of the apron, which is insufficient to maintain its original

high elocity, and a jump will occur. An obstruction can be introduced on the apron to

force the jump to form at the desired position but, if this is not done, the position at whichthe jump will occur naturally can be estimated using *1 to determine the possible

conjugate upstream and downstream depths for the "nown upstream and downstream

conditions.

!f the discharge and elocity at the foot of the spillway are "nown and thedownstream depth &' is fixed, the conjugate depth &1 can be determined. From equation

*1, if q is the discharge per unit width,

2,+31*14'

1 7

'

'

'1 gDq D D ++−= *7

6tarting from the "nown conditions at the foot of the dam, the distance along theapron at which the depth of flow will hae increased to &1 can be calculated.



Fig. '

!f the downstream conditions are not fixed, the position of the jump can still bedetermined by applying equation *1 to find the conjugate depths which are compatible

with the surface profiles upstream and downstream. For example, if the slope of the

channel changes from steep to mild *as in Fig. 7, the jump may occur either upstream or downstream of the brea" in the slope. To decide which of the two alternaties is possible,

first determine the normal depth for the upstream and downstream slopes. 8onsidering

the possibility of the jump occurring upstream of the brea", calculate the conjugate depthcorresponding to the normal depth on the downstream slope, the jump will form on the

upstream slope and be followed by an 6' cure leading to the normal depth downstream.

9n the other hand, if it is greater than the normal depth on the downstream slpe, the jump

cannot occur upstream of the brea" *as in Fig. 7*a, the depth after the jump beingnormal on the downstream slope and the corresponding conjugate depth & 1/ occurring

immediately before the jump. An :7 cure is formed upstream of the jump.

Fig. 7

34 location of hydraulic jump 374s24

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