Hydraulic Jumps and The Momentum Principle

Hydromechanics VVR090

ppt by Magnus Larson; revised by Rolf L Feb 2014

Hydraulic Jump

(photo taken in M. Larson’s kitchen sink)

SYNOPSIS

1. Specific Momentum2. The Hydraulic Jump 3. Submerged Jump4. Energy Loss in a Hydraulic Jump5. Hydraulic Jump Length6. Example

1. Specific Momentum

' ' ' ' ' '1 3 2 2 2 1 1fF F F P Q u u

g

Horizontal momentum equation is applied to a control volume in an open channel (x-direction):

For small slopes:

1 1 2 2 2 1fz A z A P Q u ug

2 2

1 1 2 21 2

1 2

2

f

f

P Q Qz A z AgA gA

PM M

QM zAgA

Continuity equation:

1 21 2

,Q Qu uA A

Rearranging the momentum equation:

(where M = specific momentum)

Specific Momentum Curve

Analogous to the specific energy (E) curve.

For a given value on M there are in general two possible water depths (sequent depths of a hydraulic jump).

The minimum value of M corresponds to the minimum value of E.

M

y

2. The Hydraulic JumpA hydraulic jump results when there is a conflict between upstream and downstream control.

If upstream control induces supercritical flow and downstream control induces subcritical flow.

=> flow has to pass from one regime to another creating a hydraulic jump

Significant turbulence and energy dissipation occurs in a hydraulic jump.

Applications of the Hydraulic Jump

1. Dissipation of energy in flows over dams, weirs, and other hydraulic structures

2. Maintenance of high water levels in channels for water distribution

3. Increase of discharge of a sluice gate by repelling the downstream tailwater

4. Reduction of uplift pressure under structures by raising the water depth

5. Mixing of chemicals

6. Aeration of flow

7. Removal of air pockets

8. Flow measurements

Hydraulic Jump Dependence on Froude Number

Hydraulic Jump in a Sink

Governing Equation for a Hydraulic Jump I

Apply the momentum equation for a finite control volume encompassing the jump and let Pf = 0:

1 22 2

1 1 2 21 2

M M

Q Qz A z AgA gA

For a rectangular section:

1 1 2 2

1 1 2 2

1 1 2 2

,/ 2, / 2

Q u A u AA by A byz y z y

Governing Equation for a Hydraulic Jump II

Momentum equation (rectangular channel):

2 21 2

2 12 2y y q u u

g

Momentum equation for rectangular section:

2

2 22 1

1 2

1 1 12

q y yg y y

Solutions:

221

1

212

2

1 1 8 12

1 1 8 12

y Fry

y Fry

Downstream Froude number (Fr2) usually small so:

221 8 1 0Fr

Sometimes useful to do a Taylor series expansion:

2 2 4 62 2 2 21 8 1 4 8 32 ...Fr Fr Fr Fr

Substituting into the hydraulic jump equation:

2 4 612 2 2

2

2 4 16 ...y Fr Fr Fry

2 31 1 11 1 ...2 8 16

x x x x ( )

3. Submerged Jump

If the downstream water level is high enough (> sequent depth y2) the jump might be submerged. The depth of submergence (y3) is then a crucial unknown.

Could occur downstream gates etc.

Submergence of Hydraulic Jump

Govinda (1963) proposed the following formula for submerged jumps (horizontal, rectangular channels):

Chow (1959):

1/ 22

2 2 23 11

1

4 1

2

221

1

21 21

1 1 8 12

y FrS Fry S

y ySy

y Fry

1/ 2

23 44

4 1

1 2 1y yFry y

4. Energy Loss in a Hydraulic Jump I

Dissipation in a horizontal channel:

1 2E E E

Rectangular section:

32 1

1 2

222 1 1 1 2

21 1

3/ 22 21 11

2 22 1 1

4

2 2 / 1 /

2

8 1 4 1

8 2

y yE

y y

y y Fr y yEE F

Fr FrEE Fr Fr

Energy Loss in a Hydraulic Jump II

2 21 2

1 2

2 21 1

1 222

22 12 1 1 2

2 1

2 2

12

44

u uy y Eg g

u yE y yg y

y yE y y y yy y

Energy equation across the jump:

32 1

1 24y y

Ey y

Example

Given a channel of rectangular cross-section. If a hydraulic jump occurs when at the downstream section depth y = 0.5 m and velocity u = 0.7 m/s then

what isa) The depth upstream of the jump ?b) The amount of energy (m) lost in the jump ?c) The power (W) lost ?

5. Hydraulic Jump Length I

Jump length hard to derive through theoretical considerations.

Length of the jump (Lj) is defined as the distance from the front face of the jump to a point on the surface immediately downstream the roller associated with the jump.

Hydraulic Jump Length II

Empirical equation for jump length given by Silvester (1964) for rectangular channels,

1.0111

9.75 1jLFr

y

and for triangular channels:

0.69511

4.26 1jLFr

y

Submerged jump:2

4.9 6.1jLS

y

Hydraulic Jump in Sloping Channels

A number of different cases have to be considered.