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The Momentum Principle

Hydromechanics VVR090

Hydraulic Jump

(photo taken 2008-02-24 in M. Larson’s kitchen sink)

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Specific Momentum

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

gγ

+ − − = β −β

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

Small slopes:

( )1 1 2 2 2 1fz A z A P Q u ugγ

γ − γ − = −

2 2

1 1 2 21 2

1 2

2

f

f

P Q Qz A z AgA gA

PM M

QM zAgA

⎛ ⎞ ⎛ ⎞= + − +⎜ ⎟ ⎜ ⎟γ ⎝ ⎠ ⎝ ⎠

= −γ

= +

Continuity equation:

1 21 2

,Q Qu uA A

= =

Rearranging the momentum equation:

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Specific Momentum Curve

Analogous to the specific energy 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.

The Hydraulic Jump

A hydraulic jump results when there is a conflict between upstream and downstream control.

Example: upstream control induce supercritical flow and downstream control 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.

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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

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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

= == == =

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Governing Equation for a Hydraulic Jump II

Momentum equation (rectangular channel):

( )2 21 2

2 12 2y y q u u

gγ γ γ

− = −

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

= + −

= + −

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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+ = + − + +( )

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.

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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

⎡ ⎤= + φ − +⎢ ⎥+ φ⎣ ⎦

−=

φ = = + −

1/ 2

23 44

4 1

1 2 1y yFry y

⎡ ⎤⎛ ⎞= + −⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦

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Energy Loss in a Hydraulic Jump I

Dissipation in a horizontal channel:

1 2E E EΔ = −

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

−Δ =

⎡ ⎤− + −Δ ⎣ ⎦=+

+ − +=

+

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

+ = + + Δ

⎛ ⎞Δ = − + −⎜ ⎟

⎝ ⎠

− ⎡ ⎤Δ = + −⎣ ⎦

Energy equation across the jump:

( )32 1

1 24y y

Ey y−

Δ =

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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,

( )1.011

1

9.75 1jLFr

y= −

and for triangular channels:

( )0.6951

1

4.26 1jLFr

y= −

Submerged jump:2

4.9 6.1jLS

y= +

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Hydraulic Jump in Sloping Channels

A number of different cases have to be considered.