Rapidly Varied Flow The Hydraulic JumpUse of hydraulic jumpsSpecific force revisitedSequent or conjugate depthsTypes of jump

Uses of Hydraulic JumpsHydraulic Jumps can and do occur naturally for example, in and after rapids in riversThey are also deliberately created:To dissipate energy after a spillway etc and so prevent scouringTo recover head downstream of an obstructionTo induce mixing and aeration

Specific Force RevisitedOver a short reach of channel, external friction forces can be ignored and if the slope is mild the downstream self-weight body forces can also be ignored.

Thus.

and

giving

Sequent or Conjugate Depths

These terms refer to the depths at sections 1 and 2, before and after a hydraulic jump

1

2

Evaluation of Conjugate DepthsIn general form:

and F1 = F2, referring to diagram on previous slide.

For a wide or rectangular channel, A=By and Hence, dividing by B:

Which leads to:

These equations can be used to determine the depths upstream or downstream of a hydraulic jump.

The Froude Number Fr is used to characterize a jump.

Undular Jump

Fr1 = 1.0 to 1.7y1 near ycry2/y1 lowLittle energy dissipationAs Fr1 increases towards 1.7 the first undulation becomes more pronounced and will eventually break.

Weak Jump

Fr1 = 1.7 to 2.5Little energy dissipationSurface fairly smoothBreaking of first undulation gives small rollers on the face of the jump.

Oscillating Jump

Fr1 = 2.5 to 4.5High velocity entering jump.This oscillates from bed to surface and back, producing waves of irregular period downstreamAvoid in design.

Steady Jump

Fr1 = 4.5 to 9.0Jet oscillation disappearsSurface roller increases in length till it ends where the jet reaches the surface45-70% energy dissipationAim for in design

Choppy Jump

Fr1 = >9.0Surface roller highly aeratedJet penetrates for a long distance downstream, requiring a long deep stilling basin.

ImportantYou have to use Specific Force for hydraulic jumps because the friction loss in the fluid due to the eddying is significantSpecific Energy does not work because it neglects frictionSpecific Force wont work on weirs etc because it does not consider the force on the weir itself.Get this right!!!!!

ExampleFlow enters a channel through an undershot sluice gate. q=2.5m3/s/m, n=0.025 and S0=0.001039. The depth upstream of the gate is 5.5m. Determine where the hydraulic jump forms. Assume no energy losses, and that the channel is wide compared to its depth.

Locating a Hydraulic Jump

The location of the hydraulic jump is controlled by the downstream depth (typically normal depth).If y2 < yconj, the flow profile downstream of the gate will expand in an M3 or H3 profile until the conjugate depth to y3 is reached.The jump will then occur.

But, if y2 = yconj, the jump forms at the vena contractaWhilst if y2 > yconj, the jump moves upstream and may break against the gate.

Downstream Depth ControlledDownstream depth may be greater than normal depth due to a further restriction (a weir, sluice gate, overfall etc)If downstream depth > y3, then jump moves upstream (F3 > F2)If downstream depth < y3 then jump moves downstream (F2 > F3)In either case, jump forms when yconj related to the actual downstream depth is reached.

Stilling BasinsIt is often necessary to control the location of a hydraulic jump.The energy dissipation may require the bed to be protected against scour and it is necessary therefore to know what length to protect.A step in the bed is used to control the position of the jump, ensuring that y3 is artificially raised to the required level to force the jump.

Typical Stilling Basin Design

Expansions and Contractions

Vertical ContractionConsidering a vertical contraction, the momentum change between sections 2 and 3 may be written as:

P3a can be obtained from hydrostatics.

Then, for a wide channel:

This is a step-up; use it to move or fix jump upstream

Vertical ExpansionSimilarly:

Step-down; use to move jump downstream

Horizontal Expansions and Contractions

Similar principles apply.

Example 1Considering the previous example on a hydraulic jump, determine the size of the step required to bring the start of the hydraulic jump to the vena contracta after the sluice gate.

Example 2Re-evaluate the previous problem with the slope S0 = 0.000129. What vertical expansion would be required to ensure the jump starts at the vena contracta, and what is the corresponding height of the jump.

Control StructuresCritical Depth MetersSluice Gates to control dischargeStilling BasinsBaffle Blocks in Stilling Basins

Critical Depth MetersAny structure at which critical flow occurs gives a fixed relationship between depth (stage) and discharge.

cf Lab experiment three types of weir.

Broadcrested weir:

This type of weir is cheaper to construct and more resilient than sharp-crested or V-notch, but V-notch is more sensitive at low flows.Can connect to chart recorder or satellite etc.

Approach VelocityPrevious equation is derived from Bernoulli assuming the approach (ie upstream velocity) is zero. If it isnt:

Various values and formulae for CD, which can vary particularly at low flows when viscosity and surface tension come more into play

SubmergenceModular flow is flow when downstream level is too low to affect flow over weir you design for this.Modular limit occurs when downstream level rises above this limit and flow over weir is affectedCan use Villemonte formula if downstream depth y3 can be estimated.

Sluice GatesGiven upstream water levelDetermine required gate opening to pass a particular discharge

Energy principle applies:

y2 can be expressed in terms of the gate opening yg: y2 = Ccyg where Cc is a coefficient of contraction, typically 0.61.

Also, the coefficient of discharge Cd can be defined as:

Hence:

ExampleA vertical, underflow sluice gate 3m wide is opened to 0.25m above the bed of a 3m wide rectangular channel. The depth upstream of the gate is 2.5m, and the channel bed slope is 0.002 with Mannings n of 0.028. Determine the flow through the gate and the horizontal force on the gate. Take Cc = 0.61.

Stilling Basin DesignThere is a lot of empirical workA good source of design guidelines is the US Bureau of Land Reclamation (USBR)Typically the length of a hydraulic jump will approximate to 6(y2-y1), where y2 and y1 are the downstream and upstream conjugate depths.Baffle blocks are often used to increase energy dissipation.

ExampleIf, for the arrangement sketched, y1 = 0.15m and y2 = 4.0m, and the discharge per unit width q = 4 m3/s/m, determine the force on the baffle blocks.

(note: force on baffle blocks affects d/s specific force and so the height of the jump)

ReferencesChadwick and Morfett pages 142-149Also pages 405-408; 427-430 and 430-435.

Chapter 13 on Hydraulic Structures is all worth a read.

CulvertsDefinitionDesign of culvertsAnalysis of existing culverts

Definition of CulvertsCovered channel or pipe carrying a watercourse under an obstruction such as a road.May extend for many kilometres in cities.Typically made from precast sections.

Definition Sketch

Design Principlesadequate size to pass debris thus avoiding need for a trash screenself-cleansing (debris and silt removal)avoid slope or cross-section changes which might reduce capacity, catch debris or lead to silting upvisual appearanceease of constructionlow maintenancerisk to children, drunken students etc.

Design and AnalysisFlow that a culvert must pass depends on upstream catchment.Determine tailwater depth for this flow based on d/s channel.Headwater depth depends on allowable depth eg to avoid overtopping.Various types of flow can exist as follows.

The six possible types of flow as shown.Free flow and surcharged conditions.Type varies with flow so multiple analysis required.Aim for Type III in designSurcharging increases head loss and can lead to upstream flooding.

ExampleAn existing box culvert is 1.5m wide, 1.2m high and 35m long, and has a slope of 0.005. There is a trash screen at the inlet with 12 20mm diameter bars in a 1.9m wide rectangular channel. The downstream channel is rectangular, 3.5m wide and 1.0m deep with a bed slope of 0.0007 and n=0.025. The maximum allowable headwater depth is 2.1m.

Determine the capacity of the culvert.

ReferenceNone for this part.