Chapter 3

Channel Controls

CE 473

Open-Channel Flow

3.1 Introduction A control is any channel feature, which fixes a relationship between depth and discharge in its neighborhood. It may be natural or human-made. Also we may have:

•Overflow structures -→spillways, weirs, free falls

•Underflow structures-→sluice gates, gates

Why use need control structures?

1.Flow profile computation-provides the boundary condition.

2.To measure discharge;

As an engineer we are concerned with the functioning of the control itself, such as the ability of a spillway to discharge floodwaters at the required rate.

We have rapid flow in the vicinity of control structures (sections).Since the streamlines are highly curved near the control sections we can not solve the equations analytically, Therefore we go to empirical relations based on experiments.

Vena Contracta

• Vena contracta is the section where the flow becomes parallel or it is

the section where area of the jet becomes constant and minimum.

Coefficient of velocity,

Coefficient of contraction

3.2 Flow through orifices and short tubes

Typical examples of orifices and short tubes are shown. As the flow issues

out vena contracta will occur.

A knowledge of the laws of the flow through them is necessary in

determining the discharge through sluiceways and the entrances to

conduits.

•

Orifices

• If the entrance is not properly shaped, a contraction of the jet occurs as

shown in sketch of vena contracta.

• The area of the jet is not as great as the area of the orifice or tube. For

properly rounded approaches to orifices, and the constant diameter short

tubes, the diameter of the jet is equal to the area of the orifice or tube.

• In case of short tubes without rounded entrances, the contraction does

occur; but the jet expands again, with certain exceptions, a partial

vacuum occurring just inside the entrance.

Let

• H = the head of water on the center line of a freely flowing orifice or

tube, or the difference in water level for a submerged orifice or tube

A= the area of the orifice and tube

V= the theoretical velocity corresponding to head H;

g= the acceleration of gravity

Energy equation between points R and N:

HR=HN

gH=U g2

U=H

Therefore

H=z and ,0=U P=P=P

,g2

U+

γ

P+z=

g2

U+

γ

P+z

N

2N

RRatmNR

2NN

N

2RR

R

⇒

H

R

N datum

Q= the discharge

Cc=the coefficient of contraction

Cf= the coefficient of friction, the reduction in total head due to friction

Cv= the coefficient of velocity

Cd= the coefficient of discharge

The general equation for the velocity of the spouting water is

Considering the friction, the actual velocity due to the head H is

The discharge is equal to the product of the actual velocity and the area

of the jet; and the area of the jet is

orif

jet

A

A

gHV 2

gH2C=V or gH2C=V factualvactual

AC=A cj

Therefore

•In experiments conducted to determine the discharge through orifices

and tubes, the coefficient of friction and the contraction coefficient are

combined and the general equation is given as:

Cd= discharge coefficient → it depends on the shape of the orifice

and tube, it is not greatly affected by the submergence. If we do not

have submergence and enough friction contracted jet does not expend

in tube. It shoots out as contracted without touching the walls.

gH2ACC=AC V=Q cvcact

cvdd CC=C where,gH2AC=Q

Orifices and Their Nominal Coefficients

Cd

•Let’s consider the following figure;

According to values of L, W and roundness, it can be a weir or sharp

crested weir.

•The discharge coefficient, Cd, will be a function of:

•→ 8 variables

•→ 3 basic dimensions, M,L,T → 5 dimensionless quantity

Dimensional Analysis

( )σ,ρ,μ,g,L,W,H,Vφ=C od

H

W

V0

L

If we choose ρ, Vo, H as primary (repeating variables); the dimensional analysis would give us:

Therefore, the discharge coefficient is a function of:

Reynolds and Weber numbers

become important at small values of H.

W,F,R,L

H,

W

Hf=C red

number Weber=HVρ

σ=W

number Froude=gH

V=F

number ynoldsRe=μ

HVρ=Re

2o

0r

0

W,RL

H,

W

Hf=C eD

dd C of measuredirect number Froude =gH2A

Q=C ⇒

A sharp crested weir normally consists of a vertical plate mounted at right

angles to the flow and having a sharp-edged crest, as shown in the figure

Sharp-crested weirs are commonly used as means of flow measurement.

3.3 Sharp-Crested Weirs

W

H V0

EGL g2V 2

0

P/g 45o

g2V 20

C h

A

B

D

• Sharp-crested weir has an importance in Open Channel Hydraulics,

simply because, its theory forms a basis for the design of spillway.

Because, the edge is sharp. → no boundary layer development only

on the vertical surface on which the velocities are small

• Therefore no viscous effects → no energy dissipation

• Let’s consider the simplest form of weir; consisting of

• a plate set perpendicular to the flow in a rectangular channel,

• its horizontal upper edge running the full width of the channel

• 2 dimensionality → no lateral contraction effects.

• Since the lateral contraction effects are suppressed by the channel

sidewalls, this type of weir is sometimes termed the SUPPRESSED

weir.

Let’s make an elementary analysis by assuming;

• no contraction over the weir

• pressure it atmospheric across the whole section AB.

• The total head at a point C is:

• The discharge per unit width then;

gh2=Vg2

V+

γ

P=h

2

0=

⇒

Hgo

o

o

o

v

gv

Hgv

gv

dhghvdhq

2/2

2

2

22/

2/

2/

2

• The discharge per unit width becomes;

• Here h is measured downwards from the EGL, not from the free

surface.

• Now, the effect of the flow contraction may be expressed by a

contraction Cc, leading to the result:

2/32

2/32

222

3

2

g

vH

g

vgq oo

2/32

2/32

222

3

2

g

vH

g

vgcq oo

c

• We can make this expression more compact by introducing a

discharge coefficient Cd;

• where;

• we should expect both Cc and the ratio of to be dependent

on the boundary geometry alone, in particular on the ratio of H/W;

• when W is large Vo is small

2/323

2Hgcq d

2/32

2/32

221

gH

v

gH

vcc oo

cd

gH2V 2o

cd

2o

CCgH2

V⇒⇒

REHBOCK found experimentally that

As W becomes very large → Cd=0.611, and

since becomes very small,

Cc≈Cd≈ 0.611.

This happens to be the numerical value of

shown last century by Kirchoff to be the

contraction coefficient of a jet issuing without

energy loss, and with negligible deflection by

gravity from a long rectangular slot in a large

tank, a two-dimensional problem.

H1000

1+

W

H08.0+611.0=Cd

Shows the viscous and the surface tension effects. Neglect unless H is very small. H is in meter.

gVo 2/2

2

2+π

π=Cc

This is the only fluid flow problem in ideal fluid for which the contraction coefficient is obtained theoretically.

• By many investigators the value 0.611 have been conformed for high weirs.

• On the other hand, when W becomes small, H/W becomes large, and this formula

cannot be true for high values of H/W. In fact, experimental work has shown that it is

true only for the values of H/W up to approximately 5. Therefore:

• For ,

• Cd begins to diverge from the value given by the formula, reaching a

value of 1.135 when H/W=10.

• If W vanishes completely, so that H/W becomes infinite, we have the

case of free over fall. In this case H=yc, critical depth, and q is

determined accordingly.

5W

H

H1000

1+

W

H08.0+611.0=Cd

10 W

H 5

• In fact, it has been shown experimentally that critical flow also occurs

just upstream of a very low weir. In this case the weir is called a sill,

i.e in the range H/W > 20.

W

H V0

EGL g2V 2

0

h

W+H=yc ccc Vy=qcc gy=V

( ) ( ) ( ) 2/32/3

2/3c H

H

W+1=W+Hg=W+HgW+H=q

• The range has not yet been completely explored.

• But Kanda Swany and Rouse showed that:

• Max. Cd occurs at H/W=10, which is the border line for weir and sill.

2/32/3

2/3d H

H

W+1g=Hg2C

3

2=q

2/3

d H

W+106.1=C

2010 W

H

Cd

4 8 0.04 0.08 10 W/H H/W

• For completely free overfull, W = 0, the equation for Cd becomes

• Cc=0.715 will be discussed later at free overfall.

1.837

2/32

2/32

221

gH

v

gH

vcc oo

cd

0.354

cd C484.1=C

715.0404.1

06.106.1106.1

2/3

cd c

H

Wc

• Two important features of the pressure distribution down the vertical

section ABC;

• Pressure distribution is nonhydrostatic, because of distinct curvature

of the streamlines in the vertical plane,

• Pressure is not atmospheric, contrary to the assumption in the

elementary analysis that :

• the pressure > the atmospheric pressure

• Consistent with the contraction and acceleration downstream of AB.

Clearly a pressure force is necessary to cause this acceleration in

the region of atmospheric pressure.

• It has been assumed throughout that the pressure is atmospheric

along the lower surface of the jet, or “nappe” as well as upper

surface if this jet is just confined between parallel walls downstream

of the weir, air will be trapped and this all will be gradually removed

by the flow, pressure in this region is reduced.

• For a given head as Q increases, there is risk of cavitation, then

ventilation is necessary.

• Any other type of weir will be three dimensional because of the

lateral contraction from the sides, as well as in the vertical plane.

Typical examples are the “contracted” rectangular weir and the

triangular weir.

Contracted Weirs

• A contracted weir involves a 3-dimensional flow problem, because of:

• the contraction from the sides as well as in the vertical plane.

• Typical examples:

• contracted rectangular weirs

• triangular weir

• Such weirs are commonly used for flow measurement, normally in

tanks large enough to be effectively infinite, so that the contraction

coefficient, Cc,has its minimum value. In this case, general equations

are available relating discharge to head; but for small tanks weirs

should be calibrated for each particular problem.

Contracted Rectangular Weirs

• Francis found experimentally that the amount of lateral contraction at

each end of the contracted weirs was equal to 0.10 H, provided that

the length L of the weir was greater than 3H. On the basis of this

result, it is commonly accepted that the discharge Q is given by;

L > 3H, Ls > 4H and W > 3H

Ls Ls

W

( ) 611.0=C ,Hg2H2.0LC3

2=Q c

2/3c

The triangular weir (V-Notch)

• The triangular weir can be analyzed in the same elementary way as

the suppressed rectangular weir leading to the result:

• The most commonly used value of the notch angle α is 90o. For this

case Cc=0.585 →somewhat less than for rectangular weir.

2/5c Hg2

2

αtanC

15

8=Q

a

• Although the value of Cc is near to 0.59 in general, it is affected by viscosity, surface tension, and weir plate roughness;

• A comprehensive study of triangular weir flow has been made by Lenz, He used many liquids in order to discover the effects of viscosity and surface tension on weir coefficients, thus extending the utility of the triangular weir as a reliable measuring decree.

• For α = 90o, Lenz proposed that

• Applicable to all liquids providing that the failing sheet of liquid does not ding to the weir plate and that H > 0.06 m, Re > 300, W > 300.

• As Re and Weber number decrease, Cc will increase.

170.0165.0

70.056.0

WRCc

V-notch weir

A V-notch weir with stilling wells is shown in figure below

Compound Weirs • Unusual situations may require special weirs. For example, a V-notch

weir might easily handle the normal range of discharges at a structure; but occasionally, much larger flows would require a rectangular weir. A compound weir, consisting of a rectangular notch with a V-notch cut into the center of the crest, might be used in this situation.

• The compound weir, as described, has a disadvantage. When the discharge begins to exceed the capacity of the V-notch, thin sheets of water will begin to pass over the wide horizontal crests. This overflow causes a discontinuity in the discharge curve (Bergmann, 1963). Therefore, the size and elevation of the V-notch should be selected so that discharge measurements in the transition range will be those of minimum importance.

• Determining discharges over compound weirs has not been fully investigated either in the laboratory or in the field. However, an equation has been developed on the basis of limited laboratory tests on a 1-ft-deep, 90-degree V-notch cut into rectangular notches 2, 4, and 6 ft wide to produce horizontal extensions of L=0, L=2, and L=4 ft, respectively (Bergmann, 1963). The weirs were fully contracted, and heads up to 2.8 ft above the notch point were used. The equation is as follows:

• where:

• Q = discharge in ft3/s

• h1 = head above the point of the V-notch in ft

• L = combined length of the horizontal portions of the weir in ft

• h2 = head above the horizontal crest in ft

• When h1 is 1 ft or less, the flow is confined to only the V-notch portion of the weir, and the standard V-notch weir equation is used.

• Further testing is needed to confirm this equation before it is used for weirs beyond the sizes for which it was developed.

5.12

72.11 Lh3.3+5.1h9.3=Q

Compound weir with 90-degree notch and suppressed rectangular

crest used by U.S. Forest Service.

North Fork 120-degree compound V-notch weir and sampling bridge

• The finish of the edge and upstream surface of a weir is important,

since the roughness of the surface or rounding of the edge tends to

suppers the lateral components of flow, increase Cc and hence

increase the discharge.

• H should be measured 3-4H upstream of weir.

3.4 Free Fall

• In this situation, flow takes place over a drop which is sharp enough that the lowermost streamline separates from the channel bed.

yc

EGL

3-4yc

yb

0.215yb Actual pressure distribution

A B

C

It is a special case of sharp-crested weir; W = 0 but it needs a special treatment, because of its use as

• a form of spillway

• a means of flow measurement because of unique relationship between the brink depth and Q.

Clearly, an important feature of the flow is the strong departure from hydrostatic pressure distribution which exist near the brink, induced by strong vertical components of acceleration in the neighborhood.

The form of this pressure distribution at the brink B will evidently be somewhat as shown in the figure, with a mean pressure considerable less than hydrostatic.

At some short distance back from the brink, the vertical accelerations will be small and the pressure will be hydrostatic. This is experimentally verified that from A to B there is pronounced acceleration and reduction in depth as shown in figure.

• Consider a long channel of two sections, one of mild upstream and one of steep slope downstream. If the upstream channel is steep, the flow at A will be supercritical and determined by the upstream conditions.

• If on the other hand the channel slope is mild, horizontal or adverse, the flow will be subcritical at A.

• Recall that the transition from mild (horizontal or adverse) slope to a steep slope.

NDL

CDL

y01 yc

O A

B

M2

S2

Sf > S0

Sf < S0

Sf = S0

yc

The flow will gradually change from subcritical at a great distance upstream to supercritical at a great distance downstream passing through critical state at same intermediate point.

In the transition region upstream of (0),

Similarly, downstream of (0)

If we consider the point 0 to be a short curve joining two long slopes,

there must be some point on this curve at which Sf=S0 and since

in this neighborhood, it follows that Fr=1. Therefore flow is critical.

0f00 S>S ⇒ V>Vy<y →

0f00 S<S V<V y>y ⇒→

0dx

dy

( )f0

2r SS=

dx

dyF1

• Imagine now that in this case the steep slope is gradually made even steeper, until the lower streamline separates and the overfall condition is reached. The critical section cannot disappear, it simply retreats upstream into a region of hydrostatic pressure i.e to A.

• The local effects of the brink is confined to the region AB; experiments shows this section to quite short, of the order 3-4 times yc.

The Head of the free overfall

• The simplest case is that of a rectangular channel with side walls

continuing downstream on either side of the free jet, so that we have

the effect of atmosphere only on the upper & lower streamlines, not

to the sides. This is a 2-D case and it is only in this form of the

problem that many investigators worked on for a theoretical solution.

• Consider the section C, a vertical section thru the jet far enough

downstream for the pressure throughout the jet to be atmospheric,

and the horizontal velocity to be constant.

• Assume that

• -the channel bottom is horizontal and

• -no resistance

• Apply momentum eq. b/w A and C, along the x direction.

11x21

21 AV=Q ,QVρ=QVρ+byγ

2

1

x211121

21 VAVρ=AVρ+byγ

2

1

y1

3-4yc

yb

A B

C

a

V2x

y’2 y

From Equation of Continuity:

x211111

2

1

2

1 VbyVbyVby2

1g

by′V=byV 2"211 αcosy=y′ 22

2x22211 yV=αcosyV=yV

V2x

12

1x2 V

y

y=V

------------------- Eq.(1)

Substituting into Eq.(1) 2

121

211 y

yV=V+yρ

2

1

Dividing by y1

1y

2

1

1

21

1

21

y

y

yρ

V=

yρ

V+

2

1

• Or rearranging we get or 2

12

1

2

1y

yFF

2

1

2

1

2

1

2

1

F2

F21

y

y

2

1

2

1

1

2

F21

F2

y

y

12 yy

1F1 if the flow is critical at section A, then y1=yc, then we have and

,

3

2

y

y

c

2 sets a lower limit on the brink depth yb:

since there is some residual pressure at the brink,

yb must be greater than y2; it follows that

1y

y

3

2

c

b

The experiment of Rouse showed that the brink section has

a depth of 0.715 yc.

• Rouse also pointed out that combination of the weir Eq.

• with the critical flow equation:

• Therefore

g2

VH+

g2

Vg2C

3

2=q

2/32o

2/32o

c

gycy=yV=q cccc coc V=V,y=H0=w →

2/3

2/3

c

2c

2/3

c

2c

cc ygy2

V1+

gy2

Vg2C

3

2=q

2/3

c

2/3

c

2/3

2/3

cc ygy2

15.1g2c

3

2q

715.0=Cc

cccb y715.0=yCy

• There have been may research on this and the conclusion suggested

by all these investigations is that a brink depth yb=0.715yc can safely

be used for flow measurements in rectangular channels with a likely

error of only 1 to or 2 %.

• We have seen that the control structures are necessary in design of

hydraulic structures, because they fix a relationship between the

depth and the discharge.

• We have seen weirs and free-overfall as control structures. But in

certain hydraulic problems, they may not be useful due to certain

disadvantages they have.

Two important disadvantages of weirs are:

1.they involve fairly large head loss when the available head may be very important to be conserved.

2.The existence of a dead-water region behind the weir where silt can accumulate and greatly change the head-discharge relationship of the weir.

Also we have seen that the critical flow establishes a fixed relationship between the depth and the discharge. But in order to apply this principle, it is necessary to create some device or use some feature that sets up critical flow at a known section in its vicinity. Then the measurement of the depth at this section enables the discharge to be calculated.

In case of free-overfall, in which, we have seen that the critical section retreats upstream to some ill-defined location; at the brink the depth is a well-defined fraction of critical depth but the rapid variation in depth calls for precise location of the depth-measuring device.

•Before considering any particular device, we first consider certain general principles. We know that the critical flow occurs at the change of channel slope at point O, provided that the pressure distribution is hydrostatic.

This will be true if the downstream slope, although steep, is not excessively so-say of the order 0.01. In this case, section O would an ideal critical-depth meter, the depth here is definitely critical, and is not changing so rapidly that the slight errors in locating the depth-measuring device would give rise to serious errors in estimates of the depth.

NDL

CDL

y01 yc

O A

B

M2

S2 yc

However, the long downstream slope is not usually available in

practice. Then we have make the downstream slope short but steep but this

brings the inconveniences of free-overfall, in which we have seen

that the critical section retreats upstream to some ill-defined region;

at the brink the depth is a well-defined fraction of critical depth, but

the rapid variation in depth calls for precise location of the depth-

measuring device.

3.5 Broad-Crested Weir • A broad crested weir is more robust than a sharp crested weir for open

channel flow measurement. Hence broad crested weirs are widely used for

flow measurement and regulation of water depth in rivers, canals and other

natural open channels. Broad crested weir flow rate calculations can be

made with a rather simple equation if the weir height is great enough to

cause critical flow over the weir crest. If there is critical flow, then the unit

discharge is:

• Then the total discharge is Q = 1.705 Cd L H3/2, where Q is the open

channel flow rate in m3/s, Cd is thw discharge coefficient, L is the weir

length (channel width) in m, and H is the head over the weir in m, as shown

in the diagram below.

2/32/3

d

2/32/3

c

705.13

2

3

2

as loss head eaccount th into take tointroduced bemay Ct coefficien dischargeA

705.13

2

3

2

and 3

2y hence

2

3H

neglected is loss head if handother On the

HCHgCq

HHgq

Hy

gyyq

dd

c

ccc

General Broad Crested Weir Configuration

A broad crested weir is normally a flat topped obstruction which extends across the

entire channel, as shown in the picture below. A longitidunal section of a typical broad

crested weir is shown in the diagram below. The head over the weir, H, the height of

the weir, P, the approach velocity and depth, V1 and y1, and the velocity and depth on

the weir crest, V2 and y2, are all shown in the diagram. A key feature of a properly

operating broad crested weir is critical flow over the weir crest. This is shown in the

diagram by noting that V2 = Vc and y2 = yc. The next section discusses determination

of the minimum weir height to ensure that critical flow will occur over the weir crest.

y1 y2=yc

Minimum Height Requirement for a Broad Crested Weir

The water velocity will increase whenever the flow in an open channel

passes over an obstruction like a broad crested weir, because of the

decrease in cross-sectional flow area. Within limits, the higher the

obstruction, the greater the water velocity will be going over the

obstruction. If the approach flow is subcritical, then the flow over a

broad crested weir will become critical at some particular weir height.

That height needed to give critical flow over the weir crest can be

calculated using some basic hydraulics equations, Using the

terminology in the broad crested weir figure above, along with B for

the width of the channel, the energy equation becomes:

•y1 + V12/2g = y2 + P + V2

2/2g

From the definition of average velocity in an open channel, assuming

that the channel is approximately rectangular:

V1 = Q/y1B and Vc = Q/ycB

From the fact that the specific energy is a minimum for critical flow

conditions:

yc = [Q2/gB2]1/3

If the channel width, B, the flow rate through the channel, Q,

and the approach depth, y1, are known, the depth required to

give critical flow over the broad crested weir crest can be

calculated with the above three equations. If a broad crested

weir has the minimum height needed for critical flow, then the

simple equation, Q = 1.6 L H3/2, can be used for flow rate

calculations over that broad crested weir.

This is illustrated with example calculations in the next section.

In order to ensure critical flow over the weir for all flow

conditions, the maximum anticipated flow rate through the

channel should be used to calculate the required weir height, P.

Example Minimum Height Calculation

Problem Statement: Consider an open channel,5 m wide, with a

maximum anticipated flow rate of 10 m3/s at a flow depth of 1.5 m.

What is the minimum height needed for a broad crested weir to

ensure critical flow over the weir crest?

Solution: Using the equations from the previous section, with Q = 10 m3/s,

y1 = 1.5 m, and B = 5 m:

m. 0.48 P E

:(2) and (1) Sections between Energy Eq.

m 59.12

Em 11.12

3E

m 742.0 //m 25

10

1

2

1

2

11c

3

23

c

c

c

EP

gy

qyy

g

qymsq

3.7 Undeflow Gates

Underflow gates are used for many purposes such as:

• Controls at the crest of an overflow spillway,

• Control at the outlet from a lake to a river or irrigation channel.

Flow in underflow gates might be

1. free outflow or,

2. submerged (drowned) outflow.

Undeflow gates

Vertical Gates Radial Gates

(Tainter Gates)

Drum Gates

Free Outflow -Vertical Gates:

g2U2

2

45o

y1

y2

g2U2

1

(1) (2)

EGL

)1(C where2q

1

d1

ywC

CgywC

c

cd

1d21

2121

221

2

22

2

221

2

121

gy2wC=)y+y(

g2yy=q or

)y+y(

1)yy(g2=q

:q for Solving 2gy

q+ y=

2gy

q+ y E=E ⇒

Drown Outflow -Vertical Gates:

(1) (2)

y1 Q=? y3

y w

y2

Consider the longitudinal section of flow shown below. The depth

y2 is produced by the gate and the depth y3 is produced by some

downstream control. If y3 is greater than the depth conjugate to y2

(the depth needed to form a hydraulic jump with y2 ), then the gate

outlet must become “ drowned “ as shown in figure.

The effect is that the jet of water issuing from beneath the gate is

overlaid by a mass of water which, although strongly turbulent, has no

net motion in any direction.

While there will be some energy loss between sections (1) and (2), a

much greater proportion of the loss will occur in the expanding flow

between sections (2) and (3). We therefore assume as an

approximation that all the loss occurs between sections (2) and (3),

that is:

Note that the piezometric head term at (2) is equal to the total depth y,

not the jet depth y2. Between (2) and (3) we can use the momentum

equation F2=F3 Noting that at (2), the hydrostatic thrust term is based

on y, not y2.

2

2

2

2

1

2

12122

y E

gy

qy

gy

qE

3

22

3

2

22

322

y

2

y F

gy

q

gy

qF

The Radial Gate (Tainter Gate)

The Tainter gate is a type of radial arm floodgate used

in dams and canal locks to control water flow. It is named

for Wisconsinstructural engineer Jeremiah Burnham Tainter.

A side view of a Tainter gate resembles a slice of pie with the curved

part of the piece facing the source or upper pool of water and the tip

pointing toward the destination or lower pool. The curved face or

skinplate of the gate takes the form of a wedge section of cylinder.

The straight sides of the pie shape, the trunnion arms, extend back

from each end of the cylinder section and meet at a trunnion which

serves as a pivot point when the gate rotates.

Pressure forces acting on a submerged body act perpendicular to the

body's surface. The design of the Tainter gate results in every

pressure force acting through the centre of the imaginary circle of

which the gate is a section, so that all resulting pressure force acts

through the pivot point of the gate, making construction and design

easier.

When a Tainter gate is closed, water bears on the convex (upstream) side.

When the gate is rotated, the rush of water passing under the gate helps to

open and close the gate. The rounded face, long radial arms and trunnion

bearings allow it to close with less effort than a flat gate. Tainter gates are

usually controlled from above with a chain/gearbox/electric motor assembly.

A critical factor in Tainter gate design is the amount of stress transferred from

the skinplate through the radial arms and to the trunnion and the resulting

friction encountered when raising or lowering the gate. Some older systems

have had to be modified to allow for frictional forces for which the original

design did not plan.

The Tainter gate is used in water control dams and locks worldwide.

The Upper Mississippi River basin alone has 321 Tainter gates, and

the Columbia River basin has 195. A Tainter gate is also used to divert the

flow of water to San Fernando Power Plant on the Los Angeles

Aqueduct. The Tainter gate was invented and first implemented

in Menomonie, Wisconsin.

(1)

y1 q

r

w Y2 =Ccw

a

(2)

1d21

2121

221

2

22

2

221

2

121

gy2wC=)y+y(

g2yy=q or

)y+y(

1)yy(g2=q

:q for Solving 2gy

q+ y=

2gy

q+ y E=E ⇒

We can write the Energy Equation between sections (1) and (2):

o

2

1

d1

90 where

36.075.01

)1(

C where2q

q

c

c

cd

C

ywC

CgywC

This equation gives results which are accurate to within percent,

provided that: 5

1 q