Hydraulic Flow Calcualtion

Hydraulics Prof. B.S. Thandaveswara

Indian Institute of Technology Madras

35.2 Stepped or Cascade Spillways (Fig. 35.4) Recent advances in technology have led to the construction of large dams, reservoirs

and channels. This progress has necessitated the provision of adequate flood disposal

facilities and safe dissipation of the energy of the flow, which may be achieved by

providing steps on the spillway face. Stepped channels and Spillways are used since

more than 3000 years. Stepped spillway is generally a modification on the downstream

face of a standard profile for an uncontrolled ogee spillway. At some distance in the

downstream of the spillway crest, steps are fitted into the spillway profile such that the

envelope of their tips follows the standard profile down to the toe of the spillway. A

stepped chute design increases higher energy dissipation and thus reduces greatly the

need for a large energy dissipator at the toe of the spillway or chute.

Spillway Stepped Spillway

Step height Sh

Length of the step ls

Figure 35.4 - Definition Sketch of a Stepped Spillway Stepped spillway was quite common in the 19th century and present practice is

confined to simple geometries ( e.g. flat horizontal steps in prismatic chutes). Generally,

a stepped channel geometry is used in channels with small - slope: for river training, in

sewers and storm waterways and channels downstream of bottom outlets, launder of

chemical processing plants, waste waterways of treatment plants and step -pool

streams.



Detailed investigation into its various elements started only about 1978 with the

comprehensive laboratory tests by Essery and Horner (1978).

During the 19th century and early 20 th century, Stepped waste - waterways ( also

called ' byewash' ) were commonly used to assist with energy dissipation of the flow

(CHANSON 1995, "Hydraulic design of stepped Cascade channels, Weirs and

Spillway", pergamon UK, 292 pages Jan 1995). Now a days stepped spillways are often

associated with roller compacted concrete ( RCC ) dams.

The stepped geometry is appropriate to the RCC placement techniques and enhances

the rate of energy dissipation compared to a smooth chute design. A related application

is the overtopping protection of embankments with RCC overlays ( e.g. ASCE Task

Force Report, 1994.

Alternatives for over topping protection of Dam - Task force Commitee on over topping

protection, 139 pages).

35.2.1 Suitability Energy dissipation below hydraulic structures is accomplished generally by single -fall

hydraulic jump type stilling basins, roller buckets or trajectory buckets. However, when

the kinetic energy at the toe of the spillway would be high. The tail water depths in the

river are often inadequate. Then first two devices, cannot be used as in the case of high

head dams.

In narrow curved gorges consisting of fractured rocks, buckets cannot be used. In such

situations, a system of cascading falls down the side of a valley, with a stilling basin in

the downstream, can be used as an alternative spillway. Cascade spillways can be

used for any type of dam irrespective of the material of construction.

The only disadvantage with stepped spillway is that at large discharges, as the jet is not

aerated for some distance downstream of the spillway, low pressure may occur and

lead to cavitation damage.



35.2.2 Physical Modelling of Stepped Spillway

Free surface flows are commonly modelled using Froude similitude. The various flow

elements,

(1) the role of the steps in enhancing turbulent dissipation as well as their interaction

with other adjacent steps and,

(ii ) the effect of aerated flow make it difficult to model.

35.2.3 Classification of Flow

The concept of stepped spillway was used as early as 1892 - 1906 in New Croton dam.

Lombardi and Marquenent were first to consider stepped spillway consisting of concrete

drop spillway and intermediate erodible river reaches. The slopes of these reaches were

such that a hydraulic jump occurred at the base of each drop. However, the

experimental studies revealed three types of flows over a stepped spillway, namely,

nappe flow, partial nappe flow (intermediate(transition)) and skimming flow.

A stepped chute consists of a open channel with a series of drops in the invert. For a

given chute profile, the flow patten may be either nappe flow at low flow rates, transition

flow for intermediate discharges or skimming flow at larger flow rates.

Nappe Flow

This type of flow occurs for small discharges. The flow cascades over the steps, falls in

a series of plunges from one step to another in a thin layer that clings to the face of

each step, with the energy dissipation occurring by breaking of the jet in the air, impact

of jet on the step, mixing on the step, with or without the formation of a partial hydraulic

jump on the step. The step height sh must be relatively large for nappe flow. This

situation may apply to relatively flat stepped channels or at low flow rates.

The depths can be determined from the expressions,

Following equations to be checked for notations:



1 1

1 1

3h h

3h h

0.4252y1 0.54 (35.1)S S

0.272y1 1.66 (35.2)S S

qg

qg

⎛ ⎞⎜ ⎟=⎜ ⎟⎝ ⎠

⎛ ⎞⎜ ⎟=⎜ ⎟⎝ ⎠

1 13

h h

0.222yc (35.3)S S

qg

⎛ ⎞⎜ ⎟=⎜ ⎟⎝ ⎠

However, the steps for a nappe flow or plunge pool type of flow need to be relatively

large. In otherwords, tread requires to be larger than the depth of flow. This requires

downstream slope of dam face to be relatively flatter. Chanson observes that if slope of

downstream face is greater than 1 : 5, the nappe flow system becomes uneconomical

except in case of embankment type structure or steep rivers.

Partial Nappe Flow (Fig.35.5) In this type of flow, the nappe does not fully impinge on the step surface and it

disperses with considerable turbulence. Flow is super - critical down the length of the

spillway.

yp

yp

yc

Figure 35.5 - Partial nappe flow



For a given step geometry, an increase in flow rate may lead to intermediate flow patten

between nappe and skimming flow - the transition flow regime also called a partial

nappe flow. The transition flow is characterised by a pool of circulating water and often

accompanied by a very small air bubble (cavity), and significant water spray and the

deflection of water jet immediately downstream of the stagnation point. Downstream of

the spray region, the supercritical flow decelerates upto the downstream step edge. The

transition flow pattern exhibits significant longitudinal variations of the flow properties on

each step. It does not present the coherent appearance of skimming flows.

Skimming Flow (Fig. 35.6) In skimming flow regimes, the water flows down the stepped face as a coherent stream,

skimming over the steps and cushioned by the recirculating fluid trapped between them.

The external edges of the steps form a pseudobottom over which the flow skims.

Beneath this, recirculating vortices form and are sustained through the transmission of

shear stress from the water flowing past the edge of the steps. At the upstream end, the

flow is transparent and has glossy appearance and no air entrainment takes place. After

a few steps the flow is characterised by air entrainment similar to a self -aerated flow

down a smooth invert spillway. In case of the skimming flow, at each step, whether air

entrainment occurs or otherwise, a stable vortex develops and the overlying flow moves

down the spillway supported by these vortices, which behave as solid boundary for the

skimming flow, and the tips of the steps. There is a continous exchange of flow between

top layer and vortices formed on steps. The flow rotates in the vortex for a brief period

and then returns to the main flow to proceed on down the spillway face. Similarly, air

bubbles penetrate and rotate with the vortex flow, when aeration takes place.

Transition from one type of flow to another is gradual and continuous, as a result both

the nappe flow and the skimming flow, appear simultaneously in a certain range, one of

them on some steps and other on the remaining, both changing spatially and

temporarily.



yc

sh

l

V0Recirculating flow

Figure 35.6 - Fully developed skimming flow

35.2.4 Transition from Crest to Initial Steps Sorensen found the free surface jet to be smooth down to the point of inception of air

entrainment. This point of inception moves progressively upstream as the discharge

decreases. However, for very small discharge, the jet after striking the first step was

redirected outward and skips several steps before it strikes the spillway face again

several steps further down. This could be overcome by introducing few smaller steps on

upper reaches of the spillway.



35.2.5 Basic Equation for Skimming Flow Consider a skimming flow in which dominant feature is the momentum exchange

between the free stream and the cavity flow within the steps. Basic dimensional analysis

yields ( Figure 35.6 ),

1

1

h

h

f ( V , y , S , , k , g, θ , , )= 0 (35.4)s1-1for horizontal steps, θ = tan (S / ).

Using Buckingum pi- theorem equation can be written as

s

s

l

l

µ ρο ο ο

ο

1

1

h

h

SV V y f , , , ,θ 0 (35.5)2 Sgy s

ksl

ρο ο ο οµο

⎡ ⎤⎢ ⎥ =⎢ ⎥⎣ ⎦

Figure 35.7 - Hydrodynamic feature of a skimming flow

Velocity Distribution

Shear layer edges

Cavity (bubble)

Mixing layer

V0

y0

sh

ls

__

δ

While deriving the above equation the interaction of adjacent steps and the effect of air

entrainment has not been taken into account. Hence, Froude number similitude alone

cannot describe the complexity of stepped spillway flows completely.

Chanson showed that Froude number has no effect on flow resistance and that

Reynolds number might not have a substantial effect and that the form drag was related

primarily to step cavity geometry. It was also reported that in case of small scale models

the developing flow regimes and flow resistance were not correctly reproduced.



35.2.6 Onset of Skimming Flow Onset of skimming flow occurs when the space between the water surface at the two

consecutive edges of the steps is filled up with water, there by, creating a smooth

surface of water parallel to the average slope of the spillway face - the condition very

difficult to establish analytically. Therefore, empirical equations have been proposed by

many investigators for the delineation of the skimming flow from nappe flow over

stepped spillway.

Essery and Horner reported that it is very difficult to distinguish between nappe and

skimming flow for flatter slopes having 0 4<1h sS / l . .

Based on available data Rajaratnam found the skimming flow to occur for 1

0 8>c hy / S . .

On the other hand, Stephenson introduced a term called Drop number, 1

3⎡ ⎤⎣ ⎦

2hD= q / gS

to distinguish between nappeflow [ D< 0.6 ] and skimming flow [ D > 0.6] Peyras, et al. studied gabion dams consisting of four step element each 0.2 m high. It was found that the transition from nappe to skimming flow occurs for a discharge of

approximately 1.5 m3 / s /m or at 0 5<1c hy / S . while Degoutte found the onset of

skimming flow on gabion steps to occur at 0 74=1c hy / S . for 0 33=

1h sS / l . and at 0.62 for

1 0=1h sS / l . . Based on the available data, Chanson developed a regression equation

for the onset of skimming flow, namely. 2/ 3b 1c

h 2 3/ 2 bb 1

1

F kys cos1 2F (k ) 1

kα

>⎛ ⎞

+ −⎜ ⎟⎜ ⎟⎝ ⎠

In which 1 b2b

1k 1 , FF

= + is the Froude number at the brink of the step and bα is the

streamline angle with the horizontal. This equation is applicable to the accelerated flow

and may predict jet deflection at the first step of the cascade.



Figure 35.8 - Onset of skimming / Nappe flow

Essery and Horner PEYRAS et al.STEPHENSONBEITZ and LAWLESSMONTESKELLSRU et al. (1994)HORNER (1969)ELVIRO and MATEOS

-20% Band+20% BandTransition fully / partially developed jumpHORNER [NA1/NA2]

0 0.2 0.4 0.6 0.8 1 1.20.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

yc / sh

Skimming Flow

Nappe Flow

sh / ls

35.2.7 Prediction of the flow regime The type of stepped flow regime is a function of the discharge and step geometry.

Chanson has reanalysed a large number of experimental data related to change in flow

regimes. Most of the data were obtained with flat horizontal steps .

Overall the result suggest that the upper limit of nappe flow may be approximated as:

h

h s

h

y Sc =0.89 - 0.4 (35.6)Sin which y is the critical depth, S is the step height, and c is the step length. The above equation indicates the transition s

l

l

h

h s

of flow from nappe to transition flow regime.While the lower limits of skimming flow may be estimated as.y Sc 1.2 0.325 (35.7)Son set of skimming flow is giv

l= −

h

h s

en by y Sc 1.057 0.465SFurther the equation 2 indicates the change of flow from transition flow to skimming flow region.Two issues must be clearly under stood.

l> −



Eqations 35.6 and 35.7 were fitted for flat horizontal steps with Sh/ls ranging from 0.05 to

1.7 ( i.e 3.4° <θ ° < 60o ); there is no information on their validity outside of that range

and their accuracy is no better than ± 10 %; Eqations 35.6 and 35.7 characterise a

change in flow regime for uniform or quasiuniform flows only.

For rapidly varied flows, the results are not valid. For example, at the upstream end of a

stepped chute, the accelerating water may flow as thick free – falling nappes before

changing to a skimming flow regime further down stream. However, Peyras et al. data,

which are for gabion (which is pervious in nature) stepped spillway, and may have

different characteristic of flow, which requires to be established. Later on, Chanson also

presented an analytical approach for the prediction of the onset of skimming flow

expressing 1c hy / S as a function of Froude number at brink of the step angle of striking

jet on the tread of the step assuming that the angle of striking jet is equal to the

downstream slope of spillway at the onset of skimming flow. However, no guidelines for

prediction of Froude number at the step edge has been provided which renders the

estimation of 1c hy / S difficult.

Tatewar and Ingle studied the energy dissipation capacity of an inclined spillway and

developed the following regression equation using available data

with range of 1h sS / l from 0.4 to 0.85 and 0θ from 0o to 20o to , to distinguish between

nappe and skimming flow.

1

10

h

yc 0.888 0.00385 0.01195 (35.8)S

They found that for slopes steeper than 0.9, the possibility of nappe flow reduces considerably.

All the

h

s

Sl

θ⎛ ⎞

= − − ⎜ ⎟⎝ ⎠

1

10

h

ycdata when plotted in terms of Z = [( ) + 0.00385 ] and represented the S

region of skimming flow as shown in Figure 35.9.

h

s

Sl

θ⎛ ⎞⎜ ⎟⎝ ⎠



Rajaratnam

0.4 0.5 0.6 0.7 0.8 0.90.6

0.7

0.8

0.9

Figure 35.9 - Onset of skimming flowsh1/l

Tatewar and Ingle

Chanson

35.2.8 Coefficient of Friction Noori studied in detail stepped steep open channel flows and reported a drag coefficient

of 0.19 for (1h sS / l = 0.2 and

M = 62 [ { y + (1hS / 2 ) for

1hS > 6 ] and, 0.17 for (1h sS / l ) = 0.1 and M= 100 [ { y + (

1hS /

2 ) for 1hS > 10 ].

In this, the value of y can be estimated at any point on the spillway as,

[ ]qy = 0.52g(z -H)φ

in which z is the vertical distance below the crest measured to the water surface at the

point where y is to be determined.

The value of φ for a stepped block was found to be considerably smaller than that for

smooth spillway for large value of (ls/ yc);



1hS is the length of the step) and slope of the spillway, and hence, a considerable

energy loss at the toe of the stepped spillway.

Based on the avilable data, Rajaratnam suggested the following equation for the

variation of coefficient of friction, fc , for aerated skimming flow.

30 0

f 22y gsc =

q

The value of fc was found to be 0.18 as compared to 0.0065 for smooth spillway, while

Christodoulou (1993) found fc to vary from 0.076 to 0.89 and , fc being higher down

the steps.

Tozzi evaluated the friction factor on stepped chutes of slope 1:2 ( V: H ) by analysing

the energy loss of air flowing in a closed conduit with roughness elements designed to

simulate the slope,

The value of is found to be f = 0.09. It was noted that the value is overestimated if

uniformly aerated flow conditions are not attained. Matos and Quentela concluded that a

value of f = 0.1 can be safely considered for the preliminary hydraulic design of stepped

spillway for slopes around 1 : 0.75 ( V: H ), typical of concrete gravity dams.

35.2.9 Energy Loss on Stepped Spillway When an overflow is smoothly directed to an outlet structure by the chute where a

concentrated energy dissipation takes place, the cascade corresponds to a distributed

dissipator. Hence, the terminal structure has only smaller area of energy to dissipate,

and would be significantly smaller. A quantitative comparison between the conventional

system chute - stilling basin and the spillway cascade is shown in figures. The latter

type is suited for small and medium discharges and has recently gained some

popularity with Roller Compacted Concrete dams.



Point ofinception

boundary layer

Growth of boundary layer

PI

PI = Point of Inception

Energy Line

RVF GVFDZ

UAF

RVF = Rapidly Varied FlowGVF = Gradually Varied FlowDZ = Developing Zone

UAF = Uniformily Aerated Flow Region

PHJ

PHJ = Pre-entrained Hydraulic Jump

Skimming Flow

T A



The typical geometry of the stepped spillway with the standard crest geometry and

increasing step height up to the point of tangency T are shown in the above figure.

The free surface profile is smooth upto crest inspite of the development of vortex in

each step. The transition to rough surface flow occurs beyond point A where the air

entrainment is initiated. The hydraulic features of the cascade spillway as compared to

chute flow are:

• the flow depth is much larger than in a chute due to the highly turbulent cascade

flow, and higher sidewalls are required,

• more air is entrained and the spray action may become an important issue.

• abrasion can be a serious problem for flows with sediment or with floating debris.

In cascade spillways two flow types may occur as shown in Figure.

• Nappe flow: is the flow from each step hits the next step as a falling jet;

• Skimming flow: the flow remains coherent over the individual steps.

• The onset of skimming flow occurs for yc /sh > 0.8, where yc is the critical depth

and sh is the height of the step. When uniform cascade flow occurs in long

channels, skimming flow dissipates more energy than nappeflow. However,

nappe flow is more efficient for a short cascade than skimming flow (Chanson,

1994) the energy dissipated hf relative to the drop height Hodepends on the drop

Froude number and the slope of the spillway.

Stephenson ( 1991 ) expressed the relative energy loss as

∆H 0.84 -1/3F (1)ο0.25H0 θ

⎛ ⎞= ⎜ ⎟

⎝ ⎠

in which ∆H is the energy loss over a height H0, F0 is the

0.5

30

qFroude Number = gH

⎡ ⎤⎢ ⎥⎣ ⎦

, and θ is expressed in degrees in the above equation.

The energy dissipated ∆H relative to the drop height H0 depends on the drop Froude

number ( )3o oF q gH= and θ slope of the spillway.



Accordingly, the effect of slope is small, where as the dam height has considerable

influence on the head loss. Christodoulou (1993) studied the effect of number of steps N

on the energy dissipation ∆ H / H0. He introduced the parameter hc= yc / ( Nsh ) with yc =

(q2 /g) 1 / 3 as critical depth sh as the step height and found for hc < 0.25

H 2exp( 30 h ) (2)H c

ο

∆= −

By Increasing the number of steps the energy dissipation can be increased and hence

the performance of the stepped spillway.

For a long cascade, above 90% of mechanical energy is dissipated along the cascade

and only a small Portion of energy must be dissipated in the stilling basin.

According to Stephenson ( 1991 ) the efficiency of the cascade spillway depends mainly

on its height and the specific discharge and marginally on the slope.

The cascade flow may reach a state of nearly uniform flow (subscript n) which may be

approximated with ls , as step length ( Vischer and Hager. 1995).

n1 / 24 6 3h 0.23 [ / ( g )] (3)hl q ss=

Diez-Cascon et al. (1991) conducted experimental investigation on a cascade spillway

of step size ls / sh = 0.75 and slope θ = 53°, followed by a horizontal stilling basin . The

sequent depth ratio varied with the approach Froude number F1 as

r 2 /3 Y = 2.9 F1

The resulting tailwater depth is higher than for the classical hydraulic jump, however,

the value F 1 for cascade flow is much smaller. The above equation is valid for uniform

approach flow. One may derive the following equation.

r

2 / 3 1/ 3F = 7. 3 ( h / l ) n sn1 / 3and Y = 21.1 ( h / l ) (4)n s

The sequent depth ratio varies slightly with the uniform flow depth relative to the step

height.



According to Chanson ( 1994) the onset of nappe flow occurs for *c cy y> where

h

h

*y sc 1.057 0.465s ls

= −

In the transitional regime between nappe and skimming flow hydraulic instability occurs

which should be avoided to prevent the problems with vibration of structures.

For skimming flow, the resistance characteristics are governed by the distance between

two adjacent step edges, protruding into the flow. Even though Chanson (1994)

analysed the hydraulics of skimming flow, there is inadequate data to describe uniform

cascade flow. A basic investigation is needed to obtain further information.

Though, it was clearly stated as early as 1970 that the adavantage of steps is to

dissipate energy a little at a time but this is true only at low flow rates.

Where as, the energy dissipation occurs due to jet breakup in the air, jet mixing on the

step, with or with out the formation of a partial hydraulic jump on the steps in case of

nappe flow; the energy dissipation in skimming flow occurs due to the momentum

transfer to the recirculating fluid. Hence, the methods required for estimation of the

energy loss need to be different for the two types. In general, about 88% to 94 %

reduction in kinetic energy was noted from the velocity measurements at the spillway

toe without and with steps. For isolated nappe flow, Peyras et al., presented an

equation (see table) for determining the energy loss below the stepped gabion



Authors Remarks Energy loss equation Peyras et al. (1992)

Isolated nappe

2

h c 21

qE = Ns + 1.5 y - 2gy

⎛ ⎞∆ ⎜ ⎟⎜ ⎟

⎝ ⎠

Rajarathnam, 1990

skimming flow

88.89 % of self aerated flow

Tatewar and Ingle (1999)

Relative loss (upper limit of energy loss)

0 2 cd

dam

E 1E y1 1.25 C

H Head over spillway

∆=

⎡ ⎤+ ⎢ ⎥+⎣ ⎦

Chamani and Rajarathnam

Relative loss ( ) ( )i

N 1c

f fh i 1

0 c

h

f

c h

h s

y1 α 1 1.5 1 αsE 1

E yN 1.5 s

in which α is the proportion of energy loss per step a function

y sof and s l

−

=

⎡ ⎤⎡ ⎤− + + −⎢ ⎥⎢ ⎥∆ ⎣ ⎦⎢ ⎥= − ⎢ ⎥⎡ ⎤

+⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠

∑

Tatewar and Ingle (1966)

Regression analysis for

fα

c hf

h s

y sα = -0.1169 - 0.8221 log + 0.0675 log θ - 0.5481 log s l

Chanson in terms of friction factor f

1/ 3 2 /3

0

dam0

c

f fcos θ + 0.5 8 sin θ 8 sin θE E = 1 - HE 15 +

y

−⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟− ⎝ ⎠ ⎝ ⎠

It was found that actual dissipation could be 10 % more in case of gabions as compared to concrete steps due to factors like infiltration in to the gabions, difference in surface roughness and spillway slope. It was also found that their equation is valid within 10% in case of partial nappe flow. Rajaratnam, found the ratio of energy dissipation by skimming flow to the energy contained in the flow down a smooth spillway is about 89%. He has assumed that the flow is uniform skimming flow which implies a high spillway, with many steps. The residual energy varies from 9 % to 12 % depending on the discharge. Christodoulou in 1993 studied the effect of number of steps on energy dissipation in case of skimming flow.



Using his own data for 1h sS / l = 0.7 and for N = 15 (number of steps) as well as the

avilable data for hc < 25, where hc = ( yc / NSh ), he showed for the same discharge the

energy dissipation increases with the increase in the number of steps. The dissipation

may be significantly less on moderately stepped spillway when compared to the uniform

flow on high spillway. As he has not consider the effect of 1s hl / S and, the energy loss

by the steps in the curved portion is likely to be more, the results of Christodoulou is not

applicable to prototype. Using the weir formula to express discharge over the spillway in

terms of head over the spillway including velocity of approach head, Tatewar and Ingle

derived a simplified expression for energy loss (upper limit of energy dissipation) using

the equation for the discharge over the weir and is given by

0 2

1918

∆=

⎛ ⎞+ ⎜ ⎟+⎝ ⎠

cd

dam

EE yC

H H

Chamani and Rajaratnam established a relation for the energy dissipation in jet flow.

Tatewar and Ingle based on regression analysis fitted an equation for the proportion of

energy per step for the range of θ from 5° to 20° S /1 slh from 0.421 to 0.842 and

yc( )S

1h from 0.05 to 0.833.

They concluded that energy dissipation is more in case of inclined steps and fα

increases marginally for steeper slope and that the increase of fα is comparatively

larger for flatter slopes.



Variation of X with yc/sh1

yc/sh10.05 0.1 0.5

0.150.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

θ = 100

θ = 50

θ = 200

θ = 150

X

}Inclined steps

θ = 00

Horizontal steps

35.2.10 Effect of Air Entrainment For higher discharges the point of inception of air entrainment occurs past the end of

the spillway section and move progressively up as the discharge decreases.

Typically, the depth decreases from the crest inception point, beyond which, owing to

the bulking of the flow, the depth continuously

increases towards the spillway toe. At very low Reynolds number, the nappe does not

break and energy loss is affected.

Aeration of cascades The Quality of waters of rivers, streams, creeks etc is often expressed in terms of the

dissolved oxygen content ( DOC). Low dissolved oxygen value often does not allow the

development as well may cause the death of aquatic life forms and indicates some form

of pollution associated with excessive waste water inflows. In natural streams, obtain

the DOC from the aeration of the free surface.



Stepped cascades are characterised by a large amount of self aeration and it may be

used to reoxygenate depleted waters. In rivers, artificial stepped cascades and weirs

have been built to enhance the DOC of polluted or eutrophic streams.

Stepped cascades are also built in the downstream reach of large dams to re-oxygenate

water.

Example: Labyrinth weir crest length of 640 m, single drop 2.3 m, design discharge 14

to 68 m3/s at South Houlston weir, USA and the two-step labyrinth drop structure (2

drops of 2 m height and design discharge of 110 m3/s) of 640 m buit by the French

Electricity Commission downstream of the Petit -Saut Dam ( the Petit-Saut Dam is a

RCC construction) installed with an overflow stepped spillway. The downstream

stepped cascade is designed to re- oxygenete the tailrace waters of power station (

depleted in oxygen ). Further, there is a series of five aeration cascades built along the

Calumet waterway in Chicago. The waterfalls are designed to re-oxygenate the polluted

canal and combine flow aeration and aesthetics to create recreation parks.

Stepped cascades could be used to reduce the dissolved nitrogen content also. In the treatment of drinking water, cascade aeration is used to remove dissolved gases (

e.g. chlorine).

35.2.11 Air entrainment in Nappe Flow Regime Typical air concentration profiles based on the experimental investigation by Chanson

and Toombes are shown in the figure.



Air Cavity

Impact PointSpray Rebound Reattachment

Longitudinal Variation of Air Concentration (isocons) along the nappeCentre line of Step 2 qw = 0.150 m2/s after Chanson and Toombes,June - 1997

90 % 50 %

The un-ventilated air cavity, the impact point and the spray region are also indicated.

The main features of the air - water flow on step for q = 0.15 m2s-1 are:

• the large air-cavity beneath the nappe,

• the sidewall standing waves and the spray ( i.e. rebounding waters ),

• the large amount of flow aeration in the spary region,

• the de- trainment at the spray re-attachment, and

• the substantial free-surface aeration at the end of the step ( i.e. C = 19% ) The flow patterns of the air - water flow on a down stream step : for the same flow rate

at the cascade ( step No.9). (Sh = 0.143, ls = 2.4 m, θ = 3.4°, 10 steps over 25 m long

flume of 0.5 m width. In the absence of steps θ = 4.0 °).

• in the absence of ventilation the air cavity had disappeared completely and

recirculating water occupies the space beneath the nappe,

• the introduction of a splitter ( into the nappe ) ventilates the nappe and induces

the formation of a sizeable air cavity; as a result the nappe trajectory increases,

• the spray region is important,

• the amount of entrained air is basically identical at the upstream and downstream

ends of the step ( i.e. C = 17% to 19%); the mean air content is maximum at the

downstream of the nappe impact ( in the spray region ) and minimum at the

downstream end of the step . An interesting difference is the presence of a small

bubble (cavity) between the nappe and the re - circulating water . The



introduction of a splitter helps in occurance of larger air cavity and enlarges the

nappe trajectory.

In comparison the mean air content at the downstream end of the smooth chute was

about 0.08 and the maximum mean air concentration was about 0.12 at a section

located 4 m downstream.

Overall the stepped chute flow is significantly has higher aeration than the smooth chute

flow for the same flow rate. The air - water flow with the stepped channel is three -

dimensional in nature unlike the smooth chute flow which is two - dimensional.

35.2.12 Air Entrainment in Skimming Flows Modern concrete stepped spillways operate in a skimming flow regime. at the upstream

end, the free surface is clear and transparent.

However, a turbulent boundary layer develops along the chute invert. When the outer edge of the boundary layer emerges to the free surface, air entrainment

commences.

The distance to the inception point of air entrainment and the flow depth at inception are

correlated by :

The location where free surface aeration occurs is called the inception point of air

entrainment. Its characteristics are the distance Li from the crest

(measured along the invert) and the flow depth yi measured normal to the channel

invert.

Model and prototype data were re - analysed by Chanson in 1994. The dimensionless distance Li/ ks and depth yi /ks are plotted as functions of the

dimensionless discharge.



hk is the step depth per unit width ( normal to the flow direction ), s is the step height. empirical correlations for stepped chutes:- The dimensionless distance from crest L / and flow depth i

yI

s

Ks

( )( )

( )

( ) ( )

( )( )

w* 3

h

h

w* 3

s

0.7130.0796I*

s

0.592I*0.04

s

q/ k increase with increasing dimensionless discharge F = sg sin θ s cos θ

k can be written as s cos θ sqThus F =

g sin θ kL 9.719 sinθ F ky 0.4034 F k sinθ

=

=



BaCaRa [1:10] (53 deg.)

BEITZ and LAWLESS (50 deg.)

BINDO (51 deg.)

FRIZELL (27 deg.)

HORNER (36.4 deg.)

SORENSEN (52 deg.)

TOZZI (53.1 deg.)

ZHOU (51.3 deg.)

Given equation for 52 deg.

Trigomil (51.3 deg.)PROTOTYPE

Trigomil dam

1.00 10.00 100.00

1

10

100

1000

BaCaRa [1/10 \ 53 deg. ]

BINDO (51 deg.)

FRIZELL (27 deg.)

HORNER (36.4 deg.)

SORENSEN (52 deg.)

TOZZI (53.1 deg.)

ZHOU (51.3 deg.)

Given equation (52 deg.)

F

1.00 10.00 100.00

1.00

10.00

0.10

LI / ks

*

2

5

20

50

500

Normalised Inception length as a function of dimensionless dischargeafter (Chanson and Toombes)

Iy ks ___

Normalised Inception depth (y ) as a function of dimensionless dischargeafter (Chanson and Toombes)

I

F*

Inception on smaller steps

LI___ks

= 9.719(sin ) 0.0796 (F*)0.713θ

yI___ks

=(sin )

(F*)0.592

θ_______0.4034

0.04

boundary layer growth rate is grater on stepped channels than on smooth chutes.

Chanson in 1995 based on the reanalysed data, concluded that the experimental results

are basically independent of the type of crest profile.

The reanalysed data included the types of crest profile were included smooth ogee

crest profiles follows by stepped chute ( with or without smaller first steps) and broad -

crests followed by stepped chute.



Above Equations may be used for estimating Li and yi. It may be noted that one

prototype observation (Trigomil dam) fills the equation 1.

35.2.13 Aeration in Fully - Developed Skimming Flow

Downstream of the inception point of air entrainment, the flow becomes fully -developed

and a layer containing a mixture of both air and water extends gradually through the

fluid. Far downstream the flow becomes uniformly aerated. This region is defined as the

uniform equilibrium flow region. The air concentration profiles are compared with a

simple diffusion model by CHANSON 1995 and validated with prototype and model

smooth - chute data . The following equation describes the air concentration distribution.

y2C 1 tanh K2 D y90

⎛ ⎞′= − −⎜ ⎟⎜ ⎟′⎝ ⎠

in which C is the air concentration, D' is a dimensionless turbulent diffusivity and K ' is

constant of integration. D' and K' are functions of the mean air concentration C , y90 is

the distance from bed at which 90% air concentration occurs.

after BAKER (1994)

Brushes Clough dam spillway

Air concentration 'C'

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

y

Measurements at Brushes Clough dam spillway (BAKER 1994) - Inclined downward steps (Sh = 0.19 m, is δ = - 5.6 deg.), θ = 18.4 degrees

C = 0.235 - Step 50

C = 0.178 - Step 30

C = 0.15 - Step 10

C = 0.20 - Step 73

Theory: C = 0.15

Theory: C = 0.235

__y

90

__

__

__

__

__

__



Theory: C = 0.25

Theory: C = 0.33

x = 14.8 m - C = 0.25

x = 13.8 m - C = 0.31

x = 26.8 m - C = 0.33

after RUFF AND FRIZELL (1994)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

y__y

90

Air concentration 'C'

x is the distance

Air concentration distribution in prototype observation after Baker and Ruff and Frizell (Chanson 1994)

(qw = 2.6 m2/s, θ =26.6 , inclined downward steps, Sh = 0.154 m )0

__

__

__

__

__

Table: Variation of the turbulent diffusivity and constant of integration with C .

C ( 1 )

D' ( 2 )

K' ( 3 )

0.01 0.007 68.70 0.05 0.037 14.00 0.10 0.073 7.16 0.15 0.110 4.88 0.20 0.146 3.74 0.30 0.223 2.57 0.40 0.311 1.93 0.50 0.423 1.51 0.60 0.587 1.18 0.70 0.878 0.90

90y1C = C dy

y 090∫

The analysis of model and prototype data showed that the air concentration profiles in

skimming flows down a stepped chute have similar shape as those in smooth chute

flows. Further the observed values of mean air concentration over stepped chute flow

are very nearly same as the mean air concentration of the fully developed flow over

smooth chutes : i.e., ( )Ce= 27% and 36% respectively for = 18.4o and 26.6o.

The data of Baker (1994) yielded C ranging between 15% and 23% with 18.4o slope

and the data of Ruff and Frizell indicate C of 33% at the end of the 26.6o slope channel.



Free-surface aeration causes the bulkage of the flow and thus reducing the risks of

cavitation damage and enhanaces the air -water gas transfer (e.g. re- oxygenation of

the water). Futher, the presence of air nearer to the bed induces a reduction of drag and

results in decrease in friction factor . The drag reduction effect and the associated

reduction in flow resistance may have a significant impact on the rate of energy

dissipation on stepped spillway. The above analysis [ of energy dissipation ] by chanson

neglects the effects of air entrainment. The friction factor and the energy dissipation are

affected significantly by the rate of free- surface aeration. The effects of air entrainment

on the residual energy cannot be neglected for [ channel ] slope larger than 30 degrees

" and " the residual energy is strongly underestimated if the effect of air entrainment is

neglected. It is most important that design engineers to take into account aeration of

flow to estimate the residual enegy and to dimension stilling basins downstream of

stepped chutes".

35.2.14 Rapidly Varied Flow at the Inception Point

The flow properties rapidly vary next to and immediately downstream of the inception

point obervations suggest that some air is entrapped in the step cavity (ies).

Immediately upstream the flow is extremely turbulent and the free surface is oscillating.

At irregular time intervals, a water jet impinges on the horizontal step face and air is

trapped in the step cavity. An instant later, a rapid unsteady flow bulking is observed

downstream. Velocity measurements indicate that, immediately upstream of the

inception point, the turbulent velocity fluctuations are large, with dimensionless

fluctuations u' / V of about 15 - 18 % and normal longitudinal and lateral components of

turbulent velocity, respectively. Observed values of u' = 0.14 m /s next to the free

surface are large enough to initate air bubble entraiment. Immediately downstream of

the inception point, time-averaged air concentration data showed an increased aeration.

For example, increase in mean air concentration C∆ = 25% along a distance x∆ = 6.5 yc

down a 30o slope for yc / sh = 5.2 ; an increase in mean air concentration C∆ = 55 % in

18 step heights down a 53o slope for yc / sh < 2 ; an increase in mean air concentration



C∆ = 32% in 2 step heights down a 22o slope for yc / sh = 1.1, (where C∆ is the mean

air concentration).

Table : Increase in Mean Air Concentration Over Stepped Spillway

Increase in

concentration C%∆

I∆ Bed slope S0 (in degree)

c

h

ys

Remarks

25 6.5 yc 30 5.2 55 * 53 < 2.0 * Over 18 steps32 * 22 1.1 Over 2 steps

heights Reference 1. BaCaRa, "Etude de la dissipation d' energie sur les evacuateurs a marches", (study

of the energy dissipation on stepped spillways) Rapport d' Essais, Project National

BaCaRa, CEMAGREF-SCP, Aix -en-provence, France, October 1991, 111 pages.

2. BaCaRa. "Roller compacted concrete: RCC for dams. "Presses de l' Ecole Nationale

des Ponts et Chausse'es, Paris, 1997.

3. Baker, R. "Brushes clough wedge block spillway - progress report no. 3" SCEL Proj.

Rewp. No. SJ542-4, University of Saford, U.K, 1994.

4. Boes R.M, "Physical model study on two - phase cascade flow" , Proc 28th IAHR

Congress, Graz, Austria, Session S1, 6 pages, 1991.

5. Chamani, M.R., and Rajaratnam N. "characteristics of skimming flow over stepped

spillways". J. Hydraulic Engineering, ASCE, 125 (5), 1999, 500 - 510. Discussion by

Robert M. Boes; Chanson H; Jorge Matos; Ohtsu I, Yasuda Y and Takahashi; Tatewar

S.P, Ingle R.N, Porey P.D and closure, ibid, November 2000, page 860 - 873.

6. Chanson H and Toombes Luke "Flow aeration at stepped cascades", Research

report number CE155, Department of Civil Engineering, Research Report series,

University of Queensland, June 1997.

7. Chanson, H. "stepped spillway flows and air entrainment" Can. J. Civil Engineering,

Ottawa, 20 (3), 1993, 422 - 435.



8. Chanson, H. "Discussion of model study of a roller compacted concrete spillway", J.

Hydraulic Enginnering, ASCE, 123 (10), 1997b, 931 - 933.

9. Chanson, H. "Hydraulics of Nappe flow regime above Stepped chutes and Spillways",

Aust. Civil Engineering Trans., I.E. Aust., Vol. CE36, No. 1, Jan., 1994, pp.69-76.

10. Chanson, H. "Hydraulic Design of Stepped cascades, Channels, Weirs and

Spillways", Pergamon, Oxford, UK, Jan., 292 pages, 1995 .

11. Chanson, H. "Air Bubble Diffusion in super critical open channel flow, Proc. 12th

Australasian Fluid Mechanics Conference AFMC, Sydney Australia, R.W. Bilger Ed.,

Vol. 2, 1995 , pp. 707 - 710.

12. Chanson, H. " Prediction of the transition nappe / skimming flow on a stepped

channel", Jl of Hydraulic Res., IAHR, Vol. 34, No. 3, 1996, pp. 421 - 429.

13. Chanson, H. "Air bubble entrainment in free surface turbulent shear flows",

Academic Press, London, UK, 1997, 401 pages.

14. Chanson, H., and Whitmore, R.L. "Investigation of the gold creek dam spillway,

Australia. "Research Report No. CE153, Department of Civil Engineering, University of

Queensland, Australia, 1996, 60 pages.

15. Chanson H. "Stepped Spillways Parts 1 and 2", Journal of Physcial Science and

Engineering Periodical, TA1 17526, Volume 5, No. 4, December 1997, Engineering

update, technical paper number 10, page no. 7 to 12 and Journal of Physcial Science

and Engineering Periodical, TA1 17256, Volume 6, No. 1, January - March 1998,

Engineering update, Technical paper No. 2, page no. 9 to 14.

16. Geoffrey G.S. Pegram, Andrew K. Officer and Samule R. Mottram, "Hydraulics of

skimming flow on Modeled Stepped Spillways", Journal of Hydraulic Engineering, May

1999, Volume 125, No. 5, Paper 3557, Discussion by Robert M. Boes; Jorge Matos;

Ohtsu I, Yasuda Y and Takahashi M; Tatewar S.P, Ingle R.N, Porey P.D; ibid, and

closure December 2000, page 947 - 953.

17. Goubet, A. "Evacuateurs de Crues en Marches d' Escalier" (stepped spillways) La

Houille Blanche, No. 2/3, pp. 247 - 248 , 1992.

18. Hans - Erwin Minor and Willi Hager editors "Hydraulic of Stepped Spillways",

Balkema, Rotterdam, The Netherlands, 2000; 201 pages.



19. Houston, K.L. "Hydraulic Model Studies of Upper Stillwater Dam stepped Spillway

and Outlet works". Report No. REC - ERC - 87 - 6, US Bureau of Reclamation, Denver

Co, USA, 1987.

20. Ruff. J.F. and Frizell, K.H, "Air concentration measurements in highly turbulent flow

on a steeply sloping chute", Proceeding Hydraulic Engineering Conference, ASCE, New

York, Vol. 2, 999 - 1003.

21. Stephenson, D. "Energy dissipation down stepped spillways" Water Power and Dam

construction, September, 27 - 30, 1991.

22. Tatewar S.P. Ingle R.N., Nappe Flow on Inclined Stepped Spillways, Journal of The

Institution of Engineers (India), Volume - 79, Page- 175 - 179, Feb. 1999.

23. Tatewar S.P., and Ingle R.N, Resistance to skimming flow over stepped spillway,

Proceeding International Seminar on Civil Engineering Practices in 21st Century,

Roorkee, India, 1039 - 1048.

24. Tozzi, M.J. "Residual energy in stepped spillways. "International water Power and

Dam construction, 1994, 46 (5), 32 - 34.

25. Virender Kumar, Stepped Spillway - a State of the art, Journal of The Institution of

Engineers (India), Volume - 82, Page- 217 - 223, Feb. 2002.

26. Vischer D.L. and Hager W.H. "Dam Hydraulics", John Wiley and Sons, 1997.

27. Yildiz, D., and Kas, I, "Hydraulic performance of stepped chute spillways",

Hydropower and Dams, 1998, 5 (4), 64 - 70.

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