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OPEN CHANNEL HYDRAULICS FOR ENGINEERS

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Chapter 4: NON-UNIFORM FLOW 71

Fig. 4.1. Examples of non-uniform flow

4.1.2. Accelerated and Retarded flow

An idealized section of a reach of a channel with accelerated and retarded flow

conditions is shown in Fig. 4.2a and Fig. 4.2b, respectively. As flow accelerates, with the

rate of flow constant, the depth h must decrease form point 1 to point 2, and a water

surface profile as shown in Fig. 4.2a results. Retarded flow will produce water surface

profiles as shown in Fig. 4.2b.

Significant in each one of the above cases is the fact that now the water surface is a curved

line and not longer parallel to the channel bottom and the energy line, as was the case for

uniform flow. The following points are made in connection with the above observations.

upstream control downstream control

control sluice

gate

rapid

varied

flow

hydraulic jump sharp-crested

weir

rapid

varied

flow

rapid varied

flow

gradually

varied

flow

gradually

varied

flow

upstream control

downstream

control hydraulic

jump

supercritical

flow subcritical

flow

critical

depth

overflow

(critical depth)

control

hc

hc

rapid varied

flow

rapid

varied

flow

rapid

varied

flow

graduallyvaried

flow gradually

varied

flow

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The water surface, as will be shown later, can have a concave or a convex shape.

The energy line is not necessarily a straight line; however, it is assumed that the

energy gradient is constant along the length of a reach and the energy line will be

represented and considered to have a slope ie = HL /L.

As was done in the case of uniform flow, it is here also accepted that the depth of

flow, h, is equal to the pressure head in the energy equation. Obviously, this applies

only when the slope of the channel bottom is small. For very steep slopes,

allowances for this discrepancy must be made.

i

Lz1 z2

water surface

HL

11

ph

22

ph

2

1V

2 g 2

2V2 g

energy-head line

hydraulic grade line

datum

Fig. 4.2a. Accelerated flow

i

Lz1 z

2

water surface

HL

11

ph

2

2

ph

2

1V

2 g2

2V

2 g

energy-head line

hydraulic grade line

datum

Fig. 4.2b. Retarded flow

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4.1.3. Equation of non-uniform flow

Fig. 4.3. Non-uniform flow

Consider a non-uniform flow in an open channel between section 1-1 and section 2-2, inwhich the water surface has a rising trend (i.e. the energy-head gradient is less than the bed

slope) as shown in Fig. 4.3.

Let V = velocity of water at section 1-1;

h = depth of water at section 1-1;

V+dV = velocity of water at section 2-2;

h+dh = depth of water at section 2-2;

ib = slope of channel bed;

ie = slope of the energy grade line;

dl = distance between section 1-1 and section 2-2;

b = average width of the channel,Q = discharge through the channel,

zb = change of bottom elevation between section 1-1 and section 2-2, and

he = HL, change of energy grade line between section 1-1 and section 2-2.

Since the depth of water at section 2-2 is larger than at section 1-1, the velocity of water at

section 2-2 will be smaller than that at section 1-1.

Applying Bernoulli’s equation at section 1-1 and section 2-2:

2 2

b e

V (V dV)z h (h dh) h

2g 2g

(4-2)

22

b e

V dVVi .dl h h dh i .dl

2g 2g

(4-3)

b e

V.dVi .dl dh i .dl

g , neglecting

2(dV)

2g(small of second order) (4-4)

or b e

dh V.dVi i

dl g.dl (dividing by dl) (4-5)

1

1

2

2

ie

ib

dl

flow

he

water surface

2V

2g

2(V+dV)

2g

h h+dh

zb

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b e

dh V.dVi i

dl g.dl (4-6)

We know that the quantity of water flowing per unit width is constant, therefore

q = V.h = constant (4-7)

dq

0dl

(4-8)

ord(Vh)

0dl

(4-9)

Differentiating the above equation (treating both V and h as variables),

V.dh h.dV0

dl dl (4-10)

dV V dh

dl h dl

(4-11)

Substituting the above value of dV

dlin Eq. (4-6), yields

2

b e

dh V dhi i

dl gh dl (4-12)

2

b e

dh V1 i i

dl gh

(4-13)

b e

2

i idh

dl V1gh

(4-14)

Notes: The above relation gives the slope of the water surface with respect to the bottom

of the channel. Or in other words, it gives the variation of water depth with respect to the

distance along the bottom of the channel. The value of dh/dl (i.e. zero, positive or negative)

gives the following important information:

If dh/dl is equal to zero, it indicates that the slope of the water surface is equal to

the bottom slope. Or in other words, the water surface is parallel to the channel

bed.

If dh/dl is positive, it indicates that the water surface rises in the direction of flow.

The profile of water, so obtained, is called backwater curve.

If dh/dl is negative, it indicates that the water surface falls in the direction of flow.

The profile of water, so obtained, is called downward curve.

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Example 4.1: A rectangular channel, 20 m wide and having a bed slope of 0.006, is

discharging water with a velocity of 1.5 m/s. The flow is regulated in such a way that the

slope of the water energy gradient is 0.0008. Find the rate at which the depth of water will

be changing at a point where the water is flowing 2 m deep.

Solution:Given: width of the channel: b = 20 m

bed slope: ib = 0.006

velocity of water: V = 1.5 m/s

slope of energy line: ie = 0.0008

depth of water: h = 2 m

Letdh

dlbe the rate of change of water depth. Using equation in (4-14):

b e

2

i idh

dl V

1 gh

= 0.0059 Ans.

4.2. GRADUALLY-VARIED STEADY FLOW

4.2.1. Backwater calculation concept

Gradually varied flow is a steady, non-uniform flow in which the depth variation in

the direction of motion is gradual enough to consider the transverse pressure distribution as

being hydrostatic. This allows the flow to be treated as one-dimensional with no transverse

pressure gradients other than those due to gravity.

For subcritical flows the flow situation is controlled by the downstream flow conditions. Adownstream hydraulic structure (e.g. bridge piers, gate) will increase the upstream depth

and create a “ backwater” effect. This concept has been introduced shortly in section 4.1.3.

The term “ backwater calculation” refers more generally to the calculation of the

longitudinal free-surface profile for both subcritical and supercritical flows. The backwater

calculation is developed assuming:

a non-uniform flow

a steady flow

that the flow is gradually varied , and

that, at a given section, the flow resistance is the same as for a uniform flow with

the same depth and discharge, regardless of trends of the depth.

4.2.2. Equation of gradually-varied flow

In addition to the basic gradually-varied flow assumption, we further assume that

the flow occurs in a prismatic channel, or one that is approximately so, and that the slope

of the energy grade line can be evaluated from uniform flow formulas with uniform flow

resistance coefficients, using the local depth as though the flow were locally uniform.

Referring to Fig. 4.4., the total energy head at any cross-section is

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2

VH z h

2g (4-15)

in which z = channel bed elevation; h = water depth, = kinetic-energy correction

coefficient as introduced in Chapter 2, and V = mean flow velocity.

Fig 4.4. Definition sketch for gradually-varied flow

If this expression for H is differentiated with respect to x, the coordinate in the flow

direction, the following equation is obtained:

dx

dEii

dx

dHbe with

2V

E h2g

(4-16)

in which ie is defined as the slope of the energy grade line; ib is the bed slope (= - dz/dx);

and E is the specific-energy head (i.e. the energy head with respect to the bottom). Solving

for dE/dx gives the first form of the equation of gradually varied flow:

eb iidx

dE (4-17)

It appears from this equation that the specific-energy head can either increase or decrease

in the downstream direction, depending on the relative magnitudes of the bed slope and the

slope of the energy grade line. Yen (1973) showed that, in the general case, ie is not the

same as the friction slope if (= 0 / R, this equation will be introduced again in Chapter 7)

or the energy dissipation gradient. Netherless, we have no better way of evaluating this

slope than applying uniform-flow formulas such as those of Manning or Chezy. It is

incorrect, however, to mix the friction slope, which clearly comes from a momentum

analysis, with terms involving , the kinetic-energy correction (Martin and Wiggert, 1975).

Note: The bed slope ie and the friction slope if are defined as:

oe f

z Hi = sin tan and i

x x R

respectively, where H is the mean total energy-head, z is the bed elevation, is the channel

slope and o is the bottom shear stress.

bed

h

2V

2 g

z

slope of energy grade line, ie

datum

dH

H

bed slo e i

dx

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The second form of the equation of gradually-varied flow can be derived if it is recognized

thatdE dE dh

dx dh dx and that, applying equation (4-11),

2dE1 Fr

dh , provided that the

Froude number is properly defined. Then, equation (4-17) becomes:

b e

2

i idh

dx 1 Fr

(4-18)

The definition of the Froude number in equation (4-18) depends on the channel geometry.

For a compound channel, it should be the compound-channel Froude-number, while for a

regular, prismatic channel, in which d /dh is negligible, it assumes the conventional

energy definition given by Q2B/gA

3.

The ratio dh/dx in Eq. (4-18) represents the slope or the tangent to the water surface at any

point along the channel. This relationship therefore indicates whether at any point along

the channel the water surface is rising (backwater condition) or dropping (drawdown

condition). Immediately the following deductions can be made:

Whendh

0dx

, the slope of the water surface is dropping in the downstream

direction and the depth decreases downstream.

Whendh

0dx

, the slope of water surface is parallel to the channel bottom and

uniform flow exists. This can be readily seen from Eq. (4-18) since, for this

condition, ib = ie must equal zero.

Whendh

0dx

, the slope of water surface rises in the downstream direction and the

depth h increases downstream.

Whendh

dx , which requires that 1 – Fr

2= 0 or Fr = 1, the slope of the water

surface must theoretically be vertical. This flow occurs when the flow changes

from subcritical to supercritical, or vice versa, as indicated by the value of the

Froude number. The formulas derived do not actually apply any longer due to the

assumptions made. A vertical water surface also does not occur in reality; however,

a very noticeable change in the water surface takes place. This is especially so

when the flow changes from below hc to above hc. In such instance a phenomenon

known as the hydraulic jump occurs.

4.3. TYPES OF WATER SURFACE PROFILES

4.3.1. Classification of flow profiles

From the foregoing, it is evident that the relationship expressed in Eq. (4-18)

provides a considerable amount of information as to the shape of the water surface profile

in an open channel. Investigation of this formula yields the following results:

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1. The relationship between the slope of the channel bottom and the slope of the

energy grade line determines whether the numerator of the equation is positive or

negative.

2. The denominator of the equation is positive if Fr < 1.0 and vice versa. In other

words, if the flow is subcritical (Fr smaller than 1) the denominator is positive, and

if the flow is supercritical (Fr greater than 1) the denominator is negative.

The conditions at which flow in an open channel can take place and the possible

relationships between the observed depth ho, the normal depth at which flow is uniform hn,

and the critical depth hc are illustrated in Fig. 4.5. It is evident from this figure that there

are three zones of channel depths at which flow can be observed:

Zone 1, with ho greater than hn and hc (i.e. ho > hn > hc)

Zone 2, with ho between hn and hc (i.e. hn > ho > hc)

Zone 3, with ho less than hn and hc (i.e. hn > hc > ho)

Fig.4.5. Three zones of channel depths

The relative bottom slope defines whether uniform flow is subcritical or supercritical.

Determine the associated Froude-number Fre.

2 22 e e e

e

e e e

V R VFr

gh h gR

where R is the hydraulic radius of the open channel flow. Subcript e denotes the

equilibrium flow. The bottom/wall shear stress is defined as:

2

o f e e f c V gR i

2

e bf

e f f

V ii

gR c c (the friction slope if = the bed slope ib )

2 e b

e

e f

R iFr

h c

We have: Ae = Be.he e e ee

e e

A P .Rh

B B , where Pe is the equilibrium wetted perimeter.

ho hn

hc

ho > hn > hc

hohn

hc

hn > ho > hc

ho

hn

hc

hn > hc > ho

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2 e b

e

e f

B iFr

P c

In case of turbulent flow: 1.

For two-dimensional flow: e e

e e

R B1

h P

So, a proper approximation for Fre is: 2 be

f

iFr

c

If ib < cf , we have a mild slope (M – type)

The “uniform flow” is subcritical: Fr e2

< 1, he > hc.

If ib > cf , we have a steep slope (S – type)

The “uniform flow” is supercritical: Fr e2

> 1, he < hc.

If ib = cf , we have a critical slope (C – type) Fre2 = 1, he = hc

Note: It can easily be derived that

12 3

f 2

g

c n gRC

, where C is Chezy coefficient and n

is Manning’s.

Two conditional channel bottom conditions or slopes exist. These do not really constitute

open channel flow, but gravity flow can take place along them. They are as follows:

If ib < 0, we have an adverse slope (A – type)

If ib = 0, we have a horizontal slope (H – type)

It should be noticed that hn = he.

Note: The actual flow depends on the boundary condition, i.e.“mild

”,“steep

”, etc. does

not tell us anything about the actual flow.

4.3.2. Sketching flow profiles

In theory, for each of the five slope descriptions above there are three zones in

which flow can be observed. It follows then that a total of 15 theoretical water surface

profiles are possible, presented in Table 4.1. These profiles, together with illustrations of

practical applications, are shown in Fig. 4.6.

While this figure is for the most part self-explanatory, the following observations and

explanations are presented for further clarification. Mild slope (ib < cf ). The M1 curve is generally very long and asymptotic to the

horizontal and the line representing ho. The M2- and M3-curves end in a sudden

drop through the line representing hc and a hydraulic jump, respectively.

Critical slope (ib = cf ). Since hc = hn in this case, there is no zone 2, and only two

water surface profiles exist, C1 and C3. The C2-curve coincides with the water

surface that corresponds to uniform flow at critical depth.

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Steep slope (ib > cf ). All curves are relatively short. S1 is asymptotic to the

horizontal, whereas S2 and S3 approach ho.

Horizontal slope and Adverse slope channels. In this case, hn is infinitely large and

uniform flow cannot take place. Hence there are no H1- or A1-profiles.

Table 4.1. Types of flow profiles in prismatic channelsDesignation Relation of ho

to hn and hc Channel

slope Zone

1

Zone

2

Zone

3

Zone

1

Zone

2

Zone

3

General type

of curve

Type of flow See

Fig.

M1 ho > hn > hc Backwater Subcritical

M2 hn > ho > hc Drawdown Subcritical

Mild

Fre2

< 1,

he > hc M3 hn > hc > ho Backwater Supercritical

4.6.a

C1 ho > hc = hn Backwater Subcritical

C2 hc = ho = hn Parallel to

channel bottom

Uniform

critical

Critical

Fre

2

= 1,he = hc C3 hc = hn > ho Backwater Supercritical

4.6.b

S1 ho > hc > hn Backwater Subcritical

S2 hc > ho > hn Drawdown Supercritical

Steep

Fre2

> 1,

he < hc S3 hc > hn > ho Backwater Supercritical

4.6.c

None ho > hn > hc None None

H2 hn > ho > hc Drawdown Subcritical

Horizontal

ib = 0

H3 hn > hc > ho Backwater Supercritical

4.6.d

None ho > (hn*) > hc None None

A2 (hn*) > ho > hc Drawdown Subcritical

Adverse

ib < 0

A3 (hn*

) > hc > ho Backwater Supercritical

4.6.e

hn*

in parentheses is assumed a possitive value.

CDL = critical-depth line; NDL = normal-depth line

horizontal

Mild slope

M1

M2

M3hc

hn

NDL

CDL

Fig.4.6.a. Mild slope (0 < ib < cf ) and examples of flow profiles

M1M1

M2 section of

enlargementM2

M3M3

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Fig.4.6.b. Critical slope (ib = cf > 0) and examples of flow

critical slope

C1

C3hc = hn

CDL = NDL

horizontal

C1

C3

steep slope

S1

S2

S3

hn

hc

NDL

CDL

horizontal

Fig.4.6.c. Steep slope (ib > cf > 0) and examples of flow profiles

S1

S1

S2

section of

enlargement

S2

S3

S3

H2

H3hc CDL

horizontal slope

horizontal

Fig.4.6.d. Horizontal slope (ib = 0) and examples of flow profiles

H2

H3

A2

A3

hc

CDL

adverse slope

Fig.4.6.e. Adverse slope (ib < 0) and examples of flow

A2

A3

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4.3.3. Prismatic channel with a change in slope

This channel is equivalent to a pair of connected prismatic channels of the same

cross section but with different slope. Several typical flow patterns along a prismatic

channel with a break or discontinuity in slope are shown in Fig. 4.7.

Fig. 4.7. Flow profiles with a change in slope

The profiles in Fig. 4.7 are self-explanatory. However, some special features should be

mentioned:

The profile near or at the critical depth cannot be predicted precisely by the theory

of gradually varied flow, since the flow is generally rapidly varied there.

In passing a critical line, the flow profile should, theoretically, have a vertical

slope. Since the flow is usually rapidly varied when passing the critical line, the

actual slope of the profile cannot be predicted precisely by the theory. For the same

reason, the critical depth may not occur exactly above the break of the channel

bottom and may be different from the depth shown in the figure.

NDL

NDL

mildmilder (very long)

CDL

M1 M1

M2hn1

hn2

hc NDL

S2

S2S3 CDLhc

hn1

hn2

steep (very long)

steeper

NDL

mild

steep

CDL

M2

NDL

S2hn1

hn2

hc

NDLsteep

CDL

NDL

S2H

hc

reservoir

NDL

mild (short)CDL

NDLM2

H

hn

reservoir

hc

tailwater

CDL

reservoir

NDL

mild (very long)

M2H

hn

hc

free overfall

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4.3.4. Composite flow profiles with various controls

Channels with a number of controls will have flow profiles that can be composed

from the different types of flow profiles presented in the previous section. The ability to

sketch the composite profiles is in many cases necessary for understanding the flow in the

channel or for determining the discharge. In all cases it is necessary to identify firstly the

controls operating in the channel, and then to trace the profiles upstream and downstreamof these controls.

Two simple cases are shown in Fig. 4.8; in the first case the slope is mild, in the second

case steep. The curves for the mild-slope situation are self-explanatory, since they

incorporate many of the features already discussed. For the steep-slope situation we have

already seen that the critical flow must occur at the head of the slope – i.e. at the outflow

from the reservoir; thereafter there must be an S2-curve tending towards the uniform-depth

line. There must be an S1-curve behind the gate, and the transition from the S2- to the S1-

curve must be via a hydraulic jump. Downstream of the sluice gate, the flow will tend to

the uniform condition via an S2- or S3-curve; thence it proceeds over the fall at the end of

the slope. In this case there is nothing that impels the flow to seek the critical condition.

In Fig. 4.8 two profiles are drawn in dashed lines above the M3- and the S2-curve. These

are loci of depths conjugated to the corresponding depths on the underlying real surface

profiles, and are therefore known as “conjugate curves”. Obviously a hydraulic jump will

occur where such a curve intersects the real (subcritical) surface profile downstream; the

conjugate curve therefore provides a convenient means of determining the location of a

hydraulic jump.

Fig. 4.8. Examples of composite longitudinal profiles

M1

M2

M3

conjugate curve

hn

hc jump overfall

reservoir

mild slope

S1

S2 S3

conjugate curve

hn

hc

jump

overfall

reservoir

steep slope

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4.4. DRAWING WATER SURFACE PROFILES

4.4.1. Direct-step methodThe computation of a flow profile by a step method consists of dividing the channel

into short reaches and determining reach by reach the change in depth for a given length of

a reach. In principle, the direct-step method could be applied to either Eq.(4-17) or Eq. )4-

18), but usually is associated with the former. Eq. (4-17) is put into finite-difference form

by approximating the derivative dE/dx with a forward difference and by taking the mean

value of the slope of the energy grade line over the step size x = (xi+1 – xi) in which thedistance x and the subscript i increase in the downstream direction. The result is:

eb

i1ii1i

ii

EExx

(4-19)

where ei is the arithmetic mean slope of the energy grade line between section i and section

i + 1, with the slope evaluated individually from Manning’s equation at each cross section.

The variables Ei+1, Ei and ei on the right hand side of Eq. (4-19) all are functions of the

depth h. The solution proceeds in a stepwise fashion in x by assuming values of depth h

and therefore values of the specific-energy head, E. As Eq. (4-19) is written, x increases inthe downstream direction. In general, upstream computations utilize Eq. (4-19) multipliedby ( – 1), so that the current value of the specific-energy head is subtracted from the

assumed value in the upstream direction and x becomes (xi+1 – xi), which is negative.Therefore, if the equation is solved in upstream direction for an M2-profile, for example,

the computed values of x should be negative for increasing values of h. Decreasing

values of h should result also in negative values of x for an M1-profile. For an M3-profile, which is supercritical, increasing values of depth in the downstream direction

correspond to decreasing values of the specific-energy head, and Eq. (4-19) indicates

positive values of x, since ei > ib.

Although the direct-step method is the easiest approach, it requires interpolation to find the

final depth at the end of the profile in a channel of specified length. Some care must betaken in specifying starting depths and checking for depth limits in a computer program. Inan M2-profile, for example, the starting depth should be taken slightly greater than the

computed critical depth, if it is a control, because of the slight inaccuracy inherent in thenumerical evaluation of critical depth. In addition, the M2-profile approaches the normal

depth asymptotically in the upstream direction, so that some arbitrary stopping limit must

be set, such as 99% of the normal depth.

Example 4.2: A trapezoidal channel has a bottom width b of 8.0 m and a side slope ratio

of 2:1. The Manning’s n of the channel is 0.025 m-1/3s, and the channel is laid on a slope of

0.001. If the channel ends in a free overfall, compute the water surface profile for a

discharge of 30 m3 /s.

Solution:

Given: bottom width: b = 8.0 m

side slope ratio: m:1 = 2:1

Manning’s n: n = 0.025 m-1/3

s

bed slope: ib = 0.001discharge: Q = 30 m

3 /s

Compute the water surface profile.

b

hn

m = 2

1

B

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First, the normal depth and the critical depth must be determined. Manning’s equation

reads as:

21

b3

2

i.R.A.n

1Q

with trapezoidal channel cross-section:n nA y (b mh )

and hydraulic radius:n n

2

n

A h (b mh )

R P b 2h 1 m

So, the Manning equation can be rewritten as:5 52

3 332n n3

2 2 13 3 22

bn

A A [h (b mh )] Q.nA.R A.

P iP b 2h 1 m

or,

53

8 8n n 3 3

2 13 22

n

h (8.0 2y 30 0.025m 23.72 m

0.0018 2h 1 2

hn = 1.754 m

From the Froude formula:1

2

32

V Q QBFr

gD A A gA g.

B

where B = b+ 2mh; D = A/B, the hydraulic depth. In case of critical flow:

1 332 22c c cc

1 132 22

c cc

QB h b 2hA QFr 1

gA g b 2mhB

or

32

5c c 21

2c

h 8 2h 30m

9.818 2 2 h

hc = 1.03 m

Due to hn > hc, this is a mild slope (ib = 0.001): we have an M2-profile that has a critical

depth at the free overfall as boundary condition.

The direct-step method, as applied to Example 4.2, can be solved in a spreadsheet(Microsoft Excel) as formatted in Table 4.2. The values of h are selected in the first

column (1). The formulas for determining the specific-energy head E, column (5), and theslope of the energy grade line ie, column (6), for a given depth, are presented below. The

arithmetic mean of ie (iebar = ei ) is computed in column (7), and the change in specific-

energy head E, DelE, in the upstream direction is shown in column (8). Formulas applied

in the spreadsheet:

2

A y(b mh)

P b 2h 1 m

Q QR ; V

P A

2

2

e 43

e1 e2ebar e1 e2

VE h

2g

(nV)i

R

(i i )i (i i ) / 2

2

2 1

b ebar

e1 e2

E Del.E E E

Del.Ex Del.x

(i i )

x Del.E /(i i )

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Then, the equation of gradually varied flow in finite difference form is solved for the

distance step x, aseb ii

Ex

= - 0.028 m in the first step.

Table 4.2. Water surface profile computation by the direct-step method.h A R V E ie iebar Del.E Del.x Sum Del.x

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)1.03 10.362 0.822 2.895 1.457 6.80E-03 0.00

1.04 10.483 0.829 2.862 1.457 6.58E-03 6.69E-03 1.62E-04 -0.028 -0.03

1.06 10.727 0.842 2.797 1.459 6.15E-03 6.36E-03 1.23E-03 -0.229 -0.26

1.08 10.973 0.855 2.734 1.461 5.75E-03 5.95E-03 2.35E-03 -0.476 -0.73

1.1 11.220 0.868 2.674 1.464 5.39E-03 5.57E-03 3.40E-03 -0.743 -1.48

1.12 11.469 0.882 2.616 1.469 5.06E-03 5.23E-03 4.36E-03 -1.032 -2.51

1.14 11.719 0.895 2.560 1.474 4.75E-03 4.90E-03 5.26E-03 -1.346 -3.85

1.16 11.971 0.908 2.506 1.480 4.47E-03 4.61E-03 6.09E-03 -1.687 -5.54

1.18 12.225 0.921 2.454 1.487 4.20E-03 4.33E-03 6.86E-03 -2.057 -7.60

1.2 12.480 0.934 2.404 1.495 3.96E-03 4.08E-03 7.58E-03 -2.460 -10.06

1.22 12.737 0.947 2.355 1.503 3.73E-03 3.84E-03 8.24E-03 -2.898 -12.96

1.24 12.995 0.959 2.309 1.512 3.52E-03 3.63E-03 8.87E-03 -3.377 -16.33

1.26 13.255 0.972 2.263 1.521 3.32E-03 3.42E-03 9.45E-03 -3.901 -20.23

1.28 13.517 0.985 2.219 1.531 3.14E-03 3.23E-03 9.99E-03 -4.474 -24.71

1.3 13.780 0.998 2.177 1.542 2.97E-03 3.06E-03 1.05E-02 -5.105 -29.81

1.32 14.045 1.010 2.136 1.553 2.81E-03 2.89E-03 1.10E-02 -5.800 -35.61

1.34 14.311 1.023 2.096 1.564 2.67E-03 2.74E-03 1.14E-02 -6.568 -42.18

1.36 14.579 1.035 2.058 1.576 2.53E-03 2.60E-03 1.18E-02 -7.419 -49.60

1.38 14.849 1.048 2.020 1.588 2.40E-03 2.46E-03 1.22E-02 -8.368 -57.97

1.4 15.120 1.060 1.984 1.601 2.28E-03 2.34E-03 1.26E-02 -9.430 -67.40

1.42 15.393 1.073 1.949 1.614 2.16E-03 2.22E-03 1.30E-02 -10.624 -78.02

1.44 15.667 1.085 1.915 1.627 2.06E-03 2.11E-03 1.33E-02 -11.975 -90.00

1.46 15.943 1.097 1.882 1.640 1.96E-03 2.01E-03 1.36E-02 -13.514 -103.51

1.48 16.221 1.110 1.849 1.654 1.86E-03 1.91E-03 1.39E-02 -15.279 -118.79

1.5 16.500 1.122 1.818 1.668 1.77E-03 1.82E-03 1.41E-02 -17.324 -136.111.52 16.781 1.134 1.788 1.683 1.69E-03 1.73E-03 1.44E-02 -19.715 -155.83

1.54 17.063 1.146 1.758 1.698 1.61E-03 1.65E-03 1.47E-02 -22.546 -178.37

1.56 17.347 1.158 1.729 1.712 1.54E-03 1.57E-03 1.49E-02 -25.945 -204.32

1.58 17.633 1.170 1.701 1.728 1.47E-03 1.50E-03 1.51E-02 -30.099 -234.42

1.6 17.920 1.182 1.674 1.743 1.40E-03 1.43E-03 1.53E-02 -35.283 -269.70

1.62 18.209 1.194 1.648 1.758 1.34E-03 1.37E-03 1.55E-02 -41.925 -311.63

1.64 18.499 1.206 1.622 1.774 1.28E-03 1.31E-03 1.57E-02 -50.732 -362.36

1.66 18.791 1.218 1.596 1.790 1.22E-03 1.25E-03 1.59E-02 -62.951 -425.31

1.68 19.085 1.230 1.572 1.806 1.17E-03 1.20E-03 1.60E-02 -81.023 -506.33

1.7 19.380 1.242 1.548 1.822 1.12E-03 1.15E-03 1.62E-02 -110.433 -616.77

1.72 19.677 1.254 1.525 1.838 1.07E-03 1.10E-03 1.63E-02 -166.658 -783.42

1.74 19.975 1.266 1.502 1.855 1.03E-03 1.05E-03 1.65E-02 -316.826 -1100.25

1.745 20.050 1.269 1.496 1.859 1.02E-03 1.02E-03 4.14E-03 -171.079 -1271.33

Note that at least three significant figures should be retained in E to avoid large round-off

errors when the differences are small in comparison to the values of E. In the last column,

the cumulative values of x are given, and these represent the distance from the starting

point where the specified depth h is reached. After the first step, uniform increments in

depth h, with h increasing in the upstream direction, are utilized. The values of h are

stopped at the finite limit of 1.745 m, which is 99.5% of the normal depth. The length

required to reach this point is 1271 m, which is the length required for this channel to be

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considered hydraulically long, but that length varies, in general. The depth increments can

be halved until the change in profile length becomes acceptably small. Alternatively,

smaller increments in depth can be used in regions of rapidly changing depth, and larger

increments may be appropriate in regions of very gradual depth-changes. A portion of the

computed M2-profile is shown in Fig. 4.9.

M2 water surface-profile

computed by the direct-step method

0

0,6

1,2

1,8

2,4

-1200 -1000 -800 -600 -400 -200 0

distance upstream (m)

d e p t h ( m )

Fig. 4.9. M2-curve drawn in example 4.2.

Example 4.3: A trapezoidal channel with a bottom width of 5 m, a side slope of 1:1, and a

Manning n of 0.013 m-1/3

s carries a discharge of 50 m3 /s at a bed slope of 0.0004. Compute

by the direct-step method the backwater profile created by a dam that backs up the water to

a depth of 6 m immediately in font of the dam. The upstream end of the profile is assumedat a depth equal to 1% greater than the normal depth.

Solution:

Given: bottom width: b = 5.0 m side slope ratio: m:1 = 1:1

Manning’s n: n = 0.013 m-1/3s bed slope: ib = 0.0004

discharge: Q = 50 m3 /s water depth: h = 6.0 m (in front of dam)

Compute the water surface profile.

Similar to Example 4.2: the normal depth and the critical depth are:5

3n n

2 13 22

bn

[h (b mh )] Q.n

ib 2h 1 m

hn = 2.87 m

M1

6 m

NDL

CDL hn hc ib = 0.0004

A

Dam

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32

c c

12

c

h b 2h Q

gb 2mh

hc = 1.57 m

Because hn > hc the channel slope is mild. The profile lies in zone 1 and therefore it is anM1 curve. The range of depth is 6m at the downstream end and (101% x 2,87) = 2.90 m at

the upstream end. Students should try to make a table computation, which is self-explanatory and draw an M1 curve as Fig. 4.10 below:

M1 water surface-profile

computed by the direct-step method

0

1

2

3

4

5

6

7

-8000 -7000 -6000 -5000 -4000 -3000 -2000 -1000 0

distance upstream (m)

w a t e r d e p t h ( m )

i n f r o n t o f d a m

Fig. 4.10. M1-curve drawn in example 4.3.

4.4.2. Direct numerical integration methodThe direct integration method is applicable to prismatic channels only. This method

uses Eq. 4-18 as governing equation:

b e

2

i idh

dx 1 Fr

2

b e

1 Frdx dh

i i

(4-20)

In its integrated form, Eq. 4-20 becomes:i 1 i 1 i 1

i i i

x h h2

i 1 i

b ex h h

1 Frdx x x dh g(h)dh

i i

(4-21)

The integrand on the right hand side of Eq. (4-21) is a function of h, g(h), which can be

integrated numerically to obtain a solution for x, as shown as in Fig. 4.11.

Fig. 4.11. Water surface-profile computation by direct numerical integration

g(h)

hho hi hi+1hn

area = xi+1 - xi

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Chapter 4: NON UNIFORM FLOW 89

A variety of numerical integration techniques are available, such as the trapezoidal rule and

Simpson’s rule, which are commonly used to find the cross-sectional area of a natural

channel, for example. Simpson’s rule is of higher order in accuracy than the trapezoidal

rule, which simply means that the same numerical accuracy can be achieved with fewer

integration steps. Application of the trapezoidal rule to the right-hand side of Eq. (4-21) for

a single step produces:

i 1 ii 1 i i 1 i

g(h ) g(h )x x h h2

(4-22)

To determine the full length of a flow profile, (xl – xo), multiple application of the

trapezoidal rule results inl 1

o l i

i 1l o

g(h ) g(h ) 2 g(h )

L x x h2

(4-23)

where L is the profile length and h = (hi+1 – hi) is the uniform depth-increment. Because

the global truncation error in the multiple application of the trapezoidal rule is of order

(h)2, halving the depth increment will reduce the error in the profile length by a factor ¼.

By successively halving the depth interval, the relative change in the profile length can be

calculated with the process continuing until the relative error is less than some acceptable

value.

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