Open Channel Flow g1b5z

Philadelphia University Faculty of Engineering Civil Engineering Department First Semester, 2011/2012

HYDRAULICS (670441)

Lecture Title

Water Flow in Open Channels

Introduction An open channel is a duct in which the liquid flows with a free surface. Open channel hydraulics is of great importance in civil engineers, it deals with flows having a free surface,

for example: •

Channels constructed for water supply, irrigation, drainage, and

•

Sewers, culverts, and

•

Tunnels flowing partially full; and

•

Natural streams and rivers.

3

Pipe Flow and Open Channel Flow Pipe Flow • The liquid completely fills the pipe and flow under pressure. • The flow in a pipe takes place due to difference of pressure (pressure gradient),

• The flow in a closed conduit is not necessarily a pipe flow.

Open Channel Flow • Flow takes place due to the slope of the channel bed (due to gravity). • The flow must be classified as open channel flow if the liquid has a free surface.

4

Pipe Flow

Open Channel Flow 5

For Pipe flow (Fig. a): • The hydraulic gradient line (HGL) is the sum of the elevation and the pressure head (connecting the water surfaces in piezometers). • The energy gradient line (EGL) is the sum of the HGL and velocity head. • The amount of energy loss when the liquid flows from section 1 to section 2 is indicated by hL.

For open channel flow (Fig. b): • The hydraulic gradient line (HGL) corresponds to the water surface line (WSL); where it subjected to only atmospheric pressure which is commonly referred to as the zero pressure reference. • The energy gradient line (EGL) is the sum of the HGL and velocity head. • The amount of energy loss when the liquid flows from section 1 to section 2 is indicated by hL. For uniform flow in an open channel, this drop in the EGL is equal to the drop in the channel bed. 6

6.1 Classifications of Open Channel Flow Classification based on the time criterion: 1. 2.

Steady Flow (time independent) (discharge and water depth do not change with time) Unsteady Flow (time dependent) (discharge and water depth at any section change with time)

Classification based on the space criterion: 1.

2.

Uniform flow (are mostly steady) (discharge and water depth remains the same at every section in the channel) Non-uniform Flow (discharge and water depth change at any section in the channel)

7

Non-uniform flow is also called varied flow ( the flow in which the water depth and or discharge change along the length of the channel), it can be further classified as: •

Gradually varied flow (GVF) where the depth of the flow changes gradually along the length of the channel.

•

Rapidly varied flow (RVF) where the depth of flow changes suddenly over a small length of the channel.

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a)

Uniform flow are mostly steady

c) Steady varied flow (over a spillway crest)

b) Unsteady uniform flows are very rare in nature

d) Unsteady varied flow (flood wave) e) Unsteady varied flow (tidal surge)

9

10

6.2 Uniform Flow in Open Channel 1.

2.

Uniform flow in an open channel must satisfy the following main features: The water depth y, flow area A, discharge Q, and the velocity distribution V at all sections throughout the entire channel length must remain constant. The slope of the energy gradient line (Se), the water surface slope (Sws), and the channel bed slope (S0) are equal.

Se = Sws = S0

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This is possible when the gravity force (W sin q) component equal the resistance to the flow (Ff)

W sin q  F1  F2  Ff  0 F1  F 2  Hydrostatic forces at 2 ends sin q  tan q  S0 W sin q  (AL) sin q  ALS0 Ff   0 PL  ( KV 2 ) PL

 0  resisting force per unit area of channel, K  cns tan t of proportionality

   A . .S0   K  P

ALS0  ( KV 2 ) PL  V  

V  C Rh Se The Chezy Formula

C

 K

 Chezy cons tan t

A Rh   Hydraulic Radius P

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• Rh = hydraulic radius or hydraulic mean depth

Rh 

areaof flow( wettedarea) A  wetted perimeter P

• C = Chezy coefficient (Chezy’s resistance factor), m1/2/s, varies in relation of both the conditions of channel and flow.

• Manning derived the following empirical relation:

1 1/ 6 C Rh n where n = Manning’s coefficient for the channel roughness

See the next table for typical values of n. 13

14

Manning’s formula • Substituting into Chezy equation, we obtain the Manning’s formula for uniform flow:

1 2/3 V  Rh n

Se

OR

1 2/3 Q  VA  A Rh Se n

Where: • Q in m3/sec, • V in m/sec, • Rh in m, • Se in (m/m), • n is dimensionless

15

Example 1

1 23 V  Rh Se n A  0.5  3  9 )  1.5  9 m 2 P  2 3  1.5 )  3  9.708 2

1.5m

Open channel of width = 3m as shown, bed slope = 1:5000, d=1.5m find the flow rate using Manning equation, n=0.025.

1 2

2

3.0m

A 9 Rh    0.927 P 9.708 2 1 V  0.927 3 1  0.538 m/s 5000 0.025 Q  VA  0.538  9  4.84 m 3 / s 16

Example 2 The cross section of an open channel is a trapezoid with a bottom width of 4 m and side slopes 1:2, calculate the discharge if the depth of water is 1.5 m and bed slope = 1/1600. Take Chezy constant C = 50.

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Example 2 Open channel as shown, bed slope = 69:1584, find the flow rate using Chezy equation, take C=35.

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V  C Rh Se 2.52  5.04 0.72  2.52 A  2.52  16.8   3.6  0.72  150  162.52 m 2 2 2 P  0.72  150 

1.8

2

 3.62 )  16.8 

2.52

2

 5.042 )  177.18 m

A 162.52 Rh    0.917 P 177.18 0.69 V  35 0.917   0.7 m/s 1584 Q  VA  0.7  162.52  113.84 m3 / s 19

6.3 Hydraulic Efficiency of open channel sections Based on their existence, an open channel can be natural or artificial: • Natural channels: such as streams, rivers, valleys, etc. These are generally irregular in shape, alignment and roughness of the surface. • Artificial channels: built for some specific purpose, such as irrigation, water supply, wastewater, water power development, and rain collection channels. These are regular in shape and alignment with uniform roughness of the boundary surface. 20

Based on their shape, an open channel can be prismatic or nonprismatic: • Prismatic channels: the cross section is uniform and the bed slop is constant.

• Non-prismatic channels: when either the cross section or the slope (or both) change, the channel is referred to as non-prismatic. It is obvious that only artificial channel can be prismatic. • The most common shapes of prismatic channels are rectangular, parabolic, triangular, trapezoidal and circular; see the next figure.

21

Most common shapes of prismatic channels

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• Most economical section is called the best hydraulic section or most efficient section as the discharge, ing through a given cross-sectional area A, slope of the bed S0 and a resistance coefficient, is maximum. • Hence the discharge Q will be maximum when the wetted perimeter P is minimum.

A 1 Q AV  AC Rh Se  AC Se  const. * P P

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Economical Rectangular Channel A  B  D, P  2D  B A P2D  D

dP 0 P should be minimum for a given area; dD dP A BD B  A   2   2   0  2  2  2 2  dD D D D D 

B D 2

D A BD 2 D D 2 D 2 Rh     Rh  P B2 D 2 D2 D 4 D 2 So, the rectangular channel will be most economical when either: the depth of the flow is half the width, or the hydraulic radius is half the depth of flow. 24

Economical Trapezoidal Channel A(BnD )D

or

A B  nD D

PB2D 1n 2 A P (  nD )  2 D 1n 2 D

dP A A dP 2 2 2 1  n  n    n  2 1  n  0  0 2 2 D dD dD D (BnD)D B2nD 2 1n  n 2 D D 2

D 1n 2 

B2nD 2

P  B  B  2 n D  2( B  n D ) A (B  n D ) D Rh   P 2(B  nD)

D Rh  2

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Other criteria for economic Trapezoidal section • When a semi-circle is drawn with the trapezoidal center, O, on the water surface and radius equal to the depth of flow, D, the three sides of the channel are tangential to the semi-circle”. • To prove this condition, using the figure shown, we have:

1 B OF  OM sin   ( B  2 n D ) sin   (  n D ) sin  2 2 sin  

OF B nD 2

using triangle KMN, we have:

sin  

MK D  MN D 1  n2

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OF 

(B 2)  n D 1  n2

B  2n D using equation D 1  n  to replace the numerator , we obtain: 2 2

OF 

D 1  n2 1  n2

OF  D

Thus, if a semi-circle is drawn with O as center and radius equal to the depth of flow D, the three sides of a most economical trapezoidal section will be tangential to the semicircle. 27

The best side slope for Trapezoidal section n

when

1

 q  60



3

B2nD D 1n   B  2 D ( 1  n2  n) 2 2

A PBB2nD2(BnD) B   n D D A  n D  2 D ( 1  n 2  n) D D  2

A 2 1 n2  n

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Now, from equations:

P  2( B  n D ) B

A  nD D

A P2 D

A 2 P  4 ( )  4 A ( 2 1  n 2  n) D 1  dP dP  0 2P  4 A [(1  n2 ) 2 * (2n)  1] dn dn 2n  1  4n2 1n2  n  1 1  3  tanq  q  60 1  n2 3 n 1 squaring both sides

2

The best side slope is at 60o to the horizontal, i.e.; of all trapezoidal sections a half hexagon is most economical. However, because of constructional difficulties, it may not be practical to adopt the most economical side slopes

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Circular section In the case of circular channels, the area of the flow cannot be maintained constant. Indeed, the cross-sectional area A and the wetted perimeter P both do not depend on D but they depend on the angle .

Referring to the figure shown, we can determine the wetted perimeter P and the area of flow A as follows:

d 2

d2 A  sin2 4 8

P  2 r   d Thus in case of circular channels, for most economical section, two separate conditions are obtained: 1. Condition for maximum discharge, and 2. Condition for maximum velocity. 30

1. Condition for Maximum Discharge for Circular Section:

Q  AV  A C

Rh S  C

A3 S P

3  A Q2  C 2 S    P

(Using the Chezy formula)

  154

D  0.95 d

(Using Manning’s formula)

  151

D  0.94 d

dQ 0 d

2. Condition for Maximum Velocity for Circular Section:

V C

Rh S  C

  128.75

A S P

 A V C S   P 2

2

dV 0 d

D  081 . d 31

Variation of flow and velocity with depth in circular pipes

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6.4 Energy Principle in Open Channel Flow The total energy of a flowing liquid per unit weight is given by:

V2 Total Energy  Z  y  2g

If the channel bed is taken as the datum, then the total energy per unit weight will be: 2

V Especific  y  2g

Specific energy (Es) of a flowing liquid in a channel is defined as energy per unit weight of the liquid measured from the channel bed as datum. It is a very useful concept in the study of open channel flow.

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2

V Es  y   E p  Ek 2g Q2 Es  y  2 g A2

Ep = potential energy of flow = y 2 V Ek = kinetic energy of flow = 2g

Valid for any cross section

Specific Energy Curve: It is defined as the curve which shows the

variation of specific energy (Es ) with depth of flow y. 34

Specific Energy Curve (Rectangular channel) Consider a rectangular channel in which a constant discharge q = discharge per unit width =

Q = constant ( since Q and B are constants) B

Q Q q V   Es  y  A B y y

2

q  E p  Ek 2 2g y

Ep

EK EP

Es

yc 35

Sub-critical, critical, and supercritical flow The criterion used in this classification is what is known by Froude number, Fr, which is the measure of the relative effects of inertia forces to gravity force:

V Fr  g Dh

2 Q T 2 Fr  3 Ag

T

V = mean velocity of flow of water, Dh = hydraulic depth of the channel

Dh 

Area of Flow (Wetted Area) A  Water Surface Width T Fr

Flow

Fr < 1

Sub-critical

1 = Fr

Critical

Fr >1

Supercritical

T

36

Referring to the energy curve, the following features can be observed:

1. The depth of flow at point C is referred to as critical depth, yc. (It is defined as that depth of flow of liquid at which the specific energy is minimum, Em in  yc The flow that corresponds to this point is called critical flow (Fr = 1.0). 2. For values of Es greater than Emin , there are two corresponding depths. One depth is greater than the critical depth and the other is smaller then Es1  y1 and y1' the critical depth, for example; These two depths for a given specific energy are called the alternate depths.

3. If the flow depth

y  yc

the flow is said to be sub-critical (Fr < 1.0).

In this case Es increases as y increases. 4. If the flow depth

y  yc

the flow is said to be super-critical (Fr > 1.0).

In this case Es decreases as y increases. 37

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Critical depth, yc for rectangular channel Critical depth, yc , is defined as that depth of flow of liquid at which the specific energy is minimum, Emin, The mathematical expression for critical depth is obtained by differentiating energy equation with respect to y and equating the result to zero;

q2 Es  y  2g y2 dE d q2 q2  2 0  (y  ) 1 ( 3 )0 2 dy dy 2g y 2g y 2

q 1 0  3 gy

2

q 3 y  g

q yc   g

2

  

1

3

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Critical velocity, Vc for rectangular channel 2 q yc3  , g

Q Q q V   A B y y OR

2 c

2 c

V y y  Vc  g 3 c

q Vc  yc g  yc

Vc  1  Fr g  yc 40

Minimum Specific Energy in of critical depth

Emin

q2  yc  2 g yc2

E min

2

q y  g 3 c

E min

3 yc  2

OR

yc  yc  2

2 E min yc  3

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Critical depth, yc , for Non- Rectangular Channels

dEs d Q2 2 Q 2 dA 0  ( y )1 ( )0 2 3 dy 2g A 2 g A dy dy Q2 dA OR 1  (constant discharge is assumed) ( )  0 g A3 dy dA/dy = the rate of increase of area with respect to y = T (top width). 2

QT Q2 A3 condition must be satisfied for the flow 1  0  3 g T at the critical depth. gA Q2 A 2 Recalling that D   A Dh  h g T V2 Dh

The equation may also be written in of velocity 

2g



2

The velocity head is equal to one-half the hydraulic depth for critical flow.42

Q2 A Es  y  Es  y  2 2T 2g A OR

1 A Ec  yc  ( ) 2 T

This equation represents the critical state

The general equation for the specific energy in critical state applicable to channels of all shapes.

Rectangular section

Trapezoidal section

3 yc Ec  2

Ec 

Circular section

Ec 

d d ( 2  sin 2 ) ( 1  cos )  2 16 sin 

( 3B  5n yc ) yc 2 ( B  2n yc )

Triangle section

5 E c  yc 4

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Constant Specific Energy The specific energy was varied and the discharge was assumed to be constant. Let us now consider the case in which the specific energy is kept constant and the discharge Q is varied.

Q2 Es  y  2 2g A

Q A

2 g ( Es  y )

Q2  A2 (2 g ) ( Es  y)  2 gA2 Es  2 gA2 y The discharge will maximum if

dQ 0 dy

 dQ dA dA 2  Q  2 g Es (2 A )  2 g (2 y A  A ) dy dy dy   dA/dy = T

 2g Es (2 AT )2g (2 yAT )2gA 0 2

44

4 EsT  4 yT  2 A  0

2T ( Es  y )  A  but

Es  y 

A 2T

Q2 Es  y  2 g A2 2

2

3

Q A Q A  y  y  2 g T 2g A 2T

Thus for a given specific energy, the discharge in a given channel is a

maximum when the flow is in the critical state. The depth corresponding to the maximum discharge is the critical depth.

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6.5 Hydraulic Jump • A hydraulic jump occurs when flow changes from a supercritical flow (unstable) to a sub-critical flow (stable). • There is a sudden rise in water level at the point where the hydraulic jump occurs. • Rollers (eddies) of turbulent water form at this point. These rollers cause dissipation of energy. •A hydraulic jump occurs in practice at the toe of a dam or below a sluice gate where the velocity is very high.

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General Expression for Hydraulic Jump: In the analysis of hydraulic jumps, the following assumptions are made: (1) The length of hydraulic jump is small. Consequently, the loss of head due to friction is negligible. (2) The flow is uniform and pressure distribution is due to hydrostatic before and after the jump. (3) The slope of the bed of the channel is very small, so that the component of the weight of the fluid in the direction of the flow is neglected.

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Location of hydraulic jump Generally, a hydraulic jump occurs when the flow changes from supercritical to subcritical flow. The most typical cases for the location of hydraulic jump are: 1. Jump below a sluice gate. 2. Jump at the toe of a spillway. 3. Jump at a glacis. (glacis is the name given to sloping floors provided in hydraulic structures.)

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•The net force in the direction of flow = the rate of change of moment in that direction

Q



g

(V2 V1)

The net force in the direction of the flow, neglecting frictional resistance and the component of weight of water in the direction of flow, R = F1 - F 2 . Therefore, the impulse-moment yields

F1  F2 

Q g

(V2  V1 )

Where F1 and F2 are the pressure forces at section 1 and 2, respectively.

 A1 y1   A2 y2 

Q

(V2  V1 )

g  Q2 1 1  A1 y1   A2 y2  (  ) g A2 A1

Q2 Q2  A1 y1   A2 y2 gA1 gA2 y = the distance from the water surface to the centroid of the flow area

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Q2 Q2  A1 y1   A2 y2 gA1 gA2

Comments: • This is the general equation governing the hydraulic jump for any

shape of channel. • The sum of two is called specific force (M). So, the equation can be written as: M1 = M2 • This equation shows that the specific force before the hydraulic jump is equal to that after the jump.

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Hydraulic Jump in Rectangular Channels

A1  B y1

y1 y1  2

A2  B y2

Q2 Q2  A1 y1   A2 y2 gA1 gA2 Q using q  B

y2 y2  2

Q2 y1 Q2 y2  ( By1 )( )   ( By2 )( ) g B y1 2 g B y2 2

q 2  y2  y1  y22  y12   g  y1 y2  2 2 q2  y1 y2 ( y2  y1 ) g

, we get

2 2 q y2 y12  y22 y1  0 g

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This is a quadratic equation, the solution of which may be written as:

y1 y2    2

 2q 2   y1       2  g y1 

y2 y1    2

2  y 2 q  2      2  g y2 

2

2

y2 1   1   y1 2 

8 q 2  1 g y13 

y1 1   1   y2 2 

8 q 2  1 g y23 

where y1 is the initial depth and y2 is called the conjugate depth. Both are called conjugate depths. These equations can be used to get the various characteristics of hydraulic jump.

52

2 q But for rectangular channels, we have yc3  g 3     y 1 y 2 Therefore,    1 18 c   y1 2 y1     

 y1 1   1  y2 2  

3

 yc   1  8    y2   

These equations can also be written in of Froude’s number as:



y2 1 2   1 18F1 y1 2 y1 1   1  y2 2 

1

)

8 F22  

F1

F2 

V1 g y1 V2 g y2 53

Head Loss in a hydraulic jump (HL): Due to the turbulent flow in hydraulic jump, a dissipation (loss) of energy occurs:

HL  E  E1  E2

Where, E = specific energy For rectangular channels: E s

q2 y 2 g y2 2 

q q     y2  hence, H L  y1  2 3 2 g y1  2 g y2  After simplifying, we obtain

2

( y2  y1 )3 E  H L  4 y1 y2 54

Height of hydraulic jump (hj): The difference of depths before and after the jump is known as the height of the jump,

hj  y2  y1 Length of hydraulic jump (Lj): The distance between the front face of the jump to a point on the downstream where the rollers (eddies) terminate and the flow becomes uniform is known as the length of the hydraulic jump. The length of the jump varies from 5 to 7 times its height. An average value is usually

taken:

Lj  6hj 55

6.6 Gradually Varied Flow • Non-uniform flow is a flow for which the depth of flow is varied. • This varied flow can be either Gradually varied flow (GVF) or Rapidly varied flow (RVF).

• Such situations occur when: - control structures are used in the channel or, - when any obstruction is found in the channel, - when a sharp change in the channel slope takes place.

56

Classification of Channel-Bed Slopes The slope of the channel bed is very important in determining the characteristics of the flow.

Let • S0 : the slope of the channel bed , • Sc : the critical slope or the slope of the channel that sustains a given discharge (Q) as uniform flow at the critical depth (yc). • yn is is the normal depth when the discharge Q flows as uniform flow on slope S0.

57

The slope of the channel bed can be classified as: 1) Critical Slope C : the bottom slope of the channel is equal to the critical slope.

S0  Sc

or

yn  yc

2) Mild Slope M : the bottom slope of the channel is less than the critical slope.

S0  Sc

or

yn  yc

3) Steep Slope S : the bottom slope of the channel is greater than the critical slope.

S0  Sc

or

yn  yc

4) Horizontal Slope H : the bottom slope of the channel is equal to zero.

S0  0.0 5) Adverse Slope A : the bottom slope of the channel rises in the direction of the flow (slope is opposite to direction of flow).

S0  negative 58

59

Classification of Flow Profiles (water surface profiles) • The surface curves of water are called flow profiles (or water surface

profiles). • The shape of water surface profiles is mainly determined by the slope of the channel bed So. • For a given discharge, the normal depth yn and the critical depth yc

may be calculated. Then the following steps are followed to classify the flow profiles: 1- A line parallel to the channel bottom with a height of yn is drawn and is designated as the normal depth line (N.D.L.) 2- A line parallel to the channel bottom with a height of yc is drawn and is designated as the critical depth line (C.D.L.) 3- The vertical space in a longitudinal section is divided into 3 zones

using the two lines drawn in steps 1 & 2 (see the next figure) 60

4- Depending upon the zone and the slope of the bed, the water profiles are classified into 13 types as follows: (a) Mild slope curves M1 , M2 , M3 . (b) Steep slope curves S1 , S2 , S3 . (c) Critical slope curves C1 , C2 , C3 . (d) Horizontal slope curves H2 , H3 . (e) Averse slope curves A2 , A3 . In all these curves, the letter indicates the slope type and the subscript indicates the zone. For example S2 curve occurs in the zone 2 of the steep slope. 61

Flow Profiles in Mild slope

Flow Profiles in Steep slope 62

Flow Profiles in Critical slope

Flow Profiles in Horizontal slope

Flow Profiles in Adverse slope

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Dynamic Equation of Gradually Varied Flow Objective: get the relationship between the water surface slope and other characteristics of flow. The following assumptions are made in the derivation of the equation

1. The flow is steady. 2. The streamlines are practically parallel (true when the variation in depth along the direction of flow is very gradual). Thus the hydrostatic distribution of pressure is assumed over the section. 3. The loss of head at any section, due to friction, is equal to that in the corresponding uniform flow with the same depth and flow characteristics. (Manning’s formula may be used to calculate the slope of the energy line) 4. The slope of the channel is small. 5. The channel is prismatic. . 6. The velocity distribution across the section is fixed. 7. The roughness coefficient is constant in the reach. 64

Consider the profile of a gradually varied flow in a small length dx of an open channel the channel as shown in the figure below.

The total head (H) at any section is given by:

V2 HZy 2g

Taking x-axis along the bed of the channel and differentiating the equation with respect to x:

dH dZ dy d V 2       dx dx dx dx  2 g  65

• dH/dx = the slope of the energy line (Sf). • dZ/dx = the bed slope (S0) . Therefore,

dy d V 2     S f   S0   dx dx  2 g 

Multiplying the velocity term by dy/dy and transposing, we get

dy dy d  V 2     S0  S f  dx dy dx  2 g 

dy  dx

S0  S f

or

dy dx

 d V 2       S0  S f  1 dy  2 g   

d V 2    1 dy  2 g 

This Equation is known as the dynamic equation of gradually varied flow. It gives the variation of depth (y) with respect to the distance along the bottom of the channel (x).

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The dynamic equation can be expressed in of the discharge Q:

S0  S f dy  dx Q2 T 1 g A3 The dynamic equation also can be expressed in of the specific energy E :

dy dE / dx  dx Q2 T 1 g A3 67

Depending upon the type of flow, dy/dx may take the values:

dy 0 (a) dx

(b)

(c)

dy  positive dx

The slope of the water surface is equal to the bottom slope. (the water surface is parallel to the channel bed) or the flow is uniform.

The slope of the water surface is less than the bottom slope (S0) . (The water surface rises in the direction of flow) or the profile obtained is called the backwater curve.

dy  negative The slope of the water surface is greater than the bottom dx slope. (The water surface falls in direction of flow) or the profile obtained is called the draw-down curve.

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Notice that the slope of water surface with respect to horizontal (Sw) is different from the slope of water surface with respect to the bottom of the channel (dy/dx). A relationship between the two slopes can be obtained:

• • • •

Consider a small length dx of the open channel. The line ab shows the free surface, The line ad is drawn parallel to the bottom at a slope of S0 with the horizontal. The line ac is horizontal.

bc cd  bd  The water surface slope (Sw) is given by Sw  sin   ab ab Let q be the angle which the bottom makes with the horizontal. Thus

cd cd S0  sin q   ad ab

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The slope of the water surface with respect to the channel bottom is given by

dy bd bd   dx ad ab dy S w  S0  dx

This equation can be used to calculate the water surface slope with respect to horizontal.

dy  S0  S w dx 70

Water Profile Computations (Gradually Varied Flow) • Engineers often require to know the distance up to which a surface profile of a gradually varied flow will extend. • To accomplish this we have to integrate the dynamic equation of gradually varied flow, so to obtain the values of y at different locations of x along the channel bed. •The figure below gives a sketch of calculating the M1 curve over a given weir.

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Direct Step Method • One of the most important method used to compute the water profiles is the direct step method. • In this method, the channel is divided into short intervals and the computation of surface profiles is carried out step by step from one section to another. For prismatic channels: Consider a short length of channel, dx , as shown in the figure.

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dx

Applying Bernoulli’s equation between section 1 and 2 , we write:

V12 V22 S0 dx  y1   y2   S f dx 2g 2g or

or

S0 dx  E1  E2  S f dx E2  E1 dx  S0  S f

where E1 and E2 are the specific energies at section 1 and, respectively. This equation will be used to compute the water profile curves. 73

The following steps summarize the direct step method: 1. Calculate the specific energy at section where depth is known. For example at section 1-1, find E1, where the depth is known (y1). This section is usually a control section. 2. Assume an appropriate value of the depth y2 at the other end of the small reach. Note that: y 2  y1 if the profile is a rising curve and,

y2  y1

if the profile is a falling curve.

3. Calculate the specific energy (E2) at section 2-2 for the assumed depth (y2). 4. Calculate the slope of the energy line (Sf) at sections 1-1 and 2-2 using Manning’s formula

1 2/3 V1  R1 n

Sf1

and

1 2/3 V2  R2 n

And the average slope in reach is calculated

S fm 

Sf2

Sf 1  Sf 2 2

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5. Compute the length of the curve between section 1-1 and 2-2

L1,2

E2  E1  dx  S0  S fm

or

L1,2 

E2  E1  Sf 1Sf 2  S0    2  

6. Now, we know the depth at section 2-2, assume the depth at the next section, say 3-3. Then repeat the procedure to find the length L2,3. 7. Repeating the procedure, the total length of the curve may be obtained. Thus

L  L1, 2  L2,3 .......  Ln1,n where (n-1) is the number of intervals into which the channel is divided.

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