Lecture 4

Graphical Integration.
Direct and Standard step method
Numerical methods.
GRADUALLY VARIED FLOW COMPUTATION

 The process of determination of the length of the surface profiles is known as the

computation of gradually varied flow.

 The various available procedures for computing GVF profiles can be classified as:

1. Graphical integration method.

2. Numerical integration method.

3. Direct integration method.

4. Direct Step method.

5. Standard step method.

GRAPHICAL INTEGRATION METHOD

 This is a simple and straight forward method and is applicable to both prismatic and

non prismatic channels of any shape and slope.

Flow profile

Y1 Y2

O
X1
X 2-X1 X
X2

Figure 3.22: Graphical integration method

in figure above. The depths of flow are y1 and y2 respectively; Let, L= x2-x1, we have

= − =∫ =∫ on simplifying

⎧ ⎫
⎪ − ⎪
= =
⎨ ⎬
⎪ − ⎪
⎩ ⎭

The above equation can be graphically integrated for a given channel and its discharge by

plotting the value of on x-axis for various values of y plotted on x-axis.

y2
y1

y
dy

Figure 3.23: Graphical integration method

y1 and y = y2) L can be determined. The area can also be determined by computing

the ordinates for different values of y and then, calculating the area between the

adjacent ordinates. Summing these areas, one can obtain the desired length L.

DIRECT STEP METHOD

 The direct step method is applicable to prismatic channels for which channel

characteristics are invariant with distance along the channel.

 In this method the channel is divided into short reaches. The computation is carried

out step by step from one section to the other.

2
Energy grade line

Y1 2
Water surface

Y2

SO dx
Channel bed

dx
a b
Direct step method
Figure 3.24:- Direct step method
 Calculate the specific energy at the control section a-a in the small reach, where the depth

of flow and other hydraulic parameters are known.

 Assume an approximate value of the depth y2 at the other end of the small reach. Assume

the depth y2 more than that at section a-a, if the profile is a rising curve and less than that

at section 1-1, if the profile is a falling curve.

 Calculate the specific energy at section b-b using the assumed y2.

 Using the Manning•s equation, calculate the slope of the energy line at section a-a and b-

b.

1 1
= ; =
/ / / /

 Find the average slope in the reach, i.e.,

+
=
,
2

 Find the length of the curve between sections a-a and b-b using,

−
=
,
− ,

 Knowing the depth at section b-b, assume the depth at section c-c, and repeat the

procedure to get L2,3. The total length of the curve is given by, = , + , + ⋯……

 In the case of subcritical flow, the computation is carried out from d/s to u/s. and in the

case of super critical flow, the computation is carried out from u/s to d/s.
STANDARD STEP METHOD

 This method is applicable to both prismatic and non-prismatic channels. Here the

computation is carried out step by step as in the direct step method.

 In this method, the distance between the sections and the depth of flow at one of the

sections are known and the method is applied to determine the depth of flow at the

other section, and requires trial and error procedure.

2
Energy grade line

Y1 2
Water surface

Y2

SO dx
Channel bed

dx
1 2
Standard step

Figure 3.25:- Standard step method

 Considering the two sections 1-1 and 2-2 of a short channel reach and applying the

energy equation between these two sections we have, + + = + + +

ℎ +ℎ where, hf and he are the loss due to friction and form/eddy loss due to

variation in cross sectional area of the channel respectively.

direction. For any given discharge, the depth of flow (y2) and other hydraulic

parameters would be known at a control section.

 It is required to calculate the depth of flow (y1) at the section immediately u/s of the

control section. For that, assume the depth of flow (y1) at this section and the total

energy is computed using, + +

 The depths at the two ends of this reach are known; therefore, h f and he may be

calculated as follows: ℎ = , ∗ Where, , = and, ℎ =

 Substituting these values in the energy equation the value of + + is

computed and compared with the corresponding value computed for the assumed y1.

If these two values match, the assumed value gives the desired depth y1. Otherwise,

another suitable value for y1 is assumed and the procedure is repeated. The same

procedure is repeated similarly for other sub – reaches.

NUMERICAL METHODS

 The basic differential equation of GVF can be expressed as = ( ). In which ( )=

and is a function of y only for a given So, n, Q and channel geometry. Above

equation is non-linear and a class of methods, which is, particularly suitable for numerical
solution of the above equation is the Runge-Kutta method.
 In Runge-Kutta methods the value y is evaluated at (x+Δx) using a given y at x. Using the
notation yi = y(xi) and xi+Δx = xi+1 and hence yi+1 = y(xi+1), the different types of Runge-
Kutta method methods for solution of equation are as:

1
a) Standard Fourth Order Runge-Kutta Methods (SRK)
= + ( +2 +2 + )
6
Where = ( )

= ( + )
2

= ( + )
2
= ( + )

b) Kutta Merson Method (KM)

= + ( +4 + )
Where = ( )

1
= ( + )
3
1
= ( + + )
3 2 2
1 3 9
= ( + + )
3 8 8
1 3 9
= ( + − +6 )
3 2 2

 Channel is divided into N parts of unknown length interval Δx. starting from the known
depth; the depths at other sections are systematically evaluated.

 For a known yi and Δx, the coefficients K1, K2...etc is determined by repeated calculations
and then by substituting in appropriate main equation the value of yi+1 is found.
 The SRK method involves the determination of F(y) four times while the KM method
involves F(y) to be evaluated five times for each depth determination. These two methods
are direct methods and no iteration is involved.