1- flow in natural rivers and... · 2018. 1. 19. · lect.no.7 asst.prof.dr. jaafar s. maatooq 2nd...

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Lect.No.7 Asst.Prof.Dr. Jaafar S. Maatooq

2nd Semester Flow Dynamics in Open Channels and Rivers 1 of 19

Rivers and artificial channels, like canals, convey water with a free surface, that is, the surface of water being exposed to air . In this lecture , the behavior and corresponding mathematical description of flow in open channels are reviewed in order to utilize them in designing water resources systems.

1- Flow in natural rivers

Figure1 shows a river carrying a low discharge . When the water surface of the river just touches its banks, the discharge flowing through the river at this stage is called the “bank full discharge”. It is also sometimes called the “dominant discharge”. If the discharge in the river increases, the water will overflow the banks and would spill over to the adjacent land, called the flood plains (Figure 2).




The amount of discharge flowing through the river is of interest to the water resources engineer it cannot be measured directly by any instruments. Rather, an indirect method is used which requires knowledge of the velocity distribution in a river or an open channel.

If we plot the velocity profile across a river, it would actually vary in three dimensions. Figure 4 shows the variation of velocity , It may be observed that velocity is highest at the center of the river but is zero at the banks.

If we now take a look at the variation of velocity in a vertical plane within a river, and we plot them along different vertical lines across the river, then we may find the velocity profiles similar to those shown in Figure 6.




In order to measure the discharge being conveyed in a river, the velocity profile or the average velocity at a number of equally spaced sections are measured, as in Figure 6. The total discharge is then approximately taken equal to the sum of the discharges passing through each segment.

It has been observed through experiments that a plot of velocity in the vertical plane would show that the maximum velocity occurs slightly below the surface (see Figure 8) for a typical river flow.

It has further been observed that an equivalent average velocity is almost equal to the actual velocity measured at 0.6 depth.




2-Variation of discharge with river stage

The water level in a river is sometimes called the “stage” and as this varies, there is a proportional change in the total discharge conveyed. For each point of a river, the relation between stage and discharge is unique but a general form is found to be as shown in Figure 9.

The general mathematical description for the stage-discharge relation is given as:

Q = k ( h − ho ) m …………………………….. (1)

Where h is the gauge corresponding to a discharge Q and h0 is the corresponding to zero discharge , k and m , are constants.

3-Flow along river length

The velocity in a river cross section actually varies from bank to bank and from riverbed to free water surface and hence, can be called a two dimensional variation in a vertical plane. However, for engineering purposes it is, sufficient, generally, to use an equivalent velocity in the direction of river motion (perpendicular to river cross section) which may be




obtained by dividing the total discharge by the cross sectional area. In a natural river, these flow velocities may vary from section to section (Fig. 10).

If we now consider an axis along the length of the river, the total energy (H) is given as:

H=Z + h + (V2/2g) …………………….. (2)

We may plot the total energy as shown in Figure 11, where the variables are as follows:

• Z: Height of riverbed above a datum

• h: Depth of water

• V: Average velocity at a section

• (V2/2g) Kinetic energy head




Along a river length, the depth and velocity would vary . So, if two consecutive of a river section is taken, then the variations in riverbed, water surface and the total energy may be considered as linear (Figure 12).




In Figure 12, three slopes have been marked, which are:

• S0: Riverbed slope

• S: Water surface slope

• Sf: Energy surface slope

The total energy of flowing water reduces along the river length due to friction

* “energy surface slope” is the “friction slope”.

* The energy loss in a river or an open channel occurs mostly due to :-

- the resistance at the channel sides and bed .

One of the earliest models for friction slope Sf or, in effect, the channel resistance was derived from the considerations of “uniform flow” where the flow variables and cross section are supposed to remain constant over a short reach.

If we take small volume of fluid from these two sections we may make a free body diagram of the forces acting on it (Figure 14).

The variables represented in the figure are as follows:-

• W: Weight of water contained in the control volume

• V: Inflow velocity, which is the same as the outflow velocities

• θ: Angle of slope river bed, which is also equal to that water surface and

friction slopes

• τ 0 : Shear stress due to friction acting on the control volume of fluid from

the river bed and along the sides , though in Figure 14 only the resistance due to the riverbed is shown.




Equating forces resulting :-

τ 0 P L = W sinθ = ρ g A L sinθ ……………… (3)

Where the variables are:

• P: wetted perimeter

• A: Cross section of flow area

• L: Length of control volume

After simplifying and re-arranging , it conclude to “Chezy equation” , take a form :-

V = C √R S …………………………………………. (4)

The constant C in equation (4) actually varies depending on boundary roughness and “R” is called a hydraulic radius , where :-

R= A/P ………………………………………………… (5)




4-Uniform steady flow and Manning’s Equation

When discharge remains the same and depth does not change then we have uniform steady flow. In this condition the surface of water is parallel to the bed of the channel (S = So ) ,

The slope of the channel can be expressed as :-

- An angle = 1 degrees

- As percent = 1%

- Or as fraction = 0.01 or 1 in 100

Velocity of flow (v) in a channel can be computed using an empirical equations called “Mannings equation “ , where ;




V= 1n

R 2/3 S 1/2 …………………………… (6)

This the SI units form of the equation with “ V “ in (meters/sec) and “ R “ (meters).

Where “n” is the Manning’s coefficient (dimensionless) values developed through experimentation .

Possible n values for various channel surfaces as shown in Table below ;

The hydraulic radius “ R=A/WP “ for various channel shapes can be shown in figure below;




5-Types of open channel flows

Steady flow – when discharge (Q) does not change with time .

Uniform flow – when depth of fluid does not change for a selected length or section of the channel . Uniform steady flow – when discharge does not change with time and depth remains constant for a selected section . Varied steady flow – when depth changes but discharge remains the same . Varied unsteady flow – when both depth and discharge change along a channel length of interest. Rapidly varying flow – depth change is rapid . Gradually varying flow – depth change is gradual .




See Figure below ;

Section 1 – rapidly varying flow

Section 2 – gradually varying flow

Section 3 – hydraulic jump

Section 4 – weir and waterfall Section 5 – gradually varying

Section 6 – hydraulic drop due to change in channel slope 6-Energy flow Flowing water contains energy in two forms, Potential Energy (PE) and Kinetic Energy (KE) . The potential energy at a particular point is represented by the depth of the water plus the elevation of the channel bottom above a convenient datum plane. The kinetic energy , is represented by the velocity head . Where , Energy at a particular point in the channel = PE + KE , this is called a specific energy .




E = Y + V 2

2g ………………… (7)

Where Y is the depth of flow and V is the velocity .

7-Specific Energy Diagram

The specific energy can be plotted graphically as a function of depth of flow.

E = Es + Ek

Es = y (static energy)

Ek = Q2/2gA2 (kinetic energy)

Relationship between y and Es & Ek as shown in Figure below ;

When combining the two relationships the resulting diagram is a specific energy diagram , see figure below ;




Key points from the diagram :-

1. The diagram applied for a given cross section and discharge .

2. As the depth of flow increases, the static energy increases, and the kinetic energy decreases .

3. The total energy curve approaches the static energy curve for high depths and the kinetic energy curve for small depths.

4. The specific energy is minimum (Emin) for a particular depth it happens to be the critical depth for which the Froude’s number = 1.0 and velocity = Vc.

5. At “ Emin “ there is only a corresponding a singular depth that is , a critical depth , yc .

6. For all other energy values there are two depth associated , one greater than the critical depth and one less than the critical depth.




7. Depths less than the critical depths considered “ supercritical flow“ Froude Number > 1.0 and V > Vc.

8. Depths greater than the critical depths considered “ subcritical flow “ Froude Number < 1.0 and V < Vc.

9. The two depths associated with the same energy values are referred to as Alternate depths .

10. As discharge increases, the specific energy curves move to the upper right portion of the chart. ( see fig. balow ) .




8-Non uniform flow

Flow that varies in depth and velocity along the channel is called non-uniform , that occur because changes :-

• In channel section, • Slope, • And or roughness .

cause the depths and average velocities of flow to vary from point to point along the channel, and the water surfaces will not be parallel to the streambed.

The Figure below shows an example of non uniform flow from channel to lake

Also :-

- The change from “ subcritical “ to “ supercritical “ is non uniform flow (Gradually Varied) flow , Fig.A below .

- The change from “ supercritical “ to “subcritical” is non uniform flow (Rapidly Varied) flow , Fig.B below .




Fig.A

Fig.B

9-Hydraulic Jump

The hydraulic jump consists of an abrupt rise of the water surface in the region of impact between rapid and tranquil flows. Flow depths before (supercritical depth, d1) and after (subcritical depth, d2) the jump are less than and greater than critical depth, respectively. The depth d1 is calculated based on the hydraulics of the channel. The depth d2 is calculated by using the “ Balenger Equation “ , where ;

y1y2

= (-1 + √√√√1 + 8F1 2) ………… (8)




The zone of impact of the jump is accompanied by large-scale turbulence, surface waves, and energy dissipation. The hydraulic jump in a channel may occur at locations such as:

a. The vicinity of a break in grade where the channel slope decreases from steep to mild.

b. A short distance upstream from channel constrictions such as those caused by bridge piers.

c. A relatively abrupt converging transition.

d. A channel junction where rapid flow occurs in a tributary channel and tranquil flow in the main channel.

e. Long channels where high velocities can no longer be sustained on a mild slope.

The geometry of the most practical regular open channels can be shown in figure below ;

1- flow in natural rivers and... · 2018. 1. 19. · lect.no.7 asst.prof.dr. jaafar s. maatooq 2nd...

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