Notes on Flow through Open Channel: Formula, Alternative Methods and Examples!
Introduction to Open Channel:
An open channel means a passage through which water flows with its surface freely exposed to the atmosphere. The water at its surface is at atmospheric pressure throughout. The channel may be open or may be provided with a cover or top. In case the channel is provided with a cover or top, it must not be running full and the air above the water surface is at atmospheric pressure. A pipe in which the water is not running full also acts as a channel.
The section of a channel may be uniform or non-uniform. For example, a canal, a sewer, an aqueduct, etc. are channels of uniform section, while rivers and streams are channels of non-uniform section.
We know in the case of flow through pipes, the flow of water is caused by a pressure difference. But in the case of flow through channels no pressure difference is built up. The bed of the channel is provided with a slope in the direction of flow, and it is this slope which causes the flow. In other words, the flow is due to gravity. The longitudinal profile of the water surface is the hydraulic gradient line for the open channel.
Open channels may be either natural channels or man-made channels. All water ways formed by natural causes are natural channels. Channels constructed for various purposes are artificial channels. For example, canals, flumes, culverts etc. are artificial channels.
Rectangular, trapezoidal and circular sections are the usually adopted sections for channels. Though for some special reasons other geometrical sections may be used. For sewers for example, oval or egg shaped sections are often selected, because in these cases there are large fluctuations in the rate of discharge.
It is desirable that the velocity should be kept high when the discharge rate is low, to prevent deposition or sediments. It is also desirable the velocity should not be excessive at high discharge to prevent abrasion of the channel bed and sides. Oval sections have these advantages.
Steady and Unsteady Flows:
Steady flow in a channel implies a steady rate of flow, i.e., the discharge rate is constant. The sectional areas of flow at different sections may be different, and obviously the mean velocity of flow at different sections may be different. If the rate of flow is different at different times the flow is said to be unsteady.
Uniform and Non-Uniform Flows:
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In a uniform flow, the depth of flow is the same at all sections. This means the velocity of flow is the same at all sections. The water surface is parallel to the bed of the channel.
In a non-uniform flow, the depth of flow is different at different sections. The mean velocity is different at different sections. The water surface is not parallel to the bed of the channel.
On the basis of the explanations given above, we will consider certain combinations of flow which are:
(i) Steady uniform flow – This means the discharge rate is constant and the section of the channel is the same at all sections. Ex – Flow in artificial channels.
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(ii) Steady non-uniform flow – This means the discharge is constant. The average velocity and depth of flow are different at different sections. The depth of flow may gradually change in which case the flow is called gradually varied flow. Ex – Flow with back water curve on the upstream side of a dam or weir.
If the depth changes abruptly, the change is called Local phenomenon. Ex – A hydraulic jump is a local phenomenon.
(iii) Unsteady uniform flow – This means the rate of flow is not constant with respect to time. But at one instant the mean velocity of flow is the same at all sections.
(iv) Unsteady non-uniform flow – This means the rate of flow is different at different times. At one instant the velocities are different at different sections.
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In any channel with a constant rate of flow, uniform flow will get established which will continue in the entire region of constant slope and cross-section. In other words, in any channel of given roughness and cross-section and slope there exists one distinct depth of flow for a given rate of flow. Changes in bed slope, irregularities of cross-section, varied roughness, obstacles etc. disturb the uniformity of flow producing a non-uniform flow.
See Fig. 14.2 from A to C the flow is accelerating non-uniform flow and becomes a uniform flow in the reach CD. Between D and E there is a violent retardation because of the change or slope. Beyond some section the flow again becomes a uniform flow. Between B and C there is acceleration since the gravity component of the force is greater than the boundary shear resistance.
The increasing velocity results in greater boundary shear resistance until at section C onwards the gravity component and the boundary shear resistance balance producing a uniform flow. The depth of flow during the uniform flow is called the normal depth.
Uniform Flow-Chezy’s Formula:
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Consider a steady uniform flow of water in a channel whose bed is at a slope i. Let A be the uniform area of flow section, and P the wetted perimeter. Let v be the mean velocity of flow. Consider the water body between the sections 1-1 and 2-2, l apart. Weight of water between the two sections = wAl.
Alternative Method:
Consider the body of water between the two sections 1-1 and 2-2. Weight of water between the two sections = wAl
Component of the weight of this water body along the direction of motion
= wAl sin i = wAl i since i is small
Frictional resistance against the motion of the water body
= f’ x Contact area x v2 = f’ Plv2
Since the depth of water is the same throughout the pressure forces on the two sections cancel out.
For the condition of uniform motion (i.e., motion with no acceleration), the resultant force in the direction of motion is equal to zero.
where, K is a constant depending on the nature of channel surface. The following table shows the values of K for various types of channel surfaces.
where, N = Kutter’s constant or roughness coefficient depending on the roughness of the channel surface.
The following table shows the values of N for various types of channel surfaces.
Shear Stress at the Channel Walls:
We know for the channel between two sections, frictional resistance provided by the channel surface,
Conveyance of a Channel Section:
Hydraulic Mean Depth for Some Particular Channel Sections:
In most of the cases trapezoidal sections are provided. The side slope depends on the nature of soil met with. Side slopes in earthen banks may not exceed 1 vertical to 1.5 horizontal. In the case of sandy soil the side slope may not exceed 1 vertical to 3 horizontal. In rocky sites, vertical sides may be provided.
Most Economical Section of Channel or Channel of Best Section:
A study of the formulae given above readily indicates that for a given roughness and slope the velocity increases with the hydraulic mean depth. Hence given an area of the channel section the discharge of the channel will be a maximum when the hydraulic mean depth is maximum or in other words the wetted perimeter must be a minimum. For a given value of the sectional area, the particular section that provides the minimum wetted perimeter is the most efficient cross-section.
Channel sections may have various geometric shapes. Of all the geometric figures, the circle provides the least perimeter for a given area. The hydraulic mean depth for a circle as well as that of the semicircle is the same (equal to half the radius). Of all geometric figures, the discharging capacity of the circular section is greatest for a given area of flow.
Circular and semicircular channels are found to be impracticable using the normally used materials of construction. Semicircular open cannels of course are built of steel. Wooden flumers have been made of rectangular section. Canals which are excavated in earth have mostly trapezoidal section with side slopes not exceeding the angle of repose of the banking material.
Once a certain geometrical shape is chosen, then for a given value of the sectional area, the dimensions of the section can be determined to have the minimum hydraulic mean depth to make the channel of best section.
Definition:
A channel of best section is a channel whose sectional dimensions are designed so as to provide the maximum discharge for a given amount of excavation.
1. Rectangular Channel of Best Section:
Consider a rectangular channel of area A. For this given area A let us determine the width and depth for the condition of maximum discharge. If b and d be the breadth and depth,
2. Trapezoidal Channel of Best Section:
A specific property of the channel of best section.
Consider the trapezoidal channel of best section shown in Fig. 14.9. Let θ be the angle of slope of sides with the horizontal.
Let O be the middle point of top width.
Draw OE perpendicular to AD.
Considering the triangles OAE and DAF.
Hence, if a semicircle is drawn with O as centre and d as radius, it will touch the three sides of the channel.
Thus, the three sides of the channel of best section are tangential to the semicircle described on the water line.
Best side slope for maximum discharge –
Thus P = 3b
This means each sloping side is equal to the bed width.
3. Triangular Channel of Best Section:
For Q to be maximum, P should be minimum and for this condition, since A is constant, tan θ + cos θ should be minimum.
Wide Channels:
In the case of channels of very great width (relative to the depth), the influence of the sides of the channel on the velocity in the central region becomes ignorable. Observations have shown that such a flow condition will prevail when the width of the channel exceeds ten times the depth of flow. Accordingly such wide channels may be treated as rectangular channels of infinite width. It is therefore convenient to make the relevant computations considering unit width of the channel.
The computations can be approximated as shown below:
Channel of Circular Section:
Consider a channel of circular section of radius R.
Let the depth of flow of water be d. Let the water surface subtend and angle 20 with the axis of the channel.
Channel of Circular Section-Condition for Maximum Velocity:
Consider water flowing in a channel of circular section of radius R and laid at a slope i. Let AB represent the water surface. See Fig. 14.40. Let the angle subtended by AB at the axis of the channel be 2θ.
Channel of Circular Section:
(A) Condition for Maximum Discharge Based on Chezy’s Formula:
We know the discharge in the channel is given by –
(b) Condition for Maximum Discharge Based on Manning’s Formula:
We know, the discharge in the channel is given by-
Isosceles Triangular Channel Section – Condition for Maximum Velocity (Sides at 45° with the Base):
Let the width at the bottom be b
Let the depth of flow be d.
Top width of the flow section = b – 2d
Area of the flow section, A = d(b – d)
Wetted perimeter P = b + 2√2 d
Isosceles Triangular Channel Section (Sides at 45° with the Base):
Specific Energy Head—Critical Depth:
The specific energy head at any section of a channel is the sum of the depth of flow and the kinetic head at that section. If d is the depth of flow and v is the velocity at a section then the specific energy head E is given by –
The idea of specific energy was conceived by Bakhmeteff in 1911, leading to simplified analysis of open channel flow problems. In the case of steady uniform flow, the depth of flow is the same at all sections; and the velocity is the same at all sections. Hence the specific energy head is the same at all sections.
Let us now consider a steady, non-uniform flow. Let the width of the rectangular channel be b. Let Q be the steady rate of flow. If q be the discharge per unit width, then –
E1 is called static energy head and E2 is the kinetic energy head.
Note that the specific energy head does not make any reference to datum.
Fig. 14.46 shows the static energy head, the kinetic energy head, the specific energy head, plotted for a definite quantity of flow, for various depths of flow.
The line OA represents the static energy head for various values of d. This line is a straight line through the origin at 45° with the x-axis.
The line MN represents the kinetic energy head for various values of d. The specific energy head curve BCD is obtained by combining the horizontal ordinates of the diagrams OA and MN.
Studying the specific energy head curve BCD, we find:
(i) The specific energy head first becomes lesser and lesser as the depth increases and reaches a minimum value at the point C.
(ii) For further increase in depth the specific energy head increases.
The depth dc corresponding to the point C at which the specific energy head is a minimum is called the critical depth. This is the depth of flow for a given discharge rate at which the specific energy head is a minimum. For each value of the specific energy head there are two possible depths of flow.
Consider the condition corresponding to the line GHI. The specific energy head for this condition is equal to OG, but the depth of flow may be either GH or GI. The depth GH = d1 is less than the critical depth dc while the depth GI = d2 is greater than the critical depth. The two possible depths d1 and d2 corresponding to the same specific energy head are called alternate depths.
When the depth of flow is greater than the critical depth, the flow is called streaming flow or tranquil flow or subcritical flow. When the depth of flow is less than the critical depth, the flow is called shooting flow or rapid flow or supercritical flow.
When the depth of flow is equal to the critical depth, the flow is called critical flow. The velocity at critical flow is called critical velocity.
The critical depth corresponding to a given rate of flow in a rectangular channel can be determined as follows:
Specific Energy Head Curves for Various Discharges:
Suppose for different discharges q1, q2, q3 … per unit width, the specific energy head curves are drawn. See Fig. 14.47.
Each specific energy head curve will meet asymptotically the static head line and the x-axis asymptotically. The points O, C1, C2, C3 … are points on the specific energy head curves corresponding to critical flow. If Ec1, Ec2, Ec3… are the minimum specific energy heads corresponding to the critical depths dc1, dc2, dc3 … we know,
Depth of Flow for Maximum Discharge in a Rectangular Channel at a Given Specific Energy:
Hence for maximum discharge condition, the Froude number = 1, which is also the condition that the depth of flow is the critical depth. Thus the critical depth may also be realised as the depth required to produce a maximum discharge for a given specific energy.
Non-Rectangular Section:
Consider a non-rectangular channel conveying a discharge Q.
Influence of Gravity and Viscosity on the Flow in a Channel:
Gravity and viscosity are the main factors influencing the flow in an open channel. The value of the characteristic dimensionless constant viz. the Froude number gives an indication of the type of flow such as critical, subscritical or super critical flow.
The Froude number is expressed as:
Celerity of Gravity Waves and Critical Velocity:
Fig. 14.67 (a) shows a small wave of height Δd moving at a velocity (or celerity) c towards left. This condition, for analysis purposes may be regarded as equivalent to a wave standing still and the fluid is moving steadily to the right at a velocity v1 = c (→) as shown in Fig. 14.67 (b).
If in this condition the average velocity at the section 2.2 is say v2.
Open Channel Transitions:
A channel transition refers to a certain stretch of channel in which the cross section of the channel varies. For example a transition is provided to connect two open channels of different widths. In the transition part the bed level may be horizontal or sloping.
We will consider the following types of channel transitions, conveying a constant discharge. Energy losses will be neglected.
Channel Transition from a Wider to a Narrower Channel without Change in Bed Level:
Consider a transition in which the width of the channel is decreased from B1 to B2. Let the depths of flow accordingly decrease from d1 to d2 Let v1 and v2 be the velocities in the wider and narrower sections respectively. Consider sections 1-1 and 2-2 corresponding to the wider and narrower sections.
Gauging Flumes:
A gauging flume is a device for gauging discharge in a channel. A gauging flume consists of a zone of restricted width called the throat provided in a channel.
Types of Gauging Flumes:
Gauging flumes are of two types namely:
(i) Non-modular flume or the venturi flume.
(ii) Modular flume or the standing wave flume.
The non-modular flume or the venturi flume is so devised that the velocity of flow at the throat is less than the critical velocity so that no standing wave will occur in the flume. But the modular flume or the standing wave flume is so devised that the velocity of flow at the throat is greater than the critical velocity resulting in the formation of standing wave in the flume.
(i) Non-Modular Flume or the Venturi Flume:
Fig. 14.90 shows the elevation and plan of a venturi flume. Let B, H, V be the normal breadth, depth of flow and velocity at entrance to the flume. Let b, h, v be the breadth, depth of flow and velocity at the throat.
At the throat, the velocity v is greater than V. Hence there will be a drop in water level at the throat as the total energy head practically remains constant.
This is the theoretical discharge through the venturi flume. In the actual cases the discharge obtained is slightly less than the above value due to losses in the flume.
Hence for the actual cases,
The throat dimensions of the venturi flume are such that the velocity of flow at the throat is less than the critical velocity and hence a standing wave will not be formed in this flume.
(ii) Modular Flume or the Standing Wave Flume:
This is similar to the venturi flume, except that in this case a standing wave is formed in the flume. This means the depth of flow in the throat region will reach a value less than the critical depth h. Hence at some section in the throat the depth of flow will be equal to the critical depth hc.
Considering an upstream section and a section at the throat, if the depths of flow at these sections are H and h respectively,
To reach the above condition for maximum discharge, the flow section at the throat section is considerably reduced to ensure that somewhere within the throat the critical depth hc is reached, which is ensured by the occurrence of a standing wave on the downstream side. See Fig. 14.95.
Generally it is found convenient to provide a hump at the throat to further reduce the flow section.
Fig. 14.96 shows the standing wave flume provided with a hump of height hp at the throat.
The equations are valid for this case also. But in this case.
H = height of upstream water surface above the hump.
Discharge = Q = 1.705 b H13/2
where H1 = H + (V2/2g)
Surface Profile Calculations in Prismatic Channels the Step Method:
Consider sections 1-1 and 2-2 of a prismatic channel, δl apart. Let the slope of the channel bed be i and the mean slope of the energy line be jm. Let d1 and d2 be the depths of flow at sections 1-1 and 2-2. Let the velocities at these sections be v1 and v2.
Applying Bernoulli’s equation to sections 1-1 and 2-2.
The length or reach of the channel may be divided into short regions or steps and the value of δl for each step may be calculated. The total length of the channel in which the required difference in depths of flow will occur = Σδl.
Surges in Channels:
A sudden change in the discharge or depth of flow or both produces a fast moving surge wave in a channel. For example, when a sluice gate provided to a channel is raised or lowered surge waves are produced. A surge wave produced due to the interaction of a tide with a river is called a tidal bore.
Surges may be classified into positive and negative surges. A positive surge is a surge of increasing depth, while a negative surge is a surge of decreasing depth. When the sluice gate of a channel is suddenly raised, a positive surge moving downstream and a negative surge moving upstream are produced. If the sluice gate is suddenly lowered a positive surge moving upstream and a negative surge moving downstream are produced.
Positive Surge Moving Upstream:
Suppose the sluice gate provided to the channel shown in Fig. 14.118 is suddenly lowered. Let us consider the consequent positive surge which moves upstream.
Let 1-1 be the section considered before the arrival of the surge. Let 2-2 be the section considered after the passage of the surge. Let the velocity of the surge be C. Let v1 and v2 be the velocities at sections 1-1 and 2-2. We will assume that if the velocities of water with respect to the surge are considered then the flow may be regarded as steady.
Positive Surge Moving Down-Stream:
Now let us consider the case when the gate is suddenly raised. A positive surge moving downstream is produced. As before 1-1 and 2-2 refer to sections before the arrival of the surge and after the passage of the surge.
