Basic about OCF and pipe flow

1. ADVANCED HYDRAULICS
CE-612

2. INTRODUCTION
Definition
• It is defined as the study of the mechanical behavior of water in physical systems (Henry M. Morris and James
M. Wiggert).
• It analyzes how surface, and/or subsurface flows move from one point to the next.
• A hydraulic analysis is used to evaluate flow in rivers, streams, storm drain networks, water aqueducts, water
lines and sewers, etc.
2
OCF
1. Have free surface
2. Subject to atm. pressure.
3. The pressure at the free surface remains constant.
4. Usually, no fixed cross-section.
5. Flow Driven by Gravity
6. The maximum velocity occurs at a little distance
below the water surface
7. Surface roughness varies with depth of flow.
8. HGL coincides with the water surface line.
Pipe Flow
1. No free surface, but confined in a closed conduit
2. Not direct atm. pressure but hydraulic pressure only.
3. Pressure in the pipe is not constant
4. Fixed cross-section (generally circular).
5. Flow Driven by Pressure
6. The maximum velocity occurs at the center of the pipe.
7. Surface roughness varies with the type of pipe
material.
8. HGL do not coincide top surface of the water.

3. 3
z1
z2
𝒗𝟏
𝟐
𝟐𝒈
𝒗𝟐
𝟐
𝟐𝒈
hL
y1
y2
Datum
𝒗𝟏
𝟐
𝟐𝒈
𝒗𝟐
𝟐
𝟐𝒈
hf
𝑷𝟏/𝜸
z1
z2
Pipe
Datum
𝑷𝟐/𝜸
Fig. 1.1 OCF & Pipe flow
TYPES OF OPEN CHANNEL
• Open channels are both natural or manmade
conveyance structure which has a free surface at
atmospheric pressure. For example, flow in rivers,
streams, flow in sanitary and storm sewers flowing
partially full.
• Manmade OC usually have fixed cross-section (may be
trapezoidal, triangular, rectangular, circular, etc.)
whereas natural OC have no fixed cross-section. Fig. 1.2 Trapezoidal shaped open channel

4. 4
OTHER EXAMPLES OF OC
• Flume is the channel made of wood, metal, concrete or masonry usually supported on or above to carry out water
across a depression.
• A chute is a channel having steep slopes.
• A drop is similar to chute but the change in the elevation is effected in a short distance.
• A culvert when partially full is a covered channel installed to drain water through highways or railways
embankment.
PRISMATIC AND NON-PRISMATIC CHANNELS
• A channel in which the cross sectional shape, size and the bottom slope are constant is termed as prismatic
channel.
• All natural channels generally have varying cross section and consequently are non-prismatic.
• Most of the man-made channels are prismatic channels over long stretches. The rectangle, trapezoid, triangle and
circle are commonly used shapes in manmade channels.
RIGID AND MOBILE BOUNDARY CHANNELS
• Rigid channels are those in which the boundary is not deformable. The shape and roughness magnitudes are not
functions of flow parameters. For example, lined canals and non-erodible unlined canals.
• In rigid channels the flow velocity and shear stress distribution will be such that no major scouring, erosion
or deposition will take place in the channel and the channel geometry and roughness are essentially constant
with respect to time.

5. 5
• When the boundary of the channel is mobile and flow carries considerable amounts of sediment through
suspension and is in contact with the bed. Such channels are classified as mobile channels.
• In the mobile channel, not only depth of flow but also bed width, longitudinal slope of channel may undergo
changes with space and time depending on type of flow.
• The resistance to flow, quantity of sediment transported and channel geometry all depends on interaction of flow
with channel boundaries.
• A general mobile boundary channel can be considered to have four degree of freedom. In rigid channel we have
one degrees of freedom.
FLOW REGIMES
1. Steady and unsteady flows:
• A steady flow is one in which the conditions (velocity, pressure and cross- section) may differ from point to point
but do not change with time.
• If at any point in the fluid, the conditions change with time, the flow is described as unsteady flow.
• Flood flows in rivers and rapidly varying surges in channels are some examples of unsteady flow.
2. Uniform and non-uniform flows:
• If the velocity at a given instant of time is same in both magnitude and direction at all points in the flow, the flow
is said to be uniform flow.
• If the velocity changes from point to point in a flow at any given instant of time, the flow is described as non-
uniform flow.

6. 6
Combining above four types of flow it can be further be classified into one of following four sub-types:
a). Steady Uniform Flow
• Flow characteristics do not change with position in the stream with time. An example is the flow of water in a
pipe of constant diameter at a constant velocity.
b). Steady Non-uniform Flow
• The Flow characteristics vary from point to point in the stream but do not change over time. For example, the
fluid flows in a taping pipe with a constant velocity in the inlet – the velocity will change as you move along the
length of the pipe toward the exit.
c). Unsteady Uniform Flow
• At a given moment, the Flow characteristics are the same at every point in a given time but will change with
time. For example, a pipe of constant diameter connected to the pump pumping at a constant rate is then closed.
d). Unsteady Non-uniform Flow
• Every state of the flow may change from point to point and over time at every point. For example waves in a
channel.
Fig. 1.3 Uniform and Non-uniform flow

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3. One, two and Three-dimensional Fluid Flow
• Although in general, all fluids flow in 3-D form, varying in all directions with pressure and velocity and other
flow properties, in many cases the largest change occurs in only two directions or even only in one.
• In these cases, changes in the other direction can be effectively ignored so as to make the analysis more
straightforward.
3(a). 1-D Flow
• The flow is 1-D if the flow parameters like velocity, pressure, depth, etc. change only in the direction of flow at a
given moment and not across the cross-section.
• The flow may be unstable, in this case, the parameter alters in time but is still not across the cross-section. For an
example of a one-dimensional flow is the flow in a pipe.
• All flow parameters can be expressed as functions of time and space coordinates only.
• Single space coordination is normally distance measured along the centre-line (not significantly straight) in
which the fluid is flowing.
• In fact, the flow is never one dimensional because the viscosity reduces the velocity to zero at the solid
boundaries.
• However, valuable results can often be obtained from “one-dimensional analysis” if the non-uniformity of the
actual flow is not very great.
• The average values ​​of the flow parameters on any section (perpendicular to the flow) are supposed to be used to
the whole flow in that section.

8. 8
3(b). 2-D Flow
• The flow is 2-D if it can be assumed that the flow parameters differ in this direction in the direction of flow and
in one direction in the right angles.
• In this type of flow, the streamlines are curved lines on one plane and are same on every parallel plane. An
example is flowing over a weir for which typical streamlines.
o All flow parameters are purposes of time and two space coordinates (say x and y).
o No variation in the z-direction.
o Similar streamline patterns are found in all planes for the z-direction in any instantaneous direction..
3(c). 3-D Flow
• In the three-dimensional fluid flow, the hydrodynamic parameters are functions of three space coordinates and
time.
• The motion of a fluid element in space simultaneously has three distinct characteristics.
o Translation
o Rate of deformation
o Rotation
4. Rotational & Irrotational flow
• The angular momentum of the fluid elements is analysed to classify any flow as rotational or Irrotational. If the
angle within the two intersecting lines of the boundary of the fluid element turns while moving in the flow, then
the flow is a Rotational Flow.
• But if the fluid element rotates as a whole and there is no change in angles within the boundary lines, the flow
cannot be rotational flow, so it is irrotational flow.

9. 9
• This means that in a rotational flow there must be some deformation in the fluid element. Such deformation of
the fluid element or shear strain is certainly caused by tangential forces and shear stresses.
• The shear stress is caused by viscosity, hence the flow of viscous fluid is rotational. Although this does not mean
that the flow of non-viscous or ideal liquids is always Irrotational. The flow of typical fluids can be rotational by
external work or heat interaction.
• Thus, it can be defined as-
o Rotational flow is the type of flow in which fluid particles also rotate on their own axis while flowing along
the flow lines.
o Irrotational flow is a type of flow in which fluid particles do not rotate about their own axis when they
flow along the flow lines.
5. Laminar or Turbulent Flow
5(a). Laminar Flow
• It is defined as a type of flow in which fluid particles move along a well-defined path or flow, and all streamlines
are straight and parallel.
o Thus the particles move in luminous or layers gliding smoothly over the adjacent layer. This type of flow is
also known as viscous flow.
o In Laminar flow, viscous shear stresses act among these layers of fluid which defines the velocity
distribution within these layers of flow. The shear stress is defined by Newton’s equation for the shear stress
in the laminar flow.

10. 10
5(b). Turbulent flow
• It is the type of flow in which fluid particles move in a zigzag manner. Due to the movement of fluid particles in
a zigzag manner, eddies are formed which are responsible for the high energy loss.
o As the flow speed of otherwise cool layers increases, these smoothly moving layers begin to increase
randomly, and with further increase in flow velocity, the flow of fluid particles becomes totally random and
no such laminar layer exists anymore. Shear stresses in turbulent flow are higher than those in laminar flow.
o A dimensionless parameter, the Reynolds number (Re), is described as the ratio of inertial and viscous
forces to identify these two types of flow patterns. With an increase in flow velocity, the initial forces
increase and so the Reynolds Number.
o Pipe Flow: For laminar/medium flow, Re <20000 and for turbulent flow, Re> 2300, for the transition zone,
the Re varies between 2000 and 4000.
o OCF: For laminar, Re <500 and for turbulent flow, Re> 2000, for the transition zone, the Re varies between
500 and 2000.
6. Compressible or Incompressible Flow
• All fluids are compressible – even water – their density will change as pressure slightly changes. Under steady
conditions, and provided that the change in pressure is small, it is possible to simplify the analysis of the flow and
assume that it is incompressible and has a constant density.
• Usually, compressing liquids is quite difficult so under most stable conditions they are considered ineligible. In
some unstable conditions, there may be very high-pressure differences and these need to be taken into account for
liquids as well.
• Glasses, by contrast, are very easily compressed, in most cases requiring that they are treated as compressed,
taking into account changes in pressure.

11. 11
• Overall, these two fluid flows can be defined as-
o Compressible flow is a type of fluid flow in which the density of the fluid varies from point to point or you
can say, the density (ρ) is not constant for the fluid.
o Incompressible flow is the type of flow in which the density for fluid flow is constant. Liquids are
generally incompressible while gases are compressible.
7. Spatially varied flow (SVF)
• If some flow is added to or subtracted from the system, the resultant varied flow is known as SVF.
• SVF can be steady or unsteady.
o In steady SVF the discharge while being steady varies along the channel length.
o The flow over a side weir is an example of steady flow.
• SVF in open channels occurs in a large variety of hydraulic structures as well as road/bridge surface drainage
channels.
Fig. 1.5 Spatially varied flow in OCF

12. 12
CHANNEL GEOMETRY
• The term channel section means cross section of channel taken normal to the direction of the flow.
• A vertical channel section is the vertical section passing through the lowest or bottom section.
• The naturel channels sections are in general very irregular in shape whereas, artificial channels are usually
designed with sections of regular geometry shapes..
• The trapezoid is a commonly used shape; the rectangle and triangle are special case of trapezoid. Since the
rectangle has vertical sides, it is commonly used for channels built of stable materials, such as lined masonry,
rocks, metal or timber.
• The depth of flow (y) is the vertical distance from the water surface to the lowest point of the channel section.
• Stage is the elevation or vertical distance of the free surface above a datum.
• Top width (T) is the width of the natural section at the water surface.
• Cross sectional area (A) area of the flow measured normal to the direction of the flow.
• The wetted perimeter (P) is the length of the line of intersection of the channel wetted surface with the cross
sectional plane normal to the direction of the flow.
• Hydraulic radius (R) is the ratio of the water area to its wetted perimeter. R=A/P.
• Hydraulic depth (D) is the ratio of the water area to the top width D=A/T.
• Section factor (Z) for critical flow computation it is the product of the water area and the square root of
hydraulic depth. [𝑍 = 𝐴 𝐷].
• Section factor for uniform flow section is given by 𝑍 = 𝐴 𝑅2 3

13. 13
1. Rectangular Channel
Breadth = B
Depth = y
Area A = B. y
Perimeter P = (B+2y)
Top width T = B
Hydraulic Radius, 𝑅 =
B. y
B+2y
2. Triangular Channel
Depth = y
Side slope m:1
Area 𝐴 = 𝑚𝑦2
Perimeter 𝑃 = 2𝑦 1 + 𝑚2
Top width T = 2my
Hydraulic Radius, 𝑅 =
𝑚𝑦
2 1+𝑚2

14. 14
3. Trapezoidal Channel
Depth=y
Slope m:1
Area 𝐴 = 𝑦(𝐵 + 𝑚𝑦)
Perimeter P= B+2𝑦 1 + 𝑚2
Top width T= B+2my
Hydraulic Radius, 𝑅 =
𝑦(𝐵+𝑚𝑦)
B+2𝑦 1+𝑚2
4. Circular Channel
Case I (When flow depth is below centre of circle at chord MN)
Area of flow (MNQ) is given by, 𝐴 = 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑠𝑒𝑐𝑡𝑜𝑟 𝑂𝑀𝑄𝑁 − 𝐴𝑟𝑒𝑎 𝑜𝑓 ∆𝑂𝑀𝑄𝑁
=
2𝜃
2𝜋
× 𝜋𝑟2
−
1
2
× 2 𝑟𝑠𝑖𝑛𝜃 × 𝑟𝑐𝑜𝑠𝜃
∴ 𝐴 = 𝒓𝟐
𝜽 −
𝟏
𝟐
𝒓𝟐
𝒔𝒊𝒏𝟐𝜽 =𝒓𝟐
(𝜽 −
𝟏
𝟐
𝒔𝒊𝒏𝟐𝜽)
Perimeter (MNQ), 𝑃 =
2𝜃
2𝜋
× 𝟐𝜋𝑟 = 𝒓 × 𝟐𝜽
Hydraulic mean radius, 𝑅 =
𝒓𝟐(𝜽−
𝟏
𝟐
𝒔𝒊𝒏𝟐𝜽)
𝒓×𝟐𝜽

15. 15
Top width, 𝑇 = 2𝑟𝑠𝑖𝑛𝜃
Hydraulic Depth, 𝐷 =
𝐴
𝑇
=
𝒓𝟐(𝜽−
𝟏
𝟐
𝒔𝒊𝒏𝟐𝜽)
2𝑟𝑠𝑖𝑛𝜃
Section Factor, Z = A 𝐷 = 𝒓𝟐
(𝜽 −
𝟏
𝟐
𝒔𝒊𝒏𝟐𝜽)
𝒓𝟐(𝜽−
𝟏
𝟐
𝒔𝒊𝒏𝟐𝜽)
2𝑟𝑠𝑖𝑛𝜃
Case II (When flow depth is above centre of circle at chord MN)
Area of flow (MNP) = 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑐𝑖𝑟𝑐𝑙𝑒 − 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑠𝑒𝑔𝑚𝑒𝑛𝑡 (𝑀𝑄𝑁𝑀)
Area of segment= 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑠𝑒𝑐𝑡𝑜𝑟 𝑂𝑀𝑄𝑁 – 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒(𝑂𝑀𝑁)
= 𝒓𝟐𝜽 −
𝟏
𝟐
𝒓𝟐𝒔𝒊𝒏𝟐𝜽
= 𝒓𝟐(𝜽 −
𝟏
𝟐
𝒔𝒊𝒏𝟐𝜽)
∴ Area of flow (MNP) = 𝜋𝑟2
- 𝑟2
𝜃 −
1
2
𝑠𝑖𝑛2𝜃 = 𝒓𝟐
(𝝅 − 𝜽 +
𝟏
𝟐
𝒔𝒊𝒏𝟐𝜽)
Perimeter (MPN), 𝑃 =
2𝜋−2𝜃
2𝜋
× 𝟐𝜋𝑟 = 𝟐𝒓(𝝅 − 𝜽)
Hydraulic radius, 𝑅ℎ =
𝒓𝟐(𝜽−
𝟏
𝟐
𝒔𝒊𝒏𝟐𝜽)
𝟐𝒓(𝝅−𝜽)
=
𝒓(𝜽−
𝟏
𝟐
𝒔𝒊𝒏𝟐𝜽)
𝟐(𝝅−𝜽)
Top width, 𝑇 = 2𝑟𝑠𝑖𝑛𝜃
Hydraulic Depth, 𝐷 =
𝐴
𝑇
=
𝒓𝟐(𝜽−
𝟏
𝟐
𝒔𝒊𝒏𝟐𝜽)
2𝑟𝑠𝑖𝑛𝜃
=
𝒓(𝜽−
𝟏
𝟐
𝒔𝒊𝒏𝟐𝜽)
2𝑠𝑖𝑛𝜃
Section factor, Z= 𝐴 𝐷 = 𝒓𝟐(𝝅 − 𝜽 +
𝟏
𝟐
𝒔𝒊𝒏𝟐𝜽)
𝒓(𝜽−
𝟏
𝟐
𝒔𝒊𝒏𝟐𝜽)
2𝑠𝑖𝑛𝜃

16. 16
VELOCITY DISTRIBUTION IN OPEN CHANNEL
• Owing to the presence of a free surface and to the friction along the channel wall, the velocities in channel are
not uniformly distributed in channel section.
• The measured maximum velocity in ordinary channels usually appears to occur below the free surface at a
distance of 0.05 to 0.25 of the depth.
• The velocity distribution in a channel section also depends on factors as the unusual shape of the section, the
roughness of the channel and the presence of bends.
• In a broad, rapid and shallow stream or in very smooth channel, the maximum velocity may often be found
very near to the free surface.
• The roughness of the channel will cause the curvature of the vertical velocity distribution curve to increase.
• On a bend, the velocity increases greatly at a convex side, owing to centrifugal action of the flow.
• Surface wind has very little effect on velocity distribution.
• Velocity component in transverse direction is usually small and insignificant in comparison to longitudinal
velocity components.
• In a long and uniform reach, remote from the entrance, a double spiral motion will occur in order to permit
equalization of shear stress on both sides of the channel. The pattern will include one spiral on each side of
centerline, where the water level is highest.
• Surface velocity 𝑣 is related to average velocity 𝑣 as 𝑣 = 𝑘𝑣 where k = coefficient with value between 0.8 and
0.95.

17. 17
• In open channel flow the kinetic energy transfer per unit time is given by-
• 𝑅𝑎𝑡𝑒 𝑜𝑓 𝐾𝐸 𝑡𝑟𝑎𝑛𝑠𝑓𝑒𝑟 =
𝜌
2 𝐴
.
𝑣3
𝑑𝐴
• Where,𝑣 stands for the point velocity which varies over the channel
section and ρ is density of fluid. However, in terms of average cross
sectional velocity 𝑉 the K.E rate transfer is defined as-
• 𝑅𝑎𝑡𝑒 𝑜𝑓 𝐾𝐸 𝑡𝑟𝑎𝑛𝑠𝑓𝑒𝑟 = 𝛼
𝜌
2
𝑉3A = 𝛼
𝜌
2
𝑄𝑉2
• Where α is kinetic energy correction coefficient, it accounts for the non-
uniform point velocity distribution within the section. For regular
channels the value of α is set as 1.0
• The non-uniform distribution of velocities over a channel section affects the computation of the momentum.
• Momentum of the fluid passing through a channel section per unit time is expresses by 𝛽𝛾𝑄𝑉 𝑔 ,where 𝛽 is
momentum coefficient, V is the mean velocity and Q is the discharge. The value of 𝛽 for a straight prismatic
channel varies from 1.01-1.12.
• The value of energy and momentum correction factor is computed from following expression.
• α =
𝑣3𝑑𝐴
𝑉3𝐴
≈
𝑣3∆𝐴
𝑉3𝐴
• 𝛽 =
𝑣2𝑑𝐴
𝑉2𝐴
≈
𝑣2∆𝐴
𝑉2𝐴
• Where ∆𝐴 is the elementary area; v is flow velocity and V is mean velocity

18. 18
• Approximate values of energy and momentum correction factor is computed form the following formula:
𝛼 = 1 + 3𝜀2
− 2𝜀3
𝛽 = 1 + 𝜀2
𝜀 =
𝑣𝑚
𝑉
− 1
Where 𝑣𝑚 is the maximum velocity and V is mean velocity.
Wide-open channels
• In very wide-open channels the velocity distribution in the central region of the section is essentially the same as
it would be in a rectangular channel of infinite width.
• The sides of the channel have practically no influence on the velocity distribution in the central region.
• A wide-open channel can safely be defined as a rectangular channel whose width is greater than 10 times the
depth of flow.
• For either experimental or analytical purposes, the flow in the central region of a wide- open channel may be
considered the same as the flow in a rectangular channel of infinite width.
• Hydraulic radius for a very wide open-channel flow is just about equal to the flow depth.

19. 19
• Consider a rectangular channel; for a rectangular channel we know that-
𝑅 =
𝐴
𝑃
=
𝐵𝑦
𝐵+2𝑦
• For a wide rectangular channel, the denominator-
𝑅 =
𝐴
𝑃
=
𝐵𝑦
𝐵
= 𝑦