# Surface Exposed To Atmospheric Pressure Biology Essay

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Flow is said to be open channel flow when there is a surface exposed to atmospheric pressure. Water flow in rivers, irrigation canals, culverts and floods and partial flow in pipes also consider as open channel flow because of their exposing surface to atmospheric pressure.

Water is assumed to be incompressible fluid because of its higher modulus of compressibility. The bulk modulus can be defined that the pressure needed to compress a unit volume of a fluid. The bulk modulus for water is roughly 2Gpa and it can be defined mathematically as,

Bulk Modulus= dp/(dv/v)

The forces, mainly influence in a flowing liquid are viscosity and gravity. In case of water, viscous forces are neglected and gravity is considering the main force causing the water to flow freely in an open channel. Viscosity is the friction force between molecules of a liquid considered. There are two types of viscosities

Dynamic Viscosity (Âµ) which is defined as shear stress/Rate of shear strain

Kinematic viscosity (Î½) which is defined as dynamic viscosity/density=Âµ/Ï

## 1.2 Types of flow

According to the changing conditions in channel dimension, channel surface and driving forces, the flow may vary to different categories, such as,

## Steady uniform flow

Flow depth and velocity are constant with the time and space along the channel. It commonly occurs in laboratory channels and can not expect such kind of flow in an open channel flow due to irregularity of channel cross section and change in driving force.

## Steady gradually varied flow

Flow depth does not change at a point with time but it changes along the channel. This kind of flow may occur in the upstream of the following conditions. Such as, channel contractions, sluice gates in dams or any kind of obstructions along the flow. This is the flow we need to concentrate in order to understand about backwater.

## Steady rapidly varied flow

Flow depth does not vary at a particular point with time but it often changes with the distance. Hydraulic jumps and other adverse conditions may occur in this kind of flow. A considerable amount of region in the flowing water which commencing from sluice gate in a dam fall in this category. Because its potential height and surface roughness and other driving forces cause this kind of flow.

## Unsteady flow

Flow depth changes with time and space. Flow is unpredictable here. Storm water flow comes under this category. Storm water flow in an uneven surface and it might have ditches, rocks, lawns, bushes, man made obstacles and other.

## 1.3 What is Back Water?

The increase in water surface elevation in the upstream of a water body due to any obstruction in downstream such as dam, culverts etc., called the backwater rise. This water level rise may continue for a quite long distance.

Open channels are categorized as man made or natural channels and the manmade channels further divided into prismatic and non prismatic channels. Almost every natural channels are non prismatic. Prismatic channels are having a definite cross section along its whole length, so calculations become easier and predictions also easier than for non prismatic. In reality no any channels are prismatic except the laboratory channels. But in our calculations we consider the cross section of the channel remains same all over its length.

There are two popular methods to calculate water level rises and predicting the shape of the water surface profile in the upstream by any obstruction in an open channel flow.

## 1.3.1 Direst step method

This method is used to predict water surface profile in prismatic channels in the upstream by backwater effect. In this method we should calculate the distance between the known water depths.

## Disadvantages in using the direct step method rather than standard step method

In order to find the water depth at a specified location, interpolation should be used. As we all know interpolation gives an approximate value as a solution. Precision of the result may be affected because of this.

It is only been used for prismatic channels, so it is inconvenient to use for an unknown location because it has unknown cross sectional area.

## 1.3.3 Standard step method

This method is widely used for non prismatic and most kind of other channels. In this method we should find the water depths which satisfy the chainage and the distance between them. Trial and error procedure should be used to find the water depths at given positions.

## Aim and Objectives

The aims of this project are:

The calculation of the backwater profile using two different methods called direct step method and standard step method, and also the numerical modeling for different types of channels to make the calculations easier.

The objective of this project is:

Analyze the water surface profile for rectangular, trapezoidal and circular channels from the results obtained from the above calculations

## Outline of the thesis

The second chapter explains the literature behind this study. It describes about open channel hydraulics, backwater formation and the types of calculation methods. The third chapter explains what are the calculations used to predict the profile of backwater and how these calculations are performed. And it is clearly explained step by step. And also the water surface profiles in graphical format. The fourth chapter explains the procedures followed to obtain the results from the beginning. The fifth chapter explains about the practical applications of backwater and the importance of the accurate calculations and the sixth chapter concludes this thesis.

## CHAPTER 2

## 2. Literature Review

Fluid mechanics is the major part dealing with engineering aspects of fluids which are related to almost every discipline in engineering. There are number of subsections developed from this major section. Hydraulics is the main branch in fluid mechanics concentrates on the study of liquids. Civil and environmental engineers are widely concerned with one liquid called water. The development of the industrial society rests on civil and environmental engineers to provide and maintain adequate water services such as water lines, water treatment plants, flood control etc.

Water is inevitable in our life. In the ancient days Egyptians and Babylonians created canals for irrigation and around castles for safety purposes. Ancient people cultivate their food by their own and satisfy with what they have grown. They needed canals for irrigation because they have not got machineries for watering the plants. Actually those people didn't know or aware about the principles in hydraulics. Greeks enunciated the laws behind buoyancy and hydrostatics.

Change in cross sectional area of the channel or in discharge of the flow could lead to different kinds of flows. There are mainly four kinds of flows:

## 2.1.1 Steady uniform flow

Flow depth and velocity are constant with the time and space along the channel. It commonly occurs in laboratory channels and can not expect such kind of flow in an open channel flow due to irregularity of channel cross section and change in driving force.

## 2.1.2 Steady gradually varied flow

Flow depth does not change at a point with time but it changes along the channel. This kind of flow may occur in the upstream of the following conditions. Such as, channel contractions, sluice gates in dams or any kind of obstructions along the flow. This is the flow we need to concentrate in order to understand about backwater.

2.1.3 Steady rapidly varied flow

Flow depth does not vary at a particular point with time but it often changes with the distance. Hydraulic jumps and other adverse conditions may occur in this kind of flow. A considerable amount of region in the flowing water which commencing from sluice gate in a dam fall in this category. Because its potential height and surface roughness and other driving forces cause this kind of flow.

2.1.4 Unsteady flow

Flow depth changes with time and space. Flow is unpredictable here. Storm water flow comes under this category. Storm water flow in an uneven surface and it might have ditches, rocks, lawns, bushes, man made obstacles and other.

## 2.2 Properties of open channels

There are two major types of open channels in civil engineering categorized according to the type of construction and the physical properties. They are:

## Artificial or man made open channels

This name came because these kinds of channels are planned and constructed by human beings. These channels include irrigation canals, navigation canals, sewers, culverts, spillways and drainage ditches. They are usually constructed in a regular cross section throughout and are called prismatic channels. It means they do not widen or get narrower along the channel. In general they are constructed of concrete, steel or earth and have surface roughness reasonably well defined. Although this roughness can vary with age, particularly the grass lined channels. Analysis of flow in such well defined channels will give reasonably more accurate results.

2.2.2 Natural channels

Natural channels can be very different. They are neither regular nor prismatic and the materials of construction can vary widely. Although they are constructed mainly of earth, this can contain many different materials and many kinds of properties. The surface roughness will often change with distance, time and even with elevation. It is quite obvious that obtaining a reasonably accurate result from natural channels is difficult task compared to artificial channels. This may be more complicated if the channel boundary is not fixed. That means it is opened for erosion, sedimentation, etc.

## 2.3 Geometric properties necessary for analysis

For any kind of mathematical analysis in a channel, the geometric properties of the channel are required. For man made channels these can usually be defined using simple algebraic equations given by the depth of flow.

The commonly needed geometric properties are shown in the figure below inside the table and defined as:

Depth (y) - The vertical distance from the lowest point of the channel section to the free surface

Stage (Z) - The vertical to the free surface to an arbitrary datum

Area (A) - The cross sectional area of flow, normal to the direction of flow

Wetted Perimeter (P) - The length of wetted surface measured normal to the direction of flow

Surface Width (B) - Width of the channel section at the free surface

Hydraulic Radius (R) - The ratio of Area to Wetted perimeter (A/P)

Hydraulic mean depth (Dm) - The ratio of Area to Surface width (A/B)

## Table 2.1 Geometric properties for some typical channels

Rectangle

Trapezoid

d

D

B

Ñ„

Circle

Shape

y

b

y

b

B

N

1

Area, A

by

(b+Ny)y

(Ñ„-sinÑ„)

Wetted Perimeter, P

b+2y

(b+2y)

Ñ„D

Top Width, B

b

b+2Ny

(sin)D

Hydraulic Radius, R

(1-)D

Hydraulic Mean Depth, Dm

y

()D

## 2.4 Principles used to formulate the equations

Hydrology deals with the cycling of water with environment and hydraulics deals with the physical attributes of flowing water. In order to predict the attributes of the flowing water some equations should be formulated.

Some principles in physics used to formulate the equations in hydraulics. They are:

## Conservation of matter

Matter can not be created or destroyed but it can be transformed from one stage to another.

## Conservation of energy

Energy can not be created or destroyed but it can be transformed from one kind of energy to another. For example, potential energy can be transformed to kinetic energy through some kind of transformation process. Some literatures say that energy losses in friction. Actually it is transformed into heat energy when friction arises.

## Conservation of momentum

A body in motion can not gain or lose momentum unless there is an external force is applied. Newton's second law of motion based on this principle,

Force= Rate of change in momentum

## 2.5 Formulation of equations

## 2.5.1 The continuity equation

The above title indicates the conservation of mass concept. For any control volume during the small time interval áµŸt the principle of conservation of mass implies that the mass of flow entering the control volume minus the mass of flow leaving the control volume equals the change of mass within the control volume.

If the flow is steady and the fluid is incompressible then there is no change of mass within the control volume. So, the mass entering is equal to mass leaving.

For the time interval áµŸt,

Mass flow entering = Mass flow leaving

A1, V1

A2, V2

## Figure 2.1 Control volume of a channel section

Considering the control volume above which is a short length of open channel of arbitrary cross section then, if Ï is the fluid density and Q is the volume flow rate then, mass flow rate is ÏQ and the continuity equation for steady incompressible flow can be written as,

ÏQ entering = ÏQ leaving

As flow rate Q is the multiplication of velocity and cross sectional area normal to the direction of flow,

Q entering = A1V1

Q leaving = A2V2

So, the continuity equation can be written as,

## A1V1 = A2V2

## 2.5.2 Derivation of Bernoulli Equation from Energy Conservation Theory

âˆ† L P+âˆ†p, A

P'

Ï´

z

P,A p'

mg

## Figure 2.2 An elemental cylindrical stream tube

Pressure forces

Upstream end PA

Downstream end - (P+âˆ†p) A

Circumference 0 (pressure forces p' cancel)

Weight force

-mg CosÏ´=-Ï A âˆ†L g CosÏ´

Newton's second law,

PA-(P+âˆ†p) A-Ï A âˆ†L g CosÏ´=Ï A âˆ†L (du/dt)

Setting âˆ†L CosÏ´=âˆ†z and cancelling the common terms,

-A âˆ†p- Ï A g âˆ†z= Ï A âˆ†L (du/dt)

Dividing by Ï A âˆ†L, and in the limit

(1/ Ï) dp/dL+ du/dt + g dz/dL=0

For steady flow, velocity(u) only varies with distance(L), Therefore,

du/dt=(dL/dt)(du/dL)=udu/dL

Substituting for du/dt in the above equation,

(1/ Ï) dp/dL+ udu/dL + g dz/dL=0

For incompressible fluids the above Euler's equation may be integrated to yield,

p/Ï + u2/2 +gz = constant

Dividing by g,

p/Ïg +/ u2/2g +z = constant

This is the Bernoulli's equation

Note:

In the derivation of the Bernoulli equation it was assumed that no energy lost in the control volume. That means the fluid is frictionless. To apply to some non frictionless fluids there must be an energy loss term included.

The dimension of each term in Bernoulli equation has the dimension of length. For this reason these terms are called as head individually.

P/Ïg = Pressure head

V2/2g = Velocity head

Z = Potential head

Although above we derived the Bernoulli equation between two sections it should strictly speaking be applied along a stream line as the velocity will differ from the top to bottom of the section. However in engineering practice it is possible to apply the Bernoulli equation without reference to the stream line.

## 2.5.3 The momentum equation

For this also consider the same control volume in the time period áµŸt,

A1, V1

A2, V2

## Figure 2.3 Control volume of a channel section

Momentum entering = ÏáµŸQ1áµŸá¿žtV1

Momentum leaving = ÏáµŸQ2áµŸá¿žtV2

From the continuity concept,

áµŸQ1 = áµŸQ2 = áµŸQ

From Newton's second law of motion,

Force = Rate of change of momentum

áµŸF=

áµŸF = Ï áµŸQ (V2-V1)

If we integrate the above equation in x-direction,

Fx = ÏQ (V2x-V1x)

Above momentum equation is applied for a region of uniform velocity and having steady flow.

## 2.6 Energy and momentum coefficients

In deriving the momentum and energy equations it was noted that the velocity must be constant over the whole cross section or along the stream line considered. But it is not the case in practice. Still these equations can be used for non uniform flow by introducing energy and momentum coefficients Î± and Î².

These coefficients can be defined as:

Î±=

Î²=

The Bernoulli equation can be rewritten in terms of mean velocity V,

+ + z = constant

And the momentum equation becomes,

Fx = ÏQÎ² (V2x-V1x)

The values of Î± and Î² must be obtained from the velocity distribution across the cross section. These values are normally greater than 1, but only by a small amount. So, it can be assumed to be 1 confidently but not always and their occurrence should be remembered whenever use the energy or momentum equation. For turbulent flow in regular channel, Î± will not go more than 1.15 and Î² will not go 1.05.

## 2.6.1 Velocity distribution in open channels

The measured velocity in an open channel will vary across the channel cross section because there must be friction along the boundary. In contrast to the pipe flow, the velocity distribution is not an axisymmetric because of free surface. It might be expected to find the maximum velocity at the free surface because the shear force is zero at the free surface. Actually it is not the case. Due to the secondary currents which are circulating from boundaries towards the section centre and the resistance at the air-water interface, the maximum velocity occurs just below the free surface.

Figure below shows some typical velocity distributions across some channel cross sections. The number indicates the percentage of maximum velocity.

## Figure 2.4 Velocity Distribution

## 2.7 Laminar and turbulent flow

The flow can be categorized as laminar or turbulent in every type of flows. It can be a pipe flow or can be an open channel flow.

Reynolds number is the determining factor of telling the type of flow.

For pipe flow the Reynolds number can be written as,

Re=

In order to differentiate the types of flow Reynolds number has different numerical values for the limits,

Laminar: Re<2000

Turbulent: Re>4000

If we take the characteristic length as the hydraulic radius R=A/P then for pipe flowing full R=D/4 and,

Re, channel= = =

Laminar: Re,channel<500

Turbulent: Re,channel>1000

In practice the limit of turbulent flow in an open channel is not well defined. So, it is usually taken as 2000.

Darcy-Wiesbach formula can be used to calculate the friction loss in a pipe in turbulent flow due to friction effects from pipe lining and other,

hf=

And make the substitution for hydraulic radius R= D/4

And if we put the bed slope So= L/hf then,

So=

And Î»= f=

The Colebrook-White equation gives the f-Re relationship for pipes, putting R=D/4 the equivalent equation for open channel is,

= -4log10( +

Where Ks is the effective roughness height

## 2.8 Uniform flow and the development of the friction formulae

The flow is said to be uniform when the gravitational forces are balanced by the frictional resistance forces, which are acting along the boundary line in the direction opposite to fluid flow.

L

V Surface

Bed Ï„ Weight

## Figure 2.5 Forces on a channel length in uniform flow

From the above diagram, gravity forces resolved in the direction of the flow,

Gravity force= ÏgALSinÏ´

Boundary shear force resolved in the direction of flow,

Shear force= Ï„oPL

Boundary shear force balances gravity force since the flow is said to be uniform.

So, Ï„oPL= ÏgALSinÏ´

The slope can be considered as small for uniform and gradually varied flow and it rarely exceeds 1 in 50,

So, SinÏ´= TanÏ´= So

Ï„o = = ÏgRSo

## 2.8.1 The Chezy Equation

An assumption can be made that shear force is proportional to the velocity squared,

Ï„oâˆžV2

Ï„o=KV2

Substituting Ï„o from the above equation,

V=

If we equate the constants in the above equation to a different constant then the equation will be,

## V=C

This is called the Chezy Equation and the C is the "Chezy C"

C is not constant because K is not constant and depend on the Reynolds number and boundary conditions.

The relationship between C and f can be achieved from Chezy and Darcy-Wiesbach equations,

C=

## 2.8.2 The Manning equation

Engineers have done several kinds of studies to evaluate the value of C and finally came to an end with the following formula,

C=

The above equation is known as Manning's formula and the "n" is called the Manning's "n"

\Substituting C in the Chezy equation,

V=

For calculation purposes we should be aware about the Manning's n for different kinds of channels

## Table 2.2 Typical Values of Manning's n

## Channel Type

## Surface Material and Form

## Manning's n Range

River

Earth, Straight

0.02-0.025

Earth, Meandering

0.03-0.05

Gravel (75-150mm), Straight

0.03-0.04

Gravel (75-150mm), Winding

0.04-0.08

Unlined Canal

Earth, Straight

0.018-0.025

Rock, Straight

0.025-0.045

Lined Canal

Concrete

0.012-0.017

Laboratory Models

Mortar

0.011-0.013

Perspex

0.009

## 2.9 Gradually Varied Flow

Cross section

## Figure 2.6 Cross-Section of Trapezoidal Channel

GVF profiles

## Figure 2.7 Gradually Varied Flow Profiles

2.9.1 Gradually varied flow profile determination

If normal depth greater than the critical depth, then the slope is said to be Mild (M). If normal depth less than the critical depth, then the slope is said to be Steep (S).Â If normal and critical depths are equal, then the slope is called Critical (C) slope. Â These three basic slopes can be subdivided through the flow pattern.

## 2.9.1.1 Mild Slopes

If water depth greater than the normal depth, then the flow pattern is an M1.Â The gradually varied flow calculations start from downstream and finishes at the upstream.Â The water depth reaches to the normal depth at a farther upstream point.

If the water depth is greater than the critical depth and less than the normal depth, then the flow profile is an M2.Â The gradually varied flow calculation starts from downstream and continues up to the upstream.Â The water depth reaches to the normal depth at the upstream.

If the critical depth greater than the water depth, then the flow profile is an M3.Â This is an unstable gradually varied flow condition since the water depth begins below both the normal and critical depths.Â Since the slope is Mild, a hydraulic jump might occur.Â Hydraulic jump is a feature when the flow is rapidly varies, so this can not be explained through gradually varied flow calculations.

## 2.9.1.2 Steep Slopes

If the water depth is greater than the critical depth, then the flow profile is an S1.Â This is an unstable gradually varied flow condition since the water depth begins above both the normal and critical depths.Â Since the slope is steep, the water depth will have to pass through the critical depth in order to reach the normal depth.Â This kind of crossing line is a rapidly varied flow condition and can not be explained using gradually varied flow equations.

If the water depth less than the normal depth and greater than the critical depth, then the flow profile is an S2.Â The gradually varied flow calculation starts from the upstream and continues to downstream.Â The water depth reaches to the normal depth at a farther downstream point.

If the normal depth greater than the water depth, then the flow profile is an S3.Â The gradually varied flow calculation starts from the upstream and finishes at downstream.Â The water depth reaches to normal depth as the calculation continues to far downstream.

## 2.9.1.3 Critical Slopes

If water depth greater than the critical depth, then the flow profile is a C1.Â The gradually varied flow calculation starts from the downstream and continues to upstream.Â The water depth reaches to the normal depth, as the calculation continues to upstream.

If the critical depth greater than the water depth, then the flow profile is a C3.Â The gradually varied flow calculation starts from upstream and finishes at downstream.Â The water depth reaches to the normal depth, as the calculation continues to downstream.

There is no such thing as a C2 profile - when the critical and normal depths are equal and the water depth can not be between them.

We can simply show the characteristics of water surface profiles which we discussed in the above paragraph through the following table,

## Table 2.3 Characteristics of the Water surface profile

## Class

## Bed Slope

## Depth

## Type

## Classification

Mild

So>0

Y>Yo>Yc

1

M1

Mild

So>0

Yo>Y>Yc

2

M2

Mild

So>0

Yo>Yc>Y

3

M3

Critical

So>0

Y>Yo=Yc

1

C1

Critical

So>0

Y<Yo=Yc

3

C3

Steep

So>0

Y>Yc>Yo

1

S1

Steep

So>0

Yc>Y>Yo

2

S2

Steep

So>0

Yc>Yo>Y

3

S3

Horizontal

So=0

Y>Yc

2

H2

Horizontal

So=0

Yc>Y

3

H3

Adverse

So<0

Y>Yc

2

A2

Adverse

So<0

Yc>Y

3

A3

In case of rapidly varied flow the channel slope and surface roughness are not so important for predicting the flow characteristics because flow region is vary rapidly and within short distances. So channel slope and surface roughness are not going to play a vital role. But in case of gradually varied flow, the surface roughness and channel slope are determining the flow characteristics. To predict the flow characteristics there must be a general equation to be derived or identified.

The derivation of the general equation for gradually varied flow can be showed as follows:

## 1 2

v12/2g Energy Line sf âˆ†x

v22/2g

y1 Water Surface

y2

Channel Bottom

z1 z2

Datum Line

âˆ†x

## Figure 2.8 Open channel flow profile

Applying Bernoulli's equation for 1 and 2,

H1= z1 + y1 + v12/2g

H2= z2 + y2 + v22/2g

There is a loss in the energy head due to the channel friction, so the equation can be written by considering the loss as well by following way,

H1 = H2 + sf âˆ†x

H2 - H1 = - sf âˆ†x

dH/dx = - sf

## CHAPTER 3

## 3. Calculations

Since this project is mostly rely on the calculation part, we need to look at the calculations involved in gradually varied flow is essential to verify the different types of flows and flow patterns occur in an open channel. Flow pattern and flow characteristics may vary according to boundary conditions and the channel characteristics. As we looked earlier the backwater feature occurs when the flow is gradually varied.

There are two popular methods to be used in order to predict the water surface profile or to find the unknown water depth. They are:

Direct Step Method and

Standard step Method

3.1 Direct Step Method

This method is more useful to find the chainage between the known water depths in prismatic channels. If we take a rectangular channel for sample calculations,

Rectangular channel

A=b*y

P=b+2y

R=A/P

V=Q/A

EG=y+(v^2/2g)

y

b

## Figure 3.1.1 Cross section of a rectangular channel

## Input Data

Flow Rate 600 m3/s

Manning's n 0.04

Bed Slope, So 0.002

Base Width, B 50 m

Initial Depth 6.5 m

## Table 3.1.1 Direct Step Method Calculation for a Rectangular Channel

## Y(m)

## A(m2)

## P(m)

## R(m)

## V(m/s)

## EG(m)

## Sf

## Sf mea

## So-Sf mean

## DEL E(m)

## DEL X(m)

## X(m)

6.5

325

63

5.2

1.8

6.7

0.0006

## Â

## Â

## Â

## Â

0.0

## Â

## Â

## Â

## Â

## Â

## Â

## Â

0.0006

0.0014

-0.189

139.08

## Â

6.3

315

62.6

5.0

1.9

6.5

0.0007

## Â

## Â

## Â

## Â

139.1

## Â

## Â

## Â

## Â

## Â

## Â

## Â

0.0007

0.0013

-0.188

145.29

## Â

6.1

305

62.2

4.9

2.0

6.3

0.0007

## Â

## Â

## Â

## Â

284.4

## Â

## Â

## Â

## Â

## Â

## Â

## Â

0.0008

0.0012

-0.186

153.21

## Â

5.9

295

61.8

4.8

2.0

6.1

0.0008

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## Â

## Â

## Â

437.6

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## Â

0.0009

0.0011

-0.185

163.62

## Â

5.7

285

61.4

4.6

2.1

5.9

0.0009

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## Â

601.2

## Â

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## Â

0.0010

0.0010

-0.183

177.81

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5.5

275

61

4.5

2.2

5.7

0.0010

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779.0

## Â

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0.0011

0.0009

-0.181

198.16

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5.3

265

60.6

4.4

2.3

5.6

0.0011

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977.2

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0.0012

0.0008

-0.179

229.54

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5.1

255

60.2

4.2

2.4

5.4

0.0013

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1206.7

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0.0014

0.0006

-0.176

283.80

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4.9

245

59.8

4.1

2.4

5.2

0.0015

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1490.5

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0.0016

0.0004

-0.173

399.00

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4.7

235

59.4

4.0

2.6

5.0

0.0017

## Â

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1889.5

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## Â

0.0018

0.0002

-0.170

802.02

## Â

4.5

225

59

3.8

2.7

4.9

0.0019

## Â

## Â

## Â

## Â

2691.5

Record the following parameters across the top of Table 3.1.1:

Q = design flow

n = manning's n value

SO = channel bottom slope

Using the desired range of flow depths, y, recorded, compute the cross-sectional area, A, the hydraulic radius, R, and average velocity, v, and record results.

Compute specific energy, delta E, in metres, by summing the velocity head and the depth of flow. Record the result.

Compute the change in specific energy, delta E, between the current and previous flow depths and record the result (not applicable for row 1).

Compute the friction slope using

Sf = (n2v2)/R4/3 (Eq. 3.1.1)

Where: Sf = friction slope (m/m)

n = manning's n value

v = average velocity (m/s)

R = hydraulic radius (m)

Record the result.

Determine the average of the friction slope between this depth and the previous depth (not applicable for row 1). Record the result.

Determine the difference between the bottom slope, SO, and the average friction slope, Sf, (not applicable for row 1). Record the result.

Compute the length of channel between consecutive rows or depths of flow using the equation:

âˆ†x=âˆ†E/( SO - Sf) (Eq. 3.1.2)

Where:

âˆ†x = length of channel between consecutive depths of flow (m)

âˆ†E = change in specific energy (m)

SO = bottom slope (m/m)

Sf = friction slope (m/m)

Record the result.

Sum the distances from the starting point to give cumulative distances, x, for each dept and record the result.

If we plot depth against the distance we found using the direct step method to check whether it has or not the backwater features.

## Figure 3.1.2 Water Surface Elevation against Distance for Rectangular channel

From the graph we have plotted, we can predict the water height within a reasonable limit and we do not need to go for the calculation or interpolation to predict the desired water height in an specified location for any construction purposes.

If we look at a channel having trapezoidal cross section,

Trapezoidal channel

A=(b+y)y

P=b+2âˆš2y

R=A/P

y

b

## Figure 3.1.3 Cross section of a trapezoidal channel

Flow rate 600 m3/s

Manning's n 0.04

Side slope 1:1

Bed slope 0.002

Base Width 50

Initial Depth 6.5

## Table 3.1.2 Direct Step Method Calculations for a Trapezoidal Channel

## Y(m)

## A(m2)

## P(m)

## R(m)

## V(m/s)

## EG(m)

## Sf

## Sf mea

## So-Sf mea

## DEL E(m)

## DEL X(m)

## X(m)

6.5

367

68.4

5.37

1.63

6.64

0.0005

## Â

## Â

## Â

## Â

0

## Â

## Â

## Â

## Â

## Â

## Â

## Â

0.0005

0.0015

-0.190

125.1

## Â

6.3

355

67.8

5.23

1.69

6.45

0.0005

## Â

## Â

## Â

## Â

125

## Â

## Â

## Â

## Â

## Â

## Â

## Â

0.0005

0.0015

-0.189

129.0

## Â

6.1

342

67.3

5.09

1.75

6.26

0.0006

## Â

## Â

## Â

## Â

254

## Â

## Â

## Â

## Â

## Â

## Â

## Â

0.0006

0.0014

-0.188

133.8

## Â

5.9

330

66.7

4.95

1.82

6.07

0.0006

## Â

## Â

## Â

## Â

388

## Â

## Â

## Â

## Â

## Â

## Â

## Â

0.0007

0.0013

-0.187

140.0

## Â

5.7

317

66.1

4.80

1.89

5.88

0.0007

## Â

## Â

## Â

## Â

528

## Â

## Â

## Â

## Â

## Â

## Â

## Â

0.0008

0.0012

-0.185

148.1

## Â

5.5

305

65.6

4.66

1.97

5.70

0.0008

## Â

## Â

## Â

## Â

676

## Â

## Â

## Â

## Â

## Â

## Â

## Â

0.0008

0.0012

-0.183

159.1

## Â

5.3

293

65.0

4.51

2.05

5.51

0.0009

## Â

## Â

## Â

## Â

835

## Â

## Â

## Â

## Â

## Â

## Â

## Â

0.0010

0.0010

-0.181

174.6

## Â

5.1

281

64.4

4.36

2.14

5.33

0.0010

## Â

## Â

## Â

## Â

1010

## Â

## Â

## Â

## Â

## Â

## Â

## Â

0.0011

0.0009

-0.179

197.9

## Â

4.9

269

63.9

4.21

2.23

5.15

0.0012

## Â

## Â

## Â

## Â

1208

## Â

## Â

## Â

## Â

## Â

## Â

## Â

0.0013

0.0007

-0.176

236.9

## Â

4.7

257

63.3

4.06

2.33

4.98

0.0013

## Â

## Â

## Â

## Â

1444

## Â

## Â

## Â

## Â

## Â

## Â

## Â

0.0014

0.0006

-0.173

313.5

## Â

4.5

245

62.7

3.91

2.45

4.81

0.0016

## Â

## Â

## Â

## Â

1758

The only difference between the calculation of a rectangular channel and the trapezoidal channel in the direct step method is the calculation of the cross sectional area and the wetted perimeter.

Cross sectional area and wetted perimeter are calculated using the side slopes also from a trapezoidal channel. But in case of a rectangular channel it is not.

## Figure 3.1.4 Water surface elevation against distance for a trapezoidal channel

## Circular Channel

A=1/8(Ñ„-sinÑ„)D2

P=1/2Ñ„D

R=A/P

y=D/2(1-cosÑ„/2)

Ñ„=2(cos-1(1-2y/D))

## Figure 3.1.5 Cross sectional area of a circular channel

Flow Rate 600 m3/s

Manning's n 0.04

Bed Slope 0.002

Diameter 9 m

Initial Depth 6.5 m

Interval 150 m

Length 1950 m

## Table 3.1.3 The Direct Step Method calculations for a circular channel

## Y(m)

## Ñ„

## A(m2)

## P(m)

## R(m)

## V(m/s)

## EG(m)

## Sf

## Sf mea

## So-Sf mea

## DEL E(m)

## Del X(m)

## X(m)

6.5

4.1

49.2

18.3

2.69

12.20

14.08

0.064

## Â

## Â

## Â

## Â

0.00

## Â

## Â

## Â

## Â

## Â

## Â

## Â

## Â

0.066

-0.064

0.418

6.51

## Â

6.3

4.0

47.6

17.8

2.67

12.61

14.41

0.069

## Â

## Â

## Â

## Â

6.51

## Â

## Â

## Â

## Â

## Â

## Â

## Â

## Â

0.072

-0.070

0.458

6.55

## Â

6.1

3.9

45.9

17.4

2.64

13.07

14.81

0.075

## Â

## Â

## Â

## Â

13.06

## Â

## Â

## Â

## Â

## Â

## Â

## Â

## Â

0.079

-0.077

0.502

6.54

## Â

5.9

3.8

44.2

17.0

2.60

13.57

15.29

0.082

## Â

## Â

## Â

## Â

19.60

## Â

## Â

## Â

## Â

## Â

## Â

## Â

## Â

0.087

-0.085

0.551

6.50

## Â

5.7

3.7

42.5

16.6

2.56

14.12

15.87

0.091

## Â

## Â

## Â

## Â

26.11

## Â

## Â

## Â

## Â

## Â

## Â

## Â

## Â

0.096

-0.094

0.605

6.43

## Â

5.5

3.6

40.7

16.2

2.52

14.73

16.56

0.101

## Â

## Â

## Â

## Â

32.54

## Â

## Â

## Â

## Â

## Â

## Â

## Â

## Â

0.107

-0.105

0.667

6.33

## Â

5.3

3.5

39.0

15.7

2.47

15.40

17.38

0.113

## Â

## Â

## Â

## Â

38.88

## Â

## Â

## Â

## Â

## Â

## Â

## Â

## Â

0.121

-0.119

0.736

6.21

## Â

5.1

3.4

37.2

15.3

2.42

16.13

18.36

0.128

## Â

## Â

## Â

## Â

45.08

## Â

## Â

## Â

## Â

## Â

## Â

## Â

## Â

0.137

-0.135

0.815

6.05

## Â

4.9

3.3

35.4

14.9

2.37

16.95

19.54

0.145

## Â

## Â

## Â

## Â

51.14

## Â

## Â

## Â

## Â

## Â

## Â

## Â

## Â

0.156

-0.154

0.906

5.88

## Â

4.7

3.2

33.6

14.5

2.31

17.85

20.94

0.167

## Â

## Â

## Â

## Â

57.01

## Â

## Â

## Â

## Â

## Â

## Â

## Â

## Â

0.180

-0.178

1.010

5.68

## Â

4.5

3.1

31.8

14.1

2.25

18.86

22.63

0.193

## Â

## Â

## Â

## Â

62.69

Record the following parameters across the top of Table 3.1.3:

Flow Rate

Manning's n

Bed Slope

Diameter

Initial Depth

Interval

Length

Using the desired range of flow depths, y, recorded, compute the angle Ñ„.

Calculate cross sectional area A, wetted perimeter P, the hydraulic radius, R, and average velocity, v, and record results.

The rest of the calculation procedures are same for the one we did to the rectangular channel.

## Figure 3.1.6 Water Surface Elevation against the distance in a circular channel

## Standard step method

Standard Step Method calculations are more useful in practical and are used to find the water depths at known locations. Trial and error procedure should be used to find the most suitable water depth at given locations. Direct step method can be used only for the calculation on prismatic channels. But standard step method can be used for prismatic and non prismatic channel calculations.

In general there must be a frictional loss due to the roughness of the boundary layers. So, it is essential to take the energy coefficient into consideration for the calculations. The water depth obtained by trial error procedure will be more accurate than obtain depth by neglecting the coefficient.

If we look at the Standard Step Method calculation for a trapezoidal channel:

A=(b+y)y

P=b+2âˆš2y

R=A/P

EG=y+(v^2/2g)

y

b

## Figure 3.2.1 The cross sectional area for a trapezoidal channel

Flow rate 19 m3/s

Manning's n 0.017

Side slope 1:1

Bed slope 0.0015

Base Width 3

Normal Depth 1.75

Energy coefficient 1.1

## Table 3.2.1 The Standard step Method Calculation for a Trapezoidal Channel

## x(m)

## Z

## y

## A

## V

## V2/2g

## H(1)

## P

## R

## Sf

## Sf mea

## Del x

## hf

## H(2)

0

4.000

4.000

28.00

0.68

0.026

4.026

14.31

1.96

5E-05

## Â

## Â

## Â

## Â

100

4.002

3.882

26.72

0.71

0.028

4.030

13.98

1.91

6E-05

6E-05

100

0.0058

4.032

200

4.005

3.705

24.84

0.76

0.033

4.038

13.48

1.84

7E-05

7E-05

100

0.0068

4.037

300

4.008

3.558

23.33

0.81

0.037

4.045

13.06

1.79

9E-05

8E-05

100

0.0082

4.046

400

4.012

3.412

21.88

0.87

0.042

4.054

12.65

1.73

1E-04

1E-04

100

0.0097

4.055

500

4.017

3.267

20.47

0.93

0.048

4.065

12.24

1.67

1E-04

1E-04

100

0.0115

4.066

600

4.039

3.116

19.06

1.00

0.056

4.095

11.81

1.61

2E-04

1E-04

100

0.0139

4.079

700

4.061

3.067

18.61

1.02

0.058

4.119

11.67

1.59

2E-04

2E-04

100

0.0157

4.110

800

4.085

2.935

17.42

1.09

0.067

4.152

11.30

1.54

2E-04

2E-04

100

0.0177

4.137

900

4.102

2.870

16.85

1.13

0.071

4.173

11.12

1.52

2E-04

2E-04

100

0.0202

4.172

1000

4.126

2.660

15.06

1.26

0.089

4.215

10.52

1.43

3E-04

2E-04

100

0.0248

4.198

1100

4.138

2.502

13.77

1.38

0.107

4.245

10.08

1.37

4E-04

3E-04

100

0.0324

4.248

1200

4.156

2.351

12.58

1.51

0.128

4.284

9.65

1.30

5E-04

4E-04

100

0.0413

4.286

1300

4.164

2.196

11.41

1.67

0.155

4.319

9.21

1.24

6E-04

5E-04

100

0.0533

4.337

1400

4.188

2.188

11.35

1.67

0.157

4.345

9.19

1.24

6E-04

6E-04

100

0.0606

4.380

1500

4.242

1.992

9.94

1.91

0.205

4.447

8.63

1.15

9E-04

7E-04

100

0.0742

4.419

1600

4.311

1.911

9.38

2.02

0.230

4.541

8.40

1.12

1E-03

9E-04

100

0.0948

4.541

1700

4.397

1.847

8.95

2.12

0.253

4.650

8.22

1.09

1E-03

1E-03

100

0.1092

4.650

1800

4.500

1.800

8.64

2.20

0.271

4.771

8.09

1.07

1E-03

1E-03

100

0.1221

4.772

1900

4.618

1.768

8.43

2.25

0.285

4.903

8.00

1.05

1E-03

1E-03

100

0.1325

4.904

The following parameters are recorded above the table

Flow rate

Manning's n

Side slope

Bed slope

Base Width

Normal Depth

Energy coefficient

Record the location of measured channel cross sections and the trial water surface elevation, z, for each section. The trial elevation will be verified or rejected based on computations of the step method.

Determine the depth of flow, y, based on trial elevation and channel section data. Record the result.

Using the depth from the above step and section data, compute the cross-sectional area, A and hydraulic radius, r. Record the results.

Divide the design discharge by the cross-sectional area from the above step to compute the average velocity, v. Record the result.

Compute the total head, H(1), by summing the water surface elevation, z, and the velocity head which is multiplied by the energy coefficient. Record the result.

Compute the friction slope, Sf, using equation written under direct step method for rectangular channel and record the result.

Determine the average friction slope, Sf, between the sections in each step (not applicable for row 1). Record the result

Record the distance between two consecutive depths (Del x) in a column

Compute the head loss by multiplying the above two columns.

Calculate the total head (H2) again from energy head H1 and head loss hf

If the difference between H(1) and H(2) are lesser than 0.001 then the assumed water depth is ok. Else the water depth must be assumed again and the calculations should be repeated until obtain a reasonable result.

The trial and error procedure is used in the above step.

## Figure 3.2.2 H(1) Vs H(2)

The above graph shows a very much closer relationship to Y=X graph. So, we can say that H(1) and H(2) are pretty much closer. And the assumed water depths are correct.

## CHAPTER 4

## 4. METHODOLOGY

As I explained in chapter 1, my thesis concentrates on the calculations related to backwater. First the broad study on hydraulics was done. Then the types of flows are identified in an open channel for different boundary conditions. Hereafter the flow which is forming the backwater was taken into account. After that two different types of calculation methods were identified. Then the above both calculation procedures are used to calculate the backwater profile in different kinds of channels. Finally, the results obtained from these calculations were checked from the water surface profile obtained by plotting in excel sheet.

## CHAPTER 5

## 5. DISCUSSION

The backwater is a rise in water level in the upstream due to any obstruction in the direction of flow. Considering the backwater effect is important for any constructions across the flow path for the safety and other environmental effects. Such as, flooding, soil erosion etc. Forces comes from the upstream water acts on the structures constructed across the flow path such as dams, spillways etc. So, these structures should withstand the forces comes from the upstream water as well. Backwater calculations are necessary in case of any implementation of a new structure in upstream. So, these calculations should be accurate in order to prevent any failures.

The direct step method can be used to calculate the distance between two consecutive known depths in prismatic channels. So, another method must exist to do the calculations in non prismatic and natural channels. It is the standard step method. From this method the water depths are calculated in particular stations using trial and error procedure as well.

These calculations can be used to predict the water surface profiles. From that the designers can come to a decision whether or not any construction can be done.

## CHAPTER 6

## 6. CONCLUSION

The importance of the backwater calculations is much clear from the studies made to complete this thesis. People rely on water supply in their day to day life. It is not possible to get water directly from a stream or any water sauce for whole year. There must be some massive storage like dams across the stream flow to stagnate water and prevent it from going to sea in whole. In order to construct these kind of massive structures there must be some kind of design calculation to be done prior to the construction. We can not imagine the damages it cause if a massive structure fails. So, the calculations should be perfect to save the structure and the environment.