OPEN CHANNEL FLOW OVER A PERMEABLE RIVER BED CYRIL PROJECT.pdf · 2015-09-04 · Udeogu for their...

Agboeze Irene E.

OPEN CHANNEL FLOW OVER A PERMEABLE RIVER

BED

i

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Agboeze Irene E.

PHYSICAL SCIENCES

MATHEMATICS


BED

UDOGU CYRIL IFEANYICHUKWU

PG/M.Sc/07/42989

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OU = Innovation Centre



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TITLE PAGE

OPEN CHANNEL FLOW OVER A PERMEABLE

RIVER BED

BY


PG/M.Sc/07/42989

DEPARTMENT OF MATHEMATICS

UNIVERSITY OF NIGERIA

NSUKKA

SUPERVISOR: PROF. G.C.E. MBAH

APRIL, 2013.

iii

CERTIFICATION PAGE

Udogu Cyril Ifeanyichukwu, PG/M.Sc/07/42989, a Post Graduate student in

the Department of Mathematics, University of Nigeria, Nsukka has

satisfactorily completed the requirement for the award of degree of Master

of Science (M.Sc) in Mathematics, by presenting this research work which is

original and has not been submitted in part or full for any other diploma or

award of any degree to this University or any other.

____________________ ___________________

UDOGU CYRIL IFEANYICHUKWU PROF. G.C.E. MBAH

PG/M.Sc/07/42989 Supervisor

Student.

_____________________ ______________

PROF. F.I NJOKU External Examiner

Head of Department

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DEDICATION

This Research work is dedicated to My Mother Ezinne Grace Udogu.

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ACKNOWLEDGEMENT

I am greatly indebted to a number of persons, whose contributions,

encouragement, advice and good wishes led to the successful completion of

this work.

First and foremost I give thanks to Almighty God for preserving my life

throughout my stay in the University.

I owe my project supervisor, Professor G.C.E. Mbah a great debt for his

patience, tolerance, advice and mentorship without which completion of this

work would have not been real.

My profound gratitude also goes to my darling wife Mrs. Callista Udogu, for

her perseverance and encouragement.

I sincerely appreciate Dr. Cosmas Anyanwu and his wife for their

benevolence.

I also would have to thank my sister Bibian Udeogu and my brother Nestor

Udeogu for their contributions towards the completion of this programme.

Indeed, I appreciate my lecturers in the Department of Mathematics for their

dedication.

This section of the work will be incomplete if I fail to appreciate my fellow

P.G. Students 2008/2009 session.

Finally, I thank my son Nelson Chiduziem Ndiezekwem for assisting greatly in

typing this work.

vi

LIST OF FIGURES, TABLES AND GRAPHS.

Figure 3.1 Shear stress in fluid dynamics

Figure 3.2 Structure of flow

Figure 4.2 Mass Conservation

Table 5.1 Effect of Permeability on velocity

Table 5.2 Effect of height of a channel on velocity

Table 5.3 Effect of Permeability on pressure gradient

Figure 5.1 Graph of velocity against permeability

Figure 5.2 Graph of velocity against height

Figure 5.3 Graph of pressure gradient against permeability

vii

ABSTRACT

We have modelled an open channel flow through a porous media (River). In

the model, we considered water as an incompressible fluid; the flow as steady

and uniform; the system is assumed to be isothermal and the flow, also a

laminar flow. We have solved the resulting equation using analytical method.

By some mathematical operations, the momentum partial differential equation

(PDE) was reduced to ordinary differential equation (ODE) and the resulting

equations are solved analytically using the technique for solving linear

equations with constant coefficients-method of variation of parameters. The

analysis of the result was done and plotted on graph using MATLAB to show

the effect of permeability on flow parameters such as velocity, pressure

gradient and the height of the channel. Recommendations were made to

control and manage the flow of rivers.

viii

TABLE OF CONTENTS

Title page -------------------------------------------------------------------------- i

Certification page --------------------------------------------------------------- ii

Dedication ------------------------------------------------------------------------ iii

Acknowledgement -------------------------------------------------------------- iv

List of figures, tables and graphs -------------------------------------------- v

Abstract ---------------------------------------------------------------------------- vi

Table of Contents ---------------------------------------------------------------- vii-viii

Open Channel flow ------------------------------------------------------------- ix

CHAPTER ONE

1.1 Introduction -------------------------------------------------------------- 1-3

1.2 Aims and Objectives of the study ------------------------------------ 3-4

1.3 Scope of the study ------------------------------------------------------- 4

1.4 Relevance of the study ------------------------------------------------- 4

1.5 Limitations of the study ------------------------------------------------ 5

CHAPTER TWO

2.0 Literature Review --------------------------------------------------------- 6-14

CHAPTER THREE

3.0 Theory of Channel flow

3.1 Properties of fluid ------------------------------------------------------- 15-26

ix

3.2 Types of flow in Channels (structure) ---------------------------------- 26-27

3.2.1 Steady flow and unsteady flow -------------------------------------- 28-29

3.2.2 Uniform and Non-uniform (or varied flow) ---------------------- 29-30

3.2.3 Laminar flow and Turbulent flow ---------------------------------- 30-31

3.2.4 Subcritical flow, critical flow and supercritical flow ------------- 31-32

3.3 Volumetric flow rate ----------------------------------------------------- 32-33

3.4 Permeability of fluid during flow -------------------------------------- 33-35

CHAPTER FOUR

4.0 Model Equation in open channel flow over a permeable flow

4.1 Mass Conservation ----------------------------------------------------- 36-39

4.2 Conservation of Momentum ------------------------------------------ 39-41

4.3 Equation of an open channel flow through porous media with porosity φ

---------------------------------------------------------------------------------------- 41-42

4.4 Method of solution of modelled equation ------------------------ 43-51

CHAPTER FIVE

5.0 Analysis and discussion of Results ---------------------------------- 52-58

5.1 Conclusion ---------------------------------------------------------------- 59

5.2 Recommendations ------------------------------------------------------ 60

REFERENCES ----------------------------------------------------------------------- 61-62

1

CHAPTER ONE

1.1 INTRODUCTION

As a result of the importance of open channel flow, and the occurrence of

porous media in a wide variety of important practical applications, the overall

objective of this work is to provide analytical technique for treating problems

which involve flow of an open channel through porous media. Specifically flow

through a permeable river bed.

The flow of water in an open channel is a familiar sight, whether in a natural

channel like that of a river or an artificial channel like that of an irrigation ditch.

Its movement poses a difficult problem when everything is considered

especially with the variability of natural channels. However, in many cases the

major features are expressed in terms of only few variables, whose behaviour

can be described adequately by a simple theory.

The principal forces at work during fluid flow are those of inertia, gravity,

viscosity and pressure gradient Calvert (2003).

In this study, water is considered as the fluid with a known density of 1g/cc and

does not vary significantly for the temperature and pressures that would be

considered in this work.

2

It has been observed that the total volume of water transported downstream

by a river, is a combination of the free water flow together with a substantial

contribution to flow through subsurface rocks and gravels that underlie the

river and its flood plain. It has also been observed that some rivers flow

intermittently. They only flow occasionally and can be dry for several years at a

time. This has been attributed to geological conditions such as highly

permeable river bed (Wikipedia, the free encyclopaedia).

Although almost all natural channels have permeable beds such as gravel Bed

Rivers, not much research has been undertaken in order to study the effect of

channel bed permeability on the mean and instantaneous flow. In common

practice a permeable bed has usually been treated analogously to an

impermeable bed, and flow resistance coefficient and velocity distributions

derived irrespective of bed porosity. Thorsten et al (2007), observed that

depending on the permeability of the subsurface, significant interaction

processes occur between the flow above the porous bed and the subsurface

area. The effects of this interaction are a non-zero velocity at the permeable

boundary. They also observed that the driving force which is responsible for

the exchange processes between the pore layer and the upper flow is the

presence of local pressure gradients.

3

According to Radiom et al (2007), fluid flow in a porous media shows some of

the characteristics of flow in the absence of rigid matrix and in such a flow

regime, the inertia and fluid shear stress effects not included in the Darcy

model becomes significant affecting the flow characteristics. Flow in solid

media or flow in the absence of a rigid matrix is governed by some

fundamental laws based on the conservation of mass, momentum and energy.

It has become pertinent to study open channel flow over a permeable river

bed so that in managing or controlling a river to make it more useful or less

disruptive to human activity, the effect of permeability would be considered.

When some rivers dry up as a result of permeability this will result to drop in

energy supply, affect habitat conservation and some other uses of the river.

The permeability of a river may contribute to the overflow of its bank and

consequently lead to over flooding.

1.2 AIM AND OBJECTIVE OF THE STUDY

River as an example of open channel flow is very important; they provide us

with food; sometimes source of drinking water; source of energy;

transportation etc.

The main purpose of this study is to provide further insight into open channel

flow over permeable beds and to enhance the understanding of the effect of

4

bed permeability on the mean and instantaneous flow, as well as other flow

parameters such as viscosity, pressure gradient and height of the channel.

1.3 SCOPE OF THE STUDY

The study which is aimed at presenting the effect of permeability on the mean

flow, pressure gradient, height and viscosity of a flowing river considers the

flow of a river between the point where water from the mountain joins the

channel and the point where the channel distributes water to the sea, ocean

etc.

The study models the laminar open-channel flow over a permeable river bed,

solves the resulting equation using analytical method.

Solution obtained is used to determine: the effect of permeability on the

velocity of the flow; the effect of permeability on the ability of the river to

resist flow; the effect permeability on the height of the river; etc. The

chemistry of rivers which depends on inputs from the atmosphere is not

considered in the study.

1.4 RELEVANCE OF STUDY

This study will help in the management and control of rivers to make them

more useful, or less disruptive, to human activity.

5

1.5 LIMITATIONS OF THE STUDY

Time constraint will not allow for an in dept coverage of all the concepts

connected with the topic under study; hence the study is limited to laminar not

turbulent flow; steady and uniform flow; homogeneous and incompressible

fluid. Finance is another constraint as experimental methods involving

measurement of the flow parameters are generally costly.

6

CHAPTER TWO

LITERATURE REVIEW

As a result of importance of fluid flow in open channels a lot of work has been

done on open channel flows but not much has been done on open channel

flow through porous media. We also have much literature in porous media.

This section of our report aims at discussing the various research work done in

recent times with regard to open channel flow through porous media.

According to Rajput (1998), fluid may be defined as a substance which is

capable of flowing, or a substance which deforms continuously when subjected

to external shearing force. From the above definition it is clear that fluid has

the characteristics of conforming to the shape of the containing vessel. Hence

fluid has no definite shape and can undergo a deformation when a small

amount of shear force is exerted on it. When fluid flows in a channel it takes

the shape of the channel. An open channel is defined as a passage in which

liquid flows with its upper surface exposed to the atmosphere, Rajput (1998).

In order words open channel flows are characterized by a free surface which is

exposed to the atmosphere. The pressure on this boundary thus remains

approximately constant irrespective of any changes in the water depth and the

flow velocity. In open channels the flow is due to gravity; thus the flow

conditions are greatly influenced by the slope of the channel.

7

These free-surface flows occur commonly in engineering practice, and include

both natural channels and artificial channels. The natural channels include

large scale geophysical flows such as rivers, streams, estuaries, etc which have

irregular sections of varying shapes, developed in a natural way, while artificial

channels such as irrigational channels, drainage channels and sewers have

cross-sections with regular geometrical shapes (which usually remain same

throughout the length of the channel).

In general, channel flows may be steady or unsteady; uniform or varied;

laminar or turbulent. Rajput(1998). The flow in an open channel may be

laminar or turbulent depending upon the value of Reynolds number defined as

Re=µ

ρVR

where Re is Reynolds number, ρ is fluid density, �is average velocity of flow in

the channel, R is the hydraulic radius (defined as the ratio of the area of flow to

the wetted perimeter) and µ is the dynamic viscosity of the fluid.

When Re < 500 flow is laminar

Re > 2000 flow is turbulent

500 < Re < 2000 flow is transitional.

8

The velocity of stream flow is controlled primarily by gravity and friction.

Gravity pulls the water from high elevations at the river’s source to low

elevations at the river’s mouth. The steeper the elevation gradient, the greater

the velocity. Friction tends to hold the water in place. Thus friction occurs in

two places. First there is frictional drag between one layer of water with those

above and below it. Second, there is frictional drag between the water and the

bed and sides of the channel through which the water flows. The interaction of

frictional and gravitational forces creates a boundary layer, where the force

induced by gravity is retarded by forces induced by friction. The change in

velocity between the bed and the top of the boundary layer is referred to as a

velocity gradient.

The manner by which water flows in this boundary layer changes from lower

velocities up stream to higher velocities downstream. Initially where water

moves slowly, only the lower most layers of the water near the streambed

show a velocity gradient. Viscous forces between water layers and between

the lowest water layer and streambed cause a decrease in velocity of these

layers towards the streambed. Each layer moves in parallel horizontal layers.

This is called laminar flow. The boundary layer undergoing laminar flow at first

thickens down flow. But as the velocity increases water layers become

unstable [i.e. a vertical component becomes part of the direction of flow]. First

9

the layers of water become more wave-like, rising and falling in the water

column. Finally, the vertical component of flow result in the formation of

eddies. Eddy flow is referred to as turbulent flow. The kind of turbulent flow

changes from streaming turbulent flow to shooting turbulent flow as velocity

increases. [The kind of turbulent flow is assessed using the Froude Number

gd

vF = where v is velocity, g is the gravity of water and d is water depth.

Froude numbers less than 1 suggest streaming turbulent flow while numbers

greater than 1 suggest shooting turbulent flow].

As the water in the boundary layer accelerates, it exerts a shear stress on the

underlying streambed and along the channel sides. The greater the velocity the

greater the shear stress. It is the shear stresses exerted by the water in the

channel that overcome the cohesiveness and weight of the grains that make

up the stream bed and side of the channel, thereby creating erosion.

As mentioned before the greater the velocity, the greater the shear stress.

Stream velocity is however controlled by several parameters including

hydraulic Radius (R) stream slope (S) and roughness of the streambed and

channel sides (n). The manning equation summarizes this relationship.

.

2

13

249.1SR

nv =

10

HOW A RIVER FLOWS

River is a natural water course, usually fresh water flowing towards an ocean, a

lake, a sea or another river. The water in a river is usually confined to a channel

made up of a streambed between banks.

Throughout the course of the river, the total volume of water transported

downstream will often be a combination of the free water flow together with a

substantial contribution flowing through sub-surface rocks and gravels that

underlie the river and its flood plain (called the hyporhporheic zone)

[WIKIPEDIA, THE FREE ENCYCLOPEDIA].

For many rivers in large valleys, this unseen component of flow may greatly

exceed the visible flow.

Rivers always flow downhill. A stream or a river is formed whenever water

moves downhill from one place to another. This means that most rivers begin

high up in the mountains where snow from the winter or ancient glaciers is

melting. On their way down the sea, they collect water from rain, and from

other streams. There is another place where rivers rise up: from springs, where

ground water seeps up unto the surface, it may form a lake or pool or it may

start running downhill and eventually ends up in a river.

11

There is also an intermittent river which only flows occasionally and can be dry

for several years at a time. These rivers are found in regions with limited or

highly variable rainfall; or can occur because of geologic condition such as

having a highly permeable river bed.

THE USES OF RIVER

Rivers have been a source of food since pre-history. They can provide a rich

source of fish and other edible aquatic life. Organisms in the riparian zone of a

river respond to changes in river channel location and pattern of flow.

Fast flowing rivers and waterfalls are widely used as source of energy.

Rivers are major source of fresh water which can be used for drinking and

irrigation.

It is also used for transport.

The coarse sediments, gravel and sand generated and removed from rivers are

extensively used in construction.

In some circumstances it can destabilize the river bed and the course of the

river and cause severe damage to spawning fish population which rely on

stable gravel formation for egg laying.

BOUNDARY CONDITION

12

For the impermeable wall, that is the river banks, it is observed that the fluid

immediately next to the walls remains at rest Tritton (1977), a fact known as

the no-slip condition. Hence the velocity of the river at its banks is zero and

non-zero elsewhere. This condition accounts for rivers being used for bathing,

washing of clothes and even source of drinking water.

POROUS MEDIA

Having established that the bed of a river is permeable, we now look at

literature on porous media.

A porous media is a solid containing void spaces (pores) either connected or

unconnected, dispersed within it in either a regular or random manner. These

so called pores may contain a variety of fluids such as air, water, oil etc. If the

pores represent a certain portion of the bulk volume, a complex net work can

be formed which is able to carry fluids URPO (2005).

Permeability is defined as the conductivity of fluid through a porous media or

material, URPO (2005). Permeability is given by the coefficient of linear

response of the fluid to a non-zero pressure gradient in terms of the flux

induced. This is mathematically summarized in Darcy’s law.

Darcy’s law

13

pk

u ∇−=µ

where u the volumetric fluid flow through the (homogeneous) medium and k is

the permeability coefficient that measures the conductivity of fluid flow

through the porous material, p is the pressure and µ is viscosity of the fluid.

Urpo (2005) also stated that Darcy’s law is presented as a particular simplified

form of the average stokes equation within its assumption. Hence fluid flow in

porous media shows some of the characteristics of flow in the absence of a

rigid matrix and in such a flow regime, the inertia and fluid shear stress effects

not included in the Darcy model, become significant, affecting the flow

characteristics. In addition, Darcy’s law is incompatible with the imposition of a

non-slip condition on the solid boundary wall and the interface between

porous media and open channel M. Parvazinia et al (2006).

Thorsten Stoesser et al (2007) also observed that depending on the

permeability of the subsurface of the porous bed, that significant interaction


area which results to non-zero velocity at the permeable boundary and the

existence of turbulent exchange of mass and momentum between the two

flow regions. These exchange processes are responsible for additional shear

stresses near the boundary. They also made references to Lovera Kennedy

14

(1969), Zaghi and Smith (1976), and Zippe and Graf (1983), who in their

separate works showed that the overall friction loss in a flow over a permeable

bed is larger than over an equivalent impermeable bed.

The driving force which is responsible for the exchange processes between the

pore layer and the upper flow is the presence of local pressure gradient.

M. Radiom et al (2007) in their work “numerical and analytical calculation of air

flow through an open channel linked to a porous media” came to the

conclusion that by decreasing the Darcy number due to increase in resistance

against the flow in the porous region the difference between the maximum of

velocity profile in porous region and open channel increases. They also

observed that there is a penetration of flow from porous media to open

channel due to resistance against flow in porous region and this penetration

increases by decreasing the Darcy number. This penetration adjusts the mass

flow in open and porous regions in order to have identical pressure difference.

The resistance to fluid flow gives rise to pressure drop in the fluid.

In a similar manner this present study is aimed at modelling mathematically,

the effect of permeability on flow parameters such as velocity, viscosity height

of channel and pressure gradient in a permeable river bed.

15

CHAPTER THREE

3.0 THEORY OF CHANNEL FLOW

FLUID

A fluid is a substance which deforms continuously when subjected to external

shearing force. In particular, we are interested in water flow over an open

channel.

CHARACTERISTICS OF A FLUID

1. It has no definite shape of its own, but conforms to the shape of the

containing vessel.

2. A small amount of shear force exerted on a liquid/fluid will cause it to

undergo a deformation which is continuous as long as the force

continues to be applied.

3.1 PROPERTIES OF FLUID

The properties of water are of much importance because the subject of

hydraulics is mainly concerned with it.

Some important properties of water are: density, specific gravity, viscosity,

vapour pressure, cohesion, adhesion, surface tension, capillarity,

16

compressibility etc. fluid flow is affected by myriad of factors or variables such

as:

(I) Velocity of flow

(II) Flow rate

(III) Density of fluid

(IV) Temperature of the fluid

(V) Compressibility of the fluid

(VI) Time of flow

(VII) Pressure gradient

(VIII) Viscosity/kinematics viscosity

(IX) Shear stress

(X) Volume of fluid

(XI) Mass of fluid

(XII) Gravity

(XIII) Inertia force

(XIV) Cohesion

(XV) Adhesion

(XVI) Surface tension

(XVII) Capillarity etc.

Velocity of flow:

17

The flow velocity u of a fluid is a vector field ),( txuu = which gives the velocity

of an element of fluid at a position x and time t . The flow velocity of a fluid

effectively describes everything about the motion of a fluid. For the present

work, we suppose that the velocity profile is the same at all distances down

the channel; that is all x. This velocity profile )( yu also known as distribution of

velocity is a function of y only. Thus )(),( yuyxu ≈ as will be seen in chapter four

of this work. This means that the velocity is dependent on depth and not on

distance from the source.

Flow rate:

Volumetric flow rate, (also known as volume flow rate, rate of fluid flow or

volume velocity) is the volume of fluid which passes through a given surface

per unit time. The SI unit is 13 −sm .

Density of fluid:

The density of a liquid is defined as the mass per unit volume

v

mat a

standard temperature and pressure. It is usually denoted by ρ (rho). Its units

are 3/ mkg , and 3/ cmg .

Weight density:

18

The weight density (also known as specific weight) is defined as the weight per

unit volume at standard temperature and pressure. It is denoted by w

gw ρ= .

Viscosity/Kinematics viscosity:

Viscosity may be defined as the property of a fluid which determines its

resistance to shearing stresses. It is a measure of the internal fluid friction

which causes resistance to flow. It is due to cohesion and molecular

momentum exchange between fluid layers, and as flow occurs, these effects

appear as shearing stresses between the moving layers of fluid. This shear

stress is proportional to the rate of change of velocity with respect to y. It is

denoted by τ (tau).

Mathematically, dy

du∝τ

dy

duµτ =

Where, =µ constant of proportionality and is known as co-efficient of

dynamic viscosity or only viscosity; dy

du is rate of shear stress or velocity

gradient.

19

Thus dydu

τ

µ =

We can therefore, also define viscosity as the shear stress required to produce

unit rate of shear strain.

Kinematics viscosity: is defined as the ratio between the dynamic viscosity

and density of fluid. It is denoted by ν (called nu).

Mathematically, ρµν ==

Density

ityViscos.

Temperature of the system:

Temperature is the measurement in degrees of how hot or cold a thing or

system is. Temperature affects viscosity of a fluid. Viscosity is the property of

the fluid that resists the flow of the fluid like liquids and gases. Understanding

the effect of temperature on the viscosity of the fluid is very important. The

viscosity of liquids decreases but that of gases increases with increase in

temperature. This is due to the reason that in liquids the stress is due to the

inter-molecular cohesion which decreases with increase of temperature. In

gases the inter-molecular cohesion is negligible and the shear stress is due to

exchange of momentum of the molecules. In this study, temperature is

approximately constant and does not affect the viscosity of water.

20

Pressure/pressure gradient:

Pressure is the ratio of force to the area over which that force is distributed

(the symbol is p). When a fluid is contained in a vessel, it exerts force at all

points on the sides, bottom and top of the container. The force per unit area is

called pressure. The pressure of a fluid on a surface will always act normal to

the surface, Rajuput (1998).

In water flow over an open channel, the pressure varies with x, obviously, but

is constant across the channel at each x hence x

p

∂∂

= constant. Further the

assumption of an unchanging velocity profile makes the dynamical processes

the same at all stations downstream; the pressure force per unit volume-i.e.

the pressure gradient is independent of x . Hence,

,Gdx

dp

x

p =−=∂∂− Say, Tritton (1977).

Compressibility of fluid:

A compressible flow is that flow in which the density of fluid changes during

flow as a result of pressure or temperature change. An incompressible flow is

flow in which the pressure variation does not produce any significant density

variation. The variation in volume of water, with variation of pressure, is so

small that for all practical purposes it is neglected. Thus water is considered to

21

be an incompressible liquid in this work water is assumed an incompressible

fluid.

Inertia:

Property of a substance by which it stays still or if moving, continues moving in

a straight line unless it is acted on by a force outside itself. In the absence of

forces, “body” at rest will stay at rest, and a body moving at a constant velocity

in a straight line continues doing so indefinitely.

In Navier-strokes Equation, which is used to model fluid flow, the second term

uu ∇⋅ is known as the inertia term and its ratio to the viscosity term,

determines whether the flow is laminar or turbulent. In this work we assume

that the viscous force is predominant over the inertia force. Hence we consider

a laminar flow.

Force of gravity:

Gravitation or gravity is a natural phenomenon by which physical bodies

attract each other with a force proportional to their masses.

Gravitation we know is the agent that gives weight to objects with mass and

causes them to fall to the ground when dropped. Gravitation is responsible for

keeping the moon in its orbit around the earth for the formation of tides and

22

for natural convection, by which fluid flow occurs under the influence of

density gradient and gravity.

In modelling fluid flow, a body force term is included to allow for the effect of

gravity. In the Navier-Stokes Equation,

gF ρ=

where F = body force

ρ = density

g = gravity

Almost every flow will take place in a gravity field. If density is uniform, the

gravitational force is balanced by a vertical pressure gradient which is present

whether or not fluid is moving and which does not interact with any flow. This

hydrostatic balance can be subtracted out of the dynamical equation and the

problem reduces to one without body forces. In the current work, gravity is

negligible because density is assumed to be constant.

Shear stress in fluid dynamics: Shear stress, denoted by τ (tau), is defined as

the component of stress coplanar with a material cross section. In other words,

shear stress arises from the force vector component parallel to the cross

section.

23

In fluid dynamics, it is due to cohesion and molecular momentum exchange

between fluid layers, and as flow occurs, these effects appear as shearing

stresses between the moving layers of fluid. This shear stress is proportional to

the rate of change of velocity with respect to y. the shear stress, for Newtonian

fluid, at a surface element parallel to a flat plate, at the point y, is given by

y

uy

∂∂= µτ )(

where:

µ is the dynamic viscosity of the fluid;

u is the velocity of the fluid along the boundary;

y is the height above the boundary.

The wall shear stress is defined as:

0)0(

=∂∂==≡

yy

uyw µττ

Although the viscous stress depends on the first spatial derivative of the

velocity, the viscous force on a fluid element depends on the second

derivative.

This can be shown below.

24

Fig.3.1

FFF

The net force on the element is the small difference of the viscous stresses on

either side of it.

In the diagram above, we show this, per unit area perpendicular to the y-

direction, force yyy

u

δµ

+

∂∂

andyy

u

∂∂− µ act in the x-direction on a region

between planes AB and CD. The net force on our element is

zxyy

u

yyy

u δδµ

δµ

∂∂−

+

∂∂

zxyy

u δδδ

µ

∂∂=

zyxy

u

yδδδ

µ

∂∂

∂∂= (When yδ is small enough)

25

xyzy

u δδδµ2

2

∂∂

(When µ is constant)

Therefore the viscous force per unit volume is

2

2

y

u

∂∂µ

In the present work which is a viscous flow, and in which u is independent of x

and z, this may be written as

2

2

dy

udµ

Cohesion:

Cohesion means intermolecular attraction between molecules of the same

liquid. It enables a liquid to resist small amount of tensile stresses. Cohesion is

a tendency of the liquid to remain as one assemblage of particle.

Adhesion:

Adhesion means attraction between the molecules of a liquid and molecules of

a solid boundary surface in contact with the liquid. The property enables a

liquid to stick to another body.

Surface tension:

26

Surface tension is caused by the force of cohesion at the free surface. A liquid

molecule in the interior of the liquid mass is surrounded by other molecules all

around and is in equilibrium. At the free surface of the liquid there are no

liquid molecules above the surface to balance the force of the molecules below

it. Consequently, there is a net inward force on the molecule. The force is

normal to the liquid surface. At the free surface a thin layer of molecules is

formed. It is because of this film that a thin small needle can float on the free

surface (the layer acts as a membrane).

In the foregoing work, the effects of surface tension are negligible.

Capillarity:

Capillarity is a phenomenon by which a liquid (depending upon its specific

gravity) rises into a thin glass tube above or below its general level. This

phenomenon is due to the combined effect of cohesion and of liquid particles.

Capillary effects are also negligible in the current work.

In this work, the above variables are expressed in terms of only few variables.

In open channel flow, the principal factors at work are inertia, gravity,

viscosity, pressure gradient and velocity.

3.2 TYPES OF FLUID FLOW

27

(I) Compressible and incompressible flows

(II) Steady and unsteady flows

(III) Uniform and non-uniform flows

(IV) One, two and three dimensional flow

(V) Rotational and non-rotational flows

(VI) Laminar and turbulent flows

CHANNEL FLOW

A channel is a passage in which liquid flows. Some channels are open while

others are closed like in the case of pipes. An open channel is a passage in

which liquid flows with its upper surface exposed to the atmosphere. Rajput

(1998). In other words open channel flows are characterized by a free surface

which is exposed to the atmosphere. In open channels the flow is due to

gravity, thus the flow conditions are greatly influenced by the slope of the

channel. The pressure distribution at all control sections is assumed

hydrostatic irrespective of any changes in the water depth and flow velocity.

These free surface flows occur commonly in engineering practice, and include

both natural channels and artificial channels.

The natural channels include large scale geophysical flows such as rivers,

streams, estuaries etc which have irregular sections of varying shapes,

developed in a natural way, while artificial channels such as irrigational

28

channels, drainage channels and sewers have cross-sections with regular

geometric shapes (which usually remain same throughout the length of the

channel).A channel with constant bed slope and the same cross-section along

its length is known as prismatic channel.

The velocity distribution in an open channel is such that the maximum

velocity occurs at a little distance below the water surface. The shape of

velocity profile is dependent on the channel roughness.

3.2 TYPES OF FLOW IN CHANNELS (STRUCTURE)

• Steady flow and unsteady flow

• Uniform flow and non-uniform (or varied) flow

• Laminar flow and turbulent flow

• Sub critical flow, critical flow and supercritical flow

3.2.1 STEADY FLOW AND UNSTEADY FLOW

• When the flow characteristics (such as depth of flow, flow velocity and

the flow rate at any cross-section) do not change with respect to time,

the flow in a channel is said to be steady.

Mathematically, 00,0 =∂∂=

∂∂=

∂∂

t

Qor

t

v

t

y

29

Where vy, and Q are depth of flow, velocity and rate of flow

respectively.

• The flow is said to be unsteady when the above flow parameters vary

with time.

Mathematically, 00,0 ≠∂∂≠

∂∂≠

∂∂

t

Qor

t

v

t

y

3.2.2 UNIFORM AND NON-UNIFORM (OR VARIED) FLOW

• Flow in a channel is said to be uniform if the depth, slope, cross-section

and velocity remains constant over a given length of the channel.

Mathematically, 0,0 =∂∂=

∂∂

t

v

t

y

• Flow in a channel are said to be non-uniform (or varied) when the

channel depth varies continuously from one section to another.

Mathematically, 0;0 ≠∂∂≠

∂∂

t

v

t

y

Varied flows are further classified as:

(I) Rapidly varied flow (R.V.F). Here the depth of flow changes abruptly

over a comparatively small length of channel leading to a hydraulic

drop or hydraulic jump.

30

(II) Gradually varied flow (G.V.F). In this case the change in depth of flow

takes place gradually in long length of the channel.

3.2.3 LAMINAR FLOW AND TURBULENT FLOW

The flow in an open channel may be characterized as laminar or turbulent

depending upon the value of Reynolds number defined as:

forcesviscous

forcesinertiavR ≈=µ

ρRe

Where v = Average velocity of flow in the channel, and

=ρ Density

R=Hydraulic radius (defined as the ratio of the area of flow to wetted

perimeter) and µ = the viscosity of the fluid

When Re < 500 flow is laminar

Re > 2000 flow is turbulent

500 < Re < 2000 flow is transitional

22

2

,L

Uvuv

L

Uuu ≈∇≈∇⋅

Hence Recos2

=≈=∇

∇⋅v

UL

forceityvis

forceinertia

uv

uu

31

The Reynolds number thus indicates the relative importance of two dynamical

processes. At a general point within the flow, the ratios of these two terms will

not be exactly equal to the Reynolds number, but their characteristics

magnitude will be in this ratio.

When the Reynolds number is much smaller than unity the viscous force

dominates over the inertia force so much that the latter plays a negligible role

in the flow dynamics. This is the case of a laminar flow which we are concerned

with in this work.

Corresponding argument for high Reynolds number indicate that the viscous

force is so small compared with the inertia force that it can be neglected. The

flow in this case is said to be turbulent.

3.2.4 SUBCRITICAL FLOW, CRITICAL FLOW, AND SUPERCRITICAL FLOW

Gravitational force is a predominant force in a channel flow therefore Froude

number gd

vFr = (where v and d are the mean velocity of flow and the

hydraulic depth of the channel respectively) is an important parameter for

analyzing open channel flow.

(I) When )(1 gdvorFr << the flow is described as sub critical (or

tranquil or streaming).

32

(II) When :1=Fr the flow is said to be in a critical state.

(III) When :1>Fr the flow is said to be supercritical (or rapid or shooting

or torrential).

STRUCTURE OF FLOW

Fig.3.2

Among these different types of channel flow, this work will be interested in

incompressible, steady, uniform, two dimensional and laminar flows.

3.3 VOLUMETRIC FLOW RATE

Volumetric flow rate, also called discharge, volume flow rate and rate of water

flow, is the volume of water or fluid which passes through a given cross-section

of the river channel per unit time. It is measured in cubic meters per second; it

is sometimes also measured in litres or gallons per second.

33

Volumetric flow rate can be thought of as the mean velocity through a given

cross-section, times that cross-sectional areaA

QvorvAQ == .

Mean velocity can be approximated through the use of the law of the wall,

which states that “the average velocity of a turbulent flow at a certain point is

proportional to the logarithm of the distance from that point to the wall or the

boundary of the fluid region” Theodore Von Karman (1930). Generally velocity

increases with the depth (or hydraulic radius) and slope of the river channel,

while the cross-sectional area scales with the depth and the width: the double

counting of depth shows the importance of this variable in determining the

discharge through the channel.

3.4 PERMEBILITY OF FLUID DURING FLOW

Throughout the course of a river, the total volume of water transported

downstream will often be a combination of the free water flow together with a

substantial contribution flow through sub-surface rocks and gravels that

underlie the river and its flood plain (called the hyporheic zone).

It has also been observed that some rivers flow intermittently. They only flow

occasionally and can be dry for several years at a time. These rivers are found

in regions with limited or highly variable rainfall, or can occur because of

geological conditions such as having a highly permeable river bed. [Wikipedia

34

the free encyclopaedia]. Hence flow throughout the course of a river is an

open channel flow over a permeable river bed.

POROUS MEDIA

A porous medium is a solid containing void spaces (pores) either connected or

unconnected, dispersed within it in either a regular or random manner. These

pores may contain a variety of fluids such as air, water, oil etc; if the pores

represent a certain portion of the bulk volume, a complex network can be

formed which is able to carry fluids. Permeability is therefore defined as the

conductivity to fluid flow of the porous material Urpo (2005). Permeability is

given by the coefficient of linear response of the fluid to a non-zero pressure

gradient in terms of the flux induced, while porosity of a porous medium is

defined as the ratio of the volume of pores to the total bulk volume of the

media. It is usually expressed as fraction or percentage. In this work it is

denoted by φ

At low Reynolds number, the most important relation describing fluid

transport through porous media is Darcy’s law.

µpk

u∇−=r

35

Where ur is the volumetric fluid flow through the (homogeneous) medium and

k is the permeability coefficient that measures the conductivity to fluid flow of

porous material, p∇ is pressure gradient and µ is viscosity of the fluid. Darcy’s

law is presented also as a particular simplified form of the averaged stokes

equation within its assumption Urpo (2005).

36

CHAPTER FOUR

4.0 MODEL EQUATION ON OPEN CHANNEL FLOW OVER A PERMEABLE BED

Fluid motions in porous media are governed by the same fundamental laws

that govern their motion in solid medium. These laws are based on the

conservation of mass, momentum and energy. From a practical stand point,

these basic laws are not applied directly to the problems of flows in porous

media; instead, a semi empirical approach is used where Darcy’s law is

employed instead of the momentum equation.

4.1 MASS CONSERVATION

Fig.4.1

F

FLOW THROUGH A POROUS MEDIA

Let xm& be x - component of the mass flux vector (mass per unit area per unit

time) of fluid of density ρ (single phase, single component).

Therefore the mass inflow across the control volume surface at x over a time

interval t∆ is

37

tAmxx ∆& .

And the mass out flow across the control volume surface at xx ∆+ is

tAmxxx ∆

∆+&

The difference between inflow and outflow must be equal to the sum of

accumulation of mass within the control volume. Mass accumulation due to

compressibility over a time interval t∆ is

tvt

∆

∆∂∂

)(ρφ

Where ρ is density, φ is porosity of the medium, v is volume of fluid,

and the removal of mass from the control volume that is, mass depletion

(accumulation) due to sink of strength q (mass per unit volume per unit time)

over a time interval t∆ is

tvq ∆∆

We have,

( ) ( ) tvqtvt

tAmm xxxxx ∆∆+∆

∆∂∂=∆− ∆+ ρφ|| && 4.1

Dividing through by tv ∆∆ , considering the fact that xAv ∆=∆ and taking limit as

38

0→∆x We have

)( qtx

m x +∂∂=

∂∂

− ρφ&

4.2

Note that q is negative for a source, since we have assumed it to be positive

for a sink.

It is possible to express mass flux in terms of superficial (or Darcy velocity)

xx um ρ=& 4.3

Where xU is a velocity in the .x direction.

Substituting 4.3 into 4.2 we obtain

( ) qtx

ux r+∂∂=

∂∂

− ρφρ 4.4

The corresponding equation for three dimensional flows in a porous medium

of arbitrary shape, for the Cartesian system of coordinate may be written as:

( ) ( ) qt

u +∂∂=∇− ρφρ. 4.5

For a river with impermeable banks 0=q and

( ) ( ) 0. =∇+∂∂

ut

ρρφ 4.6 (compressible fluid)

For an incompressible fluid

39

0. =∇ u 4.7

4.2 CONSERVATION OF MOMENTUM

Fluid flow in porous media shows some of the characteristics of flow in the

absence of a rigid matrix and in such a flow regime, the inertia and fluid shear

stress effects not included in Darcy model become significant, affecting the

flow characteristics. To obtain an equation for an open channel flow through a

porous media, the momentum equation for an open channel flow is derived

first, and then this equation would be modified to accommodate the porosity

of the open channel.

Newton’s second law states that the net force on a particle is equal to the

time rate of change of its linear momentum in an inertia reference frame.

( )dt

umdF = 4.8

dt

udmF = m= constant 4.9

When all the factors affecting the flow of a river is considered, especially with

the variability of natural channels, the movement of fluid in open channels

becomes a difficult problem. However in open channels, the major variables

can be expressed in terms of only few variables. The principal forces at work in

open channels are those of inertia, gravity, viscosity and pressure gradient. The

40

viscous force on a fluid element depends on the second derivative. Applying

Newton’s second law of motion of a fluid element, equation 4.9 becomes:

( ) gupdt

ud ρµρ +⋅∇+−∇= 2 4.10

( ) guvpdt

ud +⋅∇+∇−= 21

ρ 4.11

Where ρµν =

( )zyxtuu ,,,=r

z

z

uy

y

ux

x

ut

t

uu

δδδδδ ⋅

∂∂+⋅

∂∂+⋅

∂∂+⋅

∂∂=

t

z

z

u

t

y

y

u

t

x

x

u

t

t

t

u

t

u

δδ

δδ

δδ

δδ

δδ ⋅

∂∂+⋅

∂∂+⋅

∂∂+⋅

∂∂=

Taking limit as 0→tδ

uut

u

dt

ud ∇⋅+∂∂= 4.12

Substituting 4.12 in 4.10, we have

( ) gupuut

u ρµρ +∇+−∇=

∇⋅+∂∂

.2 4.13

The above is stokes equation for an incompressible, homogeneous fluid.

=u Velocity vector

41

=ρ Density of fluid

=p Pressure

=v Kinematics viscosity

=g Force of gravity.

4.3 EQUATION OF AN OPEN CHANNEL FLOW THROUGH POROUS MEDIA

WITH POROSITY φ

Depending on the permeability of the subsurface, significant interaction


area. The effect of this interaction is a non-zero velocity at the permeable

boundary. The driving force which is responsible for the exchange processes

between the pore layer and the upper flow is the presence of local pressure

gradients Thorsten stoesser et al (2007). Hence stokes equation for an

incompressible flow in a permeable channel becomes: )

( ) uk

gupuut

u φµρµρ −+⋅∇+−∇=

∇⋅+∂∂ 2 4.14 (a)

( ) uk

gupuut

u φµρ

νρ

11 2 −+⋅∇+∇−=∇⋅+∂∂

4.14 (b)

If we assume the following:

(I) Incompressible/homogeneous flows

42

(II) Steady/uniform channel flow

(III) Laminar flow

(IV) Effect of gravity is negligible

(V) Hydro-static pressure

(VI) Isothermal system

(VII) Density of water is 1g/cc and

(VIII) Two dimensional flow

The equation simplifies to

x- Component

02

2

2

2

=

∂∂+

∂∂+−

∂∂−

y

u

x

uu

kx

p µφµ 4.15

Y- Component

02

2

2

2

=

∂∂+

∂∂+−

∂∂−

y

v

x

vv

ky

p µφµ 4.16

43

4.4 METHOD OF SOLUTION OF MODELLED EQUATION

Flow is in the x- direction. Hence the equation

02

2

2

2

=

∂∂+

∂∂+−

∂∂−

y

u

x

uu

kx

p µφµ

The velocity profile is the same at all distances down the channel; that is at all

x. therefore it is a function of y only. Thus ).(),( yuyxu ≈

The viscous force is along the y-axis

02

2

=∂∂+−

∂∂−

y

uu

kx

p µφµ 4.17

In fully developed region, pressure drop in porous media and open channel are

equal as flows

Gtconsopendx

dp

porousdx

dp === tan 4.18

In water flow over an open channel, pressure varies with x obviously, but is

constant across the channel at each x. the assumption of an unchanging

velocity profile makes the dynamical processes the same at all stations

downstream; the pressure force per unit volume-i.e. the pressure gradient-

must be independent of x. Hence

,Gdx

dp

x

p ==∂∂

44

Hence equation 4.17 involves only one independent variables i.e. y.

02

2

=+−−dy

udu

kG µφµ

4.19

Gukdy

ud =− φµµ2

2

4.20

Where G is pressure drop dx

dp

Modelling fluid flow in open- porous channel presents difficulty to solve the

governing equations of open channel and porous media simultaneously; one

possible approach is to divide the medium into two regions.

(I) The porous region and (2) the open region and solve the equations of the

two regions separately and then apply the boundary conditions, M. Radiom et

al (2007)

02

2

=+−−dy

udu

kdx

dp µφµ For porous media 4.21

02

2

=+−dy

ud

dx

dp µ For open channel 4.22

45

BOUNDARY CONDITIONS

(I) At the inlet face a uniform velocity is specified

u= u inlet, v= v inlet=0 4.23

(2) At the outlet, velocity is extrapolated to the exit plan

00 =∂∂=

∂∂

exitx

v

exitx

u 4.24

(3) At the sides with impermeable walls no slip condition is enforced:

u=0, v=0 4.25

(4) At a section in which the flow is fully developed along the duct, the

shear stress equality is enforced as:

)()( intint −+ ==

= heighterfaceydy

du

heigherfaceydy

du µµ 4.26

Where the interface height is

2

H

SOLUTION

02

2

=+−−dy

udu

kG µφµ

Where ==dx

dpG constant.

Let βφ =k

46

Gudy

ud =− βµµ2

2

4.27

µβ G

udy

ud =−2

2

4.28

Applying technique for a constant coefficient second order ODE

The complementary solution is

ycycu ββ sinhcosh 21 +=

The particular solution

Method of variation of parameters

Let )()( 2211 ykcandykc ==

ykyku ββ sinhcosh 21 += 4.29

ykykykyku ββββββ coshsinhsinhcosh 2121

11

1 +++=

0sinhcosh 12

11 =+∴ ykyk ββ 4.30

ykykykyku ββββββββ sinhcoshcoshsinh 2121

11

11 +++= 4.31

Substituting 4.29 and 4.31 in 4.28

We have:

47

µββββ G

ykyk =+ coshsinh 12

11 4.32

Solve equations 4.30 and 4.32, applying Cramer’s rule

yG

k ββµ

sinh11 −= 4.33

11 cosh cyG

k +−=∴ βµβ

4.34

yG

k ββµ

cosh12 =

22 sinh cyG

k += βµβ

4.35

Substituting for 1k and 2k in 4.29

yG

yG

ycycu βµβ

βµβ

ββ 2221 sinhcoshsinhcosh +−+= 4.36

µβββ G

ycycU porous −+= sinhcosh 21 Solution (1) 4.37

When u=0, y=0 and

µβG

c =1

µβββ

µβG

ycyG

u −+=∴ sinhcosh 2 4.38

48

At the interface:

µβββ

µβGH

cHG

U face −+

=2

sinh2

cosh 2int

+

−

=∴

µββ

µββ

GHGU

Hc erface 2

cosh

2sinh

1int2

( )µβ

βµβ

βµββ

βµβ

Gy

GHGU

Hy

Gu erface −

+

−

+=∴ sinh

2cosh

2sinh

1cosh int

4.39

For open channel, without porosity included,

02

2

=+−dy

ud

dx

dp µ

Gdy

ud =2

2

µ 4.40

µG

dy

ud =2

2

4.41

∫ ∫ += 12

2

cG

dy

ud

µ 4.42

1cyG

dy

du +=µ

4.43

21

2

2cyc

Gy

dy

du ++=∫ µ 4.44

49

21

2

2cyc

Gyu ++=

µ 4.45

When y = 0, u=0

20 c=

ycGy

u 1

2

2+=

µ Solution (2) 4.46

28 1

2

int

Hc

GHU face +=

µ 4.47

28 1

2

int

Hc

GHU face ==

µ 4.48

2

8

2

int

1 H

GHU

cerface µ

−=∴ 4.49

yH

GHU

Gyu

erface

open

−

+=

2

82

2

int2 µµ

4.50

Linear pressure drop dx

dpG = in open channel is as follows:

Equation for open channel is

02

2

=+−dy

ud

dx

dp µ 4.51

50

From Darcy’s law,

4.52

dx

dpku

µ−= 4.53

Substituting Equation 4.53 in Equation 4.51 we have

02

2

=

+dx

dp

dy

dk

dx

dp 4.54

Put dx

dpG = then we have

02

2

=+ Gdy

Gdk 4.55

02

2

=+k

G

dy

Gd

yk

Byk

AG1

sin1

cos +=∴ solution(3) 4.56

yk

Byk

Adx

dp 1sin

1cos +=⇒ 4.57

=dx

dp Pressure gradient of the channel

=k Permeability of the channel

pk

u ∇−=µ

51

=y Height of the channel.

52

CHAPTER FIVE

5.0 ANALYSIS AND DISCUSSION OF RESULTS

The following data were generated from the analytical solutions presented

here in chapter four

�� =��cosh yk

φ + �sinh y

k

φ-

µφGk

(solution1).

In table 5.1 below, velocity (��) was computed with varied values of

permeability (k), while height of channel (y), porosity (φ ), pressure gradient

(G) and viscosity ()were all kept constant. We also noted that if velocity is

in cm/s, viscosity in centipoises and pressure gradient in atm/cm, then the

unit of k is Darcy.

Y =1cm, φ =0.5, G =1atm/cm, µ =0.01 centipoises, k is varied.

53

TABLE 5.1

The Effect of Permeability on Velocity

Permeability (Darcy) Velocity (cm/s)

0.001 16.1

0.002 11.3

0.003 9.0

0.004 7.6

0.005 6.6

0.006 5.7

In table 5.2

Velocity ( ��) was computed with varied values of Y (height of the

channel), while Parameters in the solution were kept constant.

φ =0-5, K=0.001 Darcy, G = l atm/cm, Viscosity = 0.01 centipoise

54

Table 5.2

The Effect of Height of the Channel on Velocity

Height of Channel (Ycm) Velocity (cm/s)

1 16.1

2 116.2

3 317.4

4 863.1

5 2346.4

6 6378.2

In table 5.3, the data were generated from the analytical solution:

��

��=G=Acos y

k

1+ BSin y

k

1 solution (3)

Y is kept constant at 1cm, while K is varied

55

Table 5.3

Effect of Permeability on Pressure Gradient

Permeability (K) (Darcy) Pressure gradient (G) [ atm / cm]

0.001 32.6

0.002 23.4

0.003 19.3

0.004 16.8

0.005 15.2

0.006 13.9

Figure 5.1

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

x 10-3

4

6

8

10

12

14

16

18

permeability[darcy]

velo

city

[cm

/s]

Graph of velocity against permeability

56

Figure 5.2

Figure 5.3

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

1000

2000

3000

4000

5000

6000

7000

Height[cm]

Vel

ocity

[cm

/s]

Graph of velocity against height

1 1.5 2 2.5 3 3.5 4 4.5 5

x 10-3

10

15

20

25

30

35

permeability[darcy]

Pre

ssur

e gr

adie

nt[a

tm/c

m]

Graph of pressure gradient against permeability

57

Figure 5.1 shows the effects of permeability on the velocity of flow in an open

channel over a permeable riverbed.

It can be seen that velocity decreases with increase in permeability. When the

porous media is highly permeable, the behaviour of the porous media does not

differ too much from the open channel. However, when permeability is low the

deviation of velocity of the porous media from the open channel is clear.

Figure 5.2 shows the effect of velocity on the depth of the river. Solution 1

shows that velocity increases with increase in the height of the river.

This also indicates that as the depth of the river increases, the effect of

permeability also decreases.

Figure 5.3 shows the effect of permeability on pressure drop or pressure

gradient. It can also be seen that pressure gradient decreases with increase in

permeability. When pressure gradient decreases, there is a decrease in the

resistance against the flow in the porous media, which causes a penetration

from the open channel to the porous media or we say the penetration is from

the flow above the porous bed to the subsurface area.

Hence high permeability accounts for some rivers drying up intermittently. In

the other hand, when permeability is low, pressure gradient is high and

resistance to flow in the vertical direction is high. Hence there is a penetration

from the subsurface area to the flow above the porous bed. This adjusts the

mass flow in open channel and porous media in order to have identical

pressure drop in X-direction. This again accounts for the early statement that

the total volume of water transported downstream by a river, is a combination

58

of the free water flow together with a substantial contribution to flow through

subsurface rock gravels that underlie the river and its flood plain.

59

5.1 CONCLUSIONS

Open channel flow over a permeable riverbed is studied by analytical solution

within the fully developed region. Solution (1) is for channel flow over the

permeable bed.

Solution (2) is for open channel flow, while solution (3) is obtained by

substituting the value of velocity vector in porous media, into the solution for

open channel flow. It is observed that by decreasing the permeability in Darcy,

due to increase in resistance against the flow in the porous region, the

difference between the maximum of velocity profile in Porous region and open

channel increases. This is as was predicted physically. It is also shown that

there is a penetration of flow from porous media to open channel due to

resistance against flow in the porous media and this penetration increase by

decreasing the permeability. This penetration adjusts the mass flow in open

and porous regions in order to have identical pressure difference in X-direction

for both the open channel and the porous media. In the other hand, when

permeability is high, there is a penetration from the open channel to porous

media due to decrease in resistance against flow. Hence high permeability

accounts for rivers meandering up and down. This becomes obvious in desert

areas, where the river beds are sandy and drought is a problem.

As the level of a desert stream drops, places where it was at the bottom of its

‘wave’ dry up, and all you can see is sand in the river bed. But if you walk

downstream a way, you will come to where it was at the top of its ‘wave’ and

you can still see water in that section. Water is flowing in both places, of

course. Where you saw it was dry, it is now flowing only underground.

60

5.2 RECOMMENDATIONS

When the permeability of a river is high, flooding in that area is not

catastrophic. Rainfall and snow melt will cause temporary and gradual

flooding.

Create ‘hard’ or impermeable, surfaces like roads, parking lots, sidewalks,

patios, and roofs. This is called urban runoff. Instead of soaking into the ground

and recharging the aquifer, rainfall begins to simply rush over these hard

surfaces, running directly into streams.

The result is that aquifer levels go down, lowering the level of the streams,

while run off during rainfall increases creating sudden, unseasonal flooding.

Wet lands like swamps and marshes should not be filled with dirt. Without

wetlands, rain water enters the river directly and causes catastrophic flooding.

Dams or weirs may be built to control the flow, store water or extracts energy.

The management of river is a continuous activity as rivers tend to ‘undo’ the

modifications made by people.

61

REEFERENCES

Batchelor, G.K (1967). An Introduction to fluid Dynamics. Cambridge

University Press New York Pp. 223-224.

Calvert, J.B. (2003). Open channel flow http://mysite.du..edu/-

etuttle/tech/opench.htm.

How a river flows – http://chamisa.freeshell.org//htm.From

Wikipedia, the free encyclopaedia. Accessed 24th

September, 2002.

Murray R. Spiege (1983). Advanced Mathematics for Engineers and

Scientists. Mc Graw-Hill Book Company, Singapore Pp. 74-83.

Nabovati A., Sousa, A.C.M. (2007). Fluid simulation in Random Porous

Media at Pore level using the Lattice Boltzmann Method. Journal of

Engineering Science and Technology pp. 226-237.

Obikwelu, D.O. N (2003). Differential Equations and Application.

AMDI-Nel publishers Ogidi, Nigeria.pp.60-61

Parvazinia, M. Nassehi, V.and Wakeman, R.J. (2006) Multiscale finite

element modelling of laminar steady flow through highly permeable

media. Chemical Engineering Science, pp583-596

Radiom, M. Khakbaz, M. and Kermani, M.J (2007). Numerical and

Analytical Calculation of Air flow through an open channel linked to a

porous media. Department of Mechanical Engineering, Amirkabir

University of Technology (Tehran Polytechnic) Tehran, Iran, 15875-4413.

Rajput, R.K. (1998). Fluid Mechanics and Hydraulic Machines. Schand

and Company, Ram Nagar, New Dehi.pp.3-29,867-871.

River.http://enWikipedia.org/wiki/river.FromWikipedia,thefree

encyclopaedia. Accessed 1st

March, 2012.

62

Thorsten Stoesser, Jochen Frohlich and Wolfgang Rodi (2007).

Turbulent Open Channel Flow over a Permeable bed. In proc of 32th IAHR

Congress, Venice, Italy, July 1-6.

Tritton, D.J. |(1977).Physical Fluid Dynamics. Van Nostrand Reinhold

(uk)’ Pp. 3-12.

Urpo Aaltosalmi(2005). Fluid flow in porous Media with the Lattice –

Boltzmann Method. Academic dissertation for PHD Pp. 5-8.

Wang, J. Gao, Z. Gan, G. and Wu, D. (2001). Analytical Solution of flow

Coefficients for a uniformly distributed porous channel. Chemical

Engineering Journal pp.1-6.

Zagni, A.F.E., Smith, K.V.H. (1976). Channel Flow over permeable

beds of graded spheres. ASCE, J. of the Hydraulics Division, vol-102, No. 2.

Pp. 207-222.

Zippe H.J Graf. W.H. (1983) Turbulent Boundary Layer flow over

permeable and Non permeable Rough surfaces J. of Hydraulic Research

vol. 21 No. 1. Pp. 51-65.

Zoltan E. Heinemann (2005). Fluid flow in porous media. Petroleum

Engineering Science. Vol. 1, Pp. 6-12.

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