CHAPTER 11.1 INTRODUCTIONComputeranalysisof
steadyflowinopenchannelsisavaluabletool for hydraulicengineers. It
enablesdetailedpredictionsof flowcharacteristics (surface profiles
most significantly) for a particular water course (in this case
openchannels) under certainconditions; without theneedtotakefield
measurementswhichcanbetimeconsuming, expensiveandsubject to
instrumental and human errors.The demand for efficient and accurate
software that can deal with problems faced by hydraulic engineers
has lead to numerous hydraulic packages in the softwaremarket.
Inthepast someof themoresophisticatedsoftware available required
the use of powerful computers and long run times, as a result of
the level of complexity involved. However this situation has
improved with the advances in computer hardware. 1.2 OPEN CHANNEL
HYDRAULICSAn open channel is a conduit in which water flows with a
free surface usually under atmospheric pressure. The classification
of open channel flow is made according to the change in flow depth
with respect to time and space.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 is a difficult problem when everything is considered,
especially with the variability of natural channels, but in many
cases the major features can be expressed in terms of only a few
variables, whose behavior can be described adequately by a simple
theory. The principalforces at work are those of inertia, gravity
and viscosity, each of which plays an important role.Flow in an
open channel is said to be "steady" if the depth of flow does not
change or if it can be assumed to be constant during the time
interval under consideration. The flow is "unsteady" if the depth
changes with time.1Open channel flow is said to be "uniform" if the
depth and velocity of flow is the same at every section of the
channel. Hence it follows that uniform flow canonlyoccur
inprismaticchannels.A uniformflowmaytheoretically be steady or
unsteady, depending on whether or not the depth changes with time.
Theestablishment of unsteadyuniformflowrequiresthat thewater
surface fluctuate with time while remaining parallelto the
channelbottom. Sinceit isimpossibleforthisconditiontooccur
withinachannel, steady uniformflowis the fundamental type of
flowtreated in open channel hydraulics.Flow is "varied" if the
depth of flow changes along the length of the channel. Varied flow
may be either steady or unsteady. Since unsteady uniform flow is
rare, the term "unsteady flow"is used to designate unsteady varied
flow exclusively.Variedflowmaybefurther
classifiedaseither"rapidly"or"gradually" varied.
Theflowisrapidlyvariedif thedepthchangesabruptlyover a
comparatively short distance; otherwise, it is gradually varied.
Rapidly varied flowisalsoknownasalocal phenomenon; anexampleof
whichisthe hydraulic jump.With these varying conditions, open
channel hydraulics can be very complex, encompassingmanydifferent
flowconditionsfromsteadyuniformflowto unsteady rapidly varied flow.
Most of the problems in storm water drainage involveuniform,
graduallyvariedor rapidlyvariedflowsituations. Inthis project
however computational procedures for uniform, gradually varied are
presented.1.2.1 UNIFORM FLOW2For a given channelcondition of
roughness, discharge and slope, there is onlyone(1)
possibledepthformaintainingauniformflow. Thisdepthis referred to as
normal depth.The Manning's Equation is used to determine the normal
depth for a given discharge.And is given by the expression;Q = A
R2/3 S1/2 /n, Where,Q = Total discharge, m3/sn = Roughness
coefficientA = Cross-sectional area of channel, m2R = Hydraulic
radius of channel, m (R=A/P)S = Slope of the frictional gradient,
m/mP = Wetted perimeter, m1.2.2 GRADUALLY VARIED FLOWThemost
commonexampleof gradually variedflowinurbandrainage systems occurs
in the backwater of bridge openings, culverts, storm sewer inlets
and channel constrictions.Under these conditions, gradually varied
flow willbe created and the flow depth will be greater than normal
depth in the channel (sub critical flow).1.2.3 RAPIDLY VARIED
FLOWRapidly varied flow is characterized by abrupt changes in the
water surface elevation for a constant flow. The change in
elevation may become so abrupt that the flow profile is virtually
broken (discontinuity), resulting in a state of highturbulence.
Somecommoncauses of rapidly variedflowinurban drainage systems are
side-spill weirs, weirs and spillways of detention basins.1.2.4
MANNINGS ROUGHNESS COEFFICIENT3Because several primary factors
affect the roughness coefficient, a procedure has been developed to
estimate this value, n. By this procedure, the value of n may be
computed by:n = (n0+ n1+ n2+ n3+ n4) m The grain roughness
component, n0 has a lower limit, which accounts for the smooth
boundary condition. Its value isn0 = 0.015 dm1/6 Where;dm =
d50which is the particle-diameter (mm) at which 50% (by mass) of
the material is larger than that particle-diameter.The d50 value
can be used provided that d50 > 0.05 mm. If d50 < 0.05 mm, a
minimum value of 0.05 mm should be used.n0is a basic n value for a
straight, uniform, smooth channel in the natural materials
involved, n1 is a value added to n0 to correct for the effect of
surface irregularities; n2is a value for variations in shape and
size of the channel cross section; n3 is a value for obstructions;
n4 is a value for vegetation and flow conditions; and m is a
correction factor for meandering of the channel.The Mannings
roughness coefficient is used in conjunction with the
Manningsequationwhichthemost widelyusedequationfor calculating
uniform flow in open channels. It was published in 1889, and later
modified to read (in metric units),n S AR Q / ) (2 / 1 3 / 2
Where,Q= Total discharge, m3/sn= Roughness coefficientA =
Cross-sectional area of channel, m2R = Hydraulic radius of channel,
m (PAR )S = Slope of the frictional gradient, m/mP = Wetted
perimeter, m4TheU.S. Bureauof Reclamation(1957)
publishedagooddescriptionof channels, with their suitable n value,
based on the work of Scobey. As shown below, this description gives
good information if n remains below about 0.030n = 0.012.For
surfaced, untreatedlumber flumes inexcellent condition; for short,
straight, smooth flumes of unpainted metal; for hand-poured
concrete of the highest gradeof workmanshipwithsurfaces as smoothas
atroweled sidewalk with masked expansion joints; practically no
moss, larvae, or gravel ravelings; alignment straight, tangents
connected with long radius curves; field conditions seldom make
this value applicable.n = 0.013Minimum conservative value of n for
the design of long flumes of all materials of quality described
under n = 0.012; provides for mild curvature or some sand; treated
wood stave flumes; covered flumes built of surfaced lumber, with
battens included in hydraulic computations and of high-class
workmanship; metal flumes painted and with dead smooth interiors;
concrete flumes with oiled forms, fins rubbed down with troweled
bottom; shot concrete if steel troweled; conduits to be this class
should probably attain n = 0.012 initially.n = 0.014Excellent value
for conservatively designed structures of wood, painted metal, or
concrete under usual conditions; cares for alignment about equal in
curve and tangent length; conforms to surfaces as left by
smooth-jointed forms or well-broomed shot concrete; will care for
slight algae growth or slight deposits of silt or slight
deterioration.n = 0.0155Rough, plank flumes of unsurfaced lumber
with curves made by short length, angular shifts; for metal flumes
with shallow compression member projecting into section but
otherwise of class n = 0.013; for construction with first-class
sides but roughly troweled bottom or for class n = 0.014
construction with noticeable silt or graveldeposits; value suitable
for use with muddy gravel deposits; value suitable for use with
muddy water for either poured or shot concrete; smooth concrete
that is seasonally roughened by larvae or algae growths take value
of n = 0.01 5 or higher; lowest value for highest class rubble and
concrete combination.n = 0.016Forliningmadewithrough
boardformsconveyingclearwaterwithsmall amount of debris; class n =
0.014 linings with reasonably heavy algae; or maximum larvae
growth; or large amounts of cobble detritus; or old linings
repaired with thin coat of cement mortar; or heavy lime
encrustations; earth channels in best possible conditions, with
slick deposit of silt, free of moss and nearly straight alignment;
true to grade and section; not to be used for design of earth
channels.n = 0.017For clear water on first-class bottom and
excellent rubble sides or smooth rockbottomandwoodenplanksides;
roughlycoated, pouredliningwith unevenexpansionjoints;
basicvaluefor shot concreteagainst smoothly trimmed earth base;
such a surface is distinctly rough and will scratch hand;
undulations of the order of 0.025 m.n = 0.018About the upper limit
for concrete construction in any workable condition; very rough
concrete with sharp curves and deposits of gravel and moss; minimum
design value for uniform rubble; or concrete sides and natural
channel bed; for volcanic ash soils with no vegetation; minimum
value for large high-class canals in very fine silt.6n = 0.020For
tuberculated iron; ruined masonry; well-constructed canals in firm
earth or fine packed gravel where velocities are such that the silt
may fill the interstices in the gravel; alignment straight, banks
clean; large canals of classes n = 0.0225.n = 0.0225For corrugated
pipe with hydraulic functions computed from minimum internal
diameter; average; well-constructedcanal inmaterial whichwill
eventually have a medium smooth bottom with graded gravel, grass on
the edges, and averagealignment withsilt depositsat bothsidesof
thebedandafew scattered stones in the middle; hardpan in good
condition; clay and lava-ash soil. For the largest of canals of
this type a value of n = 0.020 will be originally applicable.n =
0.025For canals where moss, dense grass near edges, or scattered
cobbles are noticeable. Earth channels with neglected maintenance
have this value and up; a good value for small head ditches serving
a couple of farms; for canals wholly in-cut and thus subject to
rolling debris; minimum value for rock-cut smoothed up with shot
concrete.n = 0.0275Cobble-bottom canals, typically occurring near
mouths of canyons; value only applicable where cobbles are graded
and well packed; can reach 0.040 for large boulders and heavy
sand.n = 0.030Canalswithheavygrowthof moss,
banksirregularandoverhangingwith
denserootlets;bottomcoveredwithlarge fragmentsof rockorbedbadly
pitted by erosion.7n = 0.035For medium large canals about 50
percent choked with moss growth and in bad order and regimen; small
channels with considerable variation in wetted cross section an
biennialmaintenance; for flood channels not continuously
maintained; for untouchedrockcutsandtunnelsbasedonpaper cross
section.n = 0.040For canals badly choked with moss, or heavy
growth; large canals in which large cobbles and boulders collect,
approaching a stream bed in character.n = 0.050 - 0.060Floodways
poorly maintained; canals two-thirds choked with vegetation.n =
0.060-0.240Floodways without channels through timber and
underbrush, hydraulic gradient 0.20 to 0.40 m per 1000 m.Artificial
channelsareusuallytakentohaveuniformsectional roughness, while the
roughness of natural channels is usually represented by a composite
sectional roughness.1.3 STEADY FLOW IN OPEN CHANNELSAll
computations will be based on the conservation of mass, momentum
and energy (in the form of Bernoulli's theorem), and the Mannings
formula for frictional resistance. Theassumptions,
methodsandformulasthat arepresent hereworkfor problems of water
flows on the usual engineering scale. The flows are long compared
to their cross-sections, so that a single velocity can describe the
situation adequately at any cross-section.81.4 ARBITRARY
SECTIONSThe sectional geometry of an artificial channel is in most
cases regular, while that of natural channelsvariesunder most
circumstances; thesefactors contributegreatly tothecharacteristics
of steady flowinbothtypes of channels.GivenS andQ, and the shape of
the channel, depth y and velocity V can be found; therefore given S
and y, we can findQ. If the given channel is one of arbitrary
shape,Q(total discharge) must be used instead ofq (discharge/unit
width), and y is usually the depth from the lowest point in the
channel. The areaAoccupied by fluid and the wetted perimeter P are
functions of y, andmay beexpressedanalytically for channels of any
geometric shape. The value of y corresponding to a givenQand S is
called the normal depth for those conditions.1.5 IMPORTANCE OF
SURFACE PROFILE COMPUTATIONSThis program can be put into diverse
use, typical instances where it can be applied include;i.Studies
involving dam break failuresii.Flood alleviation schemesiii
Elevations for;a. pumping plantsb. canal head worksc. Energy
dissipatersiv. Sediment transport estimationv.River
rehabilitation.vi.Tail water ratings for hydroelectric power
plantsvii.Grade lines fora. Highwaysb. Bridges and culverts1.6
DEFINITION OF TERMS 9In order to understand the meaning of some
important terms associated with steady flow in open channels, as
applied in this project, a brief discussion of these basic terms is
in order1.6.1 FROUDE NUMBER FrIt is defined as:Fr= V/(gDm)1/2The
knowledge of the Froude number of an open channel flow gives the
idea of the conditions of flow.When Fr>1, the resulting flow
condition is as follows: Supercritical flow Water velocity >
wave velocity Disturbances travel downstream Upstream water levels
are unaffected by downstream controlWhen Fr = 1 critical flow
resultsWhen Fr< 1, the resulting flow condition is as follows:
Sub-critical flow Water velocity < wave velocity
Disturbancesdont travel downstreamandupstreamwater levelsare
affected by downstream control1.6.2 CONTROL SECTION The control of
flow is the establishment of a definitive relationship between the
stage(depth) andthedischargeof theflow. Whenthecontrol of flowis
achieved at a section of the channel, this section said to be a
control section.Thecontrol sectioncontrolstheflowinsuchawaythat it
restrictsthe transmission of the effects of changes in flow
condition either in the upstream 10direction or in a downstream
direction, depending on the state of flow in the channel.1.6.3
PRISMATIC CHANNEL AND NON- PRISMATIC CHANNELSPrismatic channels are
channels that do not change in section longitudinally. The
computation of surface profiles in such channel is a function of
the stage (flow depth).Fabricated channels are mostly prismatic.
The non-prismatic channels on the other hand, are such channels
that vary in section longitudinally. The computation of backwater
in the non prismatic channels is a function of its sectional
dimensions. Natural courses arecategorizedas nonprismatic
channels.1.7 OBJECTIVE OF STUDYThis project is aimed at producing
software that can compute the depths of flow at selected sections
in an open channel and thus display the surface profile of the
channel of concern. This would be carried out with particular
emphasis on arbitrary sectioned and non prismatic open
channels.Theprogramwill bewrittenwiththeaidof MATLAB,
myreasonforthis selection being that;i, it is a proficient
programming environment, this allows for less voluminous programs
and thus enables faster development of the required program.ii, it
is an excellent means for production of programs where the display
of graphics is required.11CHAPTER 22.1 LITERATURE REVIEWThis
chapter contains an overview of background information and
scientific literature relevant to this research.2.2 BACKGROUND OF
BACKWATER ANALYSISThis aspect of hydraulics has been researched
upon by a lot of scientists in the past, this is evidently due to
the fact that it was discovered that a thorough insight of the
backwater phenomenon enabled a good prediction to be made of the
flow pattern in most open channels. The fact that a good knowledge
of the backwater equations serves as a great tool in the analysis
of gradually varied steady flow in open channels cannot therefore
be overemphasized. Many hydraulicians (mostly inthe18thcentury)
inthelikes of Bresse, Ruhlmann, Tolkmitts, Schaffernak, Francis,
Saint-venant, and so many others have contributed immensely to the
development of better techniques in the analysis of gradually
varied flow in open channels.The theories developed follow the
basic assumption that, the head loss at any particular section is
identical as for a uniform flow having identical flow velocity and
hydraulic radius as the section.The computation of backwater curves
based on this hypothesis has been proven satisfactory by many
experiments. The experimental verification indicates the validity
of the 12assumption for practical purposes and has been proven to
be a reliable basis for design.The basic assumption has been
extensively applied in the dynamic equation of gradually varied
flow. However, the computation of gradually varied flow profiles
involves basically the solution of the dynamic equation of
gradually varied flow. This equation here is developed and its
methods of application are also discussed. The solution methods of
the dynamic equation include;(i) Graphical integration method(ii)
Direct integration method(iii) Numerical methods2.3 ASSUMPTIONS FOR
GRADUALLY VARIED FLOWImportant assumptions made in the analysis of
gradually varying flow include: The flow is steady Streamlines are
virtually parallel ensuring hydrostatic pressures Friction losses
at a section for gradually varied flow is the same as those for
uniform flow Channel bed slope is small There is no air entrainment
Depth of flow is equivalent to depth of section The channel is
prismatic with constant alignment and slope Velocity distribution
coefficient equals one (1) The Mannings roughness coefficient is
independent of the depth or length of the channel considered.2.4
THE EQUATIONS OF GRADUALLY VARIED FLOWThe basic assumption in the
derivation of this equation is that the change in energy with
distance is equal to the friction losses.fSdxdH 13The Bernoulli
equation is:H zgVy + +22Differentiating and equating to the
frictionfSdxdzgVydxd
,_
+22orf oS SdxdE 2.1Where oS represents the bed slope andfS
represents the friction slope of the channel under considerationThe
equation below represents the change in specific energy with
depth2321 1 FrgAB QdydE 2.2Where B represents the surface width of
the channel under consideration;The combinations of equations 2.1
and 2.2 yields21 FrS Sdxdyf o 2.3This is the basic equation of
gradually varied flow. It describes how the depth,y, changes with
distance x, in terms of the bed slope,oSfriction slopefSand the
discharge, Q, and the channels shape (encompassed inFrand fS )2.5
METHODS OF ANALYSISAs already stated, therearethreemajor ways
inwhichthebackwater equation i.e. equation 2.3 can be analyzedThese
three methods will be discussed in more details,They include;14i)
Graphical integration methodii) Direct integration methodiii)
Numerical methods2.5.1 THE GRAPHICAL INTEGRATION METHODThis method
integrates the back water equation graphically. The method is
applicabletobothprismaticandnonprismaticchannels. Consideringtwo
channelsections at distances1L and2L respectively, from a chosen
origin andthecorrespondingdepthsof flowy1andy2, thedistancealongthe
channel floor can be given by 212 1yydydydxL L L2.41516practically
impossible. Many attempts have been made to solve the equation for
a few special cases or to introduce assumptions that make amenable
to 8mathematical integration. Most earlymethodsusedChezysformula,
but have been dropped in favor of the Mannings formula in more
recent times.Among the early contributors of Bakhmeteff (1932), who
carried out integration by short by short range steps with the
varied flow function. He considered the critical slope in the small
range of the varying depth in each reach to be constant. This is
true only for prismatic channels.Monodobe (1938) made an effort to
improve Bakhmeteffs method. He took the effects of velocity change
and friction head into account integrally without the necessity of
dividing the channelinto successive reaches. His method affords a
more direct and accurate computation procedure where by results
canbeobtainedwithout recoursetosuccessivesteps. Ademerit of this
methodliesinthedifficultyof usingaccompanyingcharts, whicharenot
sufficiently accurate for practical purpose.Later, Lee(1947)
andVonSeggern(1950) suggestednewassumptions, which resulted in more
satisfactory solutions. Von introduced a new varied flow function
in addition to that of Bakhmettef, hence, an additional table for
the new function is necessary in the method unlike that of
Lee.Combiningprecedingconsiderationsforadirectsolutiontothebackwater
equation, Vent TeChow(1955) cameupwithaprocedureforthedirect
solution of the equation.Nn ny C K12 2.5Ny C K122.6Mc cy C Z22
2.7My C Z222.8Where;17N and M are hydraulic exponents for critical
flow and uniform flow;1Cand 2Care coefficients.T A Z /3 2.9Z
representsthesectionfactorforflowcomputationfordischargeQat depth
y222ZZFrc2.1022KQSf 2.1122noKQS 2.1222KK SSn of2.13Krepresents the
conveyance for flow Q at depth yIf these expressions are
substituted into equation 2.3, the gradually varied flow equation
becomes( )( ) McNnoy yy ySdxdy/ 1/ 1 2.14Equation 2.14 can be
manipulated to yieldconst J v FNJyyN u F u A xMnc+11]1
,_
+ ) , ( ) , (2.15WhereonSyA 2.16NJyyBMnc
,_
2.17nyyu 2.19J Nu v/ 2.2011+ M NJ 2.2118uNuduN u F01) ,
(2.22vJvdvJ u F01) , (2.23Where ) , ( N u Fand ) , ( J v Fare
varied flow functions.According of him the length of flow profile
between two consecutive sections 1 and 2 is given as)]} , ( ) , ( [
)] , ( ) , ( [ ) {(1 2 1 2 1 2 2 1J v F J v F B N u F N u F u u A x
x L + 2.24The solution of the equation 2.24 can be simplified by
the use of the varied flow function table. The table gives us the
value of ) , ( N u Fand ) , ( J v Ffor given values of N .2.5.3
NUMERICAL METHODSAs earlier stated a direct solution of
differential equation 2.3 for all sections is practically
impossible. The solution method developed by Chow is difficult to
model for computer programming. At best direct integration is only
applicable to prismatic channels. However, numerical techniques are
available to proffer an approximate solution of acceptable
accuracy. By utilizing simple numerical methods all types of
gradually varied flow may be quickly and easily solved using a
microcomputer.When equation 2.3 is applied to prismatic channels,
the flow gradient is a function of the flow depth y, since the
section is uniform through out the length of the channel; the
equation is therefore in the form;) ( y fdxdy 2.25When the equation
is applied to a non prismatic channel, the flow gradient becomes a
function of x and y, since the channel width and flow depth varies
along the length of the channel. The equation is in the form;19) ,
( y x fdxdy 2.26In this research equation 2.26 is of special
concern.There are two basic numerical methods that can be used i.
Direct step method (distance from depth) ii. Standard step method
(depth from distance)2.5.3.1 THE DIRECT STEP METHOD (DISTANCE FROM
DEPTH)This method will calculate a distance for a given change in
the surface height. It involves the solving the reciprocal of
equation 2.32.5.3.2 THE STANDARD STEP METHOD (DEPTH FROM
DISTANCE)The method will calculate a depth for a given distance up
or down stream by solving equation 2.32.6 CLASSIFICATION OF
PROFILESBefore attempting to solve the backwater equation, a great
deal of insight is required of the profiles possible;For a given
discharge,fSand 2Fr are functions of depth( )3 / 10 3 / 4 2 2/ A P
Q n Sf2.27( )3 2 2/ gA B Q Fr 2.28A quick examination of these two
expressions reveal that they both decrease with increase in A i.e.
increase in yAlso for uniform flow,fS = oSandy =nySofS > oSwhen
y < nyfS < oSwhen y > nyAnd 202Fr >1 when y < cy2Fr
cyFrom these inequalities it can be seen how the sign of dx dy
/i.e. the surface slope changes for different slopes and Froude
numbers.Taking an example of a mild slope, treating the flow as to
be in three zonesi. Zone 1, above the normal depthii. Zone 2,
between the normal and critical depthiii. Zone 3, below critical
depthThe directions of the surface inclination may thus be
determinedZone 1y >ny>cyfS < oS2Fr < 1 dx dy /is
positive, surface risingZone 2ny > y > cyfS > oS2Fr < 1
dx dy /is negative, surface fallingZone 3ny > cy > yfS >
oS2Fr > 1 dx dy /is positive, surface risingTheconditionat
theboundaryof thegraduallyvariedflowmayalsobe determined in a
similar mannerZone 1As y thenfS and 2Fr0 and dx dy / oSHence water
surface is asymptotic to a horizontal line for its maximumAs yny
thenfS oS 0 and dx dy / 0Hence the water surface is asymptotic to
the line y = ny i.e. uniform flowZone 2As for zone 1 as y
approaches the normal depth:21As yny thenfS oS 0 and dx dy / 0Hence
the water surface is asymptotic to the line y = nyBut a problem
occurs when y approaches the critical depth:As y cy then 2Fr 1 and
dx dy / This is practically impossible but may be explained by
pointing out that in this region the gradually varied flow equation
is not applicable because at this point the fluid is in rapidly
flowing regime.Zone 3As for zone 2 a problem occurs when y
approaches the critical depth:As y cy then 2Fr 1 and dx dy / Again
the same physical impossibility holds with the same explanation and
again in reality a very steep surface slope will occur.As y0 then
dx dy / oS the slope of the bed of the channelThe gradually varied
flow does not hold here because it is clear what occursIn general,
normal depth is approached asymptotically and critical depth at
right angles to the channel bed. The possible surface profiles
within each zone can be drawn from the above considerations.All
possiblesurfaceprofilesforall thepossibleslopesof thechannel are
shown in the figure below.22Figure 123Figure 22.7 VELOCITY
DISTRIBUTION IN OPEN CHANNELS Themeasuredvelocityintheopenchannel
will alwaysvaryacrossthe channel cross section because of friction
along the boundary. Neither is the 24velocity axisymmetric (as in
pipe flow) due to the existence of the free surface. It might be
expected to find the maximum velocity at the free surface where
theshearforceisequal tozero, but thisisnot thecase. Themaximum
velocity is usually found just below the surface. The explanation
for this is the presenceofsecondary currents whichare
circulatingfrom theboundaries towards the section center and the
resistance of air/water interface. These have both been found in
both laboratory measurements and three dimensional (3D) numerical
simulations of turbulence.Thefigurebelowshowstypical
velocitydistributionacrossarectangular channel cross section.
Figure 325CHAPTER 3SOLUTION TECHNIQUES3.1 FOREWORDThis chapter
deals with the numerical methods employed for the computer
analysisof surfaceprofileof graduallyvariedflow.
Therearevarietiesof numerical methods for solvingthedifferential
equations, whichgoverns backwater flow. Each of these methods must
be critically studied in order to arrive at the selection of the
most suitable solution method.In addition, the numerical
computation of critical and normal depths is also discussed.
Thedeterminationof critical andnormal depthsservesasthe mean of
knowing when flow approaches the critical and normal depths. This
knowledge helps in recognizing the condition at which the gradually
varied flow equation becomes unstable as earlier pointed out in
chapter two, so that necessary steps may be taken to forestall
inaccurate computation.Computer analysis of backwater involves
adynamic exchangeof data resulting from different computations,
such that if these data are not properly handled it may lead to
accumulation of errors, which invariably will affect the final
output. It is therefore important at this juncture to highlight
various errors associated with the numerical solution of backwater
equation.3.2 INACCURACIES IN NUMERICAL ANALYSIS OF STEADY FLOW IN
OPEN-CHANNELFinal results of numerical computations of unknown
quantities are mostly by approximations. They are exact but involve
errors. Sources of errors include machine round-off, experimental,
truncating (local or global) and data-entry (human error).
Round-off errors result from the discarding of all decimals from
some decimal on, while the experimental errors are inherent in
given data. Truncating errors result from premature of breaking
off. This is common when double-precision data are treated as
single-precision data. Truncating error is 26local when it is
related to the values of successive steps and global when it is
related to the true value. It is also possible to make mistake by
supplying wrong data to the computer from the input device. This is
human error, and it is almost inevitable in time shearing
program.Furthermore, methods of computation adopted for the
analysis can as well contribute to the error, which exists in the
finalresults. Method associated errors must therefore be taken into
consideration in the selection of numerical methodof analysis
andalgorithm. Thefollowingaresomeerror terms associated with
numerical solutions: ABSOLUTE ERROR, |E0|:This refers to the
absolute value of errors that exists either between consecutive
terms or between a term and the exact value. Theknowledgeof
thelaterhelpsindeterminingacriterionforthe stopping of iteration,
especially when a method, which converges very slowly, is used.
ERROR OF THE APPROXIMATION, EO: This is the value of the difference
between the exact value and the approximate value. This is mostly
as a result of the approximate nature of the physicallaw or
principalupon which the solution is based. RELATIVE ERROR, RO: It
is the error derived from dividing the error of the approximation
by the actual value. Relative error gives the percentage error,
andcanbeadaptedtodeterminethenumber of significant digitsinthe
approximation. ERRORDUETOLOSSOFSIGNIFICANCE:Therecanbealoss of
significant digits due to round-off error. In this case the
function value p(xk) and the correction term p(xk) / p(xk)in
Newtons method can failto have enough accuracy to make xk+1closer
to the root than xk.27 ERROR DUE TO INSTABILITY: A problem is said
to be unstable when small changes in a parameter results in large
changes in the computed root.This instability relates to the
particular way in which the calculation is made. In essence it is
dependent on the stability of the numerical method and algorithm
adopted.In the case of algorithm stability an algorithm is admitted
to be stable when it terminates after a number of sequences of
operation, otherwise it is unstable. The general idea is that the
accuracy of the answer increase with the number of steps
performed.In the case of the stability of numerical methods, it is
important to note that there exist many stable problems that have
unstable numerical solutions. This arises from a solution involving
auxiliary intermediate number calculated for distance or depth, as
the case may serve, which become input numbers for subsequent steps
in the calculation. 3.3 NUMERICAL SOLUTIONTECHNIQUES FOR
FIRST-ORDER DIFFERENTIAL EQUATIONTheequationthat
governstheanalysisof graduallyvaryingflowinnon-prismatic open
channels is a first-order differential equation, as shown before.
It is of the form:) , ( y x fdxdyOwing to the fact that the channel
properties (geometry and roughness) and
flowconditionsareexpectedtovaryalongthechannels, adirect model
solutionfor thisequationisdifficult. However, thereaboundanumber of
numerical methods, which evolve fairly accurate solutions. These
techniques include:I. Taylors Method,28II. Eulers Method,III. Heuns
MethodIV. Runge-Kutta Methods, andV. Predictor-corrector
Methods.One of these methods is required to develop a model
computer program for the solution of the differential equation.
Since each of these methods has its peculiar weaknesses and merits
as regards the present area of application, there is need for a
critical analysis of these methods. This analysis must be based on
the following key considerations:i. Computational efforts required
for each methodii. The desired precision of computationiii Program
run-time, which also bears adequate check on speed of
convergenceiv. Applicability of the method of non-prismatic
channels of varying characteristicsv. Stability of solution method
vi. Compatibility of the method and its algorithm with other parts
of the same program3.4 ANALYSIS OF SOLUTION TECHNIQUESThe purpose
of this analysis is to reveal the merits and the demerits of each
numerical method listed above as regards arriving at a satisfactory
solution of the gradually varied flow differentialequation. This
willresult in the proper selection of the solution method to
apply.3.4.1 TAYLORS METHODThe method of Taylor is of general
applicability and it gives a way to compare the accuracy of various
numerical methods for solving Initial Value Problem (I.V.P.). It is
of the form:y(x+h) = y(x) + y (x)h + y(x)h2 / 2! + y(x)h3 / 3!
+Fornumerical purposesafiniteseriesof orderNisfixedtoapproximate
y(x+h). This gives:29y(x+h) = y(x) + y (x)h + y(x)h2 / 2! +y(N)
(x)hN /N!The method can be constructed to have a high degree of
accuracy, depending on the value of N. if the order N is fixed; it
is possible to determine athestepsizeh, sothat theglobal
truncatingerror will beassmall as desired.However,an
N-ordersolution requires anN-degree ofdifferential equation. In the
present case it is difficult to have a general model solution when
N>1. Therefore, for this method to be of any use here, it is
required that only the first two terms where N 0.02yDOWNSTREAM
FINAL LOCATION REACHED?DISPLAY PROFILEDISPLAY PROFILEPLOT CROSS
SECTIONDISPLAY NUMERICALLY AND GRAPHICALLY yn &
ycSTOPSTARTNONOYESSTOPFIGURE 5. ALGORITHM FOR PRISMATIC
CHANNELSNOYES44STARTINPUT X & Y COORDINATES FOR SECTION 1PLOT
CROSS SECTION 1INPUT X & Y COORDINATES FOR SECTION 2PLOT CROSS
SECTION 2INPUT X & Y COORDINATES FOR SECTION 3PLOT CROSS
SECTION 3COMPUTE yn+1 = yn +hf(x, y)COMPUTE Sm and FmCOMPUTE
y(n+1)m = yn +hf(xm, ym)xm= (xn+ xn+1)/2ym= (yn+ yn+1)/2|y(n+1)m+1-
y(n+1)m |< eCOMPUTATION BETWEEN SECTION 1& 2SET FINAL
y(n+1)m =ynYESNOYESFIGURE 6. ALGORITHM FOR NON-PRISMATIC
CHANNELSCHAPTER 5SELECTED EXAMPLES45PLOT PROFILESTOPCOMPUTATION
BETWEEN SECTION 2 & 3NOYESFigure 7: The MATLAB command
window46Figure 8:The MATLABEditor windowFigure 9: Open Channel
Application GUI (Graphical user interface)QUESTIONS471) Plotsuper
critical flow and sub critical flow profiles (below normal depth)
of a prismatic channel with coordinates
(0,2),(0.5,0.75),(1,0.5),(1.5,0),(3.5,0),(4,0.5),(4.5,0.75),(5,2);Flow
rate =8 m3-/s, bed slope=0.001, Mannings roughness
Coefficient=0.03Also detect the normal and critical depths for the
flow. AnswersNormal Depth=1.9243m Critical Depth=0.96615m 48Figure
10: Profile plot for super critical flowFigure 11: Profile plot for
sub critical flow (below normal depth)2) Plot a sub critical flow
profile (above normal depth) of a prismatic channelwith coordinates
(0,3),(0.5,0.75),(1,0.5),(1.5,0),(3.5,0),(4,0.5),(4.5,0.75),(5,3);Flow
rate =10 m3/s, bed slope=0.001, Mannings roughness
Coefficient=0.03Also detect the normal and critical depths for the
flow49AnswerNormal Depth=2.2658m Critical Depth=1.0835mFigure 12:
Profile plot for sub critical flow (above normal depth)3)
Computation Time TestsComparecomputationtimesfor Euler,
HeunsandRungeKuttamethodsusinga prismatic channel with the
following co-ordinates. (0, 6), (3, 0), (5, 0), (8, 6).Flow rate=
20m3/s Slope = 0.001Mannings roughness=0.02y (1) = 3.2mx (1) = 0mx
(last) =3000mAnswers:50FOR RUNGE KUTTAN = 21y (last) =
4.67771mRuntime = 0.015625secsFOR HEUNSN = 559y (last) =
4.67771mRuntime = 0.078125secs51FOR EULERN = 350000Y (last)
4.67771mRuntime =
7323.32secsOBSERVATIONTheRungeKuttamethodconverges after 20steps
withacomputationtimeof 0.015625 seconds.The Heuns method converges
after 558 steps with a computation time of 0.078125 seconds.The
Euler method converges after 349999 steps with a computation time
of 7323.32seconds.From the above stated results it is easy to
conclude that the Runge Kutta method offers thebest meansfor
thecomputationof flowprofilesinprismaticchannels; asit
convergeswiththeleast numberof stepsforagivenreachunderthesameflow
conditions with the shortest computation time.524) Compute the flow
profile for a non prismatic channel whose cross sectional
co-ordinates at 5 reaches are given below, using Euler, Heuns and
Runge Kutta methods and compare the results obtained.Data for the
channel properties are stated below;CO-ORDINATESReach 1 X Y0 1550
1050 3150 0250 0600 31000 102500 15Reach 2 X Y0 1440 1055 3120 0500
0600 31000 102500 14Reach 3 X Y0 1545 1042 2120 0250 0500 21200
102400 15Reach 4 X Y0 1545 1045 2120 0250 0500 21200 102400 15Reach
5 X Y0 1545 1045 3120 0250 0500 31000 102500 15n(1) = 0.02 So(1) =
0.001 h(1) = 100m53n(2) = 0.02 So(2) = 0.001 h(2) = 100mn(3) =
0.025 So(3) = 0.001 h(3) = 100mn(4) = 0.026 So(4) = 0.001 h(4) =
100mn(5) = 0.026 So(5) = 0.001 h(5) = 100mProfile obtained using
Euler Method y end = 10.2674 m x end = 400mProfile obtained using
Heuns Method y end = 10.2637 m x end = 400m54Profile obtained using
Runge-Kutta Method y end = 10.2636 m x end = 400mComparison of
ResultsThis would be difficult to carry out because it would
involve the inputting of a very large number of reach data, which
the written program cannot handle.55CONCLUSION AND
RECOMMENDATIONCONCLUSIONComputer analysis of the surface profile of
steady flow in open-channels has been the main focus in this work.
A computer program has been developed to compute and plot backwater
curves using MATLAB. Unquestionably, writing a
precision-sensitiveprogram ofthissort is ratherchallenging. On the
other hand, the hurdles encountered in the process of creating the
program have increased my interests in programming and hydraulics
in general.The major difficulty of the project lies in the proper
arrangement and handling of the geometric properties of the channel
sections to achieve programmable computations. For computational
purposes natural courses are represented approximately with smooth
side lines of trapezoids and the irregularities of sectional shapes
are compensated for by adequate selection of Mannings roughness
coefficient. The program developedcan handleirregular (prismatic
andnonprismatic sections (linearly and gradually varying)) channels
provided horizontal coincidingYcoordinatesandequivalent
Manningsroughnesscoefficients can be provided.The fourth order
Runge Kutta method provides accuracy sufficient enough for this
project (prismatic channels, page 50).56MATLABenabled a
comparatively easier development of the required software thanks to
more sophisticated functions like the solve function which was very
useful in the computation of normal and critical depths.Finally,
this project has proven that the computation of the surface profile
of flow from a reach of known depth in both prismatic and non
prismatic sections can be done with good speed, ease and precision
with the aid of a computer program that is executable on a
micro-computer.RECOMMENDATIONThis research deserves more probing,
there is still a lot more to be done on it.If given the opportunity
I would love to explore all the possible conditions and develop
software that would be capable of handling these
possibilities.Datafeedingwasreducedtotheminimum,
futureresearchcanhelpin discovering better ways of entering data
(e.g. through satellite connections) to provide more a flexible and
efficient programIt is a project that is worthy of government
sponsorship, as stated earlier on in the write up it is useful in
flood alleviation planning which is a major problem we face in
various locations of the country, development of software of this
nature will go along way in contributing to solving of this
problem.57BIBLIOGRAPHY Aiyesimoju.K.O.: River Engineering Lecture
Notes, CEG 519, University of Lagos, Akoka, Yaba, 2006. Boolos,
George and Jeffrey, Richard: First Edition, Computability and
Logic,Cambridge University Press, London, 1999, 262 pages. Boris A.
Bakhmeteff: Hydraulics of Open Channels, McGraw Hill Book Company,
Inc., New York, 1932. pp 143-215 Douglas J. F. John Gasiorek, John
Swaffield, 5th Edition, Fluid Mechanics, Prentice Hall, Upper
Saddle River, New Jersey, (9 Nov, 2005), 992 pages. Erwin Kreyszig
8th Edition Advanced Engineering Mathematics, John Wiley &
Sons, Inc. Hoboken, New Jersey, (11 march, 1999), 1296 pages Irving
H. Shames: 4th Edition, Mechanics of Fluid McGraw Hill Book
Company, Inc., New York, 08/02/2002, 896 pages. J.A. Ch Bresse:
Cours de mecainque appliqu, 2e partie, Hydraulique, (Course in
Applied Mechanics, pt. 2, Hydraulics), Mallet- Bachelier, Paris,
1860. Larry W:Mays First Edition, Water Resources Engineering John
Wiley & Sons, Inc. Hoboken, New Jersey, (11 Aug, 2000) 780
pages. Markov A.A.: Theory of Algorithms.[Translated by Jacques J.
Schorr-Kon and PST staff] Imprint Moscow, Academy of Sciences of
the USSR, 1954, 444 pages. MATLAB Release 14, Help Files, M. E. Von
Seggern: Integrating the equations of non uniform flow,
Transactions, Proceedings American Society of Civil Engineers, vol
115. pp 71-88, 1950. Microsoft Inc. Microsoft Encarta 2007, Ming
Lee: Steady gradually varied flow in uniform channels on mild
slopes, Ph. D. thesis, University of Illinois, Urbana, 1947.58
Nagaho Monodobe: Back-water and Draw drop-down curves for uniform
channels, Transactions, American Society of Civil Engineers, vol
103. pp 950-980, 1938 Norman Bruton Webber: Metric 2 Revised
Edition, Fluid Mechanics for Civil Engineers, Spon Press, London,
U.K. (1971) 372 pages. Stroud K.A. Dexter J. Booth: 4thRevised
Edition Advanced Engineering Mathematics,Industrial Press, Inc.
U.K. (April 2003) 1280 pages. Tolkmitt G.: Grundlagender
Wasserbaukunst, ( Fundamentals of Hydrualic Engineering,), Ernst
& Sohn, Berlin, 1898. Ven Te Chow: international Edition,
Open-Channel Hydraulics Mc-Graw Hill Education, Colombus, Ohio (31
Dec 1959) 692 pages. http://en.wikipedia.org/wiki/Algorithm
examples APPENDIXPROGRAMLISTING5960
