Uniform flow approximation for flood discharge

Uniform flow approximation for flood discharge


Very good morning to everyone, we are back
into our lecture series on advanced hydraulics. We are for last few classes going through
the module two on uniform flows. In the last class, we discussed on the channels having
compound sections, how the compound section was defined, how to evaluate mean flow velocity
for compound sections; that was also discussed in the last class. We also suggested on how
to evaluate normal slope, suppose if normal, if discharge and normal depth if it is given
for a channel, how to evaluate the normal slope of the channel, that was also discussed. Today, we will see on how to compute flood
discharge, using uniform flow approximation. To compute flood discharge, using uniform
flow approximation. So let us just go back, before going into the concept what is meant
by flood discharge and all, and you have to use the uniform flow approximation and all.
Let us recall in the uniform flow computations and all, what are the various variables, or
what are the various things you have gone through. If you want to use manning’s uniform
flow approximation and all, what are the various quantities in those equations? I will just
briefly enumerate them, you have already seen normal discharge, we gave it as symbolically
Q. We have seen the uniform flow velocity; we gave it as v bar. We have seen the concept
called normal depth y n, manning’s coefficient of roughness. In the manning situation, you
have also seen the channels slope, or bed slope, or energy slope, coming into picture
s naught, or s f coming into the picture, the geometric shape of the channel, that give
rise to cross sectional area A, hydraulic radius R etcetera.
So, if these parameters, if these six quantities are related in manning’s equation for uniform
flow computation, so for example, if you are given the, all other parameters, say normal
depth y n, coefficient of roughness n, channels slope s naught, and the geometric shape of
the channel a and r. If they are provided to you, you can easily compute the normal
uniform flow velocity, as well as the normal discharge Q in the channel. Similarly, if
normal discharge Q, if it is given to you, you can and if the other parameters normal
depth, coefficient of roughness, and geometry of the channel if they are also provided,
you can easily compute the normal slope of the channel. So these, some of the problems
related to them also we have computed and all. So you have to recall them, you have
to use those concepts. Now, to evaluate flood discharge, or to compute
flood discharge, to compute flood discharge in a channel, what do you mean by that, what
do you mean by flood discharge in a channel? You have to take a considerable reach of the
channel. Earlier we were dealing only with one cross sectional area of the channel. Now
you may need to take the considerable reach of the channel, and along with reach of the
channel, where ever it is flooded, or where ever it is over flowed, although all those
quantities how you can measure the corresponding discharge and all, you need to take into account.
Going back into the uniform flow computation, you have seen that.
Generally for uniform flow, your bed slope, your slope of the water surface, and the energy
slope, all of them are equal. Approximation is quite valid for general shift channels,
or even for laboratory experiment channels and all. However if you go through the natural
streams rivers and all, you may seldom see, where all this parameters are equal, means
exactly equal. To approximate uniform flow in those natural streams, or rivers and all,
you need to approximate, you have to suggest that your bed slope is approximately equal
to you energy slope, or like that or it is approximately equal to the water surface slope
and all; like that you need to use certain approximation, then only the uniform flow
approximations can be employed for flow natural So as you have seen in compound section, compound
sections channels and all, most of the natural streams may be having compound sections. So
the velocity, so the velocity pattern along with the same cross section of the channel,
it may not be uniform, it may be having different winds, along the flood plains the velocity
may be quite different, along the main driver channel, or main channel of the carrier channel,
main portion of main body of the channel, it may be having different velocity, so that
we have seen in the last class, we have already gone through them. So you have to now think
on certain approximation here, as velocity is may be rarely equal along compound sections,
along the same cross sections in a compound section and all.
You may have to give some approximation, certain approximation. Definitely the computation
of energy is low, means if an a if along a cross section veloc… along a cross section
of compound section, are rarely equal; therefore, energy slope may not be same as bed slope,
it is quite possible in natural stream. Then you need to incorporate correction factors,
or we have seen certain things and all, even if you do that, you will see the energy slope
and bed slope, they are somewhat not exactly equal and all, so you need to device certain
method. So here comes the portion, rather than taking only one section of the channel.
For computing flood discharge now, you have to think in terms of the stretch, or reach
of the channel, along which the flooding is getting occurred.
See if you have say a channel reach like this, so it is cross section may be like this, and
it is flooding along these portions also, let us approximate it like this. So in such
a type channel portion, along the flood plains, or along the over side channel portion channel
and all, you may have to incorporate difference velocity, in the main body it will be different
velocity. Subsequently along this entire reach, how much quantity, how much discharge is available
from the flood, flood plain, as well as along the main body; the combined discharge and
all, how to evaluate them. You are going to take the stretch or reach of the channel;
you have to do certain averaging along the reach, and then try to evaluate the discharge.
Let us go through them. So for such type of figure, you may see that,
you require now to take energy slope, rather than the bed slope. You need to think in terms
of energy slope first, because the energy parameters energy head and all, it is quite
different in the upstream. For example, here this is the upstream portion,
and this is the downstream portion. So the energy head, the difference in the energy
head, divided by the length of the reach; that will give you the energy slope, the bed
slope correspondingly. And if they are not equal, you have to first evaluate the energy
slope; you have to think in terms of energy slope here first. So the difference between
the total heads at the, you may calculate the total head at the upstream, you may calculate
the total head at the downstream, subsequently the difference in head divided by. So, difference
in total head at upstream and downstream, divided by length of reach. This will give
you the energy slope, I hope you know what is mean by total energy head; that is it is
consisting of pressure head, section head or datum head, what you say is datum head,
as well as pressure head, datum head and velocity head. So, most of the uniform flow computation,
you were able to avoid the energy, sorry velocity head to compute the energy head, because then
we know velocity in most of the cases if it is a regular section and all, they are quite
equal. And the magnitude of velocity head at upstream
and downstream, they would have been almost same, and it may not be creating any change
in the energy slope; that is not going to cause you any change in the energy slope.
However, for such compound sections, or such flood discharge flood causing channels and
all, you may need to take into account the total energy at the upstream, total energy
at the downstream, you have to evaluate the energy slope first, then you have to compute
the corresponding discharge. So, here then you will see that, during the flood when it
enters the flood plain and all, the velocity will be drastically changed along the flood
plain, in the main body it will be different. So the flood stage and the discharge, they
may be… if you want to if you are, if you are, if you are able to correlate the stage
in the flood plains; that is a flood stage, and corresponding discharge.
So if you are able to correlate flood stage and corresponding discharge, and if they are
having some gradual changes, or gradual changes and all if they are there; that is if the
flood stage if it is changing gradually, then the discharge is also, flood discharge along
the flood plains; that is also changing gradually, or like that if you are able to correlate
them. Then you will be able to give uniform flow approximation for the flood discharge
also. Otherwise for natural streams, where ever flood is occurring flood is components
are there. It is quite difficult to give uniform flow approximation, so this is quite essential.
That is this portion suggesting that, you have to correlate flood stage, and it is corresponding
discharge, if you are able to do that, then that is fine. Then uniform flow approximations
hold good. So what are some of the methods which you can use, how you can see. So mostly
for such this thing, when we suggested that, if there is a gradual change in flood with
respect to the flood discharge, with respect to the stage of the flood and all. You are required to have historical note on
flood marks in the channel. So, various historical, no means you have to give elaborate, you have
to note the various flood marks in the channel and all. So at the during the peak rainy season,
and during the peak flow of the river, this was the mark of the flood at that time, during
the lean period this is the mark, or means at various situations and all, what are the
different flood marks and all, you can just either capture them in photograph, or you
can note it, you do the survey related to the marks and all, you can just obtain it
as a data. So once you obtain the corresponding data, then you can use; say for example, slope
area method, which we will see today, to evaluate flood discharge. Then there is another method
called, contracted opening method. So here as we mentioned in the earlier portion; so
in such a channel portion such a type of channel listing, in the upstream and downstream. The
upstream and downstream, you are directly applying the energy equation. Here in the
contracted opening method, in the downstream and all, principle of energy conservation
and all is being directly employed, and you are trying to evaluate the corresponding discharge,
so that we are not going to see in today’s lecture and all, definitely we are going to
explain you on the slope area method. So, the thing is that, we require information
on, for this slope area method. We first require information on the energy slope; we require
certain informations, so that is energy slope in the reach. You may also require, or your
average cross sectional area of the reach, length of the reach, roughness coefficients.
So, like this some of the parameters are required, so we will just briefly see, what is the thing? Say, if the channel reaches like this, the
upstream portion downstream portion, length of the reach is l; length of the reach. Then
the methodology involves, you have to understand, which is first identified, which is upstream
portion, and which is downstream portion, that you need to understand first, by seeing
the direction of flow and all, based on that, you have to now first evaluate. So the procedure
for the slope area method it in was the first step is, identify area of cross section, both
at upstream and downstream. So in the first method, you will be identifying areas of the
cross section, not only that you will be identifying areas of cross section at upstream, area of
cross section at downstream, you are able to, you have to evaluate
the hydraulic radius at upstream, hydraulic
radius at downstream. You can see if manning’s coefficient if it is same at upstream and
downstream, well good you can take it like that n itself, or you can suggest n at upstream
and n at downstream, like this you can suggest them. Now using these parameters what you
have to do is that, you have to compute conveyance factor, both at upstream and downstream. The
conveyance factors at upstream and downstream, you need to evaluate them. I hope you are aware how to obtain conveyance
factor, this is 1 by n AR to the power of 2 by 3. So, for the upstream common section
or simple section, what will be, you evaluate the corresponding conveyance factor, in the
downstream of the reach what is the conveyance factor ,both you need to take into account.
Now for the entire reach, you are. The second step is, you are now obtaining average conveyance
factor, for the entire reach. So this average conveyance factor for the entire reach, it
can be given as k average, this is nothing but, the geometric mean of the conveyance
factor at upstream and downstream respectively. This is the form of formula for geometric
mean, so that has been directly employed here. So, once you get the average conveyance factor,
the third steps involved is, you have to first assume; that is, in this method first you
are assuming, velocity head is not having prominent role in energy head computation,
just it does not brief assumption, initial assumption, or you can as you might have seen
in solution of various differential equations, with respect to time and all, initial conditions
or like that. So this just a initial approximation, we are suggesting that, velocity head initially
we are assuming that, it is not having prominent role in the computation of energy head, or
rather then that you can see that, it is not having prominent role in computation of energy
slope, so that will be better terminology, not having in energy slope, this is a initial
assumption. In that way you can find slope of energy s f, this is nothing but equal to
del y by l, I will tell you what is mean by del y by l. So, the reach, if I just draw the front view
of the thing, and if this is the channel distinct; say this is the upstream section, this is
the downstream section. Now whatever drop in water surface is there, we are now approximating
that. You know that for uniform flow, water surface slope, bed slope, or energy slope,
all of them are approximately equal in general channels, or in regular shape channels and
all. Right now we have come into dealing with the natural streams, where flood discharges
occur. So there is a difference in level of water surface at the upstream and downstream,
that difference is given by del y. So this del y by l, whatever is there, we can initially
approximate this as energy slope; that is not correct, but that is the initial
approximation, we are suggesting that, the component of velocity head in this case is
very negligible, in determining the energy slope; therefore, the energy slope s f is
given by del y by l. If this is the case, del y is equal to fall in water surface, from
upstream to downstream in the reach. Now based on this particular value of slope,
based of this particular value of slope based on this particular value of slope you can
evaluate, and you give a first approximation for the discharge in the entire channel q.
The discharge in the entire channel reach, from the entire channel reach the discharge,
it can be given as Q is equal to k average into s f to the power of half or root of s
f. So please note that, we had given an approximation for the energy slope, and you are substituting
it. So let me give this as a first approximation, and this is not the correct discharge actually
from the, correct flood discharge from the channel, this is just a first approximation.
Now using this value of q, you can easily now evaluate what are the velocities at the
upstream and downstream; that is called possible. Uniform flow means generally the discharge
is same at upstream and downstream, so you can easily evaluate. So the fourth step is, using the first approximation
of discharge. Now evaluate velocities at upstream and downstream, you evaluate them, as shown
in the figure here, at the upstream, and at the downstream, whatever are the velocities,
how will you compute them, you know the formula for that. So Q is equal to a v, or v is equal
to Q by a, directly substitute them. So we will get, at upstream location, you will get
the corresponding velocity v u s, at the downstream section you will get the corresponding velocity
v d s, based on this you can easily now compute. Velocity heads at upstream and downstream,
just compute them. For example, if alpha of u s is energy head correction factor; that
is energy correction factor, for velocity at upstream section. Similarly, alpha d s
energy correction factor at downstream. So if you have this data, then you can easily
evaluate. So at upstream the section, the velocity head
will be now alpha u s v u s square by 2 g. Similarly, at downstream section, the corresponding,
section corresponding head will be alpha d s v d s square by 2 g. So once you have these
quantities, now you require to use these two heads, in the energy correction factor, in
the energy slope. You have already determine the energy slope earlier, so that was the
first step approximation. Now you know, dropping water surface; that is a measured quantity,
so that is not going to change. Therefore, energy slope for the entire reach,
can be given as s f, this is equal to the change, in this change in energy head h f,
drop or drop in energy head, total energy head h f by the length of the channel. This
h f includes now, here drop in water surface, plus a factor small k into alpha u s v u s
square divided by 2 g minus v d s square by 2 g. Once you obtained this corresponding
form of the energy drop, this is the energy drop in the entire channel reach, and head
of energy drop in the entire channel reach. This k factor, it is an empirical, if there
is obtain form empirical this thing and various experimentation and all, its scientist has
suggested that, k can be give as one, if your channel reach is contracting. If your channel
reach is contracting that means, v upstream is less then v downstream, if you channel
reach is contracting, in that case you can give k is equal to one. If v upstream is greater
than v downstream, channel is expanding you have to take k is equal to 0.5. This also
there many scientist experimentally obtain those things and all, or experimentally doing
some analytic and all, so we will just incorporate them in our analysis here. So once you get thing quantity, now you get
a, you are now getting a modified energy slope, as mentioned earlier. So s f, let me give
this as 1. This is equal to s f of 1 is equal to now del y plus k into alpha upstream v
upstream square by. Based on this, you can evaluate the modified discharge, or you can
give a new value for discharge, Q is now equal to k root s f; that has been modified. So
this you can give as the second approximation for discharge. Once you get this second approximation
for discharge, again go back. You just go back using, so using now Q 2 evaluate
v upstream, and that also you can give it as two v downstream, to evaluate s f 2. Now
so once you get s f 2, the same procedure as I adopted. You can compute Q 3, this is
nothing but, ketians root of s f 2. So once you get Q 3 go back, again evaluate, just
check it that is we are going back again, because we are finding that Q 2 and Q 3 they
are not same. Similarly, Q in the first approximation, Q in the second approximation obtain, they
are not same. So we are going back like this, till two of the conjugative approximations,
are giving you the same discharge. So evaluate like this, till say Q in the i eth approximation,
and Q in the i minus one th approximation are same. So once you get this thing, the
corresponding values of s f has to be noted, the corresponding values of v in the upstream,
and v in the downstream, those things also have to be noted, and now you have got the
discharge, flood discharge. So this is how you compute, means this is how you compute
the flood discharge, using uniform flow approximation for the natural streams. So as I mentioned earlier, you are dealing
with the entire channel reach. So if you are having many such reaches, if there are many
such reaches of different lengths; say this is L 1 this is L 2 L 3 L 4 L 5 etcetera, and
like this you have many reaches. Then for the entire natural stream, you can now average
the discharges. So using the slope area discharge method, you have evaluated the corresponding
Q form this reach. Similarly you can give it as Q 1, what is the slope using slope area
method, what is the average discharge form this thing Q 2, Q it is not average, it is
the slope area method by approximation, means by iterating you have arrived at Q 2 as the
discharge form this reach, Q 3 also discharge from this reach, like that several reaches
are there. You can average the discharges, for the entire length of stream; that is having
different reaches. like this you can submit for the entire this thing, and you can just
average it and get the average discharge for the entire natural stream; that is the why
you can compute you can use appropriate weights also to compute the discharge. Next we will just go through what is meant
by, uniform surface flow. You have heard the word uniform surface flow, or especially during
the rainy season and all, you may see various, mean in on the overland flow of water occurs
form the overland, in the form of thin sheet. Means the depth of flow will be considerable
small, but still there is some flow of water in a sheet form, through the surface, whether
it will be the same, says if whether it will be like this along the sheet, it may be going
like these and all, various type of flow you might have observed it in nature. You can
even give approximations for overland flow, through uniform flow computations; that is
also quite possible. Say this is your land surface, so before the water reaches, when
the rain fall of occurs, afer some quantity getting infilterated, and before it reaches
the main channel, or befere it reaches any channel, it flows along the ground in a form
of thin sheet. So, this depth, it is quite small, so let
me say this depth as y m, sums very small depth y m. And it may not go beyond this depth
also, mostly it is a very thin this thing. So in this case, if you observe that, again
your basic furead mechanism principles and all are coming into picture, the depth of
flow is small of course. So it may be having some velocity distibution of this form, with
respect to depth. You will observe that, for such type of flow, friction or the resistance
due to viscosity and all, is prominent in such type of flow. So viscosity is important
in overland flow. So if viscosity is important, how will you compute the uniform flow now.
So you have to use the newtons law of viscosity, i hope you know what is newtons law of viscosity. We suggest that, the shear stress along with
the bed of the channel, or shears stress along with the bed of the pipe, is nothing but praportional
to the gradient of velocity, and the prapostionality constant is called, a dynamic coefficient
of viscosity view. So the same thing we need to apply here, so therefore let me just give
the overland flow in a magnefied form. Here this is your depth y m, so it is having a
velocity distirbution. So in such cases, just take a small elementry strip from the top
surface, from the top surface of water, you just take the small elementry strip, and this
strip is at height y from the bottom of the ground. Now in this strip, whatever is there,
means you will see, that what happenes is there, for such type of overland flow, viscosity
is from suggested that, so there will be a considerable amount of friction, that causes,
that friction force will be acting in the apposite direction, and that causes, or that
opposes the motion of flow in the down stream direction. So, how the viscus flow affects
these things, should take this elementry strip, it has its weight acting down wards.
So the component in the flow direction, whatever is there. Now that is now being componsated,
or you can suggest that is being now delt with the frictional force, along the bed,
so you have to equate it now. So what is the frictional force along this direction now,
what is a frictional force. You will see that, the frictional force here. Please note that,
we are taking laminar flow approximation. So the frictional force per unit area along
the perpandicular direction of the screen here given to you per unit area, whatever
frictional force if it is there. Let us consider that, as your shear streff. You know that
force per area, gives you the stress component in many of the situations. So are stress having
the units of force per area, so let us give that as shears stress. This is nothing but,
rho g into, see what is the thing here. The depth here is y m minus y, and you know that
rho g, is a weight of the liquid in this thing, within this particular depth. So rho g y m
minus y into the slope, means you need to take into account the slope here. So we are
equating the component of the gravity force in this direction, with respect to the friction
force; that is what we are doing it here, that can be given it like this. So this is nothing but, if you again go through
the same in equation rho g y m minus y s naught this is equal to you use the newtons law of
viscosity. You are equating with with respect to newtons law of vescosity. From this equation,
just rearrange the terms so d v by d y is nothing but equal to rho g s naught by mu
y m minus y, or you can use this relationship now to obtain the intergril, say v the velocity
in that portion of the small, portion of that overland flow and the. It is nothing but integral
row g s naught by mu y m minus y into d y, you will see that all these contities are
concerned rho g s naught by and mu are constant with respect to y, you are going to get this
equation as v is equal to row g s naught by mu y into y m minus y square by 2. So, you
are getting a quadratic expression for the velocity, in the overland flow. You can now
compute the average flow, this is with respect to any height y. So you can compute average velocity, at any
section, for overland flow, can give this as v bar, this is 1 by y m, is the depth of
the overland flow, v d y. You just subtitute the quantity of v whichever we are given it
here, you will get this as rho g s naught by three times mu into y m square. So we are
this portions and all, we have refered the text from venticher on open channel hydraulics,
so this average overland flow velocity is computed in this way. So this is as good as
your uniform flow, means this velocity is same as uniform flow. You can also suggest
that the discharge per unit width; discharge, this is velocity, so discharge per unit width,
in the uniform for the overland flow, you can give this as equal to q, and this is given
as some coefficient C L into y m cube isent it. What we have done, we have just C L into
y m cube, because here this is velocity, and velocity into the width, means which ever
we discharge your talking about discharge per unit width, so you have to take the sections
appropriately, rather than unit area, you are taking unit width, so C L into y m cube,
it will be a third degree equation with respect to your depth of overland flow. So your C
L is nothing but, the coefficient seeing coefficient rho g s naught by 3 mu, it may be obsrerved
like that also, you may see such phenomanaon and all. Like this way, you can compute uniform
flow for overland flow cases also. So today, we will like to stop it here, as
a portion of interest or curieousity, I can just ask you one question. You have seen in
road sections, you have seen in various road section, roads and all. So the road is having
a gradual slope, and at the two end portions of the road; say this is your road, and at
the two end portions, you may see small gutters on the road, at the this portion, and at this
portion, like this you may see gutters in the road. Now you can use your uniform flow
approximation to compute flow along this gutter; that is also quite possible. You can also
use your overland flow approximations along these portions, to find the discharge, or
discharge per unti width along this thing, and then arrive at the discharge quantity
here, then subsequantly corporate uniform flow along this main triangular gutter chanel
and all. If I just elabrate it, if I just show it in a elabrated way, it may look like
this, something like this, like this, then like this. So this is your triangular main
channel, water may be there. A question, as a part as todays quiz, today in the todays
quiz, we are just asking you one question only. For the same triangular gutter section,
for the triangular gutter section, there is depth is given as y, and the slope is given
as 1 isto b, if this is provided to you how will you evaluate, or how will you evaluate
discharge Q, in the triangular section. So, you use uniform flow approximations, so
you shuold be aware with the geometry of of the triangular section, then you try to compute
it.

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