# 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.

## One Reply to “Uniform flow approximation for flood discharge”

thanks to so much sir….