Hydraulics

Dr. Arup Kumar Sarma

Department Of Civil Engineering
Indian Institute of Technology, Guwahati

Module No. # 03
Energy and Momentum Principle
Lecture No. # 04
Non Uniform flow:
Gradually Varied Flow

Welcome you all to this new topic that is non uniform flow. Till now that we have
discussed all about uniform flow condition and hydraulics of uniform flow. And today
we shall be starting our discussion on non uniform flow, well. And under a non uniform
flow of course, in our very basic introduction to the hydraulic engineering, we were
discussing or to our you know, introduction to the open channel flow, we are talking
about non uniform flow, there are different type of non uniform flow; that is say
gradually varied flow is one of that, where the flow depth varies gradually along the
channel section, along the channel reach.

And then, there is another class that we call as rapidly varied flow, where the flow depth
varies quickly, flow depth varies very quickly from within a small channel reach, and
there can be another non uniform flow that we term very specially varied flow; here say
discharge is changing from section to section, because of the fact that some side channel
may contribute some discharge to the main channel; again some of the discharge from
some channel can go part of the discharge go out from the main channel to a side channel
that way also there can be some changes in the discharge or that we once the discharge
changes, flow parameters are changing, velocity, depth all can change along the channel
reach. So that way, we can have due to different condition, we can have different type of
non uniform flow. And today, we will be discussing on one of the non uniform flow,
which is called gradually varied flow. So, our discussion will be on gradually varied flow
which is a type of non uniform flow.
(Refer Slide Time: 03:24)

Well, now to start with let us just briefly give definition of gradually varied flow. In our
initial discussion also we did discussed that, just to rephrase it, let us say; when say
uniform flow is occurring, suppose uniform flow is occurring, then force the component
of gravity force that is when open channel flow we are talking about, it is gravity force
we are dominating flow. So, the component of gravity force that is acting in the direction
of flow, and the force opposing that force flow, that is the resistance force r, suppose in
balanced condition, and we are getting a uniform flow in that channel.

Now, if somehow this balance is disturbed, and that this disturbance can be due to
different reason; if this balance is disturbed, then the flow earlier it was moving with a
uniform flow condition means, with a uniform velocity it was moving. Now, when this is
disturbed, then the velocity will change either it will accelerate or it will decelerate. And
then, when the velocity is changing along the channel, then its depth will also change.
So, that way, we get gradually varied flow.

So, you can concentrate on to the slide, and we can define it like that; if the balance
between component of gravity force in the directional flow, always we are talking about
the component of gravity force in the direction of flow, and the resistance flow opposing
the flow is disturbed, if this balance is disturbed, then the flow will either accelerate or
decelerate causing change in depth from section to section along the length, along the
length of the channel. And this lead to formation of gradually varied flow well. So, this is
how we can define gradually varied flow and this is how we define the gradually varied
flow.

(Refer Slide Time: 05:32)

Let us now see, some of the practical cases or some of the practical situation, where we
can have this sort of gradually varied flow. And the one example, the first example we
can take, this is flow profile on upstream of a barrier, flow profile on upstream of a
barrier; in a natural channel or you mean in a manmade canal also in a irrigation canal
also, many a time we give some barrier for utilizing the water for our bettering for our
need, that means, for better utilization of water, we can say.
(Refer Slide Time: 06:20)

How say in a river, the water is flowing in this suppose, the uniform flow is flowing;
uniform flow is of occurring, like that well. Now, we want carry the water from this river
to some agricultural field for our irrigation; then now when the water is at this level, it is
within the bank, and we may not be having any other device to carry the water from here
to the irrigation field to the agriculture field; sometimes we may be doing pumping,
sometimes we may use wheel to draw the water. But another way of doing is that if we
put some barrier here, if we put some barrier or may be a large area also some times in
case of dam, and varies like that; then what will happen? The water cannot flow through
this level, and it will be flowing like this, it will be, it will start flowing like this; and
then it is moving like this; on the downstream, it is again flowing this way.

So, that way what we have achieved that we have raised the water level to this level; now
our height is increasing that is water is going to a higher level; and from this higher level
as our existing level is high, we can carry this water to a much far distance maintaining
the height; that means, say when water will be flowing from this level, we can carry the
water, because we can draw a side, let me draw a section, cross section, suppose this is
the river, generally the slope will be in this direction, and say our water is here. Now if
our water level is here, we cannot carry the water after certain distance here; but if our
water level we increase up to this much height, then we can carry by gravity suppose
making a channel, it can be carried up to this much of level, where our level meet this
one. So, we can carry to a half distance.
So that way, this is one way when we use some barrier and we raise the water level, so
that it can facilitate our irrigation in agricultural field. Now this water surface this water
surface that we are having here is not a uniform flow, because this depth is gradually
increasing from section to section; here depth is y, and suppose on the upstream, it is
normal depth say, here it is normal depth on the upstream, we can have normal depth,
and then it is then water is gradually increasing, then it is gradually increasing like this.

So, at this point it is normal depth, but after that it is gradually increasing, and we are
having different depth y 1, y 2, y 3, y 4 like that; and this flow is nothing but one
example of gradually varied flow well. In canal also for which is made for agricultural
purpose, we sometimes increase the depth to carry it to the side channel or side canal or
field channel, so that way also this water level is increased and we get a gradually varied
flow like that. Again this gradually varied flow can be of different type, that will be
discussing later, but this one kind of gradually varied flow.

And this sort of curve is raising profile, it is rising, it is gradually rising; so this is rising
and this has another name that we call it as a back water profile. Then another example is
name itself is backwater during tide that we have experienced and seen that all the water
that is coming is we refer as backwater of course, but then during tide what happen
during tide say it is a sea here, let me draw the sea level; suppose sea level is here, and
we are having uniform flow condition like that, uniform flow condition like that; and
when during tide the water is rising up to this much level, then water from here will be
rising like that, it will have to be continuous; so, it is rising like that. And then, this is a
gradually varied profile, we write it simply as G V F. So, this is a Gradually Varied
Profile.

And one point we should remember; though a many a time, we draw gradually varied
profile suppose, a giving a barrier, we are giving this barrier, and we are drawing a line
like this; apparently, we get this filling that, depth here is, less depth here is more; then a
question may come into our mind that how the water is flowing from lower depth to
higher depth, because here the depth is higher, and the discharge is moving in this
direction; but of course, total energy at the upstream level is always more than the total
energy at downstream level, and that is why the water is flowing. And in reality, there
will be some slope, there will be some slope like this, this barrier is there, and then
suppose depth of flow is there, and it is it is rising like this, it is rising like this.
But if I draw a horizontal line here, then energy line if I draw, then you will be finding
that energy at this level is higher than the energy at this level. So, though the depth is
increasing, depth is increasing, but if I draw from a particular datum here, then this is z,
then this is y, then again due to velocity you will be getting v square by twice c, here it
will z 2 say this is the slope, this is z 1, y 1, then v 1 square by twice c, it is z 2, then y 2
plus v square by twice c, again v square by twice c, suppose this much, then we will be
getting energy line will be lighter, energy is low.

So, there will always have to be energy more here than energy here, and then only this
flow is occurring. And in reality, it is like that, and this change in depth is so gradual, the
change in depth is so gradual that when we go to a field, this is not that much of visible,
that means, on the upstream of a valleys or upstream of a (( )) if we just observe the flow,
we will never feel that water depth is rising, water level is rising, it is. So, this can be
computationally checked, we can do the computation, and we can check; and this is very
important; suppose, for some reason we are making a barrier here, for whatever may be
the reason we are making a barrier here, and water depth will be rising.

And say bank level is here, let me draw a phrase diagram; say our (( )) is this one, and
bank level is this one, bank, this is ban level. And then we are suppose when uniform
flow is occurring with maximum discharge, then it is set this level. So, there is no
question of our flooding; of course, during flood time, there may be flood, but generally
the flood is not there, flooding is not occurring. Now to raise this to increase this depth,
we are putting a barrier here. So, the water level from here will be rising like that, and it
is flowing like that.

Now, if we do not do the calculations before hand that is to what extent the water is
raising, then what will happen that if you give a barrier here, water will be coming here,
and it will be over flooding, it will be crossing the bank. So, it will create problem, our
target will not be achievable, water will not be rising to that level, it will be flowing out
from the bank. So, we will have to construct some marginal embankment or on the sight,
we will have to raise this bank. So, this is the bank level, so depth we will have to also
rise like this, this is the bank level we have to rise; so to what extent? Now, we cannot
rise this bank, suppose we know at this point you have bank height will have to be this
much; but that we cannot carry or we not carry the same height same raising up to this
point, because water is rising at this point.
So, we know that up to this point from this point water level is crossing the bank. So, we
can increase our bank height like this, we can increase our bank height like this. And that
way we can save our expenditure in raising the bank, and we achieve economy projects.
So, for those purpose we also need to know how the gradually varied profile will form;
to what extent it will go; and this sort of calculation will be becoming that will be
coming in computation of gradually varied flow, well. This is just we are putting some
example, where gradually varied flow can occur.

(Refer Slide Time: 16:19)

Then let me take another example that is confluence of two rivers that is also another
case where gradually varied flow can occur. Confluence of two river say one river is
coming like this mainstream, and another river is another tributary is joining this way,
and it is flowing like that. Well, this is control coming from some discharge here; now at
this counter, sometimes say this river has this catchment on the upstream of this
particular section, and this river is having this catchment; catchment means whatever is
occurring that is coming to this particular point; whatever rainfall is coming, it is
occurring from this particular point to this particular river.

So, if suppose heavy rainfall is occurring on upstream of this river, then the depth or
discharge in this stream in the main stream can increase significantly; and then depth of
this stream can increase. Now, earlier suppose if I draw it like this, this is the main
stream, and this the river - side river, this river is this one, it is carrying Q 1, and this
river is carrying a discharge of Q here, it is flowing like that; now earlier suppose it was
in this form; earlier it was like this. Now, when this level will be increasing, suppose due
to increased rainfall in the upper catchment, this level is increasing; then on this channel,
the water level will be rising like this, and this is not a uniform flow from section to
section, this depth is increasing.

So, at confluence of two river, this is what, this is the confluence of two river, and to
upstream of the tributary, there can be gradually varied flow profile there can be
gradually varied profile. And knowing this is important for various activities, because
when it when the water level rises, that means well, we understand that energy level,
energy difference is there, but energy difference will definitely reduced from that of the
original condition.

So, the flow velocity in this part will be less, in the tributary will be less; and this can
happen suppose, it is a river, and it is a drain, then also this situation can happen; and
then flow velocity here will be less means, less amount of water will be released or this
channel will be carrying very less amount of water. So, then it was carrying earlier,
because of the reduction in the energy level; and that sometimes can lead to flood
congestion in the upstream area here also in this channel; and so, these are important in
various analysis we use, and the concept of very gradually flow in this sort of channel
conference. Then there can be another situation on upstream of canal drop; many a time,
we have some canal, we have some canal like this, and there can be… suppose in a very
steep slope, suppose in a very steep slope…
(Refer Slide Time: 19:51)

Let me use in a very steep slope, we can have this sort of situation say, natural channel is
very steep. And we want to make a canal in this direction; then it may become difficult
due to many reasons for constructing a channel, a canal of this slope. So, what we do?
Because, a very if you construct a very steeper channel, then it may cause and then flow
velocity will be very high, and then with high flow velocity, there may be erosion or
there may be damage to the channel. So, what we do that suppose we construct it like
that, then we cut some portion here, we make a canal drop, and then we remove this
portion, and then we make another channel portion like that, this part can be deposited
here, and then again there may be canal drop like that, and that way, we are negotiating
the slope.

So ultimately, the canal will be carrying water like this, and then there is fall, then this is
going like that, then there is a fall, then it is going like that. So water is flowing like this,
though our original slope was that one. So, in this sort of canal drop, if I just draw one
canal drop like this, then what will happen? Water will be flowing, suppose this is
uniform flow depth for a particular discharge Q, for this bed slope; for this particular
slope, we have this much of uniform flow, then when water is falling, it will be coming
like this, and then it will be falling like this. And then, there may be some other
phenomenon here we are not discussing that part right now, but we are talking about
what is happening on upstream of this one.
So, this water surface is gradually coming down, and the flow depth is changing section
to, from section to section; here it is normal depth y n, then it is y 1, y 2 y 3. Here the
depth is gradually decreasing here the depth is gradually decreasing, because of the
change in the flow. Now this sort of flow is also gradually varied flow, and that flow we
call as draw down curve we call as draw down curve. So, basically it is draw down
profile; earlier, we have seen some of the cases where the channel depth was - flow depth
was rising, but here the flow depth is falling or we can call falling profile, this is a falling
profile falling profile.

So, there may be two class; one is rising profile, another is falling profile. And in the
falling profile, we name this as draw down curve; raising profile, we name it as
backwater curve. Then let us see some other example; due to some other change in bed
slope, due to change in bed slope, this is somewhat similar to that one, where we are
talking about canal drop; but here say actually canal drop is not there, but there can be
some change in the bed slope, say bed slope is changing this way, bed slope is changing
this way.

(Refer Slide Time: 23:04)

Well, now from our understanding of uniform flow, we had seen till now, but suppose a
discharge Q is coming here. Then for this slope, suppose our uniform flow level is this
much; NDL 1 suppose in this section 1, section 2, section 3. Now here, as the slope is
increasing in the channel 2, as the slope is increasing, so obviously, the depth of flow or
the uniform depth of flow will be lesser than that of the uniform depth of flow for section
1, for the channel reach 1, because discharge remaining same slope is increasing here in
the 2, then 1. So, uniform flow will be becoming lower. So, this is suppose my NDL, this
is NDL in this (( )) normal depth line 2.

In the third section again as the depth of flow, as the slope is flatter, then even the first
one - depth of flow, normal depth of flow will be again increasing NDL. But these are all
about the normal depth line we are writing; at the flow profile that the surface of water
will not be moving like this, and then suppose straight moving this way, then dropping
here, again moving like that, then rising here, not moving like that. But in a channel,
when we allow the water to flow freely, it will always try to attain the normal depth
condition; of course, if we make some disturbance there, then it will be disturbed; and if
we allow it to flow continuously for a long time without any obstruction, without any
drop, then it will try to attain the normal flow condition.

So, what will happen? Here if we assume the flow is coming like this, at this normal
depth, but here it will try to meet this normal depth; and so, it will gradually come down,
and then it will be it will be coming like this, and then of course, further detail is
necessary that is whether this flow is becoming super critical or whether these, all these
flow are remaining sub critical, those issues are there, and right now we are just simply
considering that this is changing like this. And then from here, it will be again moving
onto this particular side. Let me just talk about this part, because further detail will be…
to know further detail, we will be requiring some knowledge of the super critical, sub
critical combination, and that will be discussing later.

But when the change in channel slope is there, this flow depth are changing; here also the
depth is, here uniform flow depth, and then it is changing gradually in this part y 1, y 2, y
3, and here it is uniform flow say y n. If this length is sufficient to form the uniform flow
depth, then ultimately it will come to uniform flow depth, and then in this part, it is rising
in this part it is rising; say we can write y 1, y 2, y 3, y 4 like that. For a second section,
here it is normal depth then it is coming down like that.

But all these part are gradually varied flow; and when we talk about different type of
gradually varied flow, then this part will be one G V F, this part will be another
Gradually Varied Flow; this is a draw down curve, this is also a draw down curve, but it
is if the name of this gradually varied flow profile, and name of this gradually varied
flow profile, two gradually varied flow profile may be different. But right now, just we
need to know that gradually varied flow profile can form on the earth surface in a
channel due to change in slope, this is one example.

(Refer Slide Time: 27:44)

Then another example is say, on downstream of the sluice gate. So, what is sluice gate?
Many a time, we provide sluice gate to control or to regulate our flow; say water depth
normally is flowing in this level, let me use a dotted line; normally suppose, it flows
uniform flow occur at this level. Now somehow for reason, we want to stop this water, so
that we are using a sluice gate here; and of course, when we are using a sluice gate, and
then we will be releasing the water also when required. So, when we block this one,
water level here will be rising like that.

And then when if we close it completely, then of course, water from this side, suppose in
some situation, where say there is a downstream channel and on this side; and what this
is a tributary, and this is a main river; and from the main river, suppose the at high flood
period, water enters into the tributary, we want to avoid that. So, we are putting a sluice
gate here; so that when main channel water rises, then we are closing this, and the water
cannot come into the sub channel. So, this is one example, where we are using this sluice
gate. And there can be different example, and just one example I am stating.
And then when this water level will come down, after flood time, when this main
channel river water has come down, then we again release the water, which again,
because in this channel, in the small channel also water is was flowing, and there will be
a some deposition of water due to rain water and all in this side. So, that water we need
to release, once the water level comes down here. So, suppose we are leaving the water,
so then we are just opening this gate, and then we are allowing the water to flow.

So, this water will be flowing, and in this channel as we know that its normal depth is
this line, NDL is this line; so this water will try to reach the normal depth line from this
level and of course, as I have explained or as I have stated to other, as I have not
explained, as I have stated that the flow condition here will depend, here means in this
portion will depend on the condition whether it is super critical, and that part is sub
critical or whether both the part is super critical or whether both this part is sub critical,
what is combination depending on that? Sometimes we can have hydraulic jump in this
portion, and sometimes we can have gradually varied profile.

And sometimes we can have a gradually varied flow profile rising, and then hydraulic
jump is so kind. So, different conditions may be there, but one point is there. On the
downstream of this sluice gate, we can always a gradually varied flow profile. It may be
followed by a hydraulic jump or sometimes there may not be a hydraulic jump or
sometimes instantaneously just after using from here, without any gradually varied flow
profile also we can have a hydraulic jump like this. In that case, you will not be getting
gradually varied flow profile. So, those details will be coming later, but this is one
condition, where we can have gradually varied flow profile. Well, that way we have
discussed some of the practical situations, where we can have gradually varied flow
profile in our day to day life, in our day to day engineering activity. And then we can
move on to the theory of gradually varied flow, well.
(Refer Slide Time: 31:23)

And when we talk about theory of gradually varied flow, then we must take the name of
Belanger, who first develop theory in 1828. In 1828, the theory of gradually varied flow
was first given by Belanger; and of course, it was for simple situation, and lot of
assumptions we meet, and then still this equation is very much useful. Well, now
whenever we talk some theory, whenever we talk about some theory, then to develop the
theory that means, to develop a mathematical expression for a practical problem or a
practical situation like gradually varied flow procedure one one one problem, and we
want to derive one equation for that, we always need to make lot of assumption; because
if we try to develop one equation for representing a particular flow phenomenon,
considering all the practical situation, then it becomes so complex that it may not be
possible or we may not have complete understanding of all the processes involved in
that.

And why we make those assumptions on you to all those assumptions also; after
simplifying the problem also, after simplifying the problem even whatever answers we
get that have lot of, that has lot of practical importance; that is with this understanding
itself, we can do lot of engineering work. And as such we are always we are always
considering some assumption, and then we are developing those theories. But we should
be very much aware of those assumptions, because once an equation is given once an
equation is given normally we should not use it, and when in tactical field, they apply it;
and sometimes they find that what calculations they are doing is not matching with the
practical situation.

Then they must know that what are the assumptions that were there in that derivation of
the equation? and then only there is an understanding that is okay, this sort of differences
may be there. And when we know the assumptions, then we know the limitation of that
particular governing equation. And then we will be able to use it more judiciously, more
wisely for our practical application; well that is why, I am just giving some of the
important assumptions that are suppose very obvious; and of course, apart from these
assumptions, some smaller assumptions are always there, which we are not mentioning
here.

First assumption that you can concentrate on the slide that is the head loss is same as that
of the uniform flow; well, head loss is same as that of the uniform flow; that is the
uniform flow formula like manning’s or chassis formula are applicable. Already we have
discussed that manning’s formula, chassis formula all these are uniform flow formula.
Now these uniform flow formula basically deals with the fiction loss that is indirectly
coming as head loss. So, these head losses are same as uniform flow, because in
gradually varied flow, the depth is changing, and there will be difference in the head
losses. But we are considering that head loss is similar to that of uniform flow, and that
why for some of the calculations, we will be still using manning’s formula and chassis
formula in the computation of gradually varied flow. Well, this is one of the
assumptions.

Then second assumption is that the channel is prismatic; in that derivation, we are
considering that channel is prismatic. Now what we mean by prismatic channel that
perhaps we remember, because we have discussed this already that when channel section
does not change in the… it does not change in the entire channel reach or along the
channel, then and then if thus channel slope is also not changing, so when channel cross
sectional slope does not change, do not change, along the section, along the channel
reach then we call this as a prismatic channel. So, in the derivation of gradually varied
flow formula governing equation in the before for getting the final form, it was assumed
that the channel is prismatic; but in real situation, channel will not be prismatic, because
a channel can be always like this, channel can be always like this.
(Refer Slide Time: 36:44)

Now say a formula is given, and we want to know this is the plain view I am drawing,
this is the top view of the channel, where the channel with here is this much, here is this
much, here it is this much, again it is increasing to that much, well. Now if we use this
formula directly, and we are suppose computing gradually varied flow in this channel;
depth here, and depth here, we want to know; depth here and depth here. But if we do not
remember that there is an assumption that channel is prismatic, then we will be ending
up doing wrong calculation, because if we just try to use that formula, and if you
compute the flow depth between these center it will not (( )), because in this portion
channel is not prismatic here, it is changing.

So, what we can do? For our practical purpose, we can consider that from here to here, a
small reach we can take; and from here to here were it is gradually even we can make it
smaller, and we can consider that from here to here this channel is prismatic, because it
is a distance, and change in the for soft cross section is not that much significant, we can
take average section. Then from up to this much suppose we are talking like this, and
from here to here you can see the channel is more or less of uniform size, and if slope is
also not changing, this portion we can consider as prismatic; and from here to here, we
can again consider as prismatic; from here to here, we can consider as prismatic. Then
again in this expanding portion, we can some smaller section, and we can consider that
channel is prismatic within these sections.
So, when we will be using the formula, the governing equation of gradually varied flow,
between this section and that section say, between this section and this section, then it
will be giving more or less characteristics. We will be using governing equation between
this section and that section again, so this will be giving us correct results. And we can
use this section and that section directly; then we can apply the same equation between
this section and that section, because more or less this part is prismatic.

So, that understanding is very important what assumption we have, this is what the
channel is prismatic; then another assumption is that energy coefficient is one; we were
talking about alpha below energy coefficient I mean, the velocity coefficient; within
velocity coefficient, we have energy coefficient, and momentum coefficient, this energy
coefficient alpha is 1; that is the variation of velocity within the section, within the cross
section is suppose we are computing gradually varied flow, the velocity may be different
here, but we are assuming that this is uniform.

So, in the actual field, if this difference is quite significant or computed result maybe
varied from the external result that we need to know. And we need to know that if this
variation is small, then well, we are applying with more confident our equation; and if it
is quite high, then of course, we need to consider the value of alpha. In the final form, we
may not be having that alpha, and if this variation is very high, we need to consider this
alpha value. Then another important assumption that this assumption roughness
coefficient n, that is a Manning’s roughness coefficient or it may be chezy’s roughness
coefficient as well, each independent of depth.

Now, when we were discussing our uniform flow formula, then we very specifically
pointed out that Manning’s roughness coefficient varies with the depth; if our depth
increases, then our roughness value changes. But for computing the gradually varied
flow between two sections, if you apply that equation, we will be considering a single
and valued or single roughness parameter; means, that we are violating that very basic
condition of Manning’s roughness, that is it changes with depth. But however, we are
assuming the change in the roughness parameter is not that significant for the variation
that is occurring within the gradually varied flow portion, and that is why we are
considering that n value is not changing.
And then of course, if we are aware about that then what n value will have to use, that is
the n value for a smaller depth if we use, then it may not be applicable for a higher depth.
And similarly if we use the n value for a higher depth that may not be representing very
correctly for a very smaller depth; then we can use the n value intermediate between
these two. And so, knowing these two assumptions, we should walk; that assumption is
there.

Then another assumptions that bed slope is very small, bed slope is very small; generally
in natural channel, when we find this gradually varied flow in most of the cases of
course. Some exceptional cases will be there, but in most of the cases when we find this
gradually varied flow problem, as we have already discussed some of the cases, the bed
slope is generally not dead high and we can consider this to be very small. And when we
consider bed slope to be very small means, theta is small, and some other considerations
are automatically coming like if we remember we talking about the pressure, water
pressure, hydro static pressure then also a pressure of the flowing fluid. In steep slope,
the pressure is different, the pressure is different, but in flat slope pressure theta if we
neglect, this expression is different. So, that is the hydrostatic pressure condition we
give.

Now this ultimately leads to hydro static pressure prevails, it leads to the condition that
hydro static pressure prevails; and the pressure correction is not coming into picture, but
in a flow situation where slope is very high or even, suppose in a gradually varied flow
in this portion, suppose this is very small, but the surface slope is very steep, this portion.
Here also, in fact, the flow is moving as a curved flow as we discussed in our earlier
classes, and there hydrostatic pressure may not be exactly valid; so, we mainly to apply
connection here. And similarly when the slope is steep, we may have to apply the
correction progression; and if we do not apply, at least we should know that some
amount of (( )) we are introducing layer. So, this assumption is also important.

And then we will be deriving this equation for flow depth, considering the flow as one-
dimensional and depth averaged; depth averaged means the velocity that we are talking
about is the velocity considering the average depth, with the average velocity for the
entire depth, because we know the depth velocity here will be more, here will be less,
and there may be a velocity profile like this of course, highest velocity may not be at the
surface, it will be, generally highest velocity will be little lower than this, that we have
already discussed, but that velocity variation we are not considering in this particular
derivation.

And then we are having depth, for the entire depth we are considering a single velocity
V, which we refer as depth averaged velocity; so velocity averaged over the depth. And
of course, this equation is one-dimensional, we are considering two-dimensional, but we
can have two-dimensional expression also. Well, with these assumptions being on the
background, and understanding these assumptions, now we can move to derivation of the
governing equation, well.

(Refer Slide Time: 45:08)

So, let me draw a channel like this, and let us see that this flow is moving like that; and
let me write the expression for a section here, and of course, we can have a section here,
and this is the datum of course, say uniform, they are changing like that. Then let us see
what will be the energy, total energy H, total energy H in a section 1. As we have already
written that part, say this is z 1, this is say y cos theta, if this slope is theta, in reality it is
y cos theta means, depth if I draw like this. And then it is alpha V square by twice g that
part we have already discussed in one of our classes. So, I am not going to discuss this
again, but this is alpha V square by twice c.

So, total heat, we can write as z 1 plus y cos theta plus alpha V square by twice g; this is
the total heat, this is the total heat at any section, well. To derive that equation, what we
can do? We want to see that how this heat is changing with respect to the flow direction.
If flow direction we put here as x, then we can write what is d H d x that is how this H is
changing with respect to x. So, we can write it as say d H d x, there is derive taking
derivatives with respect to x, this is equal to what we will be getting the d z d x; I am not
using the term one, because it is I am writing in general for a particular section, then plus
we can write say d d x of y cos theta plus say the d d x of alpha V square by twice g.

Now in this expression, this d H by d x term that is the heat; now if I draw here z 2 y 2
cos theta, this is y 1 cos theta, and alpha say V 2 square by twice g, then here I am
getting a total energy level is 2; and then if I join this, then I am getting a line, which we
call as a friction slope S f, and this theta in fact, we write as bed slope S b; now this line
should be horizontal of course, this is bed slope is b, and this is the slope of the fiction
line S f. So, rate of change of heat with respect to x that is nothing but slope of this
particular line, because we are drawing this joining the total heat at two different points.

And one point again here we need to mention that if our section, we are considering 2,
and heat here, and heat here, we are joining, we are getting a slope. If I take another
section in between total heat will be different; and S f between this point to that point and
from again this point to that point, both may not be same. So, here we are taking
basically average friction slope, because friction slope at this point; and friction slope at
this point may be different, but it is varying as our section will be smaller, as this section
should be smaller, we can consider this as average friction slope.

And as heat is falling in this direction, heat is falling, because of heat is more, so heat is
falling in this direction; so, we are writing this as minus S f minus S f. Well, if we want
to give the value of S f somehow, then that numerical value of S f, we need to be
positive; because in the equation, we are using the negative sign to indicate that it is
falling. So, in the equation itself, we are using this sign concept and so, numerical value
when we calculate for S f, this we should calculate positive, this S f, because already we
are assigning the sign to that particular, considering the direction.

Similarly, this d z d x again z 1 is dropping to z 2. So, it is also falling, we are

considering that it is falling from this direction to that direction; of course, there may be
just adverse slope also, but here we are considering that it is falling from upstream to
downstream. So, this d z d x is nothing but bed slope, and this also we are assigning a
negative sign. So, numerical value of S b when we are considering positive, when it is
falling, because we are already putting a sign; so numerical value we need to put positive
for falling one; sign is already there; and if it adverse slope suppose the bed is rising like
that, numerical value itself we have to put negative; if then only with this minus sign it
will be coming as positive that is the rising one.

So, these things we should be careful well; in the derivation what we are doing that if we
know then we will be putting the sign correctly. And then we can write this as cos theta,
this has nothing to do with x, if we consider the slope is not changing in this portion;
then cos theta we can just bring here and we can write d y by d x, plus this particular
expression we can write in a form that, because this V square velocity is changing,
velocity is changing from section to section here it is V 1, here it will be V 2, velocity is
changing from section to section. But these velocity change, we can directly get in terms
of y, because depth is changing means depth is changing means we can get it the
discharge remaining same, Q we are talking about the same Q. So, we can have with the
change of depth, it is velocity is changing.

So, we can get it in the form that we can write it as plus say, let us write this change in
velocity in terms of y we can calculate, because x of course, in the direction x it is
changing, but how it is changing with x depth, it is not known to us right at this moment.
So, let us see how it is changing with y, then we can take it, again multiply it, how y is
changing with x. So, that will ultimately give us the d d x of this part. So, what we will,
write d d y of alpha v square by twice g into say d y by d x, because we are differentiated
with respect to y. So, let us differentiate y with respect to x again, original differentiation
our target is with respect to x. So, we are writing this.

Now this equation we can have in this form that S b minus S f S b I am bringing on this
side, this equal to say cos theta into d y d x cos theta into d y d x plus say d d y of alpha b
square by twice g into d y d x. And this we can further simplify now putting all those
assumption; well, this let me write this d y d x, because our target is to write one
expression for gradually varied flow profile well; gradually varied flow profile means, if
we can get how y is changing, how this depth are changing with respect to x, how this
depth are changing with respect to x, that is d y d x; how these are changing.

And then if we can get these in terms of quantity on the right hand side, then definitely
we are getting our answer; we are getting a equation, and from that equation, if we solve
it, then we will be getting we can interrogate it, we can solve it by different way; if you
solve it, we are definitely getting depth at different section. Once d y by d x is known,
once d y by d x is known as function or some known parameter, if we solve we will be
getting y at different x; and that is what our target; if we know that, then we will be able
to solve what type of profile it will be, how the profile is changing.

(Refer Slide Time: 55:27)

So, let me write it in this form dy dx into cos theta plus d dy of alpha V square by twice
g. Well from these dy dx will be, we can write that dy dx, dy dx is equal to S b minus S f
divided by 1 sorry cos theta plus cos theta plus d dy of alpha V square by twice g. Now,
we can just introduce some of the assumptions that we had, just before derivation of this
equation. So, those assumptions are first the theta is very small - for small value of theta
for small value of theta. We can have cos theta is equal to. Then again for alpha equal to
one, that is we are considering say variation neglecting, variation of velocity within the
channel section.

So, neglecting variation of velocity, variation of velocity within the channel section
within the channel section, what we can have within the channel section, if we neglect
this part, then our alpha is equal to 1. So, our equation reduced to say dy dx is equal to S
b minus S f divided by 1 plus d dy of V square by twice g, d dy of V square by twice g.
Now, we can further simplify this expression, let us see how we can write d dy of V
square by twice g; so V square, again this V is nothing but depth average velocity.

So, as we know the Q, this V can be written as d dy of say Q square twice g by A square.
Now, V is varying with depth, but Q is not varying with sorry V is varying with x and y,
but Q is not varying with y. So, Q is constant. So far, this derivation is concerned, so we
can write this as d dy of sorry, we can write Q square by twice g is a constant. So, d dy of
1 by A square. Now, this part if we just do the derivation, we can write Q square by
twice g, first A square A is definitely varying with y, but of course, we can derive it with
respect to A, and we can write dA dy. It is do that. So, that it will be A to the power
minus 2.

So, it is minus 2 into A to the power minus 3 that derivation, and then we can write dA
dy. And as we know in a section that, if this is the area for a small increase in y, if the
change in area is dA, then if top which is T, we can write dA actually dy into top with T
is nothing but dA. So, dA by dy is nothing but T. So, putting this as T, what we can write
this expression. All these are coming as Q square, this T will be T, 2 and 2 is getting
cancel minus sign is remaining here, Q square T by g A cube.

So, putting this expression for d dy of V square by twice g. We can write say, dy dx is
equal to S b minus S f divided by 1 minus Q square d by g A cube. And this is, what is
called as governing equation of gradually varied flow. This equation number, I am just
putting as A, this is very very important expression, and this equation is used for various
practical purposes. And of course, this Q square T by g A cube as we did earlier also, we
have done several times. This can be expressed in terms of fraud number also, and that is
why we can write say dy dx, we can write it in the form of dy dx is equal to S b minus S
f divided by 1 minus that is Q square by A square will become say V square. So, we can
write it V square, and A by T that will become d. So, it is g d, and that can be written as
S b minus S f divided by 1 minus F r square.
(Refer Slide Time: 1:01:17)

So, that is also another form of governing equation that dy dx is equal to S b minus S f
divided by 1 minus F r square. So, this is also one popular expression, I am writing S b.
So, that way this particular governing equation, we can have in different form. And this
is, this expression for a rectangular channel or for a wide rectangular channel, we can
again express it in a different form; in terms of normal depth, and critical depth. So, that
will be coming, and that will be discussing many a time, we write this expression as dy
dx is equal to…

Suppose S b, we are bringing here and then we write 1 minus S f by S b divided by 1

minus. Let me write it as Q square T by g A cube, and then we can have this S f by S b in
terms of as a function of y c and y n. And this also, we can have and in terms of y c and y
n no sorry. This this this we can have in terms of y and y n; in this we can have in terms
of y and y c. Like that, we can for rectangular channel, these are for rectangular channel
we can have it, for wide rectangular channel we can have it.

So, for different channel, we can simplify this particular equation into different form.
And then, we can from that we can derive lot of our important understanding or we can
see how the flow can be classified - how the gradually varied flow can be computed, and
can be classified well. One important point just I want to mention before going to all
these, it is this equation dy dx is equal to S b minus S f divided by 1 minus Q square T by
g a cube, it is a non-linear differential equation. Because this S f is also a function of y,
and from these expression you cannot separate out you cannot separate out, your y value.

That is we cannot solve it directly by integration, if this solution of this equation by

direct integration is which occurs, (( )) it is possible, but we cannot solve it directly. And
that is why for solution of this equation, we generally take recourse to numerical method,
and some other approximate solutions are there. All those different methods existing for
solution of gradually varied flow equation, computation of gradually varied flow that we
will be discussing in the next class, and as we have already mentioned, there are different
classes or classification of gradually varied flow are also there.

So, when the flow is occurring on a mild slope, and we are getting a gradually varied
flow then we name it in a different way. When the gradually varied flow is occurring on
a steep slope, then we name it in a different way. And sometimes it may happen when we
are talking about uniform flow, then we found that when it is a uniform flow occurring,
if it is on a steep slope. Then this uniform flow depth y n is always less than the CDL
line; NDL is always than the CDL line, but in gradually varied flow we are putting some
obstruction here. Though the slope is steep slope, water may be rising like that. And so,
it is going to this level, then it is a depth of flow at any point may be more than our
critical depth though the slope is steep.

So, based on all those condition, that if A, flow is occurring above critical depth in a
steep slope the name will be different. So, in a very systematic way, we classify the
gradually varied flow considering all these different different concepts, and then we
name them. And then after that we will be going for computation of gradually varied
flow. So, next class, we will be doing all those for today this much is sufficient, thank
you very much.