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Lecture – 08: Specific Energy, Specific Force and Critical Flow

Welcome all of you to these lectures on river engineering. As we discussed in the last class, hydrodynamic equations like Navier-Stokes equations and Saint-Venanat equations and we have also seen there are the solvers like HEC-RAS models for solving Saint-Venanat equations and continuity equations, but still, there are the concepts like specific energy, specific force and the critical flow that what we can use for river flow analysis. Today, I am going to talk these simple techniques which you may have read it in open channel hydraulics, the same concept we can use for river engineering that is the points what I will highlight today. So we are going to talk about specific energy, specific force and the critical flow. Those are things also we can use for solving this complex river flow problems when we have very, very limited flow geometry data, the hydrodynamics data with certain assumptions. So today, I will highlight it when you use the concept of critical flow, specific energy and specific force. What are the assumptions we do? Are the assumptions correct for that conditions that is what a river specialist work on that whether that assumptions what you have chosen there in specific energy, specific force and critical flow are valid for that cases, so that is the reasons I am going to teach you specific energy, specific force and the critical flow in the context of river engineering. (Refer Slide Time: 02:39)So looking that we can look at the books what we have added is open channel hydraulics with Hanif Chaudhry books long with other books. So we have added new book which we are talking about the critical flow, specific energy and the specific force that the concept we are highlighting through from referring some of the context from open channel flow authored by Hanif Chaudhry.

So looking that let us come back today what we are going to discuss it with specific energy
concept, channel transactions, hydraulic jump, then we can go up to non-rectangular channel
looking for the specific energy. So that is what with today’s lecture we will cover and next
series of lecture also will cover the critical flow concept.
(Refer Slide Time: 03:42)Looking that let me go back to real conditions of the river when you have a river in the hilly
regions, you can have series of the boulders. So if you can see this there are the series of
boulders and there is sometimes position of the boulders is the higher than the flow depth, it
can happen like that. So if you look at that if this is the conditions and if I consider the simple
assumption is the steady flow that means my discharge is constant.
So if I am assuming that the flow is a steady flow and considering that the same amount of
discharge is happening or we are not bothering about this as variability in a time domain or
that is not significant for us, so if that is the conditions if I just try to draw it hypothetically
typical behaviors of what could be the velocity variations, what could be the water depth
variations?
It could be like this because of presence of these big boulders you can have a stagnation
point, the point where the pressure is equal to zero. You can have a drop of the velocity head
the , so that is what the velocity head and also you can look at that I am just plotting this
water depth, this is the bed and plus the is the velocity head is equal to the energy head. So
hydraulic gradient line and energy gradient line.
So if you can look it there are the hydraulic gradient line and the energy gradient lines and as
these boulders passes through that again the V1 velocity flow happens, the stagnations point
is here, after that the flow again happens with V1 velocities, again increases and because ofthe presence of these boulders there are certain energy losses happening that is what is
reflected in the energy gradient line.
The energy gradient line, the slopes indicate because of presence of these boulders how much
of energy losses happen that is what we can quantify using Manning’s equations. So we can
have presence of the boulders, because of that there is energy dissipations that is what is
highlighted through the energy lines which typically it is indicating that. If you look at
another bigger boulders are here the formations of hydraulic jumps, as you know it there is
the change of the flow from subcritical to supercritical.
If you look at this energy dissipation what it happens because of the presence of this smaller
boulders that is the energy dissipations happens and that is what we can clearly know it from
the slope of this energy lines. Slope of this energy line we can always quantify how much
energy dissipated because of presence of the boulders. When you come into this location
where the flow changes, it reaches to a critical depth.
And there is a change from supercritical to subcritical, the hydraulic jump formations happen.
There will be hydraulic jump formations happen. The vortex formations will happen, the
energy dissipation occurs. So, because of that there will be the change of the energy losses
here. So if you look at a typical sections like this, there are the stagnation point of formations,
there are the energy dissipations because of bed roughness.
And there is a positions if more obstruction is there like this boulders it can have a lot of
hydraulic jump formations and because of the hydraulic jump formations you can have a
large drop in energy due to hydraulic jump formations that is what happens in open channel
flow. It also happens in the rivers. As you look at the typical river cross sections, you can see
like this, so the energy loss is happening.
Now the point is coming how we have to quantify it, how do know that there is the change
where the hydraulic jumps are going to happen? What type of energy dissipations are going
to happen? Those are the things we can analyze for conditions of steady flow. That means we
are assuming it the flow is a steady flow and this assumption is quite valid.When you have a high flow condition or design flow conditions what could be the extreme
conditions in terms of energy dissipation, in terms of velocity variation, in terms of the flow
depth variations. That is what we try to know from that. That is the reason many of the steady
flow analysis also enough for us to solve these problems. So that is what we do a steady flow
analysis in terms of specific energy, specific force and critical depth analysis that is what in
these figures we are very clear cut showing it.
How the hydrodynamics vary in rivers with having a different type of big boulder, smaller
boulders. There could be hydraulic jump formation, there will be stagnation point, there could
be energy losses because of bed roughness all you can understand if you sketch this figure
where hydraulic gradient line, energy gradient line is present and how we can use the concept
of specific energy and specific force for this. Let us go forward to the next slide.
(Refer Slide Time: 09:52)
Next slide what I am talking about is we have discussed about 2 basic principles, mass
conservations and the linear momentum equations. If you look at these 2 equations beyond
that we have energy equations. So we can apply this energy conservation equation in control
volumes. So if you can consider a control volume, you can apply the energy equations. As
you know it you have the 3 types of energy.
Potential energy, kinetic energy and the energy due to pressure difference that is the pressure
force, pressure energy per unit weight. So that is the reasons if you know it very basic fluid
mechanics that we use the Bernoulli’s equations the same versions we can use for the open
channel flow where the total energy per unit weight, we can define as a potential energy. Thisis energy due to the pressure per unit weight, this is the energy per unit weight as a kinetic
energy.
So this is the sum of three energy as you know it is a part of Bernoulli's equation, the same
total energy available at that particular cross section we can divide into 3 energy components.
That means if I have any channels, so I can have this hydraulic gradient line and I can find
out the energy gradient line and the h from certain datum and this is z, this is the h and this is
.
So you can easily have these 3 components from this datum, this potential energy per unit
weight that is what is the z value, this is because of the pressure difference, the energy per
unit mass because of the pressure and this is the kinetic energy per unit mass that is what is
. Many of the times also we use α as kinetic energy correction factor.
In that time, we consider the flow is not uniform. In the case of the flow is not uniform, we
adopt α value which consider the flow velocity variations in terms of computing kinetic
energy per unit weight. So this is very simple concept as you see from the fluid mechanics
that is what is idea we have brought it that total energy or total energy per unit weight which
is called as total head that is what will have a three compositions.
Potential energy, the flow energy or it is called the energy due to the pressures per unit
weight, same way you know the kinetic energy per unit weight. So these things we can easily
get it for a particular smaller control volumes as a reach we can consider it and we can
compute these 3 energy components which represent our total energy. So connecting the total
energy point is called energy gradient lines.
So how the energy variability is going in that is what you look at. So what we do is we make
it a bit easier for us that with certain assumptions is that we consider it.
(Refer Slide Time: 14:15)The first simple assumptions where we put it that if I consider is the surface is a horizontal.
and the flow is coming. If I consider this horizontal surface as my datum, so z = 0, River bed
if I consider as a datum, then z becomes 0. So I have only these two components , the energy
components will have 2 components.
Now this flow energy due to the pressure difference and there will be kinetic energy that is
what is because of the flow is there. So if you have these two energy concept and if you
consider is pressure distribution is hydrostatic, that means you are considering the pressure
distribution as flow at the rest, so hydrostatic pressure distributions. The pressure distribution
is hydrostatic.
It only varies with the depth, pressure distribution a linear triangle and it varies with the
depth, with increase in depth you will have pressure increases, it is a linear function as
discussed in any of the fluid mechanics. So if you have a hydrostatic pressure distributions
and if I assume it the channel bottom as a datums, this energy equation is very simplified it as
this form.
Now the energy that means what I am looking at, we are just getting the energy per unit
weight for channels where the pressure distributions is hydrostatic and channel bottom if I
consider is as a datum for me, then just that energy we call is specific energy. So it is a
simplified version with these assumptions. So we have an energy, specific energy E as a
function of flow depth the y, v is the velocity, .So if you look at, the very simple functions what we got it as energy calculations. So many of
the times we use the specific energy instead of this equation because these 2 assumptions we
can validate for many of the cases. This assumption can be considered for many analysis to
know it how the energy dissipations happening. So that is the reasons we compute the
specific energy.
That is the assumption, you try to understand it and that is what the specific energy is, total
energy per unit weight where the channel bottom is consider is datum and pressure
distribution is hydrostatic that is the specific energy, it is a total energy in an open channel
flow in a river flow.
(Refer Slide Time: 17:33)
So considering that now if you look at these equations, it is a very simple way, many of the
time we can consider the river as a rectangular channel, it has assumptions now. If I consider
river is a simple rectangular channel, so having a B width and y is the depth of flow, I can
consider channel is a rectangular channel, these are assumptions, it is also quite valid.
If that is the condition, now these equations again we can simplify it that if we have the Q
amount of discharge passing through the steady discharge over this, I can compute the q
which is the discharge per unit width that means Q divided by B, discharge per unit width q.
If I put it that then I have the y as a variables, so I can easily write this velocity v= q/y that is
very simple thing, discharge is equal to velocity into area.Here we have considered this discharge per unit width divided by the depth, this will be the
velocity. So that way if I can substitute it, I am getting this equation, again with some
assumptions with writing the specific discharge equation in terms of discharge per unit width.
If I consider that, my equation comes like this. So in this equation if you look at that I have 3
variables and q.
So I have specific energy, the flow depth, I have the discharge per unit width. So 3 variables I
have as established with these 3 equations. Now, I am not going to detail minutely, look at
this equation and the derivations which you can follow from Hanif Choudhry books, but let
me have a very conceptual understanding that that we have the specific energy y and the q, 3
variables, a relationship of specific energy relationship we got it.
If we got this relationship, now we are looking it, if I make E is a constant, we try to know
what could be the relationship between the y and q. Again I am going to repeat it. If I
consider E is constant, that means I am considering reach of the river where energy
dissipations are not happening that order, so that means energy remains constant at that reach
of the river.
If that is the condition, what is the relationship between the y and small q, we can establish it
using these equations. Same way, if you try to understand it if q is a constant, that means the
river width is not changing, whatever the Q discharge is coming in per unit width, width if it
does not change then smaller q, the discharge per unit width does not change.
So, you can have another conditions that one condition is that E constant we look at what
could be the relationship between y and the small q, that means this is the cases where energy
does not dissipate that much, so we can consider a reach, we can assume it there energy
dissipation is not that significant order we can find out a relationship between y and q. Same
way if I can assume the Q does not vary.
That means my Q amount of discharge what is coming in is a constant, it is a steady flow and
my width of the river is not changing. I am considering a rectangular river and the width of
the river does not change. If that is the conditions q also does not change. In that case, I am
looking at what could be the relationship between E versus y. So that is the concept and the
other derivations and all the things are the basic things, but try to understand it.There can be two conditions you can have that one is the E constant, energy dissipation is not
that significant order, second is that discharge per unit width is a constant and not varying
that significant order. If that is the condition, what is the relationship between q and y or in
case of q constant what is the relationship between E and y because the flow behaviors
happen like that.
So with this relationship, we can understand different type of the flow. So, if you look at that,
the first part what we have done is that we assumed this q is constant. So, if you have
considered q is a constant, see if you can write it the equations becomes like this, simple way.
So we can rewrite this equations when the q is a constant, you can write it like this forms.
So we can get this is a cubic equation and you can get the solutions of the y. So that means if
q is a constant, we can get solutions of y, E for this and or we can develop the solution curve
for this cubic equation which are very mathematically if you look it is a cubic equation where
we can get the behavior of the curves.
(Refer Slide Time: 23:59)
That is what is specific energy curve which discuss more in open channel flow that is what is
called specific energy curve that is what give a establishing between now if you try to
understand as q is constant, so discharge per unit width does not vary significantly. So if you
consider that, then there is a relationship between E and y that is what will happen, it is
cubical and nowadays you can solve these equations, it is not big issue.But let us see what is the behavior of these equations, the solutions behaviors that if I plot E
versus y, E is the specific energy, y is the depth, so that is what is called specific energy
curve. The equation of E-y will follow this curve, so follow this curve and third one will be
negative values which will come here. So there are 3 solutions, y1, y2, and y3 for this q equal
to constant one will be negative which we should not consider.
It is not physically possible to have a river flow depth is negative, so we cannot consider that
part. So there are the two depths y1 and y2 which is called alternative depth. That means what
we try to understand it from the specific energy curve. For a constant q, you will have the
specific energy variations with respect to y, it vary like this. So in these points if you look at
from this specific energy curve, very interestingly there will be a point where E is minimum,
beyond that the flow is not possible.
That means for a particular q value, we will have the E minimum, the minimum specific
energy, beyond that flow is not there. That minimum specific energy if a flow is happening
and that corresponding depth is called critical depth, that corresponding flow depth is called
critical depth. That means a critical depth where at that particular discharge per unit width the
critical depth occurs the point where specific energy is minimum that is the critical, beyond
that this flow is not possible.
The flow cannot be possible for this q value beyond this E minimum value, so we cannot
have that, the flow cannot be possible, it can go to the E minimum value of this. This is what
we call the critical flow. So we divide the flow into two regions, subcritical and supercritical.
So the supercritical is this part and this part will be the subcritical. If you look at this
behavior, what it actually happened between the subcritical and supercritical?
No doubt you know it that we can compute the flow Froude numbers in fluid mechanics or
open channel flow to know it whether the flow is subcritical or supercritical based on the
flow Froude numbers, as flow Froude numbers lesser than 1 that is what will be the
subcritical flow and flow Froude numbers greater than 1 will give a supercritical flow that is
by definitions you know it, let us understand physically what it actually happens.
In subcritical flow if you look at the flow depth is more, velocity head is less. In supercritical
flow, flow depth or the pressure energy is less but kinetic energy is more. Just try tounderstand it, I know it you all know about flow Froude numbers and you know it by
definition what will be subcritical, supercritical, but try to understand it that if the flow is
happening, it has two energies now as we have discussed earlier that specific energy
representing the two energy.
One is the pressure energy per unit weight another one is the kinetic energy per unit weight.
So if you look at these two energy, these two energy in a subcritical and supercritical what is
the difference? The difference is that in case of subcritical the pressure energy is quite
significant as compared to your kinetic energy, but when you go for supercritical flow the
kinetic energy is much higher as compared to the pressure energy.
That is the reason is a higher value. If you look at this graphically, it is a higher value.
Here the pressure energy, the kinetic energy lesser. That is the way try to understand it, it is
not like flow Froude number is lesser than and greater than one to define in the subcritical,
supercritical flow, but try to analyze in terms of energy available to the flow, pressure energy
and kinetic energy.
So the pressure energy what we found is quite significant high in case of subcritical flow as
compared to the supercritical flow. In supercritical flow the kinetic energy is more dominant
as compared to the pressure energy that is what the point we should try to understand instead
of just remember it the flow Froude number is lesser than 1 or greater than 1. So that is the
point what I want to highlight you.
So whenever you get a problem try to understand it, what are the energy components where
using it and how these energies varies with respect to the flow depth is a specific energy
curve, very interestingly indicate to us there is minimum points which is the minimum
specific energy happens, beyond that flow is not possible that is the point is called the critical
flow and that is the flow depth represents the critical depth.
Beyond that we have two divisions we have divided, subcritical and supercritical regions and
if you look it in terms of flow Froude numbers we can also find out flow Froude numbers
greater than 1 is supercritical, lesser than 1 subcritical and flow Froude number is equal to 1
at the point of the critical depth. So that is the concept we should understand it.(Refer Slide Time: 31:20)
Let us understand the specific energy concept in a river point of view. Let me consider the
river width variation are happening. The same discharge of q is coming it with variations
there. Because of that as the width increases, the q will decrease, smaller q, the discharge per
unit width will be decreasing. So if that is the conditions and that means if I have the
conditions the discharge width is varying it from q1 to q2.
So that means q2 is higher than q and q is there, so this is the order of increasing. If that is
there and if you draw the specific energy curve that is the same equation if you consider the
different q value, the q2 will follow like this curve, q will follow this curve and q1 will follow
this curve, the relationship in E and y space. In any y space, we can find out how the
solutions of this value.
The possibility of the flow can happen different E and y for this constant q values that is what
we have drawn it q2, q, and q1. Now let me analyze this curve. Very interesting fact they are
showing in this curve as I say that these points are reaching point of this critical flow. The
point where we have specific energy is minimum that point is shifting. So that means as the q
increases that we need to have a higher minimum specific energy minimum to maintain that
flow.
But if the q is lesser you can have a minimum or specific energy minimum is a lesser as
compared to these ones. So that way, we can understand it that as the river width increases or
decreases, how this minimum energy concept also changes the critical flow depth alsochanges it. The critical flow depth also changes it and the minimum energy concept is
changing.
So it is a very simple way to trying you the two energy partitioning between E and y with a
different value of the q and how the regions of subcritical and supercritical, that is the
subcritical and this is what the supercritical regions with a different q value. We can plot it
and we can analyze it how the flow can happen if I have the variations of the q as the q
variations shown in these graphs. So try to understand these things for river analysis.
(Refer Slide Time: 34:30)
Now if you come into that river having bed slopes, it is not the flat. So in that case that means
if the river is having this ϴ angle of the slope that is the real condition the river can have a
very gentle slope or the mild slope like the Brahmaputra river the slope is in Assam valleys is
about 1 by 10,000 or 1 by 8000 slope that is very gentle slope, but if you go to the
Himalayans the slope could be an order of 1 by 20.
The slope variability will be there. So the slope has decreasing trend as you go from hilly to
the flood plain areas. So looking that things if I consider it there is a slope in the channels in
the river and apply the same specific energy concept for q. Now we are writing in terms of Q,
so because of that the v = Q by A and also we are considering the α which is the kinetic
energy correction factor.
So we have now used the concept of the velocity distributions which is as we discussed it will
be logarithmic velocity distributions in an open channel flow. For that we can compute whatcould be the kinetic energy correction factor. So that is the reason we have introduced α value
here. Considering the velocity distributions, we can find out what is the kinetic energy
correction factors, the kinetic energy per unit width.
But being a slope here we cannot result the flow depth here and the vertical distance in terms
of d cosϴ and in terms of y we can get it y cos2ϴ into this part. This is a very simple way,
just we have used the combo, an inclined channel or inclined river having a ϴ angle okay that
is what we can assume. The slope of the river does not change that significantly.
So we can assume it that it is having angle ϴ and as it is having angle theta what could be the
specific energy equations if I consider the velocity variations in terms of α and Q by A, Q is
the discharge, area is the flow area and we can write the specific energy more detailed form
of a real for a river conditions where it is having angle ϴ, velocity distributions are there as
well as we consider it the y in terms of the normal depth, the projections part.
So simple geometry if you look at that, we will get it the specific energy y cos2ϴ α Q2/2gA2.
Nothing else, it is an equation energy, specific energy equations for a channel with having
angle ϴ and also considering velocity distributions in terms of α. The α is kinetic energy
correction factor. So that is what we can write to get this formula.
(Refer Slide Time: 38:08)
Now let us coming into to a real problem is channel transition. Many of the time as I say that
width of the river can increase and decrease. Same way in the river, we can have a rise and
the fall because of presence of the boulders you can have increase of bed levels or there couldbe a natural fall. So there could be a rise and fall, expansion and contraction. The river width
can increase and decrease.
Because of that let us use that is what is called channel transitions that means the cross
section of the channel changes such as channel width, channel bottom slope. You can have
width of the river can change it or channel bottom slope can change it and there could be a
rise or there could be fall and that is what is transitions and most of the times this energy
losses due to the frictions at these transition locations can be considered too negligible.
It is not that significant order. Also many experiments are conducted that it is not that
significant order because if have a gradually transitions happens that is what we can consider.
If you have that is the consideration let me try to understand it there is a rise. This rise, the
increasing the bed level maybe you consider the boulder presence or be maybe a small weir
construction by engineers.
So if you have that conditions, then what will be the flow behaviors that let us understand it.
So if that is conditions, there is a flow coming in which is having the flow depth y1, specific
energy as we can consider the channel is horizontal you can find out the specific energy and
the total energy is the same H1 = E1 and you have y as a flow depth and ∆z as the rise depth,
energy equivalent rise if I consider it.
Now the point is coming if I draw this, that means what will be the specific energy at the
section 1 and section 2. If i compare it and if channel width is a constant, discharge is q
which is the same for us. That means q is a constant for us and we are drawing the specific
energy curve for that. We are considering specific energy curve if you look at that where the
q is a constant, we can draw specific energy curve.
That means if section 1 is here, then what it actually we are computing if you look at very
simple things.
(Refer Slide Time: 41:19)If you look at that as we have considered there is no energy losses, so energy gradient lines
can be like this. So H1 = H2, but the specific energy at this E2 because energy gradient this
flat is here so specific energy will be at the two locations will be E1 - ∆z. So if the ∆z is a
rise, the specific energy drop will be there that is what will be the E2 = E - ∆z, very simple
thing because of increasing the level.
So the specific energy is going to reduce here as E1 - ∆z. If that is the conditions that means
in a specific energy diagrams one point lies here, the two solutions will lie either two point
here or two dash here, is it correct? So that means the E1 is energy at the upstream point, E2 is
energy at the rise point and that could have two solutions, one is two is here, two dash is here.
So if you look it that way, whether the solution is two or two dash that is what we have to try
to understand here that as the flow is subcritical, so one lies here, the flow is subcritical. The
two is a possible solution for us because the flow will try to remain on the same regime, the
subcritical regimes. As we discussed earlier that when the flow changes from supercritical to
subcritical, there will be some sort of energy dissipations will happen or it will go through the
critical flow depth.
It cannot happen suddenly at points with flow will be transit from the flow will change from
the subcritical to supercritical. So if you look at this curve if the flow is at the one point, the
flow is subcritical, then the solution