Lecture - 10: Flood Wave Celerity and Loop Rating Curve
Good morning all of you for these lectures. It is quite interesting lectures on flood wave celerity loop rating curve and at the introductions level of the sediment transport. If you look at that whenever you have a flash floods when you have a dam break and the conditions where you have a glacier lake outburst floods if that the floods is propagates like a wave and what is the celerity of that flood propagations? That is what today we will discuss it along with we will discuss about loop rating curves. And also we will give a descriptions on introduction levels descriptions on sediment transport. So mostly, I am following these books okay. And the partly we are following this book for these lecture materials. If you look at what we are going to talk about as I said it the river is morphological active when you have extreme floods. The extreme flood happens because of a flash flood happening because of glacial lake outburst or flash flood is happening because of dam break. Those flood events, how these flood wave propagates it. What is the celerity of the floodway propagations? That is what we discuss mathematically as well as the graphically we should understand it. Then we will talk about loop rating curve. So rating curve what we get it, it is not a constant functions. It has a loop activity. Then we will have a case studies which we conducted studies, long back glacier lake outburst floods propagations in Tawang Rivers. Then I will give you a brief presentations on sediment transports in river.
(Refer Slide Time: 02:24)Now if you look at next part as I discussed earlier is that whenever you create a
disturbance like this, okay and we derived the class that we will have a celerity that
means, the wave velocity with respect to velocity of medium with which the wave
traveling. So we are talking about what the wave velocities moving it when you create
a disturbance of that.
(Refer Slide Time: 02:51)
And the last class also we discussed that, when you have it that conditions it depended
upon with a simple derivations, we can see that the wave celerity is a functions of the
flow depth and it depends upon whether the flow is subcritical, supercritical or the
critical. So in case of the subcritical flow, the wave will propagates in the both
directions both upstream as well as the downstream directions.But in case of supercritical flow, both the components of the celerity of this part,
which will be indicates the flow the wave propagations in towards the downstream
only. But in case of critical which is the rare situations it happens it that we will have
only this downstream directions. Because based on this concept, we try to look at
whether is upstream controls or the downstream controls.
That is what we discussed in the last class. Let us today go for further that if they
reach a flood wave, how it propagates it. How we can derive the celerity for that.
(Refer Slide Time: 03:59)
If you look at the next slides, what we are talking about, what is a wave celerity. We
if I considering a one dimensional equations okay, we are simplifying it. So we have a
one dimensional channels. We are looking at the flood wave propagations along the
longitudinal directions. If dQ is functions of space and time, I can consider the dQ
variabilities along these lines, you will have a dQ/dx.
It is just a simple calculations that you can define it. What is the celerity? The celerity
is the points where this is what the propagates it. That is the reasons by definitions,
c=dx/dt. That is where they define the locations where the flow become steady, when
Q equal to dQ equal to 0. That is what it comes.
The dQ = 0. So if I just considering the Q variabilities and the definitions of the
celerity I can write c is a function of these ones. This simply the substituting these
ones and if I consider is one dimensional flow mass conservations equations as dQ/dx,if there is no lateral flow as we derive in the classes that if there is no lateral flow we
can derive the change of the storage equal to the rate of change of volumetric flux
along the x direction.
That part if I equate it, I am getting c = dQ/dA. C in terms of Q and flow across
sections. That is what is called flood wave celerity which is Kleitz-Seddon law. Now
if you look at this part which I will discussing with you this derivations will do in
latter part, but let me look at that the concept wise understandings of this part.
Dynamic wave, kinematic wave, diffusive wave.
So if you look at this part when you have a dynamic waves, that means what it
actually happens is that you have a conditions when you have the wave like this
propagations is coming it. So when you have these type of wave propagations in that
case the you will have celerity at this point and if you take the two sections A and B if
I compute the celerities I will see that cB will be lesser than cA.
The celerity at the point of A and B which is a distance l apart if I look at that as sf is
greater than so and these case sf will be less than so values. So, cA will be the lesser
than the cA. In that case, what will happen it after certain distance if this wave travels,
the travels flood wave amplifications happens. That means, this will be traveling
faster than the B point.
Because of that, the peak will be increasing it, amplified it. Will be amplified it the
distance l here in the same AB locations l2 will be lesser than l1. That is the conditions
happens if you look at higher flow numbers. If you look at the diffusive wave, this is a
kinematic wave. If you look at the diffusive wave cases, what it actually happens it
just reverse of that.
In this case, if you look at the flood wave which is having a two point A and B apart
from a distance l1 the cA and cB, cB is larger than the cA. cB the celerity will be
larger than the cA. In that case, the length will be increased between A and B. So the
flood will be attenuated. The peak will be reduced. So that is what It is happens for
the dynamic wave.But in case of kinematic wave which maybe sometimes happens is that where is very
simplified momentum equation is sf the friction slope is equal to the bed slope so. If
that is the conditions what it happens is that the cA is equal to cB celerity point of
view which is l1 distance the same way it is propagated same distance entity.
So if you look at the graphically if you have a different type of waves like a dynamic
wave, diffusive wave and kinematic wave, the flood wave can attenuate it or
amplified it all it depends upon the celerities. So now let us derive that celerity part in
the next slides.
(Refer Slide Time: 09:18)
If we look at the celerity part if I consider that wide rectangular channels, if I consider
the channels which is much larger, big, wide. That definitions is quite valid when you
talk about the rivers. So we have a wide rectangular channels, you can define these
area per unit width can have a functions with a power functions with the flow depth.
That is what we can define is a power functions of the flow depth.
Whereas the power coefficients α and β are unknown for us. That what can vary from
river to river. So we can define this unit discharge of the Q in terms of the flow depth.
If I use that same concept, okay? This is what will have a basic definitions, because
we are considering h is the flow depth and h is the function of the space and time.
So if you can look it and derive this dh/dt, which should be equal to 0. That
substitution if you could do it. And that what will come it back to this the samederivations what you can do it. And substituting this power functions, you can see that
celerity is a β times of P. Okay, celerity is a functions for a wide rectangular channels.
Celerity can consider as a multiplications β times of V, and β is the power exponent of
the discharge and the β values.
So that reasons, the wave celerity is always faster than the flow velocities because
when β is greater than 1. That is what you can see this very simple relationship. The
celerity of the flood wave increases with the flow depth. That is already we derived it.
The larger the flood wave we have a larger flow depth. That is what is propagates
faster than the smaller flood wave.
So larger the depth, so you have more the celerities. And that is the reasons it
propagates faster. Its travel time takes lesser. The smaller flood wave, it takes larger
time steps. It is as the celerity is less. So that what we should try to understand it,
what it happens for a larger flood wave, what it happens as a smaller flood wave.
Because of that, there is a nonlinear propagations of the flood wave. So what we use
the linear techniques based on these superpositions concept like a unit hydrograph it
does not valid for the river, where you have extreme flood events happens it, when
you have the flash floods happens it. The wave celerity is quite large. So it indicates
that method of isochrons we used in hydrology is also not applicable for the river.
So like for example, if I talk about the Himalayan rivers where the large flash floods
occurs it, the simple concept of the linear superimpositions and this isochrones
methods does not valid for us. So we should try to locate that what it happens the
flood wave, how propagation happens. Like if you talk about the celerity of the flood
wave increase with the flow depth, and that the case is a larger flood wave takes the
lesser time to propagate from A to B locations.
But the same locations when you have a smaller flood wave it takes the longer time.
So it make a nonlinearity functions. That is the reasons we cannot use the linear
hypothesis what we have in hydrology. That is, we do this flow routings based on the
linear concept but that is what it is not valid when you talk about the flood wave.And as a morphologist as we look at the river mechanics, as you know it the flood
waves play the major role for the change of the morphologies. That is the reason its
extreme conditions. That is try to understand that how the flood waves are happening.
(Refer Slide Time: 13:46)
Now if we look at the next part if I like try to look it the derivations, which is there in
the book of Julien book, if then V can be defined in terms of functions of h and β-1
and then V can define like this. If I look it write it again with some simplifications of
St. Venant equations, what we derive for the dynamic conditions the friction slope
will have a bed slope plus there is a components will come it which is functions of
beta and the flow Froudes numbers.
And you have a temporal components here. So if we combine a continuity equations.
And dV is this ones then you get it the so is these values . So that derivations please do
it. So what you can see is very interestingly that we have a flood wave diffusivity
terms, okay. dh is there and these term is defined is a flood wave diffusivity, which is
a functions of β, flow Froude numbers.
So if we look at that if β=1 and flow Froude numbers equal to one, which is the
critical flow conditions. And in that case you may have this value comes out to be 0
and you will have only diffusive factors is comes to be 1. So what I am telling it if is
β=1 so you will have the Froude diffusivity value equal to 1. Or you can show this, it
is a functions of Fr.So if Fr is the flow Froude numbers is close to the 1 then it only depends upon the β
value. But if you use the Manning’s equations, which is the functions of the velocity
and hydraulic radius and so sf of if you use that values and combining it you will get
these functions like this.
So we are not going step by step derivations please follow this the book which is there
by Pierre Julien’s so that where Froude wave diffusive terms it depends upon the
value of β and the flow Froude numbers. So these ratio between these terms which is
defined is the flood wave attenuations at the low value of the flow Froude numbers.
(Refer Slide Time: 16:44)
Now if you try to understand it if I define in another is called the diffusivity ratios,
okay which can be written is that how much of diffusivity ratio is happening it. That
is what also can be written in terms of dh/dx will be a functions of dQ/dt with the β
functions, so functions, α and β value. And Manning equations if you consider β =5/3.
The diffusivity ratio is comes out to this. So, the rapid change in the discharge
increase the flood wave diffusivity. That is correct. That means if I suddenly increase
the flow in temporal domains i.e. suddenly from 3000 cumecs of discharge, it goes up
to the 10,000 cumecs of discharge. The sudden dQ/dt will be much larger value.
So it will increase the flood wave diffusivity, increase the flood wave. That is what it
happens in flash floods. Suddenly the discharge which is much lesser, it just jump it to
one order or two orders. So that is the reasons it started the flood wave diffusivity. Sogiven the flood wave the Q and dQ, diffusivity increase with the Mannings of
coefficients of n and decreases with the channel slope.
So decreases with the channel slope, like the Himalayan rivers the channel slope is
much larger. So you can see that it can decreases with channel slope. Channel
straightening higher bed slopes channel lining the lower and decreases the flood wave
diffusivity of the natural slope. So why you do the channel straightening. So we are so
what it happens it? Change the bed slope.
The higher the bed slopes or channel lining, why do we do it? Lowering these
roughness values, Manning's roughness coefficients. That what also helps to decrease
the flood wave diffusivity or natural channels.
(Refer Slide Time: 19:04)
That is we should have also components what you look it. If I look it in terms of the
velocity applying this the basic equations of Manning’s equations and substituting
these sf value what we derive it, you will get it that. It is a very simple derivations. If
you look it that you will have s is that. The celerity will be β times of V, you will have
this part. Now let us interpret it this part which is there in the text.
The flood wave diffusivity play the dominant role alterations of the flood waves. That
is correct, okay. Dynamic waves in a steep channels when you have the channel is
very steep, okay. ϴ is very high for the channel slope. In that case ϴ is very high,
channel slope tends to form a pulsating flow or surges rolling waves. When you havea laminar flow the roll waves can theoretically form when the flow Froude number
greater than 0.5.
The steep flows are possible for the subcritical flow when the flow Froude numbers
greater than 0.7. In turbulent flow, I am not emphasizing laminar flow as the river we
have the turbulent flow. So in turbulent flow when you consider β is 5/3 the roll
waves develop very steep smooth channels under the supercritical flow.
Just try to understand it what it happens in a rivers when you have turbulent flow and
you have the flow Froude numbers more than 1.5, it can create a roll waves, okay.
The wave will have a roll waves will be there. Roll waves supercritical flow should be
avoided in open channel flow. That is we should try to understand it because it causes
the surface disturbance.
And the cross waves incurred by perturbation of the bank and the bed. So when you
go for the supercritical flow, it is will have the roll waves and supercritical flow
should be avoided from open channel design, but in case of the river does it happens
it? It can be best to achieve increase the boundary roughness to extend the flow,
which should be remained at the subcritical flow.
So basically if there is a boundary roughness which will make it the subcritical. But in
a reach of the river, if there is a supercritical and the flood wave propagates it which
generate the flood roll waves and that the surface is interact with the instability and
the cross waves. That is the reasons sometimes the river flood is so chaos.
So turbulent structures is created, because of maybe the localized formations of
supercritical flow, localized formations of supercritical flow with the flood wave, it
can create the roll waves and can propagate like a tsunami type of the waves which
will be totally destroyed it instability and cross waves. So it happens it but localized
conditions, but it can talk about the extreme conditions what it happens it
(Refer Slide Time: 22:38)Now if you look it that if our kinematic waves when you have flow Froude numbers
of these values, in this case, the bed and friction slopes are identical, wave and
discharge increases slowly on to the flow depth. In most of the flood flow is
subcritical, okay. Flow routing is generally described by the diffusive wave
propagations of St.Venant equations.
But in localized conditions, extreme conditions like the flood propagations in
Himalayas where large scale of the bed elevations variations are there. So in that case,
flow may go for the critical conditions. Otherwise, flow is a subcritical. That is the
reasons we use the St. Venant equations and try to solve it. In such case the wave
celerity and discharge do not vary with the flow depth.
But also depends on the gradient of the slope flow depth in the downstream
directions. That is flow wave attenuation is most effective the flow Froude number is
less dh/dx is large compared to the bed slopes. So that is the understanding you should
have to know it what it actually happens when you have the flood waves.
(Refer Slide Time: 24:03)Now if you look it the next components as you go it is called the loop rating curves.
Just you try to understand it that the relationship between the discharge and the
elevations we call the rating curve. Are they it is a power functions or it is a loop?
That is what actually happens it is a loop rating curve. If you look at this rating curve
of a Mississippi rivers at a particular locations the x axis is the discharge y axis the
What is actually happens it as the discharge increases it follow this path, then come
back it like that. So if you look at it this way, when you have again, I am to repeat it
that the loop rating curves. That means as the discharge is increasing in channels it
increases with the elevations, but after the maximum discharge reaches it when its
falls is comes it, it have a higher flow depth as compared to the rising stage.
Why does it happens it. So when if you look at that when you have a rising stage you
have a dh/dt is increasing trend. And we have the decreasing trend when you have the
delta the temporal variations of h is a negative directions. So if you look at this the
discharge if I look it in terms of the same equations, we are writing this as with
multiplying the velocity into area we know this hydraulic depth that is dh/dx, then this
is a flood diffusivity constants and these value.
Now if you look it when you have a rising stage sf not equal to so. That is what it
happens it. When you have falling levels of the water hydrograph the loops are
induced by the rating curve relationship, okay. So if you look it that, it depends uponthis rating curve relationships if you look at during the rising period as a
counterclockwise loop is often between a relations between stage and discharge.
The elevations versus in most of the river channels you would have received. The
maximum discharge reached before the maximum flow depth okay. But bed shear
stress if I can compute it, which is governed by the shear stress equal to the unit
weight of the waters hydraulic radius and the friction slope. If you just substitute the
friction slope I will be getting like this.
It is a functions of dh/dx, okay. Functions of flow Froude numbers, functions of so.
Function of flow Froude numbers and functions of so. If I look it this way, what it
actually happened the shear stress at a given flow depth, the shear stress and bed
sediment transport are larger than the rising limb than the falling limb, okay.
So what it actually happen it when you have the larger rising limbs if you try to
understand these equations you will have a more sediment transport okay as the bed
load transport than the falling limb. So the Meyer-Peter formulas, which we will
discuss later on, which is a giving a reasons between the access stress between the
applied shear stress and the critical shear stress as a power functions with a bed load.
That is what. So bed loads also depends upon that as your asset value is changing
between the rising limb and the falling limb, you will have a more bed load sediment
transport in case of the rising limb as compared to the falling limb, okay. So more
detail you try to understand these situations and you can interpret it that how does it
(Refer Slide Time: 28:29)Now let it look at the sediment rating curves, what it actually happens the relationship
between sediment discharge versus the water discharge, okay. Qs increases faster than
the Q. That is what if you look at that if you have a kinematic waves, just look the
figures okay, kinematic wave case, you if you look at that, if we have a red color is a
t1 time and the green color after dt time the flood wave will have a change from the
red to the green, okay.
So you can have the kinematic wave. So the peak will not unrouted it, but the
sediment will go like the same way if you look at that, sediment discharge versus
time. But in case of the dynamic wave, you can see that there will be the
amplifications. That is the red color is upstream hydrographs, green color is a
downstream hydrographs and you can see these amplifications happens it.
Because of these amplifications of things, you will have a bed degradations will
happen it, bed erosions process will happen it during these amplification process. And
you can see that the downstream and upstream sediment graphs with the times which
will change it. The same way if you look it if you have a diffusive wave the flood is
attenuated. The peak decreases and the length of the base increases.
In that case what will happen the channel bed aggradations will happen, the
aggradations will happen the depositions will be started, when you have the flood
attenuation process that. The effects of the dynamic kinematics on the discharge andbed load transport is can be shown it and we should try to understand it how the
sediment transport happens during these flood wave propagations, river flood wave.
(Refer Slide Time: 30:29)
Let I go it a very a simple case studies what we have done it for glacial lake outburst
floods which is the publications here. If you look it that, what you have done it for a
lake which is a danger is quantifications of discharge potential lake. And that is what
we have used a mathematical models and use a dam break and hydraulic flow routing
(Refer Slide Time: 30:56)
So if you look at that part, so what is there we have done the identifications of glacial
lake, vulnerability of the glacier lakes which identify lakes for detecting a critical lake
and you use the lake location as a dam break and graph assessment using ahydrodynamic routings. That is what if you look at this the study area false color
(Refer Slide Time: 31:21)
And if you try to look it that this is the lake locations and we are looking at these
outer locations what will be the flood hydrographs. So it is showing it the water sets,
and you can see that how the velocity varying it. It goes up to the 8 m/s is much more
higher, okay. And you can see that the lake side hydrographs and the hydrograph at
the project sides, okay, you can see that.
The flood attenuations are happen it and that is what we got it. And this can go as
high as 2000 cumecs and at the dam site, the project side it can reduce to the 1500
cumecs and most often if you look at this the velocity variations which is 8 m/s,
which is much larger, okay. You can try to understand it in Himalayan regions if have
the GLOF type of conditions the velocity of the GLOF is a range of the 8 m/s.
So this is the case studies what we did it and it is a quite interesting to you with all
these recently geospatial database, we have used to derive what could be the
celerities, what could be the flood attenuations because of GLOF, Glacier Lake
(Refer Slide Time: 32:49)Now let me I come back to very interesting topic as introductions level I will talk it
today and more detail will go later on. See if you look it that there are bigger issue of
the sediment transport the issue 1950s onwards. It was started that we should try to
understand the sediment transport mechanism conducting the channel flow. As you
have seen this is a channel flow with a sinuous channel or you can have a non-sinuous
And if you increase the flow depths as you know that it will have the more the shear
stress. As you increase the flow depth and you have a more the shear stress the bed
particles which would be there, they will start motions, okay start moving from the
bed. That is the conditions we call incipient motions.
And if you further increasing the flow depth then what will happen it, a layer of the
bed will be move it, a bed particles will move it and they can go for a different
process okay with a rolling, jumping bed particles will go it and you can have a the
bed particles can contribute a partly to layers which is moving along the bed and there
will be part which is above of that which remains as a suspended conditions.
All these things are happens it we look it in a very microscopic scales because the last
two decades there are lot of study has been done it look it the very microscopic level
at the near bed conditions. What it does it happens. What is the hydraulic conditions
happen the near bed? What is the turbulent structures are there? Why the bed particles
are detaching from the bed?What the hydraulic condition also it is depend upon the sediment characteristics. If
you try to understand this case and if you look at this flow behaviors and if you put
the color dyes and it shows that how the flow things are happening it okay. So it is
easy nowadays. If you have the flow, you can conduct the experiment to know it the
quantifications of turbulent structures, incipient motions, the bed load, the suspended
All these experiment we can conduct it and we can try to establish the relationship
between the hydrodynamic characteristics, sediment characteristics and with a
sediment transport processes.
(Refer Slide Time: 35:16)
Now one of the process is what is done it cases if you look at the bed load transport in
a rivers, which is looking at very microscopic scale, if you look at these figures okay.
If you say that the uc is the velocity is coming which is at the incipient motions the
conditions at the bed okay the bed particles are just starting the moving incipient is a
average velocity uc is coming it at the particle levels can have a the velocity which is
The particle levels which is a critical velocity just push that sediment particle from the
bed one positions to other positions or it can go to the suspended loop. More detail
enlarge view or microscopic if you look it at the sand particles microscopy are theseyou can see that there will be the velocity components and these particles are having
the trajectory detaching from that, okay detaching from that.
That is what is there the flow velocity depends upon the flow velocity, it depends on
the mean flow velocity. It depends upon the sediment particle sizes, the d50 d90s. It
depends upon subscript stands for the threshold conditions of motion of the sediment
particles is incipient motions. And typically if you look it the target particles are
moving. So nowadays people are looking that level how these target particles are
Same way if you look it that, how the shear stress distributions are happening it. If
you look it, it has two component. The decomposition of the fluid is bed shear stress
into dispersive particle shear stress, interfacial fluid particle stresses. That is what is
happening it. If you look it that, it has two components. One is dispersive of particle
shear components, okay τo which is high as you go up it reduces.
And interfacial the fluid shear stress particles. And if you look at these particles, the
bed layers okay, it is a very good and the bed compositions if you look it try to
understand it and there is eddies formations and the rolling, sliding will happen it. So
more microscopically if you look it that how the shear stress acting it along the rivers,
along the channels.
In that case you can clearly interestingly look it that the particle shear stress, which is
much more higher, that is what is going to decrease along the depth where is the
interfacial fluid shear stress which generally in fluid we talk about that that is what is
also have a increasing trend. And we should know it and more detail if you can follow
these recent publications on scaling large of the sediment transport.
(Refer Slide Time: 38:13)Not only that, the people nowadays use the CFD solvers to know the sediment
particles how it moves it. You see these velocity is there. CFD solutions are there and
the particles are considered a solid rigid circular balls and each force components are
there and we can use the CFD as a multi-phase flow concept. Try to know it how the
sediment particles are dethatching from the bed, remains and suspension states are
falling to the bed, how this happens in this process.
So if you look it that, at the microscopically we can know it nowadays because we
have a tools to measure the velocity distributions and all as well as we have also the
computational capacity now to run a CFD softwares very detail to try to for a
multipath flow, try to know it that how the particles are dethatching from that or
starting the incipient motions, how long it will remain as a suspension stage.