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Lecture – 07: Saint Venant Equation and Solver

Welcome all of you to river engineering course and in the last class we have derived SaintVenant equations. Today I will derive the next versions of the Saint-Venant equations in terms of discharge and also we will talk about how we can solve these Saint-Venant equations. It is quite interesting lectures today and focusing on Saint-Venant equations and its solver. If you look at this book, we have Applied Hydrology by V. T. Chow and D. R. Maidment and Mays book which we are following for the Saint-Venant equations derivation in terms of discharge Q and the flow depth and also we have other books what we have been following it as part of the river engineering course. Now, let me look at going back to the contents what we are talking about that, we will talk about flow and geometry of the open channel flow or the rivers, how things are different. Then we will talk about how we can consider this contraction, expansion zone and how we can write a new momentum outflow or the new momentum equations for the control volumes, Rewrite again Saint-Venant equations and more detail we will discuss in it how we can do a classification for distributed flow routing models. Then, we have a solved examples and we will show demonstrations of HECRAS river models, which is the solver of Saint-Venant equations and the continuity equations. (Refer Slide Time: 02:34)Let me go for very basic diagram. If you look at a river, it looks like this, you will have a
thalweg lines, the deepest depth, where the deepest depth will be there, that is what we call
the thalweg line. The regions where we are having the perimeter is wetted that is what the
wetted perimeters or symbolically we represent P and you have the flow depth and this is
what is showing the top width B of the channel.
If you look at that when you have a river, its flow varies, so you may have the low flow here,
this is low flow, you can have the flow at the top width which will indicates is the bank full
discharge and sometimes you can have the flow in this level, which includes the river and the
floodplain, maybe this is 10 year return period flood. So, we have the flow variabilities.
Because the flow varies from low flow, bankfull discharge to 10 years return period
discharge when you have the river as well as the floodplain.
So, there will be the left bank, there will be the right bank and when you go for the flood
situations if you look at that there will be different roughness will come it because of the
floodplain. There is a floodplain, there will be different flow resistance as compared to the
main rivers which generally have bed materials like a gravel, sand but when you come to the
floodplain regions you can have trees, bushes and floodplain area can have a different land
use land cover.
So, if we look at that as the flow varies from the bankfull discharge to 10-year return period
flood, when it comes, it also occupies the space in the floodplain as well as in the river and as
you go for the low flow the channel is confined within this flow. So, the flow variations are
there from low flow to 10-year return period flood or 100-year return period flood. So, you
can imagine it that the flow depth variability is there.
The basic parameters of these in channels like flow resistance, flow depth, the discharge, the
velocity varies with time as well as the space. So, that is indicating it, the velocity varies with
the space and the time, here I can say that area average velocities and the flow depth can vary
with the space and time, but most of the cases we simplified it to the one-dimensional flow.
So, we simplified it to one dimensional flow that is what we have done for the Saint-Venant
equations.Please remember that when you have the river flow, not only the floodplain and the river also
interact with, there is interaction between the groundwaters and the surface water. There is
interaction between surface water and the groundwater. There will be interactions, the water
from the river to the ground waters or groundwater to the river, so there are interactions of the
lateral flow from the groundwater to the surface water.
So, there are the mechanisms working it with a different flow, different discharge, different
flow resistance, mostly we are looking in terms of mass conservation equations, momentum
conservation equations, similar way we can look for also energy conservation equation. So,
we have 3 basic principal equations what we can derive, but those are all I can say is
approximations when you look for real conditions like these figures is indicating for, the real
conditions are much more complex.
We simplified it and we tried to write in terms of mass conservation, momentum
conservation or the energy conservation equations. So, still we have the assumptions, still we
have a lot of simplifications in terms if you look at the complexity of a river systems.
(Refer Slide Time: 07:33)
Now if you look at the next figure which is very interesting figure, it shows that the river is
not as straight channels. We are not having the canals which are straight channels with
constant slope, in this case slope variability will be there, bed material variability will be
there. There are the regions can have the island formations like this, Because of that the
velocity distributions changes.Look at this, these are our major velocity distributions, that is all the velocity distributions
along the channel and here the river has the curvatures, the meanders, but in this case there
were expansion and contraction sections. There is expansions and the contractions of the river
is happening. There are the river bend formations, there is an island, the bar formations are
there.
So, if you look at this river is not a simple channel like canal, where you have a constant
slope and main bed channels, it has the natural channels, because of that there is expansion,
contractions, river curvatures or the river meanders formations will be there. That is a reason
if you look at this major velocity distributions, which indicating is a very complex process
what is happening in terms if you just look the velocity distribution.
If you look at these figures, which are very interesting figure for us, and we look at to
simplify the river flow problems. Like if we look at that I have just a stone hit here and we
have a very gentle bed slope S0, very gentle bed slopes and assuming that there are some bed
perturbations or bed materials like the stone formations are here, what is going to happen?
There will be part where the flow will be uniform flow, means the flow value like velocity
and the depth, the variations of the velocity and the depth with respect to the x. That means
this is if I define x is longitudinal direction that remains very close to the 0, in that case we
can consider the uniform. In other way around the flow depth and the average velocity does
not change with the space or does not change with x coordinate directions.
If it is that, then we call uniform flow and this is a very simplified case when you have the
uniform flow, but there are the cases because of the stone formations you will have an effect
that what will be the gradually and rapidly. Under the varied flow conditions, we will have a
gradually variations that means these variations is there, but this quantity is very small.
You can see that slope of water levels variations is there, but variations is not that large, it is
very small. If it is that condition, we call gradually varied flow, but if varies considerably
high then we call rapidly varied flow. So, these are the reasons we are just here if there couldbe a formation of hydraulic jump. There could be a formation of hydraulic jump and there
could be formations of Eddies.
So, if you look at this way, if we just conceptualize with simple channel, if you put some
obstructions, you can see there are different type of flows are happening or flow reaches we
can approximate it for analysis point of view as uniform flow, gradually varied flow or
rapidly varied flow. After the rapidly varied flow, there would be a gradually varied flow like
free surface change will be there, but that change is not significant.
As the free surface flow depth is a changing it, the similar way the velocity also changes as
you know from basic mass conservations equations. So, if you look at that way there is a
uniform flow, gradually varied flow, rapidly varied flow. As water resource specialist or a
river engineer specialist, we need to look at what type of flow approximations we can do
when you try to solve real-life problems.
Otherwise it is not necessary to solve very complex equations for uniform flow or gradually
varied flow. So, we need to try to understand the river flow at the reach scales, also at the
larger scales like beyond the reach scales what is happening that is what we try to understand
it with this longitudinal velocity variations as well as if we look at this the flow classifications
in open channel flow.
(Refer Slide Time: 12:58)Now, we will come back to the basic concept what we have derived in Saint-Venant equation
in terms of 3 force components. One is the gravity force, another is the friction force. These
two forces are more and that what is equal to mass multiplied by accelerations that is what we
have done it, but as we discussed earlier, river is not a constant width channel. There will be
expansion and the contractions.
If it is that, let me sketch the simple control volume for you. So, if I look it, so I will have the
expansion of the flow, which having Q amount of the flow is coming in that dx distance using
the Taylor series concept, I can make it Q variability like this. So, this is my control volume
and I can consider small q will be the lateral flow per unit length.
That is what is the flow coming in this direction, this is my control volumes and the distance
is dx. If this is control volume, then if you have Q is the amount of volumetric flow coming
into the control volume, we can approximate using the Taylor series the flow variability at the
surface will be this way. The same way I can write about ρβVQ is a momentum flux.
So, β is considering the velocity distribution momentum corrections factor. So, ρVQ will be
the momentum flux and that way I can also write it, ρ if comes out β VQ plus the variation of
ρVQ in dx distance. So, more detail derivations you can look in applied hydrology text book.
So, that means you can have the control volumes, you can write the mass flux in terms of Q
and the momentum flux in terms of ρβVQ that is what you can consider.
If it is that, there expansions are happening. Because of this expansion if you look at that
streamline, it will be expanded, these are the streamlines. These are the streamlines that will
be expanded and there are the formations of eddies. The eddy formations will be there and
because of eddy formations, there will be energy losses. There will be formations of eddies
and that is what will be conducting energy losses there.
So, we are not going into seeks up how the eddies formations are happening and how
quantifying of energy, we are just quantifying energy as we are following this hypothesis of
pipe flow. In a pipe flow if we look at this eddy formations and all in a pipe expansion joint,
we considered as minor losses and we try to establish the energy losses in terms of velocity
head.The exactly same way for these channel expansions, we are following this pipe flow minor
losses concept. We are considering the energy losses is related to the change of velocity head
that is what we are looking. It is proportional to the velocity head through the length of the
course what is causing it. That means what we are talking about that we have considered as
equivalent hypothesis that as the river is expanding as equivalent to a pipe expansion.
As equivalent to a pipe expansion, you will have a formation of eddies and these eddies if
you know in a pipe flow, it is related to the energy losses related to eddies in the pipe flow to
the velocity head. The same concept we are using here to quantify the energy losses due to
the formation of eddies as the river expanses, as the width increases that what will be related
to the velocity head.
And if that is considered as the eddy energy loss slope, Se stands for energy loss slope, this Se
equivalent to a frictional slope. So, as equivalent concept if you consider the force because of
this eddy generated the energy losses on these control volumes can be defined as similar to
the friction slope concept. That you just try to understand it. Because of this energy losses,
what will be the additional stresses which is going to act on the surface, on the bed as well at
the surface?
That as equivalent if you will consider it as we have done for the friction slopes, the same
way we can consider it and we can compute what will be the drag force because of the eddy
formations, because of river expansions, and Se is a gradient of this that is the reasons we
have a partial derivative of the velocity head and is a proportional constant, Ke is a
proportional constants which from experiment we can get it what will be the Ke value.
So, whenever you set up river models, it asks what will be the expansion contraction
coefficient that is depending upon your rivers, your physical model information you can
include it what will be the Ke value for a particular river system. So, that way if you look it
there will be non-dimensional expansion and contraction coefficient, it is negative for the
channel expansion, positive for the contractions of the channels.
So, basically, the eddy energy losses because of expansion of the river or the contractions that
what we consider with these 2 equations as an energy eddy loss slope and the drag force due
to the eddy losses.(Refer Slide Time: 20:04)
Now if you look at the next one what I want to just revisit you that when you consider that
the same control volumes okay. Same control volumes, I have the Q amount of the flow is
coming in and there are lateral flow small q amount of the flow is coming okay and this is the
x direction. So, definitely this is making the velocity components, this lateral flow also is not
along this q directions, there could be a Vx the velocity component along this x direction.
This is the lateral flow and all the things. So, now if we consider this is the control volume
where we have the lateral flow what is coming in and this is the control volume in and out,
then we can write it mass influx like this way. You can find out this Q and the qdx this is the
lateral flow per unit length. So, mass influx you can write it and corresponding momentum
will be the ρVQ.
This ρVQ multiplied by β which is momentum correction factor, to get momentum flux. Here
we have used the momentum flux components because of Q discharge which is coming, it is
making a Vx component along the x direction. So, here we are considering this Vx, the lateral
the velocity x components into this part into qdx. So, this is the momentum flux due to the
lateral flow which is not coming along this Q direction, it has the inclination.
Because of that, let us consider the Vx amount of momentum flux velocity is coming as
equivalent the momentum flux we can get it. So, you know it and we can find out what will
be the moment of flux going out from this control volume that is what will be the ρVQ and its
gradient that is what is the Taylor series. If I try to look at net momentum crosses across thecontrol volumes, just looking at influx and outflux, we are considering the sign conventions
of positive and negative.
For outflux is a positive and influx is negative, To understand this conventions please follow
Reynolds transport theorems which has discussed in any fluid mechanics lectures, also I have
some lectures on fluid mechanics. So, please go through it, why do we have a positive and
negative sign for the moment of fluxes. So net momentum fluxes will come out to this. So,
now if we look at that moment flux what earlier we derived, now that will be the different
part.
(Refer Slide Time: 23:17)
So, for more detailed derivations and all you just look at in applied hydrology book, I am just
skipping that part of more details, but if we write the momentum equations Saint-Venant
equations, you will finally find out like this form. So, if we look at that the additional
components we are getting because this expansion contractions point of view, this is because
of lateral flow, this is because additional part we call it wind shear factors.
So, many of the times also when you have a heavy wind, big reservoirs are there, we can
consider the wind force also acting on that, that part please do the self-readings, which is
there in applied hydrology book. So, basically what I am trying to do is that in we have
derived the Saint-Venant equations now in terms of Q and h and the A. So, we have derived
this equation now the Saint-Venant equation in terms of Q, A and h variables.And we have included the Ke is the expansion and contractions energy loss gradient, we have
considered the moment of flux because of lateral flow, we also consider the force part or the
energy what is acting if there will be the wind force acting over the rivers or mostly this is
significant when you have big reservoirs. So, there will be certain commentative during the
high wind speeds that will be also give a force acting on that.
So we can include this wind shear factor here to estimate that. See if you look at this, again
we are getting the Saint-Venant equations in terms of Q, h and A and in terms of partial
derivative x and the t, but still it is a nonlinear partial differential equation, still we have that
forms. Now, next we are going to discuss that how we can simplify the equations.
(Refer Slide Time: 25:47)
Now, as I discuss with you that we could look at the Saint-Venant equations is a nonlinear
partial differential equation, but solving this nonlinear partial differential equation is not that
difficult, but most of the times is difficult to give appropriate river geometry. As I said it in
very first that river is quite dynamic, low flow, bankfull discharge, 10-year return period flow
systems or 100-year flow return period systems all the floods.
That is the reason we need to give a lot of input data into a river model, even if we solve this
Saint-Venant equation. That is the reason and most of the times all elsewhere in the world
also getting so detailed data along the rivers, the river depth, the land use land covers, more
details it is not possible. So, we try to do a simplification, we try to look the dominancy
behavior, whether we need to compute as a Saint-Venant equations or we need to do some
simplifications on that.That is what we have to do under this classification of distributed flow routings. So, if you
look at this Saint-Venant equations, it has different terms like there are the local acceleration
term, convective acceleration term, presser force terms, gravity force terms, friction force
terms and others what we have included expansion-contraction terms, the wind force terms
and the momentum because of the lateral flow.
So, if you look at that way which are the significant order, are all these terms are the same
magnitudes or some are not significant like maybe some cases the local accelerations may not
be significant, more or less the flow is steady, the discharge variability is not there. So we can
drop this local acceleration term. Similar way can we drop these convective acceleration
term, in which case we can do it?
Can you drop these expansion and contractions part, this is depending on the rivers, if the
expansion and contractions is not there, you can drop that part. So the basic idea is to locate
the dominancy behaviors because each term we have to try to know what is the order of
magnitudes, is it a significant? If it is not significant, you can neglect that part. That is the
approach we will follow it.
This equation has to be simplified into one dimensional distributed routing as I said that, but
many of the causes you will have to try to understand what it actually happens. Like this
example what I said it earlier if we have the flow, the flow can change from critical,
subcritical or supercritical. More detail about the critical flow we will discuss later on, but
you try to understand as you know from basic hydraulics that subcritical and supercritical.
When in subcritical, the flow Froude number is lesser than 1. So in that conditions what it
happens is if you do any disturbance, it propagates in both upstream and the downstream.
That is what it happens that, that is what is called backwater effect. So that will be the
backwater effect, in case of the subcritical flow we will have a backwater effect.
Basically when you look at the Saint-Venant equation, the classifications means we try to
look at all these terms in Saint-Venant equations like local acceleration, the convective
acceleration, the pressure force terms, gravity force and friction force term which are the
dominated component, which are the significant? Like for example, in case of the steadyflow, the local acceleration terms will not be that significant, so we can drop that part, we can
consider other 4 terms.
The basic idea comes here to know it that how we will be dropping or we will be
approximating some of the terms in the Saint-Venant equation so that we can easily solve it
with a limited river geometry data set that is the basic idea. When you have a very limited
river geometric data, you can do an order analysis of each terms and find out which are the
terms that are the significant and which are the terms that are not significant.
Based on that, you can drop the terms. For example, if you have the subcritical flow as you
can understand from basic hydraulics books that the subcritical flow it matters for us to know
it whenever you do a disturbance, it affects both upstream and downstream and that is the
cases you can have this local acceleration component and convective and pressure terms. For
these conditions, the Lumped routing methods will not be suitable, when you have the
subcritical flow when you have a backwater effect.
(Refer Slide Time: 31:15)
Now, if you will look at basically what we try to do when you talk about these Saint-Venant
equation solutions when you go for the hilly area, the floodplain area, the slopes considerably
change, the gravity force components considerably change. So we try to locate the force
dominancy part. Many of the times we neglected the local accelerations, the convective
acceleration and the pressure terms.Then we have a kinematic wave models where it is very simple model, the bed slope is equal
to the friction slope, It is very simplified now if you are considering that local accelerations,
convective accelerations and the pressure terms are not that significant. So that means you
have only 2 terms, the friction slopes and the bed slope, that is what will comes out to be the
Saint-Venant equations approximation that is the approximation when do it we talk about
kinematic wave models.
That means S0 = Sf, it is a very simple model and we can solve the cases very easily. Even if
in Microsoft Excel level we can develop a river model when we have kinematic wave models
S0 = Sf, but there are the cases we can neglect only the local and convective accelerations, not
the pressure terms. You can consider neglecting only this local and convective accelerations,
then you call diffusion wave model.
And if you consider all the terms then we call the dynamic wave models. So, we have now
classify the 3 different approximations. The first approximations we consider the local
acceleration, the convective acceleration terms and the pressure terms they are not that
significant order. So, if that is the conditions like hilly area where the slope is much larger, so
we can approximate the kinematic wave models to do the river routings.
But when you come back to the regions where the pressure variations are there, water depth
variations are there, but local acceleration and convective acceleration are not that significant,
then you can use diffusion wave model. If you consider as a Saint-Venant equation, then we
call the dynamic wave models. So, whenever you do the flow routings using the Saint-Venant
equations, we can look at the other whether we were doing it kinematic wave models,
diffusion wave model or the dynamic wave model.
(Refer Slide Time: 34:08)Now in equations form if you look at that again coming back to this, we define in 2 terms is a
conservative and non-conservative. If you look at this first continuity equations which have
conservative form because we in terms of mass flux we do it, so that is the reasons you do not
have any approximations when you do the conservative form of equations, but in nonconservative form, you write in terms of velocity and the flow depth.
There is not exactly in the mass form there is approximation that the discharge is equal to
area into the velocity that approximation which gives a non-conservative form. That is the
reasons we put it a non-conservative form because maybe some cases these conservations of
mass may not hold good. So, that is the reasons we call it non-conservative forms as we have
a conservative equations and non-conservative equations like this.
The same equations only we have put it here and there is no lateral flow. Same way if you
look at that, you can have a conservative form which will be in terms of Q and
nonconservative form in terms of the V. If we look at the velocity and the flow depth, and
there are certain assumptions like if we are considering these two parts. Again just trying to
summarize that you will have a kinematic wave approximations.
If you consider these 3 components, then we call the diffusion wave models, if I consider all
these components then you will have a dynamic wave. So, if you look at this, these are the
equations in Q and A or the Q, V and y, y is the flow depth. All this only 2 equations we can
use it to solve the equation, 2 equations for the 2 dependent variables like the discharge and
area or the velocity and depth.If you solve it you can do it, but as I said it, it is not possible to solve these equations
mathematically. It is getting the flow geometry data and channel geometry data and the flow
resistance all are not easy for natural rivers. So, that is the reason we do certain degree of
approximations like kinematic wave models, diffusions wave model or the dynamic wave
model.
(Refer Slide Time: 36:51)
Now let us come solutions to example problems. In a river the average velocity and the water
depth are measured at the three sections okay. The velocities given V1 h1, V2 h2, V3 h3 and
that the velocities and flow depth were measured at 1 hour interval. Justify the applicability
of the diffusion wave approximations for the reaches. So, looking at this velocity and depth
measurements, we can justify it whether the applicability of diffusion wave approximation is
okay for us.
So basically, we try to look at all the terms of this Saint-Venant equation and try to look at
whether in this case diffusions wave models if you consider is it okay for not considering the
velocity and the flow depth data, this is the measured flow depth data. So, if you look at the
velocity data and flow depth data, so if you look at these variations and u look at this, so there
is a velocity.
So, if you look at this equation, we try to look at the gradient of h along the x direction,
gradient of U in the x directions and the gradient of U in the t direction. So, we try to look atthe velocity, the temporal gradient of that velocity, the partial gradient of velocity and .
Then we try to look for each term how the significant order of each terms, if they are in not
significant order, then you can approximate it as a diffusion wave approximations.
(Refer Slide Time: 38:48)
Now, if you look at what we have done it for real river cases, we try to find out the gradient
okay, is the gradient that is what we follow the Central Difference Scheme. If I follow it
we know the U3 U2 by dx, here is 2 meters, I can compute what will be the gradient. Same
way at the time t I can know the gradient. So, I can compute what will be the average at
Central Difference Scheme, I can compute that part.
Same way, we can find out what is a temporal gradient of the U values. That is what we can
put it and we remember we can put it in terms of minutes, 60 minutes and you can get these
values.
(Refer Slide Time: 39:40)Now, if I look at and I look at values and that is what I can compute for the t = 0 as also
t same way we have in the Central Difference Scheme to find out the gradient h with respect
to the x that can be approximated as delta U here as well as it is simple central differencing
methods to approximate what will be the gradient. We have got the gradient like 1 by 20.
(Refer Slide Time: 40:17)
So, if you look it this way now for each part, we just substitute this value. So, each part we
just substitute gradient in terms of the bed slope and all these terms. If you try to look at that
you can easily find out this x4 is much lesser, the x4 part is much, much lesser than x1. So, if
you look at these two terms, these two terms are much, much lesser than these two terms. So,
we can always tell it this is not significant as compared to x1, x2.As compared to this x1, x2 these terms are not that significant. If I am considering that part,
then I can assume it because this term is very less, the x4 terms are very less, we can easily
tell that we can use diffusions wave approximations for this case. So, these examples are
indicating that you can have a velocity, you can measure the depth, and from that we can find
out what type of models we can use for Saint-Venant equations.
You need not to know its total dynamic wave mode