# Mod-2 Lec-5 Uniform Flow in Mobile Boundary Channel

Friends welcome you all to this class of uniform

flow in mobile boundary channel. In the last few classes we have discussed about uniform

flow, various aspects of uniform flow, but all those that we made those discussions were

on rigid boundary channel, that means, when the channel boundary is fix by boundary here

we mean that either the bed or the bank I mean everything is fix, so on in that context

only we were discussing uniform flow. Now, today class we will be basically concentrating

on the issue that if the channel boundary is not fix, it is mobile like say in a river,

it can happen particularly when the river is flowing through a alluvial soil with say

fine sand or may be little coarser sand, but still suppose the water can erode that sand,

then there can be change in the bed form; and then similarly, suppose, if we talk about

manmade canal also say in irrigation canal, we can have sometimes due to economy we may

not it may not be possible to provide a lining to all those channels. In field level channel

it may not be possible. Suppose, for irrigation in the main canal

it may be possible to provide a lining, but say the minor canal or the distributaries

or that is going into the field which is ultimately carrying the water it may not be possible

to provide a lining; and if the soil type is of the type that it get eroded or the water

can erode it, then this makes a different cases of uniform flow, basic concept is same,

but I mean we need to see whether all the theories that we applied for rigid boundary

channel, all the formula whether we can apply those as it is or we need to consider some

special issues in that. Well, before going to this we have done in

the last class, say, uniform flow that we have discussed very clearly. Well, let us

recapitulate what we did in our last class. Well, we studied uniform flow. That already

I have explained the how uniform flow occur and all these things we have studied in the

last few classes; then of course we did uniform flow computation in rigid boundary channel,

and then we also studied that mobile boundary channel there can be scouring and silting,

this when mobile boundary channel is there in our very introductory class and in subsequent

classes also we have discussed that there can be scouring, there can be silting. Now, our point is that should we consider

these factors in analyzing uniform flow. Well, now, before going to various expression, various

relation of a forces, and that influences the flow in mobile boundary channel, we need

to know some of the fundamental relation that when the water flow, then how the water exert

force on the bed or what sort of stresses can develop in the bed. And that is why what

I feel that some of the pre requisite that I feel that is necessary before studying the

mobile boundary channel, we will be discussing first; well, then one of that is that expression

for the bed shear stress tau 0, now you can concentrate to the slide that we generally

use the symbol this tau 0 for the bed shear stress.

Well, now, what can be the expression for this tau 0, bed shear stress means, when the

flow is moving, then how much shear stress the flow is exerting on the bed, and then

for that in uniform flow how we can have the expression for this tau 0 well. Now, let me take a simple channel like this,

and say uniform flow is flowing; now, we are concerned about uniform flow, channel section

may be anything like that, but it is flowing uniform flow; now, if we take two section,

if we take two section, 1 and 2, 1 and 2, and we recall the expression for our momentum,

and whether if we write the momentum equation. Then we can write that what are the forces

acting on these water, what are the forces acting on this section the water between the

existing between the section 1 and 2;. Well, that we can write first, say, the force the

pressure force working in this direction; and if say x bar is the distance up to the

center then this pressure force we can write that gamma into x bar or sometimes we write

this as rho g in to x bar, that is the pressure at that point multiplied by the area multiplied

by the area we get the pressure force, that way some pressure force is acting from this

side. And another pressure force I mean in this

body, the pressure force on the from the downstream side that will be say P 2 and weight of the

liquid that is W, and then component of this weight which is acting in this direction,

that is in the direction of flow is W sin theta. In fact, this sort of forces we have

written several time; now, normally we used to write earlier that this force that is the

resisting force that is acting on the…, that is opposing the flow as, say, f; now,

here we will be writing it in a different term, that is say tau 0.

If tau 0 is the shear stress if tau 0 is the shear stress, then if we consider the tau

0 is the shear average, shear stress means, shear stress in the bed at different point

in reality it can be different, but say we are assuming, and on the side also it can

be different. But let us assume that, this is this can be considered uniform shear stress;

and then if this is the perimeter, and if this is the length if this is the length,

then we can write that shear stress at or this force resisting force is equal to total

surface area is equal to perimeter multiplied by the length and multiplied by the shear

stress. So this is the shear stress which is generated here and is opposing actually

frictional resistance, when the flow is moving always we know that opposing force will be

just be sufficient to just obstruct the flow, and that way the this tau 0 is we are writing

that bed shear stress, and it will be the four generated will be proportional to this

tau 0, and it will be equal to rho into L. Well, now, this force what we can write, say,

total force, and suppose, the Q is the discharge moving Q is the discharge moving in this direction,

then we can write the equations as say P1 is the force from this direction then minus

P 2 is the force from that direction this direction, and then plus W sin theta then

minus tau 0 into P into L, these are the forces that are acting here, and this is equal to

rate of change of momentum; now, we know that when we talk about momentum, we need to put

the momentum coefficient when we are writing this in terms of average velocity. So, Q we

are writing say as a into V or whatever it is, so we will be using the momentum coefficient

P, and let us write it as Q is the discharge flowing, then multiplied by rho is the mass,

and the change in velocity say V 2 minus V 1; so, ultimately this expression is giving

us the rate of change of momentum, and that way these are the applied force and these

are the rate of change of momentum. Now, this expression we have written earlier

also just to refresh our mind and to write it in the form of tau 0 we are writing it

again. Well, now, this is the case of uniform flow this is the case of uniform flow; now,

when we talk about uniform flow our depth is same; suppose, whatever depth we are having

here, whatever depth is here, these two are same means section 1 and 2 depth are same.

Now, if this depth are same, and if the channel is prismatic, that is this section, and that

section we are having that is section 1 and 2 our sectional area is also same. Then this

pressure force from the upstream side and pressure force from the downstream side, these

two will be equal this two will be equal; and similarly, we know that in uniform flow

this V 2, that is the velocity at the section 2 and the V 1 that is the velocity at the

section one will be again equal. Well, that way what we can see that if this V 2 and V

1 are equal to 0 in uniform flow, so this will become 0, and then if this P 1 P 2 are

also equal, then we can simply write that w sin theta W sin theta minus say tau 0 into

P L equal to 0 or this two we can just equate like this, that is say W sin theta is equal

to tau 0 into P into L. Now, what is this weight? The weight is already

we were drawing several times, this weight is nothing but if the unit weight of the fluid

is say w or sometimes we write it as rho g, rho g is the unit weight of the fluid, sectional

area is say a, and length is L, then this is the total weight of the fluid. And then that is why here we can write it

as W, we can write it as rho g A into L and sin theta this is equal to tau 0 into P into

L tau 0 into p into L; now, this sin theta again for small value of theta for small value

of theta, what we can do for small value of theta, we can always write sin theta equal

to tan theta equal to theta, so that means, this is equal to the slope S fine; and then

this equation you can again write in the form that rho g L and L would get cancel, so A

into S is equal to tau 0 into P. So, from this what we can write that our basic objective

is to get this tau 0. So, tau 0 is equal to we can write rho g then

A by P into S, now this by P term A by P term we know that it is nothing but the hydraulic

radius of the open channel, so we can write the tau 0 is equal to rho g R into S, and

many a time this rho g we can write as W R S – W means this is unit weight unit weight;

sometimes it is written as gamma also. So, any symbol can be used W R S, so that is what

our tau 0 expression for the shear stress; and then using this expression for shear stress.

Now this is the very basic expression which will be using for finding, that is if the

stress is more than certain value then there will be erosion if the stress is less than

certain value then there may not be erosion or scouring, and this shear stress again we

can write it, we can express a different value like that is tau 0 by rho tau 0 by rho.

And then if we take a square root of that if we take a square root of that, this particular

expression is said as u star is written as u star, what this is it is called shear velocity

shear velocity; you can see this root over tau 0 by rho means we are getting root over

g and root over R S; now, again the velocity formula also we know that root over R S comes,

so it is in the similar form and then its unit is also similar to velocities, so it

is called shear velocity. So, with this first introduction to shear velocity, now we can

go to another important parameter or another important understanding of the sheer force;

now shear force or shear stress on the bed is one issue.

Then again another issue is that, whether the bed is smooth or rough, means, when the

erosion process start that is in the mobile boundary channel suppose the erosion process

is starting, then there may be some undulation in the bed; and otherwise also, of course,

it is a general in general term also there will be some roughness in the bed; now, that

roughness how much it is rough whether it is smooth that is very relative terms, that

is very relative term, say, one can say this is rough, another can for another person it

may be smooth; and when we come in the context of water say in a river bed there may be some

undulation, and then suppose depth of flow is very high.

Now, for that high depth of flow a smaller undulation may not be considered as rough,

but say for a small depth the same undulation of the bed may be considered as rough; similarly,

suppose boundary river is there, that is the size of the bolder, and that size of the bolder

whether we call these as a rough bed or a smooth bed; well, there also again that question

come; if basically the point is coming in the form that, if that roughness is not creating

turbulence in the main flow that roughness is not going into the turbulent zone and creating

turbulence, then we refer this as a smooth. So, I mean there is a specific definition

of smooth and rough boundary when we talk in terms of hydraulics. So, we call that as

a hydraulically smooth and rough boundary well. So, we can now concentrate into the

slide again, we will be just drawing how we can define this hydraulically smooth and rough

boundary. Well, that basically when we studied, probably

you have studied already or we have seen in our fluid mechanics also that when a flow

is coming from this side, and say it is moving over a boundary, now in channel, of course,

this is always a boundary I mean our bed itself is a boundary. So, when it is moving over

a boundary, then the flow get retarded near the boundary and so the zone where the flow

is getting retarded is increasing like that, say, in this portion the flow is retarded,

and then it is going up to certain extent, then in this portion the flow as it is getting

retarded, and the flow will be laminar in this portion, and then it is moving like this;

that means, this is extending the zone of influence that is extending, and then here

the flow may be turbulent, flow will be turbulent; and then but still we get a layer here below

which, that means, in this portion the flow is still laminar.

So, that layer which is suppose a flow is turbulent, in general flow is turbulent, but

under the turbulent flow there is a layer which is again continuing as a laminar, so,

that suppose depth if I write as delta dash, then it is called laminar sub layer, so this

layer is called laminar sub layer. Now, in open channel flow say let us consider there

is a up to infinite extent there is the channel bed and it is coming from this side also there

is a continuation of the channel bed, so we are concerned about this part only, that is

we know that there is a laminar sub layer, there is a laminar sub layer; and of course,

this expression for laminar sub layer is also given by different experimental study and

this is can be written as u star like that 11.6 u star; well, this laminar sub layer

has something to do with the roughness. Now, let me draw one boundary, say this, now

this boundary may have some roughness, this boundary may have some roughness, again as

we understand that when we talk about a roughness in a boundary, it is very difficult to say

or very difficult to define that say roughness if I write as some value K, suppose roughness

is or we can use a suffix K s it is a surface roughness K, then definitely the roughness

at different area will be different, and exactly defining a diameter or dimension of that roughness

is very difficult, and that is why we use a term that is called equivalent sand grain

roughness; this roughness can be expressed in terms of equivalent sand grain roughness

K s. Well, what we mean by equivalent sand grain

roughness equivalent sand grain roughness, that information is also required. In fact,

Nicholas says he conducted experiments, so Nicholas says conducted experiment with different

material with different material say he did used different material, and say making pipe,

and then he found the, he was observing how this material is behaving say may be head

loss how it is how the head loss is occurring there, and then he did use different sand

grain, collected different sand grain, and say of uniform size sand grain he was collecting,

and then he took the similar diameter pipe, and then the inside of the pipes were coated

by different type of sand grain, means, different sizes of sand grain, say, one pipe is being

coated with a very fine sand grain, another pipe is coated with a little coarser sand

grain; and so, that way now for those pipe where it was coated for those pipe which were

coated rather with known sand grain, that are that that are known, that is this pipe

is coated with this sand grain means, its roughness is the diameter of that particular

sand grain, and diameter of that particular sand grain is known; now, he conducted experiment

with those pipe, and the pipe of different material, then he found that yes this material

is behaving similarly to the material to the pipe coated with a particular size of sand

grain. So, now, he has two things in his hand, one

is a particular material of the pipe, another is a pipe coated with a particular size of

sand grain which are behaving in the same way. So, then for that particular material

suppose steel for the steel he has a sin is hand with a particular type of sand grain,

so he said that this has sand grain roughness of this much well. So, that way for different

materials sand grain roughness were given, and then for a for a particular material we

can considered this K s, of course, on principle or in general we can say that this is a roughness

or suppose for some material there can be where we may not have this sort of value. Now, this is the roughness, you take it like

that, this is the roughness; and see the laminar sub layer depth is suppose here, here is the

laminar sub layer depth say this is the laminar sub layer we generally go from the average

height average of this thing, so the laminar sub layer height and depth and this say delta

dash; now, if this K s is very very small as compared to this laminar sub layer depth,

then we call this as a smooth; now, again in engineering if we say very very small these

are sometimes become big, so we need to express it in terms of some definite value.

Well, that definite value definite value was given as if K s divided by say delta dash,

if this is less than equal to 0.25 means delta dash is say 4 time greater than K s; if it

is less than equal to 0.25, then we call that particular surface as smooth surface that

particular surface as smooth surface; well, that means, there is another point important

point that it is not only the flow depth, but a depth of laminar sub layer, and this

depth of laminar sub layer will depend on this u star that is shear velocity, so that

way and viscosity of this fluid So, this entire factor will influence, and

finally we are getting that a particular surface can be considered as rough for a situation;

and for another situation the same surface can behave as a smooth also, so it is always

dependent; now, this if this is the value, then we can call this as a smooth boundary

smooth boundary; smooth means, hydraulically smooth; and then if we have that value that

K s by delta dash, say, another case here, say, roughness are very large, and this is

our roughness K s. And then if I calculate the say delta, this

is smaller than this delta dash, delta dash is smaller than this, that means, the roughness

will move into the or rather roughness will penetrate through the laminar sub layer, and

then it will go to this turbulent layer, this part is turbulent, this part is turbulent,

this part is of course laminar, this part is laminar, and this is going laminar; and

that laminar portion which is remaining below turbulent we are naming as laminar sub layer.

Well, so, this roughness when it is moving or when it is going inside the turbulent then

this will influence this sort of flow. When the roughness is going like that, then

this is influencing this sort of flow; otherwise, if it is stop going into the turbulent zone,

then whatever rough it may be it is not influencing this flow, so that way this flow will be behaving

as if it is moving over a smooth surface, and that is why we call this as a smooth.

Now, when this is when this value is greater than equal to 6 greater than equal to 6, that

is K s is 6 time larger than the delta dash, laminar sub layer depth, then we call this

as a rough, we call this as a rough. Well, so, we have got the smooth hydraulically smooth

boundary and hydraulically rough boundary, then in between there can be always transition

like that, if it is point 0.25 less than equal to say we can write K s by delta dash, and

it is less than equal to 6, then in between this part we will call that the boundary is

in transition; now, why we are talking about all this rough, smooth, and what is our basic

objective. Basically, we will start with some derivation of equation, and these in these

derivation we will have to use some approximation or some theory that has come from observed

value, and say those relation will give us some relation which are valid only for smooth

not some relation which are valid only for rough, so we need to know what is basically

smooth and rough and on which these things will be valid, and because of that we are

starting with this an just introducing what is hydraulically smooth and hydraulically

rough boundary. Well, already we have defined that; then we

need to know another fundamental another fundamental relationship which relate which relate the

shear stress tau shear stress tau, we were talking about bed shear stress which we were

writing as tau 0, well, but there may be a situation that we need to know we need to

know the bed shear stress, but exactly at bed, what is happening, it may not be possible;

and sometimes our say boundary material or the bed material size is little larger or

say we need to actually consider, say, if it is the bed if it is the bed then say boundary

materials are actually like this; and then we need to consider shear stress to move this

particular shear stress is acting at this surface. So, it is not really acting at this

point, but it is acting at some height that height is nothing but equal to the diameter

of that particular. So, that way we need to consider those things and for understanding

all those we need to know this Prandtl von Karman universal velocity distribution equation.

Well, now, say we need to start from one relation that we call as a Prandtl turbulent mixing

length theory; well, we will not go into much detail of these things how it is coming. But

we will start with taking this expression as it is, that is if a in a channel in a channel

say this is the surface, and then this velocity distribution we are interested, how this is

actually velocity distribution is occurring, and very near to bed say at any level here

at any level, so the u is changing and at depth y say rate of change of this is d u

d y d u d y. So, rate of change of velocity with respect

to y we are just writing like this, and then shear stress at any level, this is we are

not mentioning this shear stress to be the shear stress in the bed; it is valid that

shear stress at any level can be expressed at as say shear stress at any level is equal

to rho that is the density of the fluid, then K square, this constant is called von Karman’s

constant. And then it is y square means K square y square is the depth from the bed

y square and d u d y whole square. So, that expression we are taking directly;

and Von Karman found that after experimenting in a in this particular phenomenon he found

that this K can be expressed as a constant value, this K can be expressed as a constant

value, that value he suggested is point four; however, this now a days you can say that

the K we cannot actually considered as point four, it will be influenced by several other

parameter; anyway right at this moment we are considering that K to be constant and

let me take as K as 0.4, and then how we will consider the shear stress at very near to

the bed this is our bed . So, shear stress at bed, if we assume that assuming that shear

stress near the boundary near the boundary as equal to tau 0, means, tau 0 is basically

bed shear stress and our interest is to have some expression for tau 0; and if that shear

stress we consider that very near to bed is we are considering as tau 0, that is the shear

stress. With that if we start with that if we start,

then we can ultimately derive an expression for velocity distribution, and that velocity

distribution expression we will be finding that u can be expressed in term of logarithmic

equation, and that is why that we term as logarithmic velocity distribution equation;

well, you can concentrate into the slide that as…. Now, we are talking about bed shear and we

are we are writing say tau 0; well, we can write a small diagram here also, that this

is the bed, and we are writing this as a tau 0, tau 0 is equal to rho K square y square

into d u d y whole square d u d y whole square; then we know that tau 0 by rho already we

have expressed this root over this root over this is equal to we can write K y, then d

u d y K y d u d y; well, now, tau 0 by rho that we can write as shear velocity u star,

and this is equal to K y d u d y; now, we can we are in a position that we can separate

the variable and we can interrogate it. So, we can from here we can write that 1 by u

star into d u is equal to 1 by K one by y into d y.

Then integrating this integrating this what we can get, integrating we can get, this will

be u by u star u by u star, so u is the velocity at any point we are getting interrogation

of d u and u star is remaining here, this is equal to 1 by K, then this will become

log of y 1 by K and then we are getting log of y and plus say c log of y plus c, this

is say c dash, I am writing this c dash as a constant of integration. Well, now, from

this we need to find out what this value of c, and how we can write this. In fact, from

this integration we are almost getting any relation that if we try to know what the velocity

u, what the velocity u at any depth we can find with respect to y. But and then here

we have the von Karman constant, that value also we can take as 0.4, but what this constant

of interrogation that will be; so, that we can find from the boundary condition that

if for, say, y equal to let a very small distance c at y equal to c say u is equal to 0.

So, u will become somewhere 0, if I draw the velocity diagram if I draw the velocity diagram

y will become somewhere 0, but very near to the bed say at y equal to c u is equal to

0. So, with that if I write, then we will be getting that from the boundary condition

0 is equal to say 1 by K log of c is equal to plus c dash, so, we can write that c dash

is equal to minus 1 by K log of c; so, finally, we can write u by u star is equal to 1 by

K then log of y and c, in place of c dash we can write minus 1 by K log of c; well,

then this we can write that u by u star is equal to 1 by K log of y by c well. So, this

is one relation we are getting this is one relation we are getting; well, now, we know

that this 1 by K we can express K we can write as0.4, and this logarithmic actually it is

e base. So, we can write from this expression. Let

me write in a take a fresh page can write u by u star that we can write as 1 by K, K

can write as 2.5, 1 by K can be written as 2.5 and then it is log e y by c, then if we

convert it to log 10, then we can write u by u star is equal to 2.5 into 2.3 converting

it to 10 base, so 10 y by c, so that will give us ultimately u by u star is equal to

this if we combine them we are getting 5.75, 5.75 log of 10 y by c; well, this equation

is in the logarithmic form, and of course, we are having problem with this c value again

that we need to find experimentally, otherwise this equation we can call as logarithmic equation.

Then for smooth boundary, that is why we were discussing what is smooth and what is rough

to get a complete expression for this equation; now, for smooth boundary, smooth boundary

means we are talking about hydraulically smooth boundary, we are talking about hydraulically

smooth boundary, again for smooth boundary Von Karman’s, he conducted experiment and

on the basis of his data he could get that C can be expressed as this C, this C can be

expressed as nu by 9 u star 9 u star; now, if we put this value of C, then we can write

it in the form that u by u star is equal to 5.75, then log of ten, and this 1 by C, so

it will be 9 we will be writing separately, so it will be say u star y by nu; then of

course, we can write plus 5.75 into log of 9, we are writing separately in this; now

this u star y means, y is what, y is the depth at any point, y is depth at any point, we

are talking about a depth at any point y. So, at any depth y when we are having that,

we want to find the velocity u at any depth y, then we can find out this u do not confuse

this is not the u star. So, we want to find this u, so, u we can find in terms of y of

course, for that we need to know this shear velocity u star, and to know this shear velocity

we need to know rho density of water, of course, we will be knowing, but we need to know the

shear stress tau 0. So, once we know the tau 0 we can find it, and we can have this we

can have this equation, and u by u star of course we have a different name for this expressions

75 log of 10 base, this u star let me write this u star y right now and then nu and then

plus this value if we simplify it is coming as 5.5.

Well, that means, these are constant, all u star y by nu; we know that what is the expression

for Reynolds number; expression for Reynolds number is u v, that means, velocity at a particular

point, and if we try to calculate the Reynolds number we know that u v by nu, nu is the kinematic

viscosity, well, here nu is kinematic viscosity, nu is kinematic viscosity. So, as we know

that u star or say u y by nu is the expression for Reynolds number, but here our this velocity

is not the velocity that we are having at a particular point or something like that

it is called shear velocity. So, this Reynolds number this Reynolds number

is also called shear Reynolds number, and that is why we can write it in the form that

u by u star is a function of; that means, all these we can just club as function of

u star y by nu u star y by nu. So, this is one important expression that we will be using

in our subsequent analysis; then this is what we are doing 1 point we are knowing that,

see till now we are expressing tau 0 as some value root over sorry u star as root over

tau 0 by rho, and tau 0 we got one expression that W R S, if you remember just now we did

that tau 0 is equal to w means unit weight sometimes we can write it as gamma also, so

gamma R S. Now, all these things are applicable when it is a case of uniform flow as we did

cancel all these things; in fact, if we apply it to some other cases we will have to make

it very small, we will have to discredited it and we need to consider that this part

is smooth and that way only we can express that things.

Well. So, understanding of these things are required; now, this if it is not a smooth

boundary if it is not a smooth boundary then also we have an expression for this C, that

also let us keep that also. Let us keep that is if it is not a smooth

boundary for rough boundary for a rough boundary, of course, that may not be coming right now

into our application in deriving under subsequent relation, but still as information we can

do that; in terms of Nicolas experiment again that sand grain roughness K s, in terms of

sand grain roughness K s, c can be given as K s by 30 K s by 30, and using that if we

go for this expression then we will be getting u by u star, that is replacing C value by

K s by thirty we will be getting 5.75 log of ten, then it is y by K s it is y by K s

plus 8.5. So, that means, this here for rough boundary our velocity distribution is not

dependent from the Reynolds number rather it is depending on the depth and the roughness

this ratio this value. So, that way these things are this information’s

will be useful and for, of course, this we are talking about that is rough boundary and

smooth boundary, if the boundary is in transition, if the boundary is in transition then also

in a different books in the books of Ranga Raju, K G Ranga Raju also he has given that

his experiments compiling the experimentation done by different people at different time,

he gave that value that this particular value can be changed as c 1, and I mean some constant

and for transition transition, means, again this transition depends on the ratio how much

is the ratio K s by delta. So, depending on different value of K s by delta depending

on different value of K s by delta you he gave some value of this parameter say c 1

some coefficients, and then for that can be used for transition boundary also; well, now,

with this with these information’s now we can proceed further we can proceed further.

Well, so, till now what we know is that when a fluid is moving when a fluid is moving then

it is exciting some stress at the bay, other that we will called as boundary. At the boundary

it is exacting some stress. And that stress, we know that this can be express as, if you

write it as tau zero then we get zero velocity u star and then that stress if it is exist

a particular limit then there will be erosion, there will be erosion. And then considering

that erosion means ah that again we use a concept that is called as incipient motion

condition, that means ah the force required, force required take the particle or the velocity

required to move the particle. This is one issue and then the particle has its own weight,

so it will be having its own resistance, it will try to stand in its position. Now when

the force or the velocity are just sufficient to move the particle then particle is just

in the of motion, just in the motion. So that situation we called as incipient motion condition,

that situation we called as incipient motion condition.

Well, now for deriving a relationship, for its incipient motion condition there are different

approaches or different approaches are possible. Well, one approach is call that competent

velocity, competent velocity. And based on these approach also we derive some relationship,

competent velocity means, we talk about velocity and one point you should always remember,

because the sediment movement, these are very complex, these are very complex. It is not

that straight simple like in boundary channel. And because the reason is there that sediment

what we are talking about say in a , although we give in a particular size of sediment in

nature we cannot have that uniform size of sediment. It will be mixture of various size

of sediment. And sometimes we talked at only frictional resistance between the sediments

are existing that is to move the sediment from its position, say frictional resistance

between two sediment are working and then that is why or sometimes it is situate on

the responsible, this is moving. Suppose, if some forces moved on that way,

it can be moved. But sometimes, in sediment there can be finer material as and that way

there can be some cohesive force between the sediment also. So as whole in the sediment,

we can have say cohesive force. Well, by sediment we are including here, say bolder also small

bolder or or say very fine sand to very core sand to fine sand all these we are including.

And most of the study generally consider that sediment to be cohesion less, that sediment

to be cohesion less and that with some expression at derived. And so these expression there

lot of assumptions always. Well, and then so when they are lot of assumptions exist

analytical solutions are analytical expression of a particular ah value is not possible in

it. If we go by some analytical approach, there will be lot assumptions and what will

be derive ultimately that may not be giving as exact result. So, generally recourse are

made to experimentation, recourse are made to experimentation. Lot of laboratory experiment

has been conducted already by different scientist all about world at different places and then

similarly again there is another point, that is the experimentation that is conducted in

the laboratory is than in a small scale. I mean, you cannot represent a river exactly

as it is. You will have to reduce the scale. And the scale even you reduce, then the sediment

size if you reduce to that particular in the same scale, suppose a hundred meter wide channel

we are expressing representing by a one meter width channel then there is scale is one is

to hundred. Now they are some sediments are flowing in the river, if we need to reduce

it that sediment size, this will be very small. If you reduce it to one is to hundred scale

that will be powder and will be completely same, so we cannot do that, because it will

be completely cohesive in that if you reduce it to that size. So that will lot of complexity

are there in experimentation also. But still which our knowledge of hydraulic model that

hydraulic model study that will be talking as different topic in our later part of this

course hydraulic engineering. And there will be discussing all those things

in details, but what I want to emphasize at this point that this sort of analysis on sediment,

I really very complex. We normally, we always need to go for experimentation and a part

from the experimentation as they are difficulties in experimentation also from the understanding

what we derived in experimentation we need to observe in the real field also. So observation

in the real field also I necessary. And considering all these things some theory

has been developed. And using those theory whatever engineering purpose we have seen

that being little conservative, being remaining in safer side we can always design our structure,

design our structure. Of course just and all these things have still active research area.

We are yet to come out with a very concrete decision about these sort of in many cases

well. And then say what we we were discussing is that there can be different approaches

for having some analytical understanding of incipient motion condition. And one is competent

velocity approach there we can make an assumption or we take it like that. There will be a competent

velocity there will be a competent velocity which will be just sufficient which will be

just sufficient to remove or to move the sediment particle in the .

So, that velocity you can express somehow and you get a relation. But another approach

is the critical force approach, we can call it is critical force approach or competent

force. So, there will be a force, which will be sufficient it which will be a just sufficient

to move the particle from it is position, and then we will try to equate the force required

to particle, and then what is the actually force exerted by the flow on the particle.

There are two things; one is the force when the fluid is moving it will be exacting some

force, and then we need to see the force, when it is just equal to the force required

to move the particle or just any more then that the force term, the particle start moving

will be getting a insufficient motion condition. Again when we talk about force, when we talk

about force there can be two concepts. One is the lift concept, and other is the drag

concept. Well, lift concept we know that when a fluid is some of Z is moving into it is

moving around object then will be lift force acting on the particle on drag force acting

on the well. Now, in to know the move the force can be

significant. And that way sometimes you consider it from the… That means it can be consider

that lift force you can consider or sometimes you can consists see the drag force. That

in this sort of analysis, when we talk about hydraulics, then the particle will be in the

beat. It is restring under beat, another that when the flow is moving over the particle,

drag force will be significant, drag force will be significant, and that way generally

this drag force approach is . Generally this drag force approach is adopted and lead as

to a term that is called critical tractive force; that is called critical tractive force.

So, we will be discussing the how this critical tractive force, we can explicit an using this

critical tractive force, how we can find that will be move in a beat this flow will be different

form the flow in a rigid boundary channel or in that critical tractive force is below

particular value, then we can consider this as a as similar as that of a rigid boundary

channel well. Then shield this is a very popular name in the sediment shield, that is shields.

Shields gave a semi theoretical analysis for computing critical tractive force, shields

gave a semi empirical semi theoretical . Why because in lease analysis, he initially came

with the understanding of various forces, and he equated this different forces, and

theoretical basics is there. But when that theoretical forces were equated,

then in fact to this force, directly it was not possible, because the sediment sizes are

different. Suppose we want to know how much force will be require for lifting or for dragging

the particle. It will depends on the particle. Now, when we talk the weight of the individual

particle, then we need to know what is the volume of that particle?

Now, in the beam in the particle would not be objectively , it will be of different type,

different shape. So, to find out volume of the particle, we need to make some based on

the shape of the particle volume we can consider that, it will be proportional to the diameter

cube. If the diameters of the particle is g then we can call it is proportional to the

diameter cube. However, exactly how much is not possible, so that way he by his analytical

understanding he derived the relation that is can be equal to, this things this this

value can be related. Then these he collected lot of data, conducted lot of experiment,

and collected lot of data and those data were used to relate this sort of parameter.

Parameter means, the parameters for which he has know the analytical these are related

or this can be related. This is one is function of other, so that way some non-dimensional

parameter whose were related, and then some and then from there critical , but it can

be that was analyze. Of course, later on some other scientist has also then on the direction

and has modified, he is analysis or other he is data and as given some other value,

of course more or less close to the value what was given by shield, well. So, that we

will be discussing in our next class, that how’s can be used, and then based on the will

be having on understanding, what sort of shear stress will be sufficient, to make up particle

moving, and if to particle start moving, then we will need to know that how the break form

will change, how the break form will change? And then the question coming, if the break

form change then how the resistance value, that is we were using for resistance parameter,

whether that will be valued or we need to take some more spatial understanding or some

more collection to this value, well. That will be discussing in the next class.

Thank you very much.

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