 Myself, Deshmukh Sachin, working as an assistant professor in the department of civil engineering of wall chain stuff technology, SOLAPOL. Today we are going to learn competition of gradual varied flow profile by using step method. At the end of this session, you will be able to understand what is the step method and how to calculate the length of the channel. We just see the figures, photographs. We can observe here is a, you can say it is a gradually varied flow. What is a gradually varied? It is defined as a steady non-uniform flow in the channel in which there are gradual changes in the water depth. Now we can see here it is a steady and non-uniform flow here also and particularly here also. These three photographs of the reverse I have chosen where we can see it is a GVF that is a gradually varied flow. The another is rapidly varied where we can say abrupt change of the different characteristics took place. Here gradual changes are there. Now in this take a control volume. Now to start this first of all just go through this figure. A section is taken control section where you can say energy slope line is there, water surface line is there and channel bottom is there. Here it is Y1 that is above channel bottom up to water surface. The depth is Y1, Y2. This is a you can say depth of water at section 1, depth of water at section 2. This is a kinetic head V1 square upon 2g here also the kinetic head. Now remember this figure we are going to calculate this delta x that is a length. Length of this particular channel this control section if the flow condition on either side of length delta x to be computed this method is generally adopted. In the figure a short reach of the channel length delta x is considered. Now by applying Bernoulli's theorem to section 1, 1 and section 2, 2. Here you can see S0 delta x Y1 and V1 square upon 2g here Sfm is you can say average because energy line is fluctuating. So Sf1, Sf2, Sf3 different slopes may occur. So we have to take average of that that is Sfm delta x is a vertical line divided by dx is equal to y by x from that we can calculate this also and this also. So addition of this equal to addition of remaining on the second section S0 delta x plus Y1 plus V1 square upon 2g is equal to Y2 plus V2 square upon 2g plus Sfm delta x. So Sfm it is average mean slope of the energy line which is equal to Sf1 plus Sf2 upon 2. Now put this value what is it Y2 plus V2 square upon 2g and Y1 plus V1 square upon 2g are nothing but the specific energies specific energies where we are not going to take the datum we are directly taking this Y value or this depth of water plus kinetic head. So delta x is we are getting S2 minus S1 specific energy at section 2 minus specific energy at section 1 upon S0 minus Sfm. Now Sfm is Sf1 plus Sf2 upon 2. Thus if the whole channel is split into such short steps because it is a length is bigger one it is a huge length is there. So you can split it into short steps and the length delta x for the each step is determined starting from the control section by using the above equation that is S2 minus S1 upon S0 minus Sfm the we can find out the length the accuracy of this method depends upon the depth of the increment selected and for better accuracy the values of Sfm at the end of the step should not vary appreciably this method is used for prismatic channel that is particularly slope is and the cross section is same along with this there are some other methods geometric method is there arithmetic method is there graphical method is there but this method is particularly which is more accurate than the other methods. Now we will solve one problem on this step method so that we will come to know how we can find out the length it is a problem a rectangular channel carrying a discharge of 30 meter cube per second has a width of 10 meter and the bed slope of 1 in 5625 and Manning's n as 0.02 at a particular section the depth of the flow is 1.6 meter determine how far upstream or downstream of the section the depth of flow will be 2 meter use step method and take two steps you we can go for three steps we can go for four steps we can go for many steps to get more accuracy now putting these values putting these values okay normal depth is 2.97 and critical depth is 0.97 since normal depth is greater than critical depth it is a mild surface profile and y 0 that is normal depth is greater than actual depth and which is greater than critical depth this is a m2 type of profile we can say it is a draw down curve see here 2.97 is greater than 2 which is greater than 1.6 2.97 which is greater than 2 meter which is greater than 1.6 meter so it is a m2 type of profile m sorry it is a critical depth is 0.97 0.97 and section is from 1.6 meter to 2 meter now this is a section given to us now this section we are now we are going to find out this length of this section that is from 1.6 meter to 2 meter by step method now particularly from my point of view if we can solve this step method by making observation table clearly and putting the values in the appropriate column we can we can get easily or it is a simple method I can say just see here from serial number to depth of the flow then area of the section mean velocity velocity head specific energy hydraulic radius then slope then mean slope then delta E s etcetera up to delta x that is total length if we can go through this observation table 1 by 1 then it is very easy now 1.6 meter that is again I will show you it is 1.6 meter to 2 meter we have to travel and they ask us to go use the two steps take two steps so 1.6 to 1.8 meter and from 1.8 to 2 meter these are the two steps which I have taken see here 1.6 to 1.8 meter and 1.8 to 2 meter this is as one step and this is the another step okay now area of the section that is 10 meter wide so 1.6 into 10 that is 16 1.8 into 10 18 2 into 10 20 like this velocity v is equal to q upon a if we have calculated this one then velocity head or kinetic head that is v square upon 2g specific energy is nothing but the depth of water plus kinetic head this we have calculated the hydraulic radius r is equal to a by p area upon weighted perimeter that is calculated 1.2 1.3 1.4 1.21 1.3 2 sorry and 1.43 is hydraulic radius then slope of energy line now this particular column just concentrate on this how it comes SF is equal to n square v square r s to 4 by 3 just now we have studied the Manning's formula v is equal to 1 upon n r s to 2 third and s s to half from this formula this SF is equal to n square v square upon r s to 4 by 3 this comes this comes from Manning's formula then mean slope this you are going to take this slope first of and average of these two here and average of these two here then e s 2 minus e s 1 find out e s 2 minus e s 1 it is 0.16 to 0.173 SF m minus SF m minus SF 0 this values and finally delta x is equal to e s 2 minus e s 1 that is here upon SF minus SF m delta x it is 1.3 4 5 meters like this we can go for any channel section these are the questions let us go through these questions objective type of questions these are the answers for these questions and you can refer many books and some papers are there on Google that you can thank you.