Gradually Varied FlowsSurface profiles

Surface profilesM-CurvesM1, M2, M3S-CurvesS1,S2,S3C-CurvesC1, C3H-CurvesH2, H3A-CurvesA2, A3

Surface Profiles Dynamic equation for gradually varied flows :-

Type M ProfilesM1- Profile: This is a backwater curve and it lies in zone-1 of mild channel.The given depth can have limiting value as yyn on u/s And y on d/s

This profile occurs where the D/S end of a long mild channel is submerged in a reservoir to a depth greater than normal depth of flow in the channel.

Obstructions to flow, such as weirs, dams, bends etc. produces M1 backwater curves.

These extend to several kms u/s before merging with the normal depth

Type M ProfilesM2-Profile: This is a draw down curve and it lies in Zone-II of a mild channel.

The given depth can have limiting value as yyn on u/s And y yc on d/s

For example, profile at D/S end of a mild channel having free over fall, at sudden drop in the bed of the channel, at constriction type of transitions etc.

Type M ProfilesM3-Profile: This profile is a backwater curve which lies in Zone III of a mild channel

It starts from the upstream channel bottom and terminates with a hydraulic jump at the D/S end. It occurs when a supercritical flow enters a mild channel.

For example: Profile in a stream below a sluice gate and profile after the change in bottom slope from steep to mild.

Type M ProfilesCompare to M1 and M2 profiles, M3 curves are of relatively short length.

Type S ProfilesS1-Profile: This is a backwater curve, which lies in Zone-1 of a steep channel.

It begins with a jump at the U/S and becomes tangent to the horizontal pool level at the D/S end.

For example: when flow from a steep channel is terminated by a deep pool created by obstruction, such as dam or weir.

Type S ProfilesS2-Profile: This is a draw down curve which lies in Zone-2 of a steep channel.

It is usually very short and rather like a transition between a hydraulic drop and uniform flow. Since it starts U/S with a vertical slope at the critical depth and is tangent to the normal- depth line at the D/S end.

For example, profiles formed at entrance region of a steep channel leading from a reservoir, on the D/S side of an enlargement of channel section and on the steep slope side as the channel slope changes from steep to steeper.

Type S ProfilesS3-Profile: This is a backwater curve, which lies in Zone-III of a steep channel. For example, profile below a sluice with the depth of the entering flow less than the normal depth on a steep slope, profile on steep slope as the channel slope changes from steep to milder steep.

Type S Profiles

Type S Profiles

Type C ProfilesC-Profiles: Since, S0 =Sc and yn=yc, these profiles represent the transition condition between M and S profiles.For a wide rectangular channel if Mannings formula is used, C1 and C3 profiles are used but when Chezys formula is used, both these profiles are horizontal lines, since, dy/dx = S0 for yn = yc

Type C ProfilesC1-Profile: This is a backwater curve, which lies in Zone-I of a critical sloped channel.For example, profile formed on the critical slope side of the channel having a break in the bottom slope in which the critical slope changes to a mild slope.

Type C ProfilesC2-Profile: This represents the case of uniform critical flow.GVF equation cannot be used.

Type C ProfilesC3-Profile: This is a back water curve which lies in Zone-III of a critical sloped channel.The U/S end of the curve starts theoretically from the channel bottom. At the D/S end the profile terminates with a hydraulic jumpOccurs where a super critical flow enters a critical sloped channel.For example, Profile in a stream below a sluice gate provided in a critical sloped channel.

Type H ProfilesH-Profile: A horizontal channel can be considered as the lower limit reached by a mild slope as its bed slope becomes flatter.There is no region 1as yn

Type H ProfilesH2-Profile: it is a drawdown curve, which lies in Zone-II of a horizontal channel.The U/S end of the H2 profile tends to approach horizontal line tangentially while the D/S end of the profile tends to meet the C.D.L perpendicularly, thus ending in a hydraulic drop.For example, profile at D/S end of a horizontal channel having a free over fall.

Type H ProfilesH3-Profile: It is a backwater curve which lies in Zone-III of a horizontal channel.The U/S end of H3 profile starts theoretically from the channel bottom while the D/S end of the profile terminates with a hydraulic jump occurs when a supercritical flow enters a horizontal channel. For example, profile in a stream below a sluice gate provided in a horizontal channel

Type H Profiles

Type A ProfilesA1-ProfilesA1-Profile is impossible, since the value of yn is not real.Adverse slopes are rather rare and A2 and A3 curves are similar to H2 and H3 curves respectively.These profiles are of very short length.

Type A ProfilesA2-Profile: It is a draw down curve which lies in Zone-2 of an adverse sloped channel.At the U/S end A2- profile tends to approach a horizontal line tangentially, while the D/S end tends to meet the C.D.L. perpendicularly, ending in a hydraulic drop.For example, profile at the D/S end of an adverse sloped channel having a freely discharging weir.

Type A ProfilesA3-Profile: It is a backwater curve which lies in Zone-3 of an adverse sloped channel.The U/S end of A3 profile starts theoretically from the channel bottom while the D/S end of the profile terminates with a hydraulic jump occurs when a super critical flow enters an adverse sloped channel.For example, profile in a stream below a sluice gate provided in an adverse sloped channel.

Control SectionsControl section is defined as a section in whichfixed relationship exists between the discharge and depth of flow.Well defined sections where the properties are known in priorWeirs, spillways, sluice gates are some typical examples of structures which give rise to control sections.The critical depth is also a control point. However, it is effective in a flow profile which changes from subcritical to supercritical flow. In the reverse case of transition from supercritical flow to subcritical flow, a hydraulic jump is usually formed bypassing the critical depth as a control point.Any GVF profile will have at least one control section.In the synthesis of GVF profiles occurring in a serially connected channel elements, the control sections provide a key to the identification of proper profile shapes.It may be noted that subcritical flows have controls in the downstream of flow (e.g. spillway), while supercritical flows have control sections in the upstream end of the flow.

In Figs. (a) and (b) for the M1 profile, the control section (indicated by a dark dot in the figures) is just at the upstream of the spillway and sluice gate respectively.

In Figs. (b and d) for M3 and S3 profiles respectively, the control point is at the vena contracta of the sluice gate flow.

In subcritical flow reservoir offtakes (Fig.c), even though the discharge is governed by the reservoir elevation, the channel entry section is not strictly a control section. The true control section will be at a downstream location in the channel. For the situation the critical depth at the free overflow at the channel end acts as the downstream control.For a sudden drop (free overflow) due to curvature of the streamlines the critical depth usually occurs at distance of about 4yc upstream of the drop.This distance, being small compared to GVF lengths, is neglected and it is usual to perform calculations by assuming yc to occur at the drop.

For a supercritical canal intake (Fig.e), the reservoir water surface falls to the critical depth at the head of the canal and then onwards the water surface follows the S2 curve.The critical depth occurring at the upstream end of the channel is the control for this flow.

Surface ProfileProcedure to determine Class of surface profile:-Compute yn and yc for given discharge and plot lines representing channel bed and CDL and NDL

By comparing yn and yc determine whether the channel slope is mild, critical, steep adverse or horizontal.

Knowing the normal depth and depth at control section, determine the type of the surface profile.

If in any reach the supercritical depth has to meet the subcritical depth, i.e., when the stream of water has to cross the critical depth line then there will be a hydraulic jump developed in between

Break in GradesA channel carrying a gradually varied flow can in general contain different prismoidal channel cross-sections of varying hydraulic properties.There can be a number of control sections at various locations.To determine the resulting water surface profile in a given case, one should be in a position to analyze the effects of various channel sections and controls connected in series.Simple cases are illustrated to provide information and experience to handle more complex cases.

In Fig. (a), a break in grade from a mild channel to a milder channel is shown.It is necessary to first draw the critical-depth line (CDL) and the normal-depth line (NDL) for both slopes.Since yc does not depend upon the slope for a taken Q = discharge, the CDL is at a constant height above the channel bed in both slopes.The normal depth y01 for the mild slope is lower than that of the of the milder slope (y02).In this case, y02 acts as a control, similar to the weir or spillway case and an M1 backwater curve is produced in the mild slope channel.

Various combinations of slopes and the resulting GVF profiles are presented in Figs.

In some situations there can be more than one possible profiles.For example, in Fig. (e), a jump and S1 profile or an M3 profile and a jump possible. The particular curve in this case depends on the channel and its flow properties

In the examples indicated the section where the grade changes acts a control section and this can be classified as a natural control.It should be noted that even though the bed slope is considered as the only variable in the above examples, the same type of analysis would hold good for channel sections in which there is a marked change in the roughness characteristics with or without change in the bed slope.A long reach of unlined canal followed by a lined reach serves as a typical example for the same.A change in the channel geometry (the bed width or side slope) beyond a section while retaining the prismoidal nature in each reach also leads to a natural control section.

Composite water surface profilesIn real life, a channel may have variable cross section or bottom slope. And it may have various control sections.Steps outlined below are followed to sketch the composite water surface profiles in a series Channel system.Compute normal and critical depths for each reach of the channel system based on specified flow rate, roughness coefficient, slope of the reach, and the channel cross section.Plot the channel bed, the normal depth line (NDL) and the critical depth line (CDL) for each reach in the system.Mark the control sections, both imposed as well as natural controls.Starting from each control point, sketch the appropriate water surface profile depending on the zone in which the depth at the control section falls and the nature of the slope.If in any reach the supercritical depth has to meet the subcritical depth, i.e., when the stream of water has to cross the critical depth line then there will be a hydraulic jump developed in between.

Problems:-A rectangular channel with a bottom width of 4.0 m and a bottom slope of 0.0008 has a discharge of 1.50 m3/sec. In a gradually varied flow in this channel, the depth at a certain location is found to be 0.30 m. assuming n = 0.016, determine the type of GVF profile.

Problems:-Question:-Identify and sketch the GVF profiles in three mild slopes which could be described as mild, steeper mild and milder. The three slopes are in series. The last slope has a sluice gate in the middle of the reach and the downstream end of the channel has a free overfall.

Solution:The longitudinal profile of the channel, critical depth line and normal depth lines for the various reaches are shown in the Fig. The free overfall at E is obviously a control. The vena contracta downstream of the sluice gate at D is another control. Since for subcritical flow the control is at the downstream end of the channel, the higher of the two normal depths at C acts as a control for the reach CB, giving rise to an M1 profile over CB. At B, the normal depth of the channel CB acts as a control giving rise to an M2 profile over AB. The controls are marked distinctly in Fig. With these controls the possible flow profiles are: an M2 profile on channel AB, M1 profile on channel BC, M3 profile and M2 profile through a jump on the stretch DE.

Problems:-A trapezoidal channel has three reaches A, B, and C connected in series with the following physical characteristics. For a discharge Q = 22.5 m3/sec through this channel, sketch the resulting water surface profiles. The length of the reaches can be assumed to be sufficiently long for the GVF profiles to develop fully.

Solution: The normal depths and critical water depths in the various reaches are calculated as:

Reach A is a mild slope channel as y0A = 2.26 m > yc = 1.32 m and the flow is subcritical.Reach B and C are steep slope channels as y0B, y0C < yc and the flow is supercritical on both reaches.Reach B is steeper than reach C.The CDL is drawn at a height of 1.32 m above the bed level and NDLs are drawn at the calculated y0 values.The controls are marked in the figure.Reach A will have an M2 drawdown curve, reach B an S2 drawdown curve and reach C a rising curve as shown in the figure.

Gradually varying flow computationsAlmost all major hydraulic engineering activities in free surface flows involve the computation of GVF profilesConsiderable computational effort is required in the analysis ofDetermination of the effects of a hydraulic structure on the channelsInundation of lands due to dam or weir constructionEstimation of flood zones

GVF ComputationsProcess of determination of the length of the surface profile is known as the computation of Gradually Varied flowThe various available procedures for computing the GVF profiles are classified as:-Graphical Integration methodDirect Integration methodNumerical methodDirect Step methodStandard step method

Analysis of GVF profilesThe calculations must proceed upstream in subcritical flows and downstream in supercritical flows to keep the errors minimum.

The steps need not have the same increment in depth. In fact it is desirable to decrease the depth increment as the flow profile approaches the asymptotic value. Calculations are terminated at : y= (10.01)yn

The accuracy would depend upon number of steps chosen.

1. The Graphical-integration Method. This method is to integrate the dynamic equation of gradually varied flow by a graphical procedure. Consider two channel sections (Fig. 1a) at distances x1 and x2, respectively, from a chosen origin and with corresponding depths of flow y1 and y2. The distance along the channel floor is:

Assume several values of y, and compute the corresponding values of the right-side member of a gradually-varied-flow equation.

A curve of y against dx/dy is then constructed (Fig. 1b).

According to Eq. (1), it is apparent that the value of x is equal to the shaded area formed by the curve, the y axis, and the ordinates of corresponding to y1 and y2. This area can be measured and the value of x determined.

This method has broad application. It applies to flow in prismatic as well as non-prismatic channels of any shape and slope. The procedure is straight forward and easy to follow. It may, however, become very laborious when applied to actual problems.

ProblemExample A trapezoidal channel having b = 20 ft, m=2, So = 0.0016, and n = 0.025 carries a discharge of 400 cfs. Compute the backwater profile created by a dam which backs up the water to a depth of 5 ft immediately behind the dam. The upstream end of the profile is assumed at a depth equal to 1% greater than the normal depth. The energy coefficient = 1.10.

Solution First of all find yc and yn by the already discussed methods.The critical and normal depths are found to be yc = 2.22 ft and yn = 3.36 ft, respectively. Since yn is greater than y, and the flow starts with a depth (5ft) greater than yn, the flow profile is of the M1type.

The section factor =

The conveyance =

For simplicity, the channel bottom at the site of the dam is chosen as the origin and the x value in the upstream direction is taken as positive. The computation of dx/dy by means of Eq. (1) is given in Table 1 for various values of y varying from 5ft to 1% greater than yn or 3.40 ft. For instance, when y = 5.00 it, the values in other columns of the table areT = 40.00 ftA = 150.00 ft2R = 3.54 ft R2/3 = 2.323

Table 1 Computation of the flow profile by graphical integrationQ = 400 cfs n = 0.025 So = 0.0016 yc = 2.22 ft yn = 3.36 ft = 1.10

Values of y are then plotted against the corresponding values of dx/dy (Fig. 2).The increments in area are planimetered and listed in the table.

Fig. 2 A curve of y vs dx/dy

According to Eq. (1), the cumulative values of should give the length x of the flow profile.Finally, the backwater profile is obtained by plotting y against x (Fig. 3).

Fig. 3 M1 flow profile computed by graphical-integration method

Method of Direct IntegrationThe differential equation of GVF for a prismoidal channel is a non linear, first order, ordinary differential equation.Can be solved using any solving procedure for the differential equation.Since this is first order so we require one boundary condition to solve this differential equation (mathematically)In field conditions, control sections can be used as boundary condition for GVF computation, as there the flow properties are known in prior.

Chow MethodBased on certain assumptions but applicable with a fair degree of accuracy to a wide range of field conditions.By this method, the hydraulic exponents are expressed in terms of the depth of flow.Let it be required to find y= f(x) in the depth range y1 to y2. Following two assumptions are made:-The conveyance at any depth is given by: Conveyance at normal depth:The section factor Z at any depth y is given by: Section factor at critical depth is given by:

Where,C1 and C2 are coefficientsM is first hydraulic exponent, N is second hydraulic exponent.

The gradually-varied-flow equation becomes:

-----------(2)-----------(3)

This equation can be integrated for the length x of the flow profile. Since the change in depth of a gradually varied flow is generally small, the hydraulic exponents may be assumed constant within the range of the limits of integration. In a case where the hydraulic exponents are noticeably dependent on y within the limits of a given reach, the reach should be subdivided for integration; then the hydraulic exponents in each subdivided reach may be assumed constant. Integrating Eq. (3),

The first integral on the right side of the above equation is designated by F(u,N), or ,

which is known as the Chow varied-flow function-----------(4)-----------(5)

The second integral in Eq. (4) may also be expressed in the form of the varied-flow function.

This is a varied-flow function like F(u,N), except that the variables u and N are replaced by v and J, respectively.

-----------(6)-----------(7)

-----------(8)-----------(9)Using the notation for varied-flow functions, Eq. (4) may be written as:----------10)

Equation (10) contains varied-flow functions, and its solution can be simplified by the use of the varied-flow-function table. This table gives values of F(u,N) for N ranging from 2.2 to 9.8. Replacing values of u and N by corresponding values of v and J, this table also gives values of F(v,J).In computing a flow profile, first the flow in the channel is analyzed, and the channel is divided into a number of reaches. Then, the length of each reach is computed by Eq. (10) from known or assumed depths at the ends of the reach. The procedure of computation is as follows:Compute the normal depth yn and critical depth yc from the given data Q and So.Determine the hydraulic exponents N and M for an estimate average depth of flow in the reach under consideration. It is assumed that the channel section under consideration has approximately constant hydraulic exponents. Compute J by

Compute values of at two end sections the reach.

From the varied-flow-function table, find value F(u,N) and F(v,J). Compute the length of the reach by Eq. (10).

ProblemEXAMPLE: A trapezoidal channel has a bed width B = 5.0m,So = 0.0004, side slope m = 2 horizontal: 1 vertical and n = 0.02. The normal depth of flow yn = 3.0 m. If the channel empties into a pool at the downstream end and the pool elevation is 1.25 m higher than the canal bed elevation at the downstream end, calculate and plot the resulting GVF profile. Assume = 1.0.

The calculations commence from the downstream control depth of y = 1.69 m and are terminated at y = 2.97 m, i.e. at a value of depth which is 1 per cent less than yn.

In nominating the depths in the first column, it is advantageous if u values are fixed at values which do not involve interpolation in the use of tables of varied-flow functions and the corresponding y values entered in the first column.

If the value of u or v is not explicitly given in the varied-flow function tables, it will have to be interpolated between two appropriate neighboring values.

The last column indicates the distance from the downstream end to the various sections. It can easily be appreciated that the necessary interpolations in the use of the varied-flow function table not only make the calculations laborious but are also sources of possible errors. Another source of error is the fixing of constant values of N and M for the whole reach.

Step MethodIn general a step method is characterized by dividing the channel into short reaches and carrying the computation step by step from one end of the reach to other. There is a great variety of step methods. Some methods appear superior to others in certain respects, but no one method has been found to be the best in all applications. The step methods are classified as:Direct step methodDoes not involve any trial and error procedure, and is applicable to a prismatic channel only. Standard step methodInvolves a trial and error procedure, and is applicable to both the prismatic as well as non-prismatic channels.

The Direct Step MethodIt is a simple step method applicable to prismatic channels.Consider the differential energy equation of GVF:Writing in finite difference form:

Here Sf,mean = average friction slope in the reach

And between two sections 1 and 2

Here,

Note that, in either the direct step method or the standard step method which will be described later, the step computation should be carried upstream if the flow is subcritical and downstream if the flow is supercritical. Step computations carried in the wrong direction tend inevitably to make the result diverge from the correct flow profile

ProblemA trapezoidal channel having a bed width b = 7 m, side slope z = 2, bed slope So = 0.0016 and Manning's n = 0.025 carries a discharge of 12.26 m3/s. Compute the back water profile created by a dam which backs up the water to a depth of 2 m immediately behind the dam by Direct Step Method. The upstream end of the profile is assumed at a depth equal to 1 % greater than the normal depth. Assume the energy coefficient = 1.0.

Solution: For the given data, following are first calculated: yn = 1.0m yc =0.64m

As yc

col. 1. Depth of flow (m), arbitrarily assigned from 2.00 to 1.01 m.col. 2. Water area (m2), corresponding to the depth y in col. 1.col.3. Hydraulic radius (m) corresponding to y in col. 1.col. 4. Four - thirds power of the hydraulic radius.col. 5. Mean velocity (m/s) obtained by dividing 12.26 m3/s by the water area in col. 2.col. 6. Velocity head (m) - Energy Coefficient is taken as unity.col. 7. Specific energy (m) obtained by adding the velocity head in col. 6 to the depth of flow in col.1. col. 8. Change of specific energy (m), equal to the difference between the two consecutive E-values in col. 7.

col. 9. Friction slope Sf which is equal to where, n = 0.025 and V is the mean velocity obtained in col. 5.col. 10. Average friction slope equal to the arithmetic mean of the friction slopes, considering the two consecutive sections.col. 11. Difference between the bottom slope So (= 0.0016) and the average friction slope Sfa.col. 12. Length of the reach in between the consecutive steps, computed by dividing the value of in col 8 by the corresponding value in col.11.co1.13. Distance from the section under consideration to the darn site given by the cumulative sum of the values in col. 12.

Fourth order Runge-Kutta, methodThe direct-step method is based on truncation of the Taylor's series (of specific energy, E) after the linear term and hence is only first order accurate. In other words, the error in the numerical solution would vary linearly with the step size, and we may need to use a very small step size to obtain an acceptable accuracy. Higher order methods, therefore, have been proposed and the fourth order Runge-Kutta method, which is probably the most commonly used method for numerical solution of a first-order differential equation, is described next.

For prismatic channel, dy/dx is not a function of x. However, we write it as f(x, y) to elaborate the difficulties associated with the solution for a nonprismatic channel. The fourth order Runge-Kutta (R-K) method, starting from a known depth, yi, at the location xi, computes the depth, yi+1 at a distance , computes the depth yi+1 at a distance , from xi as:

where the k's represent the slope, dy/dx, i.e., f(x, y), at different locations as follows:

It may be noted that for prismatic channels, the k's are only functions of depth and not x

ProblemA trapezoidal channel, with bed width of 20 m and side slopes of 2H:1V, has a bed slope of 0.0005 and roughness coefficient of 0.015. Uniform flow occurs at a flow depth of 3 m. A small dam across the channel raises the flow depth to 5 m at that section. How much length upstream of the dam will be affected by the backwater?

SolutionThe discharge in the channel is obtained from Manning's equation as:

The corresponding critical depth is = 2.05m.Since the normal depth is more than the critical depth, the channel has a mild slope and the profile would be M1.A rough estimate of the backwater profile length is obtained by dividing the flow depth at the dam by the bed slope, i.e. =10,000 m.

Use 10 steps of 1000 m each. Table below shows the computations with = 1000 m and the required length (where the depth is 1 % more than the normal depth) is obtained as a little more than 8000 m.

One may do a linear interpolation to obtain the location where the depth is exactly 3.03 m (computed as 8167 m). However, since this limit of 1.01 yn is itself an approximation, we can say that the profile length is 8 km (or 9 km, if we want to be on the safer side).

The Standard Step MethodThis method is applicable to both prismatic and non-prismatic channels. In non-prismatic channels, the hydraulic elements are no longer independent of the distance along the channel. Thus, in natural channels, it is generally necessary to conduct a field survey to collect the data required at all sections considered in the computation.The computation is carried on by steps from station to station where the hydraulic characteristics have been determined. In such cases the distance between stations is given, and the procedure is to determine the depth of flow at the stations. Such a procedure is usually carried out by trial and error.

In explaining this method it is convenient to refer the position of the water surface to a horizontal datum. In Fig. 5, reproduced here, the water-surface elevations above the datum at the two end sections are:

and

The friction loss is, where the friction slope is taken as the average of the slopes at the two end sections, or as .

Substituting the above expressions in eq.

the following may be written:

where he is added for the eddy loss, which may be appreciable in non-prismatic channels.

No rational method of evaluating eddy loss is available. The eddy loss depends mainly on the velocity head change and may be expressed as a part of it, or where k is a coefficient.

For gradually converging and diverging reaches, k = 0 to 0.1 and 0.2, respectively.For abrupt expansions and contractions, k is about 0.5. For prismatic and regular channels, the eddy loss is practically zero or k = 0.For convenience of computation, he may sometimes be considered part of the friction loss and Manning's n may be properly increased in computing hf. Then, he is zero in the computation.

The total heads at two end sections are

Therefore, Eq. becomes

This is the basic equation that defines the procedure of the standard step method.The standard step method is best suited to computations for natural channels.

ProblemA small stream has a cross-section which can be approximated by a trapezoid. The cross-sectional properties at three sections are as follows:

Section A is the downstream-most section. For a discharge of 100.0 m3/s in the stream, water surface elevation at A was 104.500 m. Estimate the water-surface elevation at the upstream sections B and C. Assume n = 0.02 and = 1.0 at all sections.

Solution: The calculations are performed in Table below. In this k= 0.3 for expansion, so that:

Col. 14 is obtained by adding col. 12 and 13 to the value of H at section 1 d/s.

Col. 14 is obtained by adding col. 12 and 13 to the value of H at section 1 d/s.

in the ith trial

y2 is chosen such that HE = HE

ProblemA trapezoidal channel having a bed width b = 7 m, side slope z = 2, bed slope So = 0.0016 and Manning's n = 0.025 carries a discharge of 12.26 m3/s. Compute the back water profile created by a dam which backs up the water to a depth of 2 m immediately behind the dam by Standard Step Method. The upstream end of the profile is assumed at a depth equal to 1 % greater than the normal depth. Assume the energy coefficient = 1.0.

Solution: For the given data, following are first calculated:yn = 1.0m, yc =0.64mAs yc

The step computations, arranged in a tabular form in above Table are explained as follows:col. 1 Section identified by station number or the distance (x) from the first station (m),col. 2 Water surface elevation at the station: A trial value is first entered in this column to be verified or rejected on the basis of the computations made in the remaining columns of the table. For the first step, this elevation must be given or can be assumed. Since the elevation of the dam site is 100 m (above M.S.L.) and the depth of water at dam site is 2 m, the elevation of the water surface is 102 m. When the trial value used in the second step has been verified, it becomes the basis for the verification of the trial value in the next step, and so on,

col. 3 Depth of flow in metres corresponding to the water surface elevation in col. 2 for instance, the depth of flow at a station is equal to water surface elevation minus the elevation of the bed,col. 4 Water area corresponding to y in col. 3,col. 5 Mean velocity of flow equal to the given discharge divided by the water area in col. 4.col. 6 Velocity head corresponding to the velocity in col. 5.col. 7 Total head computed equal to the sum of Z in col.2 and the velocity head in col. 6,col. 8 Hydraulic radius, corresponding to y in col.3.

col. 9 Four-thirds power of the hydraulic radius,col. 10 Friction slope computed using the values of n, V & R4/3coI. 11 Average friction slope through the reach between the two consecutive sections (in each step): approximately equal to the arithmetic mean of the friction slopes (computed in col. 10) of this and previous step,coI. 12 Length of reach between the sections as determined on the basis of site survey,col. 13 Friction loss in the reach given by the product of the values in col. 11 and 12coI. 14 Evaluation of the total head (m)

The standard step method has an advantage while applying to natural channels. The water surface elevation at the initial section, where a flow-profile computation starts, may not be known: and, if the computation proceeds with an assumed elevation that is incorrect (for a given discharge) the resulting flow profile will get more and more correctly adjusted after every step, provided the computation is carried in the right direction. Therefore, if no elevation is known within or near the reach under consideration, an arbitrary elevation may be assumed at a distant location quite far away (upstream or downstream) from the initial location. By the time the step computation is carried to the initial section the assumed elevation would stand corrected to the appropriate value. Now, a check may be made by performing the same computation with another assumed elevation at the distant section. The computed elevation al the initial section would be taken as the relevant elevation if the second computed value agrees with the first computed value. The two values usually agree if the distant location is sufficiently away

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