K Subramanya is a retired Professor of Civil Engineering . This third edition of Flow in Open Channels marks the silver jubilee of the book which first appeared . Flow in Open Channels-K Subrahmanya - Ebook download as PDF File .pdf), Text File .txt) or read book online. Flow in Open Channels-K Subrahmanya. Flow In Open Channels by K aracer.mobi - Ebook download as PDF File . pdf), Text File .txt) or read book online.

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In this Flow in Open Channels By K Subramanya, the scope of the book is defined to provide source material in the form of a Text book that would meet all the. K. Subramanya in the area of HYDRAULIC AND WATER RESOURCES Flow in Open Channels 4th Edition, McGraw Hill Education (India) Private Limited. Flow in Open Channels book. Read reviews from world's largest community for readers.

Goodreads helps you keep track of books you want to read. Want to Read saving…. Want to Read Currently Reading Read. Flow in Open Channels. Other editions. Error rating book. Refresh and try again. Open Preview See a Problem? Details if other: Thanks for telling us about the problem.

The second edition of the book which came out in had substantial improvement of the material from that of the first revised edition and was very well received as reflected in more than 25 reprints of that edition.

This third edition is being brought out by incorporating advances in the subject matter, changes in the technology and related practices. Further, certain topics in the earlier edition that could be considered to be irrelevant or of marginal value due to advancement of knowledge of the subject and technology have been deleted.

In this third edition, the scope of the book is defined to provide source material in the form of a textbook that would meet all the requirements of the undergraduate course and most of the requirements of a post-graduate course in open-channel hydraulics as taught in a typical Indian university. This has resulted in inclusion of detailed coverage on Flow through culverts Discharge estimation in compound channels Scour at bridge constrictions Further, many existing sections have been revised through more precise and better presentations.

These include substantive improvement to Section Additional worked examples and additional figures at appropriate locations have been provided for easy comprehension of the subject matter. Major deletions from the previous edition for reasons of being of marginal value include Pruning of Tables 2A. Chapters 1 and 2 contain the introduction to the basic principles and energy-depth relationships in open-channel flow.

CR of the specificenergy curve. We thus get an important result that the critical flow corresponds to the minimum specific energy and at this condition the Froude number of the flow is unity. Other channel properties such as the bed slope and roughness do not influence the critical-flow condition for any given Q.

In the upper limb CR'. Discharge as a Variable In the above section the critical-flow condition was derived by keeping the discharge constant. Thus differentiating Eq. The specific-energy diagram can be plotted. This region is called the supercritical flow region. At the lower limb. Different Q curves give different intercepts. It is seen that for the ordinate PP'. The dotted line in Fig. Any specific energy curve of higher Q value i. Consider a section PP' in this plot.

Since by Eq. In this figure. The difference between the alternate depths decreases as the Q value increases. Solution From Eq. Example 2. Calculate the alternate depths and corresponding Froude numbers. What is the specific energy of the flow? Hence by Eq. Rectangular Section For a rectangular section. By Eq. Empirical relationships for critical depth in circular channels Sl.

Energy—Depth Relationships 49 Substituting these in Eq. Straub W O Ref. This table is very useful in quick solution of problems related to critical depth in trapezoidal channels. Rewriting the right-hand side of Eq.

The nondimensional representation of Eq. As a corollary of Eq. It is found that generally M is a slowly-varying. Energy—Depth Relationships 51 Further the specific energy at critical depth.

Note that the left-hand side of Eq. The variation of M and my for a trapezoidal channel is indicated in Fig. If M is constant between two points Z1. For a trapezoidal channel. If the specific energy E is kept constant. The above value of M can also be obtained directly by using Eq. For rectangular and triangular channel sections. Tables 2A. With the general availability of computers. The graphical solutions and monographs which were in use some decades back are obsolete now.

Energy—Depth Relationships 55 in problems connected with circular and trapezoidal channels respectively. Examples 2. I V carries a certain discharge. Energy—Depth Relationships 1. Solution a Rectangular Section The solution here is straightforward.

Solution a At critical depth. Triangular channel. Trapezoidal channel. Circular channel. If the critical depth is 1. Energy—Depth Relationships Since by Eq. At Section 2 Fig. The minimum depth is reached when the point R coincides with C. The principles are nevertheless equally applicable to channels of any shape and other types of transitions.

Let the flow be subcritical. To illustrate the various aspects.

At this point the hump height will be maximum. The upstream depth has to increase to cause an increase in the specific energy at Section At this condition.

The energy Eq.. Ec2 is less then E2. It will decrease to have a higher specific energy E'1. Energy—Depth Relationships 63 point R' to depth at the Section 2. Solution Let the suffixes 1 and 2 refer to the upstream and downstream sections respectively as in Fig. Neglect the energy loss. Calculate the likely change in the water surface. Up to the critical depth. The upstream depth y1 will increase to a depth y'1.

Solution a From Example 2. The minimum specific energy at the Section 2 is greater than E2.

Energy—Depth Relationships 65 b Here. Energy loss. Calculate the minimum height of a streamlined. By use of Eq. Since there are no losses involved and since the bed elevations at Sections 1 and 2 are same. In the specific energy diagram Fig. At Section 1. At the Section 2 the channel width has been constricted to B2 by a smooth transition.

The flow will not. At this minimum width. If B2 is made smaller. At Section 2. As the width B2 is decreased. Any further reduction in B2 causes the upstream depth to decrease to y'1 so that E1 rises to E'1.

The variation of y1. It is proposed to reduce the width of the channel at a hydraulic structure. Solution Let suffixes 1 and 2 denote sections upstream and downstream of the transition respectively. This onset of critical condition at Section 2 is a prerequisite to choking.

Assuming the transition to be horizontal and the flow to be frictionless determine the water surface elevations upstream and downstream of the constriction when the constricted width is a 2. The upstream depth y1 will increase to y'. Details about subcritical flow transitions are available in Ref. Many complicated transition situations can be analysed by using the principles of specific energy and critical depth. At a downstream section the width is reduced to 3. In subcritical flow transitions the emphasis is essentially to provide smooth and gradual changes in the boundary to prevent flow separation and consequent energy losses.

The transitions in supercritical flow. M Transitions 1. The upstream depth will therefore rise by 0. N and Chiranjeevi. J of Hyd.

Civil Engineering. General Miscellaneous concepts 2. March Energy—Depth Relationships 73 Hence the contraction will be working under choked conditions.. P K and Basak. The upstream depth must rise to create a higher total head. If it is desired to keep the watersurface elevation unaffected by this change. Determine the specific energy and alternate depth. The width beyond a certain section is to be changed to 3. For Part c use Table 2A. Use the trial and error method.

For a critical depth of 0. Energy—Depth Relationships 75 2. Obtain an expression for the relative specific energy at the critical flow. Show that for these two conditions to occur simultaneously. Find the diameter of the conduit such that the flow is critical when the conduit is running quarter full. Estimate the discharge and the specific energy. Estimate the value of yc. If the depth of flow at a section where the flow is known to be at a critical state is 0.

Critical depth is known to occur at a section in this canal. Estimate the discharge and specific energy corresponding to an observed critical depth of 1. Use Eq. At a section there is a smooth drop of 0. What is the water surface elevation downstream of the drop? At a certain section it is proposed to build a hump. Calculate the water surface elevations at upstream of the hump and over the hump if the hump height is a 0.

Assume no loss of energy at the hump. A flat hump is to be built at a certain section. Assuming a loss of head equal to the upstream velocity head, compute the minimum height of the hump to provide critical flow. What will happen a if the height of the hump is higher than the computed value and b if the energy loss is less than the assumed value? A contraction of the channel width is required at a certain section. Find the greatest allowable contraction in the width for the upstream flow to be possible as specified.

A contraction of width is proposed at a section in this canal. Calculate the water surface elevations in the contracted section as well as in an upstream 2. Neglect energy losses in the transition. If the depth in the contracted section is 0. At a certain section of the channel it is proposed to reduce the width to 2. At certain section the width is reduced to 1. Will the upstream depth be affected and if so, to what extent?

If the width is to be reduced to 2. Neglect the loss of energy in transition. What maximum rise in the bed level of the contracted section is possible without affecting the depth of flow upstream of the transition?

If at a section there is a smooth upward step of 0. At a certain section the width is reduced to 2. The energy losses in the contraction can be neglected. At a section the channel undergoes transition to a triangular section of side slopes 2 horizontal: If the flow in the triangular section is to be critical without changing the upstream water surface, find the location of the vertex of the triangular section relative to the bed of the rectangular channel.

Assume zero energy loss at the transition. The specific energy head in m is a 3. The critical depth in m for this flow is a 2. The Froude number of the flow is a 0. The critical depth in m is a 0. If the depth of flow is 1. The Froude number of flow is a 0. If, after building the hump, it is found that the energy losses in the transition are appreciable, the effect of this hump on the flow will be a to make the flow over the hump subcritical b to make the flow over the hump supercritical c to cause the depth of flow upstream of the hump to raise d to lower the upstream water surface 2.

If the width is expanded at a certain section, the water surface a at a downstream section will drop b at the downstream section will rise c at the upstream section will rise d at the upstream section will drop 2. If the flow is subcritical throughout, this will cause a a rise in the water surface on the rack b a drop in the water surface over the rack c a jump over the rack d a lowering of the water surface upstream of the rack.

Table 2A. At the critical depth yc' Thus for example 0. As mentioned earlier, the term uniform flow in open channels is understood to mean steady uniform flow. The depth of flow remains constant at all sections in a uniform flow Fig. Thus in a uniform flow, the depth of flow, area of cross-section and velocity of flow remain constant along the channel.

It is obvious, therefore, that uniform flow is possible only in prismatic channels. The trace of the water surface and channel bottom slope are parallel in uniform flow Fig. As such, the slope of the energy line Sf , slope of the water surface Sw and bottom slope S0 will all be equal to each other. By applying the momentum equation to a control volume encompassing Sections 1 and 2, distance L apart, as shown in Fig. Since the flow is uniform, Also,. R is a length parameter accounting for the shape of the channel.

It plays a very important role in developing flow equations which are common to all shapes of channels. Equation 3. The coefficient C is known as the Chezy coefficient. From the time of Prandtl — and.

Von Karman — research by numerous eminent investigators has enabled considerable understanding of turbulent flow and associated useful practical applications. The basics of velocity distribution and shear resistance in a turbulent flow are available in any good text on fluid mechanics1,2.

Only relevant information necessary for our study in summed up in this section. Pipe Flow A surface can be termed hydraulically smooth, rough or in transition depending on the relative thickness of the roughness magnitude to the thickness of the laminar sub-layer. The classification is as follows: In the transition regime, both the Reynolds number and relative roughness play important roles.

The extensive experimental investigations of pipe flow have yielded the following generally accepted relations for the variation of f in various regimes of flow: The hydraulic radius would then be the appropriate length parameter and prediction of friction factor f can be done by using Eqs 3. Open Channels For purposes of flow resistance which essentially takes place in a thin layer adjacent to the wall.

Studies on non-circular conduits. Eqs 3. Due to paucity of reliable experimental or field data on channels covering a wide range of parameters. Table 3. Simplified empirical forms of Eqs 3.

Comparing Eq. Owing to its simplicity and acceptable degree of accuracy in a variety of practical applications. If Eq.

This coefficient is essentially a function of the nature of boundary surface. Many of these are archaic and are of historic interest only. A few selected ones are listed below: This formula appears to be in use in Russia. Ganguillet and Kutter Formula 1 0. Uniform Flow 91 n2. The fully developed velocity distributions are similar to the logarithmic. This equation is applicable to both rough and smooth boundaries alike.

Assuming the velocity distribution of Eq. The maximum velocity um occurs essentially at the water surface. For further details of the velocity distributions Ref. For completely rough turbulent flows. It has been found that k is a universal constant irrespective of the roughness size5.

The most important feature of the velocity distributions in such channels is the occurrence of velocity-dip. The turbulence of the flow and the presence of secondary cur-. It is zero at the intersection of the water surface with the boundary and also at the corners in the boundary. Uniform Flow 95 rents in the channel also contribute to the non-uniformity of the shear stress distribution.

Distributions of boundary shear stress by using Preston tube in rectangular. Lane9 obtained the shear stress distributions on the sides and bed of trapezoidal and rectangular channels by the use of membrane analogy. Isaacs and Macintosh8 report the use of a modified Preston tube to measure shear stress in open channels. A knowledge of the shear stress distribution in a channel is of interest not only in the understanding of the mechanics of flow but also in certain problems involving sediment transport and design of stable channels in non-cohesive material Chapter Preston tube5 is a very convenient device for the boundary shear stress measurements in a laboratory channel.

Estimation of correct n-value of natural channels is of utmost importance in practical problems associated with backwater computations. Some typical values of n for various normally encountered channel surfaces prepared from information gathered from various sources Cowan15 has developed a procedure to estimate the value of roughness factor n of natural channels in a systematic way by giving weightages to various important factors that affect the roughness coefficient.

The Darcy—weisbach coefficient f used with the Chezy formula is also an equally effective way of representing the resistance in uniform flow. The roughness coefficient. In the book. A comprehensive list of various types of channels. The photographs of man-made and natural channels with corresponding values of n given by Chow Photographs of selected typical reaches of canals. Barnes11 and Arcemont and Schnieder14 are very useful in obtaining a first estimate of roughness coefficient in such situations.

These include: According to Cowan. The selection of a value for n is subjective. It should be realized that for open channel flows with hydrodynamically smooth boundaries. These act as type values and by comparing the channel under question with a figure and description set that resembles it most. Uniform Flow 97 Table 3. Find the hydrodynamic nature of the surface if the channel is made of a very smooth concrete and b rough concrete. The channel is laid on a slope of 0.

Solution i Case a: Example 3. Uniform Flow 99 Since the boundary is in the transitional stage. Re is not known to start with and hence a trial and error method has to be adopted. The type of grass and density of coverage also influence the value of n. The procedure is sometimes also applied to account for other types of form losses.

An interesting feature of the roughness coefficient is observed in some large rivers. At low velocities and small depths vegetations. The dependence of the value of n on the surface roughness in indicated in Tables 3.

No satisfactory explanation is available for this phenomenon. Some important factors are: These relate n to the bedparticle size. Channel irregularities and curvature. For other types of vegetation. The chief among these are the characteristics of the surface. The most popular form under this type is the Strickler formula: For grass-covered channels.

For mixtures of bed materials with considerable coarse-grained sizes. This equation is reported to be useful in predicting n in mountain streams paved with coarse gravel and cobbles. The vegetation on the channel perimeter acts as a flexible roughness element. Another instance of similar. The resistance to flow in alluvial channels is complex owing to the interaction of the flow.

Canals in which only the sides are lined. This equivalent roughness. Uniform Flow drains. One of the commonly used method due to Horton and Einstein is described below. PN are the lengths of these N parts and n1. A large number of formulae.

For calculating subareas the dividing lines can be vertical lines or bisector of angles at the break in the geometry of the roughness element. The range of variation of n is about 30 per cent. All of them are based on some assumptions and are approximately effective to the same degree.

Consider a channel having its perimeter composed of N types of roughness. Detailed information on this is available in standard treatises on sediment transport Section This formula was independently developed by Horton in and by Einstein in No Investigator ne Concept 1 Horton In an economic study to remedy excessive seepage from the canal two proposals.

Total resistance force F is sum of subarea resistance forces. This list is extracted from Ref. For the sides: Solution Case a Lining of the sides only Here for the bed: Uniform Flow For a given channel.

We shall denote these channels as channels of the first kind. For example. Thus the normal depth is defined as the depth of flow at which a given discharge flows as uniform flow in a given channel. The normal depth 0. This depth is called the normal depth. This is also true for any other shape of channel provided that the top width is either constant or increases with depth. Figure 3. The channels of the first kind thus have one normal depth only.

For example 1. Circular and ovoid sewers are typical examples of this category. It may be seen that in some ranges of depth. Channels with a closing top-width can be designated as channels of the second kind. While a majority of the channels belong to the first kind. A typical example for each type of problem is given below.

Geometric elements Q. Uniform Flow As can be seen form Fig. The bed slope is 0. Problems of the types 4 and 5 usually do not have explicit solutions and as such may involve trial-and-error solutions procedures.

From among the above. The available relations are 1. Continuity equation 3. B and m for a trapezoidal channel. There can be many other derived variables accompanied by corresponding relationships. Geometry of the cross section The basic variables in uniform flow situations can be the discharge Q. Problem type 1 2 3 4 5 Given Required y0. Types of Problems Uniform flow computation problems are relatively simple.

Compute the mean velocity and discharge for a depth of flow of 3. If the bed slope is 0. The normal depth is found to be 1. The bottom slope is to be 0. In these channels. Since practically all open channel problems involve normal depth.

Such channels with large bed-widths as compared to their respective depths are known as wide rectangular channels. Considering a unit width of a wide rectangular channel. A few aids for computing normal depth in some common channel sections are given below. This is true for many other channel shapes also. This table will be useful in quick solution of a variety of uniform flow problems in rectangular and trapezoidal channels.

S0 and B in a rectangular channel.

Table 3A. Use Table 3A. Find the normal depth corresponding to discharges of i Find the depth of flow when the discharge is 2. Solution 2. In practice. The graphical plot of Eq. As noted earlier. Using this table. The advantage of using Table 2A. For convenience and ease of identification. For such hard surface lined canals the cross-section recommended by Indian Standards IS: These standard lined sections have interesting geometrical properties which are beneficial in the solution of some uniform flow problems.

Exposed hard surface lining using materials such as cement concrete. Solution For a standard lined trapezoidal canal section Fig. If a bed width of The longitudinal slope of the bed is 1 in The side slopes are to be 1.

Problem 3. Referring to Fig. With the slope. This channel section is also called the best section. More information about this seller Contact this seller. Add to Basket. Home K. ISBN Available From More Booksellers. About the Book. We're sorry; this specific copy is no longer available. AbeBooks has millions of books. We've listed similar copies below.

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