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1. Chapter Two 1 2.1. Introduction 2.2. Criterion for the critical state of flow 2.3. Calculation of the critical Depth 2.4. Section factor and First Hydraulic Exponent 2.5. Characteristics of Sub-Critical and Super-Critical flow 2.6. Transitions 2.6.1 . Channel with a hump 2.6.2 . Transition with a change in width 2.7. Choking

2. 2 .1 . Introduction For a given Q, as the specific energy is increased the difference between the two alternate depths increases. On the other hand, if E is decreased: the difference (y1` – y1) will decrease and at a certain value E = Ec, the two depths will merge with each other (at point C). No value for y can be obtained when E < Ec denoting that the flow under the given conditions is not possible in this region. Thus this condition of minimum specific energy is known as: the critical flow condition and the corresponding depth yc is known as critical depth. 2

3. 2 .2. Criterion for a critical state of flow The critical state of flow is defined as the state of flow at which: the specific energy and specific force is a minimum for a given discharge or it is the condition for which the Froude number (Fr 2 ) is equal to unity. The discharge is maximum at critical flow for a given channel section The velocity head is equal to half the hydraulic depth at critical flow Flow at the critical state is unstable F = 1, Critical Flow, F< 1, Sub Critical Flow and F >1, Super critical Flow 3

4. If the critical sate of flow exists throughout the entire length of the channel or over a reach of the channel the flow in the channel is a critical flow At critical depth, the specific energy is minimum, differentiating the equation of specific energy with respect to y (keeping Q constant) and equating to zero , 4 It is the basic equation governing the critical flow conditions in a channel.

5. It may be noted that the critical flow condition is governed solely by the channel geometry and discharge. Other channel characteristics such as the bed slope and roughness do not influence the critical flow condition for any given Q. 5 A t the critical state of flow the velocity head is equal to half of hydraulic depth.

6. 2.2. Calculation of the critical Depth Rectangular Channel 6

7. B. Triangular Channel 7

8. 8 C. Circular Channel . Let D be the diameter of a circular channel and 2θ be the angel in radians subtended by the water surface at the center.

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10. 10 The function is evaluated using critical depth T able 2A.1. However scientists try to develop empirical relationships for critical depth in circular channel.

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16. 16 D. Trapezoidal Channel For a trapezoidal channel having a bottom width of B and side slopes of m horizontal : 1 vertical Area A = (B + my)y and Top width T = (B + 2my)

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21. 21 2.4.1.

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23. 23 2.4.2. First Hydraulic Exponent (M) In many computations involving a wide range of depths in channel, such as in the GVF computations, it is convenient to express the variation of Z with y in an exponential form. The (Z-y) relationship Z 2 =C 1 y m is found to be very advantageous. In this equation C 1 = a coefficient and M= an exponent called first hydraulic exponent. It is found that generally M is a slowly –varying function of the aspect ratio for most of the channel shapes.

24. 24 2.5. Characteristics of Sub-Critical and Super-Critical flow If the critical state of flow exists throughout the entire length of the channel or over a reach of the channel, the flow in the channel is critical flow. The slope of a channel that sustains a given discharge at a uniform and critical depth is called the critical slope (So). A slope of the channel less than the critical slope will cause a slower flow of sub critical state for the given discharge, as will be shown later, and hence, is called a mild or sub critical slope. A slope greater than the critical slope will result in a faster flow of supercritical state, and is called a steep or supercritical slope.

25. 2.6. Transitions The concepts of specific energy and critical energy are useful in the analysis of transition problems. It occurs where there is a change in width, shape, slope, roughness, bottom elevation of the channel. For changes in slope and roughness we can use back water curves to evaluate the effect of the changes. For other types of smooth transitions (transitions were energy loss is minimal), we can use energy relationships to evaluate the impact of the transition. Transitions in rectangular channels are presented here. The principles are equally applicable to channels of any shape and other types of transitions . 25

26. 26 2.6.1. Channel with a Hump a) Subcritical Flow Consider a horizontal, frictionless rectangular channel of width B carrying discharge Q at depth y1. Let the flow be subcritical. At a section 2 (Fig.2. 1) a smooth hump of height ∆Z is built on the floor. Since there are no energy losses between sections 1 and 2, construction of a hump causes the specific energy at a section 2 to decrease by ∆Z . Thus the specific energies at sections 1 and 2 are Figure 2.1. Channel transition with a hump

27. 27 Since the flow is subcritical, the water surface will drop due to a decrease in the specific energy. In Fig. (2.2), the water surface which was at P section 1 will come down to point R at section 2. The depth y 2 will be given by, Figure 2.2 Specific energy diagram for fig 2.1 It is easy to see from Fig. (2.2) that as the value of ∆ Z is increased, the depth at section 2, or y2 , will decrease. The minimum depth is reached when the point R coincides with C, the critical depth . At this point the hump height will be maximum, ∆Z max , y 2 = y c = critical depth, and E 2 = Ec = minimum energy for the flowing discharge Q. The condition at ∆Z max is given by the relation,

28. 28 The question may arise as to what happens when ∆Z > ∆ Z max.. From Fig. (2.2) it is seen that the flow is not possible with the given conditions (given discharge). The upstream depth has to increase to cause and increase in the specific energy at section 1. If this modified depth is represented by y 1 ',

29. 29 Recollecting the various sequences, when 0 < ∆ Z < ∆Z max the upstream water level remains stationary at y 1 while the depth of flow at section 2 decreases with ∆Z reaching a minimum value of yc at ∆Z = ∆Z max . (Fig.2.2). with further increase in the value of ∆Z, (i.e., for ∆Z >∆Z max , y 1 will change to y 1 ' while y 2 will continue to remain y c ). The variation of y 1 and y 2 with ∆Z in the subcritical regime can be clearly seen in Fig.2.3. Figure 2.3. Variation of y1 and y2 in subcritical

30. 30 b) Supercritical Flow If y 1 is in the supercritical flow regime Fig (2.2) shows that the depth of flow increases due to the reduction of specific energy. In Fig (2.2) point P` corresponds to y 1 and point R` to depth at the section 2 . Up to the critical depth, y 2 increases to reach yc at ∆Z = ∆ Z max . For ∆ Z > ∆Z max , the depth over the hump y 2 = y c will remain constant and the max upstream depth y 1 will change . It will decrease to have a higher specific energy E 1 ` by increasing velocity V 1 . The variation of the depths y 1 and y 2 with ∆Z in the supercritical flow is shown in Fig. (2.3b). Figure 2.4.Super Critical flow over a hump

31. 31 2.6.2. Transition with a Change in Width a) Subcritical Flow in a Width Constriction Consider a frictionless horizontal channel of width B1 carrying a discharge Q at a depth y1. At a section 2 channel width has been constricted to B2 by a smooth transition. Since there are no losses involved and since the bed elevations at sections 1 and 2 are the same, the specific energy at section1 is equal to the specific energy at section2. Fig 2.4 Transition with width constriction

32. 32 It is convenient to analyze the flow in terms of the discharge intensity q = Q/B. At section 1, q 1 = Q/B 1 and at section 2, q 2 = Q/B. since B 2 < B 1 , q 2 > q 1 . In the specific energy diagram ( Fig.2.5) drawn with the discharge intensity, point P on the curve q1 corresponds to depth y 1 and specific energy E 1 . Since at section 2, E 2 = E 1 and q = q 2 , point P will move vertically downward to point R on the curve q 2 to reach the depth y 2 . Thus , in subcritical flow the depth is y 2 < y 1 . If B 2 is made smaller, then q 2 will increase and y 2 will decrease . The limit of the contracted width B 2 = B 2min is reached when corresponding to E 1 , the discharge intensity q 2 = q 2max , i.e. the maximum discharge intensity for a given specific energy (critical flow condition) will prevail.

33. 33 Figure 2.5: Specific energy diagram for Fig. ( 2.4)

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35. 35 Since B 2 < B 2min , y c2 will be larger than y cm , y c2 > y cm . Thus even though critical flow prevails for all B2 < B2min, the depth section 2 is not constant as in the hump case but increases as y 1 `and hence E 1 ` rises. The variation of y 1 , y 2 and E with B 2 /B 1 is shown schematically in Fig.2.6.

36. 36 Figure 2.6. Variation of y 1 and y 2 in subcritical flow in a width constriction

37. 37 (b) Supercritical Flow in a Width Constriction If the upstream depth y 1 is in the supercritical flow regime, a reduction of the flow width and hence an increase in the discharge intensity causes a rise in depth y 2 . In Fig. 2.5, point P' corresponds to y 1 and point R' to y 2 . As the width B 2 is decreased, R' moves up till it becomes critical at B 2 = B 2m . Any further reduction in B 2 causes the upstream depth to decrease to y' 1 so that E 1 rises to E' 1 . At Section 2, critical depth y'c corresponding to the new specific energy E' 1 will prevail. The variation of y 1 , y 2 and E with B 2 / B 1 in supercritical flow regime is indicated in Fig. 2.7.

38. 38 Figure 2.7. Variation of y 1 and y 2 in supercritical flow in a width constriction

39. 39 A transition in its general form may have a change of channel shape, provision of a hump or a depression´ and contraction or expansion of channel width, in any combination. In addition, there may be various degrees of loss of energy at various components. However , the basic dependence of the depths of flow on the channel geometry and specific energy of flow will remain the same. Many complicated transition situations can be analyzed by using the principles of specific energy and critical depth. 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.

40. 40 2.7. Choking Choking in the case of channel with a hump, and also in the case or situation of a width constriction, it is observed that the upstream water surface elevation is not affected by the conditions at Section 2 till a critical stage is first achieved . Thus in the case of a hump for all Δ Z < Δ Z m , the upstream water depth is constant and for all Δ Z > Δ Z m the upstream depth is different from y 1 . Similarly , in the case of the width constriction, for B 2 ≥ B 2m , the upstream depth y1 is constant; while for all B 2 < B 2m , the upstream depth undergoes a change. This onset of critical condition at Section 2 is a pre requisite to choking. Thus all cases with Δ Z > Δ Zm or B 2 < B 2m are known as choked conditions . Obviously , choked conditions are undesirable and need to be watched in the design of culverts and other surface-drainage features involving channel transitions.

41. The End Thank You 41