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I Horizontal transitions are required for economic and practical reasons for irrigation works like bridges, weirs and barrages in natural and artificial open channels. The transitions may be contracting or expanding suddenly or gradually while the flow is I either subcritical or supercritical. The problem is solved using the specific energy i principle. Most of the previous researches solved the problem neglecting the head loss (theoretical solution). The author presents in this research paper a new approach to solve the problem for I horizontal transition for rectangular open channels. i The specific energy equation in this paper takes a real condition in consideration for i both contracting and expanding problems. The practical horizontal transitions cases are ! solved taking into consideration the head loss due to contracting or expanding. Also, this research paper deals with the problem of channel section while contracting with amount greater than that of critical one. If the contraction of bed is increased further, the maximum b, and the specific energy were both held constant with this contraction, the discharge will be decreased until the given specific energy is equal to minimum one corresponding to the new discharge, because the energy cannot be increased without outside affect. Consequently the upstream water depth will be changed to a new value. Manuscript received from Dr. M.F. Sobeih Accepted on : 23 / 9 / 2002 Engineering Research Journal Vol 25, No 4,2002 Minufiya University, Faculty Of Engineering, Shebien El-Kom ,Egypt, ISSN 1110-1180 INTODUCTION , : : , . , . . . : , , ., . . The horizontal transitiqn problems can be solved: by the &,quition which takes the following simplest form (for a channel of very small ;lope af~d a -1.0) (Fig. (1)): 7/ 2 in which: E is the specific energy; y is the water depth; a is the energy coefficient; V is the mean velocity normal to water area; and g is the acceleration due to gravity. Since Q = A. v Equation (1) may be written as follows: in which: Q is the discharge; and A is the water cross-sectional area. Solving for Q:  Q = A Jm (2) Equation (2) is presented graphically in Fig. (2). The curve shows that, for a given discharge, there are two water depths, the low stage y~ and the high stage y*. The low stage is called supercritical flow. The high stage is called subcritical flow. The two water depths are called alternative water depths. If the specific energy changes, the discharge will be changed accordingly. It means that very specific energy has a specific curve for a certain discharge Q. Practical Horizontal Transition Problems: When the size or shape of the cross section of an open channel changes along the length through a horizontal transition, it should be considered as a link between the original and the new channel. Horizontal transitions may be either contracting or expanding, suddenly or gradually taking a short distance or a long one. This research paper is relevant to the solution of the practical problem while the head loss is considered. Thus, the solution of the problem using the specific energy equation takes the following shape: El = E2 +hi in which: El is the specific energy upstream the transition; E2is the specific energy downstream the transition; h, is the head loss due to the transition; y, is the water depth at upstream the transition; y, is the water depth downstream the transition; A, is the water cross-sectional upstream the transition; and A, is the water cross-section downstream the transition. For rectangular section Formica [I] applied the following equations to the energy loss in open channel with a horizontal transitions, as follows: Case I: For a sudden contraction. V2 hi = K 2g (4) in which: K is coefficient of the head loss due to the contraction. The approximate median value of K for sudden contraction is 0.1 and, for flared or moderate is 0.06 (Fig. (3)). Case 11: For a sudden expansion: in which: E is the coefficient of head loss due to expansion. V, is the water velocity upstream the transition; and V , is the water velocity downstream the transition. Formica obtained average values of E for eight sudden expansion shapes which varied between 0.82 and 0.44 [I]. Substituting Equation (4) in Equation (3) for practical sudden contraction problem for rectangular open channels, the following equation is obtained: in which: b, is the section breadth upstream the transition; and b, is the section breadth downstream the transition. Putting dimensionless parameters Q, = d Q2 , and y, = y2/E, the above equation 2gb,2~,3 takes the following form: Q.' l = ~ , + ( l + K ) ~ Y. Y*' = ya3 + Q , ~ + K Y , ~ Q* = y,,/ (6) in which: Q, is the dimensionless discharge; and y, is the dimensionless water depth. Equation (6) is plotted in a graphical shape in Fig. (4). The figure shows three curves for K = 0.0 (theoretical) and K =0.1 and 0.06 for real contraction. The following are displaying some problems that usually in the hydraulic design for the horizontal transitions (contracting). 1. To estimate the size of contracting just required to make the flow critical ( b ~ b , ) (which may be termed as critical contraction); 2. To predict the flow after the transition when the transition is smaller than the critical value (b2 < b,); 3. To predict the flow after the transition as well as upstream when the size of contraction is larger than the critical size @2> b,). To get the condition of maximum contracting differentiates the discharge to the depth and equate to the zero as: or: K =1-1.5y ,,  (8) Substituting Equation (7) into Equation . (6) . it can be obtained: Q., = Y?; J! Y,, 1 + I .~Y,, 3 Q*,' = 0 . 5 ~ , ~  (9) in which: Q,, is the dimensionless critical discharge; and y,, is the dimensionless critical water depth. or: 1 312 Q., = -y,,  (10) f i Demonstration of the Proposed Method Case of contracting: As mentioned before the approximate median value of K for sudden contraction is 0.1 and for flared or moderate is 0.06, then the corresponding values of y,,= 0.6, 0.627 and Q,,=0.3286, 0.351 respectively. There are three cases of contracting according to the amount of contraction 01 the breadth as following: Case: if b2 = bc. Case 11: if bz > b,. Case III: if b2 < b, Case I:( b2 = b& For given Q, b,, and yl and if bz = b,, then: To solve this problem compute the upstream specific energy El = y, + Q Z and 2 g 4 2 ~ , Z substituting the value of the dimensionless parameter Q*, = 0.3826 or 0.35laccording to the case of contraction. Then for bz=b, the following equation can be derived: Case 11: (62 1 be) For given Q, b, y!, and b2 and to derive an expression for y2 the following steps are followed: Compute the practical critical contraction using the same steps as in case I. Check that bz b, By using the values of Q, and K get y,, from the Equation (6 ) , where y,, ( y*, ( y'c if the flow is supercritical and y,,) y,, ) y,c if the flow is subcritical. From any of the two states of flow, the downstream, water depth can be calculate as following:

DIRECT SOLUTION TO PRACTICAL PROBLEM OF OPEN CHANNEL FLOW FOR HORIZONTAL TRANSITIONS