Prof. B.S. Thandaveswara. A short horizontal reach of a prismatic channel is considered. Further, the external

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1 Hdraulics 9. Speciic Force short horizontal reach o a prismatic channel is considered. Further, the external rictional orce and the eect o weight component o water can be considered as negligible. Then ( ) ( ) γ β V-βV g =P-P +Wsinθ-P I θ = 0, and P = 0 and also i β = β =, then the momentum equation simpliies can be written as γ V V = P P g The hdrostatic pressure orces P and P are respectivel P = γz and P = γz in which z and z are the distances to the centroids below the surace o low o the respective water low areas ( and ). _ z lso,v = and V =. Then, the momentum equation reduces to g centroid rom ree surace + z = + z g The two sides o the above equation are analogous and, hence, ma be generall expressed or an channel geometr as = + z g The irst term is the rate o change o momentum o the low passing through the channel section per unit weight o water, and the second term is the orce per unit weight o water. Since both terms are essentiall orce per unit weight o water, their sum is known as the speciic orce indicated as. ccordingl, it ma be expressed as =. This means that the speciic orces o sections and are equal, provided that Indian Institute o Technolog adras

2 Hdraulics the external orces and the weight eect o water in the reach between the two sections can be ignored. 9.. The momentum Function - Rectangular channels The general situation is shown in Figure in which there ma or ma not be an energ loss between sections and, and there ma or ma not be some obstacle on which there is a drag orce P. In Figure the direction o P is that o the orce exerted b the obstacle on the low. It is this orce (not the drag on the obstacle) which is to be considered in the momentum equation. P Flow P P Deinition Sketch - omentum Equation I there are an blu bod oering resistance orce (P ) to low then P - = γ The orce P should include the rictional resistance due to boundar surace, and weight o the blu bod. The ollowing are some o the particular cases that occur in the ield.energ loss E = 0, P 0 (the sluice gate). E 0, P = 0 (the simple hdraulic jump) 3. E 0, P 0 (the hdraulic jump with its ormation assisted b some obstructions in the low such as dentated sill (Forced hdraulic jump) Sequent depths o Normal Hdraulic jump Indian Institute o Technolog adras

3 Hdraulics I P = 0 then the speciic orce equations can be simpliied as ( ) q = g q ie.., = ( + g ) `The substitution q = v leads to v = ( + g ) v or = F = + g which is the well- known equation o the normal hraulic jump (NHJ). The Froude number F plas a ke role. The above equation is quadratic in /, whose solution is given b = + 8F and = + 8F In general, there are three independent quantities, and knowing two o them initiall third one can be calculated. The downstream control can create appropriate conditions to orm the jump. The corresponding depths and are known as conjugate or sequent depths. 9.. Speciic Force Diagram The diagram shows the variation o the depth against the speciic orce or a given channel section and discharge, is called speciic - orce diagram. This curve has two limbs C and BC. The limb BC approaches the horizontal axis asmptoticall toward the right. The limb C rises upward and extends indeinitel to the right. For a given value o the speciic orce, the curve has two possible depths and. These two depths constitute the initial and sequent depths o a hdraulic jump (see box). t point C the speciic orce is minimum at the critical depth (see box). Indian Institute o Technolog adras

4 Hdraulics 3 C Speciic-orce Speciic-orce diagram B C is the point o minimum speciic orce or a given discharge This corresponds to critical depth, C is the sub critical limb, BC is the super critical limb. For a given speciic energ there are two depths (Points, and 3 respectivel) known as sequent depths. The dierence between points and 3 represent minus the speciic orce at point 3. =speciic orce at point The phenomenon o the hdraulic jump occurs when low changes rom supercritical to sub critical low. Indian Institute o Technolog adras

5 Hdraulics inimum value o speciic orce: The speciic orce to be o a minimum value then the irst derivation o with d d d( z) respect to should be zero, i.e. = - + = 0 d g d d For a change die in the depth, the corresponding change d( z) click in the static moment o the water area becomes d( z) simpliies as d d = - + = 0 d g d d. Then the above equation Since, d / d = T, / = V, and / T = D. the above equation reduces to V D = g This is the criterion or the critical low condition (Froude number =). Thereore, the depth at the minimum value o the speciic orce is the critical depth. In other words the speciic orce is minimum or the given discharge at the critical state o low Comparison between speciic orce and speciic energ For a given speciic energ E, the speciic - energ curve indicates two possible depths, namel, a low stage in the supercritical low region and a high stage in the sub critical low region. For a given value o, the speciic-orce curve also indicates two possible depths, namel, an initial depth in the supercritical region and a sequent depth in the sub critical low region. I the low stage and the initial depth are both equal to. Then the sequent depth is alwas less than the high stage. Furthermore, the energ content E or the depth is less than the energ content E or the depth. Hence, in order to maintain a constant value o, the depth o low ma be changed rom to which results in loss o speciic energ is E= E E. Indian Institute o Technolog adras

6 Hdraulics 0 Sluice 0 0 gate E c 0 c c E E Hdraulic Speciic-orce jump Speciic-energ diagram diagram Hdraulic jump at sluice gate outlet E P P d _ z T d P P B C P" P Centroid C Note: c 0 E 0 45 or a channel E o zero or small E E (a) (b) (c) slope Speciic-orce curve supplemented with speciic-energ curve. (a) Speciic-energ curve; (b) channel section; (c) speciic-orce curve c P Specic energ diagram. Given E as initial depth (point P ). Initial depth is super critical depth.. Corresponding to E the alternate depth on sub critical limb P 3. The sequent depth due to hdraulic jump is and the corresponding speciic Speciic orce diagram. Corresponding to initial depth speciic orce is (Point P). Initial depth is super critical depth.. Corresponding to alternate depth the speciic orce is point P. 3. The corresponding speciic orce or the sequent depth is. In other words or Normal Hdraulic Jump, the upstream and downstream speciic orces are the same. 4. The speciic orce corresponding to sequent depth is indicated b the point P. Indian Institute o Technolog adras

7 Hdraulics In Hdraulic jump energ loss takes place. The depth corresponding to given E at high stage is known as alternate depth to and vice versa. Whereas the depths due to jump are known as sequent depths., are sequent depths., are alternate depths. Indian Institute o Technolog adras