39.1 Gradually Varied Unsteady Flow

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39.1 Gradually Varied Unsteady Flow Gradually varied unsteady low occurs when the low variables such as the low depth and velocity do not change rapidly in time and space.

1 39.1 Gradually Varied Unsteady Flow Gradually varied unsteady low occurs when the low variables such as the low depth and velocity do not change rapidly in time and space. Such lows are very common in rivers during loods and in canals during the period o slow variation in gate opening or closure. Typically two low variables, such as the low depth and velocity or the discharge and depth, deine the low conditions at a channel section. Two governing equations, known as Saint Venant equations, are used to descrine the spatial and temporal variation o the above two low variables. These equations are based on the application o conservation o mass and momentum principles to a stationary control volume such as shown in Figure Assumptions Following assumptions are made in the derivation o the Saint Venant equations: The pressure distribution in the vertical direction at any cross section is hydrostatic. The channel bottom slope is small. The velocity is uniorm within a cross section. The channel is prismatic. Steady state resistance laws are applicable under unsteady conditions. There is no lateral inlow or outlow Derivation Consider unsteady low in a channel as shown in ig Consider a control volume o length x as shown in this igure.

2 1 Water Surace V 1 A1 y 1 y1 V1 A1 y1 y 1 Flow A C.G y x x S 0 Bed x 1 x Fig. 39.1: Deinition sketch or derivation o St.Venant equations The control volume in Fig has ixed boundaries. The Reynolds transport theorem is applied to derive the continuity and momentum equations. Continuity Equation Based on the Reynolds transport theorem and treating water as an incompressible luid, Continuity equation or the control volume in Fig can be written as d dt x 1 1 Adx + A V AV = in which A = low area, V = low velocity and subscripts 1 and indicate low variables at sections 1 and, respectively. Application o Leibritz's theorem to the irst term on the let hand side o the above equation, ollowed by the application o mean value theorem yields A x x 1 AV AV ( ) + = It may be noted that both A and A are assumed continous with respect to both x and t. Similarly, treating AV and VA as continous with respect to x and t, and letting x = x x tend to zero, one can get 1

3 A AV + = x 0 ( 39.3) Noting that low rate, Q = AV. A + = x 0 ( 39.4) Equation (39.4) is the continuity equation in the "Conservation orm". For prismatic channels in which the top width, T is a continous unction o the low depth, y, Eq. (39.4) may be written as da y + dy x or = y T + = x Substitution o Q = VA in Eq and subsequent simpliication leads to y A V y + + V = T x x Momentum Equation Based on the Reynolds transport theorem, momentum equation or the control volume in ig can be written as x = d ρ + ρ ρ 39.8 Res 1 1 dt F V Adx V A V A in which F Res = resultant orce acting on the control volume in the direction o low. As in the case o continuity equation, application o Leibritz theorem and mean value theorem to Eq leads to ( AV ) F Re s = ρ x x ( AV ) Noting that low rate Q = AV,

4 FRe s = ρ x x ( QV ) Resultant orce F Res on the control volume is evaluated as ollows. Channel is assumed to be prismatic. Thereore, orces do not arise due to changes in cross section. Waves set up by the wind action are not considered here. Thereore, shear stress on the low surace due to wind is neglected. Open channel lows in canals, streams and rivers are considered. Flows in large water bodies such as estuaries and oceans are not considered here. Thereore, Coriolis orces are neglected. Net orce on Control volume comprises o (i) pressure orce at section - 1 (See Fig. 39.1), (ii) pressure orce at section -, (iii) Component o weight o water in the low direction and (iv) the rictional orce due to shear between water and the channel sides and the channel bottom. These orces are evaluated as ollows. Pressure orces at sections 1 & are given by y 1 = depth to the centroid o area A 1. y = depth to the centroid o area A. F = ρga y 39.11, F = ρ ga y 39.1 F 1 acts in the positive x direction while F acts in the negative x direction. Component o weight o water in the direction o low = x F = ρ g AS dx 3 0 ( 39.13)

5 Frictional orce x = F ρ g AS dx ( 39.14) 4 = in which S 0 = channel bottom slope and S = riction slope. Friction slope or the slope o the energy gradient line to overcome riction may be estimated using any riction loss equation such as the Manning equation. F 3 acts in the positive x-direction while F 4 acts in the negative x-direction. Substitution o equations or orces in Eq. (39.10) leads to Or Or ( 1 ) g A y A y + ga S S = + AV x ( 0 ) ( 39.15) 1 + = x ( QV ) ( gay) ga( S S ) ( QV gay) ga( S0 S ) ( 39.17) + + = x Equation (39.17) is the momentum equation in the conservation orm. For any cross section in which the top width, T is a continous unction o low depth, y 1 A y+ y + T y -Ay ( Ay) = lim y y 0 y Neglecting higher order terms, y and ( Ay) = A ( 39.19) y y ( gay) = g ( Ay) = ga ( 39.0) x y x x Substitution o Eq. (39.0) in Eq. (39.17) leads to

6 V y + + ga = ga S x x ( 0 ) ( 39.1) Substitution o Q = AV into Eq. (39.1), subsequent expansion o terms, and urther simpliication using continuity equation leads to V V y + V + g = g S 39. x x ( 0 ) Equation (39.) is usually reerred to as the "Dynamic Equation". In this equation, the irst term on the let hand side represents the local acceleration, the second term represents the convective acceleration and the third term represents the pressure gradient. The irst term on the right hand side represents weight component (eect o channel slope) while the second term represents the resistance eect due to shear between the water and the channel surace. For steady, non-uniorm lows, local acceleration is zero and Eq. (39.) reduces to d V dx g + y = S0 ( 39.3). Substitution o Q = AV leads to d dx Q A + y = S0 g Q da dy or - + = S 3 0 ga dx dx dy Q T or = 1 = 3 S0 dx ga dy S0 S or = dx QT 1 3 ga ( 39.4) Equation (39.4) is nothing but equation or steady gradually varied low when the energy correction actor α = 1.

7 For steady, uniorm lows, local and convective acceleration are zero and the low depth, y does not vary with x. Thereore, Eq. (39.) reduces to S0 S = Flood routing problem is deined as: given (i) the channel characteristics (slope, shape parameters, roughness coeicient) and (ii) the lood discharge or the stage hydrograph at an upstream section, determine the lood discharge and the stage hydrographs at any downstream section. This is same as solving or the temporal and spatial variations o Q and y given the (i) channel characteristics, (ii) initial conditions (Q and y at all points in the channel at t = 0) and (iii) Boundary condtions (Q or y variation at x = 0 or all t). Flood routing based on the solution o complete equations or mass and momentum conservation (Eqs and 39.) is termed as "Dynamic Routing". Flood rating in which the irst two terms (acceleration terms) on the let hand side o Eqs. 39. are negelected is termed as "Zero-Inertia Routing". Flood routing in which equations 39.7 and 39.5 are solved together is termed as "Kinematic Wave Routing". Many times Zero - Inertia Routing and Kinematic Wave Routing methods are adopted to avoid computational diiculties.

RESOLUTION MSC.362(92) (Adopted on 14 June 2013) REVISED RECOMMENDATION ON A STANDARD METHOD FOR EVALUATING CROSS-FLOODING ARRANGEMENTS

RESOLUTION MSC.362(92) (Adopted on 14 June 2013) REVISED RECOMMENDATION ON A STANDARD METHOD FOR EVALUATING CROSS-FLOODING ARRANGEMENTS (Adopted on 4 June 203) (Adopted on 4 June 203) ANNEX 8 (Adopted on 4 June 203) MSC 92/26/Add. Annex 8, page THE MARITIME SAFETY COMMITTEE, RECALLING Article 28(b) o the Convention on the International