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Snyder in 1938, came up with three parameters for the development of a synthetic unit hydrograph, these are base time (tb), width of the unit hydrograph at 50% and 75% of the peak discharge (W) and lag time (tpR), required peak discharge per unit drainage area (qpR), unit peak discharge (qp) and the Rainfall duration (tr). He came up with the following relations for determining the parameters of unit hydrograph with a main basin length (km) of stream (L), length (km) from the outlet to a point on the stream, nearest to the centroid of the catchment basin (Lc) and Ct (usually obtained from gaged catchment basin and ranges between 2.
0 ” 6.5). Sudhakar et al., (2015), Chow et al., (1988), Raghunath (2006):t_p (hr)=5.5t_r (2.30)t_p (hr)=0.75C_t (LL_c )^0.3 (2.31)q_p (m^3″(s.–km—^2 ))=(2.75C_p)/t_p (2.32)t_p (hr)=t_pR+(t_r-t_R)/4 (2.33)q_pR (m^3″(s.–km—^2 ))=(q_p t_p)/t_pR (2.34)t_b (hr)=5.56/q_pR (2.35)W_0.75 (hr)=1.22(q_pR )^(-1.08) (2.36)W_0.5 (hr)=2.14(q_pR )^(-1.08) (2.37) Classification of Open Channel.Open channels flow for drainage purposes, such as flood, erosion and Runoff control structures are sometimes grouped according to the type of lining materials used, these include: Vegetative Linings: This is a natural lining material and consist of grasses and swales. They are easy and cheap to design, providing habitat and water quality benefits. They also allow for infiltration and ground water recharge. However, they are easily eroded away by high velocity Runoff and require regular maintenance to prevent standing water and taller vegetation.
Flexible Linings: Rocks, riprap and rubble are the common type of flexible channels. The surfaces are rough surface and can dissipate energy and mitigate increases in erosive velocity. However, they may require the use of a filter fabric depending on the underlying soils, these linings are usually less expensive than rigid linings and have self-healing qualities that reduce maintenance. They are less susceptible to structural failure, because they can conform to the changes in the channel shape. They allow infiltration and exfiltration, and they provide habitat opportunities for local flora and fauna. The main disadvantage of flexible linings is the growth of grass and weeds, which may present maintenance problems. Also, they can only sustain limited magnitudes of erosive forces. To account for the same safe design discharge, a channel section with a flexible lining would have to be considerably larger than a section lined with a rigid material. Therefore, flexible lining can lead to higher overall channel costs although the flexible lining materials are usually less expensive than the rigid lining materials in terms of construction costs. Rigid Linings: Rigid linings are generally constructed of concrete, stone masonry, grouted riprap and soil cement. Rigid linings can resist high shear stress, provide a much higher conveyance capacity for the same cross section size, and channel slope than flexible lining. They are designed with a minimum permissible velocity of 0.6m/s, this velocity seeks to prevent the occurrence of siltation, growth of weeds or sedimentation. Other advantages include long economic life, low cost of maintenance, reduction in cross section area/land use, and stability of side slopes. However, they are susceptible to failure from structural instability caused by freeze-thaw, swelling due to high temperature, and excessive soil pore pressures. When a rigid lining deteriorates, large broken slabs may be dislodged or displaced by the channel flow. This will result in significant erosion, slope problems, and structure failures. Navier-Stokes Equations for Channel FlowTheoretically, the flow in an open channel can be described by three-dimensional hydrodynamic equations usually as partial differential equations. The Navier-Stokes equations, which are derived from the Newton’s second law of motion, describe the motion of a fluid at a given arbitrary point. The equation defines a wide range of flows such as unsteady, compressible flows to steady incompressible flow. The derivation of the governing equations for Navier-Stokes equations are well researched and documented in literature by Biruk (2018) and Ibrahim et al., (2018)The basic equations of the Navier-Stokes is represented below: (€‚v ‘)/€‚t+v ‘‹…€ ‘v ‘=-€ ‘p+·€^2 v ‘+(+·/3) € ‘(€ ‘‹…v ‘ ) (2.38) Where: € ‘=i ‘(€‚/€‚x)+j ‘(€‚/€‚y)+k ‘(€‚/€‚z)=nabla differential operator €^2 v ‘=(€‚^2 v ‘)/(€‚x^2 )+(€‚^2 v ‘)/(€‚y^2 )+(€‚^2 v ‘)/(€‚z^2 )=Laplacian velocity vector in 3D€v ‘=(€‚v ‘)/€‚x+(€‚v ‘)/€‚y+(€‚v ‘)/€‚zv ‘=iv_x+jv_y+kv_z=velocity vector=densityp=pressure·=shear viscosity=bulk viscosity Saint-Venant Equations for Unsteady Open Channel FlowIn practical field applications, the Navier-Stokes equation of open channel require considerable amount of data which are complex and vary spatially, thus they can only be approximated in the field making the three-dimensional solution susceptible to errors. However, for most practical purposes found in hydraulic channel modelling tools such as HEC-RAS, Hydraulic Tool Box, MIKE-URBAN, SWMM, SEWERGEM and STORM, uses a one-dimensional process along the longitudinal direction of flow, thus ignoring the spatial variation and transverse directions.Unsteady flows through open channels are computed using the one-dimensional full Saint-Venant’s equations. It consists of the equation for the conservation of mass (continuity equation) and momentum (momentum equation) respectively.The Saint-Venant’s equations are derived from the Navier-Stokes equations with the following basic key fundamental assumptions. Cimorelli et al., (2015), Ibrahim et al., (2017) and Kaveh et al., (2018): The flow is one-dimensional: i.e. the velocity is uniform in a given cross section and the transverse free-surface profile is generally horizontal. The streamline curvature is very small and the vertical fluid accelerations are negligible as a result, the pressure distribution is hydrostatic. The flow resistance and turbulence losses are the same as for a steady uniform flow for the same depth and velocity, regardless of trends of the path. The bed slope (bottom slope of channel) is very small enough to satisfy the following approximations: sin–‰€tan–, — — cos–‰€1— The density of water is constant (incompressible) while the channel has fixed boundaries and air entrainment and sediment motion are negligible.The Equations of the Saint-Venant are well researched and documented in literature and derived by Eugene et al., (2018), Kane et al., (2017), Kao-Hua et al., (2018), Sebastien Boyaval (2017) and Saleh et al., (2015). Following these assumptions, the Saint-Venant’s Equations for unsteady prismatic open channel flow of an incompressible fluid (with no lateral inflows and outflows) can be represented by two partial differential equations.
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