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The discharge formula, Q = A V, can be used to rewrite Gauckler–Manning's equation by substitution for V. Solving for Q then allows an estimate of the volumetric flow rate (discharge) without knowing the limiting or actual flow velocity. The formula can be obtained by use of dimensional analysis.
The Chézy Formula is a semi-empirical resistance equation [1] [2] which estimates mean flow velocity in open channel conduits. [3] The relationship was conceptualized and developed in 1768 by French physicist and engineer Antoine de Chézy (1718–1798) while designing Paris's water canal system.
The Darcy-Weisbach equation was difficult to use because the friction factor was difficult to estimate. [7] In 1906, Hazen and Williams provided an empirical formula that was easy to use. The general form of the equation relates the mean velocity of water in a pipe with the geometric properties of the pipe and the slope of the energy line.
Shear velocity also helps in thinking about the rate of shear and dispersion in a flow. Shear velocity scales well to rates of dispersion and bedload sediment transport. A general rule is that the shear velocity is between 5% and 10% of the mean flow velocity. For river base case, the shear velocity can be calculated by Manning's equation.
Manning equation – Estimate of velocity in open channel flows; Mild-slope equation – Physics phenomenon and formula; Morison equation – Equation for force on an object in sea waves; Navier–Stokes equations – Equations describing the motion of viscous fluid substances
The remaining term, known as the boundary shear velocity, approximates the flow of water downhill under the influence of gravity and has units of velocity, i.e., L/T. [8] [1] [4] From experimental data, Stickler proposed that the dimensionally homogeneous form of the Manning formula could be quantified as: [3]
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The wall shear stress τ is dependent on the flow velocity u, they can be related by using e.g. the Darcy–Weisbach equation, Manning formula or Chézy formula. Further, equation ( 1 ) is the continuity equation , expressing conservation of water volume for this incompressible homogeneous fluid.