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While a vertical velocity term is not present in the shallow-water equations, note that this velocity is not necessarily zero. This is an important distinction because, for example, the vertical velocity cannot be zero when the floor changes depth, and thus if it were zero only flat floors would be usable with the shallow-water equations.
For Reynolds number greater than 4000, the flow is turbulent; the resistance to flow follows the Darcy–Weisbach equation: it is proportional to the square of the mean flow velocity. Over a domain of many orders of magnitude of Re ( 4000 < Re < 10 8 ), the friction factor varies less than one order of magnitude ( 0.006 < f D < 0.06 ).
Superficial velocity (or superficial flow velocity), in engineering of multiphase flows and flows in porous media, is a hypothetical (artificial) flow velocity calculated as if the given phase or fluid were the only one flowing or present in a given cross sectional area. Other phases, particles, the skeleton of the porous medium, etc. present ...
The group velocity is depicted by the red lines (marked B) in the two figures above. In shallow water, the group velocity is equal to the shallow-water phase velocity. This is because shallow water waves are not dispersive. In deep water, the group velocity is equal to half the phase velocity: {{math|c g = 1 / 2 c p. [7]
For shallow water waves, such as tsunamis and hydraulic jumps, the characteristic velocity U is the average flow velocity, averaged over the cross-section perpendicular to the flow direction. The wave velocity, termed celerity c, is equal to the square root of gravitational acceleration g, times cross-sectional area A, divided by free-surface ...
The concentration of particles usually spreads out in a straight line, and the Rouse distribution works in the water column above the sheet-flow layer where the particles are less concentrated. However, velocity distribution formulas are still being refined to accurately describe particle velocity profiles in steady or oscillatory sheet flows. [2]
The quasi-geostrophic equations are approximations to the shallow water equations in the limit of small Rossby number, so that inertial forces are an order of magnitude smaller than the Coriolis and pressure forces. If the Rossby number is equal to zero then we recover geostrophic flow.
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. In the 2000s this formula was derived theoretically using the phenomenological theory of turbulence. [4] [5]