Simple Equations to Calculate Fall Velocity and Sediment Scale Parameter

This paper investigates and compares four simple, continuous equations that can potentially make useful estimates of fall velocity. These estimates span the range of laminar, transitional, and turbulent flow regimes. All equations take advantage of the fact that the fall velocity Reynolds number is a function of only one variable, the Archimedes buoyancy index. Adjusting the coefficients can make a significant improvement in the fit of these equations to data. In the commonly used beach equilibrium profile model, $h=A_p{x}^{2/3}\text{,}$ the sediment scale parameter $A_p\text{,}$ is usually tabulated as a function of grain diameter. It will be shown that the normalized sediment scale parameter $A_p/d^{1/3}$ is also a function of only the Archimedes buoyancy index. This research indicates that the Archimedes buoyancy index is the fundamental independent variable for both the fall velocity Reynolds number and the normalized sediment scale parameter. Calculating the Archimedes buoyancy index is facilitated by some simple equations, which in turn allow easy computation of fall velocity and sediment scale parameter.