Most fall velocity derivations start with balancing the gravitational force and the drag force on a particle falling through the water column. The free body diagram is included in the figure below. However, the resulting equation is circular because fall velocity is function of the drag coefficient CD, which is a function of the Reynolds number, which is itself a function of fall velocity. This self-referential quality of the force balance requires either an approximation of the drag coefficient/Reynolds number or an iterative solution. The fall velocity options in HEC-RAS are detailed in Chapter 12, pages 12-30 to 12-32, but a few brief comments on how each of these methods attempts to solve this equation (fall velocity dependence on fall velocity) are given below.
Rubey assumes a Reynolds number to derive a simple, analytical function for fall velocity. Toffaleti developed empirical, fall velocity curves that, based on experimental data, which HEC-RAS reads and interpolates directly. Van Rijn uses Rubey as an initial guess and then computed a new fall velocity from experimental curves based on the Reynolds number computed from the initial guess. Finally, Report 12 is an iterative solution that uses the same curves as Van Rijn but uses the computed fall velocity to compute a new Reynolds number and continues to iterate until the assumed fall velocity matches the computed within an acceptable tolerance.
Fall velocity is also dependent upon particle shape. The aspect ratio of a particle can cause both the driving and resisting forces in Figure 2 9 to diverge from their simple spherical derivation. All of the equations assume a shape factor or build one into their experimental curve. Only Report 12 is flexible enough to compute fall velocity as a function of shape factor. Therefore, HEC-RAS exposes shape factor as a user input variable but only uses it if the Report 12 method is selected.
