## Fall Velocity for Live-bed Contraction Scour Calculation Using HEC-18 Equation

To apply the HEC-18 **Equation 6.2** (** Figure 1**) to calculate the live-bed contraction scour depth, the bed material particle D50 fall velocity needs to be determined to estimate the coefficient

**K1**(

**).**

*Figure 2***HEC-18 Figure 6.8** can be used to estimate the fall velocity for D50 up to 10mm (** Figure 3**).

To automate the fall velocity calculation process and more importantly, to estimate the fall velocities for D >10mm, two equations are introduced here whose results are in good accordance with **HEC-18 Figure 6.8**.

**Van Rijn Equation (HEC-RAS Hydraulic Reference Manual)**

Based on Stokes’ law, Van Rijn proposed a set of 3 equations to estimate fall velocities for different sizes of particles (** Figure 4**). HEC-RAS Hydraulic Reference Manual introduced the equations, however, the manual has a typo: the diameter d’s exponent of 2 is missing in the first equation for 0.001<d=<0.1mm.

**below has added back the exponent.**

*Figure 4***Ferguson Church Equation**

Ferguson and Church proposed another equation to calculate the fall velocity (** Figure 5**).

The two equations including the suggested constants introduced above are for fall velocities of natural sediment particles at water temperature around 20°C (68°F). The results of the two equations match well especially at D<10mm (** Figure 6**).

The equations and the suggested constants are only meant to estimate the fall velocity of D50 and HEC-18 live-bed contraction scour **Equation 6.2** coefficient K1. Their application for other purposes should be carefully evaluated for suitability.

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