Froehlich’s Equation

Froehlich analyzed 170 live-bed scour measurements in laboratory flumes by regression analysis to obtain the following equation:

1)	$\begin{array}{l}\displaystyle \displaystyle y_s = 2.27 K_1 K_2 (L')^{0.43} y^{0.57}_a Fr^{0.61} + y_a\end{array}$

Symbol	Description	Units
$\begin{array}{l}y_s\end{array}$	Scour depth	ft (m)
$\begin{array}{l}K_1\end{array}$	Correction factor for abutment shape, Table 10-4
$\begin{array}{l}K_2\end{array}$	Correction factor for angle of attack ( $\begin{array}{l}\theta\end{array}$ ) of flow with abutment. $\begin{array}{l}\theta\end{array}$ = 90 when abutments are perpendicular to the flow, $\begin{array}{l}\theta\end{array}$ < 90 if embankment points downstream, and $\begin{array}{l}\theta\end{array}$ > 90 if embankment points upstream (Figure 10-1). $\begin{array}{l}K_2 = (\theta / 90)^{0.13}\end{array}$
$\begin{array}{l}L'\end{array}$	Length of abutment (embankment) projected normal to flow	ft (m)
$\begin{array}{l}y_a\end{array}$	Average depth of flow on the floodplain at the approach section	ft (m)
$\begin{array}{l}Fr\end{array}$	Froude number of the floodplain flow at the approach section, $\begin{array}{l}Fr= V_e / (gy_a)^{1/2}\end{array}$
$\begin{array}{l}V_e\end{array}$	Average velocity of the approach flow $\begin{array}{l}V_e = Q_e / A_e\end{array}$	ft/s
$\begin{array}{l}Q_e\end{array}$	Flow obstructed by the abutment and embankment at the approach section	cfs (m³/s)
$\begin{array}{l}A_e\end{array}$	Flow area of the approach section obstructed by the abutment and embankment	ft² (m²)

Note

The above form of the Froehlich equation is for design purposes. The addition of the average depth at the approach section, $\begin{array}{l}y_a\end{array}$ , was added to the equation in order to envelope 98 percent of the data. If the equation is to be used in an analysis mode (i.e. for predicting the scour of a particular event), Froehlich suggests dropping the addition of the approach depth (+ $\begin{array}{l}y_a\end{array}$ ). The HEC-RAS program always calculates the abutment scour with the (+ $\begin{array}{l}y_a\end{array}$ ) included in the equation.