Computing Pier Scour With The CSU Equation

The CSU equation predicts maximum pier scour depths for both live-bed and clear-water pier scour. The equation is:

1)	$\begin{array}{l}\displaystyle y_s = 2.0 K_1 K_2 K_3 K_4 a^{0.65} y^{0.35}_1 Fr^{0.43}_1\end{array}$

Symbol	Description	Units
$\begin{array}{l}y_s\end{array}$	Depth of scour	feet (meters)
$\begin{array}{l}K_1\end{array}$	Correction factor for pier nose shape
$\begin{array}{l}K_2\end{array}$	Correction factor for angle of attack of flow
$\begin{array}{l}K_3\end{array}$	Correction factor for bed condition
$\begin{array}{l}K_4\end{array}$	Correction factor for armoring of bed material
$\begin{array}{l}a\end{array}$	Pier width	feet (meters)
$\begin{array}{l}y_1\end{array}$	Flow depth directly upstream of the pier. This is taken from the flow distribution output for the cross section just upstream from the bridge.
$\begin{array}{l}Fr_1\end{array}$	Froude Number directly upstream of the pier. This is taken from the flow distribution output for the cross section just upstream from the bridge.

Note

For round nose piers aligned with the flow, the maximum scour depth is limited as follows:

y_s ≤ 2.4 times the pier width (a) for Fr₁ ≤ 0.8
y_s ≤ 3.0 times the pier width (a) for Fr₁ > 0.8

An optional correction factor, Kw for wide piers in shallow water can be applied to the CSU equation.

$\begin{array}{l}\displaystyle K_w = 2.58 \left( \frac{y}{a} \right) ^{0.34} F^{0.65}\end{array}$ for $\begin{array}{l}\displaystyle \frac{V}{V_c}\end{array}$ < 1

$\begin{array}{l}\displaystyle K_w = 1.0 \left( \frac{y}{a} \right) ^{0.13} F^{0.25}\end{array}$ for $\begin{array}{l}\displaystyle \frac{V}{V_c}\end{array}$ ≥ 1

Because this correction factor was developed based on limited flume data, it is not automatically accounted for in HEC-RAS. The user, however, can manually apply this factor to the computed scour depth, or can combine it with one of the user-entered correction factors (K₁ through K₄). See section 6.3 of HEC-18.

The correction factor for pier nose shape, K₁, is given in Table 10-1 below:

Table 10-1 Correction Factor, K1, for Pier Nose Shape

Shape of Pier Nose	K₁
(a) Square nose	1.1
(b) Round nose	1.0
(c) Circular cylinder	1.0
(d) Group of cylinders	1.0
(e) Sharp nose (triangular)	0.9

The correction factor for angle of attack of the flow, $\begin{array}{l}K_2\end{array}$ , is calculated in the program with the following equation:

2)	$\begin{array}{l}\displaystyle \displaystyle K_2 = \left( cos \theta + \frac{L}{a} sin \theta \right) ^{0.65}\end{array}$

Symbol	Description	Units
$\begin{array}{l}L\end{array}$	Length of the pier along the flow line	feet (meters)
$\begin{array}{l}\theta\end{array}$	Angle of attack of the flow, with respect to the pier

Note

If L/a is larger than 12, the program uses L/a = 12 as a maximum in (2). If the angle of attack is greater than 5 degrees, K₂ dominates and K₁ should be set to 1.0 (the software does this automatically).

The correction factor for bed condition, K₃, is shown in table 10-2.

Table 10-2 Increase in Equilibrium Pier Scour Depth, K₃, For Bed Condition

Bed Condition	Dune Height H feet	K₃
Clear-Water Scour	N/A	1.1
Plane Bed and Antidune Flow	N/A	1.1
Small Dunes	10 > H ≥ 2	1.1
Medium Dunes	30 > H ≥ 10	1.1 to 1.2
Large Dunes	H ≥ 30	1.3

The correction factor K₄ decreases scour depths for armoring of the scour hole for bed materials that have a D₅₀ equal to or larger than 0.007 feet (0.002 m) and a D₉₅ equal to or larger than 0.066 feet (0.020 m). The correction factor results from recent research by A. Molinas at CSU, which showed that when the velocity (V₁) is less than the critical velocity (Vc₉₀) of the D₉₀ size of the bed material, and there is a gradation in sizes in the bed material, the D90 will limit the scour depth. The equation developed by J. S. Jones from analysis of the data is:

3)	$\begin{array}{l}\displaystyle K_4 = 0.4 (V_R)^{0.15}\end{array}$

4)	$\begin{array}{l}\displaystyle \displaystyle V_R = \left[ \frac{V_1 - V_{i50}}{V_{c50} - V_{i95}} \right]\end{array}$

5)	$\begin{array}{l}\displaystyle V_{i50} = -0.645 \left[ \frac{D_{50}}{a} \right] ^{0.053} V_{c50}\end{array}$

Symbol	Description	Units
$\begin{array}{l}V_{i95}\end{array}$	$\begin{array}{l}\displaystyle 0.645 \left[ \frac{D_{95}}{a} \right] ^{0.053} V_{c95}\end{array}$
$\begin{array}{l}V_R\end{array}$	Velocity ratio
$\begin{array}{l}V_1\end{array}$	Average velocity in the main channel or overbank area at the cross section just upstream of the bridge	ft/s (m/s)
$\begin{array}{l}V_{i50}\end{array}$	Approach velocity required to initiate scour at the pier for grain size D50	ft/s (m/s)
$\begin{array}{l}V_{i95}\end{array}$	Approach velocity required to initiate scour at the pier for grain size D95	ft/s (m/s)
$\begin{array}{l}V_{c50}\end{array}$	Critical velocity for D50 bed material size	ft/s (m/s)
$\begin{array}{l}V_{c95}\end{array}$	Critical velocity for D95 bed material size	ft/s (m/s)
$\begin{array}{l}a\end{array}$	Pier width	ft (m)

6)	$\begin{array}{l}\displaystyle V_{c50} = K_u y^{1/6} D^{1/3}_{50}\end{array}$

$\begin{array}{l}\displaystyle V_{c95} = K_u y^{1/6} D^{1/3}_{95}\end{array}$

Symbol	Description	Units
$\begin{array}{l}y\end{array}$	The depth of water just upstream of the pier	ft (m)
$\begin{array}{l}K_u\end{array}$	11.17 (English Units), 6.19 (S.I. Units)

Limiting $\begin{array}{l}K_4\end{array}$ values and bed material size are given in Table 10-3.

Table 10-3 Limits for Bed Material Size and K₄ Values

Factor	Minimum Bed Material Size	Minimum K4 Value
K4	D50 ≥ 0.006 ft (0.002 m) D95≥0.06 ft (0.02 m)	0.4