Manning's Equation for Open Channel Flow in QUAL2K

QUAL2K computes the velocity, depth, and cross-sectional area of each reach using either Manning's equation or power-law rating curves, as described in the EPA QUAL2K technical documentation. Manning's equation is the default and most commonly used method for natural streams with known geometry. The computed hydraulic parameters feed directly into the reaeration coefficient and flow routing calculations.

Manning's Equation

Manning's equation relates the flow discharge $Q$ (m³/s) to the channel geometry and roughness:

Q = \frac{1}{n} \cdot A \cdot R^{2/3} \cdot S^{1/2}

Where:

$n$ — Manning's roughness coefficient (dimensionless)
$A$ — cross-sectional area of flow (m²)
$R = A / P$ — hydraulic radius (m)
$P$ — wetted perimeter (m)
$S$ — channel bed slope (m/m)

Trapezoidal Cross-Section Geometry

QUAL2K models the channel as a trapezoidal cross-section with a flat bottom of width $B$ and two (potentially different) side slopes $ss_1$ and $ss_2$ :

Figure 1: Trapezoidal channel cross-section with labeled parameters

For a given depth of flow $y$ :

A = B \cdot y + \bar{ss} \cdot y^2

where $\bar{ss} = (ss_1 + ss_2) / 2$ is the average side slope.

P = B + y \left( \sqrt{1 + ss_1^2} + \sqrt{1 + ss_2^2} \right)

Newton-Raphson Depth Solver

Given a known discharge $Q$ , the depth $y$ must be found iteratively. QUAL2K uses the Newton-Raphson method to solve:

f(y) = \frac{1}{n} \cdot A(y) \cdot R(y)^{2/3} \cdot S^{1/2} - Q = 0

The derivative is:

f'(y) = \frac{S^{1/2}}{n} \left[ \frac{5}{3} T \cdot R^{2/3} - \frac{2}{3} R^{5/3} \cdot \frac{dP}{dy} \right]

where $T = B + (ss_1 + ss_2) \cdot y$ is the top width and $dP/dy = \sqrt{1 + ss_1^2} + \sqrt{1 + ss_2^2}$ .

The iteration starts at $y_0 = 0.5$ m and converges in 5–10 iterations for typical channels. The update rule is:

y_{k+1} = y_k - \frac{f(y_k)}{f'(y_k)}

Derived Quantities

Once the depth $y$ is known, the remaining hydraulic parameters follow directly:

U = \frac{Q}{A} \quad \text{(velocity, m/s)}

t_{travel} = \frac{A \cdot L}{Q \cdot 86{,}400} \quad \text{(travel time, days)}

where $L$ is the reach length in meters.

Rating Curve Alternative

When rating curve coefficients are provided, QUAL2K uses power-law relationships instead of Manning's equation:

U = \alpha_1 \cdot Q^{\beta_1}

H = \alpha_2 \cdot Q^{\beta_2}

The cross-sectional area is then $A = Q / U$ and the width is $B = A / H$ . This method is used when field-measured rating curves are available.

Typical Manning's n Values

Manning's roughness coefficients for natural streams

Channel Type	n (minimum)	n (typical)	n (maximum)
Clean, straight, full stage	0.025	0.030	0.033
Clean, winding, some pools	0.033	0.040	0.045
Sluggish, weedy, deep pools	0.050	0.070	0.080
Very weedy, heavy brush	0.075	0.100	0.150
Mountain streams, cobbles	0.030	0.050	0.070
Mountain streams, boulders	0.040	0.070	0.100
Floodplain, pasture	0.025	0.035	0.050
Floodplain, heavy timber	0.100	0.120	0.160

Python Implementation

pythonhydraulics.py — Newton-Raphson depth solver

def calculate_hydraulics(reach, geo_method):
    q_s = reach.get('q', 0)  # Flow in m³/s
    n = reach.get('nm', 0.03)
    s = reach.get('s', 0.0001)
    bb = reach.get('BB', 1.0)
    ss1 = reach.get('SS1', 0.0)
    ss2 = reach.get('SS2', 0.0)
    ss_avg = (ss1 + ss2) / 2.0

    # Newton-Raphson iteration for depth y
    y = 0.5  # initial guess (meters)
    for _ in range(30):
        area = bb * y + ss_avg * (y ** 2)
        perim = bb + y * (sqrt(1 + ss1**2) + sqrt(1 + ss2**2))
        radius = area / perim
        f = (1/n) * area * radius**(2/3) * s**0.5 - q_s
        top_width = bb + (ss1 + ss2) * y
        df_dy = (1/n) * s**0.5 * (
            (5/3) * top_width * radius**(2/3) -
            (2/3) * radius**(5/3) * (sqrt(1+ss1**2) + sqrt(1+ss2**2))
        )
        dy = f / df_dy
        y = y - dy
        if abs(dy) < 1e-7:
            break