14.1 Open Channel Hydraulics, Manning’s Equation & Swale Design

Core Focus: Surface drainage conveyance represents the primary line of defense in site stormwater management. Landscape architects must master open channel hydraulic principles, apply Manning's Equation to design stable vegetated swales and paved gutters, select durable lining materials based on permissible shear stresses and velocities, and control street gutter spread to protect pedestrian safety and vehicular corridors.

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1. Principles of Open Channel Hydraulics & Gravity Flow

Open channel flow occurs whenever liquid moves through a conduit or swale with a free, unconfined upper surface exposed directly to atmospheric pressure. Unlike closed pipes operating under internal hydrostatic pressure, the driving force in open channel conveyance is strictly the component of gravity acting parallel to the longitudinal bed slope ($S$).

Flow Regimes and Hydraulic Properties

To analyze channel flow accurately on the LARE, candidates must understand three fundamental flow conditions:

1. Steady vs. Unsteady Flow: Flow is classified as steady if the depth and velocity at a given cross-section remain constant over time. If discharge changes over time (as during a fluctuating storm hydrograph), the flow is unsteady.
2. Uniform vs. Non-Uniform (Varied) Flow: Flow is uniform if the depth, water cross-sectional area, and velocity remain identical at every successive cross-section along the channel length. This occurs when the gravitational driving force exactly balances the frictional resistance exerted by the channel boundaries. Engineering design equations (such as Manning's Equation) assume steady, uniform flow conditions.
3. Laminar, Subcritical, and Supercritical Flow:
   • Subcritical Flow ($Fr < 1.0$): Low velocity, tranquil flow where gravitational forces dominate inertial forces. Surface waves can travel upstream. Natural swales and grass waterways are designed to operate in this stable regime.
   • Critical Flow ($Fr = 1.0$): The threshold state where specific energy is minimized for a given discharge.
   • Supercritical Flow ($Fr > 1.0$): High-velocity, torrential flow where inertial forces dominate gravity. Surface disturbances cannot travel upstream. Supercritical flow in soil channels causes violent scouring and catastrophic bed incision.

$\text{Froude Number: } Fr = \frac{V}{\sqrt{g \cdot D_m}}$

Where $V$ is mean flow velocity (ft/s), $g$ is the gravitational acceleration constant ($32.2 \text{ ft/s}^2$), and $D_m$ is hydraulic mean depth ($A / T$, where $A$ is cross-sectional area in $\text{ft}^2$ and $T$ is top water surface width in feet).

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2. Manning's Equation: Variables, Units, and Derivations

Manning's Equation is the universal empirical formulation used by landscape architects and civil engineers to calculate mean flow velocity and volumetric discharge in open channels and gravity-flow pipes.

The Fundamental Equations

In US Customary units (feet, seconds, cubic feet per second):

$V = \left( \frac{1.486}{n} \right) R^{2/3} S^{1/2}$

$Q = A \cdot V = \left( \frac{1.486}{n} \right) A \cdot R^{2/3} S^{1/2}$

In SI Metric units (meters, seconds, cubic meters per second), the conversion factor $1.486$ becomes $1.000$:

$V = \left( \frac{1.0}{n} \right) R^{2/3} S^{1/2} \qquad Q = \left( \frac{1.0}{n} \right) A \cdot R^{2/3} S^{1/2}$

Definition of Variables

• $V$ = Mean Flow Velocity (feet per second, fps): The average speed of water moving through the channel cross-section.
• $Q$ = Volumetric Discharge (cubic feet per second, cfs): The total volume of runoff passing a given point per second.
• $n$ = Manning's Roughness Coefficient (dimensionless): An empirical measure of boundary friction, surface texture, vegetation resistance, and channel irregularity.
• $A$ = Cross-Sectional Flow Area (square feet, $\text{ft}^2$): The area occupied by water perpendicular to the direction of flow.
• $P$ = Wetted Perimeter (feet, ft): The length of the channel boundary in direct contact with the water prism (excluding the air-water interface at the top surface).
• $R$ = Hydraulic Radius (feet, ft): The ratio of flow area to wetted perimeter: $R = \frac{A}{P}$ Hydraulic Radius Insight: A larger hydraulic radius indicates higher hydraulic efficiency. When $R$ increases, a greater proportion of the water volume is isolated from the drag of the friction-producing wetted boundaries, resulting in higher velocity for an identical bed slope.
• $S$ = Longitudinal Channel Slope / Energy Gradient (dimensionless, ft/ft): The vertical drop per unit horizontal length along the swale flow line (e.g., a 2.0% slope is expressed mathematically as $S = 0.020$).

+-------------------------------------------------------------------------+
|                   OPEN CHANNEL HYDRAULIC GEOMETRY                       |
+-------------------------------------------------------------------------+
|                             Top Width (T)                               |
|                  |<=================================>|                  |
|                  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~  Water Surface   |
|                   \                                 /                   |
|                    \          Flow Area (A)        /                    |
|   Wetted Perimeter  \                             /  Wetted Perimeter   |
|        (P_left)      \          Depth (d)        /       (P_right)      |
|                       \                         /                       |
|                        \           v           /                        |
|                         +---------------------+                         |
|                             Bottom Width (b)                            |
|                                                                         |
|   Wetted Perimeter P = P_left + b + P_right                             |
|   Hydraulic Radius R = A / P                                            |
+-------------------------------------------------------------------------+

Manning's Roughness Coefficients ($n$)

The value of $n$ varies significantly depending on channel lining, vegetation density, grass height, and construction quality:

LARE Exam Warning on Vegetal Retardance: Under the USDA Natural Resources Conservation Service (NRCS) classification, vegetative waterways exhibit variable retardance depending on flow depth and vegetative stiffness (Classes A through E, where A is dense tall brush/kudzu with highest resistance and E is very short mowed turf with lowest resistance). As water depth increases and bends the grass blades over, the effective roughness coefficient $n$ drops dramatically, causing flow velocity to accelerate.

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3. Geometric Cross-Sections of Swales

Landscape architects routinely shape site swales into one of three standard geometric configurations: parabolic, trapezoidal, or triangular (V-shaped).

+-------------------------------------------------------------------------+
|                    SWALE GEOMETRIC CROSS-SECTIONS                       |
+-------------------------------------------------------------------------+
| 1. PARABOLIC SWALE: Best hydraulic efficiency, easy mowing, natural look|
|                      ~~~~~~~~~~~~~~~ T ~~~~~~~~~~~~~~~                  |
|                      \                               /                  |
|                       \              d              /                   |
|                        ' .                       . '                    |
|                            ' - - - . _ . - - - '                        |
|    A = (2/3) * T * d          P = T + (8 * d^2) / (3 * T)               |
+-------------------------------------------------------------------------+
| 2. TRAPEZOIDAL SWALE: Highest volume, stable banks, wide bottom        |
|                    |<~~~~~~~~~~~~~~~ T ~~~~~~~~~~~~~~~>|                |
|                    \  z:1                         z:1  /                |
|                     \                d                /                 |
|                      \                               /                  |
|                       +-----------------------------+                   |
|                       |<------------ b ------------>|                   |
|    A = (b + z*d) * d          P = b + 2*d * sqrt(1 + z^2)               |
+-------------------------------------------------------------------------+
| 3. TRIANGULAR (V-SHAPED) SWALE: Scour prone at invert, low discharge   |
|                    \  z:1                         z:1  /                |
|                     \                d                /                 |
|                      \                               /                  |
|                       \              v              /                   |
|                        \                           /                    |
|                         +-------------------------+                     |
|                                    Apex                                 |
|    A = z * d^2                P = 2*d * sqrt(1 + z^2)                   |
+-------------------------------------------------------------------------+

1. Parabolic Swale

• Physical Properties: The channel bed follows a continuous curved parabolic function ($y = a x^2$). It seamlessly blends into surrounding site contours without sharp angle breaks.
• Mathematical Formulas: $\text{Top Width: } T$ $\text{Flow Area: } A = \frac{2}{3} T \cdot d$ $\text{Wetted Perimeter: } P = T + \frac{8 d^2}{3 T} \quad (\text{for shallow flows where } d \le 0.2 T)$ $\text{Hydraulic Radius: } R = \frac{A}{P} = \frac{\frac{2}{3} T d}{T + \frac{8 d^2}{3 T}} = \frac{2 T^2 d}{3 T^2 + 8 d^2}$
• Site Application: Ideal for residential subdivisions, public parks, and campus lawns. Eliminates sharp transitions that scalped mower blades produce and mimics natural geomorphic drainageways.

2. Trapezoidal Swale

• Physical Properties: Features a flat horizontal bottom width ($b$) and sloping planar sidewalls angled at horizontal-to-vertical ratios of $z:1$ (e.g., $3:1$ or $4:1$).
• Mathematical Formulas: $\text{Flow Area: } A = (b + z \cdot d) d$ $\text{Wetted Perimeter: } P = b + 2 d \sqrt{1 + z^2}$ $\text{Top Width: } T = b + 2 z \cdot d$
• Side Slope Criteria: Side slopes should be $\le 3:1$ or preferably $4:1$ to ensure safe maintenance with commercial riding mowers. Slopes steeper than $3:1$ (such as $2:1$) require mechanical stabilization, rock lining, or permanent groundcovers that do not require turf mowing.
• Site Application: Standard for highway medians, industrial park perimeters, and regional drainage corridors where volumetric runoff demands exceed the capacity of shallow parabolic swales.

3. Triangular (V-Shaped) Swale

• Physical Properties: Formed by two sloping planes meeting at a sharp central bottom vertex ($b = 0$).
• Mathematical Formulas: $\text{Flow Area: } A = z \cdot d^2$ $\text{Wetted Perimeter: } P = 2 d \sqrt{1 + z^2}$ $\text{Hydraulic Radius: } R = \frac{z d^2}{2 d \sqrt{1 + z^2}} = \frac{z d}{2 \sqrt{1 + z^2}}$
• Hydraulic Limitations: Concentrates total kinetic shear stress at the single bottom centerline point, making the invert highly vulnerable to rill formation and bed incision. Triangular swales are prohibited on steep slopes or for high discharges; their use is restricted to minor roadside ditches, swales behind retaining walls, and temporary perimeter swales.

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4. Maximum Permissible Velocities & Channel Linings

A stable channel is defined as an open conduit where neither bed scour (erosion) nor excessive sediment deposition occurs over time. Landscape architects evaluate channel stability using either the Maximum Permissible Velocity Method or the Permissible Tractive Force (Critical Shear Stress) Method.

Permissible Velocity Design Standards

Under the permissible velocity method, the channel is sized so that the calculated mean velocity ($V$) for the design storm does not exceed the maximum allowable threshold for the specific lining material and underlying soil type.

Minimum Longitudinal Gradients and Trickle Gutters

• Minimum Slope: Vegetated swales must maintain a longitudinal gradient of at least 1.0% to 2.0% (ideally $\ge 2.0%$) to ensure positive drainage. Gradients flatter than 1.0% cause chronic waterlogging, turf die-off, weed proliferation, and vector (mosquito) breeding.
• Trickle Gutters (Low-Flow Flumes): When a grass swale must be constructed with a slope between 0.5% and 1.0%, standard site engineering requires an integrated concrete or stone trickle gutter (valley pan) along the centerline to convey dry-weather baseflow and nuisance irrigation runoff without saturating the surrounding turf.

Velocity Reduction via Check Dams

When channel slope ($S$) produces flow velocities exceeding permissible vegetative limits (typically on slopes $> 4.0%$ to $5.0%$), landscape architects install intermediate check dams constructed from dumped rock, gravel bags, or biological fiber logs.

• Hydraulic Function: Check dams act as small broad-crested weirs that flatten the effective hydraulic energy slope of the channel between structures, forcing the flow to drop energy over hardened vertical grade-control structures.
• Check Dam Spacing Formula: $L = \frac{H}{S}$ Where $L$ is horizontal spacing between check dams (feet), $H$ is the vertical height of the check dam crest (measured from channel bed to weir overflow notch in feet), and $S$ is the natural longitudinal slope of the channel (ft/ft). Design Rule: The toe of the upstream check dam must be set at the exact same elevation as the crest (overflow weir lip) of the downstream check dam to prevent bed erosion between structures.

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