CELE Hydraulics & Fluid Mechanics — Flow in Open ChannelsSummary

The hydraulic radius is defined as the ratio of the flow cross-sectional area (A) to the wetted perimeter (P): R = A/P. The wetted perimeter includes only the channel boundary in contact with the water—the free surface is excluded. For a rectangular channel of width b and depth y: A = by, P = b + 2y, so R = by/(b + 2y). The larger the hydraulic radius, the more efficient the channel at conveying flow. This concept is central to Manning's equation and is not the geometric radius of a circular pipe.

In uniform flow (constant depth along the channel reach), the average velocity is given by Manning's equation in SI units: v = (1/n)R^(2/3)S^(1/2), where n is Manning's roughness coefficient (dimensionless), R is the hydraulic radius in metres, and S is the bed slope in m/m (expressed as a decimal, e.g., 0.001 for a 0.1% slope, not as a percentage). The discharge is then Q = Av (in m³/s). Manning's n varies with material: n ≈ 0.010–0.013 for concrete/lined channels, 0.025–0.035 for natural earth channels, and higher for very rough or vegetated channels. Uniform flow occurs when the energy slope (friction losses) equals the bed slope.

For a given cross-sectional area A and bed slope S, the discharge Q = Av is maximized when the velocity v is maximized. Since v ∝ R^(2/3) and R = A/P, for fixed A, v is maximized when P (wetted perimeter) is minimized. Therefore, the most efficient section is the one with the smallest wetted perimeter for a given area. For a rectangular channel, this optimal condition occurs when b = 2y (width equals twice the depth), yielding R = y/2. For a trapezoidal channel, the most efficient shape is a half-hexagon with sides at 60° from horizontal. For a circular conduit, maximum discharge occurs at approximately 94% of full depth (not at full flow). This principle guides channel design when economy of construction is important.

Specific energy is the total mechanical energy per unit weight of water, measured relative to the channel bed: E = y + v²/(2g), where y is the flow depth and v²/(2g) is the velocity head. For a given discharge (or unit discharge q = Q/b for rectangular channels), specific energy varies with depth. A graph of E versus y shows that two different depths can convey the same discharge with the same energy—one shallow depth (supercritical flow) and one deeper depth (subcritical flow). The energy curve has a minimum point called the critical depth (y_c), which occurs at the critical condition where the Froude number Fr = 1. At critical depth, E is at its minimum: E_min = (3/2)y_c. For a rectangular channel, y_c = (q²/g)^(1/3), and this is a unique depth determined solely by the discharge per unit width.

The Froude number quantifies the relative importance of inertial forces to gravitational forces in open-channel flow: Fr = v/√(gy_h), where v is the mean velocity, g is gravitational acceleration (9.81 m/s²), and y_h is the hydraulic depth. For rectangular channels, y_h = y (depth); for non-rectangular sections, y_h = A/T, where T is the top water-surface width. Flow is classified as: (1) Subcritical (Fr < 1)—slow, tranquil flow where depth is greater than critical depth; (2) Critical (Fr = 1)—occurs at y = y_c, the transition condition; (3) Supercritical (Fr > 1)—rapid, shooting flow where depth is less than critical depth. The Froude number indicates whether a disturbance (such as a change in bed elevation or channel contraction) will propagate upstream (subcritical) or be swept downstream (supercritical).

A hydraulic jump is an abrupt, turbulent transition from supercritical flow (shallow, high velocity) to subcritical flow (deeper, lower velocity) that occurs when the downstream flow resistance becomes too great for the supercritical flow to persist. Jumps are characterized by vigorous turbulence, air entrainment, and significant energy dissipation. The depth before the jump (y₁, called the initial or sequent depth) and the depth after the jump (y₂, called the conjugate or sequent depth) are related by the momentum equation: y₂/y₁ = [1 + √(1 + 8Fr₁²)]/2, where Fr₁ is the Froude number at the upstream section. The energy loss across the jump is ΔE = E₁ − E₂, which dissipates as heat and turbulent motion. Hydraulic jumps are used in stilling basins downstream of spillways and control structures to dissipate kinetic energy and prevent erosion.

Uniform flow occurs when the water depth, velocity, and cross-sectional area remain constant along a channel reach. This steady-state condition exists when the driving force (component of gravity along the slope) equals the resistance force (friction). At uniform flow, the energy slope S_f equals the bed slope S_b. The normal depth y_n is the depth at which uniform flow occurs for a given discharge, roughness, and slope. It is found by solving Manning's equation: Q = (A/n)R^(2/3)S^(1/2) for the depth y that satisfies this equation. The normal depth is not necessarily equal to the critical depth; the relationship depends on the slope classification. For mild slopes (S < S_c, where S_c is the critical slope for the given Q and n), y_n > y_c (subcritical uniform flow). For steep slopes (S > S_c), y_n < y_c (supercritical uniform flow). For critical slopes, y_n = y_c.

Open channels can have various cross-sectional shapes, each with different hydraulic properties. Common shapes include: (1) Rectangular—simple, often used in lined canals; A = by, P = b + 2y; (2) Trapezoidal—practical for earth channels with stable side slopes; for bottom width b, depth y, and side slope m (m:1 means horizontal:vertical), A = (b + my)y, P = b + 2y√(1 + m²); (3) Triangular—used for small ditches; A = my², P = 2y√(1 + m²); (4) Circular—used in pipes and tunnels; geometric relationships depend on the fraction of full depth. The choice of shape depends on material (concrete favours rectangular), stability (earth favours trapezoidal with 1.5:1 or 2:1 side slopes), and cost. For the same area, the semicircular shape (width = height, forming a half-hexagon in trapezoidal approximation) is most efficient.

Manning's n is an empirical coefficient that accounts for resistance to flow due to channel bed and wall friction, vegetation, channel irregularity, and other factors. It is dimensionless in SI units and varies with material, condition, and flow regime. Typical values: n ≈ 0.010–0.012 for smooth concrete; 0.013–0.015 for unfinished concrete; 0.020–0.025 for brick or stone masonry; 0.025–0.035 for natural earth channels; 0.035–0.050 for natural streams with vegetation; 0.050–0.080 for very rough natural channels. The Cowan method allows adjustment of a base value for conditions such as vegetation density, channel irregularity, and bed material size. In the Philippines, NSCP 2015 and local hydraulic standards provide guidance on selecting appropriate n values for specific channel types used in irrigation and drainage systems.

Manning's equation v = (1/n)R^(2/3)S^(1/2) shows that velocity depends on the hydraulic radius R = A/P. For a given area A, the designer can choose a shape that minimizes P (and thus maximizes R) to increase capacity. The rectangular shape with b = 2y is the most efficient because it achieves the minimum P for a given A, thereby maximizing R and thus v and Q.

For a rectangular channel, the critical depth depends only on the unit discharge q = Q/b per unit width: y_c = (q²/g)^(1/3). Once q is known, y_c is determined regardless of slope or roughness. This relationship is independent of Manning's equation and depends only on the energy balance at critical flow.

The Froude number Fr = v/√(gy_h) relates velocity and gravity to classify flow. Fr < 1 (subcritical) means the flow depth is greater than critical depth (y > y_c). Fr > 1 (supercritical) means y < y_c. The relationship is direct: if you calculate the actual depth from Manning's equation and compare it to the critical depth from the energy equation, you can determine Fr and flow regime without explicitly calculating Fr itself.

The normal depth y_n is determined by solving Manning's equation for a given Q, n, and S. The critical depth y_c is determined from y_c = (q²/g)^(1/3). By comparing y_n to y_c, the flow regime is identified: if y_n > y_c, the slope is mild and uniform flow is subcritical; if y_n < y_c, the slope is steep and uniform flow is supercritical. This three-way relationship (Manning, energy, and classification) is central to open-channel analysis.

When a supercritical flow encounters a condition that requires subcritical flow (such as a tailwater elevation or a change in slope), a hydraulic jump forms. The sequent depth y₂ is calculated from the Froude number at y₁ using the jump equation. The energy dissipated in the jump (ΔE) is the difference E₁ − E₂, and this energy loss is converted to turbulence and heat. This chain of cause-and-effect is exploited in spillway design to control erosion.

The specific energy curve E = y + v²/(2g) (or E = y + q²/2gy² for rectangular channels) shows that for a given E and q, two depths are possible: one on the subcritical branch (high depth, low velocity) and one on the supercritical branch (low depth, high velocity). The curve has a minimum at critical depth y_c where E_min = (3/2)y_c. This graphical relationship explains how a disturbance can shift the flow from one branch to the other through a hydraulic jump or a transition feature.

The critical slope S_c is the slope at which the normal depth equals the critical depth (y_n = y_c) for a given Q and n. Slopes less than S_c are mild (y_n > y_c, subcritical uniform flow). Slopes greater than S_c are steep (y_n < y_c, supercritical uniform flow). Slope classification determines whether uniform flow (if it exists) is tranquil or rapid, and influences how disturbances propagate through the channel.

Design an irrigation canal to deliver 2 m³/s to a plantation in a gentle terrain where slope is limited to 0.0008 m/m. Using Manning's equation with n = 0.025 (lined earth canal), the engineer calculates the most efficient rectangular section (b = 2y) to minimize excavation and lining cost. Once y is determined, normal depth is compared to critical depth to confirm subcritical flow (appropriate for irrigation where upstream control is possible). Side slopes are set to 1.5:1 for stability in the local soil. NSCP 2015 Section 3 provides guidance on freeboard, check structure spacing, and maintenance access width.

A municipal stormwater system in Metro Manila must convey a 10-year design storm discharge of 8 m³/s through a trapezoidal concrete-lined channel (n = 0.013). The channel bed slope is 0.002 m/m. The engineer calculates the normal depth using Manning's equation, finds the critical depth from the storm discharge per unit width, and determines whether the design depth exceeds both normal and critical depths (providing flow stability and preventing both scour and submergence). If the actual depth is less than critical, the channel will experience rapid flow and possibly a hydraulic jump at a downstream control structure, requiring a stilling basin design.

Water exits a concrete spillway with supercritical flow (high velocity, shallow depth, Fr >> 1). To prevent scour of the downstream riverbed, a stilling basin is designed using hydraulic jump principles. The basin floor is set at the tailwater level corresponding to the basin volume required to accommodate the flow. Using the jump equation, the conjugate depth y₂ is calculated from the incoming Froude number Fr₁ and sequent depth y₁. Energy dissipation across the jump (ΔE = E₁ − E₂) reduces the kinetic energy from ~v₁²/2g to ~v₂²/2g, protecting downstream reaches from erosion. The basin length is typically 5 to 7 times the jump height (y₂ − y₁).

A diversion weir in the Abra River diverts water into an irrigation system. The engineer must ensure that at the design flood discharge, flow approaching the intake is subcritical (Fr < 1) so that the water surface elevation is controlled by downstream conditions and the intake functions reliably. Using Manning's equation for the natural river channel, the normal depth is calculated. If the slope is mild and normal depth is large, subcritical flow is confirmed. Conversely, if the channel becomes steeper upstream (steep reach with y_n < y_c), flow may become supercritical, potentially creating a hydraulic jump upstream of the intake. The design must account for these transitions.

In low-lying coastal areas of the Philippines, tidal channels and canalised rivers must convey both river discharge and tidal flow without excessive flooding or saltwater intrusion. The designer calculates the conveyance of the channel for a given tide level using Manning's equation, ensuring the channel depth and shape are adequate to handle peak spring tide combined with upstream runoff. The critical depth is calculated to determine if supercritical flow can occur during high tides, which might induce upstream drawdown. Regular maintenance is required because n (roughness) increases as vegetation grows, reducing discharge capacity.

When an open-channel aqueduct is constructed to transport water across a valley (using a trestle or flume), the cross-sectional area is often constrained by structural or cost considerations. The engineer chooses a shape that maximises hydraulic radius (minimises wetted perimeter) for the given area. A rectangular section with b = 2y is preferred because it is the most efficient and easiest to form in concrete. For larger capacities, a trapezoidal section with optimal side slope (1:1 or 1.5:1) may be used. This optimisation reduces friction losses and allows smaller depths, reducing construction height and cost.

In steep mountainous regions or in urban stormwater systems, cascading channels with check dams or drop structures control erosion and provide grade control. As flow drops from one level to the next, it accelerates and becomes supercritical. At the base of each drop, a small plunge pool (or stilling basin) is provided to induce a hydraulic jump that dissipates kinetic energy before the flow enters the next reach. The engineer calculates the sequent depths and energy loss to size the basin and protect the downstream channel bed. This principle is used extensively in Philippine upland and urban drainage projects.

When a channel widens or narrows (due to land constraints or confluences), the specific energy and flow regime may change. Using the energy equation and continuity, the engineer predicts whether the transition will cause a rise or fall in water surface, and whether critical depth will be exceeded. If supercritical flow is forced into a narrowing section where y < y_c is impossible, a hydraulic jump occurs upstream of the constriction. Proper design of transitions (gradual flaring or tapered approaches) reduces energy loss and flood risk.