10.2 Open-channel flows

Types of open-channel flows

Open-channel flows are defined by the presence of a free surface in contact with the atmosphere. This distinguishes them from pipe flows, where the fluid fills the entire conduit. The behavior of flow in an open channel depends on channel geometry, roughness, slope, and the flow conditions themselves.

Flows are classified based on how they vary in time, how they vary in space, and what flow regime they fall into.

Steady vs unsteady flows

• Steady flows have constant flow properties (velocity, depth, discharge) at any given location over time.
  • Example: Flow in a canal with a constant water supply.
• Unsteady flows have flow properties that vary with time at a given location.
  • Example: Flow in a river during a flood event, where the water level and velocity change over time.
• Unsteady flows are significantly harder to analyze and typically require numerical methods (such as the Saint-Venant equations) to solve.

Uniform vs non-uniform flows

• Uniform flows have constant flow properties (velocity, depth, slope of the energy grade line) along the channel length.
  • This only happens when the channel cross-section, roughness, and slope are all constant over a sufficient length.
  • Example: Flow in a long, straight, prismatic channel with constant discharge.
• Non-uniform flows have flow properties that vary along the channel length.
  • Caused by changes in channel geometry, roughness, or bed slope.
  • Example: Flow through a channel contraction or expansion, or approaching a dam.

Laminar vs turbulent flows

• Laminar flows have smooth, parallel streamlines with minimal mixing between fluid layers.
  • Characterized by low Reynolds numbers (typically $Re < 500$ for open channels).
  • Example: Flow in a very shallow, smooth channel at low velocity.
• Turbulent flows have irregular, fluctuating velocity fields with significant mixing.
  • Characterized by high Reynolds numbers (typically $Re > 2000$ for open channels).
  • Example: Flow in a steep, rough channel at high velocity.

The Reynolds number for open channels uses the hydraulic radius $R$ as the length scale: $Re = \frac{vR}{\nu}$, where $v$ is the mean velocity and $\nu$ is the kinematic viscosity. In practice, nearly all real-world open-channel flows are turbulent.

Subcritical vs supercritical flows

The Froude number governs this classification: $Fr = \frac{v}{\sqrt{gy}}$, where $v$ is the mean velocity, $g$ is gravitational acceleration, and $y$ is the flow depth.

• Subcritical flow ($Fr < 1$): Relatively low velocity and high depth. Surface waves can travel upstream, so downstream conditions influence the flow.
  • Example: Flow in a wide channel with a mild slope.
• Supercritical flow ($Fr > 1$): Relatively high velocity and low depth. Surface waves cannot travel upstream, so the flow is controlled by upstream conditions.
  • Example: Flow down a steep chute or over a spillway.
• Critical flow ($Fr = 1$): The transition point between subcritical and supercritical regimes. At this condition, the specific energy is at its minimum for a given discharge.

Flow characteristics

Velocity distribution

The velocity in an open channel is not uniform across the cross-section. It's zero at the channel bed and walls (no-slip condition) and reaches its maximum slightly below the free surface. The maximum occurs below the surface (not at it) because of wind shear and secondary currents.

• In the turbulent region near the bed, the velocity follows a logarithmic profile.
• Channel geometry, roughness, and whether the flow is laminar or turbulent all affect the shape of the velocity distribution.
• In wide rectangular channels, the distribution is approximately one-dimensional (varying mainly with depth).

Pressure distribution

For most open-channel flows, the pressure distribution is hydrostatic, meaning pressure varies linearly with depth:

$p = \rho g h$

where $p$ is the gauge pressure, $\rho$ is the fluid density, $g$ is gravitational acceleration, and $h$ is the depth below the free surface. At the free surface, the pressure equals atmospheric pressure.

This hydrostatic assumption holds when streamline curvature is small and depth changes are gradual. Near rapidly varied flow regions (like a hydraulic jump or sharp crest), the pressure distribution deviates from hydrostatic.

Shear stress distribution

Shear stress arises from friction between the flowing fluid and the channel boundaries. In a wide channel, the bed shear stress is the dominant component and can be estimated as:

$\tau_0 = \rho g R S$

where $R$ is the hydraulic radius and $S$ is the energy slope. Shear stress is highest at the bed and decreases toward the free surface, where it approaches zero.

In non-rectangular channels (like trapezoidal sections), the shear stress varies along the wetted perimeter because the side slopes and bed have different orientations. Bed shear stress is a critical parameter for predicting sediment transport and channel erosion.

Specific energy in open-channel flows

Specific energy is the total mechanical energy per unit weight of fluid, measured relative to the channel bed:

$E = y + \frac{v^2}{2g}$

where $y$ is the flow depth and $v$ is the mean velocity. You can also write this using discharge per unit width $q$:

$E = y + \frac{q^2}{2gy^2}$

The specific energy diagram (a plot of $E$ vs. $y$ for a given discharge) is one of the most useful tools in open-channel analysis. It shows that:

• For any specific energy above the minimum, there are two possible depths: a subcritical (deeper) depth and a supercritical (shallower) depth. These are called alternate depths.
• At the minimum specific energy, the flow is at critical depth and $Fr = 1$.
• For a rectangular channel, the critical depth is $y_c = \left(\frac{q^2}{g}\right)^{1/3}$.

Equations of open-channel flows

The governing equations for open-channel flow come from the same conservation principles used throughout fluid mechanics: conservation of mass, momentum, and energy. These are simplified from the Navier-Stokes equations for one-dimensional, steady-state conditions.

Continuity equation

The continuity equation expresses conservation of mass. For steady, incompressible flow:

$Q = A_1 v_1 = A_2 v_2$

where $Q$ is the volumetric discharge, $A$ is the cross-sectional flow area, and $v$ is the mean velocity at each section. This tells you that if the channel narrows ($A$ decreases), the velocity must increase to maintain the same discharge.

Momentum equation

For a control volume between two cross-sections in steady flow:

$\rho Q (v_2 - v_1) = P_1 - P_2 + W \sin\theta - F_f$

where $P_1$ and $P_2$ are the hydrostatic pressure forces on each cross-section, $W \sin\theta$ is the component of the fluid weight along the flow direction, and $F_f$ is the friction force along the channel boundaries.

The momentum equation is especially useful for analyzing situations with abrupt changes, like hydraulic jumps, where energy losses are significant and hard to quantify directly.

Energy equation

The energy equation (Bernoulli's equation extended to account for friction losses) between two sections:

$\frac{v_1^2}{2g} + y_1 + z_1 = \frac{v_2^2}{2g} + y_2 + z_2 + h_L$

where $z$ is the bed elevation and $h_L$ is the head loss due to friction between the two sections. Each side represents the total head (velocity head + pressure head + elevation head) at that section.

This equation is the basis for analyzing gradually varied flow profiles and for tracking the energy grade line along a channel.

Uniform flow

Uniform flow is the idealized condition where depth, velocity, and cross-sectional area remain constant along the channel. This happens when the gravitational driving force exactly balances the frictional resistance. The water surface, channel bed, and energy grade line are all parallel.

The depth under uniform flow conditions is called the normal depth ($y_n$).

Chezy's equation

Chezy's equation is one of the earliest empirical formulas for uniform flow velocity:

$v = C \sqrt{RS}$

where $C$ is the Chezy coefficient (units of $\text{m}^{1/2}/\text{s}$), $R$ is the hydraulic radius ($R = A/P$, with $A$ being the flow area and $P$ the wetted perimeter), and $S$ is the bed slope.

The Chezy coefficient depends on channel roughness and can be related to Manning's $n$ by $C = \frac{1}{n} R^{1/6}$.

Manning's equation

Manning's equation is the most widely used formula for uniform open-channel flow:

$v = \frac{1}{n} R^{2/3} S^{1/2}$

where $n$ is Manning's roughness coefficient, $R$ is the hydraulic radius, and $S$ is the bed slope. This equation uses SI units; in US customary units, a factor of 1.486 replaces the 1 in the numerator.

Manning's equation is preferred over Chezy's because tabulated $n$ values are widely available and the equation is straightforward to apply.

Roughness coefficients

The roughness coefficient captures the resistance to flow from the channel boundaries. Some typical Manning's $n$ values:

• Smooth concrete: $n \approx 0.012$
• Earth channel (clean): $n \approx 0.022$
• Natural stream with vegetation: $n \approx 0.035 - 0.050$
• Floodplain with heavy brush: $n \approx 0.075 - 0.15$

The coefficient depends on surface material, irregularities, vegetation, channel alignment, and obstructions.

Computation of uniform flow

To compute uniform flow for a given channel:

1. Define the channel geometry (shape, dimensions) and calculate the cross-sectional area $A$, wetted perimeter $P$, and hydraulic radius $R = A/P$.
2. Estimate the roughness coefficient $n$ (or $C$) from tables based on channel material and condition.
3. Calculate the mean velocity using Manning's equation: $v = \frac{1}{n} R^{2/3} S^{1/2}$.
4. Calculate the discharge: $Q = Av$.

If the discharge is given and you need to find the normal depth, the process requires iteration because $A$ and $R$ both depend on $y_n$. You substitute the depth-dependent expressions for $A$ and $R$ into Manning's equation and solve (usually by trial and error or a root-finding method).

Gradually varied flow

Gradually varied flow (GVF) occurs when depth changes slowly along the channel, driven by gradual changes in channel geometry, slope, or downstream/upstream boundary conditions. The streamlines remain nearly parallel, so the hydrostatic pressure assumption still holds.

Dynamic equation of gradually varied flow

The GVF equation is derived by combining the energy and continuity equations for steady, one-dimensional flow:

$\frac{dy}{dx} = \frac{S_0 - S_f}{1 - Fr^2}$

where:

• $dy/dx$ is the rate of change of depth along the channel
• $S_0$ is the bed slope
• $S_f$ is the friction slope (computed using Manning's or Chezy's equation with the local depth)
• $Fr$ is the local Froude number

This equation tells you a lot about how the water surface behaves:

• The numerator ($S_0 - S_f$) compares gravity's driving force to friction.
• The denominator ($1 - Fr^2$) changes sign at critical flow. This is why the water surface profile behaves very differently in subcritical vs. supercritical flow.
• When $Fr = 1$, the equation predicts $dy/dx \to \infty$, which signals a transition (like a hydraulic jump) rather than a physical infinite slope.

Classification of flow profiles

GVF profiles are classified by channel slope type and the relationship between the actual depth $y$, the normal depth $y_n$, and the critical depth $y_c$. The main categories are:

Mild slope ($y_n > y_c$, meaning $S_0 < S_c$):

• M1: $y > y_n > y_c$. Depth increases downstream. Occurs upstream of a dam or reservoir (backwater curve).
• M2: $y_n > y > y_c$. Depth decreases downstream. Occurs at a free overfall or channel steepening.
• M3: $y_n > y_c > y$. Supercritical flow with depth increasing downstream. Occurs downstream of a sluice gate on a mild slope.

Steep slope ($y_n < y_c$, meaning $S_0 > S_c$):

• S1: $y > y_c > y_n$. Subcritical flow with depth increasing downstream. Occurs upstream of a hydraulic jump on a steep slope.
• S2: $y_c > y > y_n$. Depth decreases toward normal depth. Occurs at the entrance to a steep channel.
• S3: $y_c > y_n > y$. Supercritical flow with depth increasing toward normal depth. Occurs downstream of a sluice gate on a steep slope.

There are also profiles for critical slope (C), horizontal bed (H), and adverse slope (A) channels, though these are less common in practice.

Computation of gradually varied flow

To compute a GVF profile:

1. Determine the channel geometry, roughness ($n$), and bed slope ($S_0$).
2. Calculate the normal depth ($y_n$) and critical depth ($y_c$) for the given discharge.
3. Classify the flow profile (M1, M2, S1, etc.) based on the boundary conditions and the relationship between $y$, $y_n$, and $y_c$.
4. Identify the boundary condition: use the known downstream depth for subcritical profiles (compute upstream) and the known upstream depth for supercritical profiles (compute downstream).
5. Solve the GVF equation numerically using the direct step method (for prismatic channels) or the standard step method (for natural or non-prismatic channels).
6. Plot the computed water surface profile ($y$ vs. $x$) and verify that it matches the expected profile type.

GVF analysis is essential for predicting backwater effects behind dams, designing channel transitions, and understanding water surface behavior in natural rivers.

Rapidly varied flow

Rapidly varied flow (RVF) occurs when depth changes abruptly over a short distance. The hydrostatic pressure assumption breaks down in the transition zone, and significant energy dissipation, turbulence, and air entrainment are common. Analysis of RVF typically relies on the momentum and continuity equations rather than the energy equation, because energy losses in the transition are difficult to predict directly.

Hydraulic jump

A hydraulic jump is a sudden transition from supercritical to subcritical flow. It's one of the most important phenomena in open-channel hydraulics.

The Bélanger equation relates the sequent depths (depths before and after the jump) for a rectangular channel:

$\frac{y_2}{y_1} = \frac{1}{2}\left(\sqrt{1 + 8Fr_1^2} - 1\right)$

where $y_1$ is the upstream (supercritical) depth and $Fr_1$ is the upstream Froude number.

The energy dissipated in the jump is:

$\Delta E = \frac{(y_2 - y_1)^3}{4 y_1 y_2}$

The length of the jump is roughly estimated as $L_j \approx 6(y_2 - y_1)$, though this varies with the Froude number and channel geometry.

Hydraulic jumps are deliberately used in engineering for:

• Energy dissipation downstream of spillways and sluice gates (stilling basins)
• Flow mixing and aeration
• Raising the water level for irrigation diversions

Flow over spillways

Spillways release excess water from reservoirs in a controlled way. For free flow (no downstream submergence), the discharge is:

$Q = C_d L H^{3/2}$

where $C_d$ is the discharge coefficient, $L$ is the effective crest length, and $H$ is the total head above the spillway crest (including the velocity head of approach).

The discharge coefficient $C_d$ depends on spillway shape and is typically around 1.7 to 2.2 (SI units) for standard ogee spillways. Spillway types include ogee, stepped, and labyrinth designs, each with different performance characteristics.

Flow under sluice gates

Sluice gates control flow by partially obstructing the channel. For free flow conditions (supercritical jet downstream), the discharge is:

$Q = C_d a \sqrt{2g H}$

where $C_d$ is the discharge coefficient (typically 0.55 to 0.65 for sharp-edged gates), $a$ is the gate opening area, and $H$ is the upstream depth above the gate invert.

When the downstream water level is high enough to submerge the gate opening, the flow becomes submerged, and the discharge is reduced. Submerged flow requires a modified equation that accounts for the downstream depth.

Flow measurement in open channels

Accurate flow measurement in open channels is critical for water resource management, irrigation scheduling, flood forecasting, and environmental monitoring. The most common methods use structures that create a known relationship between water depth and discharge.

Weirs are overflow structures placed across the channel. The discharge depends on the head over the weir crest:

• Sharp-crested rectangular weir: $Q = C_d \frac{2}{3} \sqrt{2g} \, L \, H^{3/2}$
• V-notch (triangular) weir: $Q = C_d \frac{8}{15} \sqrt{2g} \tan\left(\frac{\alpha}{2}\right) H^{5/2}$, where $\alpha$ is the notch angle

Flumes (such as the Parshall flume) constrict the channel to create critical flow conditions. The discharge is determined from a single depth measurement upstream of the throat, using calibrated rating curves specific to the flume size.

Velocity-area methods involve measuring the velocity at multiple points across the channel cross-section (using current meters or acoustic Doppler instruments) and integrating to get the total discharge. This is the standard approach for natural streams where installing a fixed structure isn't practical.