12.3 Uniform and Gradually Varied Flow

Uniform Flow in Open Channels

Open channels carry water through rivers, canals, and drainage systems using gravity rather than pressure. Uniform flow is the simplest case: depth and velocity stay constant along the channel because gravity pulling the water downhill is exactly balanced by friction slowing it down. Gradually varied flow (GVF) is what happens when that balance breaks, causing depth to change gradually along the channel. GVF profiles describe how the water surface behaves under different slope and depth conditions.

Characteristics of uniform channel flow

For uniform flow to exist, three conditions must hold: the channel is prismatic (constant cross-sectional shape and slope), the flow is steady (nothing changes with time at a given point), and depth, velocity, and cross-sectional area remain constant along the channel length.

The driving mechanism is a balance between two forces:

• Gravitational force pulls water downstream, determined by channel slope and fluid weight
• Resistance force opposes motion through friction between the fluid and the channel boundary (think rough concrete linings vs. natural streambeds with rocks and vegetation)

When these forces balance perfectly, the energy grade line (EGL) and hydraulic grade line (HGL) both run parallel to the channel bottom. The EGL represents total energy head (elevation head + pressure head + velocity head), while the HGL represents piezometric head (elevation head + pressure head only). In uniform flow, the slopes of the EGL, HGL, and channel bed are all equal.

Chezy and Manning equations

Two classical equations relate flow velocity to channel geometry and roughness.

Chezy equation:

$V = C\sqrt{RS}$

• $V$ = average flow velocity
• $C$ = Chezy coefficient, which depends on channel roughness (units: $\text{m}^{1/2}/\text{s}$)
• $R$ = hydraulic radius, defined as cross-sectional flow area divided by wetted perimeter: $R = A/P$
• $S$ = channel bed slope (dimensionless)

Manning equation (more commonly used in practice):

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

• $n$ = Manning's roughness coefficient (units: $\text{s}/\text{m}^{1/3}$), with tabulated values for different materials. For example, finished concrete might have $n \approx 0.012$, while a natural stream with heavy vegetation could be $n \approx 0.05$ or higher.

Once you have velocity, discharge follows from the continuity equation:

$Q = VA$

where $A$ is the cross-sectional area of flow. Combining Manning's equation with continuity gives you discharge directly for a known channel geometry and normal depth.

Gradually Varied Flow in Open Channels

Analysis of gradually varied flow

Gradually varied flow (GVF) occurs when depth and velocity change slowly along the channel. The flow is still steady (no time variation), but it's non-uniform because conditions vary with position. Streamlines remain nearly parallel, so the pressure distribution across any cross-section is still approximately hydrostatic.

GVF profiles are classified by comparing three depths:

• Actual depth ($y$): the depth you observe or calculate at a given section
• Normal depth ($y_n$): the depth that would exist if the flow were uniform at the given discharge and slope
• Critical depth ($y_c$): the depth at which the Froude number equals 1, marking the boundary between subcritical and supercritical flow

$Fr = \frac{V}{\sqrt{gy}}$

When $Fr < 1$, flow is subcritical (deeper, slower). When $Fr > 1$, flow is supercritical (shallower, faster).

Profile classification depends on the channel slope category and where the actual depth sits relative to $y_n$ and $y_c$:

Each zone number (1, 2, or 3) tells you whether the actual depth is above both reference depths, between them, or below both.

Water surface profiles in prismatic channels

The gradually varied flow equation (GVFE) governs how depth changes with distance:

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

• $dy/dx$ = rate of change of water depth along the channel
• $S_0$ = channel bed slope
• $S_f$ = friction slope (slope of the energy grade line), typically computed using Manning's equation applied locally
• $Fr$ = Froude number at the section

The sign of $dy/dx$ tells you whether the water surface is rising or falling. The numerator compares gravity's pull ($S_0$) to frictional resistance ($S_f$). The denominator flips sign at critical depth ($Fr = 1$), which is why the water surface behavior changes between subcritical and supercritical regimes.

Solving the GVFE requires numerical methods because it's a nonlinear first-order ODE. Two standard approaches:

1. Direct step method: You pick increments of depth ($\Delta y$) and solve for the corresponding distance ($\Delta x$). Works well for prismatic channels.
2. Standard step method: You pick distance increments ($\Delta x$) and iterate to find the depth at each station. More flexible for natural channels with varying cross-sections.

Boundary conditions are essential to start the computation:

• A known water depth at a specific location, such as a downstream weir, sluice gate, or free overfall
• The discharge $Q$ and whether the flow is subcritical or supercritical

For subcritical flow, computations proceed upstream from a downstream control. For supercritical flow, computations proceed downstream from an upstream control. The computed profile should match the expected GVF profile type (M1, S2, etc.) based on the channel slope and boundary conditions. If it doesn't, recheck your boundary condition and flow regime assumption.