12.1 Classification of Open Channel Flows

Classification of Open Channel Flows

Open channel flows are categorized based on how the flow behaves in space and time. These classifications help engineers analyze hydraulic structures, predict flow patterns, and manage water resources. The main tools for classifying flows are the Reynolds number, the Froude number, and the concept of specific energy.

Laminar vs. Turbulent Flow

The Reynolds number determines whether open channel flow is laminar or turbulent, just as it does for pipe flow. However, the thresholds are different for open channels.

• Laminar flow occurs at low Reynolds numbers ($Re < 500$). Fluid particles move in smooth, parallel layers without mixing, producing a parabolic velocity profile across the channel. Laminar flow is rare in real-world open channels because natural flows almost always have enough velocity and depth to push $Re$ well above 500.
• Turbulent flow occurs at high Reynolds numbers ($Re > 2000$). Fluid particles move irregularly and mix across layers, which produces a more uniform velocity distribution. This is the dominant flow regime in nearly all practical open channel situations.
• For $500 < Re < 2000$, the flow is in a transitional zone where it may exhibit characteristics of both regimes.

The Reynolds number for open channels is typically defined as $Re = \frac{V R}{\nu}$, where $V$ is the mean velocity, $R$ is the hydraulic radius, and $\nu$ is the kinematic viscosity of the fluid.

Temporal and Spatial Classifications

Open channel flows are classified along two independent axes: how properties change over time (temporal) and how they change along the channel length (spatial).

Steady vs. Unsteady Flow

• Steady flow: Flow properties like depth, velocity, and discharge remain constant at a given location over time. Most design problems assume steady flow conditions.
• Unsteady flow: Flow properties change with time at a given location. Examples include flood waves, tidal flows, and dam-break scenarios.

Uniform vs. Non-Uniform Flow

• Uniform flow: Flow depth, velocity, and cross-sectional area stay constant along the channel length. This requires a prismatic channel (constant slope, roughness, and cross-section). The water surface runs parallel to the channel bed.
• Non-uniform (varied) flow: Flow properties change along the channel length. This is further divided into two subtypes:
  • Gradually varied flow (GVF): Changes in depth and velocity occur over a long distance. Backwater curves upstream of a dam are a classic example.
  • Rapidly varied flow (RVF): Changes happen abruptly over a short distance. Hydraulic jumps, waterfalls, and flow over weirs are typical examples.

A flow can be both steady and non-uniform (e.g., water flowing through a channel constriction at constant discharge), or unsteady and uniform (though this combination is mostly theoretical).

Critical Flow Classification (Froude Number)

The Froude number classifies flow based on the ratio of inertial forces to gravitational forces:

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

where $V$ is the average flow velocity, $g$ is gravitational acceleration, and $D$ is the hydraulic depth (cross-sectional area divided by the top width).

The physical meaning of $Fr$ relates to wave propagation. The term $\sqrt{gD}$ is the speed at which a small surface disturbance (a shallow-water wave) travels through the flow. So the Froude number compares the flow speed to the wave speed.

• Subcritical flow ($Fr < 1$): Gravity forces dominate. The flow is relatively deep and slow. Surface disturbances can travel upstream, so downstream conditions influence the flow. Found on mild slopes.
• Critical flow ($Fr = 1$): Inertial and gravitational forces are in balance. The flow velocity exactly equals the wave propagation speed. This condition corresponds to the minimum specific energy for a given discharge.
• Supercritical flow ($Fr > 1$): Inertial forces dominate. The flow is shallow and fast. Surface disturbances cannot travel upstream, so the flow is controlled by upstream conditions. Found on steep slopes.

Specific Energy in Channels

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

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

The first term, $y$, is the flow depth (potential energy component). The second term, $\frac{V^2}{2g}$, is the velocity head (kinetic energy component).

Since discharge per unit width $q = Vy$, you can also write specific energy as:

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

This form is useful because it expresses $E$ purely in terms of depth for a given discharge.

The Specific Energy Diagram

Plotting $E$ against $y$ for a constant discharge produces a curve with two branches:

1. The upper branch (subcritical) curves to the right as depth increases, where potential energy dominates.
2. The lower branch (supercritical) curves to the right as depth decreases, where kinetic energy dominates.

The two branches meet at the critical depth $y_c$, which is the point of minimum specific energy for that discharge. For a rectangular channel, critical depth is:

$y_c = \left(\frac{q^2}{g}\right)^{1/3}$

Why Specific Energy Matters

• It determines whether flow transitions between subcritical and supercritical states are possible for a given energy budget.
• It helps analyze hydraulic jumps, where supercritical flow abruptly transitions to subcritical flow with an associated energy loss.
• It guides the design of channel contractions and expansions, where changes in channel width force changes in depth and velocity.

For any specific energy greater than the minimum, two possible depths exist (called alternate depths): one subcritical and one supercritical. Both carry the same discharge at the same specific energy but with very different depth-velocity combinations.