7.2 Hydraulics and open-channel flow

Open-channel flow is a crucial aspect of water resources engineering. It deals with the movement of water in channels with a free surface exposed to the atmosphere, like rivers and canals.

This section dives into hydraulics principles, energy and momentum concepts, and flow regimes in open channels. It also covers channel design, gradually varied flow, and the analysis of hydraulic structures like weirs, spillways, and culverts.

Energy and Momentum in Open Channels

Energy Principles in Open Channels

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• The energy equation, derived from Bernoulli's principle, relates the total head at two different sections of an open channel
  • The total head includes the elevation head, pressure head, and velocity head
• The specific energy at a channel section is the sum of the depth of flow and velocity head
  • It represents the total energy per unit weight of water at any section of a channel measured with respect to the channel bottom
• Specific energy diagrams are used to study the relationship between depth and velocity in open-channel flows
  • These diagrams help to identify critical, subcritical, and supercritical flow conditions (critical depth, alternate depths)

Momentum Principles in Open Channels

• The momentum equation, based on Newton's second law of motion, is used to analyze forces acting on a control volume of fluid in an open channel
  • It relates the net force to the change in momentum flux between two sections
• The specific force at a channel section is the sum of the hydrostatic force and the momentum flux
  • It represents the total force per unit weight of water at any section of a channel
• Specific force diagrams are used to analyze the relationship between depth and force in open-channel flows
  • They help to determine the sequent depths in hydraulic jumps and the forces acting on hydraulic structures (stilling basins, energy dissipators)

Flow Regimes and Depths

Classification of Open-Channel Flows

• Open-channel flows can be classified based on various criteria
  • Velocity distribution: uniform (constant depth and velocity) or non-uniform (varying depth and velocity)
  • Time variation: steady (constant discharge) or unsteady (varying discharge)
  • Froude number: subcritical (Fr < 1), critical (Fr = 1), or supercritical (Fr > 1)
• The Froude number (Fr) is a dimensionless parameter that characterizes the flow regime in open channels
  • It is defined as the ratio of inertial forces to gravitational forces: $Fr = V / (gD)^{0.5}$, where $V$ is the average velocity, $g$ is the acceleration due to gravity, and $D$ is the hydraulic depth

Flow Regimes and Their Characteristics

• Subcritical flow (Fr < 1) occurs when gravitational forces dominate, resulting in slower velocities and deeper depths
  • The flow is tranquil and characterized by small surface disturbances that propagate upstream and downstream (ripples, waves)
• Critical flow (Fr = 1) represents the transition between subcritical and supercritical flow
  • It occurs when the specific energy is minimum for a given discharge
  • The critical depth can be calculated using the condition $Fr = 1$
• Supercritical flow (Fr > 1) occurs when inertial forces dominate, resulting in higher velocities and shallower depths
  • The flow is rapid and characterized by standing waves and hydraulic jumps (white water, chutes)
• Normal depth is the depth of flow in a channel when the slope of the energy grade line is equal to the slope of the channel bottom
  • It represents the equilibrium depth for uniform flow conditions and can be calculated using Manning's equation or the Chezy equation

Channel Design for Flow Conditions

Uniform Flow in Open Channels

• Uniform flow occurs when the depth, velocity, and cross-sectional area of the flow remain constant along the channel length
  • It is achieved when the channel slope, roughness, and geometry are constant
• Manning's equation is widely used to calculate the normal depth and velocity in uniform flow conditions
  • It relates the channel slope, roughness coefficient (Manning's n), hydraulic radius, and cross-sectional area to the discharge: $Q = \frac{1}{n} A R^{2/3} S^{1/2}$
• The Chezy equation is another empirical formula used to calculate the average velocity in open channels
  • It relates the velocity to the hydraulic radius, channel slope, and Chezy coefficient (C), which depends on the channel roughness: $V = C \sqrt{RS}$

Gradually Varied Flow in Open Channels

• Gradually varied flow occurs when the depth and velocity of the flow change gradually along the channel length
  • It is caused by variations in the channel slope, roughness, or cross-section (mild slopes, transitions)
• The profile of the water surface in gradually varied flow can be classified into various types
  • Mild (M), steep (S), critical (C), horizontal (H), and adverse (A) slopes
  • The type of profile depends on the relationship between the normal depth and critical depth
• The direct step method is a numerical technique used to compute the water surface profile in gradually varied flow
  • It involves dividing the channel into short reaches and calculating the depth and velocity at each section using the energy equation and continuity equation

Hydraulic Structure Analysis

Weirs for Flow Measurement and Control

• Weirs are hydraulic structures used to measure or control the flow in open channels
  • They are typically placed across the channel to create a backwater effect and raise the upstream water level (dams, diversion structures)
• Sharp-crested weirs have a thin crest and are commonly used for flow measurement
  • The discharge over a sharp-crested weir can be calculated using empirical equations
    • Rectangular weir equation: $Q = \frac{2}{3} C_d b H^{3/2}$
    • V-notch weir equation: $Q = \frac{8}{15} C_d \tan(\theta/2) H^{5/2}$
  • $Q$ is the discharge, $C_d$ is the discharge coefficient, $b$ is the weir width, $H$ is the head over the weir crest, and $\theta$ is the angle of the V-notch
• Broad-crested weirs have a wider crest and are used for flow control and energy dissipation
  • The discharge over a broad-crested weir can be estimated using the standard weir equation: $Q = C L H^{3/2}$
  • $C$ is the weir coefficient, $L$ is the weir length, and $H$ is the head over the weir crest

Spillways and Culverts

• Spillways are hydraulic structures designed to safely release excess water from dams or reservoirs
  • They are typically located at the top of the dam and can be controlled by gates or be uncontrolled (ogee spillways, chute spillways)
• Ogee spillways have a curved crest profile that closely follows the trajectory of the nappe (water jet) over the spillway
  • The discharge capacity of an ogee spillway can be calculated using the standard weir equation with appropriate discharge coefficients
• Culverts are hydraulic structures that allow water to pass underneath roads, railways, or embankments
  • They can be designed to operate under various flow conditions, such as inlet control or outlet control (box culverts, pipe culverts)
• Inlet control occurs when the culvert entrance limits the discharge capacity
  • The headwater depth upstream of the culvert can be calculated using empirical equations or nomographs based on the culvert geometry and entrance type
• Outlet control occurs when the culvert barrel or downstream conditions limit the discharge capacity
  • The headwater depth can be calculated using the energy equation, considering the losses due to friction, entrance, and exit effects