10.2 Open-channel flows

Open-channel flows involve fluid motion with a free surface exposed to the atmosphere. This topic explores various flow types, including steady vs. unsteady, uniform vs. non-uniform, and laminar vs. turbulent, as well as subcritical and supercritical flows.

Key characteristics of open-channel flows are examined, such as velocity distribution, pressure distribution, and shear stress. The chapter also covers fundamental equations, uniform flow analysis, gradually varied flow, rapidly varied flow, and flow measurement techniques in open channels.

Types of open-channel flows

• Open-channel flows are characterized by the presence of a free surface, where the fluid is in contact with the atmosphere
• The flow behavior in open channels is influenced by various factors such as channel geometry, roughness, and flow conditions
• Different types of open-channel flows are classified based on their temporal and spatial variations, as well as their flow regime and velocity

Steady vs unsteady flows

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• 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 more complex to analyze and require specialized equations and numerical methods

Uniform vs non-uniform flows

• Uniform flows have constant flow properties (velocity, depth, slope) along the channel length
  • Occurs when the channel cross-section, roughness, and slope are constant
  • Example: Flow in a long, straight, prismatic channel with a constant discharge
• Non-uniform flows have flow properties that vary along the channel length
  • Caused by changes in channel geometry, roughness, or slope
  • Example: Flow in a channel with varying cross-sections or a channel transition (e.g., a contraction or expansion)

Laminar vs turbulent flows

• Laminar flows have parallel streamlines and minimal mixing between fluid layers
  • Characterized by low Reynolds numbers (typically < 500 for open channels)
  • Example: Flow in a shallow, smooth channel with low velocity
• Turbulent flows have irregular, fluctuating streamlines and significant mixing between fluid layers
  • Characterized by high Reynolds numbers (typically > 2000 for open channels)
  • Example: Flow in a steep, rough channel with high velocity
• Most open-channel flows encountered in practice are turbulent

Subcritical vs supercritical flows

• Subcritical flows have a Froude number less than 1 and are characterized by low velocity and high depth
  • Disturbances can propagate upstream in subcritical flows
  • Example: Flow in a wide, shallow channel with mild slope
• Supercritical flows have a Froude number greater than 1 and are characterized by high velocity and low depth
  • Disturbances cannot propagate upstream in supercritical flows
  • Example: Flow in a steep, narrow channel or over a spillway
• The transition between subcritical and supercritical flow occurs at critical flow conditions (Froude number = 1)

Flow characteristics

• Open-channel flows exhibit distinct characteristics in terms of velocity, pressure, and shear stress distributions
• Understanding these characteristics is essential for analyzing and predicting flow behavior in open channels
• The specific energy concept is also crucial in determining flow transitions and critical flow conditions

Velocity distribution

• In open-channel flows, the velocity distribution is non-uniform due to the presence of the free surface and channel boundaries
• The velocity is typically highest near the surface and decreases towards the channel bed and sides
• Factors influencing the velocity distribution include channel geometry, roughness, and flow regime (laminar or turbulent)
• Example: In a wide, rectangular channel, the velocity distribution follows a logarithmic profile in the turbulent region near the bed

Pressure distribution

• In open-channel flows, the pressure distribution is hydrostatic, meaning it varies linearly with depth
• The pressure is atmospheric at the free surface and increases with depth following the hydrostatic equation: $p = \rho g h$
  • $p$ is the pressure, $\rho$ is the fluid density, $g$ is the gravitational acceleration, and $h$ is the depth below the free surface
• The hydrostatic pressure distribution is valid for flows with small streamline curvature and gradual changes in depth

Shear stress distribution

• Shear stress in open-channel flows arises from the interaction between the flowing fluid and the channel boundaries
• The shear stress distribution depends on the channel geometry, roughness, and flow conditions
• In wide, open channels, the shear stress is highest at the channel bed and decreases towards the free surface
• The bed shear stress is a key parameter in determining sediment transport and erosion in open channels
• Example: In a trapezoidal channel, the shear stress distribution along the wetted perimeter varies due to the different inclinations of the side slopes and bed

Specific energy in open-channel flows

• Specific energy is the sum of the potential and kinetic energy per unit weight of fluid in an open-channel flow
• It is defined as: $E = y + \frac{v^2}{2g}$, where $E$ is the specific energy, $y$ is the flow depth, $v$ is the average velocity, and $g$ is the gravitational acceleration
• The specific energy diagram (plot of specific energy vs. flow depth) is used to determine critical flow conditions and flow transitions
• At critical flow, the specific energy is minimum for a given discharge, and the Froude number is equal to 1
• Example: In a rectangular channel, the critical depth can be calculated using the specific energy concept and the continuity equation

Equations of open-channel flows

• Open-channel flows are governed by the principles of conservation of mass, momentum, and energy
• The continuity, momentum, and energy equations are fundamental in analyzing and predicting flow behavior in open channels
• These equations are derived from the Navier-Stokes equations and are adapted to the specific conditions of open-channel flows

Continuity equation

• The continuity equation expresses the conservation of mass in an open-channel flow
• For a one-dimensional, steady flow, the continuity equation is: $Q = A_1 v_1 = A_2 v_2$
  • $Q$ is the discharge, $A_1$ and $A_2$ are the cross-sectional areas at two different locations, and $v_1$ and $v_2$ are the corresponding average velocities
• The continuity equation states that the discharge is constant along the channel for a steady flow
• Example: In a channel with a rectangular cross-section, the continuity equation can be used to calculate the change in flow depth when the channel width varies

Momentum equation

• The momentum equation expresses the conservation of momentum in an open-channel flow
• For a one-dimensional, steady flow, the momentum equation is: $\rho Q (v_2 - v_1) = P_1 - P_2 + W \sin\theta - F_f$
  • $\rho$ is the fluid density, $Q$ is the discharge, $v_1$ and $v_2$ are the average velocities at two different locations, $P_1$ and $P_2$ are the hydrostatic pressure forces, $W$ is the weight of the fluid, $\theta$ is the channel slope angle, and $F_f$ is the friction force
• The momentum equation is used to analyze flow transitions, such as hydraulic jumps, and to determine the forces acting on hydraulic structures

Energy equation

• The energy equation expresses the conservation of energy in an open-channel flow
• For a one-dimensional, steady flow, the energy equation is: $\frac{v_1^2}{2g} + y_1 + z_1 = \frac{v_2^2}{2g} + y_2 + z_2 + h_L$
  • $v_1$ and $v_2$ are the average velocities at two different locations, $y_1$ and $y_2$ are the flow depths, $z_1$ and $z_2$ are the elevations of the channel bed, $g$ is the gravitational acceleration, and $h_L$ is the head loss due to friction
• The energy equation is used to analyze gradually varied flow profiles and to determine the energy grade line along the channel
• Example: In a channel with a mild slope, the energy equation can be used to calculate the change in flow depth along the channel length

Uniform flow

• Uniform flow occurs when the flow properties (velocity, depth, slope) remain constant along the channel length
• It is an idealized flow condition that serves as a reference for analyzing more complex flow situations
• The main equations used to describe uniform flow are Chezy's equation and Manning's equation, which relate the flow velocity to the channel characteristics and slope

Chezy's equation

• Chezy's equation is an empirical formula that relates the average flow velocity to the channel slope and hydraulic radius
• The equation is: $v = C \sqrt{RS}$
  • $v$ is the average velocity, $C$ is the Chezy coefficient, $R$ is the hydraulic radius, and $S$ is the channel slope
• The Chezy coefficient depends on the channel roughness and can be estimated using various empirical formulas or tables
• Example: In a wide, rectangular channel with a known slope and roughness, Chezy's equation can be used to calculate the average flow velocity

Manning's equation

• Manning's equation is another empirical formula widely used to describe uniform flow in open channels
• The equation is: $v = \frac{1}{n} R^{2/3} S^{1/2}$
  • $v$ is the average velocity, $n$ is the Manning's roughness coefficient, $R$ is the hydraulic radius, and $S$ is the channel slope
• Manning's equation is more commonly used than Chezy's equation due to its simplicity and the availability of tabulated values for the roughness coefficient
• Example: In a trapezoidal channel with a given slope and roughness, Manning's equation can be used to determine the normal depth (flow depth under uniform flow conditions)

Roughness coefficients

• The roughness coefficients (Chezy's C or Manning's n) represent the resistance to flow offered by the channel boundaries
• They depend on the material, surface irregularities, vegetation, and other factors affecting the channel roughness
• Tabulated values of roughness coefficients are available for various channel types and materials
• Example: A concrete-lined channel has a lower roughness coefficient compared to a natural, vegetated channel

Computation of uniform flow

• The computation of uniform flow involves determining the flow velocity, depth, and discharge for a given channel geometry, roughness, and slope
• The main steps in computing uniform flow are:
  1. Determine the channel cross-sectional area, wetted perimeter, and hydraulic radius
  2. Estimate the roughness coefficient (Chezy's C or Manning's n) based on the channel characteristics
  3. Calculate the average flow velocity using Chezy's or Manning's equation
  4. Calculate the discharge using the continuity equation ($Q = Av$)
• Uniform flow computations are essential for designing stable channels, determining the capacity of existing channels, and estimating the flow resistance in open-channel flows

Gradually varied flow

• Gradually varied flow (GVF) occurs when the flow properties (velocity, depth) change gradually along the channel length due to changes in channel geometry, roughness, or slope
• GVF profiles are classified based on the relative magnitudes of the normal depth (yn) and the critical depth (yc)
• The analysis of GVF involves solving the dynamic equation of gradually varied flow, which combines the continuity and energy equations

Dynamic equation of gradually varied flow

• The dynamic equation of gradually varied flow is derived from the continuity and energy equations for a one-dimensional, steady flow
• The equation is: $\frac{dy}{dx} = \frac{S_0 - S_f}{1 - Fr^2}$
  • $y$ is the flow depth, $x$ is the distance along the channel, $S_0$ is the channel slope, $S_f$ is the friction slope (energy grade line slope), and $Fr$ is the Froude number
• The dynamic equation describes the change in flow depth along the channel length as a function of the channel slope, friction slope, and Froude number
• The equation is solved numerically using methods such as the direct step method or the standard step method

Classification of flow profiles

• GVF profiles are classified into five main types based on the relative magnitudes of the normal depth (yn) and the critical depth (yc)
  • M1 (Mild slope, yn > yc): The flow is subcritical throughout, and the depth gradually increases downstream
  • M2 (Mild slope, yn < yc): The flow is subcritical upstream and transitions to supercritical flow downstream
  • M3 (Mild slope, yn < yc): The flow is supercritical throughout, and the depth gradually decreases downstream
  • S1 (Steep slope, yn > yc): The flow is supercritical throughout, and the depth gradually increases downstream
  • S2 (Steep slope, yn < yc): The flow is supercritical upstream and transitions to subcritical flow downstream
• The classification of flow profiles helps in understanding the behavior of GVF and in selecting appropriate boundary conditions for numerical solutions

Computation of gradually varied flow

• The computation of GVF involves solving the dynamic equation of gradually varied flow numerically
• The main steps in computing GVF are:
  1. Determine the channel geometry, roughness, and slope
  2. Calculate the normal depth (yn) and critical depth (yc) for the given flow conditions
  3. Classify the flow profile based on the relative magnitudes of yn and yc
  4. Select appropriate boundary conditions (e.g., downstream depth for M1 and M2 profiles, upstream depth for M3, S1, and S2 profiles)
  5. Solve the dynamic equation numerically using the direct step or standard step method
  6. Plot the computed flow profile (depth vs. distance) and analyze the results
• GVF computations are essential for designing transitions in open channels, analyzing backwater effects, and predicting flow behavior in natural streams and rivers

Rapidly varied flow

• Rapidly varied flow (RVF) occurs when the flow properties (velocity, depth) change abruptly over a short distance due to sudden changes in channel geometry or flow conditions
• Examples of RVF include hydraulic jumps, flow over spillways, and flow under sluice gates
• RVF is characterized by significant energy dissipation, turbulence, and air entrainment, which make the flow analysis more complex compared to gradually varied flow

Hydraulic jump

• A hydraulic jump is a sudden transition from supercritical flow to subcritical flow, accompanied by a rapid increase in flow depth and significant energy dissipation
• Hydraulic jumps occur when the upstream Froude number is greater than 1 (supercritical flow) and the downstream conditions impose a subcritical flow
• The main characteristics of a hydraulic jump are:
  • The sequent depths (y1 and y2) upstream and downstream of the jump, related by the Belanger equation: $\frac{y_2}{y_1} = \frac{1}{2}\left(\sqrt{1 + 8Fr_1^2} - 1\right)$, where $Fr_1$ is the upstream Froude number
  • The energy dissipation, expressed as a fraction of the upstream specific energy: $\frac{\Delta E}{E_1} = \frac{(y_2 - y_1)^3}{4y_1y_2}$
  • The length of the jump, estimated using empirical formulas such as $L_j = 6(y_2 - y_1)$
• Hydraulic jumps are used in hydraulic structures for energy dissipation, flow control, and aeration

Flow over spillways

• Spillways are hydraulic structures designed to release excess water from a reservoir or dam in a controlled manner
• The flow over a spillway can be either free flow (no downstream submergence) or submerged flow (affected by downstream water level)
• For free flow, the discharge over a spillway is given by the weir equation: $Q = C_dL H^{3/2}$
  • $Q$ is the discharge, $C_d$ is the discharge coefficient, $L$ is the spillway length, and $H$ is the total head over the spillway crest
• The discharge coefficient depends on the spillway geometry and flow conditions and can be determined using empirical formulas or hydraulic model studies
• Spillway design involves selecting the appropriate type (e.g., ogee, stepped, labyrinth), determining the crest elevation and length, and analyzing the flow profile and energy dissipation downstream of the spillway

Flow under sluice gates

• Sluice gates are hydraulic structures used to control the flow in open channels, such as irrigation canals or drainage systems
• The flow under a sluice gate can be either free flow (supercritical downstream) or submerged flow (subcritical downstream)
• For free flow, the discharge under a sluice gate is given by the orifice equation: $Q = C_d a \sqrt{2g H}$
  • $Q$ is the discharge, $C_d$ is the discharge coefficient, $a$ is the gate opening area, $g$ is the gravitational acceleration, and $H$ is the upstream water depth above the gate invert
• The discharge coefficient depends on the gate geometry, flow conditions, and downstream submergence and can be determined using empirical formulas or hydraulic model studies
• Sluice gate design involves selecting the appropriate gate type (e.g., vertical, radial), determining the gate dimensions and invert elevation, and analyzing the flow profile and energy dissipation downstream of the gate

Flow measurement in open channels

• Flow measurement in open channels is essential for water resource management, irrigation, an