3.5 Open Channel Flow

3.5 Open Channel Flow

Introduction to Open Channel Flow

• Open channel flow is characterized by a free surface exposed to the atmosphere, driven primarily by gravity rather than pressure.

• It is fundamental to the design and analysis of rivers, canals, storm sewers, culverts, and irrigation systems.

• This section covers the geometric properties of channels, classification of flow types, the application of energy and momentum principles, analysis of varied flow profiles, the dynamics of hydraulic jumps, and the unique considerations for channels with erodible boundaries.

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1. Geometrical Properties of Channel Sections

• These properties are essential for hydraulic calculations involving discharge, velocity, and energy.

1.1 Common Section Types

• Rectangular: Simple, common in man-made channels and laboratory flumes.

  • Width: $\boldsymbol{b}$

  • Flow Depth: $\boldsymbol{y}$

  • Area: $\boldsymbol{A = b y}$

  • Wetted Perimeter: $\boldsymbol{P = b + 2y}$

• Trapezoidal: Most common for large canals, combines stability with efficient area.

  • Bottom Width: $\boldsymbol{b}$

  • Side Slope (Horizontal:Vertical): $\boldsymbol{z : 1}$

  • Area: $\boldsymbol{A = (b + z y) y}$

  • Wetted Perimeter: $\boldsymbol{P = b + 2y \sqrt{1 + z^2}}$

• Triangular (V-shaped): Small roadside ditches, theoretical sections.

  • Area: $\boldsymbol{A = z y^2}$

  • Wetted Perimeter: $\boldsymbol{P = 2y \sqrt{1 + z^2}}$

• Circular (Partially Full): Sewers and culverts.

  • Requires trigonometric relations based on central angle $\boldsymbol{\theta}$.

1.2 Key Hydraulic Parameters

• Flow Area ($\boldsymbol{A}$): Cross-sectional area of the flow.

• Wetted Perimeter ($\boldsymbol{P}$): Length of the channel cross-section in contact with the fluid.

• Hydraulic Radius ($\boldsymbol{R}$): A measure of flow efficiency.

  $\boldsymbol{R = \frac{A}{P}}$

  Larger $\boldsymbol{R}$ indicates lower frictional resistance for a given area.

• Hydraulic Depth ($\boldsymbol{D}$): For non-rectangular sections, used in Froude number calculations.

  $\boldsymbol{D = \frac{A}{T}}$

  where, $\boldsymbol{T}$ is the top width of the flow surface.

• Section Factor for Uniform Flow ($\boldsymbol{AR^{2/3}}$): Appears in the Manning's equation. Determines discharge for a given slope and roughness.

• Section Factor for Critical Flow ($\boldsymbol{A \sqrt{D}}$ or $\boldsymbol{\frac{A^{3/2}}{T^{1/2}}}$): Used to determine critical depth.

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2. Various Types of Flows

• Open channel flow is classified based on several independent criteria.

2.1 Based on Temporal Variation

• Steady Flow: Flow depth and velocity at a section do not change with time.

  $\boldsymbol{\frac{\partial y}{\partial t} = 0, \quad \frac{\partial V}{\partial t} = 0}$

• Unsteady Flow: Flow characteristics vary with time (e.g., flood waves, tidal flows).

  $\boldsymbol{\frac{\partial y}{\partial t} \neq 0}$

2.2 Based on Spatial Variation

• Uniform Flow: Depth, cross-section, and velocity remain constant along the channel length. Achieved when gravity force exactly balances frictional resistance.

  $\boldsymbol{\frac{dy}{dx} = 0, \quad S_f = S_0}$ where $\boldsymbol{S_f}$ is the friction slope and $\boldsymbol{S_0}$ is the bed slope.

• Varied (Non-uniform) Flow: Depth and velocity change along the channel length.

  $\boldsymbol{\frac{dy}{dx} \neq 0}$

• Gradually Varied Flow (GVF): Depth changes slowly, and pressure distribution is hydrostatic. Analyzed using the GVF equation.

• Rapidly Varied Flow (RVF): Depth changes abruptly over a short distance (e.g., hydraulic jump, waterfall). Pressure distribution is not hydrostatic; momentum principle is used.

2.3 Based on Froude Number (Flow Regime)

• The Froude Number ($\boldsymbol{F_r}$) is the key dimensionless parameter in open channel flow, representing the ratio of inertial to gravitational forces.

  $\boldsymbol{F_r = \frac{V}{\sqrt{g D}}}$

  where, $\boldsymbol{D}$ is the hydraulic depth.

• Subcritical Flow (Tranquil, Streaming):

  • $\boldsymbol{F_r < 1}$

  • Velocity < wave celerity. Disturbances can travel upstream.

  • Deep, slow flow. Dominated by gravity, common in rivers and canals.

• Critical Flow:

  • $\boldsymbol{F_r = 1}$

  • Minimum specific energy for a given discharge. A control condition.

  • Unstable, used for flow measurement (e.g., Parshall flume, broad-crested weir).

• Supercritical Flow (Shooting, Rapid):

  • $\boldsymbol{F_r > 1}$

  • Velocity > wave celerity. Disturbances cannot travel upstream.

  • Shallow, fast flow. Dominated by inertia, common in chutes, spillways, and steep channels.

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3. Energy and Momentum Principles

3.1 Specific Energy

• Definition: Energy per unit weight of fluid measured relative to the channel bed.

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

  where the first term is the pressure head (depth) and the second is the velocity head.

• Specific Energy Diagram ($\boldsymbol{E}$ vs $\boldsymbol{y}$ for constant $\boldsymbol{Q}$):

  • For a given $\boldsymbol{E}$, there are two possible depths: the subcritical depth (deep, slow) and the supercritical depth (shallow, fast), except at the critical point.

  • The curve has a minimum point defining the critical depth ($\boldsymbol{y_c}$) and minimum specific energy ($\boldsymbol{E_{min}}$).

• Critical Depth ($\boldsymbol{y_c}$) Condition: Occurs when $\boldsymbol{F_r = 1}$ and $\boldsymbol{dE/dy = 0}$.

  • For a rectangular channel of width $\boldsymbol{b}$:

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

    where, $\boldsymbol{q = Q/b}$ is the discharge per unit width $\boldsymbol{[\mathrm{m^2/s}]}$. $\boldsymbol{E_{min} = \frac{3}{2} y_c}$

  • For a general channel, solve: $\boldsymbol{\frac{Q^2 T_c}{g A_c^3} = 1}$.

• Applications: Analysis of channel transitions (constrictions, steps), flow over bumps, and flow under sluice gates.

3.2 Specific Force (Momentum Function)

• Definition: Force per unit weight derived from the momentum equation for a short horizontal control volume. Used for rapidly varied flow analysis.

  $\boldsymbol{F = \frac{Q^2}{g A} + \bar{y} A}$

  where, $\boldsymbol{\bar{y}}$ is the depth to the centroid of the flow area.

• Specific Force Diagram ($\boldsymbol{F}$ vs $\boldsymbol{y}$ for constant $\boldsymbol{Q}$): Similar to the specific energy curve, has a minimum at critical depth.

• Applications: Primarily for analyzing the hydraulic jump, where external forces are negligible, and specific force is conserved between the conjugate (sequent) depths.

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4. Types of Gradually Varied Flow Profiles

• GVF describes the gradual change in water surface profile in a prismatic channel where bed slope, roughness, and discharge are constant.

• The GVF Differential Equation governs the profile:

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

  where, $\boldsymbol{S_f}$ is the friction slope (computed from Manning's or Chezy's equation).

4.1 Classification of Profiles

• Profiles are classified based on:

  1. Bed Slope ($\boldsymbol{S_0}$): Mild ($\boldsymbol{S_0 < S_c}$), Steep ($\boldsymbol{S_0 > S_c}$), Critical ($\boldsymbol{S_0 = S_c}$), Horizontal ($\boldsymbol{S_0 = 0}$), Adverse ($\boldsymbol{S_0 < 0}$).

  2. Depth Zone: Zone 1 ($\boldsymbol{y > y_n > y_c}$ or $\boldsymbol{y > y_c > y_n}$), Zone 2 (depth between $\boldsymbol{y_n}$ and $\boldsymbol{y_c}$), Zone 3 ($\boldsymbol{y < y_n < y_c}$ or $\boldsymbol{y < y_c < y_n}$).

• $\boldsymbol{y_n}$ is the normal depth (depth for uniform flow).

• $\boldsymbol{y_c}$ is the critical depth.

4.2 Common GVF Profile Families

• Mild Slope (M):

  • $\boldsymbol{y_n > y_c}$. Normal flow is subcritical.

  • M1: $\boldsymbol{y > y_n}$. Backwater curve upstream of a dam. Depth decreases downstream.

  • M2: $\boldsymbol{y_n > y > y_c}$. Drawdown curve upstream of a free overfall. Depth decreases downstream.

  • M3: $\boldsymbol{y < y_c}$. Supercritical flow downstream of a sluice gate. Depth increases downstream.

• Steep Slope (S):

  • $\boldsymbol{y_c > y_n}$. Normal flow is supercritical.

  • S1: $\boldsymbol{y > y_c}$. Backwater curve upstream of a submerged hydraulic jump. Depth decreases downstream.

  • S2: $\boldsymbol{y_c > y > y_n}$. Drawdown curve on a steep slope. Depth decreases downstream.

  • S3: $\boldsymbol{y < y_n}$. Supercritical flow on a steep slope. Depth increases downstream.

• Critical Slope (C): $\boldsymbol{y_n = y_c}$. Profiles are approximately horizontal.

• Horizontal Slope (H): $\boldsymbol{S_0 = 0}$, $\boldsymbol{y_n \to \infty}$. H2 and H3 profiles exist.

• Adverse Slope (A): $\boldsymbol{S_0 < 0}$, $\boldsymbol{y_n}$ is imaginary. A2 and A3 profiles exist (rare).

4.3 Computation Methods

• Direct Step Method: Suitable for prismatic channels. Uses finite difference form of the GVF equation to step from a known depth to find distance $\boldsymbol{\Delta x}$.

• Standard Step Method: More general, suitable for natural channels. Iteratively solves for depth at successive stations given the distance.

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5. Hydraulic Jump

• A rapid transition from supercritical to subcritical flow. Characterized by intense turbulence, energy dissipation, and a sudden rise in water surface.

5.1 Types Based on Froude Number ($\boldsymbol{F_{r1}}$)

• Undular Jump: $\boldsymbol{1.0 < F_{r1} < 1.7}$. Smooth rise with surface waves.

• Weak Jump: $\boldsymbol{1.7 < F_{r1} < 2.5}$. Smooth roller forms.

• Oscillating Jump: $\boldsymbol{2.5 < F_{r1} < 4.5}$. Unsteady, oscillating jet.

• Steady Jump: $\boldsymbol{4.5 < F_{r1} < 9.0}$. Stable, well-defined roller, good energy dissipation.

• Strong/Choppy Jump: $\boldsymbol{F_{r1} > 9.0}$. Rough, turbulent, effective but potentially erosive.

5.2 Theory for Horizontal, Rectangular Channel

• Applying the momentum principle between sections 1 (upstream, supercritical) and 2 (downstream, subcritical), assuming hydrostatic pressure and negligible friction, yields the conjugate depth relationship:

  $\boldsymbol{\frac{y_2}{y_1} = \frac{1}{2} \left( \sqrt{1 + 8 F_{r1}^2} - 1 \right)}$

  where, $\boldsymbol{F_{r1} = \frac{V_1}{\sqrt{g y_1}}}$.

• Head Loss in Jump: The energy loss due to turbulence is significant.

  $\boldsymbol{h_L = E_1 - E_2 = \frac{(y_2 - y_1)^3}{4 y_1 y_2}}$

  where, $\boldsymbol{E}$ is the specific energy.

• Length of Jump ($\boldsymbol{L_j}$): Empirical. Approximately $\boldsymbol{L_j \approx 6 y_2}$ for $\boldsymbol{4.5 < F_{r1} < 13}$.

5.3 Practical Applications

• Energy Dissipation: Below spillways and outlet works to protect channels from erosion.

• Mixing: In water treatment and aeration.

• Flow Control: To stabilize flow and create a subcritical condition downstream.

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6. Flow in Mobile Boundary Channels

• Channels with erodible beds (sand, gravel) where sediment transport and bed forms interact with the flow.

6.1 Design Principles/Approaches

• Tractive Force (Shear Stress) Method:

  • Ensures flow shear stress on the channel boundary does not exceed the critical shear stress of the bed/bank material.

  • Used for non-scouring, non-silting (regime) design in cohesive soils or lined channels.

• Regime Theory (Empirical):

  • Based on observations of stable canals in alluvial (sediment-transporting) regions (e.g., Lacey's, Kennedy's formulas).

  • Relates stable channel dimensions (width, depth, slope) to discharge and sediment characteristics.

• Permissible Velocity Method:

  • Limits average flow velocity to a value that will not cause erosion for a given bed material (tabulated values).

6.2 Inception of Motion Condition

• The condition at which sediment particles on the channel bed just begin to move.

• Governed by the balance between the destabilizing fluid force (shear stress) and the stabilizing forces (submerged weight, friction, cohesion).

• Critical Shear Stress ($\boldsymbol{\tau_c}$): The bed shear stress at the inception of motion. Depends on:

  • Sediment size ($\boldsymbol{d}$) and density ($\boldsymbol{\rho_s}$).

  • Particle shape and packing.

  • For non-cohesive sediments (sand, gravel), $\boldsymbol{\tau_c}$ increases with particle size.

  • For cohesive sediments (clay, silt), inter-particle forces dominate, making $\boldsymbol{\tau_c}$ harder to predict and often higher than for sands.

6.3 Shields Diagram

• A fundamental dimensionless graph used to determine the critical condition for sediment movement.

• Shields Parameter (Dimensionless Critical Shear Stress):

  $\boldsymbol{\theta_c = \frac{\tau_c}{(\rho_s - \rho) g d}}$

  where, $\boldsymbol{\rho_s}$ is the sediment density.

• Boundary Reynolds Number (Particle Reynolds Number):

  $\boldsymbol{Re_* = \frac{u_* d}{\nu}}$

  where, $\boldsymbol{u_* = \sqrt{\tau_0 / \rho}}$ is the shear velocity.

• The Diagram: Plots $\boldsymbol{\theta_c}$ vs $\boldsymbol{Re_*}$.

  • For low $\boldsymbol{Re_*}$ (fine sediments, viscous regime), $\boldsymbol{\theta_c \propto 1/Re_*}$.

  • For high $\boldsymbol{Re_*}$ (coarse sediments, fully rough turbulent regime), $\boldsymbol{\theta_c \approx}$ constant (typically 0.03-0.06).

• Usage: For a given sediment size $\boldsymbol{d}$ and known fluid properties, calculate $\boldsymbol{Re_*}$, find $\boldsymbol{\theta_c}$ from the diagram, then compute the actual critical shear stress $\boldsymbol{\tau_c = \theta_c (\rho_s - \rho) g d}$.

6.4 Bed Forms and Resistance

• As flow intensity increases past the critical condition, the bed develops forms that significantly affect flow resistance (Manning's $\boldsymbol{n}$):

  1. Ripples: Small, triangular forms ($\boldsymbol{d < 0.6 \ \mathrm{mm}}$).

  2. Dunes: Larger, out-of-phase with water surface. Increase resistance.

  3. Transition: Washed-out dunes.

  4. Plane Bed: No bed forms, minimum resistance.

  5. Antidunes: In-phase with water surface waves, occur in supercritical flow.

  6. Chutes and Pools: Associated with alternate bars in steep channels.

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Summary and Interrelationships

• Uniform Flow: Governed by Manning's Equation $\boldsymbol{Q = \frac{1}{n} A R^{2/3} S_0^{1/2}}$, used to find normal depth $\boldsymbol{y_n}$.

• Critical Flow: Governed by Froude Number $\boldsymbol{F_r = 1}$, defines critical depth $\boldsymbol{y_c}$ and minimum specific energy.

• GVF Profiles: Determined by the relative magnitudes of $\boldsymbol{y}$, $\boldsymbol{y_n}$, $\boldsymbol{y_c}$ and the sign of the bed slope $\boldsymbol{S_0}$.

• Hydraulic Jump: A momentum-controlled RVF connecting supercritical and subcritical states, characterized by the upstream $\boldsymbol{F_{r1}}$.

• Mobile Boundaries: Design must consider sediment transport thresholds via the Shields Parameter and account for variable resistance due to bed forms.

The analysis of open channel flow integrates geometric, kinematic, dynamic, and sediment transport principles to design stable, efficient, and safe channels for water conveyance.

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