Open channel flow is a crucial concept in hydraulic engineering, focusing on water movement in partially filled conduits. This section explores the characteristics of uniform and non-uniform flows, including their classification and analysis methods.

The Manning equation is a key tool for calculating flow velocity and discharge in open channels. We'll examine its components, applications, and limitations, as well as dive into gradually varied flow profiles and their practical implications in channel design.

Uniform vs Non-uniform Flow

Characteristics of Uniform and Non-uniform Flow

• Uniform flow maintains constant depth, velocity, and cross-sectional area along channel length
  • Occurs in prismatic channels with consistent slope and roughness
  • Water surface profile parallels channel bed
• Non-uniform flow exhibits variations in depth, velocity, or cross-sectional area
  • Results from changes in channel geometry, slope, or obstructions
  • Water surface profile may fluctuate
• Both uniform and non-uniform flows can be steady or unsteady
  • Steady flow parameters remain constant over time
  • Unsteady flow parameters change with time
• Energy grade line and hydraulic grade line slopes differ
  • Constant slopes in uniform flow
  • Variable slopes in non-uniform flow

Flow Classification and Analysis

• Froude number classifies flow regimes in both uniform and non-uniform conditions
  • Subcritical flow (Fr < 1): tranquil, slow-moving (rivers)
  • Critical flow (Fr = 1): transitional state
  • Supercritical flow (Fr > 1): rapid, shallow (steep mountain streams)
• Manning equation applies to uniform flow calculations
  • Used for velocity and discharge estimations
• Gradually varied flow equations analyze non-uniform flow profiles
  • Account for gradual changes in depth and velocity

Flow Velocity and Discharge Calculation

Manning Equation Components

• Manning equation estimates average velocity in open channel flow
  • Formula: $V = \frac{1}{n} R^{2/3} S^{1/2}$
  • V: velocity
  • n: Manning's roughness coefficient
  • R: hydraulic radius
  • S: channel slope
• Manning's roughness coefficient (n) represents channel's flow resistance
  • Varies based on surface material (concrete: 0.012, earth channels: 0.025-0.035)
  • Affected by irregularity and vegetation
• Hydraulic radius (R) calculated as ratio of flow area to wetted perimeter
  • $R = \frac{A}{P}$
  • A: cross-sectional area of flow
  • P: wetted perimeter
• Channel slope (S) approximated as bed slope for uniform flow

Discharge Calculation and Applications

• Discharge (Q) calculated using continuity equation
  • $Q = V A$
  • Q: discharge
  • V: velocity
  • A: cross-sectional area of flow
• Manning equation assumes steady, uniform flow conditions
  • Most accurate for fully turbulent flow in rough channels
• Various forms of Manning equation exist for different applications
  • Non-circular conduits
  • Partially full pipes
  • Different unit systems (SI, Imperial)

Gradually Varied Flow Profiles

Fundamentals of Gradually Varied Flow

• Gradually varied flow (GVF) characterized by gradual depth and velocity changes
  • Negligible vertical accelerations
• GVF governing equation derived from energy conservation principles
  • Relates rate of change of depth to channel and flow characteristics
• Classification of GVF profiles based on depth relationships
  • Normal depth
  • Critical depth
  • Actual depth
• 12 possible profile types identified
  • Mild slope profiles (M1, M2, M3)
  • Steep slope profiles (S1, S2, S3)
  • Critical slope profiles (C1, C2, C3)
  • Horizontal slope profiles (H2, H3)
  • Adverse slope profiles (A2, A3)

Backwater and Drawdown Curves

• Backwater curves occur when water surface is above normal depth
  • Examples: M1, S1, C1 profiles
  • Often caused by downstream obstructions (dams, weirs)
• Drawdown curves form when water surface is below normal depth
  • Examples: M2, S2, C2 profiles
  • Typically approach free overfall or sudden channel transition
• Numerical methods used to compute GVF profiles
  • Standard step method
  • Direct integration
• Control points essential for determining GVF calculation starting conditions
  • Critical depth sections
  • Known water surface elevations (gauging stations)

Channel Design for Specific Flow Conditions

Channel Design Principles

• Channel design involves selecting appropriate:
  • Cross-sectional geometry (rectangular, trapezoidal, circular)
  • Slope
  • Lining material (concrete, riprap, vegetation)
• Hydraulically efficient sections maximize flow capacity
  • Best hydraulic rectangle: width = 2 depth
  • Best hydraulic trapezoid: side slopes at 60° angles
• Critical flow conditions important for specific structures
  • Energy dissipation (hydraulic jumps)
  • Flow measurement devices (weirs, flumes)
• Freeboard consideration prevents overtopping
  • Vertical distance between design water surface and channel top
  • Accounts for wave action, flow fluctuations, safety factors

Stability and Environmental Considerations

• Channel stability analysis prevents erosion and sediment transport issues
  • Assessment of permissible velocities for various lining materials
  • Evaluation of tractive forces on channel boundaries
• Supercritical flow channel design requires special attention
  • Potential for hydraulic jumps
  • Air entrainment considerations
  • Specialized energy dissipation structures (stilling basins)
• Environmental factors influence channel design decisions
  • Aquatic habitat requirements (fish passages, riffles, pools)
  • Riparian vegetation preservation
  • Aesthetic considerations for urban waterways