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Normal depth

CE 382
Dr. Huidae Cho
Department of Civil and Environmental Engineering...New Mexico State University
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1   What is normal depth?

  • Definition
    • Depth of flow under uniform flow conditions
    • Flow depth does not change along the channel
    • Occurs when driving force balances resistance
  • Condition
    • Energy slope equals bed slope
    • $S_f = S_0$
  • Interpretation
    • Depth at which gravity and friction are in equilibrium
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2   Finding normal depth from discharge

  • Goal
    • Given discharge $Q$
    • Find normal depth $y_n$
  • Condition
    • Uniform flow
    • $S_f = S_0$
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3   Governing equation

  • Unknown
    • Depth $y$
  • Known
    • $Q, n, S, \text{geometry}$
  • Challenge
    • Equation is nonlinear in $y$
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4   Geometry for rectangular channel

  • Area
    • $A = b y$
  • Wetted perimeter
    • $P = b + 2y$
  • Hydraulic radius
    • $R = \frac{A}{P}$
  • Substitute into Manning’s equation
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5   Solution approach

  • Cannot solve explicitly for $y$
  • Use trial and error
  • Steps
    • Guess $y$
    • Compute $A, P, R$
    • Compute $Q$
    • Compare with given discharge
    • Adjust $y$
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6   Iteration logic

  • If computed $Q$ is too small
    • Increase $y$
  • If computed $Q$ is too large
    • Decrease $y$
  • Continue until convergence
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7   Example

  • Given
    • $Q = 10 \ \text{m}^3/\text{s}$
    • $b = 3.0 \ \text{m}$
    • $n = 0.015$
    • $S = 0.001$
  • Find normal depth $y_n$
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8   Trial 1

  • Assume $y = 1.0 \ \text{m}$
  • $A = 3.0$
  • $P = 5.0$
  • $R = 0.60$
  • Compute discharge
    • $Q \approx 5.6 \ \text{m}^3/\text{s}$
  • Too small
    • Increase $y$
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9   Trial 2

  • Assume $y = 1.5 \ \text{m}$
  • $A = 4.5$
  • $P = 6.0$
  • $R = 0.75$
  • Compute discharge
    • $Q \approx 9.2 \ \text{m}^3/\text{s}$
  • Still small
    • Increase $y$
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10   Trial 3

  • Assume $y = 1.6 \ \text{m}$
  • $A = 4.8$
  • $P = 6.2$
  • $R = 0.77$
  • Compute discharge
    • $Q \approx 10.1 \ \text{m}^3/\text{s}$
  • Close enough
  • Answer
    • $y_n \approx 1.6 \ \text{m}$
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11   Key points

  • Normal depth occurs under uniform flow
  • Determined from Manning’s equation
  • Requires iteration
  • Depends on
    • Discharge
    • Roughness
    • Slope
    • Channel shape
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12   Common mistakes

  • Using energy equation instead of Manning
  • Forgetting to update $R$
  • Arithmetic errors in iteration
  • Stopping too early
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13   Quick check

  • If discharge increases
    • Normal depth increases
  • If slope increases
    • Normal depth decreases
  • If roughness increases
    • Normal depth increases
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