Invitation to the Computer River Engineering

Hydraulics for Practitioners (Reiwa Edition)

• Preface

• 1. Introduction
  • 1.1. Objectives of this text
  • 1.2. The second objective of this text
  • 1.3. Acknowledgments
• 2. Basics of partial differential equations
  • 2.1. Finite differential schemes for advection equations
  • 2.2. Differences in solutions by difference method
    • 2.2.1. Backward difference scheme (The Python code and calculation results)
    • 2.2.2. The forward difference scheme (The Python code and calculation results)
    • 2.2.3. Central difference scheme (The Python code and calculation results)
    • 2.2.4. Summary
  • 2.3. Advection direction and difference schemes
    • 2.3.1. The Python code and calculation results
  • 2.4. Finite differential scheme for a diffusion equation
    • 2.4.1. The Python code and calculation results
  • 2.5. Numerical diffusion caused by upwind differential shame
• 3. The higher-oder upwind difference scheme
  • 3.1. Interpolation of variable distribution between grids
  • 3.2. The basic concept of the CIP method
  • 3.3. Method for estimating the variable distribution form between grids
    • 3.3.1. Introduction of a 3rd order function
      • 3.3.1.1. Problem (1)
    • 3.3.2. Summary of a method for estimating the distribution equation of link variables by using the CIP method.
    • 3.3.3. The Python code and calculation examples
  • 3.4. Advection equations with varying advection directions
    • 3.4.1. When u<0,
      • 3.4.1.1. Problem (2)
    • 3.4.2. When u>0,
      • 3.4.2.1. Problem (3)
    • 3.4.3. How can we make it usable for both u>0 and u<0?
      • 3.4.3.1. Example
    • 3.4.4. The Python code and calculation examples
  • 3.5. An advection equation with the source term
    • 3.5.1. Non-advection phase
    • 3.5.2. Advection phase
      • 3.5.2.1. Problem (4)
    • 3.5.3. The Python code and calculation examples
• 4. 1D unsteady flow with constant width
  • 4.1. Basic formulas
  • 4.2. Preparation for applying the CIP method (the split method)
  • 4.3. The non-advection phase
  • 4.4. The advection phase
  • 4.5. Boundary conditions
  • 4.6. Summary
  • 4.7. The Python code
  • 4.8. Calculation example
    • 4.8.1. Back water
    • 4.8.2. Drawdown
    • 4.8.3. Hydraulic jump
    • 4.8.4. Transitional flow
    • 4.8.5. Flow in a channel with a bump at the channel center
• 5. 1D unsteady flow with width variation
  • 5.1. Basic equations
  • 5.2. Preparation for applying the CIP method (the split method)
  • 5.3. The non-advection phase
  • 5.4. Advection phase
  • 5.5. Summary
  • 5.6. The Python code
  • 5.7. A calculation example
    • 5.7.1. Example of 1D flow calculation for a straight channel with a constriction
• 6. The 2D advection equation
  • 6.1. Basic formulas
  • 6.2. The upwind difference scheme
    • 6.2.1. An example of calculations by backward difference scheme
      • 6.2.1.1. The Python code and a calculation example
    • 6.2.2. The upwind difference scheme with varying advection direction
      • 6.2.2.1. The Python code and a calculation example
      • 6.2.2.2. Problem (4)
  • 6.3. The higher-order difference scheme
    • 6.3.1. Calculating the advection term by using the 2D CIP method
      • 6.3.1.1. Problem (5)
    • 6.3.2. Calculating the advection term with the 2D CIP method (generalization)
      • 6.3.2.1. Problem (6)
    • 6.3.3. Updating the differentiation amount
    • 6.3.4. The Python code and a calculation example
• 7. 2-dimensional unsteady flow with free water surface
  • 7.1. Basic formulas and the application of the split method
  • 7.2. The non-advection phase I
  • 7.3. The non-advection phase II
  • 7.4. The advection phase
    • 7.4.1. u in the advection phase
    • 7.4.2. v in the advection phase
    • 7.4.3. Updating differentiation amounts in the advection phase
      • 7.4.3.1. Problem (7)
  • 7.5. The Python code, and a calculation example
    • 7.5.1. The Python code
  • 7.6. About NaysMini
• 8. Basic knowledge on the alluvial river hydraulics
  • 8.1. Riverbed and channel plane morphology
  • 8.2. Basic features of small-scale bed morphology
    • 8.2.1. The wavelength and wave height of the dune
    • 8.2.2. Wavelength and wave height of ripples
  • 8.3. Basic features of mesoscale bed morphology
    • 8.3.1. Alternate (single-row) bars
      • 8.3.1.1. Naka River
      • 8.3.1.2. Tokachi River
    • 8.3.2. Double-row bars
      • 8.3.2.1. Sho River
      • 8.3.2.2. Satsunai River
    • 8.3.3. Multi-row bars (braided bars, scaly bars)
      • 8.3.3.1. Oi River
      • 8.3.3.2. Hii River
      • 8.3.3.3. Braided rivers around the world
    • 8.3.4. Regime criteria of the mesoscale bed morphology
    • 8.3.5. Basic features of mesoscale bed morphology
  • 8.4. Plane morphology of the meandering rivers
    • 8.4.1. Meandering caused by bars
    • 8.4.2. Menders caused by plane shape instability
  • 8.5. Free bars and point bars
    • 8.5.1. Critical meander angle for bar progress
    • 8.5.2. Gourd-shaped river channel (a river characterized by narrow and wide sections repeating periodically)
  • 8.6. Examples of meandering rivers
    • 8.6.1. Shimanto River
    • 8.6.2. Sorachi River
    • 8.6.3. Ukayari River
    • 8.6.4. Ichilo River
  • 8.7. The shear stress and the critical shear stress
  • 8.8. Sediment transport type
  • 8.9. Threshold of bedload motion
  • 8.10. Suspending and settling velocities of suspended sediments, and possible regions of their existences
  • 8.11. Bedload transport formula
    • 8.11.1. The Meyer-Peter and Muller (MPM) Equation
    • 8.11.2. Ashida and Michiue Equation
  • 8.12. Suspended sediment
    • 8.12.1. Definition of suspended sediment, and the suspended sediment volume.
      • 8.12.1.1. Suspended sediment concentration at a reference point and the equation for entrained suspended sediment
    • 8.12.2. Formula for the suspended sediment concentration distribution and the depth-averaged suspended sediment concentration
• 9. One dimensional bed deformation modeling
  • 9.1. With bedload sediment transport
    • 9.1.1. Continuity equation for bedload sediment transport
    • 9.1.2. Calculation procedure
    • 9.1.3. The Python code
    • 9.1.4. Calculation results
  • 9.2. With suspended load sediment transport
    • 9.2.1. Continuity equation for suspended sediment concentration
    • 9.2.2. Continuity equation for the bed with suspended sediment transport
    • 9.2.3. The Python codes
    • 9.2.4. Calculation results
• 10. Calculation of 2D flow in meandering channels(basic edition)
  • 10.1. Sine-generated curve
    • 10.1.1. Generating a sine-generated curve
    • 10.1.2. Making calculation grids by using a sine-generated curve
    • 10.1.3. Giving a “bar and pool” geometry to the bed
  • 10.2. Kinoshita meandering curve (meander curve proposed by Kinoshita)
  • 10.3. Calculation of the sinuosity
• 11. Calculation of 2D flow in meandering channels (practical edition)
  • 11.1. Basic equations
  • 11.2. Calculation method
  • 11.3. Dry Bed (calculation point with no water)
  • 11.4. The Python code
  • 11.5. An example of calculation in a sine-generated curve channel
  • 11.6. Periodic boundary conditions
• 12. Calculation of 2D flow and bed deformation in meandering channels (basic edition)
  • 12.1. Secondary flow and the direction of bottom shear stress
    • 12.1.1. What is a secondary flow?
    • 12.1.2. Distribution of vertical velocities for the main flow (parabolic distribution)
    • 12.1.3. Velocity distribution of secondary flows (6D distribution function)
    • 12.1.4. Secondary flow intensity
  • 12.2. Streamline curvature
    • 12.2.1. Calculation method for streamline sinuosity
      • 12.2.1.1. Cartesian linear coordinates
      • 12.2.1.2. For curvilinear coordinates
    • 12.2.2. Calculation examples for the curvature of the streamline
      • 12.2.2.1. An example of the sine-generated curve
      • 12.2.2.2. A calculation example for a straight channel with a group of spur dikes
• 13. Calculation of the flow and bed deformation in meandering channels (practical edition)
  • 13.1. Method for calculating 2D sediment transport
    • 13.1.1. The direction of the bottom shear stress resulting from secondary flow
    • 13.1.2. Correction due to the effect of gravity
    • 13.1.3. The introduction of a slope failure model
  • 13.2. The equation of continuity for sediment transport on a 2D riverbed
  • 13.3. Procedure for 2D bed deformation calculation
  • 13.4. Calculation code and calculation results
• 14. Calculations of 2D flow and bed deformation in a channel with width variation
  • 14.1. Equation of continuity
  • 14.2. A calculation example for a widened channel
  • 14.3. The Python code

• Epilogue
  • Postscript
• Appendix I (Theory and basic formulas of flow and bed deformation calculation)
  • Theoretical solutions to advection equations
  • Other higher-order difference schemes for the advection term
    • Kawamura-Kuwahara scheme (K-K scheme) [Ref:2]
      • Difference equation
      • Python codes
      • Calculation results
    • QUICK scheme [Ref:3]
      • Difference equation
      • Python codes
      • Calculation results
    • A fifth-order upwind difference scheme [Ref:4]
      • Difference equation
      • Python codes
      • Calculation results
  • Expressions of riverbed shear force
  • River bank (sidewall) friction resistance
    • Wall friction coefficient
  • Velocity distribution
    • Logarithmic velocity distribution (logarithmic law)
      • Distribution equation in the vertical direction
      • The depth-averaged velocity determined by means of logarithmic law
    • Parabolic velocity distribution
      • Vertical distribution equation
      • The depth-averaged velocity from the parabolic velocity distribution
    • Comparison of logarithmic law and parabolic velocity distribution
  • Karman-Prandtl’s equation for the near-wall turbulent flow velocity ([Ref:31])
  • Roughness of a flat riverbed
    • Expression with the Manning’s roughness coefficient
    • Regarding the relationship between the roughness and the coefficients of resistance and flow velocity
• Appendix II (Basic equations for 2D flow in the s-n coordinates)
  • Conversion of the basic 2D flow equations to the s-n coordinates
    • Equation of continuity
    • Momentum equation
      • The advection terms
      • Acceleration along the channel downstream direction
      • Acceleration in the transverse direction
    • Pressure term
    • Friction term
    • Diffusion term
    • Summary
• Appendix III (Diffusion equation in the s-n coordinates)
  • Derivation of the diffusion equation in the s-n coordinates
  • Derivation of the equation of continuity on the s-n coordinates (alternative solution)
• Appendix IV (1D non-uniform flow calculation)
  • 1D non-uniform flow calculation
• Appendix V (The main flow and the secondary flow velocity distribution by Engelund [Ref:38] )
  • Vertical velocity distribution of the main flow
  • Velocity distributions of secondary flow
  • Bottom flow velocity
    • Secondary flow intensity
• References