Applied Computational Hydraulics for Engineers

Contents
Chapter 1
Introduction to Computational Hydraulics
	1.1. Definition of Computational Hydraulics
	1.2. Basic Principles of Computational Hydraulics
	1.3. Forms of the Governing Water-Flow Equations
	1.4. Integral Equations
	1.5. Differential Equations
	1.6. Numerical Methods of Ordinary Differential Equations
		1.6.1. The Euler Method
		1.6.2. The Modified Euler Method
	1.7. Numerical Methods of Partial Differential Equations
	1.8. Terms of Hydraulic Modeling
		1.8.1. Terms Referring to Numerical Modeling
Chapter 2
Review of Governing Equations
	2.1. Introduction
	2.2. Continuity Equation
		2.2.1. General Remarks
		2.2.2. Continuity Equation in Cartesian Coordinates
		2.2.3. Continuity Equation in Cylindrical Coordinates
	2.3. Momentum Equation
		2.3.1. General Remarks
		2.3.2. Momentum Equation in Cartesian Coordinates
		2.3.3. Momentum Equation in Cylindrical Coordinates
		2.3.4. Bernoulli Equation
		2.3.5. Momentum and Forces in Fluid Flow
	2.4. Energy Equation
		2.4.1. General Remarks
		2.4.2. Energy Equation in Cartesian Coordinates
		2.4.3. Energy Equation in Cylindrical Coordinates
	2.5. Transport Equation
		2.5.1. General Remarks
		2.5.2. Convection-Diffusion Equation
		2.5.3. Convection Equation
		2.5.4. Diffusion Equation
		2.5.5. Diffusion Equation in Cylindrical Coordinates
		2.5.6. The Advection-Diffusion-Reaction Equation
	2.6. Navier-Stokes Equations in Stream  Function-Vorticity Form
		2.6.1. General Remarks
		2.6.2. Vorticity – Stream Function in Cartesian Coordinates
		2.6.3. Stream Function for Plane Two-Dimensional Flow in Cartesian Coordinates
		2.6.4. Vorticity – Stream Function in Cylindrical Coordinates
		2.6.5. Stream Function for Plane Two-Dimensional Flow  in Cylindrical Coordinates
	2.7. Saint-Venant Equations
		2.7.1. General Remarks
		2.7.2. Continuity Equation
		2.7.3. Momentum Equation
		2.7.4. Saint-Venant Equations for Steady State Flow
	2.8. Vorticity
	2.9. Potential Flow
		2.9.1. Velocity Potential Function
		2.9.2. Stream Function
		2.9.3. Laplace Equation in Cartesian Coordinates
		2.9.4. Laplace Equation in Cylindrical Coordinates
		2.9.5. Potential Lines and Streamlines
	2.10. Hydrostatic Balance and Earth’s Rotation
		2.10.1. Hydrostatic Balance
		2.10.2. Earth’s Rotation
	2.11. Scales and Dimensionless Numbers
		2.11.1. Definition of Scales
		2.11.2. The Froude Number
		2.11.3. The Richardson Number
		2.11.4. The Reynolds Number
		2.11.5. The Peclet Number
		2.11.6. The Rossby Number
Chapter 3
Some Analytical Solutions
	3.1. Introduction
	3.2. Sample Equations
		3.2.1. Some Differential Equations in Hydraulics
		3.2.2. Order and Degree of a Differential Equation
		3.2.3. Linear and Non-Linear Differential Equations
		3.2.4. Classification of a Second-Order Partial Differential Equation
	3.3. Solution of Emptying Time of a Dam Reservoir
	3.4. Solution of an Ordinary Linear Differential Equation
	3.5. Example for a Simple Lake Purification Model
Chapter 4
Applications of Numerical Analysis
	4.1. Introduction
	4.2. Newton-Raphson Method
	4.3. Least Squares Method
	4.4. Polynomial Curve Fitting
	4.5. Numerical Integration
		4.5.1. Numerical Integration Using Trapezoidal Rule
		4.5.2. Numerical Integration by Simpson’s Rule
	4.6. Applications by Numerical Analysis
		4.6.1. Calculation of Flow Velocity at Downstream of a Spillway Using Newton-Raphson Method
		4.6.2. Finding a Relationship between Temperature and Liquid Volume Using Least Squares Method
		4.6.3. Application of Polynomial Fitting
		4.6.4. Determining a Relationship between Water Depth  and Reservoir Volume Using Curve Fitting
		4.6.5. Applications of Numerical Integration Using Simpson’s Rule
		4.6.6. Determination of Emptying Time of a Reservoir  with Constant Inflow
		4.6.7. Determination of Emptying Time for a Small Reservoir
		4.6.8. Determination of Emptying Time of a Reservoir  with No Inflow
Chapter 5
Finite Difference Methods
	5.1. Introduction
	5.2. Difference Equation
	5.3. Partial Differential Equations
	5.4. Derivation of Some Basic Finite Difference Forms
		5.4.1. Taylor Series Expansions
		5.4.2. Polynomial Fitting
	5.5. Types of Finite Difference Schemes
		5.5.1. Basic Forms
		5.5.2. Geometric Interpretation of Basic Finite Difference Forms
		5.5.3. Explicit and Implicit Finite Difference Schemes
		5.5.4. Comparison of Explicit and Implicit Schemes
	5.6. Finite Difference Schemes
		5.6.1. Finite Difference Formulas for First Derivatives
		5.6.2. Finite Difference Formulas for Second Derivatives
		5.6.3. Finite Difference Formulas for Third Derivatives
		5.6.4. Finite Difference Formulas for Fourth Derivatives
	5.7. Finite Difference Formulations for Some Equations
		5.7.1. Pure Diffusion Equation
		5.7.2. Pure Advection Equation
		5.7.3. Numerical vs Physical diffusion
		5.7.4. Truncation Error and Round-off Error
	5.8. A Numerical Scheme Analysis
		5.8.1. Taylor Series Analysis
		5.8.2. Fourier Analysis
	5.9. Steps for Finite Difference Solutions
	5.10. Numerical Solutions of Differential Equations
		5.10.1. Example of Finite Difference Approximation
Chapter 6
Boundary and Initial Conditions
	6.1. Introduction
	6.2. Governing Equations
	6.3. Definition of Stream Function  and Velocity Potential Function
	6.4. Boundary Conditions for the Primitive Equations
	6.5. Determination of Velocity Components  from Nondimensional Stream Function
	6.6. Boundary Conditions for the Stream Function Equations
	6.7. Boundary Conditions for the Vorticity and Stream Function Equations
		6.7.1. General
		6.7.2. Walls in the First Mesh System
		6.7.3. Sloping Wall Boundary
		6.7.4. Symmetry Boundaries
		6.7.5. Upper Boundary
		6.7.6. Upstream Boundary
		6.7.7. Outflow Boundary
	6.8. Irregular Boundaries
	6.9. Boundary Conditions for Sharp Corners
	6.10. Initial Conditions
	6.11. Number of Boundary Conditions
	6.12. Applications of Boundary and Initial Conditions
		6.12.1. Determination of Velocity Components  from Non-Dimensional Stream Function
		6.12.2. Example for Irregular Boundary Conditions
		6.12.3. Determination of Boundary Conditions for Navier-Stokes Equations in Vorticity/Stream Function Form
Chapter 7
Solutions of Elliptic Equations
	7.1. Introduction
	7.2. Governing Equations
	7.3. Finite Difference Formulation of Laplace Equation
	7.4. Solution Methods for Elliptic Partial  Differential Equations
		7.4.1. Direct Method
		7.4.2. Jacobi Iteration
		7.4.3. Gauss-Seidel Iteration
		7.4.4. Successive Over Relaxation Method
	7.5. Applications of Elliptic Partial Differential Equations
		7.5.1. Numerical Solution to the Groundwater Flow Equation
		7.5.2. Numerical Solution of Stream Function Values, Velocities, Flow Rate for Flow, Potential Function Values and Pressures  under Sheet Piles
		7.5.3. Numerical Solution for Deflection of a Thin Plate
		7.5.4. Numerical Solution for Flow under a Dam
		7.5.5. Calculating Stream Function for a Potential Flow
		7.5.6. Calculation of Velocity Components of Flow from Stream Function Values in a Potential Flow
Chapter 8
Solutions of Parabolic Equations
	8.1. Introduction
	8.2. Classification of Parabolic Partial Differential Equations
	8.3. Schemes for Parabolic Partial Differential Equations
		8.3.1. Forward in Time and Centered Space (FTCS) Scheme
		8.3.2. Backward-Time Centered Space (BTCS) Scheme
		8.3.3. Dufort-Frankel Scheme
		8.3.4. Crank-Nicolson Scheme
		8.3.5. Leapfrog Scheme
		8.3.6. ADI Method
	8.4. Comparison of Schemes for Numerical Solution  of Parabolic Equations
	8.5. Stability for Parabolic Partial Differential Equations
	8.6. Applications of Parabolic Partial Differential Equations
		8.6.1. Solution of Parabolic Partial Differential Equation Using Explicit Finite Differences
		8.6.2. Determination of the Stability Criterion for Flow Between Two Parallel Plates
		8.6.3. Numerical Solution of Heat Flow Equation
Chapter 9
Solutions of Hyperbolic Equations
	9.1. Introduction
	9.2. Hyperbolic Partial Differential Equations
	9.3. Boundary and Initial Conditions
	9.4. Review of Some Hyperbolic Equations
		9.4.1. General
		9.4.2. The Advection Equation in One Dimension
		9.4.3. The Advection Equation in One Dimension Using a Finite Difference Method
		9.4.4. The Wave Equation in Two Spatial Dimensions
	9.5. Finite Difference Schemes in Hyperbolic  Differential Equation
		9.5.1. Finite Difference Formulation for Schemes
		9.5.2. The Courant-Friedrichs-Levy Stability Condition
	9.6. Applications of Hyperbolic Equations
		9.6.1. Solution of a Linear Kinematic Wave Model
		9.6.2. Example for Flood Routing Using Kinematic Wave Model
		9.6.3. Determination of Stability Criterion
		9.6.4. Example for Numerical Solution of an Overland Flow
		9.6.5. Numerical Solution of Shoreline Change
Chapter 10
Applications of Unsteady Open Channel Flows
	10.1. Introduction
	10.2. The Saint-Venant Equations
	10.3. Open Channel Flow Models
		10.3.1. Steady Flow
		10.3.2. Quasi-Steady Approximation
		10.3.3. Kinematic and Diffusion Wave Models
		10.3.4. Inertial and Dynamic Wave Models
	10.4. Applicability of Kinematic and Diffusion Wave Models
	10.5. Unsteady Flow Models in Open Channel Flow
		10.5.1. Summary of Saint-Venant Equations
		10.5.2. Kinematic Wave Model
		10.5.3. Finite Difference Formulation for Kinematic Wave Equation
		10.5.4. Formulation for Nonlinear Kinematic Wave Scheme
	10.6. Applications of Open Channel Flow Modeling
Chapter 11
Applications of Gradually Varied Flow
	11.1. Introduction
	11.2. Derivation of the Gradually-Varied-Flow Equation
	11.3. Determination of the Friction Slope
	11.4. Classification of Free Surface Profiles
	11.5. Formulation of the Gradually Varied Flow Equation
		11.5.1. Total-Head Form of the Gradually Varied Flow Equation
		11.5.2. Specific-Energy Form of the Gradually Varied Flow Equation
		11.5.3. Depth Form of the Gradually Varied Flow Equation
	11.6. Applications of Gradually Varied Flow
Chapter 12
Applications by Method of Characteristics
	12.1. Introduction
	12.2. Method of Characteristics in Transient Flow
	12.3. Charasteristics Method to Solve Saint-Venant Equations
	12.4. Solution of Wave Equation
	12.5. Kinematic Wave Modeling of Overland Flow
		12.5.1. General Remarks
		12.5.2. Boundary and Initial Conditions
		12.5.3. Analytical Solution Using Characteristics Method
		12.5.4. Equilibrium Case
		12.5.5. Partial Equilibrium Case
	12.6. Applications of Characteristics Method
		12.6.1. Solution of Transient Flow Equation
		12.6.2. Solution of Saint-Venant Equation
		12.6.3. Calculation of Discharge Hydrograph
		12.6.4. Solution of Advection Equation
		12.6.5. Calculation of the Time of Concentration for  a Diverging Surface
		12.6.6. Calculation of Rising Hydrograph
		12.6.7. Calculation of Time of Concentration
		12.6.8. Calculation of Concentration Time Using Kinematic  Wave Model
References
About the Authors
Index
Blank Page
Blank Page