Wave Propagation in Fluids

Vincent Guinot is professor of hydrodynamic modeling at the University of Montpellier, France. He teaches fluid mechanics, hydraulics, numerical methods and hydrodynamic modeling.

• Introduction xvChapter 1. Scalar Hyperbolic Conservation Laws in One Dimension of Space 11.1. Definitions 11.1.1. Hyperbolic scalar conservation laws 11.1.2. Derivation from general conservation principles 31.1.3. Non-conservation form 61.1.4. Characteristic form – Riemann invariants 71.2. Determination of the solution 91.2.1. Representation in the phase space 91.2.2. Initial conditions, boundary conditions 121.3. A linear law: the advection equation 141.3.1. Physical context – conservation form 141.3.2. Characteristic form 161.3.3. Example: movement of a contaminant in a river 171.3.4. Summary 211.4. A convex law: the inviscid Burgers equation 211.4.1. Physical context – conservation form 211.4.2. Characteristic form 231.4.3. Example: propagation of a perturbation in a fluid 241.4.4. Summary 281.5. Another convex law: the kinematic wave for free-surface hydraulics 281.5.1. Physical context – conservation form 281.5.2. Non-conservation and characteristic forms 291.5.3. Expression of the celerity 311.5.4. Specific case: flow in a rectangular channel 341.5.5. Summary 351.6. A non-convex conservation law: the Buckley-Leverett equation 361.6.1. Physical context – conservation form 361.6.2. Characteristic form 391.6.3. Example: decontamination of an aquifer 401.6.4. Summary 421.7. Advection with adsorption/desorption 421.7.1. Physical context – conservation form 421.7.2. Characteristic form 451.7.3. Summary 471.8. Conclusions 481.8.1. What you should remember 481.8.2. Application exercises 48Chapter 2. Hyperbolic Systems of Conservation Laws in One Dimension of Space 552.1. Definitions 552.1.1. Hyperbolic systems of conservation laws 552.1.2. Hyperbolic systems of conservation laws – examples 572.1.3. Characteristic form – Riemann invariants 592.2. Determination of the solution 622.2.1. Domain of influence, domain of dependence 622.2.2. Existence and uniqueness of solutions – initial and boundary conditions 642.3. Specific case: compressible flows 652.3.1. Definition 652.3.2. Conservation form 652.3.3. Characteristic form 682.3.4. Physical interpretation 702.4. A 2×2 linear system: the water hammer equations 712.4.1. Physical context – hypotheses 712.4.2. Conservation form 732.4.3. Characteristic form – Riemann invariants 782.4.4. Calculation of the solution 822.4.5. Summary 872.5. A nonlinear 2×2 system: the Saint Venant equations 872.5.1. Physical context – hypotheses 872.5.2. Conservation form 882.5.3. Characteristic form – Riemann invariants 942.5.4. Calculation of solutions 1052.5.5. Summary 1122.6. A nonlinear 3×3 system: the Euler equations 1122.6.1. Physical context – hypotheses 1122.6.2. Conservation form 1142.6.3. Characteristic form – Riemann invariants 1182.6.4. Calculation of the solution 1222.6.5. Summary 1262.7. Summary of Chapter 2 1272.7.1. What you should remember 1272.7.2. Application exercises 128Chapter 3. Weak Solutions and their Properties 1353.1. Appearance of discontinuous solutions 1353.1.1. Governing mechanisms 1353.1.2. Local invalidity of the characteristic formulation– graphical approach 1383.1.3. Practical examples of discontinuous flows 1403.2. Classification of waves 1433.2.1. Shock wave 1433.2.2. Rarefaction wave 1443.2.3. Contact discontinuity 1453.2.4. Mixed/compound wave 1453.3. Simple waves 1463.3.1. Definition and properties 1463.3.2. Generalized Riemann invariants 1473.4. Weak solutions and their properties 1493.4.1. Definitions 1493.4.2. Non-equivalence between the formulations 1503.4.3. Jump relationships 1503.4.4. Non-uniqueness of weak solutions 1523.4.5. The entropy condition 1573.4.6. Irreversibility 1593.4.7. Approximations for the jump relationships 1603.5. Summary 1613.5.1. What you should remember 1613.5.2. Application exercises 162Chapter 4. The Riemann Problem 1654.1. Definitions – solution properties 1654.1.1. The Riemann problem 1654.1.2. The generalized Riemann problem 1664.1.3. Solution properties 1674.2. Solution for scalar conservation laws 1674.2.1. The linear advection equation 1674.2.2. The inviscid Burgers equation 1684.2.3. The Buckley-Leverett equation 1704.3. Solution for hyperbolic systems of conservation laws 1754.3.1. General principle 1754.3.2. Application to the water hammer problem: sudden valve failure 1764.3.3. Free surface flow: the dambreak problem 1794.3.4. The Euler equations: the shock tube problem 1864.4. Summary 1924.4.1. What you should remember 1924.4.2. Application exercises 193Chapter 5. Multidimensional Hyperbolic Systems 1955.1. Definitions 1955.1.1. Scalar laws 1955.1.2. Two-dimensional hyperbolic systems 1975.1.3. Three-dimensional hyperbolic systems 1995.2. Derivation from conservation principles 2005.3. Solution properties 2035.3.1. Two-dimensional hyperbolic systems 2035.3.2. Three-dimensional hyperbolic systems 2105.4. Application to two-dimensional free-surface flow 2115.4.1. Governing equations 2115.4.2. The secant plane approach 2175.4.3. Interpretation – determination of the solution 2225.5. Summary 2255.5.1. What you should remember 2255.5.2. Application exercises 225Chapter 6. Finite Difference Methods for Hyperbolic Systems 2296.1. Discretization of time and space 2296.1.1. Discretization for one-dimensional problems 2296.1.2. Multidimensional discretization 2306.1.3. Explicit schemes, implicit schemes 2316.2. The method of characteristics (MOC) 2326.2.1. MOC for scalar hyperbolic laws 2326.2.2. MOC for hyperbolic systems of conservation laws 2416.2.3. Application examples 2466.3. Upwind schemes for scalar laws 2506.3.1. The explicit upwind scheme (non-conservation version) 2506.3.2. The implicit upwind scheme (non-conservation version) 2526.3.3. Conservative versions of the implicit upwind scheme 2536.3.4. Application examples 2556.4. The Preissmann scheme 2576.4.1. Formulation 2576.4.2. Estimation of nonlinear terms – algorithmic aspects 2606.4.3. Numerical applications 2616.5. Centered schemes 2676.5.1. The Crank-Nicholson scheme 2676.5.2. Centered schemes with Runge-Kutta time stepping 2686.6. TVD schemes 2706.6.1. Definitions 2706.6.2. General formulation of TVD schemes 2716.6.3. Harten’s and Sweby’s criteria 2746.6.4. Traditional limiters 2766.6.5. Calculation example 2776.7. The flux splitting technique 2806.7.1. Principle of the approach 2806.7.2. Application to traditional schemes 2836.8. Conservative discretizations: Roe’s matrix 2896.8.1. Motivation and principle of the approach 2896.8.2. Expression of Roe’s matrix 2906.9. Multidimensional problems 2936.9.1. Explicit alternate directions2936.9.2. The ADI method 2966.9.3. Multidimensional schemes 2986.10. Summary 2996.10.1. What you should remember 2996.10.2. Application exercises 301Chapter 7. Finite Volume Methods for Hyperbolic Systems 3037.1. Principle 3037.1.1. One-dimensional conservation laws 3037.1.2. Multidimensional conservation laws 3057.1.3. Application to the two-dimensional shallow water equations 3087.2. Godunov’s scheme 3107.2.1. Principle 3107.2.2. Application to the scalar advection equation 3117.2.3. Application to the inviscid Burgers equation 3167.2.4. Application to the water hammer equations 3197.3. Higher-order Godunov-type schemes 3247.3.1. Rationale and principle 3247.3.2. Example: the MUSCL scheme 3287.4. Summary 3307.4.1. What you should remember 3307.4.2. Suggested exercises 331Appendix A. Linear Algebra 333A.1. Definitions 333A.2. Operations on matrices and vectors 335A.2.1. Addition 335A.2.2. Multiplication by a scalar 335A.2.3. Matrix product 336A.2.4. Determinant of a matrix 336A.2.5. Inverse of a matrix 337A.3. Differential operations using matrices and vectors 337A.3.1. Differentiation 337A.3.2. Jacobian matrix 338A.4. Eigenvalues, eigenvectors 338A.4.1. Definitions 338A.4.2. Example 339Appendix B. Numerical Analysis 341B.1. Consistency 341B.1.1. Definitions 341B.1.2. Principle of a consistency analysis 341B.1.3. Numerical diffusion, numerical dispersion 343B.2. Stability 345B.2.1. Definition 345B.2.2. Principle of a stability analysis 346B.2.3. Harmonic analysis of analytical solutions 348B.2.4. Harmonic analysis of numerical solutions 352B.2.5. Amplitude and phase portraits 355B.2.6. Extension to systems of equations 357B.3. Convergence 359B.3.1. Definition 359B.3.2. Lax’s theorem 359Appendix C. Approximate Riemann Solvers 361C.1. HLL and HLLC solvers 361C.1.1. HLL solver 361C.1.2. HLLC solver 363C.2. Roe’s solver 366Appendix D. Summary of the Formulae 369References 375Index 379