{"product_id":"numerical-computation-of-internal-and-external-flows-volume-2-isbn-9780471924524","title":"Numerical Computation of Internal and External Flows, Volume 2","description":"\u003ci\u003eNumerical Computation of Internal and External Flows Volume 2: Computational Methods for Inviscid and Viscous Flows\u003c\/i\u003e C. Hirsch, Vrije Universiteit Brussel, Brussels, Belgium\u003cbr\u003e\u003cbr\u003eThis second volume deals with the applications of computational methods to the problems of fluid dynamics. It complements the first volume to provide an excellent reference source in this vital and fast growing area. The author includes material on the numerical computation of potential flows and on the most up-to-date methods for Euler and Navier-Stokes equations. The coverage is comprehensive and includes detailed discussion of numerical techniques and algorithms, including implementation topics such as boundary conditions. Problems are given at the end of each chapter and there are comprehensive reference lists. Of increasing interest, the subject has powerful implications in such crucial fields as aeronautics and industrial fluid dynamics. Striking a balance between theory and application, the combined volumes will be useful for an increasing number of courses, as well as to practitioners and researchers in computational fluid dynamics.\u003cbr\u003e\u003cbr\u003eContents Preface Nomenclature Part V: The Numerical Computation of Potential Flows Chapter 13 The Mathematical Formulations of the Potential Flow Model Chapter 14 The Discretization of the Subsonic Potential Equation Chapter 15 The Computation of Stationary Transonic Potential Flows Part VI: The Numerical Solution of the System of Euler Equations Chapter 16 The Mathematical Formulation of the System of Euler Equations Chapter 17 The Lax - Wendroff Family of Space-centred Schemes Chapter 18 The Central Schemes with Independent Time Integration Chapter 19 The Treatment of Boundary Conditions Chapter 20 Upwind Schemes for the Euler Equations Chapter 21 Second-order Upwind and High-resolution Schemes Part VII: The Numerical Solution of the Navier-Stokes Equations Chapter 22 The Properties of the System of Navier-Stokes Equations Chapter 23 Discretization Methods for the Navier-Stokes Equations Index \u003cp\u003ePreface xv\u003c\/p\u003e \u003cp\u003eNomenclature xix\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart V: The Numerical Computation of Potential Flows 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 13 The Mathematical Formulations of the Potential Flow Model 4\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e13.1 Conservative Form of the Potential Equation 4\u003c\/p\u003e \u003cp\u003e13.2 The Non-conservative Form of the Isentropic Potential Flow Model 6\u003c\/p\u003e \u003cp\u003e13.2.1 Small-perturbation potential equation 7\u003c\/p\u003e \u003cp\u003e13.3 The Mathematical Properties of the Potential Equation 9\u003c\/p\u003e \u003cp\u003e13.3.1 Unsteady potential flow 9\u003c\/p\u003e \u003cp\u003e13.3.2 Steady potential flow 9\u003c\/p\u003e \u003cp\u003e13.4 Boundary Conditions 14\u003c\/p\u003e \u003cp\u003e13.4.1 Solid wall boundary condition 14\u003c\/p\u003e \u003cp\u003e13.4.2 Far field conditions 15\u003c\/p\u003e \u003cp\u003e13.4.3 Cascade and channel flows 17\u003c\/p\u003e \u003cp\u003e13.4.4 Circulation and Kutta condition 18\u003c\/p\u003e \u003cp\u003e13.5 Integral or Weak Formulation of the Potential Model 18\u003c\/p\u003e \u003cp\u003e13.5.1 Bateman variational principle 19\u003c\/p\u003e \u003cp\u003e13.5.2 Analysis of some properties of the variational integral 20\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 14 The Discretization of the Subsonic Potential Equation 26\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e14.1 Finite Difference Formulation 27\u003c\/p\u003e \u003cp\u003e14.1.1 Numerical estimation of the density 29\u003c\/p\u003e \u003cp\u003e14.1.2 Curvilinear mesh 31\u003c\/p\u003e \u003cp\u003e14.1.3 Consistency of the discretization of metric coefficients 34\u003c\/p\u003e \u003cp\u003e14.1.4 Boundary conditions—curved solid wall 36\u003c\/p\u003e \u003cp\u003e14.2 Finite Volume Formulation 38\u003c\/p\u003e \u003cp\u003e14.2.1 Jameson and Caughey’s finite volume method 39\u003c\/p\u003e \u003cp\u003e14.3 Finite Element Formulation 42\u003c\/p\u003e \u003cp\u003e14.3.1 The finite element—Galerkin method 43\u003c\/p\u003e \u003cp\u003e14.3.2 Least squares or optimal control approach 47\u003c\/p\u003e \u003cp\u003e14.4 Iteration Scheme for the Density 47\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 15 The Computation of Stationary Transonic Potential Flows 57\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e15.1 The Treatment of the Supersonic Region: Artificial Viscosity—Density and Flux Upwinding 61\u003c\/p\u003e \u003cp\u003e15.1.1 Artificial viscosity—non-conservative potential equation 62\u003c\/p\u003e \u003cp\u003e15.1.2 Artificial viscosity—conservative potential equation 66\u003c\/p\u003e \u003cp\u003e15.1.3 Artificial compressibility 67\u003c\/p\u003e \u003cp\u003e15.1.4 Artificial flux or flux upwinding 70\u003c\/p\u003e \u003cp\u003e15.2 Iteration Schemes for Potential Flow Computations 77\u003c\/p\u003e \u003cp\u003e15.2.1 Line relaxation schemes 77\u003c\/p\u003e \u003cp\u003e15.2.2 Guidelines for resolution of the discretized potential equation 81\u003c\/p\u003e \u003cp\u003e15.2.3 The alternating direction implicit method—approximate factorization schemes 88\u003c\/p\u003e \u003cp\u003e15.2.4 Other techniques—multigrid methods 98\u003c\/p\u003e \u003cp\u003e15.3 Non-uniqueness and Non-isentropic Potential Models 104\u003c\/p\u003e \u003cp\u003e15.3.1 Isentropic shocks 105\u003c\/p\u003e \u003cp\u003e15.3.2 Non-uniqueness and breakdown of the transonic potential flow model 105\u003c\/p\u003e \u003cp\u003e15.3.3 Non-isentropic potential models 112\u003c\/p\u003e \u003cp\u003e15.4 Conclusions 117\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart VI: The Numerical Solution of the System of Euler Equations 125\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 16 The Mathematical Formulation of the System of Euler Equations 132\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e16.1 The Conservative Formulation of the Euler Equations 132\u003c\/p\u003e \u003cp\u003e16.1.1 Integral conservative formulation of the Euler equations 133\u003c\/p\u003e \u003cp\u003e16.1.2 Differential conservative formulation 134\u003c\/p\u003e \u003cp\u003e16.1.3 Cartesian system of coordinates 134\u003c\/p\u003e \u003cp\u003e16.1.4 Discontinuities and Rankine-Hugoniot relations—entropy condition 135\u003c\/p\u003e \u003cp\u003e16.2 The Quasi-linear Formulation of the Euler Equations 138\u003c\/p\u003e \u003cp\u003e16.2.l The Jacobian matrices for conservative variables 138\u003c\/p\u003e \u003cp\u003e16.2.2 The Jacobian matrices for primitive variables 145\u003c\/p\u003e \u003cp\u003e16.2.3 Transformation matrices between conservative and non-conservative variables 147\u003c\/p\u003e \u003cp\u003e16.3 The Characteristic Formulation of the Euler Equations—Eigenvalues and Compatibility Relations 150\u003c\/p\u003e \u003cp\u003e16.3.1 General properties of characteristics 151\u003c\/p\u003e \u003cp\u003e16.3.2 Diagonalization of the Jacobian matrices 153\u003c\/p\u003e \u003cp\u003e16.3.3 Compatibility equations 154\u003c\/p\u003e \u003cp\u003e16.4 Characteristic Variables and Eigenvalues for One-dimensional Flows 157\u003c\/p\u003e \u003cp\u003e16.4.1 Eigenvalues and eigenvectors of Jacobian matrix 158\u003c\/p\u003e \u003cp\u003e16.4.2 Characteristic variables 162\u003c\/p\u003e \u003cp\u003e16.4.3 Characteristics in the \u003ci\u003ext\u003c\/i\u003e-plane—shocks and contact discontinuities 168\u003c\/p\u003e \u003cp\u003e16.4.4 Physical boundary conditions 171\u003c\/p\u003e \u003cp\u003e16.4.5 Characteristics and simple wave solutions 173\u003c\/p\u003e \u003cp\u003e16.5 Eigenvalues and Compatibility Relations in Multidimensional Flows 176\u003c\/p\u003e \u003cp\u003e16.5.1 Jacobian eigenvalues and eigenvectors in primitive variables 177\u003c\/p\u003e \u003cp\u003e16.5.2 Diagonalization of the conservative Jacobians 180\u003c\/p\u003e \u003cp\u003e16.5.3 Mach cone and compatibility relations 184\u003c\/p\u003e \u003cp\u003e16.5.4 Boundary conditions 191\u003c\/p\u003e \u003cp\u003e16.6 Some Simple Exact Reference Solutions for One-dimensional Inviscid Flows 196\u003c\/p\u003e \u003cp\u003e16.6.1 The linear wave equation 196\u003c\/p\u003e \u003cp\u003e16.6.2 The inviscid Burgers equation 196\u003c\/p\u003e \u003cp\u003e16.6.3 The shock tube problem or Riemann problem 204\u003c\/p\u003e \u003cp\u003e16.6.4 The quasi-one-dimensional nozzle flow 211\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 17 The Lax\u003c\/b\u003e–\u003cb\u003eWendroff Family of Space-centred Schemes 224\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e17.1 The Space-centred Explicit Schemes of First Order 226\u003c\/p\u003e \u003cp\u003e17.1.1 The one-dimensional Lax–Friedrichs scheme 226\u003c\/p\u003e \u003cp\u003e17.1.2 The two-dimensional Lax–Friedrichs scheme 229\u003c\/p\u003e \u003cp\u003e17.1.3 Corrected viscosity scheme 233\u003c\/p\u003e \u003cp\u003e17.2 The Space-centred Explicit Schemes of Second Order 234\u003c\/p\u003e \u003cp\u003e17.2.1 The basic one-dimensional Lax–Wendroff scheme 234\u003c\/p\u003e \u003cp\u003e17.2.2 The two-step Lax–Wendroff schemes in one dimension 238\u003c\/p\u003e \u003cp\u003e17.2.3 Lerat and Peyret’s  family of non-linear two-step Lax–Wendroff schemes 246\u003c\/p\u003e \u003cp\u003e17.2.4 One-step Lax–Wendroff schemes in two dimensions 251\u003c\/p\u003e \u003cp\u003e17.2.5 Two-step Lax–Wendroff schemes in two dimensions 258\u003c\/p\u003e \u003cp\u003e17.3 The Concept of Artificial Dissipation or Artificial Viscosity 272\u003c\/p\u003e \u003cp\u003e17.3.1 General form of artificial dissipation terms 273\u003c\/p\u003e \u003cp\u003e17.3.2 Von Neumann–Richtmyer artificial viscosity 274\u003c\/p\u003e \u003cp\u003e17.3.3 Higher-order artificial viscosities 279\u003c\/p\u003e \u003cp\u003e17.4 Lerat’s Implicit Schemes of Lax–Wendroff Type 283\u003c\/p\u003e \u003cp\u003e17.4.1 Analysis for linear systems in one dimension 285\u003c\/p\u003e \u003cp\u003e17.4.2 Construction of the family of schemes 288\u003c\/p\u003e \u003cp\u003e17.4.3 Extension to non-linear systems in conservation form 292\u003c\/p\u003e \u003cp\u003e17.4.4 Extension to multi-dimensional flows 296\u003c\/p\u003e \u003cp\u003e17.5 Summary 296\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 18 The Central Schemes with Independent Time Integration 307\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e18.1 The Central Second-order Implicit Schemes of Beam and Warming in One Dimension 309\u003c\/p\u003e \u003cp\u003e18.1.1 The basic Beam and Warming schemes 310\u003c\/p\u003e \u003cp\u003e18.1.2 Addition of artificial viscosity 315\u003c\/p\u003e \u003cp\u003e18.2 The Multidimensional Implicit Beam and Warming Schemes 326\u003c\/p\u003e \u003cp\u003e18.2.1 The diagonal variant of Pulliam and Chaussee 328\u003c\/p\u003e \u003cp\u003e18.3 Jameson’s Multistage Method 334\u003c\/p\u003e \u003cp\u003e18.3.1 Time integration 334\u003c\/p\u003e \u003cp\u003e18.3.2 Convergence acceleration to steady state 335\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 19 The Treatment of Boundary Conditions 344\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e19.1 One-dimensional Boundary Treatment for Euler Equations 345\u003c\/p\u003e \u003cp\u003e19.1.1 Characteristic boundary conditions 346\u003c\/p\u003e \u003cp\u003e19.1.2 Compatibility relations 347\u003c\/p\u003e \u003cp\u003e19.1.3 Characteristic boundary conditions as a function of conservative and primitive variables 349\u003c\/p\u003e \u003cp\u003e19.1.4 Extrapolation methods 353\u003c\/p\u003e \u003cp\u003e19.1.5 Practical implementation methods for numerical boundary conditions 357\u003c\/p\u003e \u003cp\u003e19.1.6 Nonreflecting boundary conditions 369\u003c\/p\u003e \u003cp\u003e19.2 Multidimensional Boundary Treatment 372\u003c\/p\u003e \u003cp\u003e19.2.1 Physical and numerical boundary conditions 372\u003c\/p\u003e \u003cp\u003e19.2.2 Multidimensional compatibility relations 376\u003c\/p\u003e \u003cp\u003e19.2.3 Farfield treatment for steadystate flows 377\u003c\/p\u003e \u003cp\u003e19.2.4 Solid wall boundary 379\u003c\/p\u003e \u003cp\u003e19.2.5 Nonreflective boundary conditions 384\u003c\/p\u003e \u003cp\u003e19.3 The Far-field Boundary Corrections 385\u003c\/p\u003e \u003cp\u003e19.4 The Kutta Condition 395\u003c\/p\u003e \u003cp\u003e19.5 Summary 401\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 20 Upwind Schemes for the Euler Equations 408\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e20.1 The Basic Principles of Upwind Schemes 409\u003c\/p\u003e \u003cp\u003e20.2 One-dimensional Flux Vector Splitting 415\u003c\/p\u003e \u003cp\u003e20.2.1 Steger and Warming flux vector splitting 415\u003c\/p\u003e \u003cp\u003e20.2.2 Properties of split flux vectors 417\u003c\/p\u003e \u003cp\u003e20.2.3 Van Leer’s flux splitting 420\u003c\/p\u003e \u003cp\u003e20.2.4 Non-reflective boundary conditions and split fluxes 425\u003c\/p\u003e \u003cp\u003e20.3 One-dimensional Upwind Discretizations Based on Flux Vector Splitting 426\u003c\/p\u003e \u003cp\u003e20.3.1 First-order explicit upwind schemes 426\u003c\/p\u003e \u003cp\u003e20.3.2 Stability conditions for first-order flux vector splitting schemes 428\u003c\/p\u003e \u003cp\u003e20.3.3 Non-conservative firstorder upwind schemes 438\u003c\/p\u003e \u003cp\u003e20.4 Multi-dimensional Flux Vector Splitting 438\u003c\/p\u003e \u003cp\u003e20.4.1 Steger and Warming flux splitting 440\u003c\/p\u003e \u003cp\u003e20.4.2 Van Leer flux splitting 440\u003c\/p\u003e \u003cp\u003e20.4.3 Arbitrary meshes 441\u003c\/p\u003e \u003cp\u003e20.5 The Godunov-type Schemes 443\u003c\/p\u003e \u003cp\u003e20.5.1 The basic Godunov scheme 444\u003c\/p\u003e \u003cp\u003e20.5.2 Osher’s approximate Riemann solver 453\u003c\/p\u003e \u003cp\u003e20.5.3 Roe’s approximate Riemann solver 460\u003c\/p\u003e \u003cp\u003e20.5.4 Other Godunov-type methods 469\u003c\/p\u003e \u003cp\u003e20.5.5 Summary 472\u003c\/p\u003e \u003cp\u003e20.6 First-order Implicit Upwind Schemes 473\u003c\/p\u003e \u003cp\u003e20.7 Multi-dimensional First-order Upwind Schemes 475\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 21 Second-order Upwind and High-resolution Schemes 493\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e21.1 General Formulation of Higher-order Upwind Schemes 494\u003c\/p\u003e \u003cp\u003e21.1.1 Higher-order projection stages-variable extrapolation or MUSCL approach 495\u003c\/p\u003e \u003cp\u003e21.1.2 Numerical flux for higher-order upwind schemes 498\u003c\/p\u003e \u003cp\u003e21.1.3 Second-order space- and time-accurate upwind schemes based on variable extrapolation 499\u003c\/p\u003e \u003cp\u003e21.1.4 Linearized analysis of second-order upwind schemes 502\u003c\/p\u003e \u003cp\u003e21.1.5 Numerical flux for higher-order upwind schemes—flux extrapolation 504\u003c\/p\u003e \u003cp\u003e21.1.6 Implicit second-order upwind schemes 512\u003c\/p\u003e \u003cp\u003e21.1.7 Implicit second-order upwind schemes in two dimensions 514\u003c\/p\u003e \u003cp\u003e21.1.8 Summary 516\u003c\/p\u003e \u003cp\u003e21.2 The Definition of High-resolution Schemes 517\u003c\/p\u003e \u003cp\u003e21.2.1 The generalized entropy condition for inviscid equations 519\u003c\/p\u003e \u003cp\u003e21.2.2 Monotonicity condition 525\u003c\/p\u003e \u003cp\u003e21.2.3 Total variation diminishing (TVD)schemes 528\u003c\/p\u003e \u003cp\u003e21.3 Second-order TVD Semi-discretized Schemes with Limiters 536\u003c\/p\u003e \u003cp\u003e21.3.1 Definition of limiters for the linear convection equation 537\u003c\/p\u003e \u003cp\u003e21.3.2 General definition of flux limiters 550\u003c\/p\u003e \u003cp\u003e21.3.3 Limiters for variable extrapolation—MUSCL—method 552\u003c\/p\u003e \u003cp\u003e21.4 Timeintegration Methods for TVD Schemes 556\u003c\/p\u003e \u003cp\u003e21.4.1 Explicit TVD schemes of first-order accuracy in time 557\u003c\/p\u003e \u003cp\u003e21.4.2 Implicit TVD schemes 558\u003c\/p\u003e \u003cp\u003e21.4.3 Explicit second-order TVD schemes 560\u003c\/p\u003e \u003cp\u003e21.4.4 TVD schemes and artificial dissipation 564\u003c\/p\u003e \u003cp\u003e21.4.5 TVD limiters and the entropy condition 568\u003c\/p\u003e \u003cp\u003e21.5 Extension to Non-linear Systems and to Multi-dimensions 570\u003c\/p\u003e \u003cp\u003e21.6 Conclusions to Part VI 583\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePart VII: The Numerical Solution of the Navier-Stokes Equations 595\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 22 The Properties of the System of Navier–Stokes Equations 597\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e22.1 Mathematical Formulation of the Navier–Stokes Equations 597\u003c\/p\u003e \u003cp\u003e22.1.1 Conservative form of the Navier–Stokes equations 597\u003c\/p\u003e \u003cp\u003e22.1.2 Integral form of the Navier–Stokes equations 599\u003c\/p\u003e \u003cp\u003e22.1.3 Shock waves and contact layers 600\u003c\/p\u003e \u003cp\u003e22.1.4 Mathematical properties and boundary conditions 601\u003c\/p\u003e \u003cp\u003e22.2 Reynolds-averaged Navier–Stokes Equations 603\u003c\/p\u003e \u003cp\u003e22.2.1 Turbulent-averaged energy equation 604\u003c\/p\u003e \u003cp\u003e22.3 Turbulence Models 606\u003c\/p\u003e \u003cp\u003e22.3.1 Algebraic models 608\u003c\/p\u003e \u003cp\u003e22.3.2 One- and two-equation models—\u003ci\u003ek\u003c\/i\u003e–\u003ci\u003eε\u003c\/i\u003e models 613\u003c\/p\u003e \u003cp\u003e22.3.3 Algebraic Reynolds stress models 615\u003c\/p\u003e \u003cp\u003e22.4 Some Exact One-dimensional Solutions 618\u003c\/p\u003e \u003cp\u003e22.4.1 Solutions to the linear convection-diffusion equation 618\u003c\/p\u003e \u003cp\u003e22.4.2 Solutions to Burgers equation 620\u003c\/p\u003e \u003cp\u003e22.4.3 Other simple test cases 621\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 23 Discretization Methods for the Navier–Stokes Equations 624\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e23.1 Discretization of Viscous and Heat Conduction Terms 625\u003c\/p\u003e \u003cp\u003e23.2 Time-dependent Methods for Compressible Navier–Stokes Equations 627\u003c\/p\u003e \u003cp\u003e23.2.1 First-order explicit central schemes 628\u003c\/p\u003e \u003cp\u003e23.2.2 One-step Lax–Wendroff schemes 629\u003c\/p\u003e \u003cp\u003e23.2.3 Two-step Lax–Wendroff schemes 630\u003c\/p\u003e \u003cp\u003e23.2.4 Central schemes with separate space and time discretization 636\u003c\/p\u003e \u003cp\u003e23.2.5 Upwind schemes 648\u003c\/p\u003e \u003cp\u003e23.3 Discretization of the Incompressible Navier–Stokes Equations 654\u003c\/p\u003e \u003cp\u003e23.3.1 Incompressible Navier–Stokes equations 654\u003c\/p\u003e \u003cp\u003e23.3.2 Pseudo-compressibility method 656\u003c\/p\u003e \u003cp\u003e23.3.3 Pressure correction methods 661\u003c\/p\u003e \u003cp\u003e23.3.4 Selection of the space discretization 666\u003c\/p\u003e \u003cp\u003e23.4 Conclusions to Part VII 674\u003c\/p\u003e \u003cp\u003eIndex 685\u003c\/p\u003e  \u003cp\u003eCharles Sidney Hirsch was an American forensic pathologist who served as the Chief Medical Examiner of New York City from 1989 until 2013. He oversaw the identification of victims from the World Trade Center attacks in 2001.  Numerical Computation of Internal and External Flows Volume 2: Computational Methods for Inviscid and Viscous Flows C. Hirsch, Vrije Universiteit Brussel, Brussels, Belgium This second volume deals with the applications of computational methods to the problems of fluid dynamics. It complements the first volume to provide an excellent reference source in this vital and fast growing area. The author includes material on the numerical computation of potential flows and on the most up-to-date methods for Euler and Navier-Stokes equations. The coverage is comprehensive and includes detailed discussion of numerical techniques and algorithms, including implementation topics such as boundary conditions. Problems are given at the end of each chapter and there are comprehensive reference lists. Of increasing interest, the subject has powerful implications in such crucial fields as aeronautics and industrial fluid dynamics. Striking a balance between theory and application, the combined volumes will be useful for an increasing number of courses, as well as to practitioners and researchers in computational fluid dynamics. Contents Preface Nomenclature Part V: The Numerical Computation of Potential Flows Chapter 13 The Mathematical Formulations of the Potential Flow Model Chapter 14 The Discretization of the Subsonic Potential Equation Chapter 15 The Computation of Stationary Transonic Potential Flows Part VI: The Numerical Solution of the System of Euler Equations Chapter 16 The Mathematical Formulation of the System of Euler Equations Chapter 17 The Lax - Wendroff Family of Space-centred Schemes Chapter 18 The Central Schemes with Independent Time Integration Chapter 19 The Treatment of Boundary Conditions Chapter 20 Upwind Schemes for the Euler Equations Chapter 21 Second-order Upwind and High-resolution Schemes Part VII: The Numerical Solution of the Navier-Stokes Equations Chapter 22 The Properties of the System of Navier-Stokes Equations Chapter 23 Discretization Methods for the Navier-Stokes Equations Index\u003c\/p\u003e","brand":"Wiley","offers":[{"title":"Default Title","offer_id":47989702557925,"sku":"NP9780471924524","price":308.95,"currency_code":"USD","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/1842\/7735\/files\/9780471924524.jpg?v=1761785168","url":"https:\/\/k12savings.com\/products\/numerical-computation-of-internal-and-external-flows-volume-2-isbn-9780471924524","provider":"K12savings","version":"1.0","type":"link"}