Theoretical Fluid Dynamics

This textbook gives an introduction to fluid dynamics based on flows for which analytical solutions exist, like individual vortices, vortex streets, vortex sheets, accretions disks, wakes, jets, cavities, shallow water waves, bores, tides, linear and non-linear free-surface waves, capillary waves, internal gravity waves and shocks.

Advanced mathematical techniques ("calculus") are introduced and applied to obtain these solutions, mostly from complex function theory (Schwarz-Christoffel theorem and Wiener-Hopf technique), exterior calculus, singularity theory, asymptotic analysis, the theory of linear and nonlinear integral equations and the theory of characteristics.

Many of the derivations, so far contained only in research journals, are made available here to a wider public.

Auteur

Achim Feldmeier is an astrophysicist and obtained his PhD in astronomy from Ludwig-Maximilians-Universität in München in 1994. He was postdoc at the University of Kentucky in Lexington and at Imperial College in London.

Since 2000 he works at the Universität Potsdam, where he is apl professor since 2006. He gave numerous courses in hydrodynamics and his research work is on flow properties of stellar winds.

Contenu

1 Description of fluids 51.1 Euler and Lagrange picture . . . . . . . . . . . . . . . . . . . . 51.2 Lagrange derivative . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Divergence-free vector field . . . . . . . . . . . . . . . . . . . . 101.5 Fluid boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.6 Phase space fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.7 Moving fluid line . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.8 Internal fluid stress . . . . . . . . . . . . . . . . . . . . . . . . . 181.9 Fluid equations from kinetic theory . . . . . . . . . . . . . . . 291.10 Streamlines and Pathlines . . . . . . . . . . . . . . . . . . . . . 321.11 Vortex line, vortex tube and line vortex . . . . . . . . . . . . . 331.12 Vortex sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371.13 Vector gradient in cylindrical coordinates . . . . . . . . . . . . 391.14 Vector gradient in orthogonal coordinates . . . . . . . . . . . . 411.15 Vorticity equation . . . . . . . . . . . . . . . . . . . . . . . . . . 451.16 Velocity from vorticity . . . . . . . . . . . . . . . . . . . . . . . . 471.17 Bernoulli equation . . . . . . . . . . . . . . . . . . . . . . . . . . 501.18 Euler-Lagrange equation for fluids . . . . . . . . . . . . . . . . 521.19 Water waves from Euler-Lagrange equations . . . . . . . . . . 581.20 Stretching in an isotropic random velocity field . . . . . . . . 631.21 Converse Poincaré lemma . . . . . . . . . . . . . . . . . . . . . 65 2 Flows in the complex plane 792.1 Laplace equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 792.2 Green's theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 812.3 Dirichlet and Neumann boundary conditions . . . . . . . . . . 822.4 Mean value and maximum property . . . . . . . . . . . . . . . 832.5 Logarithmic potential . . . . . . . . . . . . . . . . . . . . . . . . 852.6 Dirichlet's principle . . . . . . . . . . . . . . . . . . . . . . . . . 882.7 Streamfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 922.8 Vorticity on a sphere . . . . . . . . . . . . . . . . . . . . . . . . . 942.9 Complex speed and potential . . . . . . . . . . . . . . . . . . . 962.10 Analytic functions, conformal transformation . . . . . . . . . 982.11 Schwarz-Christoffel theorem . . . . . . . . . . . . . . . . . . . . 1002.12 Mapping of semi-infinite and infinite strips . . . . . . . . . . . 1062.13 Riemann surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3 Vortices, corner flow and flow past plates 1173.1 Straight vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1173.2 Corner flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1203.3 Corner flow with viscosity . . . . . . . . . . . . . . . . . . . . . 1223.4 Flow past a flat plate . . . . . . . . . . . . . . . . . . . . . . . . 1293.5 Blasius and Kutta-Jukowski theorems . . . . . . . . . . . . . . 1323.6 Plane flow past a cylinder . . . . . . . . . . . . . . . . . . . . . 1353.7 Kármán vortex street . . . . . . . . . . . . . . . . . . . . . . . . 1373.8 Corner eddy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1503.9 Angular momentum transport . . . . . . . . . . . . . . . . . . . 155 4 Jets, wakes and cavities 1634.1 Free streamlines . . . . . . . . . . . . . . . . . . . . . . . . . . . 1634.2 Flow past a step . . . . . . . . . . . . . . . . . . . . . . . . . . . 1674.3 Complex potential and speed plane . . . . . . . . . . . . . . . . 1694.4 Outflow from an orifice . . . . . . . . . . . . . . . . . . . . . . . 1704.5 A simple wake model . . . . . . . . . . . . . . . . . . . . . . . . 1754.6 Riabouchinsky cavity . . . . . . . . . . . . . . . . . . . . . . . . 1814.7 Levi-Civita method . . . . . . . . . . . . . . . . . . . . . . . . . 1854.8 Kolscher's cusped cavity . . . . . . . . . . . . . . . . . . . . . . 1884.9 Re-entrant jet cavity . . . . . . . . . . . . . . . . . . . . . . . . . 1974.10 Tilted wedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1984.11 Weinstein theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 205 5 Kelvin-Helmholtz instability 2115.1 Kelvin-Helmholtz circulation theorem . . . . . . . . . . . . . . 2115.2 Bjerknes circulation theorem . . . . . . . . . . . . . . . . . . . 2175.3 Kelvin-Helmholtz instability . . . . . . . . . . . . . . . . . . . . 2205.4 Vortex chain perturbation . . . . . . . . . . . . . . . . . . . . . 2225.5 Vortex accumulation . . . . . . . . . . . . . . . . . . . . . . . . . 2265.6 Linear stability analysis . . . . . . . . . . . . . . . . . . . . . . 2305.7 Birkhoff-Rott equation for vortex sheets . . . . . . . . . . . . . 2355.8 Curvature singularity in evolving vortex sheet . . . . . . . . . 2395.9 Subsequent work on Moore's singularity . . . . . . . . . . . . 2545.10 Nonlinear stages of K-H instability . . . . . . . . . . . . . . . . 2575.11 Why do large eddies occur in fast flows? . . . . . . . . . . . . . 2595.12 Atmospheric instability . . . . . . . . . . . . . . . . . . . . . . . 2625.13 Rayleigh inflexion theorem . . . . . . . . . . . . . . . . . . . . . 2645.14 Kinematics of vortex rings . . . . . . . . . . . . . . . . . . . . . 2665.15 Curvature and torsion . . . . . . . . . . . . . . . . . . . . . . . . 2695.16 Helical line vortices . . . . . . . . . . . . . . . . . . . . . . . . . 2715.17 Knotted and linked vortex rings . . . . . . . . . . . . . . . . . . 2745.18 Clebsch coordinates and knottedness . . . . . . . . . . . . . . . 278 6 Kinematics of waves 2796.1 Wave basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2806.2 Group speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2816.3 Kinematic waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 2866.4 The wavefront . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2876.5 Waves and instability from a radiative force . . . . . . . . . . 289 7 Shallow water waves 2997.1 Continuity equation . . . . . . . . . . . . . . . . . . . . . . . . . 3007.2 Euler equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3037.3 Wave equation for linear water waves . . . . . . . . . . . . . . 3047.4 Tides in canals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3077.5 Cotidal lines and amphidromic points . . . . . . . . . . . . . . 3127.6 Waves of finite amplitude . . . . . . . . . . . . . . . . . . . . . . 3177.7 Nonlinear tides in an estuary . . . . . . . . . . . . . . . . . . . 3217.8 Similarity solution: dam break . . . . . . . . . . . . . . . . . . 3297.9 Non-breaking waves . . . . . . . . . . . . . . . . . . . . . . . . . 3347.10 Bores . …