x ASTR 670 • Hydrodynamics by Benedikt Diemer (University of Maryland) Updated May 10, 2021 Contents 0 About these notes4 0.1 Acknowledgments and commonly used references.................... 4 0.2 Notation........................................... 4 1 Introduction: What is hydrodynamics?7 1.1 What is a fluid?....................................... 7 1.2 Astrophysical fluids..................................... 8 2 From particles to fluids9 2.1 Averaging over particles .................................. 9 2.2 Fluid Quantities ...................................... 9 2.3 Towards the fluid equations: a schematic outline .................... 11 2.4 Kinematical concepts.................................... 12 3 The Euler equations 13 3.1 Eulerian and Lagrangian derivatives ........................... 13 3.2 The continuity (density) equation............................. 14 3.3 The velocity equation ................................... 15 3.4 The internal energy equation ............................... 16 3.5 The equation of state.................................... 17 3.6 Summary of the Euler equations ............................. 18 4 Equilibrium and steady flows 20 4.1 Hydrostatic equilibrium .................................. 20 4.2 Adiabatic flows....................................... 21 4.3 Barotropic flows and Bernoulli’s principle ........................ 22 4.4 The limits of inviscid fluid dynamics: d’Alembert’s paradox.............. 24 5 Sound waves 26 5.1 The linearized Euler equations .............................. 26 5.2 The properties of sound waves............................... 27 6 Computational hydro I: Theoretical background 29 6.1 The self-similarity of hydrodynamics........................... 29 6.2 The conservation-law form of the equations ....................... 30 6.3 The advection equation .................................. 32 6.4 A very basic taxonomy of partial differential equations................. 32 6.5 Computing in space and time: finite difference vs. finite volume............ 33 6.6 The Courant-Friedrichs-Lewy condition ......................... 34 6.7 Computational error terms ................................ 35 7 Computational hydro II: Finite-differencing schemes 36 7.1 First-order schemes for the advection equation ..................... 36 7.2 Stability analysis...................................... 38 7.3 Understanding numerical diffusion ............................ 40 7.4 Higher-order finite-difference schemes........................... 41 8 Shocks 43 8.1 Why shocks arise: Burgers’ equation........................... 43 8.2 What is a shock?...................................... 44 2 ASTR 670 • Hydrodynamics3 8.3 The Rankine-Hugoniot conditions............................. 46 8.4 Ideal gas shocks....................................... 47 8.5 Isothermal shocks...................................... 49 8.6 Supernova blast waves................................... 49 9 Computational hydro III: Finite-volume schemes 53 9.1 Godunov schemes...................................... 53 9.2 Riemann solvers ...................................... 55 9.3 The shocktube problem .................................. 59 9.4 Higher-order schemes: Reconstruction and slope limiters................ 60 9.5 Higher-order schemes: Time integration ......................... 63 9.6 Multiple dimensions .................................... 64 9.7 Popular hydrodynamics codes in astrophysics...................... 64 10 Fluid instabilities 66 10.1 The Jeans instability.................................... 66 10.2 Perturbations of a general two-fluid interface ...................... 68 10.3 Surface gravity waves ................................... 69 10.4 The Rayleigh-Taylor instability.............................. 70 10.5 The Kelvin-Helmholtz instability............................. 70 11 Magnetohydrodynamics 72 11.1 The interaction between fluids and electromagnetism.................. 72 11.2 The two-fluid approach................................... 73 11.3 Eliminating electric fields and currents.......................... 74 11.4 The equations of ideal MHD................................ 75 11.5 Basic MHD dynamics: flux freezing and dynamos.................... 77 11.6 MHD waves......................................... 78 11.7 Observing magnetic fields via the rotation measure................... 79 12 Turbulence 80 12.1 Self-similarity and the turbulent cascade......................... 80 12.2 The Kolmogorov-Obukhov law .............................. 81 A Mathematical Background 84 A.1 Vector operators ...................................... 84 A.2 Index notation and tensors ................................ 84 A.3 Vector identities ...................................... 85 A.4 Eigenvalues and eigenvectors ............................... 86 B Derivations 87 B.1 Conservation-law form of the Euler equations...................... 87 B.2 The matrix form of the fluid equations and the eigenvalue perspective . 88 B.3 Dispersion relation of perturbations at a two-fluid interface .............. 90 C Ulula: a lightweight hydro code in Python 93 References 94 ASTR 670 • Hydrodynamics4 0 About these notes These notes serve as a guide to ASTR 670, a graduate class in fluid dynamics and the interstellar medium (ISM) at the University of Maryland. Since hydrodynamics accounts for only half the course (about 12 lectures), we have to omit numerous important topics and interesting details. The goal of the class is to develop an intuitive, physical understanding for fluid dynamics rather than to give an exhaustive treatment of the field. In particular, we will skip lengthy, mathematical derivations and refer to textbooks instead. In the spirit of leaning how to tackle the kinds of hydrodynamics problems that arise in the practice of astrophysics, a significant portion of the course is devoted to modern computational methods. The numerical chapters are deliberately interspersed with physics-focused ones, but the topics could also be taught in a different order. 0.1 Acknowledgments and commonly used references My treatment of hydrodynamics is, of course, based on the much more substantial works of many colleagues. The most commonly cited text is the excellent book by Clarke & Carswell (2014, hereafterCC), while the classic text of Shu (1992, hereafter Shu) provides a comprehensive mathe- matical background. I have also made extensive use of the outstanding lecture notes by Frank van den Bosch (2020, hereafter van den Bosch; you can find the notes on his website). For the computational aspects of the course, I have built on the innovative computational hydrodynamics book by Michael Zingale (2021, hereafter Zingale; the book is publicly available on github). I have also used his Pyro teaching code (Zingale 2014) and code snippets from my colleague Derek Richardson’s computational astrophysics course. Toro (2009, hereafter Toro) provides a comprehensive reference for the mathematical underpinnings of the numerical methods discussed. Finally, I am grateful to my colleague Alberto Bolatto who taught this course before me, and whose notes provided the basis for my version of the class. I have taken lots of inspiration from Andrey Kravtsov’s graduate course at the University of Chicago, which introduced me to numerical hydrodynamics during my PhD. The image on the front page shows a simulation of the Kelvin-Helmholtz instability created with the moving-mesh code Arepo (Springel 2010), using initial conditions and a plotting script provided by Philip Mocz. 0.2 Notation It is virtually impossible to find two texts on hydrodynamics that agree on the same set of symbols. Is pressure denoted p or P ? Is m the mass of a particle or the mass in a fluid element? Is the fluid velocity denoted v or u? There are no correct answers, of course, but we will try to use the symbols listed in Tables1 and2 consistently. Throughout the notes, bold symbols denote 3-vectors, e.g., u = (ux, uy, uz); we use row and column notation interchangeably. We will try to use vector notation throughout, but sometimes the index summation notation is clearer (see §A.2). ASTR 670 • Hydrodynamics5 Sym. Units Meaning § α cfl — √ CFL number 6.6 B Tesla or 4π Gauss Magnetic field 11.1 cs cm/s Sound speed 5.2 E erg/cm3 Total energy per unit volume in a fluid element 2.2 E Volt/m Electric field 11.1 ε erg/g = cm2/s2 Thermal (“internal”) energy per unit mass in a fluid el. 2.2 η g/cm/s Dynamical coefficient of viscosity 12.1 f s−1 Frequency 5.2 F (multiple) Vector of flux terms 6.2 Φ erg/g = cm2/s2 Gravitational potential 2.2 γ — Ideal gas equation of state parameter 3.5 Γ erg/cm3/s Heating rate per unit volume 3.4 H erg/g = cm2/s2 Bernoulli constant 4.3 I — Identity matrix (usually 3 × 3) — J Ampere Electric current 11.1 k cm−1 Wave vector (with amplitude k = 2π/λ) 5.2 K g1−γ cm3γ−1 s−2 Constant in barotropic equation of state 4.2 κ cm2/s Diffusivity 6.4 λ cm Wavelength 5.2 λk (depends) Eigenvalues of a matrix A.4 Λ erg/cm3/s Cooling rate per unit volume 3.4 λmfp cm Mean free path of particles 2.1 m g Mass in a fluid element 2.2 mptl g Mass of individual particles 2.2 M — Mach number 8.2 µ — Mass of particles in units of proton mass mp 3.5 n cm−3 Number density 2.1 N — Number of some species in a fluid element 2.1 ω s−1 Angular wave frequency (ω = 2πf) 5.2 ω s−1 Vorticity, the curl of the bulk velocity 4.3 p g cm/s Momentum of particles 2.1 P dyne/cm2 = erg/cm3 Pressure in a fluid element 2.2 Π erg/cm3 Momentum flux density tensor B.1 q Coulomb Net charge of fluid 11.1 q (depends) Position in Lagrangian space 3.1 Q (unknown) Unspecified