Fluid dynamics and Plasma physics AA 363 2:0 credits (Aug-Dec 2018)
Tues & Thurs 11:30-13:00 @ Multimedia room, Physics Department, IISc
Prerequisites: Mathematical physics, vector calculus, complex analysis, thermodynamics and statistical physics, electrodynamics
Recommended resources and books
- AA 363 Aug-Dec 2017 Course summary (Fluids)
- Elementary Fluid Dynamics, D. J. Acheson, Clarendon Press, 1990
- An Introduction to Fluid Dynamics, George Batchelor, Cambridge University Press, 1967, 2000
- Fluid Mechanics (2nd edition), Landau & Lifshitz, Course of theoretical physics vol. 6, Butterworth Heineman, 1987
- Geophysical Fluid dynamics, Joseph Pedlosky, Springer-Verlag, 1979
- The physics of fluids and plasmas, A. R. Choudhuri, Cambridge university press, 1998
- Lecture notes by Prasad Subramanian, IISER, Pune
- Computational Fluid Dynamics, J. D. Anderson Jr, McGraw-Hill Education, 1995
- Elements of gas dynamics and the classical theory of shock waves. Ya. B. Zeldovich and Yu. P. Raizer, Academic Press, 1968
Grading
- Homework assignments carrying a weightage of 30%. HW assignments will be roughly 1-2 problems per lecture which will be a combination of reading, derivations, concepts and numerics, plotting etc. I prefer that assignments are always turned in as hard copies including plots.
- [Friendly advice] I strongly recommend you keep working consistently on the homework given in the class rather than working on all of it at the end of the semester.
- [04-Oct-2018] Deadline for submission of homework problems set-1
- There may be brownie point questions with each HW assignment to increase your chances of a higher grade. These brownie points can be redeemed for marks in the two exams.
Course summary
- Lecture-1 (07-Aug): Fluid as collection of classical neutral particles (hard spheres) obeying Newton's laws and undergoing elastic collisions, Derivation of Boltzmann transport equation and motivating the collisional integral. Boltzmann H-theorem and irreversability in statistical ensembles. Particle collisions as Markov processes and numerical example (distribution of candies amongst two children) of a transition from an initial non-equilibrium state to the equilibrium state to demonstrate how DH/dt < 0 . Cars on a winding road as collisionless particles. Example of lens exposure times in moving light photography to demonstrate the time scales and length scales above which the collection of particles can be treated as continuum. [Choudhuri, personal notes]
- Lecture-2 (14-Aug): Definition of density using the Boltzmann distribution function. Concepts of Eulerian and Lagrangian approaches, total and advective derivatives, conservative and non-conservative forms. Derivation of the continuity equation (mass conservation) using both Eulerian and Lagrangian approaches. Momentum conservation using Eulerian approach. [Anderson]
- Lecture-3 (16-Aug): Derivation of the equation for conservation of energy, primitive equations (velocity and entropy), Approximations depending on the behavior of equation of continuity: Incompressible, Anelastic and Boussinesq approximations, Define Mach number and how above approximations allieviate time step constraints due to sound speed. How the choice of approximation depends on the system of interest we have to numerically simulate. [personal notes]
- Lecture-4 (21-Aug): Fun problem: Can steady rain drizzle be treated as fluid? Anelastic approximation is equivalent to filtering of sound waves by assuming sound speed is infinite compared to any flow speed. Streamlines, Particle pathlines, Streaklines. Streamlines and streaklines are snapshots in time. Streamlines, Particle pathlines lines and streaklines are different for unsteady flows. Streamlines do not cross in space whereas streaklines can intersect when the origin point for one streakline lies on another streakline. Streamfunction for 2D flows: Exact differential, independent of path of integration. [Batchelor, personal notes]
- Lecture-5 (23-Aug): Streamlines as mathematical constructs whereas particle pathlines, streaklines and timelines as flow visualization methods used in laboratory. Various ways of physically understanding the concept of streamfunction (exact differential, curve to which instantaneous velocity vector is a tangent, mass flux across an area perpendicular to streamlines). Gauge freedom in defining the streamfunction. Concept of streamtube. Zero mass flux across walls of a stream tube. Introduction of line vortex and how it differs from rigid body rotation.
- Lecture-6 (28-Aug): Incompressible and invisid flows, Bernoulli's theorem, Conserved quantities along streamlines for incompressible and steady flows, Vorticity equation, Streamtubes and "Frozen-in" Vortex tubes, Kelvin's circulation theorem and Helmholtz vortex theorems. Examples of line vortex and smoke rings to differentiate between streamtube and vortex tubes in a flow. Vortex tubes are bounded by curves which are fluid "material lines". Neither streamtubes nor vortex tubes can arbitraritly end inside the volume of a fluid. They can close on themselves or end at a boundary/interface. Introduce complex potentials for 2D flows [Acheson].
- Lecture-7 (30-Aug): Vortex rings and their destruction in fluid with small (but non-zero) viscosity. The vortex stretching term in the vorticity equation plays a vital role in the stretch-twist-fold cycle of vortex ring destruction. Vortex stretching in intensification of hurricanes. Concept of kinetic helicity. The kinetic helicity is conserved for invisid flows (cf. Moffat, 1969). Complex potentials for invisid, irrotational and incompressible flows. The Milne Circle theorem and its use for flow across a cylindrical/spherical obstacle. The method of images and uniqueness of boundary conditions for the solution of Poisson equation [Acheson]. [Note: Presence of viscosity will create vortices (due to interaction with rigid boundaries) in the domain so that complex potential method is rendered ineffective.]
- Lecture-8 (04-Sep): Rotating coordinate frames, Coriolis force. Vorticity equation in rotating coordinate frame, Kelvin's circulation in intertial frame vs in rotating frame. Ertel's potential vorticity (PV) conservation theorem (which is essentially Kelvin's vorticity derived for a very special surface) [Pedlosky]
- Lecture-9 (06-Sep): Rossby number. Low and high Rossby number flows, Bathtub vortex experiment in northern and southern hemisphere. Introduce aspect ratio, D/L to compare vertical scales to horizontal scales in planetary atmosphere. Thermal wind balance equations, Taylor-Proudman constraint (No variation of velocity components perpendicular to Ω in the direction parallel to Ω. [Pedlosky]
- Lecture-10 (11-Sep): Geostrophic balance in the velocity equation is a manifestaion of the "Thermal wind balance" in the vorticity equation for low Rossby number flows. Introduction to shallow water equations under the assumption of incompressibility (implying barotropy). Applications for proto-planetary and accretion disks as well as thin planetary atmospheres and stellar internal structure/convection zone etc. [Pedlosky]
- Lecture-11 (18-Sep): Derivation of linearized shallow water equations independent of vertical coordinate, z in accordance with Taylor Proudman balance for low Rossby number flows. Derivation for expression of potential vorticity: Π=(ζ+f)/ρH, where f, is the planetary vorticity and ζ is the relative vorticity component along the vertical. Rossby waves are excited due to conservation of potential vorticity. Conceptual demostration of why Rossby waves propagate opposite to direction of rotation velocity (westward in Earth's and Sun's atmosphere). Chart for classification of waves in shallow water fluid models [Poincare, Kelvin, Rossby, Surface gravity] and deep water model [Gravity (g-modes), surface gravity (f-modes), acoustic (p-modes), Inertia-gravity, Kelvin, Rossby]. [Pedlosky, personal notes]
- Lecture-12 (20-Sep): f-plane approximation for vertical component of planetary vorticity: f=f_{0}+βy, β=2Ω/R, where R is the radius of the planet and Ω is the angular rotation frequency. Derivation of dispersion relations (eigenvalues) and spatial structure (eigenfunctions) of different modes/waves possible in a shallow water f-plane model away from equator: Poincare, Coastal Kelvin and quasigeotrophic Rossby waves. Only for Rossby waves are the winds in approximate geostrophic balance and a slight imbalance from geostrophy provokes Rossby waves. At the equator (f=0) the waves possible in a shallow water layer: Poincare, equatorial Rossby and Kelvin. No geostrophic balance possible at the equator as f_{0}=0. The equator acts as a waveguide for waves as their amplitude decreases away from the equator. The eigenfunctions for equatorial Poincare and Rossby wave are given by H_{n}(Y) e^{-Y2/2}, where Y=(β/ √ gH )^{1/2} y and H_{n}(Y) is Hermite polynomial of degree n and y is the length in meridional direction. Dispersion curves for planetary waves at the mid-latitude and equator. Rossby waves always propagate westward where as equatorial Kelvin waves propagate eastward. Important role played by equatorial Rossby and Kelvin waves in weather patterns like El Nino and Indian summer monsoon. [Pedlosky, personal notes]