Fluid dynamics and Plasma physics AA 363 2:0 credits (Fall 2017)
Tues & Thurs 11:30-13:00 @ LH2, Physics Department, IISc
Recommended books
- 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 gasdynamics and the classical theory of shock waves. Ya. B. Zeldovich and Yu. P. Raizer, Academic Press, 1968
Grading
- My portion (fluids) consists of 2/3 of the entire course and will have a similar weightage in the final grading. The following percentages are applicable to the 2/3 part only.
- 2 Homework assignments carrying a weightage of 50%. The due date, 2 weeks later is a hard deadline and any relaxation is my will alone. HW assignments will be a combination of derivations, concepts and numerics, plotting etc. I prefer that assignments are always turned in as hard copies including plots. Soft copies will be considered only in exceptional cases when the student cannot make it to the class.
- [Scheduled on Oct, 30 at 1430 - 1700 hrs, venue: LH-2] 1 final exam (for fluids alone) carrying 50% weightage. You are allowed only one A4 size cheat sheet with your hand written information.
- 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 final and only exam.
Course summary
- Lecture-1 (08-Aug): Why study fluids? Fluid dynamics as approximation to kinetic theory under certain conditions. Concepts of Eulerian and Lagrangian approaches, Substantial and convective derivatives, conservative and non-conservative forms. Derivation of the continuity equation (mass conservation) using Lagrangian approach. Fully compressible and anelastic approximations (sound speed filtering). [Anderson]
- Lecture-2 (10-Aug): Continuity equation using Eulerian approach. Velocity equation using Lagrangian approach. Incompressible flows, Bernoulli's theorem, streamlines and 2D stream function. Conserved quantities along streamlines for incompressible and steady flows. [Anderson, Acheson]
- Lecture-3 (17-Aug): Energy conservation equation using Eulerian approach, vorticity equation, vortex stretching term in 3D, vorticity conservation in 2D incompressible fluids, Prandtl-Batchelor's theorem for closed streamlines in 2D. Kelvin's circulation theorem, frozen-in vorticity, flows in rotating frames-introduction. [Acheson, Pedlosky]
- Lecture-4 (22-Aug):
21-Aug-2017 Solar Eclipse special
- Lecture-5 (24-Aug) : Flows in rotating reference frames, Rossby number, Effect of planetary rotation on large scale flows, Conservation of flux of absolute vorticity (circulation), Bathtub vortex-experiment by Austrian physicist Ottokar Tumlirz, Interesting problem about two drains in a large tank of still water in northern hemisphere (proposed by a student), Baroclinic flows, Taylor-Proudman theorem. [Pedlosky]
- Lecture-6 (28-Aug) : Ertel's potential vorticity theorem, conservation of potential vorticity under special conditions, derivation of a simple expression for potential vorticity using shallow water equations for incompressible fluids, Rossby waves due to conservation of potential vorticity for low Rossby number flows.[Pedlosky]
- Lecture-7 (31-Aug) : Shallow water equations, geostrophic balance (Coriolis force balancing horizontal pressure gradients) and thermal wind relation, Equatorial shallow water waves - Poincare waves, Kelvin waves and Rossby waves, dispersion relation and eigen functions (Hermite polynomials).
- Lecture-8 (05-Sep) : Kind of waves in the fluid not affected by rotation: Surface gravity waves, internal gravity waves, derivation of dispersion relations, derivation of Brunt-Vaisala frequency and stability of fluid stratification in presence of gravity. [Pedlosky, Acheson]
- Lecture-9 (07-Sep) :
Helioseismology special lecture
Gravity (g) modes, acoustic (p) modes and surface gravity (f) standing wave modes on the Sun. Only p and f modes are observable using "Dopplerograms". How helioseismology played a key role in resolution of the solar neutrino problem. Helioseismology is used as a probe of the internal structure and flows in the interior of the Sun. In presence of rotation, the acoustic waves travelling in prograde and retrograde directions have different frequencies (splitting) and speeds.
link to presentation - Lecture-10 (12-Sep) : Instability in stationary fluids (Rayleigh Taylor Instability, examples: Crab nebula, mushroom clouds) and fluids in shearing motion (Kelvin-Helmholtz Instablity, examples: Jupiter's spot, Jet stream). Vorticity is intensified due to the baroclinic term for unstable stratification (heavier fluid on top) but decays for the case stable stratification (heavier fluid at the bottom). Derivation of instability criteria at an interface between two fluids and inside stratified fluids. Critical Richardson number (Ri < 0.25). [Acheson] HW-1 tutorial (Q2,Q4).
- Lecture-11 (14-Sep): Summary of waves and instabilities. Definition of second rank stress tensor and the viscous stress tensor. Click this link for proof of symmetry of the viscous stress tensor wrt indices i,j.(page nos. 8 to 10). Newtonian and non-Newtonian fluids. Derivation of the form of viscous stress tensor as a linear function of the rate-of-strain tensor using symmetry, isotropy and homogenity arguments. [Batchelor, Landau & Lifshitz]
- Lecture-12 (21-Sep) Derivation of the Navier-Stokes equation, Relevance of second (or bulk) viscosity at high frequencies (vibrational or rotational excitation in molecules) and inside shocks (with large divergence of velocity), Boundary layers as consequence of viscosity and the no-slip boundary condition, Definiton of dynamic and kinematic viscosity coefficients. [Batchelor, Landau & Lifshitz]
- Lecture-13 (26-Sep) Derivation of viscous dissipation for incompressible fluids, Dependence of dynamic viscosity on square-root of temperature for gases, Fluid viscosity decreases with increasing temperature due to decrease of intermolecular forces (e.g., breaking of hydrogen bonds in water), 2D flow around a cylindrical obstacle using Milne-Thompson circle theorem and in presence of a line vortex placed at the center of the cylinder, Kutta-Joukowski Lift theorem. [Acheson]
- Lecture-14 (05-Oct) Stokes flow around a cylinder and laminar boundary layers, favorable and adverse pressure gradients, flow reversal,
boundary layer separation at Re>5,
Von-Karman vortex sheet due to interaction of shear layers and vortex shedding in the wake, Strouhal number,
Oscillations induced by flow instabilities.[Acheson, Batchelor]
Transition to turbulence at high Reynolds number by mechanism of fluid instabilities like KH instability, Rayleigh-Taylor or shock instability. Turbulent boundary layer has greater mixing due to greater momentum transfer between layers at molecular scales as opposed to laminar boundary layers where flow remains in layers or "lamina". Concepts of injection scale (largest length scale), inertial and dissipation scales in fully developed turbulence. Derivation of the Kolmogorov length scale. Turbulent eddies defined as correlation length scales in the fluid.
Turbulence causes "turbulent stresses" due to the second order correlations of fluctuation in flows (a.k.a Reynolds stresses). Reynolds stresses often modelled as proportional to the gradient of mean velocity, the proportionality constant being "turbulent viscosity". Mean field theory uses this formulation to model the effect of small scale turbulence on the mean flows (as opposed to direct numerical simulation using only molecular viscosity).
Energy cascading from larger to smaller scales in 3D turbulence due to the effect of "vortex stretching" and "dissipation term" in the vorticity equation. Absence of vortex stretching in 2D leads to inverse cascade.
- Lecture-15 (10-Oct) Convective instability revisited: Schwarzchild criteria - Adiabatic parcel rising under pressure equilibrium. Convection in fluids with viscosity: Rayleigh number and critical Rayleigh number. Arguments for existence of a critical Rayleigh number. Stellar convection, radiative flux and its dependence on temperature gradient and opacity, cool starts have convective envelope while hot starts have convective cores due to large opacity in convection zones.
- Lecture-16 (12-Oct) Rayleigh-Benard convection cells, Mixing length theory of stellar convection, mixing length is the height traversed by a convective eddy in the unstable atmosphere before giving up its heat to neighbors and dissipating, mixing length parameter defined as ratio of mixing length and pressure scale height, definition of super-adiabaticity, very small deviation from adiabatic stratification (1/10^7) needed for a sun like star to transport entire flux by convection in the top 30% radius, where as a large opacity in the solar convection zone makes it very difficult for radiation to carry the solar flux outward to the surface. The convective flux is much larger than radiative flux in the solar convection zone. Link for estimates done in class
- Lecture-17 (17-Oct) Gas dynamics, Introduction to the Method of characteristics and its use, Riemann invarients: J+ (along C+), J-(along C-). C+ and C- family of characteristics curves are normal to each other. Determination of fluid velocity and sound speed at any point in time using characteristic curves and initial conditions, For smooth fluid flow profiles curves in each family remain "parallel" to each other. Shocks are formed in the fluid when curves of the same family (say C+) intersect with each other. This is due to the non-linear nature of the advection term in the E uler's equation. Book (Zel'dovich and Raizer) parts covering the above material
- Lecture-18 (19-Oct) Emission of sound waves due to flow itself: Proof of Lighthill mechanism, Oblique shocks, maximum turning angle and detachment of shocks from object to form bow shocks. [Choudhuri, Acheson]
- Lecture-19 (24-Oct) Derivation of Rankine-Hugoniot jump conditions for normal shocks, De Laval nozzle and the twin exhaust model of Blandford and Rees, Blast waves (supernova, detonation wave), Solar wind: Parker's steady state solution [Choudhuri, Acheson]
- Lecture-20 (26-Oct) Examination tips and concept clarifications
- Lecture-21 (30-Oct) Final Examination