Turbulence is everywhere—in the interiors and atmospheres of planets and stars, galaxies, biological systems including human body, engineering flows, etc.   In this course, we will cover fundamental aspects of turbulence—Kolmogorov’s theory of turbulence in spectral and real space; two-dimensional turbulence; energy transfers; enstrophy and kinetic helicity cascades; more complex applications, such as passive scalar, turbulent thermal convection, and magnetohydrodynamic turbulence.



1.    M. K. Verma, Energy transfers in Fluid Flows, Cambridge Univ. Press, 2019


1.    M. K. Verma, Physics of Buoyant Flows, World Scientific, 2018.
2.    S. B. Pope, Turbulent Flows, Cambridge Univ. Press, 2000.
3.    Research papers


Based on homework, exam, projects (to be discussed in the first class)


Week -1 (29/7-4/8): Introduction; Basic equations of hydrodynamics in real space; Conservation laws (1,2a,2b,2c,2d)

Week -2 (5/8-11/8): Fourier Space Description of Hydrodynamics (3a-3e)

Week -3 (13/8-18/8): Fourier description contd.  Craya-Herring basis (4a-4c)

Week -4 (19/8-25/8): Instabilities (5a-5d)

Weak-5 (26/8-1/9): Saturation of nonlinearity; Patterns (5e-5f)

Week-6 (2/9-9/9): Energy transfers in fluid flows (6a-6d); 

Week-7 (2/9-13/9): Revision, Travel; 14/9: Discussion session; 

16/9-22/9: Midsem exam

Week-8 (23/9-29/9): Kolmogorov’s theory of turbulence (in Fourier space) (7)

Week-9 (30/9-6/10): Kolmogorov’s theory of turbulence (in real space) (8)

7/10:13/10: Midsem break

Week -10 (14/10-20/10): Enstrophy; Two-dimensional turbulence; Kinetic helicity (9)

Week -11 (21/10-27/10): Turbulence with a scalar; Passive scalar; (10a, 10b)

Week-12 (28/10-3/11): Turbulent thermal convection (10c, 10d)

Week -13 (4/11-10/11): Turbulence with a vector; Magnetohydrodynamic turbulence (11)

Week-14 (11/11-17/11): Revision