Math 812: Fall 2006

Intro to Stability and Turbulence in Shear Flows

Office hours: by appointment.

Prerequisites

REFERENCES: (no textbook required) I'll pick some inspiration from

• Turbulent Flows, Stephen B. Pope, Cambridge University Press (2000) (Engineering oriented)
• Hydrodynamic Instability, P.G. Drazin and W.H. Reid, Cambridge University Press (1981) (Classic reference for stability)
• Elementary Fluid Dynamics, D.J. Acheson, Oxford University Press. (Very readable, advanced undergrad intro to fluid dynamics)

SUGGESTED READING: (note this is not an endorsement of all of these papers, although most are recommended reading, some are there to test your critical thinking) [I have provided some links, but don't be too lazy and limit yourself to those! do a bit of `research' to get the other papers too]

Navier-Stokes Equations, and a few words about turbulence

• Origin, derivation, assumptions: We did it our way but most of that and some more is in
• Pope, Chapters 1, 2.
• See also Cauchy stress tensor, constitutive laws, Newtonian fluid, Kinetic Theory point of view (and a nice brief review of KT;), Kinetic Energy dissipation rate, Entropy production. Sound waves, speed of sound, Mach number; incompressible flow.

  For further information about the derivation of the Navier-Stokes equations see Chapter 6 in Acheson (especially read the 1st two pages of history) and/or the books by Batchelor (An introduction to fluid dynamics, Cambridge U. Press 1967) or Landau and Lifshitz `Fluid Mechanics', 2nd edition, Pergamon Press (1987) (usually pretty complete but a bit hard to read). The book by R.L. Panton `Incompressible Flow', Wiley 1996 is also quite good and well-known in engineering depts. The first 6 (!) chapters cover the material we reviewed in our first 3 lectures. Although it is called `incompressible flows' it contains good discussions/reviews of the continuum hypothesis, thermodynamics, indicial (or index) notation, vector calculus, etc. Any fluid mechanics book should have a derivation of NSE and a discussion of the Cauchy stress tensor. Yep, this is the same Cauchy as in Cauchy-Riemann, Cauchy's theorem, Cauchy sequences, etc.

  Some recent papers on boundary conditions for the Poisson equation for pressure in incompressible flow:

• Hans Johnston and Jian-Guo Liu, `Accurate, stable and efficient Navier-Stokes solvers based on explicit treatment of the pressure term', Journal of Computational Physics, Volume 199, Issue 1, 1 September 2004, Pages 221-259
• Dietmar Rempfer, `On Boundary Conditions for Incompressible Navier-Stokes Problems', Applied Mechanics Reviews -- May 2006 -- Volume 59, Issue 3, pp. 107-125
• Jian-Guo Liu, Jie Liu, Robert L. Pego, `On Incompressible Navier-Stokes Dynamics: A New Approach for Analysis and Computation', arxiv.org math.AP/0503212 (Mar 2005)

Part I: Turbulence Folklore, Inertial (Kolmogorov) scaling

• Pope, Chapter 6.
• K.R. Sreenivasan, `On the scaling of the turbulence energy dissipation rate' Physics of Fluids 27 , 1048-1051 (1984).
• K.R. Sreenivasan, `An update on the dissipation rate in homogeneous turbulence' Physics of Fluids 10 , 528-529 (1998).
• O. Cadot, Y. Couder, A. Daerr, S. Douady, and A. Tsinober, `Energy injection in closed turbulent flows: Stirring through boundary layers versus inertial stirring' Phys. Rev. E 56, 427-433 (1997)
• G. Gioia and F. A. Bombardelli, `Scaling and Similarity in Rough Channel Flows' Phys. Rev. Lett. 88, 014501 (2002)
• T.S. Lundgren, `A small-scale turbulence model,' Phys. Fluids A 5, 1472 (1993)
• T.S. Lundgren, `Kolmogorov turbulence by matched asymptotic expansions,' Phys. Fluids 15, 1074 (2003)
• J.L. Aragon, et al. Kolmogorov scaling in impassioned van Gogh paintings
• Local vs Nonlocal interactions
  • Pope remarks,section 6.7.1 and references therein
  • Google `turbulence nonlocal interactions', and see what research papers you find
  • Domaradzki and Rogallo, `Local energy transfer and nonlocal interactions in homogeneous, isotropic turbulence' Phys. Fluids A 2, 413 - 426 (1990)
  • Yeung and Brasseur, `The response of isotropic turbulence to isotropic and anisotropic forcing at the large scales' Phys Fluids A 3 , 884-897 (1991)
  • Waleffe, The nature of triads interactions in homogeneous turbulence Phys Fluids A 4 , 350-363 (1992)
    [SEE Sect VII. Interpretation, for a discussion of local-nonlocal interactions and the need to consider at least two triads in order to recover proper physics. That is for 3D NSE, we did it in class for Burgers and slightly differently]
  • Ohkitani and Kida, `Triad interactions in a forced turbulence' Phys Fluids A 4 , 794-802 (1992)
  • Remarks on a quasi-linear model of the Navier-Stokes equations, Fabian Waleffe, arXiv:math.AP/0409310, Sept 19, 2004. [This 4 page paper contains a derivation of the equations for the evolution of Kelvin modes on a general background flow with uniform gradients . ]
  • On transition scenarios in channel flows Mar 2005 Preprint, unfinished. Read Intro + sect 2 (general equations for stability of shear flows) + Sect 4 (Kelvin mode solution for pure shear)
• 2D Turbulence:
  • Inertial Ranges in Two-Dimensional Turbulence , R.H. Kraichnan, Physics of Fluids -- July 1967 -- Volume 10, Issue 7, pp. 1417-1423
  • Forced 2D Turbulence: Experimental Evidence of Simultaneous Inverse Energy and Forward Enstrophy Cascades Maarten A. Rutgers, Phys. Rev. Lett. 81, 2244-2247 (1998)

Part II: Turbulence in wall-bounded flows

• Pope, Chapter 7.
• Stability of shear flows:
  • Drazin & Reid, Chapter 1, Chap 4, sects 20, 21, 22, 23, Chap 7, sect 53.1 and 53.2.
  • Romanov, V.A., "Stability of plane-parallel Couette flow", Functional Anal. & its Applic. 7 , 137-146 (1973). [Proof of the linear stability of plane Couette]
  • Chapman, S.J., Subcritical transition in channel flows, J. Fluid Mech. 451, 35-97 (2002) [Lots of things I don't quite agree with, see `On transition scenarios in channel flows' above, but some nice asymptotics of the linear stability of plane Couette and Poiseuille flow]
  • Cathleen Morawetz, The Eigenvalues of Some Stability Problems Involving Viscosity, Indiana Univ. Math. J. 1 No. 4 (1952), 579-603
• Coherent structures
  • Robinson, S.K. Coherent Motions in the Turbulent Boundary Layer, Annual Review of Fluid Mechanics, 23 , 601-639 (1991)
  • Kerswell R.K. Recent progress in understanding the transition to turbulence in a pipe, Nonlinearity 18 , R17-R44 (2005).
  • Waleffe, F. Exact coherent structures in turbulent shear flows, Proceedings of the Conference on Turbulence and Interactions (TI2006), [in production].
• Spectral methods
  • An Elementary View of Euler's Summation Formula Tom M. Apostol The American Mathematical Monthly, Vol. 106, No. 5. (May, 1999), pp. 409-418.
  • The Euler-Maclaurin and Taylor Formulas: Twin, Elementary Derivations Vito Lampret, Mathematics Magazine, Vol. 74, No. 2. (Apr., 2001), pp. 109-122.