About this course

The Linear Stability and Laminar-Turbulent Transition course, offered at Technion and shared within the EuroTeQ framework, introduces linear stability theory (LST) and laminar-turbulent transition of incompressible and compressible laminar flows. For reasons of effective delivery, the full scope of the taught material is divided in two parts, Theoretical Foundations and Numerical Implementation, both delivered during the winter semester. Concepts are introduced and equations are derived in LST Theoretical Foundations, while solved examples are discussed and practiced within LST Numerical Implementation. Both Theoretical Foundations (this course) and Numerical Implementation are stand-alone modules and can be taken independently of each other.

Experimental Evidence: Tollmien-Schlichting waves. Crossflow instabilities. Görtler vortices. Mack modes. Global modes.

Theoretical Foundations: The equations of motion and their linearization. The fundamental decompositions in 1, 2 and 3 spatial dimensions; local and global LST. Fundamental theorems; Rayleigh, Fjørtoft, Howard and the generalized inflection point. Direct and adjoint linear operators. The Bi-orthogonal decomposition. Receptivity and Sensitivity. Inviscid and viscous instability. The temporal and the spatial eigenvalue problem. The eN method and the N-factor. Multiple scales and the Parabolized Stability Equations. The initial value problem and singular value decomposition. Convective and absolute instability. Impulse response / Green’s functions. Floquet theory for analysis of periodic motions; secondary linear instability. Weakly-nonlinear analysis and the Stuart-Landau equation. Global LST: The BiGlobal, PSE3D and the TriGlobal EVP. Control of linear instabilities.

A single textbook describing the course content does not exist presently, but individual topics can be studied in:

Books on Flow Stability • Criminale WO, Jackson TL, Joslin RD (2019) Theory and Computation of Hydrodynamic Stability, Cambridge University Press, 2nd Ed. • Schmid PJ, Henningson DS (2001) Stability and Transition in Shear Flows, Springer

Mathematics Textbooks

• Haberman R (2014) Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, Pearson, 5th Ed. • Kreyszig E (2006) Advanced Engineering Mathematics, John Wiley & Sons, 9th Ed.

Numerical Methods

• Boyd JP (1989) Chebyshev and Fourier Spectral Methods, Springer. Also (2000) Dover 2nd Ed. • Trefethen LN (2000) Spectral Methods in Matlab, SIAM

Articles and Matlab Codes* • * Cossu C (2014) An Introduction to Optimal Control. Appl. Mech. Rev. 66(2):02480, • Fedorov A (2011) Transition and Stability of High-Speed Boundary Layers. Annu. Rev. Fluid Mech. Vol. 43, pp. 79–95, • * Juniper MP, Hanifi A, Theofilis V (2014) Modal Stability Theory. Appl. Mech. Rev. 66(2):024804, • Luchini P, Bottaro A (2014) Adjoint Equations in Stability Analysis. Annu. Rev. Fluid Mech.Vol. 46, pp. 493–517, • Mack LM (1984) Compressible boundary layer stability theory. AGARD Rep. R-709. • * Schmid PJ, Brandt L (2014) Analysis of Fluid Systems: Stability, Receptivity, Sensitivity. Appl. Mech. Rev. Mar 2014, 66(2):024803, • Taira K, Brunton SL, Dawson STM, Rowley CW, Colonius T, McKeon BJ, Schmidt OT, Gordeyev S, Theofilis V, Ukeiley LS (2017) Modal Analysis of Fluid Flows: An Overview. AIAA Journal 55(12):4013-4041, • Theofilis V (2011) Global Linear Instability. Annu. Rev. Fluid Mech. 43:319–352,

Semester Start Date: November 3, 2024

Day & Time: TBD

Contact Hours per week: 3

Learning outcomes

On completion of the two-part course, students will have obtained understanding of physical mechanisms that underpin laminar-turbulent transition from the incompressible to the hypersonic flow regime. They will have developed the ability to describe linear modal and non-modal flow instability in precise mathematical terms and acquired practiced skills to solve the related equations numerically. Students will familiarize themselves with stability analysis of steady and time-periodic laminar states and be able to describe the entire process of different kinds of environmental perturbations being converted via receptivity into linear instabilities that may lead laminar flow to breakdown to turbulence. Predictive state-of-the-art tools will have been exposed to them, preparing students to pursue careers in academia or industry.


Student assessment will be based on preparing offline, and submitting within a week from the respective publication, three homework assignments, each counting 20% of the final grade. Each assignment will be graded out of 100, with 50 points required to pass. Submission of all three assignments is mandatory for completion of the course (i.e. an “incomplete” grade will be submitted if any of the assignments is not uploaded on time. The remaining 40% of the final grade will be obtained from one final exam, which will be published on Moodle at the end of the course, and for which a three-hour window will be given for answers to be submitted.

Course requirements

Incompressible and Compressible Fluid Mechanics, Advanced Engineering Mathematics, Numerical Analysis and Linear Algebra, Programming (compiled or interpreted computer languages).


Lectures and homework exercises

Additional information

  • Credits
    ECTS 4
  • Level
  • Contact hours per week
  • Instructors
    Prof. V. Theofilis
  • Mode of instruction
If anything remains unclear, please check the FAQ of Technion (Israel).


  • Start date

    3 November 2024

    • Ends
      2 March 2025
    • Term *
      Winter Semester 2024/2025
    • Location
    • Instruction language
    • Register between
      14 May - 29 Jul 2024
    Only 8 days to enrol
    Apply now
These offerings are valid for students of TUM (Germany)