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Tutorial

Modal Stability Theory
Lecture notes from the FLOW-NORDITA Summer School on Advanced Instability Methods for Complex Flows, Stockholm, Sweden, 2013

[+] Author and Article Information
Matthew P. Juniper

Engineering Department,
Cambridge University,
Trumpington Street,
Cambridge CB2 1PZ, UK
e-mail: mpj1001@cam.ac.uk

Ardeshir Hanifi

Swedish Defence Research Agency,
FOI, SE-164 90 Stockholm, Sweden
Linné FLOW Centre,
KTH Royal Institute of Technology,
Stockholm SE-100 44, Sweden
e-mail: ardeshir.hanifi@foi.se

Vassilios Theofilis

School of Aeronautics,
Universidad Politécnica de Madrid,
Plaza Cardenal Cisneros 3,
Madrid E-28040, Spain
e-mail: vassilis@aero.upm.es

ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (DETC2013-13115)

Not to be confused with the term mean flow/state, which is reserved for time and/or space-averaged turbulent flow.

Manuscript received August 11, 2013; final manuscript received January 20, 2014; published online March 25, 2014. Assoc. Editor: Gianluca Iaccarino.

Appl. Mech. Rev 66(2), 024804 (Mar 25, 2014) (22 pages) Paper No: AMR-13-1059; doi: 10.1115/1.4026604 History: Received August 11, 2013; Revised January 20, 2014

This article contains a review of modal stability theory. It covers local stability analysis of parallel flows including temporal stability, spatial stability, phase velocity, group velocity, spatio-temporal stability, the linearized Navier–Stokes equations, the Orr–Sommerfeld equation, the Rayleigh equation, the Briggs–Bers criterion, Poiseuille flow, free shear flows, and secondary modal instability. It also covers the parabolized stability equation (PSE), temporal and spatial biglobal theory, 2D eigenvalue problems, 3D eigenvalue problems, spectral collocation methods, and other numerical solution methods. Computer codes are provided for tutorials described in the article. These tutorials cover the main topics of the article and can be adapted to form the basis of research codes.

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Figures

Grahic Jump Location
Fig. 1

Waves often travel in packets and have a well-defined envelope. The phase velocity is the velocity at which the wave crests travel. The group velocity is the velocity at which the envelope travels.

Grahic Jump Location
Fig. 2

The base flows in this tutorial are all planar jets/wakes with top hat velocity profiles. These flows have analytical dispersion relations, which can be calculated without numerical methods.

Grahic Jump Location
Fig. 3

Left: The imaginary and real components of ω(k) calculated with a temporal stability analysis, in which k is constrained to be real. Right: The imaginary components of ω(k) for the five dispersion relations.

Grahic Jump Location
Fig. 8

Contours of ωi(k) for varicose perturbations of a confined jet flow with surface tension, showing the integration path from k = −∞ to k = +∞. The integration path passes over saddle points s1 and s3, which means that they contribute to the integral. It does not pass over saddle points s2a or s2b, however, which means that they do not contribute to the integral. This is another way to visualize the Briggs–Bers criterion, which states that a saddle point is only valid if it is pinched between a k+ and a k hill [33].

Grahic Jump Location
Fig. 9

Contours of ωi(k) for the same flow as Fig. 8, as the surface tension is reduced. This reduction in Σ causes the s1 saddle to move to lower ωi and, therefore, causes the integration path to pass over the s2a saddle point. When the surface tension tends to zero (not shown here) all the s2 saddle points move onto the integration path [33].

Grahic Jump Location
Fig. 7

The impulse response for varicose perturbations of an unconfined low density jet with Λ = 1/1.1, S = 0.1, and finite thickness shear layers. The impulse is at (x,z) = (0,0) and the resultant waves disperse at their individual group velocities in the x- and z-directions. The chart plots the growth rate, ωi, of the wave that dominates along each ray (x/t,z/t), i.e., the wave that has group velocity (x/t,z/t). We see that most of the wavepacket propagates and grows downstream (to the right) but that some waves propagate and grow upstream (to the left). The growth rate at the point of impulse (x/t,z/t) = (0,0) is positive, which means that this is an absolutely unstable flow [33].

Grahic Jump Location
Fig. 10

Eigenvalue spectrum for PPF at Re = 104, α = 1, β = 0. Solution of LNS (•) and the Orr–Sommerfeld (◻) equations. Notice that the vorticity modes are not represented by Orr–Sommerfeld equation. Results produced by run_LNS.m and run_OS.m.

Grahic Jump Location
Fig. 11

Velocity and pressure perturbations corresponding to the most unstable eigenvalue of PPF at Re = 104, α = 1, β = 0. Data produced by run_LNS.m for the most unstable mode in Fig. 10. All components are normalized with max(|û|).

Grahic Jump Location
Fig. 13

Subharmonic secondary growth rate σ as a function of the spanwise wave number β for F = ωr /Re = 1.24 × 10−4 at Branch II, Re = 606 [43]. The parameter A indicates the amplitude of the primary disturbance as a percentage of the maximum value of the streamwise basic flow velocity component.

Grahic Jump Location
Fig. 14

Neutral stability curves based on the maximum of û and v∧. The nonlocal results are compared to the DNS calculations (Berlin et al. [69]) and experimental data (Klingmann et al. [70]).

Grahic Jump Location
Fig. 15

Effect of streamwise step size on the growth rates of 2D waves in a Blasius boundary layer flow, F = 1.0 × 10−4. Computed by pse.m with stab = false.

Grahic Jump Location
Fig. 16

Effect of stabilizing term on the growth rates of 2D waves in a Blasius boundary layer flow, F = 1.0 × 10−4. Computed by pse.m with stab = true.

Grahic Jump Location
Fig. 17

Relative size of the discretized operators describing modal stability, when the same number of collocation nodes is used in one (local, PSE) and two (global, PSE-3D) resolved spatial directions (from de Tullio et al. [83]).

Grahic Jump Location
Fig. 20

Relative error in the computation of the leading eigenmode in PPF at Re = 104, α = 1, using spectral collocation (circle symbol), a suite of common high-order finite-difference methods, and the recently proposed FD-q finite-difference method [84].

Grahic Jump Location
Fig. 21

Sparsity patterns of the laplace operator (left) incompressible BiGlobal EVP (middle) and compressible BiGlobal EVP (right), discretized using the spectral collocation method

Grahic Jump Location
Fig. 22

The Arnoldi algorithm (from Ref. [27])

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