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Review Article

A Practical Review on Linear and Nonlinear Global Approaches to Flow Instabilities

[+] Author and Article Information
D. Fabre, D. Ferreira Sabino, P. Bonnefis, J. Sierra

Institut de Mécanique des fluides
de Toulouse (IMFT),
Université de Toulouse,
CNRS, INPT, UPS,
Toulouse, France

V. Citro

Dipartimento di Ingegneria (DIIN),
Universitá degli Studi di Salerno,
Via Giovanni Paolo II, 132,
Fisciano 84084, Italy;
Institut de Mécanique des fluides
de Toulouse (IMFT),
Université de Toulouse,
CNRS, INPT, UPS,
Toulouse, France

F. Giannetti

Dipartimento di Ingegneria (DIIN),
Universitá degli Studi di Salerno,
Via Giovanni Paolo II, 132,
Fisciano 84084, Italy

M. Pigou

Institut de Mécanique
des fluides de Toulouse (IMFT),
Université de Toulouse,
CNRS, INPT, UPS,
Toulouse, France

Manuscript received June 15, 2018; final manuscript received January 23, 2019; published online February 13, 2019. Editor: Harry Dankowicz.

Appl. Mech. Rev 70(6), 060802 (Feb 13, 2019) (16 pages) Paper No: AMR-18-1070; doi: 10.1115/1.4042737 History: Received June 15, 2018; Revised January 23, 2019

This paper aims at reviewing linear and nonlinear approaches to study the stability of fluid flows. We provide a concise but self-contained exposition of the main concepts and specific numerical methods designed for global stability studies, including the classical linear stability analysis, the adjoint-based sensitivity, and the most recent nonlinear developments. Regarding numerical implementation, a number of ideas making resolution particularly efficient are discussed, including mesh adaptation, simple shift-invert strategy instead of the classical Arnoldi algorithm, and a simplification of the recent nonlinear self-consistent (SC) approach proposed by Mantič-Lugo et al. (2014, “Self-Consistent Mean Flow Description of the Nonlinear Saturation of the Vortex Shedding in the Cylinder Wake,” Phys. Rev. Lett., 113(8), p. 084501). An open-source software implementing all the concepts discussed in this paper is provided. The software is demonstrated for the reference case of the two-dimensional (2D) flow around a circular cylinder, in both incompressible and compressible cases, but is easily customizable to a variety of other flow configurations or flow equations.

Copyright © 2018 by ASME
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Figures

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Fig. 1

Illustration of the usage of the stabfem software to produce an adapted mesh and study the base flowand the linear stability properties of the wake flow around a cylinder (extracted from script SCRIPT_CYLINDER_ALLFIGURES.m)

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Fig. 2

Illustration of the stucture of mesh M2 (adapted to both the base flow and structural sensitivity)

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Fig. 3

Recirculation length Lx (a) and nondimensional drag Fx (b) of the base flow over a cylinder as function of Re

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Fig. 4

Base flow for the flow over a cylinder at Re = 60. Pressure field (color or grayscale levels) and streamlines (iso-levels of the streamfunction ψ).

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Fig. 5

Growth rate σ (a) and Strouhal number St = ω/2π (b) as function of the Reynolds number

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Fig. 6

Contour plot of the streamwise velocity component: (a) (direct) eigenmode and (b) adjoint mode at Re = 60

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Fig. 7

Structural sensitivity Sw for the cylinder's wake at Re = 60. The red line represents the streamline bounding the recirculation region.

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Fig. 8

Structure of the unstable eigenmode (streamwise velocity component) for the compressible flow around a cylinder (Re = 150, M =0.2)

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Fig. 9

Structure of the unstable eigenmode (pressure component) for the compressible flow around a cylinder (Re = 150, M =0.2). Upper half: real part and lower half: imaginary part.

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Fig. 10

Illustration of the procedure for nonlinear calculations using stabfem (extract from script SCRIPT_CYLINDER_ALLFIGURES.m)

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Fig. 11

Comparison between the WNL results, the harmonic balance data (HB1), and baseflow/linear results: Strouhal number (a), mean drag (b), amplitude of oscillating lift (c), energy-amplitude of the nonlinear perturbation (d), and recirculation length of mean/base flows (e)

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Fig. 12

Structure of the mean flow over a cylinder for Re = 60, as computed by the HB1 model. Color levels: pressure; streamlines.

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Fig. 13

Illustration of the stucture of mesh M4 (adapted to both the base flow and direct eigenmode)

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Fig. 14

Illustration of the implementation of the Newton algorithm for base-flow computation (extract from freefem++ program Newton2D.edp)

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Fig. 15

Illustration of the implementation of the shift-invert algorithm for single eigenmode computation (extract from freefem++ program Stab2D.edp)

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