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Analysis of Fluid Systems: Stability, Receptivity, SensitivityLecture notes from the FLOW-NORDITA Summer School on Advanced Instability Methods for Complex Flows, Stockholm, Sweden, 2013

[+] Author and Article Information
Peter J. Schmid

Department of Mathematics,
Imperial College London,
London SW7 2AZ, UK
e-mail: peter.schmid@imperial.ac.uk

Luca Brandt

Linné FLOW Centre,
Department of Mechanics,
Royal Institute of Technology (KTH),
Stockholm SE-10044, Sweden
e-mail: luca@mech.kth.se

Manuscript received July 11, 2013; final manuscript received December 18, 2013; published online March 24, 2014. Assoc. Editor: Ardeshir Hanifi.

Appl. Mech. Rev 66(2), 024803 (Mar 24, 2014) (21 pages) Paper No: AMR-13-1049; doi: 10.1115/1.4026375 History: Received July 11, 2013; Revised December 18, 2013

This article presents techniques for the analysis of fluid systems. It adopts an optimization-based point of view, formulating common concepts such as stability and receptivity in terms of a cost functional to be optimized subject to constraints given by the governing equations. This approach differs significantly from eigenvalue-based methods that cover the time-asymptotic limit for stability problems or the resonant limit for receptivity problems. Formal substitution of the solution operator for linear time-invariant systems results in the matrix exponential norm and the resolvent norm as measures to assess the optimal response to initial conditions or external harmonic forcing. The optimization-based approach can be extended by introducing adjoint variables that enforce governing equations and constraints. This step allows the analysis of far more general fluid systems, such as time-varying and nonlinear flows, and the investigation of wavemaker regions, structural sensitivities, and passive control strategies.

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References

Figures

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Fig. 1

Sketch of the two flow configurations, the coordinate system and the base flow profiles. (a) Plane Poiseuille flow, and (b) plane Couette flow.

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Fig. 2

Energy amplification G(t) for the model problem for three different Reynolds numbers, showing monotonic decay (for Re = 2), transient growth and asymptotic decay (for Re = 25), and transient and asymptotic growth (for Re = 125)

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Fig. 6

Numerical range (red boundary), resolvent contours and spectrum (blue symbols) for plane Poiseuille flow (top) and plane Couette flow (bottom). The parameters are α = 1,β =  0.25, and Re  =  2000 for plane Poiseuille flow and Re  = 1000 for plane Couette flow. The results are obtained with the routine NumRange.m. (See the “Supplemental Data” tab for this paper on the ASME Digital Collection.)

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Fig. 5

Illustration of the numerical range using the 2 × 2 model problem for Re = 10. (a) Choosing μ = 1 results in a non-normal matrix and a numerical range (delimited by the red curve) that is detached from the spectrum (black symbols); (b) for μ = 0 we have a normal matrix with identical eigenvalues, but a numerical range that deteriorates to the convex hull of the two eigenvalues, given simply by a line connecting the two eigenvalues.

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Fig. 11

Sketch of the computational procedure to recover the optimal initial condition via a singular value decomposition

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Fig. 4

Spectrum (left column) and transient energy growth (right column) for plane Poiseuille flow (top row) and plane Couette flow (bottom row). The parameters for plane Poiseuille flow are: α = 1, β = 0.25, Re = 2000; the parameters for plane Couette flow are: α = 1, β = 0.25, Re = 1000 Results obtained with the routine TransientGrowth.m. (See the “Supplemental Data” tab for this paper on the ASME Digital Collection.)

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Fig. 3

Geometric interpretation of transient growth

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Fig. 7

Sketch of supercritical (left) and subcritical (right) bifurcation behavior. The critical parameter is indicated by the symbol. Dashed lines denote unstable branches. For supercritical behavior, finite-amplitude states exist only after the (linear) infinitesimal-amplitude state has gone unstable (right of the thin vertical line). For subcritical behavior, finite-amplitude states exist even before the (linear) infinitesimal-amplitude state has become unstable (left of the thin vertical line).

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Fig. 8

Parametric study of maximum transient growth as a function of the streamwise wavenumber and Reynolds number (α, Re) for plane Poiseuille flow (top) and plane Couette flow (bottom). The spanwise wavenumber in both cases is β = 0. The area shaded in gray (for plane Poiseuille flow) denotes the parameter space for exponential (modal) growth. The white contour line is given by a zero value of the numerical abscissa. The contour levels represent log10(Gmax) The results are obtained with the routine Neutral_a_Re.m. (See the “Supplemental Data” tab for this paper on the ASME Digital Collection.)

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Fig. 9

Parametric study of maximum transient growth as a function of the streamwise and spanwise wavenumber (α,β) for plane Poiseuille flow (top, Re = 10,000) and plane Couette flow (bottom, Re = 500). The area shaded in gray (for plane Poiseuille flow) denotes the parameter space for exponential (modal) growth. The contour levels represent log10(Gmax) The results are obtained with the routine Neutral_alpha_beta.m. (See the “Supplemental Data” tab for this paper on the ASME Digital Collection.)

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Fig. 10

Three parameter domains of qualitative stability behavior: in region I (Re < Re1) both the numerical abscissa and the growth rate of the least stable eigenvalue are negative, resulting in monotonic energy decay; in region II (Re1 < Re < Re2) the numerical abscissa is positive, but the eigenvalues are still confined to the stable half-plane, yielding short-time energy amplification followed by asymptotic decay; in region III (Re > Re2) both the numerical range and the spectrum have crossed into the unstable half-plane, giving rise to transient and asymptotic energy growth

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Fig. 12

Resolvent norm ‖(iωI-L)-1‖ for plane Poiseuille (top) and plane Couette flow (bottom), thick black line. The parameters are: α = 1,β = 0.25 and Re = 2000 for plane Poiseuille flow and Re = 1000 for plane Couette flow. The thin red line represents the resonant limit, based on the inverse of the minimal distance of the forcing frequency ω to the spectrum. The results are obtained with the routine Resolvent.m. (See the “Supplemental Data” tab for this paper on the ASME Digital Collection.)

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Fig. 14

Componentwise input-output analysis for plane Couette flow (Re = 1000). Each panel displays the maximal amplification over all forcing frequencies, as a function of the streamwise and spanwise wavenumbers. In each panel, the black symbol indicates the maximum response.

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Fig. 15

Sensitivity of eigenvalues, illustrated on the 2 × 2-model problem. (left) Superposition of 100 spectra of the perturbed non-normal system matrix with μ = 1; (right) same for the normal system matrix with μ = 0. In both cases, the norm of the perturbation matrix is ||ΔL|| = 10-2.

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Fig. 13

Componentwise input-output analysis for plane Poiseuille flow (Re = 2000). Each panel displays the maximal amplification over all forcing frequencies, as a function of the streamwise and spanwise wavenumbers. In each panel, the black symbol indicates the maximum response.

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Fig. 16

Sensitivity of eigenvalues, for plane Poiseuille (left) and plane Couette (right) flow. The unperturbed spectrum is illustrated by red symbols. A superposition of 200 spectra (in blue) is shown for α = 1,β = 0. The Poiseuille spectrum (for Re = 2000) is perturbed by random matrices of norm ɛ = 5·10-3. The Couette spectrum (for Re = 1000) is perturbed by random matrices of norm ɛ = 10-3. The resolvent norm can be displayed in the complex plane using the routine Resolvent.m (See the “Supplemental Data” tab for this paper on the ASME Digital Collection.).

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Fig. 17

Sketch of two nonorthogoonal eigendirections q1 and q2, and the corresponding bi-orthogonal adjoint modes p1 and p2. The figure displays how the adjoint mode provides the largest projection in the direction of the corresponding direct eigenvector.

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Fig. 20

Adjoint of the unstable global mode (real and imaginary part) for flow past a circular cylinder at Re = 50

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Fig. 21

Structural sensitivity (the wave maker) for flow past a circular cylinder at Re = 50. Variations of the real and imaginary part of the eigenvalue are reported in the figure.

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Fig. 18

Streamlines and velocity magnitude for the base flow past a circular cylinder at Re = 50

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Fig. 19

(Top) Spectrum of eigenvalues and (bottom) unstable global mode, real and imaginary part, for the flow past a circular cylinder at Re = 50

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Fig. 23

Sketch of the numerical procedure used to approximate a matrix L by the Hessenberg matrix H using the orthonormal basis V

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Fig. 22

Sensitivity to base-flow modifications and total sensitivity to a localized resistance force for flow past a circular cylinder at Re = 50. Variations of the real and imaginary part of the eigenvalue are reported in the figure.

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Fig. 24

Sketch of adjoint looping procedure. Power iterations are used to determine the principal eigenvector and eigenvalue corresponding to the optimal initial condition and growth for any time-dependent flow. The propagator P evolves the initial condition from t = 0 to some final time t, while PH backpropagates a terminal vector from t to t = 0 using the adjoint equation.

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