A high-level description of the turbulence process. The lower block contains the linear dynamics of the fluctuations interacting with the mean velocity profile.
A schematic of pipe geometry and nomenclature.
A block diagram showing the network of resolvents, , acting on radially varying traveling waves of different wavenumber and frequency. An input-output relationship can be written for all K, but u K ≠ 0 for a finite range of K in real flows. FT and IFT denote the Fourier transform and inverse Fourier transform, respectively. u 0 and f 0 describe the equation for the turbulent mean profile.
An illustration of the effect of matrix M on the unit circle (left plot mapped to centre plot). The singular values are the radii of the resulting ellipse and the singular vectors give the rotation of the ellipse. a i are the columns of A and b i are the columns of B. In this illustration, the matrix has the singular value decomposition M = ASB* with The rightmost plot shows the effect of the optimal rank-1 approximation to the mapping M on the unit circle, where .
Block diagram for Eqs. (11) and (12) referring to the linear equations projected onto the forcing and response modes. Diagram (a) depicts the “true” resolvent and (b) depicts the Schmidt decomposition of the resolvent. The rank-1 approximation is depicted in (c). This approximation can be performed for each lower sub-system in Figure 3 .
(a) Variation of the location of the peak streamwise velocity with increasing wavespeed, c, for the first velocity response mode with (k, n) = (1, 10) at R + = 1800 (solid line). The turbulent mean profile normalized by the centerline velocity, , at this Reynolds number is also shown in dotted gray for reference. (b) The corresponding first singular values.
Representative velocity response mode shapes for the “attached,” “attached and critical,” and “critical modes,” shown for conditions corresponding to points (i)–(iii) in Figure 6 , i.e., (k, n) = (1, 10), c = 0.25, 0.67 and 0.85, and R + = 1800. (Top) u(y +); (middle) v(y +); (bottom) uv(y +).
(a) Distribution of streamwise energy over the pipe radius for the near wall mode with . Reynolds numbers: — 75 × 103, −− 150 × 103, ⋅⋅⋅ 410 × 103, · − · 1 × 106. (b) Shape of the first singular mode representative of the dominant near wall motions. Color denotes isosurfaces of streamwise velocity (streaks), where red and blue correspond to high and low velocity, respectively, relative to the mean flow (heading into the page), and the white arrows show the sense of the in-plane velocity field.
Wall-normal variation of (a) amplitudes of the three velocity components and (b) corresponding phases in multiples of 2π for the VLSM response mode with K = (1, 10, 2/3) at R + = 1800. The lines denote −o −: u, −: v, −× −: w. Solid/dashed horizontal line shows the location of the critical layer.
Kelvin's cats' eyes (also called Kelvin-Stuart vortices): structure at the inviscid, laminar critical layer as observed by an observer moving at the critical velocity.
The periodic array of pro- and retro-grade hairpin vortices associated with near-wall velocity response modes at R + = 1800, identified by an isosurface of constant swirling strength at 50% of the absolute maximum value, color-coded with the local (model) azimuthal vorticity. Red and blue denote rotation in and counter to the sense of the classical (prograde) hairpin vortex, respectively.
Isosurfaces of constant swirling strength (33% of maximum value) for the “ideal packet” at R + = 1800, color-coded by the local (model) azimuthal vorticity. Red and blue denote rotation in and counter to the sense of the classical hairpin vortex or prograde and retrograde vortices, respectively.
An example of a morphing surface geometry capable of simple interaction with the resolvent analysis: an “active eggbox” that introduces a single K. A roughness geometry described by single wavenumbers in the wall-parallel directions (k, n) is actuated at a single frequency, ω, leading to harmonic temporal variation in amplitude.
A schematic of the arrangement of the flat plate, the roughness strip, and the diagnostic locations; not to scale. The internal layers are also marked in order to provide an idea of their relative sizes and development rates. The first internal layer, δ1 grows rapidly; the second, δ2, grows slowly. The mean boundary layer thickness (δ99) at the location of the roughness in the unperturbed flow is denoted δ0. The three key measurement locations highlighted in the subsequent discussion are marked with black lines.
Composite spectra for the unperturbed and perturbed flows. Ten contour levels, equally spaced across the color bar, are indicated. Top row: unperturbed case, measured at Re θ = 4040. Middle row: static roughness perturbation; bottom row: dynamic perturbation. For the latter two rows, left to right is the direction of increasing streamwise distance downstream from the impulse, at approximately x/δ = 3, 8, 24. The mean internal layer locations, when distinguishable, are denoted: −− first internal layer, and ⋅⋅⋅ second internal layer, following Figure 14 . The solid white lines in the dynamically perturbed case indicate the location of the (local) critical layer.
Phase-locked maps of the streamwise component of the velocity field measured from the hotwire over an average period (abscissa t ∈ [0, 2π]) in outer units (ordinate y/δ) at the three streamwise locations indicated in Figure 14 . (a) The periodic component, . (b) The rms of the fluctuating component, u, with the mean value subtracted. The mean internal layer locations, when distinguishable, are denoted: −− first internal layer, and ⋅⋅⋅ second internal layer. The solid green lines indicate the location of the critical layer, ascertained from the maximum amplitude of the streamwise velocity mode.
Phase-locked maps of the wall-normal component of the velocity field measured from the PIV over an average period (abscissa t ∈ [0, 2π]) in outer units (ordinate y/δ) at the first downstream location indicated in Figure 14 , x/δ ≈ 4. (a) The periodic component, , and (b) the rms of the fluctuating component, v, with the mean value subtracted. The mean internal layer locations, when distinguishable, are denoted: −− first internal layer, and ⋅⋅⋅ second internal layer. The solid green lines indicate the location of the critical layer, as measured from the hotwire measurements at the equivalent streamwise location.
The amplitude and phase of the velocity modes from the experiment and the resolvent analysis. The amplitude variation is shown in the middle pane. is shown on the lower-left axes: — from the resolvent analysis using the unperturbed velocity profile; −· − from the resolvent analysis using the perturbed velocity profile; x/δ = 0.1 +; 2.3 × from the experimental hotwire measurements. v K (y, t) is shown in gray on the upper-right axes: — from the resolvent analysis using the unperturbed velocity profile; — from the experimental PIV measurement with PIV window centered at x/δ ≈ 4. The left pane indicates the phase of , and the right pane indicates the phase of . The (a) marks a distinctive variation in phase which, to our knowledge, is a robust feature of all Orr-Sommerfeld type solutions. The location of the internal layers have been marked for the streamwise component: −−− the first internal layer; ⋅⋅⋅ the second internal layer.
Comparison of the resolvent calculations and phase-locked measurements. (Top) Maps of the calculated most-amplified singular mode over an average period (t ∈ [0, 2π]). and , left and right. (Bottom) The corresponding experimentally measured maps at x/δ ≈ 2.3, as in Figure 16(a) .
A sketch of the physical form of the experimentally observed mode shape, emphasizing the distortion produced by the stress bore, as well as the location of the internal and critical layers. This sketch of spatial variation can be compared with the temporal experimental observations presented in Figure 16 .
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