(a) and (b) effect of U 0 on the leading eigenmodes for the advection–diffusion–reaction model (for x max = 25 and a 0 = 1). The least stable part of the eigenvalue spectrum is shown in (a), and the leading eigenmode for each value of U 0 (represented in (a) by circles) are displayed in (b). (c) spectrum (+ symbols) and iso-contours of the pseudo-spectrum for x max = 25, U 0 = 5, and a 0 = 1 (logarithmic scale). (d) Spatial growth rate of the leading global mode (measured as ψ(0)′(10)/ψ(0)(10)) as a function of parameters a 0 and U 0. The solid contour represents the limit between growing and decaying modes, the dashed lines give the maximum value of the advection parameter for which computation is possible in a domain of a given length (indicated on the curve).
Flow configuration for (a) incompressible and (b) compressible computations. The incompressible Navier–Stokes equations are solved on the 2D domain Ω using a finite element formulation, with an inflow boundary condition (BC) on Γ i (thin solid line), a no slip BC on Γ w (thick solid line), a stress-free BC on Γ o (dashed line), and compatibility conditions on the axis Γ a (dashed-dotted line). No sponge layers are used in this case. The compressible Navier–Stokes equations are discretized using high order Finite Differences (FD) on the rectangular domain represented in (b). The shaded regions correspond to sponge layers, and the presence of an infinitely thin adiabatic wall for r = 1 and x ⩽ 0 is taken into account by means of appropriate FD schemes.
Axial velocity field of the two base flows. (a) laminar base flow, computed as a steady solution of the Navier–Stokes equations. (b) turbulent mean flow, adapted from an analytical model. 36
Spatio-temporal evolution of the optimal initial condition for m = 0 and T = 10 for the turbulent jet mean profile in the incompressible case. The value of the axial velocity along the line r = 0.9 is represented at various time steps, as indicated next to the curve.
Gains associated with the optimal perturbations for (a) the laminar and (b) the model base flows. Thick solid line: m = 0, dashed line: m = 1, dashed-dotted line: m = 2, dotted line: m = 3, and thin solid line: m = 4.
(a) Optimal perturbations for the model subsonic jet at Ma = 0.75 for m = 0 (solid line) and m = 1 (dashed line). (b) and (c) Azimuthal vorticity field for the optimal initial condition for m = 0 and T = 12, and the corresponding perturbation at t = 12. (d) and (e) Dilatation field for the optimal initial condition for m = 0 and T = 30, and the corresponding perturbation at t = 30.
Global modes computed for m = 0 on the laminar base flow. (a) eigenfrequency spectrum. (b)–(e) axial velocity magnitude of four selected modes, in logarithmic scale, as indicated in (a).
Axial velocity for global modes (c) and (d) of Figure 7 , in linear scale.
Spectra computed for various domain lengths using stress-free (a) and convective outflow (b) boundary conditions at the outlet. Crosses: x max = 40 (black). Triangles: x max = 60 (blue). Plusses: x max = 80 (green). Circles: x max = 100 (red). The dashed lines correspond to the estimated decay rate (11) .
Global spectra computed for m = 0 for the model base flow. Crosses: x max = 40 (black). Triangles: x max = 60 (blue). Plusses: x max = 80 (green). The dashed lines correspond to the estimated decay rate of the spurious branch given by (11) .
Axial velocity fields of selected global modes computed for m = 0 for the model base flow with x max = 60 and stress-free outflow boundary conditions (logarithmic scale). (a) ω = 0.22 − 0.11i, (b) ω = 1.0 − 0.17i, (c) ω = 0.98 − 0.23i.
Global modes computed for m = 0 on the model subsonic jet at Ma = 0.75. (a) eigenvalue spectrum. (b) − (d) azimuthal vorticity of three selected modes, as indicated in (a).
Optimal transient amplification of axisymmetric perturbations for the laminar incompressible jet. (Thick line) computation using the adjoint equations, as computed in Sec. IV . (Thin lines) optimal gains computed by projecting the dynamics onto the space spanned by the N least stable eigenmodes.
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