^{1,a)}, L. Lesshafft

^{1}, P. J. Schmid

^{1}and P. Huerre

^{1}

### Abstract

The linear stability dynamics of incompressible and compressible isothermal jets are investigated by means of their optimal initial perturbations and of their temporal eigenmodes. The transient growth analysis of optimal perturbations is robust and allows physical interpretation of the salient instability mechanisms. In contrast, the modal representation appears to be inadequate, as neither the computed eigenvalue spectrum nor the eigenmode shapes allow a characterization of the flow dynamics in these settings. More surprisingly, numerical issues also prevent the reconstruction of the dynamics from a basis of computed eigenmodes. An investigation of simple model problems reveals inherent problems of this modal approach in the context of a stable convection-dominated configuration. In particular, eigenmodes may exhibit an exponential growth in the streamwise direction even in regions where the flow is locally stable.

This work was supported by DGA Grant No. 2009.60.034.00.470.75.01 and by a fellowship from the EADS Foundation. Calculations were performed using HPC resources from GENCI (Grant No. 2012-026451). The authors thank F. Gallaire for his help.

I. INTRODUCTION

II. MODEL PROBLEMS: EIGENMODES OF ADVECTIVE SYSTEMS

A. Advection equation with upstream boundary forcing

B. Unforced advection–diffusion–reaction equation

C. Conclusions from model problems

III. SETUP OF THE JET PROBLEM

A. Flow configuration

1. Incompressible setting

2. Compressible setting

B. Base flows

1. Laminar steady state

2. Turbulent mean flow

C. Numerical methods

1. Incompressible setting

2. Compressible setting

IV. TRANSIENT GROWTH OF PERTURBATIONS

A. Incompressible flow

B. Effects of compressibility

V. MODAL ANALYSIS

A. Incompressible global modes

1. Spectrum of the laminar base state

2. Influence of domain truncation

3. Spectrum of the turbulent mean flow

B. Compressible eigenmodes

VI. PROJECTION OF THE TRANSIENT DYNAMICS ONTO THE SPACE SPANNED BY EIGENMODES

VII. CONCLUSIONS

### Key Topics

- Normal modes
- 52.0
- Turbulent flows
- 50.0
- Flow instabilities
- 34.0
- Laminar flows
- 29.0
- Eigenvalues
- 19.0

## Figures

(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).

(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.

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

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.

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.

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.

(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).

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) .

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) .

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.

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).

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.

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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