Wave packets and Orr mechanism in turbulent jets

Instability waves traveling within subsonic turbulent jets have a modal linear growth until approximatively the end of the potential core. At these stations it is believed that nonlinear and/or nonmodal effects become important and a mismatch appears between experimental measurements and linear models. In this paper the response of the linearized operator to nonlinearities treated here as an external forcing is found to be consistent with a simplified model of the Orr mechanism, supporting the idea that a nonmodal growth of disturbances occurs in the downstream region of the jet in response to the modeled nonlinear forcing.

Figure 1

Temporal Orr solution $\psi (x,y,t)$ with the initial condition (12) at (a) $t=-5$, (b) $t=0$, and (c) $t=5$.

Figure 2

Spatial Orr solution $\text{Re}\left[\tilde{\psi }(x,y,\omega )\right]$, with $\omega =3.77$, for (a) the Fourier transform of the temporal solution $\psi (x,y,t)$, (b) the spatial model solved numerically, and (c) the spatial model solved using Green's functions.

Figure 3

Centerline streamwise fluctuating energy at $\text{St}=0.6$ for the first POD mode of PIV measurements $(\circ )$, the observation operator for the homogeneous PSE $(*)$, the observation operator for the forced PSE $\left(\blacksquare \right)$, and the PSE solution of the forced PSE $(\square )$.

Figure 4

Infinitesimal forcing and associated response. Colors are $\text{Re}\left(\delta {\tilde{\mathit{q}}}^u\right)$ and isocontours are $\text{Re}\left(\delta {\tilde{\mathit{f}}}^u\right)$; solid lines are for positive values and dashed lines are for negative values. The thick solid line is the critical-layer position.

Figure 5

Infinitesimal forcing and associated response. Colors are $|\delta {\tilde{\mathit{q}}}^v{|}^2$ and isocontours are $|\delta {\tilde{\mathit{f}}}^v{|}^2$. The red point indicates the position where the conditions (shear, mean flow velocity, inflow profile, and local wave number) are used for building the Couette flow approximation. The dashed line is the line where the PSE and the Orr model are compared.

Figure 6

Spatial Orr solution $\left[\text{Re}\right(u\left)\right]$ with inflow calibrated on the jet wave-packet response $\delta \tilde{\mathit{q}}$ at $x/D=6$. The red point indicates the position where the conditions from the PSE (shear, mean flow velocity, inflow profile, and local wave number) are used for building the Couette flow approximation. The dashed line is the line where the PSE and Orr model are compared.

Figure 7

Comparison between the PSE response to the sensitivity forcing and the spatial Orr model. For the PSE, we take the values of ${\left|v\right|}^2$ and tilting angle along $r/D=0.35$, the critical-layer position at $x/D=6$. For the Orr model, the values are taken at $y=\omega /S\alpha =0.27$ such that the mean-flow velocity corresponds to the one at the critical layer in the PSE. The vertical dashed line is the position where the PSE has neutral growth.

Figure 8

Comparison between the linear PSE and the spatial Orr model. For the PSE, we take the values of ${\left|v\right|}^2$ and tilting angle along $r/D=0.4$, the critical-layer position at $x/D=4.6$. For the Orr model, the values are taken at $y=\omega /S\alpha =0.47$ such that the mean-flow velocity corresponds to the one at the critical layer in the PSE. The vertical dashed line is the position where the PSE has neutral growth.

Figure 9

Similar results to Fig. 7, comparing $\delta \mathit{q}$ from the PSE with the spatial Orr model for $\text{St}=0.5$.

Figure 10

Similar results to Fig. 8, comparing the linear PSE with the spatial Orr model for $\text{St}=0.5$.

Figure 11

Similar results to Fig. 7, comparing $\delta \mathit{q}$ from the PSE with the spatial Orr model for $\text{St}=0.7$.

Figure 12

Similar results to Fig. 8, comparing the linear PSE with the spatial Orr model for $\text{St}=0.7$.