Linear two-dimensional stability of a Lamb-Oseen dipole as an aircraft wake model

The dynamics of perturbed aircraft wake models is investigated in the two-dimensional limit by means of a linear stability analysis. The base flow, computed by a direct numerical simulation of the incompressible Navier-Stokes equations, is a viscous counter-rotating vortex dipole obtained from an initial condition which is either a superposition of two Lamb-Oseen vortices or a vorticity sheet with elliptical vorticity distribution. The former, referred to as the Lamb-Oseen dipole (LOD), is a model for the far field of the wake and gives rise to a family of quasisteady dipoles parametrized by their aspect ratio only. The later approaches the near-field wake of a wing during the rolling phase which eventually converges towards the LOD model at later times. First, a modal stability analysis of the LOD under the assumption of a frozen base flow is performed for aspect ratios $a/b$ ranging from 0.05 to 0.36 at various Reynolds numbers. Several families of unstable antisymmetric and symmetric modes are observed. The maximal growth rates are reached at low Reynolds numbers. The results are consistent with those obtained by Brion et al. [Phys. Fluids 26, 064103 (2014)] for the higher aspect ratio inviscid Lamb-Chaplygin dipole (LCD). However, the a posteriori verification of the validity of the frozen base flow assumption shows that, except for a few modes occurring at the highest aspect ratios and large Reynolds numbers, these two-dimensional instabilities do not survive the base flow unsteadiness due to viscous diffusion. They are thus not likely to develop in the flow. Second, we focus on the transient dynamics of the dipoles by looking for the optimal perturbations through a nonmodal stability analysis based on a direct-adjoint approach. The observed energy gains are substantial and indicate the potential of transient mechanisms. In the short time dynamics, the optimal perturbation consists of intertwined vorticity layers located within each vortex core and leading to a $m=2$ deformation Kelvin wave excited by the Orr mechanism. For moderate to large horizon times, the optimal perturbation takes the form of vorticity layers localized outside the vortex core which eventually give rise to the two-dimensional unstable mode unveiled by the modal analysis through a combination of Orr mechanism and velocity induction. The robustness of these modes is examined by considering the initial stage of the development of aircraft wakes. The optimal perturbations developing on an elliptic and a double-elliptic vorticity sheet present a similar structure and rely on the same mechanisms as the ones observed for the LOD but come with lower energy gains. It is concluded that the rolling-up of the vorticity sheet in the near-field of the wake does not influence significantly the linear development of these two-dimensional perturbations.

Figure 1

Sketch of the streamwise development of an aircraft wake. The first sectional plan lies in the near-field region where the vorticity sheet rolls-up. The second one corresponds to the far-field region where the wake takes the form of a counter-rotating vortex dipole. The scales are not representative of real wake.

Figure 2

(a) Measure of the LOD unsteadiness as a function of the descent velocity relative deviation $\mathrm{\Delta }{V}_d=V_d/V_1-1$. (b) Influence of $\mathrm{\Delta }{V}_d$ on the growth rate and frequency of the antisymmetric mode for $(a/b,\mathrm{Re})=(0.3,788)$. The symbols correspond to the different evaluations of the descent velocity: $V_2$ ($+$), $V_3$ ($\times$), $V_4$ ($◯$), and $V_5$ ($\square$).

Figure 3

Streamlines (black solid lines) and vorticity contours (color scale $E_6$, see Eq. (4) and text for details) for four LODs of increasing aspect ratio: (a) $a/b=0.05$, (b) $a/b=0.15$, (c) $a/b=0.25,$ and (d) $a/b=0.35$. A small part of the computational domain is represented, i.e., $(x,y)\in [-1.5b,1.5b]\times [-1.5b,3b]$. The dotted thick line corresponds to the Kelvin oval.

Figure 4

Evolution of (a) ellipticity $a_y/a_x$ and (b) circulations $\frac{\mathrm{\Gamma }}{{\mathrm{\Gamma }}_0}$ (solid lines), $\frac{{\mathrm{\Gamma }}_i}{{\mathrm{\Gamma }}_0}$ (dotted lines) of the vortices as a function of the dipole aspect ratio $a/b$ for various Reynolds numbers.

Figure 5

Nondimensional growth rate ${\sigma }^{*}$ (a) and frequency $f^{*}$ (b) of the most amplified antisymmetric mode as a function of the dipole aspect ratio $a/b$ for different Reynolds numbers. Oscillatory and stationary branches are, respectively, identified by the hollow and filled symbols positioned at the extremities and maximum of each branch. Growth rates and frequencies for the LCD computed by Brion et al. [14] are added for comparison.

Figure 6

Real (top) and imaginary (bottom) parts of the vorticity field of antisymmetric oscillatory unstable modes for various values of $(a/b,\mathrm{Re})$ indicated above the figures. From left to right, $\alpha =0.931,0.621,0.971,$ and 0.987. A small part of the computational domain is represented, i.e., $(x,y)\in [-2b,2b]\times [-1.5b,10b]$. The $E_4$ scale is used. The dotted thick line corresponds to the Kelvin oval while the solid and dotted thin lines correspond to ${\omega }_B=\pm 0.1{\parallel {\omega }_B\parallel }_{\infty }$.

Figure 7

(a) Definition of the orientation angle $\theta$ on a close-up view (left vortex) of the unstable mode obtained for $(a/b,\mathrm{Re})=(0.24,223)$. (b) Evolution over one period of $\theta$ for antisymmetric modes presented in Fig. 6: $(a/b,\mathrm{Re})=(0.06,223)$ ($◯$), $(a/b,\mathrm{Re})=(0.24,223)$ ($\square$), $(a/b,\mathrm{Re})=(0.24,1280)$ ($▵$), and $(a/b,\mathrm{Re})=(0.24,2500)$ ($\diamond$).

Figure 8

(a) Growth rates ${\sigma }^{*}$ and (b) frequencies $f^{*}$ of the most amplified symmetric mode as a function of the dipole aspect ratio $a/b$ for different Reynolds numbers. Same conventions as described in the caption of Fig. 5.

Figure 9

Vorticity field of symmetric stationary modes for various values of $(a/b,\mathrm{Re})$ indicated above the figures. Same conventions as described in the caption of Fig. 6.

Figure 10

Growth rates $T_{\nu }\sigma$ normalized by the dipole viscous time scale for both (a) antisymmetric and (b) symmetric two-dimensional modes as a function of $a/b$ for different $\mathrm{Re}$. Same conventions as described in the caption of Fig. 5.

Figure 11

Energy gain evolution of (a) the antisymmetric oscillatory mode and (b) the symmetric stationary mode for $(a/b,\mathrm{Re})=(0.25,5000)$, for a frozen (solid line) and a time-evolving (dashed line) base flow. The oscillatory mode is initialized at three different instants within the period: $t_m$ ($\square$), $t_{\text{med}}$ ($◯$), and $t_M$ ($▵$).

Figure 12

Time evolution of the energy gain of the (a) antisymmetric and (b) symmetric optimal perturbations obtained for a LOD of aspect ratio $a_0/b_0=0.15$ for three horizon times, $T_f/T_b\in \{0.15,1.5,9\}$, and three Reynolds numbers, $\mathrm{Re}=223$ ($\square$), $\mathrm{Re}=1280$ ($◯$), and $\mathrm{Re}=5000$ ($▵$).

Figure 13

Optimal perturbation ${\omega }_0$ and optimal outcome ${\omega }_f$ vorticity fields obtained for a LOD with $(a_0/b_0,\mathrm{Re})=(0.15,1280)$ and three different horizon times. A small part of the computational domain is represented, i.e., $(x,y)\in [-2b,2b]\times [-1.5b,10b]$ and the $E_3$ scale is used. The solid and dotted thin lines correspond, respectively, to ${\omega }_B=\pm 0.1{\parallel {\omega }_B\parallel }_{\infty }$.

Figure 14

Time evolution of the vorticity field (scale $E_5$) for an elliptic (top) and a double-elliptic (bottom) vorticity sheets at $\mathrm{Re}=1280$. A small part of the computational domain is represented, i.e., $(x,y)\in {[-1.5b,1.5b]}^2$ for $t/T_b\in \{0,0.015,0.15\}$ and $(x,y)\in {[-2b_0,2b_0]}^2$ for $t/T_b=1.5$.

Figure 15

Time evolution of the energy gain of the (a), (c) antisymmetric and (b), (d) symmetric optimal perturbations obtained for an elliptic (top) and a double-elliptic (bottom) vorticity sheet of initial width $h_e/l_0=0.05$ for three horizon times, $T_f/T_b\in \{0.05,0.15,1.5\}$, and three Reynolds numbers, $\mathrm{Re}=223$ ($\square$), $\mathrm{Re}=1280$ ($◯$), and $\mathrm{Re}=5000$ ($▵$).

Figure 16

Optimal perturbation ${\omega }_0$ and optimal outcome ${\omega }_f$ vorticity fields obtained for an elliptic vorticity sheet with $(h_e,\mathrm{Re})=(0.05,1280)$ and three different horizon times. A small part of the computational domain is represented, i.e., $(x,y)\in {[-1.5b,1.5b]}^2$ for $t/T_b\in \{0,0.015,0.15\}$ and $(x,y)\in {[-2b_0,2b_0]}^2$ for $t/T_b=1.5$. The $E_4$ scale is used. The solid and dotted thin lines correspond, respectively, to ${\omega }_B=\pm 0.1{\parallel {\omega }_B\parallel }_{\infty }$.

Figure 17

Optimal perturbation ${\omega }_0$ and optimal outcome ${\omega }_f$ vorticity fields obtained for a double-elliptic vorticity sheet with $(h_e,\mathrm{Re})=(0.05,1280)$ and three different horizon times. Same conventions as described in the caption of Fig. 16.