Absolute/convective secondary instabilities and the role of confinement in free shear layers

We study the linear spatiotemporal stability of an infinite row of equal point vortices under symmetric confinement between parallel walls. These rows of vortices serve to model the secondary instability leading to the merging of consecutive (Kelvin-Helmholtz) vortices in free shear layers, allowing us to study how confinement limits the growth of shear layers through vortex pairings. Using a geometric construction akin to a Legendre transform on the dispersion relation, we compute the growth rate of the instability in different reference frames as a function of the frame velocity with respect to the vortices. This approach is verified and complemented with numerical computations of the linear impulse response, fully characterizing the absolute/convective nature of the instability. Similar to results by Healey on the primary instability of parallel tanh profiles [J. Fluid Mech. 623, 241 (2009)], we observe a range of confinement in which absolute instability is promoted. For a parallel shear layer with prescribed confinement and mixing length, the threshold for absolute/convective instability of the secondary pairing instability depends on the separation distance between consecutive vortices, which is physically determined by the wavelength selected by the previous (primary or pairing) instability. In the presence of counterflow and moderate to weak confinement, small (large) wavelength of the vortex row leads to absolute (convective) instability. While absolute secondary instabilities in spatially developing flows have been previously related to an abrupt transition to a complex behavior, this secondary pairing instability regenerates the flow with an increased wavelength, eventually leading to a convectively unstable row of vortices. We argue that since the primary instability remains active for large wavelengths, a spatially developing shear layer can directly saturate on the wavelength of such a convectively unstable row, by-passing the smaller wavelengths of absolute secondary instability. This provides a wavelength selection mechanism, according to which the distance between consecutive vortices should be sufficiently large in comparison with the channel width in order for the row of vortices to persist. We argue that the proposed wavelength selection criteria can serve as a guideline for experimentally obtaining plane shear layers with counterflow, which has remained an experimental challenge.

Figure 1

Confined single row of point vortices.

Figure 2

Temporal dispersion relation. (a) Growth rate ${\omega }_i$ vs wave number $k$ for different confinement ratios. (b) Its value at $k=\pi$ vs confinement ratio $q$.

Figure 3

(a) Isocontours of ${\omega }_i$ in the complex $k$ plane, calculated from the dispersion relation (9) for $q=0.8$. The solid red line represents the locus of absolute wave numbers defined by (19). (b) Geometrical construction to determine the growth rate $\sigma \left(v\right)$ from ${\omega }_i(k_{0r}^v\left(k_i\right),k_i)$. The $k_{0i}^v$ pertaining to a particular $v$ is given by (18a), and the corresponding growth rate is obtained from (17). The dashed line on the left side of the plot shows the ${\omega }_i$ given by approximation (10), with the diamond indicating the inflection point determining the validity limit of the approximation (see text).

Figure 4

(a) Growth rate $\sigma \left(v\right)$ of the impulse response wave packet along spatiotemporal rays $x/t=v$ for a confinement ratio $q=0.8$. The streamwise extent of the wave packet is given by the leading- and trailing-edge velocities $v_g^{\pm }$ such that $\sigma \left(v_g^{\pm }\right)=0$. Results from imposing the transformation (21) to the exact (solid line) and approximate (dashed line) dispersion relations and from numerical simulations of the impulse response (circles). (b) Magnitude of the leading- and trailing-edge velocities $|v_g^{\pm }|$ of the impulse response wave packet versus the confinement ratio $q$. Results from the analytical dispersion relation with the graphical method (solid line) are compared with those from numerical simulations of the impulse response (circles). As $q$ increases, $|v_g^{\pm }|$ quickly approaches the theoretical value of $\pi$ for the unconfined single row of point vortices [44].

Figure 5

Spatiotemporal behavior of the $tanh$ profile (24) and the row of point vortices. Domains of absolute and convective primary instability $(tanh$ profile) are separated by the critical velocity ratio $R_{c1}$ (solid line), while in the gray region it is stable. Domains of absolute and convective instability of the SPI, separated by the critical velocity ratio $R_{c2,\lambda },$ are shown for different values of the intervortex spacing $\lambda =4\pi ,8\pi ,12\pi ,16\pi ,...,44\pi$ (dotted lines). In both cases, values of $R$ greater (lower) than the reported critical value lead to absolute (convective) instability. The $R_{c2,\lambda }$ determine the critical velocity ratio of the 2VPI to the right of ${\overline{R}}_{2C}$ (dashed line), below which the 2VPI is always convective. The inset shows a close-up around ${\overline{R}}_{2C}$ including various intersecting $R_{c2,\lambda }$ (solid lines). Results for the primary instability are from Healey [6].

Figure 6

Tentative scenario for wavelength selection. If the ratio $\lambda /h$ of intervortex spacing to midchannel width is such that the secondary instability is absolute (2A), successive vortex pairings will increase $\lambda$ and eventually bring the system in the region where the secondary instability is convective (2C), in which it will remain. According to this interpretation, a spatially developing mixing layer with counterflow $(R>1)$ can only exhibit a row of vortices if $\lambda /h$ is large enough to fall in the region 2C.

Figure 7

Isocontours of ${\omega }_i$ in the complex $k$ plane for the dispersion relation of the Kármán street (A1) with $p=0.3$ (left) and $p=0.316$ (right). The red line represents the locus ${\mathcal{K}}_0$ of absolute wave numbers defined by (19). Note that in both cases ${\mathcal{K}}_0$ contains multiple values of $k_r$ for some intervals of $k_i$.

Figure 8

Imaginary part of the frequency along the locus ${\mathcal{K}}_0$ of absolute wave numbers, expressed as a multivalued function ${\omega }_i(k_{0r}^v\left(k_i\right),k_i)$ of $k_i$ for $p=0.3$ (left) and $p=0.316$ (right).

Figure 9

Growth rate $\sigma \left(v\right)$ of the impulse response wave packet along spatiotemporal rays $x/t=v$ for $p=0.3$ (left) and $p=0.316$ (right). The streamwise extent of the wave packet is given by the leading- and trailing-edge velocities $v_g^{\pm }$ such that $v_g^{\pm }=0$.