Effects of spanwise confinement on stratified shear instabilities

We consider the influence of transverse confinement on the instability properties of velocity and density distributions reminiscent of those pertaining to exchange flows in stratified inclined ducts, such as the recent experiment of Lefauve et al. [J. Fluid Mech. 848, 508 (2018)]. Using a normal mode streamwise and temporal expansion for flows in ducts with various aspect ratios $B$ and nontrivial transverse velocity profiles, we calculate two-dimensional (2D) dispersion relations with associated eigenfunctions varying in the “crosswise” direction, in which the density varies, and the spanwise direction, both normal to the duct walls and to the flow direction. We also compare these 2D dispersion relations to the so-called one-dimensional (1D) dispersion relations obtained for spanwise invariant perturbations, for different aspect ratios $B$ and bulk Richardson numbers ${\text{Ri}}_b$. In this limited parameter space, the presence of lateral walls has a stabilizing effect, in that the 1D growth-rate predictions are almost systematically an upper bound to the 2D growth rates, which in turn decrease monotonically as lateral walls are brought together with increased spanwise confinement ($B\to 0$). Furthermore, accounting for spanwise-varying perturbations results in a plethora of unstable modes, the number of which increases as the aspect ratio is increased. These modes present an odd-even regularity in their spatial structures, which is rationalized by comparison to the so-called one-dimensional oblique dispersion relation obtained for oblique waves, characterized by a continuously varying spanwise wavenumber in addition to the streamwise wavenumber. Finally, we show that in most cases, the most unstable 2D mode is the one that oscillates the least in the spanwise direction, as a consequence of viscous damping. However, in a limited region of the parameter space and in the absence of stratification, we show that a secondary mode with a more complex “twisted” structure dominated by crosswise vorticity becomes more unstable than the least oscillating Kelvin-Helmholtz mode associated with spanwise vorticity.

Figure 1

Schematic of our confined duct flow configuration (dimensional variables).

Figure 2

Illustration of the three different spanwise profiles, $M\left(y\right)=M_p\left(y\right)$ (black dashed), $M_{2.1}\left(y\right)$ (dark gray solid), and $M_5\left(y\right)$ (light gray solid), used in the rest of the paper. The full base velocity is $U(y,z)=-sin\left(\pi z\right)M\left(y\right)$.

Figure 3

Dispersion relations of the most unstable mode of the spectrum. The left column shows the growth rate ${\sigma }_r$, and the right column shows the frequency ${\sigma }_i$. We chose a Poiseuille spanwise profile $M_p$, four different aspect ratios $B$, and three different ${\text{Ri}}_b$ (rows). Solid and dashed lines stand for the 2D problem and the lighter shade of grey corresponds to higher $B$ in the set $B=[1,3,5]$. Dash-dotted lines mark the 1D problem ($B\to \infty$). A marker, which is different for each $B$, indicates the maximum growth rate ($k=k_m$).

Figure 4

Stabilizing effects of the aspect ratio $B$, spanwise profile $M\left(y\right)$, and Re. (a) Relative error between the 1D and 2D most unstable growth rate evaluated at the most unstable 1D wavenumber $k_m^{\text{1D}}\approx 1.88$. Three different spanwise profiles and two different Re are chosen, all for ${\text{Ri}}_b=1$. A circle symbol is placed at the threshold aspect ratio $B^{5\%}$ where the error is $E_r=5\%$; (b) variation of this threshold aspect ratio $B^{5\%}$ with ${\text{Ri}}_b$.

Figure 5

Unstable part of the spectra for $\text{Re}=440$, ${\text{Ri}}_b=0.25$, and $k=k_m^{\text{1D}}=1.88$. We show two different aspect ratio flows $B=3,5$ and two different $M=M_p,M_5$. The black star symbol marks the 1D eigenvalue, the black squares are the 2D eigenvalues, and the red line is the 1D-O dispersion relation as a function of the spanwise wavenumber $\beta$. The latter is the same for the four panels as it does not consider the spanwise variations of the base flow. As a consequence of the strongly negative $z_0$, note that all eigenvalues are in the ${\sigma }_i>0$ half plane.

Figure 6

Sliced views of $\rho ,u,v,w$ in the $x\text{−}y$ plane of the three most unstable 2D eigenmodes of Fig. 5 ($H_1$, $H_2$, and $H_3$, from left to right, ordered by decreasing growth rates). The spanwise profile is $M\left(y\right)=M_5\left(y\right)$, and the parameters are $\text{Re}=440$, ${\text{Ri}}_b=0.25$, $B=3$, $k=k_m^{\text{1D}}=1.88$.

Figure 7

Sliced view on the $x\text{−}z$ plane of the three 2D most unstable eigenmodes for $k=k_m^{\text{1D}}=1.88$. The 1D most unstable mode is added at the fourth column. Modes from left to right between the first and the third columns are $H_1$, $H_2$, and $H_3$, as ordered in decreasing order of growth rates. $M\left(y\right)=M_5\left(y\right)$, $\text{Re}=440$, ${\text{Ri}}_b=0.25$, and $B=3$.

Figure 8

Growth rates of oblique modes as a function of $\beta$ (red line), and of 2D modes at $k=k_m^{\text{1D}}=1.88$ as a function of both ${\lambda }_i^{\rho ,u,w}$ (open circles) and ${\lambda }_i^v$ (solid circles). Parameters are $M\left(y\right)=M_5\left(y\right)$, $\text{Re}=440$, and ${\text{Ri}}_b=0.25$. The 1D-O dispersion relation in $\beta$ predicts the discrete 2D spanwise harmonics increasingly well as $B$ is increased and $M_5\left(y\right)\to 1$ over most of the domain.

Figure 9

Visualizations of the $K{H}_1$ and $K{H}_T$ eigenmodes for $B=1$ and $k=0.64$ where ${\sigma }_r(k=0.64)\approx 0.0440$, ($M=M_p\left(y\right),\text{Re}=440,{\text{Ri}}_b=0$), highlighting their different spatial structure. (a) $K{H}_1$, where blue and green surfaces are isocontours of equal and opposite values of ${\omega }_y$; (b) $K{H}_T$, where yellow and magenta surfaces are isocontours of equal and opposite values of ${\omega }_z$

Figure 10

Some slice views of the $K{H}_T$ mode shown in Fig. 9. Dashed lines show the locations of the planes in the other columns.

Figure 11

Growth rates ${\sigma }_r$ of $K{H}_1$ (left column) and $K{H}_T$ (right column) in the $(k,B)$ plane for $\text{Re}=440,{\text{Ri}}_b=0$. We compare $M_5\left(y\right)$ (top row) to $M_p\left(y\right)$ (bottom row). At the left of the red line, inside the gray shaded area $K{H}_T$ is more unstable than $K{H}_1$. The black line is the most unstable growth rate ${\sigma }_m\left(B\right)$ over $k$.

Figure 12

Energy contribution to the $K{H}_T$ mode shown in Fig. 9.