Stable stratification enhances transient growth in streaky shear flows

TL;DR

This paper probes how stable stratification affects transient growth in a two-dimensional streaky base flow modeled by an augmented Kelvin framework. By adding a constant stable density gradient with Brunt-Väisälä frequency $N$ to the base flow ${\boldsymbol U}_B = [y + \beta \cos{(k_z z)}]{\hat{\boldsymbol x}}$ and expanding perturbations in Kelvin modes, the authors derive evolution equations for the wall-normal velocity $v$, wall-normal vorticity $\eta$, and density $\rho$, enabling computation of the optimal energy gain $G(T, k_x, k_z, \beta, N, \text{Re}, \text{Pr})$ under vertical ($\alpha=0$) and horizontal ($\alpha=\tfrac{1}{2}\pi$) shear. They show that vertical stratification suppresses wall-normal motions, unleashing growth up to roughly three orders of magnitude larger than unstratified cases and approaching the growth of a wall-normal velocity-removed minimal model, with a predictive threshold $N_c = \tfrac{1}{\sqrt{2}}\beta k_x$ for suppression in the minimal system; conversely, horizontal stratification inhibits growth by dampening spanwise motion. Across reduced and full (up to $M=10$) models, increasing $N$ drives the stratified solutions toward the unstratified minimal behavior in the vertical-shear case while reducing growth in the horizontal-shear case, revealing distinct roles for vertical versus horizontal motions in the near-wall cycle. The work provides a simple screening criterion for stratified growth and highlights potential implications for stratified near-wall turbulence in natural environments.

Abstract

Recent work has found that the well-known `lift-up' mechanism is not important for, and may even inhibit, the transient growth possible on streaky wall-bounded shear flows which is believed an important process in the near-wall cycle for turbulent flows. Moreover, artificially removing the wall-normal velocity has been found to unleash 3 orders of magnitude more perturbation energy growth in an unbounded streaky flow model. Motivated by this, we examine the effect of introducing stable stratification which naturally suppresses wall-normal velocities (the `vertical' shear case) and find it permits the hugely enhanced linear energy growth predicted by simply removing the wall-normal velocity. Alternatively, imposing stable stratification such that the spanwise velocities are suppressed (`horizontal shear') not surprisingly inhibits transient growth by weakening the active `push over' mechanism. A formula for the critical stratification strength to completely suppress the preferred growth mechanism is determined which proves a useful predictor for what is seen in the full numerical solutions of the model. Implications for a stratified near-wall cycle are briefly discussed.

Figures (9)

• Figure 1: The vertical unbounded constant shear is shown in red, extended by horizontal (spanwise) streaks in green, with the gravity vector indicated in blue.
• Figure 2: Two sequences of contour plots of the optimal gain in wavenumber space, for parameters $T=5$, $\beta=1$, $\text{Re}=200$ and $\text{Pr}=7$. The upper row has $N=1$ and the lower row $N=20$. The columns correspond to increasing complexity in the model used: the first (leftmost) is the unstratified minimal system of OK25, the second the reduced system, and the third (rightmost) shows the full system with $M=10$. This plot shows that the reduced and full systems behave growthwise like the unstratified minimal system as the stratification gets large enough.
• Figure 3: Left: global optimal gain vs $N$ for the parameters $T=5$, $\beta=1$, $\text{Re}=200$ and $\text{Pr}=7$. The solid line is for the reduced system, the dashed line is for the full system (with $M=10$) and the dotted line is for the unstratified minimal system of OK25. Right: same as left but now for the optimising $k_x$ (blue) and $k_z$ (red). These plots demonstrate clearly that the full problem exhibits enhanced growth when stratification is increased, and that the unstratified minimal system becomes a good approximation to the full solution for large $N$.
• Figure 4: Left: potential (solid lines) and kinetic (dashed lines) energies over time for the global optimal perturbation in the full system ($M=10$) for $T=5$, $\beta=1$, $\text{Re}=200$ and $\text{Pr}=7$ for $N=1$ (blue), $N= 5$ (maroon) and $N=50$ (black). Right: A blow-up of the potential energy (solid black again) and vertical kinetic energy (i.e. that due to $v$ alone, dashed purple) for $N=50$. This shows that the vertical kinetic energy and potential energy oscillate out of phase indicating that internal waves are initially excited at $N=50$.
• Figure 5: A close-up plot of the early time potential energy evolution for $N=50$, with the solid black line showing the same data as that used for the left plot, while the transparent thick red line is used to display the data which takes $M=20$ as well as tighter integration and optimisation tolerances to confirm that the signal is real. The inset plots are of $v$ (left in each pair) and $\rho$(right) over $(z,y) \in [0,2\pi/k_z] \times [-\pi,\pi]$ at $x=0$ with the $z$ dimension magnified by a factor of 5 for clarity. The cross-shear wavenumber $k_y=1-k_x t$ vanishes at $t \approx 0.2$ here ($k_x \approx 5$ and $k_z \approx 14.7$: see figure \ref{['vshearvN']}(right) ).