Second-order sensitivity and optimal spanwise-periodic flow modifications

We perform a second-order sensitivity analysis of the linear temporal stability of a parallel mixing layer $U_0(y)$ subject to small spanwise-periodic modification $\epsilon U_1(y)\cos(\beta z)$. It is known that spanwise-periodic flow modifications have a quadratic effect on the stability properties of parallel flows (i.e. the first-order eigenvalue variation is zero), hence the need for a second-order analysis. From a simple one-dimensional (1D) calculation we compute the second-order sensitivity operator, which allows us to predict the effect on stability of any $U_1(y)$ without computing the eigenmode correction. Comparisons with two-dimensional (2D) stability calculations of modified flows show excellent agreement. From the second-order sensitivity operator we optimise the growth rate variation and compute the most stabilising and most destabilising $U_1$, together with lower and upper bounds for the growth rate variation induced by any spanwise-periodic modification of given amplitude $\epsilon$. These bounds show a local maximum for a spanwise wavenumber of order unity $\beta \simeq 1$, while they diverge like $\beta^{-2}$ as $\beta$ goes to zero (long-wavelength limit). The most stabilising (destabilising) $U_1$ determined for the most unstable streamwise wavenumber $\alpha_{max}$ is efficient in stabilising (destabilising) the flow for other values of $\alpha$ too. We also find that optimal 2D spanwise-periodic flow modifications are more efficient in stabilising (destabilising) than the optimal 1D spanwise-invariant modification provided the amplitude $\epsilon$ and wavelength $2\pi/\beta$ are large enough.