Enhancing wall-to-wall heat transport with unsteady flow perturbations

We determine unsteady flow perturbations that are optimal for enhancing the rate of heat transfer between hot and cold walls (i.e. the Nusselt number Nu), under the constraint of fixed flow power (Pe$^2$, where Pe is the P\'{e}clet number). The unsteady flows are perturbations of previously computed optimal steady flows and are given by eigenmodes of the Hessian matrix of Nu, the matrix of second derivatives with respect to amplitudes of flow mode coefficients. Positive eigenvalues of the Hessian correspond to increases in Nu by unsteady flows, and occur at Pe $\geq 10^{3.5}$ and within a band of flow periods $\tau \sim$ Pe$^{-1}$. For $\tau$Pe $\leq 10^{0.5}$, the optimal flows are chains of vortices that move along the walls or along eddies enclosed by flow branches near the walls. At larger $\tau$Pe the vorticity distributions are often more complex and extend farther from the walls. The heat flux is enhanced at locations on the walls near the unsteady vorticity. We construct an iterative time-spectral solver for the unsteady temperature field and find increases in Nu of up to 7% at moderate-to-large perturbation amplitudes.