Enhancing heat transfer in a channel with unsteady flow perturbations

We compute unsteady perturbations that optimally increase the heat transfer (Nu) of optimal steady unidirectional channel flows, for a given average rate of power consumption Pe$^2$. The perturbations are expanded in a basis of modes, and the heat transfer enhancement corresponds to eigenvalues of the Hessian matrix of second derivatives of the Nusselt number with respect to the mode coefficients. Enhanced heat transfer, i.e. positive eigenvalues, occur in a range of temporal periods $\tau$ that scale as Pe$^{-1}$. At small to moderate $\tau$Pe values the corresponding flows are chains of eddies near the walls that move as traveling waves at the steady background flow speed. At large $\tau$Pe the flows have eddies of multiple scales ranging up to the domain size. We use an unsteady solver to simulate these flows with perturbation sizes ranging from small to large, and find increases in Nu of up to 56% at Pe = 2$^{19}$. Large Nu can be obtained by eddies with small spatial/temporal scales and by eddies with a range of spatial scales and large temporal scales.