Incompressible moderate-Reynolds-number flow in periodically grooved channels is investigated by direct numerical simulation using the spectral element method. For Reynolds numbers less than a critical value R_c the flow is found to approach a stable steady state, comprising an ‘outer’ channel flow, a shear layer at the groove lip, and a weak re-circulating vortex in the groove proper. The linear stability of this flow is then analysed, and it is found that the least stable modes closely resemble Tollmien–Schlichting channel waves, forced by Kelvin–Helmholtz shear-layer instability at the cavity edge. A theory for frequency prediction based on the Orr–Sommerfeld dispersion relation is presented, and verified by variation of the geometric parameters of the problem. The accuracy of the theory, and the fact that it predicts many qualitative features of low-speed groove experiments, suggests that the frequency-selection process in these flows is largely governed by the outer, more stable flow (here a channel), in contrast to most current theories based solely on shear-layer considerations. The instability of the linear mode for R > R_c is shown to result in self-sustained flow oscillations (at frequencies only slightly shifted from the originating linear modes), which again resemble (finite-amplitude) Tollmien-Schlichting modes driven by an unstable groove vortex sheet. Analysis of the amplitude dependence of the oscillations on degree of criticality reveals the transition to oscillatory flow to be a regular Hopf bifurcation.

Modulatory heat-transfer enhancement in grooved channels is investigated by direct numerical simulation of the Navier–Stokes and energy equations using the spectral element method. It is shown that oscillatory perturbation of the flow at the frequency of the least-stable mode of the linearized system results in subcritical resonant excitation and associated transport enhancement as the critical Reynolds number of the flow is approached. The Tollmien–Schlichting frequency theory that was presented in Part 1 of this paper is shown to accurately predict the optimal frequency for transport augmentation for small values of the modulatory amplitude, and the effect of the excited travelling-wave channel modes on the resulting temperature distribution is described. The importance of (non-trivial) geometry in the forced response of a flow is discussed, and grooved-channel flow is compared to (straight-channel) plane Poiseuille flow, for which no resonance excitation occurs owing to a zero projection of the forcing inhomogeneity on the dangerous modes of the system. For the particular grooved-channel geometry investigated, resonant oscillatory forcing at modulatory amplitudes as small as 20% of the mean flow results in a doubling of transport as measured by a time, space-averaged Nusselt number.