As buoyancy can help drive a flow, the vertical heated-pipe arrangement is
widely used in thermal engineering applications. Cependant, buoyancy suppresses
and can even laminarise turbulence in the flow, thereby seriously damaging the
heat transfer, measured by the Nusselt number Nu. As buoyancy, measured by the
parameter C, is increased, three flow regimes are possible: shear-driven
turbulence, laminarised flow, and convective turbulence. In Chu et al. (2024)
we employed a variational optimisation method to investigate how the buoyancy
changes the structure of the minimal flow perturbation that triggers
turbulence. Here, we extend the method to find an optimal body force of limited
magnitude that maximises heat transfer, and examine how time-dependence of the
flow affects the optimisation in each of the three flow regimes. Optimisations
are performed at Re = 3000, and the force is found to laminarise convective
turbulence, or make it only weakly chaotic for C up to 8. Consistent with
previous computations that assume steady flow, the optimal force induces
streamwise-independent rolls, but at larger amplitude the force triggers
time-dependent turbulent flow. Transition from the laminar
streamwise-independent state to turbulent flow can either enhance Nu or reduce
Nu. For highly chaotic flows, either shear turbulence at C = 1 or convective
turbulence at C = 16, 32, optimisations place rolls closer to the wall than
calculations with the steady flow assumption. At any given force amplitude,
however, the enhanced Nu is only weakly dependent on the number of induced
rolls.

Cet article explore les excursions dans le temps et leurs implications.

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