Session G17: Forced Convection I

Asymptotic solutions for forced convection in a shrouded longitudinal-fin heat sink

We consider laminar forced convection in a fully-shrouded longitudinal-fin heat sink (LFHS), as described by the pioneering study of Sparrow, Baliga and Patankar [1978, J. Heat Trans, 100(4)]. The base of the LFHS is isothermal and the fins are assumed thin such that the fins are not isothermal but the temperature distribution from their base to their tip is considered, and coupled to the fluid flow. Whereas Sparrow et al. solved the fully-developed flow and thermal problems numerically for a range of geometries (they also considered tip clearance), we consider here the physically realistic asymptotic limit where the fin spacing is small in comparison to their height. Using asymptotic analysis, we find approximate reduced-order solutions for the flow field, temperature field (in both the fluid and along the fins), and hence the local and average Nusselt numbers. We compare the solutions to full numerical results for a range of fin spacings, and assess the practical use of our reduced model.   

Transition to branching flows in optimal planar convection

We study steady flows that are optimal for heat transfer in a two-dimensional periodic domain. The flows maximize heat transfer under the constraints of incompressibility and a given energy budget (i.e. mean viscous power dissipation). Using an unconstrained optimization approach, we compute optima starting from 30--50 random initializations across several decades of Pe, the energy budget parameter. At Pe between 10$^{4.5}$ and 10$^{4.75}$, convective rolls with U-shaped branching near the walls emerge. They exceed the heat transfer of the simple convective roll optimum at Pe between 10$^{5}$ and 10$^{5.25}$. At larger Pe, multiple layers of branching occur in the optima, and become increasingly elongated, asymmetrical, and heterogeneous. The rate of heat transfer scales as Pe$^{0.575}$, which, in this range of Pe, is very close to the Pe$^{2/3}(log , $Pe)$^{-4/3}$ behavior of self-similar branching solutions proposed earlier. Compared to the simple convective roll, the branching flows have lower maximum speeds and thinner boundary layers, but nearly the same maximum power density.   

Heat transport in fixed flux non-uniform internally heated convection

Rigorous scaling laws for turbulence driven by thermal convection can be obtained by use of the auxiliary functional method. By use of a quadratic auxiliary functional since the 1990s upper bounds on heat transport for Rayleigh-Benard convection have been known. If the boundary conditions are fixed flux bounds can be proven albeit with greater effort, however the scaling laws remain unchanged as compared to fixed temperature boundary conditions. This is not the case for internally heated convection. Further still for internal heating with fixed flux boundaries, the rigorous upper bounds cannot rule out conduction from the top to the bottom for all Rayleigh numbers. One idea for the proof of a "physical" result is to consider a non-uniformly heated plane layer. We can demonstrate that a class of non-uniform heating profiles exist, which decreases the unphysical bound on mean inverse conduction. This inverse mean conduction is zero when the non-uniform heating is localised entirely at the lower boundary.