Optimal heat transfer enhancement in plane Couette flow

Abstract

Optimal heat transfer enhancement has been explored theoretically in plane Couette flow. The velocity field to be optimised is time-independent and incompressible, and temperature is determined in terms of the velocity as a solution to an advection-diffusion equation. The Prandtl number is set to unity, and consistent boundary conditions are imposed on the velocity and the temperature fields. The excess of a wall heat flux (or equivalently total scalar dissipation) over total energy dissipation is taken as an objective functional, and by using a variational method the Euler-Lagrange equations are derived, which are solved numerically to obtain the optimal states in the sense of maximisation of the functional. The laminar conductive field is an optimal state at low Reynolds number Re ∼ 10 0 . At higher Reynolds number Re ∼ 10 1 , however, the optimal state exhibits a streamwise-independent two-dimensional velocity field. The two-dimensional field consists of large-scale circulation rolls that play a role in heat transfer enhancement with respect to the conductive state as in thermal convection. A further increase of the Reynolds number leads to a three-dimensional optimal state at Re 10 2 . In the three-dimensional velocity field there appear smaller-scale hierarchical quasi-streamwise vortex tubes near the walls in addition to the large-scale rolls. The streamwise vortices are tilted in the spanwise direction so that they may produce the anticyclonic vorticity antiparallel to the mean-shear vorticity, bringing about significant three-dimensionality. The isotherms wrapped around the tilted anticyclonic vortices undergo the cross-axial shear of the mean flow, so that the spacing of the wrapped isotherms is narrower and so the temperature gradient is steeper than those around a purely streamwise (twodimensional) vortex tube, intensifying scalar dissipation and so a wall heat flux. Moreover, the tilted anticyclonic vortices induce the flow towards the wall to push low-(or high-) temperature fluids on the hot (or cold) wall, enhancing a wall heat flux. The optimised three-dimensional velocity fields achieve a much higher wall heat flux and much lower energy dissipation than those of plane Couette turbulence.

Wall-to-wall optimal transport in two dimensions

“…We wonder exactly how large Pe must be for the higher modes to play a significant role in optimizing heat transport in a two-dimensional fluid layer. The three-dimensional computations reported in Motoki et al (2018a) show a stark difference in flow structures and transport scaling as compared to the two-dimensional flows for both stress-free boundaries from Hassanzadeh et al (2014) and no-slip boundaries from §4 of the present paper (see also Motoki et al (2018b)). Three-dimensional versions of branching appear to achieve the optimal scaling Nu ∼ Pe 2/3 already for Pe ∈ [10 3 , 10 4 ] with a prefactor within 10% of the background upper bound computations from Plasting & Kerswell (2003).…”

“…This strictly enforces the advection-diffusion equation and its adjoint at every time-step, Utilising δF δu to compute corrections to the flow field. This approach was taken in Motoki et al (2018b), and the rate limiting step is the solution of the advectiondiffusion equation and its adjoint. In the present work, we take a different approach and compute numerical solutions to (3.3) and (3.8) Utilising two different algorithms.…”

Maximal heat transfer between two parallel plates

“…Near the walls, meanwhile, selfsimilar vortical structures locally enhance heat transfer, and yield logarithmic mean temperature distribution. Our earlier optimisation for heat transfer in plane Couette flow (Motoki et al 2018) provided the optimal velocity fields in which we observed hierarchical structure consisting of a number of streamwise vortex tubes. The logarithmic mean temperature profiles as well as the ultimate scaling N u ∼ Ra 1/2 were also found in the optimal fields.…”

“…This is because G = F − (µ/2)P e 2 and thus the Euler-Lagrange equations for G are also given by (2.8)-(2.11). In our previous work on a different functional in a different configuration (Motoki et al 2018), we have developed a numerical approach to find local maxima of a functional kindred to G by a combination of the steepest ascent method and the Newton-Krylov method. Using the same procedures, we obtain an optimal state (u opt , θ opt , θ * opt , p * opt ) maximising G for given µ.…”

“…taking account of the fact that the decrease (or increase) in µ corresponds to the increase (or decrease) in P e, where ǫ is a small positive constant. Equations (2.8)-(2.11) are discretised employing the spectral Galerkin method based on Fourier-Chebyshev expansions (for more details, see section 3 and appendix A in Motoki et al 2018).…”