On optimization of natural convection flows

Two different techniques are presented for enhancing the thermal performance of natural convection cooled heat sinks. The physics is described by solving the conjugate heat transfer problem with a spectral element method. The temperature distribution is computed in two sub-domains, whose solution is then conjugated. Additionally, the contribution of the natural convection to the heat transfer is evaluated by solving the complete incompressible Navier–Stokes equations in the fluid domain.

One method focuses on the natural convection driven flow. The disturbances about a base flow are sought, which yield the maximal transient growth of a quadratic functional measuring the thermal performance. The “optimal initial condition” method is used for identifying the above mentioned perturbations. The control variable is the initial state of the perturbation and the problem is subject to the constraints enforced by the linearized governing equations. This method is validated in a simple two–dimensional setup and then applied to a periodic heat sink.

The second approach is a topology optimization of the heat sink itself. The design of the solid is optimized for maximizing the heat flux. The control variable is a so-called material distribution function that describes the presence of solid and fluid in the domain. By modifying the design of the heat sink, the flow is optimally conveyed and, by convection, it extracts the maximal possible amount of heat from the solid. The constraints are given by the governing equations, the position of the heat sink, and some manufacturing constraints (i.e., the maximal volume, or the minimal thickness). After a validation in a two– dimensional setup, the method is applied to a three–dimensional case. Complex tree-like shaped heat sinks induce an increase of the thermal performance by 5 to 16%, depending on the conditions considered.