Time-averaged algorithm for solving the topology optimization problem for unsteady laminar, turbulent and anisothermal flows

This paper proposes a new algorithm to solve topology optimization problems for laminar unsteady or turbulent flows. Instead of computing the gradient of the cost function after solving the direct and adjoint (both unsteady) PDE on the full time interval, our algorithm uses averaged physical quantities on a smaller unspecified time interval to define a (steady) Reynolds-Averaged Method (RAM) model which is then used as constraint in an optimization problem to update the design variable. Another feature of the proposed method is that the RAM model can be defined whatever the initial model and CFD turbulence models initially chosen to compute the instantaneous physical quantities. The RAM model involves turbulent quantities such as turbulent kinetic viscosity and turbulent thermal diffusivity are estimated instead of using the concept of "frozen turbulence". In contrast with the classical methods built to solve unsteady topology optimization problems, the main advantage of the proposed algorithm is that it updates the design variable by solving an auxiliary steady topology optimization problem. Three configuration cases are studied to illustrate the ability of our algorithm to optimize pressure losses and heat transfer by adding material to smooth the laminar unsteady or turbulent flows. We also calculate the number of required design parameter updates to obtain an optimized design. Thus, our algorithm overcomes three major scientific challenges in solving optimization problems in turbulence, namely leveraging efficient temporal turbulence models or a Direct Numerical Simulation (DNS) model, computational cost and data storage requirements.