This paper addresses the optimization of a two-dimensional U-bend passage of an internal serpentine cooling channel for reduced total pressure loss by means of a steepest-descent method. A steady-state incompressible flow is considered at a Reynolds number of 40,000 based on the bulk velocity at the domain inlet. The two-equation k-ε model is used for primal turbulence modeling. After only 30 design iterations, the gradient-based optimization results in a reduction of total pressure loss by 46% compared to the baseline geometry.
To obtain the required objective gradients efficiently, a continuous adjoint approach is implemented in the OpenFOAM environment. Adjoint governing equations and boundary conditions are derived from state equations for steady-state, incompressible, turbulent flows under the assumption of frozen turbulence. Two different methods are proposed for modifying the shape of internal and external curves defining the duct geometry. The first method makes use of direct displacement of boundary grid points, allowing for a wide design space. The second, novel parameterization utilizes a projection of the surface sensitivities to an underlying Bézier curve. In this case, the Bézier control points are used as design variables. A comparison of both methods demonstrates a slightly lower performance improvement by the Bézier-based approach due to the reduced design freedom. This approach has, however, several practical advantages.
Previous studies already addressed this optimization problem using gradient-free methods, but were limited in the degrees of freedom given to the shape variation. The present gradient-based optimization allows for a much larger design space and hence is used to compare the different methodologies. It shows that both optimizations result in similar shapes, although the gradient-based method allows for a slightly larger reduction in pressure loss due to the wider design space, while converging faster towards the optimum.