A multi-objective function for discrete topology optimization in labyrinth seal design problems

Abstract

Labyrinth seals are commonly used in sealing mechanisms to separate regions with different pressures and minimize leakage along their intricate fluid paths. In this paper, topology optimization is applied to labyrinth seal design via a novel multi-objective expression combining forward and backward flows. However, the traditional strategy is susceptible to the bad local minimum of fluid inlet/outlet closure and the absence of interlaced labyrinth-like solid regions in the final design. The aim is to provide a solution to both issues. In our approach, the labyrinth seal objective is defined by combining fluid flow energy dissipation with vorticity magnitude to design the flow path that should be favored in one direction (forward) while unfavored in the opposite direction (backward). Therefore, we address the optimization problem in the form of simultaneous minimization of forward energy dissipation while maximizing backward vorticity. Volume fraction is assumed as the optimization constraint. The Topology Optimization of Binary Structures (TOBS) method is used to solve the optimization problem. This is a gradient-based method that produces a sequence of linearly approximated problems and solves them via integer linear programming. The steady Navier–Stokes equations govern the fluid motion with the standard Darcy term used for topology optimization. It is demonstrated that the porous material model favors solutions with labyrinths of radial interlacing teeth for higher porosity values and axial interlacing topologies for lower values. Numerical examples are presented for two-dimensional prismatic and axisymmetric problems with real CO${}_2$ gas properties.