Curves and surfaces for computer aided geometric design: a practical guide
Curves and surfaces for computer aided geometric design: a practical guide
Mixed and hybrid finite element methods
Mixed and hybrid finite element methods
Numerics and hydrodynamic stability: toward error control in computational fluid dynamics
SIAM Journal on Numerical Analysis
Finite Element Method for Elliptic Problems
Finite Element Method for Elliptic Problems
Perspectives in Flow Control and Optimization
Perspectives in Flow Control and Optimization
Journal of Computational and Applied Mathematics
Journal of Computational Methods in Sciences and Engineering
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Hydrodynamic stability plays a crucial role for many applications. Existing approaches focus on the dependence of the stability properties on control parameters such as the Reynolds or the Rayleigh number. In this paper we propose a numerical method which aims at solving shape optimization problems in the context of hydrodynamic stability. The considered approach allows to guarantee hydrodynamic stability by modifying parts of the underlying geometry within a certain flow regime. This leads to a formulation of a shape optimization problem with constraints on the eigenvalues related to the linearized Navier-Stokes equations. In that context the eigenvalue problem is generally non-symmetric and may involve complex eigenvalues. To validate the proposed numerical approach we consider the flow around a body in a channel. The shape of the body is parameterized and can be changed by means of a discrete number of design variables. It is our aim to find a design which minimizes the drag force and ensures at the same time hydrodynamic stability while keeping the volume of the body constant. The numerical results show that a transition from an unstable design to a stable one is attainable by considering an adequate change of the body shape. The resulting bodies are long and flat which corresponds to common intuition.