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Shape optimization of flow field improving hydrodynamic stability

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Abstract

This paper presents a solution of a shape optimization problem of a flow field for delaying transition from a laminar flow to a turbulent flow. Mapping from an initial domain to a new domain is chosen as the design variable. Main problems are defined by the stationary Navier–Stokes problem and an eigenvalue problem assuming a linear disturbance on the solution of the stationary Navier–Stokes problem. The maximum value of the real part of the eigenvalue is used as an objective cost function. The shape derivative of the cost function is defined as the Fréchet derivative of the cost function with respect to arbitrary variation of the design variable, which denotes the domain variation, and is evaluated using the Lagrange multiplier method. To obtain a numerical solution, we use an iterative algorithm based on the \(H^{1}\) gradient method using the finite element method. To confirm the validity of the solution, a numerical example for two-dimensional Poiseuille flow with a sudden expansion is presented. Results reveal that a critical Reynolds number increases by the iteration of reshaping.

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Authors and Affiliations

  1. Graduate School of Information Science, Tohoku University, Aoba-ku, Aramaki Aza, Aoba 6-3, Sendai, Miyagi, 980-8579, Japan

    Takashi Nakazawa

  2. Graduate School of Information Science, Nagoya University, Chikusa-ku Hurou-chou, Nagiya, Aichi, 464-8601, Japan

    Hideyuki Azegami

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  1. Takashi Nakazawa
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  2. Hideyuki Azegami
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Correspondence to Takashi Nakazawa.

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Nakazawa, T., Azegami, H. Shape optimization of flow field improving hydrodynamic stability. Japan J. Indust. Appl. Math. 33, 167–181 (2016). https://doi.org/10.1007/s13160-015-0201-9

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  • DOI: https://doi.org/10.1007/s13160-015-0201-9

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