The continuous adjoint approach to the k–ε turbulence model for shape optimization and optimal active control of turbulent flows

Abstract

The continuous adjoint to the incompressible Reynolds-averaged Navier–Stokes equations coupled with the low Reynolds number Launder–Sharma k–ε turbulence model is presented. Both shape and active flow control optimization problems in fluid mechanics are considered, aiming at minimum viscous losses. In contrast to the frequently used assumption of frozen turbulence, the adjoint to the turbulence model equations together with appropriate boundary conditions are derived, discretized and solved. This is the first time that the adjoint equations to the Launder–Sharma k–ε model have been derived. Compared to the formulation that neglects turbulence variations, the impact of additional terms and equations is evaluated. Sensitivities computed using direct differentiation and/or finite differences are used for comparative purposes. To demonstrate the need for formulating and solving the adjoint to the turbulence model equations, instead of merely relying upon the ‘frozen turbulence assumption’, the gain in the optimization turnaround time offered by the proposed method is quantified.

Keywords:

• continuous adjoint
• adjoint turbulence models
• sensitivity derivatives
• aerodynamic shape optimization
• active flow control optimization

Acknowledgements

This research was conducted in association with ICON Technology and Process Consulting (UK) Ltd. The methods presented in this article are implemented using technology based upon OpenFOAM^®1 and made available within the CFD service ICON FOAMpro, http://www.iconcfd.com/foampro

Notes

1. OpenFOAM^® is a CFD Toolbox produced by OpenCFD Ltd and distributed by the OpenFOAM Foundation.