On the Contribution of Wall Distance Fields to the Adjoint of a RANS Model

Abstract

The adjoint method has been extensively used in many areas of CFD such as gradient-based shape optimisation. When utilising the RANS equations for simulating turbulent flows, the adjoint method requires a scrupulous differentiation of the RANS equations, including the wall distance contribution. This can be a challenging task and a potential source of inaccuracy for functional sensitivities if not correctly executed. This paper presents a formulation for including the contribution of an equation-based wall distance model to the discrete adjoint of a RANS model. The proposed formulation is tested in a gradient-based optimisation scenario and the effects of the wall distance adjoint fields on the functional sensitivities are investigated. Neglecting the contribution of the wall distance adjoint yields an error in the functional sensitivities with respect to volume mesh nodes. Including the wall distance adjoint restores the accuracy of the functional sensitivities yielding better convergence of the design optimisation.

Keywords:

• CFD
• adjoint method
• turbulence models
• wall distance
• PDE-based wall distance
• shape optimisation

Acknowledgments

The authors acknowledge and thank Prof. Mark Turner for his valuable feedback and recommendations regarding the work presented in this manuscript. The authors would also like to thank the University of Cincinnati College of Engineering and Applied Science for the funding provided for supporting this research effort. Cleared for Public Release on 13 October 2021. Case Number: AFRL-2021-3499.

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Notes

1 the direct approach exploits the derivatives of flow variables to the grid node coordinates as intermediate quantities instead of the adjoint variables, thus solving directly Equation (12(12) $\begin{array}{rl}\frac{\mathrm{\partial }\mathit{u}}{\mathrm{\partial }\mathit{x}}+\frac{\mathrm{\partial }\mathit{u}}{\mathrm{\partial }\mathit{d}}\frac{\mathrm{\partial }\mathit{d}}{\mathrm{\partial }\mathit{x}}& =-{\left(\frac{\mathrm{\partial }\mathit{R}}{\mathrm{\partial }\mathit{u}}\right)}^{-1}[\frac{\mathrm{\partial }\mathit{R}}{\mathrm{\partial }\mathit{x}}+\frac{\mathrm{\partial }\mathit{R}}{\mathrm{\partial }\mathit{d}}\frac{\mathrm{\partial }\mathit{d}}{\mathrm{\partial }\mathit{x}}]\end{array}$(12) ).