Keywords: sensitivity analysis, heat transfer, adjoint Navier-Stokes-Fourier equations, shape optimisation, Boussinesq approximation, finite volume method.

This paper presents the adjoint-based sensitivity equation and the adjoint buoyancy-driven steady-state Navier-Stokes-Fourier equations for a perfect gas. The approach is based on the augmented cost function which includes the primal governing equations, in our case the Reynolds-averaged Navier-Stokes-Fourier (RANS-F) equations as constraints, which have to be satisfied in the computational domain. Starting from the residual form of the weakly compressible, buoyancy-driven, RANS-F equations, the adjoint momentum, continuity and temperature equations are derived by building the variational formulation of the augmented cost function. Integration by parts leads to a formulation where variations of primal flow variables can be factored out. Similar approaches have been made in [1,2]. The remaining, non-zero part of the augmented cost function variation leads to the adjoint sensitivity equation. The adjoint sensitivity displays the gradient of the cost function with respect to a normal variation of the boundary. The sensitivity can readily be coupled to a simple optimisation approach, i.e. steepest descent.

In the second part of this paper the model is verified by comparing the gradient of the cost function with the results from a direct differentiation method. Herein, the gradient of the cost function is directly computed by shifting each discrete boundary element by a small distance in the normal direction and evaluating the related change of the cost function. For the verification study the laminar flow over a two-dimensional heated ellipsoid is simulated which is centrally mounted in a channel. The cost function uniformity of temperature is employed at the outlet as an example for the adjoint sensitivity analysis. As the adjoint equations are similar to the primal equations, a great portion of the primal solver can be re-used, to solve the adjoint equations. The implementation of a semi-implicit pressure-linked equation solver (SIMPLE) is used within the open source computational fluid dynamics (CFD) environment open field operation and manipulation (OpenFOAM). The adjoint sensitivity is compared to the sensitivity obtained from the direct differentiation. It can be shown that the adjoint sensitivity displays a satisfactory agreement with the result of the direct differentiation analysis. Differences can be explained by pronounced local non-linearities of the primal flow field (i.e. flow reversal). Related applications are of relevance for the optimisation of fluids-engineering devices where the temperature distribution is crucial, e.g. in thermal-comfort studies.

1

Y. Suzuki, N. Kasagi, K. Morimoto, "Adjoint Analysis of Heat and Fluid Toward Optimal Shape Design of Recuperators", Thermal Science&Engineering, 13, 2005.

2

X. Luo, "Identifikation von Wärmequellen in konvektiv getriebenen Strömungen", Diplomarbeit, Universität Hamburg/Technische Universität Hamburg-Hamburg, Germany, 2011.

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