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CivilComp Proceedings
ISSN 17593433 CCP: 100
PROCEEDINGS OF THE EIGHTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY Edited by: B.H.V. Topping
Paper 100
AdjointBased Sensitivity Analysis for BuoyancyDriven Incompressible NavierStokes Equations with Heat Transfer A. Lincke and T. Rung
Fluiddynamics and Ship Theory, Hamburg University of Technology, Germany A. Lincke, T. Rung, "AdjointBased Sensitivity Analysis for BuoyancyDriven Incompressible NavierStokes Equations with Heat Transfer", in B.H.V. Topping, (Editor), "Proceedings of the Eighth International Conference on Engineering Computational Technology", CivilComp Press, Stirlingshire, UK, Paper 100, 2012. doi:10.4203/ccp.100.100
Keywords: sensitivity analysis, heat transfer, adjoint NavierStokesFourier equations, shape optimisation, Boussinesq approximation, finite volume method.
Summary
This paper presents the adjointbased sensitivity equation and the adjoint buoyancydriven steadystate NavierStokesFourier equations for a perfect gas.
The approach is based on the augmented cost function which includes the primal governing equations,
in our case the Reynoldsaveraged NavierStokesFourier (RANSF) equations as constraints, which
have to be satisfied in the computational domain.
Starting from the residual form of the weakly compressible, buoyancydriven, RANSF 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, nonzero 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 twodimensional 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 reused, to solve the adjoint equations. The implementation of a semiimplicit pressurelinked 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 nonlinearities of the primal flow field (i.e. flow reversal). Related applications are of relevance for the optimisation of fluidsengineering devices where the temperature distribution is crucial, e.g. in thermalcomfort studies. References
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