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Entropy and Adjoint Methods

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Abstract

Aerodynamic drag can be partially approximated by the entropy flux across fluid domain boundaries with a formula due to Oswatitsch. In this paper, we build the adjoint solution that corresponds to this representation of the drag and investigate its relation to the entropy variables, which are linked to the integrated residual of the entropy transport equation. For inviscid isentropic flows, the resulting adjoint variables are identical to the entropy variables, an observation originally due to Fidkowski and Roe, while for non-isentropic flows there is a significant difference that is explicitly demonstrated with analytic solutions in the shocked quasi-1D case. Both approaches are also investigated for viscous and inviscid flows in two and three dimensions, where the adjoint equations and boundary conditions are derived. The application of both approaches to mesh adaptation is investigated, with especial emphasis on inviscid flows with shocks.

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Notes

  1. Transonic rotational flows contain an additional singularity in the form of a slip line emanating from the sharp trailing edge. However, this is a contact discontinuity (no mass flow), so it does not contribute to (48).

  2. Notice also from Fig. 14 that, as in the quasi-1D case, entropy-based local indicators serve quite effectively as well as shock detectors.

  3. In Tau, the viscous flux of a quantity \( \phi \) under the effect of viscosity \( \mu \) across the dual grid face associated with edge ij is given by \( \mu \nabla \phi_{ij} \cdot \vec{n}_{ij} \), where \( \nabla \phi_{ij} = \tfrac{1}{2}(\nabla \phi_{i} + \nabla \phi_{j} ) \) in the plane perpendicular to \( \Delta \vec{x}_{ij} = \vec{x}_{i} - \vec{x}_{j} \) and \( \nabla \phi_{ij} = (\phi_{i} - \phi_{j} )\tfrac{{\Delta \vec{x}_{ij} }}{{\Delta \vec{x}_{ij} \cdot \Delta \vec{x}_{ij} }} \) in the direction of \( \Delta \vec{x}_{ij} \). Here, \( \nabla \phi_{i} \) and \( \nabla \phi_{j} \) are the gradients of \( \phi \) in the dual grid cells i and j obtained by either Green–Gauss or least-squares reconstruction.

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Acknowledgements

This work has been supported by INTA and the Spanish Ministry of Defence under the research program “Termofluidodinámica” (IGB99001). The 2D and 3D computations were carried out with the TAU Code, developed at DLR’s Institute of Aerodynamics and Flow Technology at Göttingen and Braunschweig, which is licensed to INTA through a research and development cooperation agreement.

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  1. Computational Aerodynamics Group, National Institute for Aerospace Technology (INTA), Torrejón de Ardoz, Spain

    Carlos Lozano

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Lozano, C. Entropy and Adjoint Methods. J Sci Comput 81, 2447–2483 (2019). https://doi.org/10.1007/s10915-019-01092-0

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