Why are bundle-valued forms important for computational fluid dynamics?

  • Rashad, Ramy (University of Twente)
  • Califano, Federico (University of Twente)
  • Brugnoli, Andrea (University of Twente)
  • Stramigioli, Stefano (University of Twente)

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Structure-preserving numerical schemes aim to respect the underlying topological and geometric structures underlying the governing equations of fluid dynamics. A major drawback of standard vector and tensor calculus formulations is that topology and geometry are inextricably intertwined and thus are not suitable for structure-preserving discretization. On the other hand, exterior calculus highlights the different between metric-dependent and metric-free operations as well as orientation-dependent and orientation-independent objects. Furthermore, exterior calculus uses differential forms which are naturally associated with differentiation and integration and thus are an appropriate tool for discretization. In our presentation we highlight the importance of bundle-valued differential forms for formulating fluid dynamics using exterior calculus and how standard scalar-valued differential forms are not applicable for fluid dynamics. Our main result is that even though both formulations yield the same expression for the viscous forces in the momentum equation for incompressible flow, the resulting expressions for the dissipated power within the spatial domain and the supplied power through the boundary are different. Only using bundle-valued forms does one get the correct powers and boundary conditions. Finally, we show in our work the implications of this difference for the development of computational fluid dynamics codes using finite-element exterior calculus.