Accurately predicting turbulent fluid mechanics remains a significant challenge in engineering and applied science. Reynolds-Averaged Navier–Stokes (RANS) simulations and Large-Eddy Simulation (LES) are generally accurate, though non-Boussinesq turbulence and/or unresolved multiphysical phenomena can preclude predictive accuracy in certain regimes. In turbulent combustion, flame–turbulence interactions lead to inverse-cascade energy transfer, which violates the assumptions of many RANS and LES closures. We survey the regime dependence of these effects using a series of high-resolution Direct Numerical Simulations (DNS) of turbulent jet flames, from which an intermediate regime of heat-release effects, associated with the hypothesis of an “active cascade,” is apparent, with severe implications for physics- and data-driven closure models. We apply adjoint-based data assimilation method to augment the RANS and LES equations using trusted (though not necessarily high-fidelity) data. This uses a Python-native flow solver that leverages differentiable-programming techniques, automatic construction of adjoint equations, and solver-in-the-loop optimization. Applications to canonical turbulence, shock-dominated flows, aerodynamics, and flow control are presented, and opportunities for reacting flow modeling are discussed.

Recently Linares-Otto-Tempelmayr have unveiled a very interesting algebraic structure which allows to define a new class of rough paths/regularity structures, with associated applications to stochastic PDEs or ODEs. This approach does not consider trees as combinatorial tools but their fertility, namely the function which associates to each integer k the number of vertices in the tree with exactly k children. In a joint work with J-D Jacques we have studied this algebraic structure and shown that it is related with a general and simple class of so-called post-Lie algebras. The construction has remarkable properties and I will try to present them in the simplest possible way.