Many fluid-mechanical systems exhibit spatio-temporal multi-scale dynamics. Detailed methods that resolve the smallest and shortest scales cannot reach the larger and longer ones. Phenomenological models, on the other hand, are connected to a larger degree of uncertainty.
To overcome these limitations, hybrid models combine knowledge of the laws of physics with observational data to include mesoscopic information. Since high-fidelity data for large-scale flows are expensive to obtain, we developed the hybrid simulation technique “recurrence CFD” [1] that approximates the evolution of dynamic systems with relatively little data in a pragmatic fashion. For a given flow field, we determine the most similar reference state in a precomputed database and apply the corresponding evolution over a large time step. Upon iteration, long time series can be constructed from this reduced basis of flow states and slow transport processes simulated with little costs.
Application examples include long-term investigations of (i) heat transfer in fluidized beds [2], (ii) heterogeneous reactions in moving beds [3] and (iii) species transport in turbulent flows [4]. In these cases, the processes of interest evolve over much longer time scales than those characteristic for the rapid dynamics of these systems.
Finally, an outlook on our envisioned future research is given, which will focus on the generalizability to off-database conditions. To this end, a propagator formalism is introduced to systematically account for changes in the underlying dynamics [5].

[1] Lichtenegger T., and Pirker S. "Recurrence CFD – a novel approach to simulate multiphase flows with strongly separated time scales." Chem. Eng. Sci. 153 (2016): 394-410.
[2] Lichtenegger T., et al. "Dynamics and long-time behavior of gas–solid flows on recurrent-transient backgrounds." Chem. Eng. J. 364 (2019): 562-577.
[3] Lichtenegger T., and Pirker S.. "Fast long-term simulations of hot, reacting, moving particle beds with a melting zone." Chem. Eng. Sci. 283 (2024): 119402.
[4] Pirker S., and Lichtenegger T. "Efficient time-extrapolation of single-and multiphase simulations by transport based recurrence CFD (rCFD)." Chem. Eng. Sci. 188 (2018): 65-83.
[5] Lichtenegger T. "Data-assisted, physics-informed propagators for recurrent flows." Phys. Rev. Fluids 9.2 (2024): 024401.