A conservative Eulerian-Lagrangian decomposition principle for the solution of multi-scale flow problems at high Schmidt or Prandtl numbers

The simulation of turbulent flow that involves scalar transport at high Schmidt or Prandtl numbers is a major challenge. Enhanced Schmidt and Prandtl numbers demand an excessive increase in numerical resolution. Otherwise, the accuracy of transport would suffer substantially through unresolved information and numerical diffusion. With the aim of providing an efficient alternative for such applications, this paper presents a simulation method that is based on a novel Eulerian-Lagrangian decomposition principle (ELD) of the transported quantity. Low-pass filtering of the initial scalar quantity field separates it into a smooth low-frequency component and a fine-structured high-frequency component. The low-frequency component is represented and transported according to the Eulerian description by applying the Finite Volume Method (FVM) with a numerical resolution according to the Kolmogorov scale. The high-frequency component is translated into the Lagrangian description by the formation of particles, which are transported in parallel. By exchanging information between the two components, a re-initialisation mechanism ensures that the frequency-based decomposition is maintained throughout the simulation. Such ELD approach combines the efficiency of the FVM with the numerical stability of Lagrangian particles. As a result of the frequency-separation, the latter are by principle limited to zones of small scales and thus effectively complement the FVM. Furthermore, the particle information allows details of the scalar quantity field to be reconstructed that extend into the sub-grid level. By using a mixing layer setup, the proposed method is tested for a range of Schmidt numbers, and the numerical costs are considered and discussed. •The transport of scalar quantities at high Schmidt numbers is becoming an increasingly important phenomenon in science.•Their accurate numerical simulation is extremely costly with the methods and resources that are currently available.•The combination of FVM with particle methods is a promising approach provided that the integration is well designed.•A concentrated use of particles in regions where they provide a substantial gain in resolution ensures numerical efficiency.