Constrained multi-scale turnover Lagrangian map for anisotropic synthetic turbulence: A priori tests

Synthetic turbulence has been useful in the modelling and simulation of turbulence, and as a surrogate to understand the dynamics of real hydrodynamic turbulence. In a recently proposed Multiscale Turnover Lagrangian Map (MTLM) method, an initial random field is transformed into a synthetic field after a series of simple mappings, with moderate computational cost. It has been shown that the resulted fields reproduce highly realistic statistics on many aspects of isotropic hydrodynamic turbulence, including small-scale intermittency, geometric statistics, and pressure statistics. Thus, it is of great interests to generalize the method to model inhomogeneous turbulence. In this paper, we formulate the problem as an optimization problem, where the initial random field is taken as the control variable, and the additional features presented in inhomogeneous turbulence are taken as a target function to be matched by the synthetic fields. The goal is to find the optimal control variable which minimizes the difference between the target function and the synthetic field. Using the adjoint formulation, we derive the optimality system of the problem, which formulates a procedure to generate inhomogeneous synthetic turbulence. The procedure, named the Constrained MTLM, is applied to synthesize two Kolmogorov flows where persistent large scale structures produce nontrivial mean flow statistics and local anisotropy in small scales. We compare the synthetic fields with direct numerical simulation data, and show that the former reproduces closely the mean flow statistics such as Reynolds stress distribution and mean turbulent kinetic energy balance. They also reproduce the effects of inhomogeneity on small scale structures, which is manifested in the distributions of mean subgrid-scale energy dissipation, and the alignment between the subgrid-scale stress tensor and the filtered strain rate tensor, among others. We conclude that the method is useful to further extend the applicability of synthetic turbulence.