This study aims to propose novel solutions to the complex problem of turbulent flows using data-driven statistical and mathematical models. The proposed models reduce the huge computational cost of the direct numerical simulations and make them tractable while maintaining the important statistical features of the chaotic flows. Unlike the conventional models in the literature, the new proposed dynamic models take into account the inherent nonlocality of turbulence and predict the final statistical quantities with higher accuracy and correlations. First, we developed a novel autonomously dynamic nonlocal turbulence model for the large and very large eddy simulation (LES, VLES) of the homogeneous isotropic turbulent flows (HIT). The model is based on a generalized (integer-to-noninteger) order Laplacian of the filtered velocity field, and a novel dynamic model has been formulated to avoid the need for tuning the model constant. Three data-driven approaches were introduced for the determination of the fractional-order to have a model which is totally free of any tuning parameter. Our analysis includes both the a priori and the a posteriori tests. In the former test, using a high-fidelity and well-resolved dataset from direct numerical simulations (DNS), we computed the correlation coefficients for the stress components of the subgrid-scale (SGS) stress tensor and the one we get directly from the DNS results. Moreover, we compared the probability density function of the ensemble-averaged SGS forces for different filter sizes. In the latter, we employed our new model along with other conventional models including static and dynamic Smagorinsky into our pseudo-spectral solver and tested the final predicted quantities. The results of the newly developed model exhibit an expressive agreement with the ground-truth DNS results in all components of the SGS stress and forces. Also, the model exhibits promising results in the VLES region as well as the LES region, which could be remarkably important for the cost-efficient nonlocal turbulence modeling e.g., in meteorological and environmental applications.Afterwards, we extend the same dynamic nonlocal idea to the scalar turbulence. To this end, we formulate the underlying nonlocal model starting from the filtered Boltzmann kinetic transport equation, where the divergence of subgrid-scale scalar fluxes emerges as a fractional-order Laplacian term in the filtered advection-diffusion model, coding the corresponding super-diffusive nature of scalar turbulence. Subsequently, we develop a robust data-driven algorithm for estimation of the fractional (non-integer) Laplacian exponent, where we on-the-fly calculate the corresponding model coefficient employing a new dynamic procedure. Our a priori tests show that our new dynamically nonlocal LES paradigm provides better agreements with the ground-truth filtered DNS data in comparison to the conventional static and dynamic Prandtl-Smagorisnky models. Moreover, in order to analyze the numerical stability and assessing the model's performance, we carry out a comprehensive a posteriori tests. They unanimously illustrate that our new model considerably outperforms other existing functional models, correctly predicting the backscattering phenomena at the same time and providing higher correlations at small-to-large filter sizes. We conclude that our proposed nonlocal subgrid-scale model for scalar turbulence is amenable for coarse LES and VLES frameworks even with strong anisotropies, applicable to environmental applications.Finally, we developed a new dynamic tempered fractional subgrid-scale model, DTF, for the large and very large eddy simulation of turbulent flows. The nonlocality of the turbulent flows is the innate feature that can be seen in the non-Gaussian statistics of the velocity increments and can be addressed properly by the nonlocal models in terms of the fractional operators. Using kinetic transport, we developed a dynamic tempered fractional model that encompasses the three main characteristics of an ideal turbulence model: (i) nonlocal nature, (ii) dynamic model constant computations, and (iii) tempered and finite variance property. Several simulations of forced homogeneous isotropic and multi-layer temporal shear layer turbulent flows have been done in the a priori and a posteriori analyses. The results show that the new model is not only numerically stable and can maintain low- and high-order structures in long-range simulations, but it also provides better predictions than local models and nontempered models.
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