Understanding and predicting the dynamics of scalar turbulence using multiscale analysis, computational simulations, and stochastic models

Abstract

We investigate the dynamics of turbulent flows and scalar fields based on multiscale analysis, numerical simulations, and modeling. Specifically, we study the fundamental mechanisms of multiscale energy transfers in stratified turbulence where both the turbulent fluid flow and scalar field are present and exchanging energies (i.e., kinetic and potential energies). We also have developed a Lagrangian model which shows great capabilities for predicting the important dynamics of passive scalars in isotropic turbulence. Further evaluations and analysis of the scalar gradient diffusion term (which is approximated by the Lagrangian closure model) are also performed based on direct numerical simulation (DNS) data at higher Reynolds numbers $Re$, to potentially improve the model capability for higher $Re$.In the first part of the work, we analyze the budgets of turbulent kinetic energy (TKE) and turbulent potential energy (TPE) at different scales $\ell$ in sheared, stably stratified turbulence using a filtering approach. We consider the competing effects in the flow along with the physical mechanisms governing the energy fluxes between scales. The theoretical work of our energy budget analysis is used to analyze data from direct numerical simulation (DNS) at buoyancy Reynolds number $Re_b=O(100)$. Various quantities in the energy budget equations are evaluated based on DNS data of SSST, with detailed discussions on both the mean-field behavior of the flow, as well as fluctuations about this mean-field state. Importantly, it is shown that the TKE and TPE fluxes between scales are both downscale on average and their instantaneous values are positively correlated, but not strongly. The relative weak correlation occurs mainly due to the different physical mechanisms that govern the TKE and TPE fluxes. Moreover, the contribution to these fluxes arising from the sub-grid fields (i.e., small scales) are shown to be significant, in addition to the filtered scale contributions associated with the processes of strain-self amplification, vortex stretching, and density gradient amplification.Motivated by our findings that the average downscale flux of TKE and TPE are due to different mechanisms and that the contributions to the energy fluxes from small scale (i.e., sub-grid) dynamics are significant, in the second part we develop a Lagrangian model for studying the small-scale scalar dynamics in isotropic turbulence. It is known that the equation for the fluid velocity gradient along a Lagrangian trajectory immediately follows from the Navier-Stokes equation, and such an equation involves two terms that cannot be determined from the velocity gradient along the chosen Lagrangian path: the pressure Hessian and the viscous Laplacian; similarly, the equation for passive scalar gradients also involves an unclosed term in the Lagrangian frame, namely the scalar gradient diffusion term which needs to be closed. For the fluid velocity gradient, a recent model handles the unclosed terms using a multi-level version of the recent deformation of Gaussian fields (RDGF) closure (Johnson \& Meneveau, Phys.~Rev.~Fluids, 2017). The model is in remarkable agreement with DNS data and works for arbitrary Taylor Reynolds numbers $Re_\lambda$. Inspired by this, our Lagrangian model for the passive scalar gradients is developed using the RDGF approach. However, comparisons of the statistics obtained from this model with direct numerical simulation (DNS) data reveal substantial errors due to erroneously large fluctuations generated by the model. We address this defect by incorporating into the closure approximation information regarding the scalar gradient production along the local trajectory history of the particle. This modified model makes predictions for the scalar gradients, their production rates, and alignments with the strain-rate eigenvectors that are in very good agreement with DNS data. However, while the model yields valid predictions up to around $Re_\lambda\approx 500$, beyond this, the model breaks down.In consideration of the model failure beyond $Re_\lambda\approx 500$, the final part of work conducts further investigations via theoretical analysis and computations of more DNS data at various Reynolds numbers $Re_\lambda$. We theoretically analyzed the governing equations and identified two key mechanisms preventing the divergence of the scalar gradient magnitude. The conditional average of the scalar gradient diffusion term is also analyzed via its reduced forms which are used to test the model closure against DNS results. The model closure shows considerable errors in terms of its linear predictions of the conditional averages, in contrast to the strongly nonlinear dependencies on the condition quantities shown in DNS data. Such revealed errors potentially could be the reason why the model collapses beyond $Re_\lambda\approx 500$. Also discussed are the local relations of the scalar gradient diffusion term and various relevant quantities. It has been found that the diffusion term acts strictly to dissipate fluctuations of the scalar gradients in all regions where the scalar gradients are being either amplified or suppressed. Scalar gradients are dissipated most strongly in the regions where the straining motions are strong and the TKE are most strongly dissipated. Overall, the presented work here gains novel insights into the dynamics of scalar turbulence, reveals important implications/defects of the existing model closures, and in the meantime provides useful guidance for further improvements of existing model closures or for developing new models so that the complex scalar dynamics can be better captured in a more accurate manner.