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Turbulence closure modeling using machine learning is at an early crossroads. The extraordinary success of machine learning (ML) in a variety of challenging fields has given rise to justifiable optimism regarding similar transformative advances in the area of turbulence closure modeling. However, by most accounts, the current rate of progress toward accurate and predictive ML-RANS (Reynolds Averaged Navier-Stokes) closure models has been much slower than initially projected. The slower-than-expected rate of progress can be attributed to two reasons: high initial expectations without a complete comprehension of the complexity of the turbulence phenomenon, and the use of ML techniques in a manner that may be inconsistent with turbulence physics. To do full justice to the potential of data-driven approaches in turbulence modeling, this article seeks to identify the foundational physics challenges underlying turbulence closure modeling and assess whether machine learning techniques can effectively tackle them. Drawing analogies with statistical mechanics and stochastic systems, the key physical phenomena and mathematical limitations that render turbulence closure modeling complicated are first identified. The inherent capability of ML to overcome each of the identified challenges is then investigated. The conclusions drawn highlight the limitations of ML-based closures and pave the way for a more judicious framework for the use of ML in turbulence modeling. As ML methods evolve (which is happening at a rapid pace) and our understanding of the turbulence phenomenon improves, the inferences expressed here should be suitably modified.

Velocity gradient tensor, $A_{ij}\equiv \partial u_i/\partial x_j$, in a turbulence flow field is modeled by separating the treatment of intermittent magnitude ($A = \sqrt{A_{ij}A_{ij}}$) from that of the more universal normalized velocity gradient tensor, $b_{ij} \equiv A_{ij}/A$. The boundedness and compactness of the $b_{ij}$-space along with its universal dynamics allows for the development of models that are reasonably insensitive to Reynolds number. The near-lognormality of the magnitude $A$ is then exploited to derive a model based on a modified Ornstein-Uhlenbeck process. These models are developed using data-driven strategies employing high-fidelity forced isotropic turbulence data sets. A posteriori model results agree well with direct numerical simulation (DNS) data over a wide range of velocity-gradient features.

Hydrodynamic stability of compressible boundary layers is strongly influenced by Mach number ($M$), Prandtl number ($Pr$) and thermal wall boundary condition. These effects manifest on the flow stability via the flow-thermodynamic interactions. Comprehensive understanding of stability flow physics is of fundamental interest and important for developing predictive tools and closure models for integrated transition-to-turbulence computations. The flow-thermodynamic interactions are examined using linear analysis and direct numerical simulations (DNS) in the following parameter regime: $0.5 \leq M \leq 8$; and, $0.5 \leq Pr \leq 1.3$. For adiabatic wall boundary condition, increasing Prandtl number has a destabilizing effect. In this work, we characterize the behavior of production, pressure-strain correlation and pressure-dilatation as functions of Mach and Prandtl numbers. First and second instability modes exhibit similar stability trends but the underlying flow physics is shown to be diametrically opposite. The Prandtl-number influence on instability is explicated in terms of the base flow profile with respect to the different perturbation mode shapes.

The partially-averaged Navier-Stokes (PANS) equations are used to predict the variable-density Rayleigh-Taylor (RT) flow at Atwood number 0.5 and maximum Reynolds number $500$. This is a prototypical problem of material mixing featuring laminar, transitional, and turbulent flow, instabilities and coherent structures, density fluctuations, and production of turbulence kinetic energy by both shear and buoyancy mechanisms. These features pose numerous challenges to modeling and simulation, making the RT flow ideal to develop the validation space of the recently proposed PANS BHR-LEVM closure. The numerical simulations are conducted at different levels of physical resolution and test three approaches to set the parameters $f_\phi$ defining the range of physically resolved scales. The computations demonstrate the efficiency (accuracy vs. cost) of the PANS model predicting the spatio-temporal development of the RT flow. Results comparable to large-eddy simulations and direct numerical simulations are obtained, at significantly lower physical resolution without the limitations of the Reynolds-averaged Navier-Stokes equations in these transitional flows. The data also illustrate the importance of appropriate selection of the physical resolution and the resolved fraction of each dependent quantity $\phi$ of the turbulent closure, $f_\phi$. These two aspects determine the ability of the model to resolve the flow phenomena not amenable to modeling by the closure and, as such, the computations' fidelity.

The PANS equations model is used to predict the transitional Taylor-Green vortex (TGV) flow at Re=3000. A new form of the closure is proposed, in which the PANS procedure is applied to a variant of the BHR turbulence model. The TGV is a benchmark case for transitional flows in which the onset of turbulence is driven by vortex-stretching and reconnection mechanisms. Since these physical phenomena are observed in numerous flows of variable-density, this study constitutes the first step toward extending the PANS method to such a class of problems. We start by deriving the governing equations of the model and analyze the selection of the parameters controlling its physical resolution, fe and fk, through a-priori testing. Afterward, we conduct PANS computations at different constant physical resolutions to evaluate the model's accuracy and cost predicting the TGV flow. This is performed through simple verification and validation exercises, and the physical and modeling interpretation of the numerical predictions. The results confirm that PANS can efficiently (accuracy vs. cost) predict the present flow problem. Yet, this is closely dependent on the physical resolution of the model. Whereas high-physical resolution (fk<0.50) computations are in good agreement with the reference DNS studies, low-physical resolution (fk>=0.50) simulations lead to large discrepancies with the reference data. The physical and modeling interpretation of the results demonstrates that the origin of these distinct behaviors lies in the model's ability to resolve the phenomena driving the onset of turbulence not amenable to modeling by the closure. The comparison of the computations' cost indicates that high-physical resolution PANS achieves the accuracy of DNS (fk=0.00) at a fraction of the cost. We observe a cost reduction of one order of magnitude at the current Re, which is expected to grow with Re.

The local streamline topology classification method of Chong et al. (1990) is adapted and extended to describe the geometry of infinitesimal vortex lines. Direct numerical simulation (DNS) data of forced isotropic turbulence reveals that joint probability density function (PDF) of the second ($q_\omega$) and third ($r_\omega$) normalized invariants of the vorticity gradient tensor asymptotes to a self-similar bell form beyond $Re_\lambda > 200$. The same PDF shape is also seen at the late stages of breakdown of Taylor-Green vortex suggesting the universality of the bell-shaped pdf form in turbulent flows. Additionally, vortex reconnection from different initial configurations is examined. The local topology and geometry of the reconnection bridge is shown to be identical in all cases with elliptic vortex lines on one side and hyperbolic filaments on the other. Overall, topological characterization of vorticity field provides a useful analytical basis for examining vorticity dynamics in turbulence and other fluid flows.

Scale-space energy density function, $E(\mathbf{x}, \mathbf{r})$, is defined as the derivative of the two-point velocity correlation. The function E describes the turbulent kinetic energy density of scale r at a location x and can be considered as the generalization of spectral energy density function concept to inhomogeneous flows. We derive the transport equation for the scale-space energy density function in compressible flows to develop a better understanding of scale-to-scale energy transfer and the degree of non-locality of the energy interactions. Specifically, the effects of variable-density and dilatation on turbulence energy dynamics are identified. It is expected that these findings will yield deeper insight into compressibility effects leading to improved models at all levels of closure for mass flux, density-variance, pressure-dilatation, pressure-strain correlation and dilatational dissipation processes., Comment: 19 pages

A recent thrust in turbulence closure modeling research is to incorporate machine learning (ML) elements, such as neural networks, for the purpose of enhancing the predictive capability to a broader class of flows. Such a turbulence closure framework entails solving a system of equations comprised of ML functionals coupled with traditional (physics-based - PB) elements. While combining closure elements from fundamentally different ideologies can lead to unprecedented progress, there are many critical challenges that must be overcome. This study examines three such challenges: (i) Physical compatibility (or lack thereof) between ML and PB constituents of the modeling system of equations; (ii) Internal (self) consistency of the ML training process; and (iii) Formulation of an optimal objective (or loss) function for training. These issues are critically important for generalization of the ML-enhanced methods to predictive computations of complex engineering flows. Training and implementation strategies in current practice that may lead to significant incompatibilities and inconsistencies are identified. Using the simple test case of turbulent channel flow, key deficiencies are highlighted and proposals for mitigating them are investigated. Compatibility constraints are evaluated and it is demonstrated that an iterative training procedure can help ensure certain degree of consistency. In summary, this work develops foundational tenets to guide development of ML-enhanced turbulence closure models.

This study develops a comprehensive description of local streamline geometry and uses the resulting shape features to characterize velocity gradient ($A_{ij}$) dynamics. The local streamline geometric shape parameters and scale-factor (size) are extracted from $A_{ij}$ by extending the linearized critical point analysis. In the present analysis, $A_{ij}$ is factorized into its magnitude ($A \equiv \sqrt{A_{ij}A_{ij}}$) and normalized tensor $b_{ij} \equiv A_{ij}/A$. The geometric shape is shown to be determined exclusively by four $b_{ij}$ parameters -- second invariant, $q$; third invariant, $r$; intermediate strain-rate eigenvalue, $a_2$; and, angle between vorticity and intermediate strain-rate eigenvector, $\omega_2$. Velocity gradient magnitude $A$ plays a role only in determining the scale of the local streamline structure. Direct numerical simulation data of forced isotropic turbulence ($Re_\lambda \sim 200 - 600$) is used to establish streamline shape and scale distribution and, then to characterize velocity-gradient dynamics. Conditional mean trajectories (CMTs) in $q$-$r$ space reveal important non-local features of pressure and viscous dynamics which are not evident from the $A_{ij}$-invariants. Two distinct types of $q$-$r$ CMTs demarcated by a separatrix are identified. The inner trajectories are dominated by inertia-pressure interactions and the viscous effects play a significant role only in the outer trajectories. Dynamical system characterization of inertial, pressure and viscous effects in the $q$-$r$ phase space is developed. Additionally, it is shown that the residence time of $q$-$r$ CMTs through different topologies correlate well with the corresponding population fractions. These findings not only lead to improved understanding of non-local dynamics, but also provide an important foundation for developing Lagrangian velocity-gradient models.

Turbulence small-scale behavior has been commonly investigated in literature by decomposing the velocity-gradient tensor ($A_{ij}$) into the symmetric strain-rate ($S_{ij}$) and anti-symmetric rotation-rate ($W_{ij}$) tensors. To develop further insight, we revisit some of the key studies using a triple decomposition of the velocity-gradient tensor. The additive triple decomposition formally segregates the contributions of normal-strain-rate ($N_{ij}$), pure-shear ($H_{ij}$) and rigid-body-rotation-rate ($R_{ij}$). The decomposition not only highlights the key role of shear, but it also provides a more accurate account of the influence of normal-strain and pure rotation on important small-scale features. First, the local streamline topology and geometry are described in terms of the three constituent tensors in velocity-gradient invariants' space. Using DNS data sets of forced isotropic turbulence, the velocity-gradient and pressure field fluctuations are examined at different Reynolds numbers. At all Reynolds numbers, shear contributes the most and rigid-body-rotation the least toward the velocity-gradient magnitude ($A^2$). Especially, shear contribution is dominant in regions of intermittency (high values of $A^2$). It is also shown that the high-degree of enstrophy intermittency reported in literature is due to the shear contribution toward vorticity rather than that of rigid-body-rotation. The study also provides an explanation for the absence of intermittency of the pressure-Laplacian, despite the strong intermittency of enstrophy and dissipation fields. Overall, it is demonstrated that triple decomposition offers unique and deeper understanding of velocity-gradient behavior in turbulence.