Session X02: Turbulence: Theory II

On the higher order correction of the turbulent power-law velocity profile: comparison with the log law

For several decades, the classical log law of the wall has been widely regarded as the definitive formulation to describe the turbulent velocity profile near walls. However, in recent years, an alternative power law formulation of the turbulent velocity profile has emerged, challenging the central position of the log law. In a recent study by Anbarlooei et al. (Anbarlooei, H. R., Ramos, F. & Cruz, D. O. A. 2022), they demonstrated that an extended version of the power law expression can accurately replicate the turbulent velocity profile, even within the region where the log law is valid.

In this current study, we present an explicit expression for the turbulent mean velocity profile, which holds true in both the inner and outer regions, including the region of validity for the log law. Crucially, this new equation is derived entirely from first principles (attached eddies hypothesis) and does not rely on any turbulence model or asymptotic methods.   

Entropy and fluctuation relation in isotropic turbulence

Based on a generalized local Kolmogorov-Hill equation expressing the evolution of kinetic energy integrated over spheres of sizes in the inertial range of fluid turbulence, we examine a possible definition of entropy and entropy generation for turbulence. Its measurement from direct numerical simulations in isotropic turbulence leads to confirmation of the validity of the fluctuation relation (FR) from non-equilibrium thermodynamics in the inertial range of turbulent flows. Specifically, the ratio of probability densities of forward and inverse cascade at a certain scale is shown to follow exponential behavior with the entropy generation rate if the latter is defined by including an appropriately defined notion of “temperature of turbulence” proportional to the kinetic energy at a certain scale.

We thank the JHTDB/IDIES staff for their assistance with the database and its maintenance.    

Role of pressure on formation of extreme velocity gradients in turbulence

Turbulent flows are characterized by intermittent generation of intense velocity gradients, which are important to study in both theory and modeling. Such intense gradients result from nonlinear self-amplification and are also influenced by the nonlocal pressure field via its Hessian tensor. Prior work on the subject has have been restricted to low Reynolds numbers. Here, using direct numerical simulations (DNS) of isotropic turbulence with Taylor-scale Reynolds number in the range 140-1300, we systematically investigate how pressure Hessian affects the amplification of vorticity and strain-rate and contrast it with other inviscid nonlinear mechanisms.

Overall, H inhibits vortex stretching (VS), although the inhibition is much weaker than the amplification induced by other nonlinear mechanisms. However, in regions of most intense vorticity, the contribution from H dominates over nonlinearity, leading to an overall depletion of VS. We also observe near-perfect alignment between vorticity and the eigenvector of H corresponding to the smallest eigenvalue, consistent with the presence of intense vortex-tubes. We discuss the connection between the VS-inhibiting role of pressure, and the recently identified self-attenuation mechanism [Buaria et al. Nat. Commun. 11:5852 (2020)], whereby intense vorticity is locally attenuated through inviscid effects. We also discuss the influence of pressure Hessian on strain amplification, which is substantially weaker.    

Non-universal scaling for the direct cascade in 2D turbulent flow

For 2D turbulent flows, Kraichnan-Leith-Batchelor’s (KLB) theory predicts an energy scaling $E(k) propto k^{-3}$ for the direct cascade. However, the presence of large-scale coherent structures leads to a power-law scaling with a different -- typically fractal -- exponent. The dominant physical mechanism underlying the direct cascade involves stretching and folding of vorticity filaments in the hyperbolic regions of the large-scale flow. We show that the deviations from the KLB predictions are due to the interaction between adjacent hyperbolic regions which tends to orient vorticity filaments with respect to the expanding and contracting directions of the large-scale flow. This orientational effect leads to the emergence of self-similar structure of small-scale vorticity characterized by a fractal scaling exponent.   

Investigating the dependence of the pressure Hessian process on the Cauchy-Green tensor in turbulent flows

Tracking the evolution of the velocity gradient tensor following a fluid element in a turbulent flow provides fundamental understanding of various nonlinear processes like energy cascading, intermittency, scalar mixing etc. Thus, development of simple dynamical models of the velocity gradient tensor in turbulent flows has been an active area of research.  The time evolution equation of velocity gradient tensor has two unclosed terms - the pressure Hessian tensor and viscous Laplacian process.  In the past, some attempts have been made to model the pressure Hessian tensor in terms of the Cauchy-Green tensor. While the predictions of such models have been indeed encouraging, more improvements are still expected. With the motivation to develop deeper insights into the relationship between the two tensors (the pressure Hessian and the Cauchy-Green tensors), in this work, we employ direct numerical simulation (DNS) database of homogeneous turbulence to examine the interdependence between these tensors. Specifically, we focus on the alignment tendencies of the eigenvector-systems of the two tensors. In the light of these findings, subsequently, we identify the shortcomings of the existing models and attempt to suggest improvements. This study is performed for both incompressible and compressible flow fields.   

Scale-Locality: Insights into the energy cascade across scales in a shock

Inter-scale energy transfer (or the cascade) is of relevance to both LES modeling and turbulence theory. In incompressible homogeneous isotropic turbulence, there has been compelling theoretical and empirical support that the scale-transfer of kinetic energy (KE) is local. Here, we analyze the locality of KE scale-transfer in compressible turbulence. There is a common notion that shocks and discontinuities that pervade compressible turbulence necessarily imply a non-local scale-transfer. We show this not to be the case by demonstrating rigorous proofs of scale-locality using two examples: (i) solution to the Burgers equation and (ii) the 1D normal shock solution. Proofs of scale-locality in compressible turbulence hold in broad generality, at any Mach number, for any equation of state, and without the requirement of homogeneity or isotropy. Rather, locality rests on assumptions about the scaling of velocity, pressure, and density structure functions, which are weak and enjoy broad empirical support.   

Bridging inertial range scaling exponents of Lagrangian and Eulerian structure functions in high Reynolds number turbulence

A central question in turbulence theory concerns the inertial range scaling exponents of structure functions, which are known to depart from Kolmogorov's 1941 mean-field description due to small scale intermittency. This anomalous scaling can be studied from the Eulerian viewpoint capturing spatial intermittency, or the Lagrangian viewpoint capturing temporal intermittency. Bridging these two approaches has been a major challenge, primarily due to lack of reliable data. Using state-of-the-art direct numerical simulations (DNS) of isotropic turbulence at Taylor-scale Reynolds number of up to 1300, we extract inertial range scaling exponents for both Lagrangian and Eulerian structure functions. For the Eulerian case, we demonstrate that scaling exponents for longitudinal and transverse directions are different for high moments orders, in essential agreement with many past studies. It is further shown that the transverse Eulerian exponents saturate at ≈ 2.1 for moment orders p ≥10. The Lagrangian exponents likewise saturate at ≈ 2 for p ≥ 8.  It is further shown that Lagrangian and Eulerian transverse exponents can be related by the same multifractal spectrum, which is different from that for Eulrian longitudinal exponents. Our results suggest that Lagrangian intermittency can be solely characterized by Eulerian transverse intermittency, and not by the longitudinal or a combination of both, as previously believed. Implications for extending multifractal predictions to dissipation range are also discussed, especially for Lagrangian acceleration.   

Title: Influence of vibrational non-equilibrium on velocity gradient dynamics of compressible flows

In hypersonic flows, the phenomenon of turbulence tends to become more complex because of high levels of compressibility and elevated temperature levels. These complexities, in turn, lead to the emergence of many processes, such as vibrational energy excitation, dissociation and ionization of molecules. Such novel phenomena observed in vibrational non-equilibrium flows may affect various non-linear turbulent processes like scalar mixing, energy cascading, intermittency, and material element deformation. Tracking the evolution of velocity gradient dynamics is indeed a means to examine and understand the behaviour of these non-linear processes in a turbulent flow field. To clearly identify and understand the interaction of vibrational non-equilibrium with velocity gradient dynamics, in this work, we first derive the exact evolution equation of the velocity gradient and the pressure Hessian tensor for a flow field which is in a state of vibrational non-equilibrium. As expected, several “new” and unclosed mechanisms appear in the evolution equation of the pressure Hessian equation. To gain some more physical insights into these new mechanisms, as well as, to understand their relative importance in the dynamics of compressible velocity gradients, in the second part of this study, we employ direct numerical simulation database of compressible homogeneous turbulence. This study is expected to contribute towards the development of improved closure models for vibrationally excited compressible flows. 

 

 

Key words: Compressible flows, Pressure-Hessian tensor, Velocity gradient tensor, Vibrational non-equilibrium   

On the formulation of turbulence field statistics with lognormal multifractal velocity increments

Obtaining field statistics for fluid turbulence remains an outstanding challenge in turbulence theory and modeling despite the availability of the linear, closed Hopf equation for the characteristic functional. Multifractal models, e.g. for velocity increment distributions, are successful in capturing intermittency, but they do not contain the full statistical information of turbulent velocity fields and thus do not enjoy closed equations that can be derived from first principles. Here, we present a method that generates field statistics, in the form of a characteristic functional, that reduces to the two-point lognormal multifractal statistical model in the inertial range of scales. The functional is constructed as a superposition of Gaussian characteristic functionals each defined by a scaling exponent. A parameter transformation recovers a length scale dependence for the resulting velocity increment distributions that is consistent with multifractal theory. Applications and statistics derived from the functional are discussed, and comparisons are drawn with related approaches.   

Comparing local energy cascade rates in isotropic turbulence using structure function and filtering formulations

Two common definitions of the spatially local energy cascade rate at some scale $ell$ in turbulent flows are (i) the cubic velocity difference term appearing in the generalized Kolmogorov-Hill equation (structure function approach), and (ii) the subfilter-scale energy flux term in the transport equation for subgrid-scale kinetic energy (filtering approach). We perform a comparative study of both quantities based on direct numerical simulation data of isotropic turbulence at Taylor-scale Reynolds number of 1250. Conditional averaging is used to explore the relationship between the local cascade rate and the local filtered dissipation rate as well as filtered velocity gradient tensor properties such as its invariants. By conditioning on the local dissipation, we confirm Kolmogorov's second refined similarity hypothesis with both quantities. Conditioning on velocity gradients invariants, we find statistically robust evidence of inverse cascade when both the large-scale rotation rate is strong and the large-scale strain rate is weak. Even stronger net inverse cascading is observed in the ``vortex compression'' $R>0$, $Q>0$ quadrant where $R$ and $Q$ are velocity gradient invariants. Qualitatively similar, but quantitatively much weaker trends are observed for the conditionally averaged subfilter scale energy flux.   

Coherent structures and the direct cascade in 2D turbulence

The classical theory of Kraichnan, Leith, and Batchelor predicts a universal power-law scaling for the direct (enstrophy) cascade in 2D turbulence. While power-law spectra are indeed observed in both experiments and simulations, the scaling exponent is found to be nonuniversal due to the presence of large-scale coherent structures. The direct cascade is dominated by the hyperbolic regions of the large-scale flow, where small-scale vorticity behaves like a passive scalar. For nearly-time-periodic large-scale flows, chaotic advection aligns vorticity filaments along the unstable manifolds of the saddle points. To investigate how the tangling of stable and unstable manifolds associated with different saddles leads to the emergence of a fractal structure and a fractal scaling exponent, we investigate a model problem which involves a passive scalar advected by a prescribed time-periodic flow that is qualitatively similar to large-scale flows found in DNS of 2D turbulence. This allows us to independently control the properties of the large-scale flow and investigate their impact on the scaling exponent.   

Reynolds number dependence of moments of kinetic energy dissipation rate and enstrophy

Probability density functions (PDFs) of the kinetic energy dissipation rate and enstophy have recently been found to be stretched gamma distribution with the same stretching exponents. The prefactor of the exponential function is of the power law with the exponents 3/2 and 1/2 for the dissipation and enstrophy, respectively, that are known from the Gaussian random velocity. Under the constraints of the normalization and unity for the mean on the PDF, it is theoretically predicted that the moments of the order between 0 and 1 decrease with increase of the Reynolds number, while those with the order lower than 0 or greater than 1 increase with the Reynolds number. A set of the direct numerical simulation (DNS) confirms this trend of the moments. Implication of the Reynolds number dependence of these moments on the spectra of the kinetic energy and scalar variance will be discussed.