Session X12: Turbulence: Theory (10:45am - 11:30am CST)

General Theory of Reynolds Stress

Starting from a few years back, I formalized a Lagrangian transport equation for the Reynolds shear stress in canonical flows. This approach is generalized for the full tensor including the streamwise, lateral and shear stress components, and validated using available DNS (direct numerical simulation) data for channel, jet, and zero-pressure gradient boundary layer flows. Next level of flow effects such as adverse pressure gradient and swirl are considered, showing similar level of agreement between theory and DNS data. When viewed from this Lagrangian perspective, i.e. moving with the mean flow, the Reynolds stress is seen to follow the basic momentum principle, illustrating the intricate but intuitive balance of flux, pressure and viscous effects of turbulence momentum and energy. This formalism allows for dynamic explanations of the turbulence structure, and also a solution algorithm for simple geometries. The uniqueness of the solution to the Reynolds-averaged Navier-Stokes equation, and the lognormal form of the turbulence energy spectra, are also briefly philosophized in the context of the maximum entropy principle.   

Exploring the turbulent velocity gradients at different scales from the perspective of the strain-rate eigenframe

The behavior of the filtered velocity gradient tensor (FVGT) in turbulence is analyzed by expressing its evolution equation in the strain-rate eigenframe. This provides an insightful way to understand the nature and interplay of various dynamical processes such as strain self-amplification, vortex stretching and tilting. Using Direct Numerical Simulation (DNS) data of isotropic turbulence, we consider the importance of local and non-local terms in the FVGT eigenframe equations across different scales. The eigenframe rotation-rate (that drives vorticity tilting) is shown to exhibit highly non-Gaussian fluctuations even at large scales due to kinematic effects, but dynamically, the anisotropic pressure Hessian plays a key role. The anisotropic pressure Hessian conditioned on the eigenvalues and vorticity exhibits highly non-linear behavior, with important modeling implications. We also derive a generalization of the Lumley triangle that allows us to show that the pressure Hessian has a preference for two-component axisymmetric configurations at small scales, with a transition to a more isotropic state at larger scales. Our results provide useful guidelines for improving Lagrangian FVGT models, since current models fail to capture a number of subtle features observed in our results.   

On the probability law of turbulent kinetic energy in the atmospheric surface layer

The probability density function $p(k)$ of the turbulent kinetic energy $k$ is investigated for diabatic atmospheric surface layer (ASL) flows. When the velocity components are near-Gaussian and their squared quantities are nearly independent, the resulting $p(k)$ is shown to be gamma-distributed with exponents that vary from 0.8 to 1.8. A non-linear Langevin equation that preserves a gamma-distributed $p(k)$ but allows linear relaxation of $k$ to its mean state is proposed and tested using multiple ASL data sets. The three parameters needed to describe the drift and non-linear diffusion terms can be determined from the ground shear stress and the mean velocity at height $z$ from the ground. Using these model parameters, the Langevin equation reproduces the measured $p(k)$ with minimal Kullback-Leibler divergence. Analysis of the partial autocorrelation function is conducted to investigate the validity of the exponential decay assumption in the autocorrelation function, and numerous runs exhibit non-exponential decay. Such adjustments need not revise the linearity of the drift term and can be accommodated by assuming a relaxation time scale that is not constant or time dependent. A non-constant relaxation time is one possibility to bridge the Langevin model with super-statistics.   

The formation of large-scale streams in two-dimensional turbulence

Following the suggestion from previous Monte--Carlo experiments (Jim\'enez, J. of Turbul., 2020), that dipoles are as important to the dynamics of decaying two-dimensional turbulence as individual vortex cores, it is found that the kinetic energy of this flow is carried by elongated streams formed by the concatenation of dipoles. Vortices separate into a family of small fast-moving cores, and another family of larger slowly moving ones, which can be described as `frozen' into a slowly evolving `crystal'. The kinematics of both families are very different, and only the former is self-similar. The latter is responsible for most of the kinetic energy of the flow, and its vortices form the dipoles and the streams. A mechanism is proposed for the growth of this slow component, and it is suggested that crystallisation drives the energy cascade towards larger scales   

The inefficacy of fluid turbulence to mix passive scalars when Schmidt numbers are large

A defining property of fluid turbulence is that it mixes substances extremely well. Thus, any circumstances leading to a loss of that property is of vital importance from both theoretical and practical perspectives. We demonstrate one such instance by considering the mixing of passive scalars advected in stationary isotropic turbulence, using state-of-the-art direct numerical simulations on up to grids of $8192^3$ points. The microscale Reynolds number is in the range $1-650$ and the Schmidt number $Sc$ is in the range $1-512$. First, we show that the mean scalar dissipation rate, when suitably non-dimensionalized, decreases as $1/\log Sc$, violating the principle of anomalous dissipation in the limit of large $Sc$. One-dimensional (1D) cuts through the scalar field indicate increasing density of sharp fronts on larger scales as $Sc$ increases, which oscillate sharply between high and low scalar concentrations leading to reduced mixing. The scaling exponents of the scalar structure functions in the inertial-convective range saturate with respect to the moment order and the saturation exponent approaches unity as $Sc$ increases, qualitatively consistent with 1D cuts of the scalar.   

Investigation of internal intermittency by way of spectral moments

The analysis of turbulence data using higher-order spectral moments is relatively uncommon despite the use of such methods in other fields of physics and engineering. In this study, we investigate the use of higher-order spectral moments to infer the effects of internal intermittency in multiple turbulent flows. We present spectral moments of both turbulent velocity and passive scalar (temperature) fields and offer a comparison of their intermittent behaviour. The experimental data analysed herein includes measurements of homogeneous, isotropic high-Reynolds-number grid turbulence, heated wakes of a cylinder, a heated turbulent jet, and turbulent channel flow. We focus on third- and fourth-order spectral moments using the definitions first proposed by Dwyer (\textit{J. Acoust. Soc. Am.}, 1983), as they are sensitive to transients and provide insight into the study of internal intermittency. Moreover, we confirm the dependence of internal intermittency on Reynolds number (e.g. Sreenivasan and Antonia, \textit{Ann.~Rev.~Fluid Mech.}, 1997) and the higher degree of scalar field internal intermittency, as compared to that of the velocity field (e.g. Warhaft, \textit{Ann.~Rev.~Fluid Mech.}, 2000).   

Lagrangian velocity-gradient evolution: Closure modeling conditioned on local streamline geometry.

We develop a stochastic diffusion model for the evolution of velocity gradient tensor (A$_{\mathrm{ij}}\equiv \partial $u$_{\mathrm{i}}$/$\partial $x$_{\mathrm{j}})$ following a fluid particle in isotropic incompressible turbulent flow. Two primary challenges toward accurate modeling of turbulence velocity-gradient dynamics are (i) intermittent nature of A$_{\mathrm{ij}}$ and (ii) non-locality of pressure and viscous terms in its evolution equation. To overcome these difficulties, we factorize A$_{\mathrm{ij}}$ into its intermittent magnitude (A$\equiv \surd $(A$_{\mathrm{kl}}$A$_{\mathrm{kl}})$, streamline scale) and normalized velocity gradient tensor (b$_{\mathrm{ij}}\equiv $A$_{\mathrm{ij}}$/A) which fully determines the local streamline shape. It is first shown that the evolution of magnitude A does not require any additional closures, once b$_{\mathrm{ij}}$ equation is suitably modeled. Then, the evolution of mathematically bounded tensor b$_{\mathrm{ij}}$ is modeled using a stochastic differential equation, and the non-local pressure and viscous terms are modeled with closures conditioned on local streamline shapes. The key advantages of this approach are:(i) relative ease of modeling the bounded tensor b$_{\mathrm{ij}}$ and (ii) amenability of conditioning nonlocal processes upon local streamline shape. The model accurately captures one-time statistics of isotropic turbulence, including high-order moments of A$_{\mathrm{ij}}$, streamline-shape distribution and vorticity-strain rate alignment.   

Objective Quantification of Turbulent Particle Pair Diffusion

Turbulence consists of interacting flow structures extending over a wide range of length and time scales. But what range of turbulence length scales governs pair diffusion in close proximity? We address this question by both fine scales and larger scale coherent structures - we encounter a combination of both local and non-local interactions associated with the small and large length scales. The local structures possess length scales of the same order of magnitude as the pair separation $l$, and they induce strong relative motion between the particle pair; the non-local structures possess length scales much larger than $l$ and also induce (via Biot-Savart) significant relative motion (ignored in prior studies). This leads to the prediction of the pair diffusivity $K$ scaling as $K\sim l^{1.556}$ -- agreeing within $1\%$ of experimental data. The `Richardson-Obukhov constant' $g_l$ is shown to be not a constant, although widely assumed to be a constant. But new constants $G_K$ and $G_l$ (representing, respectively, pair diffusivity and pair separation) are identified which we show to asymptote to, respectively, $0.73$ and $0.01$ at high Reynolds numbers. Our findings are important for improving turbulent diffusion modelling.   

Velocity gradients and fluid element deformation in turbulence: a new stabilizing mechanism due to the pressure Hessian

We analyze the internal motion of a small and incompressible fluid element through the velocity gradient and deformation tensor at its center of mass. We identify a novel stabilizing effect due to the pressure Hessian, leading to a decomposition of the Hessian into a conservative and non-conservative part. Restricting the Hessian to its conservative part generates a new class of time-reversible and spin-preserving models for the velocity gradient, which also includes the inviscid tetrad and Recent Fluid Deformation closures. In contrast to those models, the presented conservative system allows controlling the smoothness of the solutions by means of its first integral of motion. Despite time-reversibility, the new model can accurately predict the vorticity alignment with the intermediate strain-rate eigenvector. The new stabilizing effect of pressure is detected and studied in steady and isotropic turbulent flows from direct numerical simulations. The presented stabilizing mechanism is directly related to the strain-rate rather than vorticity, which is instead key for the reduction of nonlinearity (Chertkov et al. 1999). Therefore, those two stabilizing mechanisms are complementary and together can lead to an enhanced understanding and modeling of non-localities in turbulence.   

Amplification and self-attenuation of intense vorticity events in turbulent flows

Turbulent fluid flows, governed by the incompressible Navier-Stokes equations (INSE), are characterized by intermittent generation of very intense vortical motions over small scales. It is well known that the generating mechanism is vortex stretching resulting from non-linear amplification of vorticity by strain. This interaction is non-local, i.e., depends on the entire state of the flow, and thus is in the way of deriving turbulence theories and in establishing the regularity of INSE. Here, we show results on the contributions of local versus non-local strain to vortex stretching. The local contribution we obtain through the Biot-Savart integral of vorticity in a sphere of radius $R$, and the non-local contribution is the residue. Analyzing highly resolved numerical simulations of stationary isotropic turbulence up to Taylor-scale Reynolds number of $1300$, we show that vorticity is predominantly amplified by the non-local strain, which can be described by linear dynamics. However, as the vorticity is amplified beyond a threshold, the local strain non-linearly acts to counteract this amplification. This self-attenuation mechanism gives support of the regularity of the INSE.   

Kinetic and potential energy cascade mechanisms in sheared, stably stratified turbulence

In Carbone \& Bragg (J. Fluid Mech., 883, 2020 , R2), we used theory and Direct Numerical Simulations (DNS) to explore the mechanisms of the energy cascade in isotropic turbulence. Concerning the average energy cascade, the analysis showed that the dominant mechanism comes from the self-amplification of the filtered strain-rate, with a smaller but significant contribution from the stretching of filtered vorticity. Here we advance the analysis to sheared, stably stratified turbulence. This is a much more complex flow, involving cascades of both kinetic and potential energy, internal waves, strong anisotropy, and the suppression of vertical fluctuations due to buoyancy. We analyze the cascades above and below the Ozmidov scale, exploring the roles of the nonlinear amplification of the filtered strain-rate, vorticity, and density gradients fields, and how these amplification processes vary in different directions of the flow. We find that the relative sizes of the contributions to the kinetic energy cascade from the self-amplification of the strain-rate and vortex stretching depends strongly upon direction, and that the sign of these terms can even change for different directions. We also explore the statistical geometry of the flow, upon which these amplification processes depend.   

Statistical geometry of material loops in turbulence

Turbulent mixing is often characterized by the statistics of one- or two-particle dispersion. An even more comprehensive characterization of the complexity of turbulent mixing can be achieved by capturing the evolution of extended material lines and surfaces. Here, we investigate the statistical geometry of material loops, i.e.~closed material lines, by combining simulations, statistical turbulence theory, and dynamical systems theory. Tracking these structures in direct numerical simulations of homogeneous isotropic turbulence reveals that, while the loops develop convoluted shapes over time, their statistical geometry approaches a stationary state. In particular, their curvature distribution forms clear power-law tails, which we analytically determine in the framework of the Kraichnan model. Dynamically, we show that the high-curvature regime is dominated by the formation of isolated folds and that the power-law exponent can be related quantitatively to finite-time Lyapunov exponents. Thereby, the statistical geometry of material lines can be traced back to their dynamical evolution.   

Self-similarity and the direct cascade in two-dimensional turbulence

The multiscale nature of fluid turbulence arises as a result of various cascades transporting momentum between scales. This talk focuses on the direct (enstrophy) cascade responsible for the emergence of small-scale structure in two-dimensional turbulence. Existing understanding of this cascade is based primarily on the Fourier space analysis of Kraichnan, Leigh, and Batchelor (KLB) and implicitly assumes interaction between large and small scales. More recent studies Eyink and coworkers suggest that the physical mechanism of this cascade is related to stretching of vorticity field by the strain-dominated regions of the flow. We show that the KLB predictions can be derived and understood more naturally by analyzing the flow in the real space. Specifically, one can find in analytic form a family of self-similar solutions to the Euler equation which make this physical mechanism explicit, with vorticity field and the straining regions of velocity field describing, respectively, the small and large scales. These self-similar solutions immediately yield the $k^{-3}$ energy scaling in the inertial range. The analysis can be extended to the Navier-Stokes equations, with viscosity yielding an exponential correction to the scaling law in the viscous sub-range.   

Emergence of scaling in compressible turbulence

Anomalous scaling in high-Reynolds number compressible flows is a result of extreme fluctuations in velocity gradients and viscous action due to shocks. Recently it was shown that velocity gradients in incompressible turbulence undergo a transition from gaussian statistics to algebraic growth with respect to Reynolds number. This transition appears at low Reynolds numbers beyond which velocity gradients present the same behavior as turbulence at asymptotically high Reynolds numbers. Most extreme fluctuations undergo the transition first and a proper rescaling of the transition Reynolds number for different order moments of velocity gradients reveals a universal transition at Reynolds number of order 10. In this work we show that compressible turbulence, both solenoidal and dilatational modes, undergo a similar transition from gaussian at low-Reynolds to anomalous at high-Reynolds numbers. The solenoidal velocity field scales similar to its incompressible counterpart. The dilatational field, with contributions of shocklets of varying power, scales differently. The transition Reynolds number in both fields is shown to depend on the turbulent Mach number, which characterizes flow compressibility. Consequences of the scaling on nature of singularities in the two fields are discussed.