Turbulence Theory Research Articles

Convective and Orographic Origins of the Mesoscale Kinetic Energy Spectrum

AbstractThe mesoscale spectrum describes the distribution of kinetic energy in the Earth's atmosphere between length scales of 10 and 400 km. Since the first observations, the origins of this spectrum have been controversial. At synoptic scales, the spectrum follows a −3 spectral slope, consistent with two‐dimensional turbulence theory, but a shallower −5/3 slope was observed at the shorter mesoscales. The cause of the shallower slope remains obscure, illustrating our lack of understanding. Through a novel coarse‐graining methodology, we are able to present a spatio‐temporal climatology of the spectral slope. We find convection and orography have a shallowing effect and can quantify this using “conditioned spectra.” These are typical spectra for a meteorological condition, obtained by aggregating spectra where the condition holds. This allows the investigation of new relationships, such as that between energy flux and spectral slope. Potential future applications of our methodology include predictability research and model validation.

Ordering of time scales predicts applicability of quasilinear theory in unstable flows

We discuss the applicability of quasilinear-type approximations for a turbulent system with a large range of spatial and temporal scales. We consider a paradigm fluid system of rotating convection with vertical and horizontal temperature gradients. In particular, the interaction of rotation with the horizontal temperature gradient drives a ‘thermal wind’ shear flow whose strength is controlled by the horizontal temperature gradient. Varying this parameter therefore systematically alters the ordering of the shearing time scale, the convective time scale and the correlation time scale. We demonstrate that quasilinear-type approximations work well when the shearing time scale or the correlation time scale is sufficiently short. In all cases, the generalised quasilinear approximation systematically outperforms the quasilinear approximation. We discuss the consequences for statistical theories of turbulence interacting with mean gradients.

A physical stochastic model of near-surface fluctuating wind fields

Modeling and simulation of atmospheric turbulence in coastal areas has emerged as a prominent area of research in recent years. This study presents a physical stochastic model to describe the horizontal coherence of fluctuating wind fields within the physical stochastic modeling frame. Based on the one-dimensional stochastic Fourier spectrum and isotropic turbulence theory, the horizontal coherence for fluctuating wind fields is expressed as a random function, with its probability information determined by the physical mechanism/background. The proposed physical stochastic model directly depicts the stochastic time series thereby enabling it to capture all probability information in detail comprehensively. The proposed model is numerically validated with the measured data obtained from an observation array constructed in Southeast China. This investigation holds significant potential for its application in wind-resistance design and reliability assessment of long-span structures.

Ground failure and soil erosion caused by bursting of buried water pipeline: experimental and numerical investigations

Pressurized water pipelines buried in an urban environment are prone to bursting failures, threatening public safety and traffic convenience. The limited studies in literature just focused on soil fluidization while few studies considered ground failure, shear strain, soil erosion and the influence of leakage locations during pipe bursts. In this study, extensive experimental tests along with a finite difference method – discrete element method (FDM-DEM) solid–fluid coupling analysis were conducted to investigate these issues. It was disclosed that the failure development during pipe bursts can be divided into three stages, i.e., seepage diffusion, erosion cavity expansion, and soil fluidization. By digital image correlation (DIC) analysis of the experimental results, a wedge-shaped displacement zone in ground was identified, with peak shear strain near its boundaries. Moreover, it was revealed that leakage locations affected the expansion origin of erosion cavity; as the burial depths increased, the ground heave range increased linearly; the maximum water outflow distance was closely related to the internal pressures of buried pipeline, which could be modeled by a square root formula based on turbulent jet theory. Mesoscopic analyses revealed that finer particles were more susceptible to erosion during pipe bursts because of the low possibility of forming strong connections with surrounding particles. The findings yielded from this study can enhance the understanding of pipe bursts and help professionals mitigate potential damage.

Scaling Law of Flow and Heat Transfer Characteristics in Turbulent Radiative Rayleigh-Bénard Convection of Optically Thick Media

Radiative natural convection is of vital importance in the process of energy storage, power generation, and thermal storage technology. As the attenuation coefficients of many heat transfer media in these fields are high enough to be considered as optically thick media, like nanofluids or molten salts in concentrated solar power or phase change thermal storage, Rosseland approximation is commonly used. In this paper, we delve into the impact of thermal radiation on the Rayleigh-Bénard (RB) convection. Theoretical analysis has been conducted by modifying the Grossmann-Lohse (GL) model. Based on turbulent dissipation theory, the corresponding scaling laws in four main regimes are proposed. Direct numerical simulation (DNS) was also performed, revealing that radiation exerts a notable influence on both flow and heat transfer, particularly on the formation of large-scale circulation. By comparing with DNS results, it is found that due to the presence of radiation, the modified Nu scaling law in small Pr range of the GL model is more suitable for predicting the transport characteristics of optical thick media with large Pr. The maximum deviation between the results of DNS and prediction model is about 10%, suggesting the summarized scaling law can effectively predict the Nu of radiative RB convection.

Dynamic and Daily Partner-Specific Processes of Relationship Uncertainty and Enacted Relationship Talk

Guided by relational turbulence theory (RTT), this intensive longitudinal study examined how within-person daily fluctuations in relationship uncertainty corresponded with individuals’ decisions to engage in daily enacted relationship talk. Using a person-specific approach, this study also examined how individuals’ attachment insecurity predicted within-person differences in month-long processes predicted by RTT. College-aged dating partners ( N = 202, between-person) reported their attachment proclivities in a pre-test survey and subsequently reported on their relationship uncertainty and enacted relationship talk once per day over a period of 30 consecutive days ( N = 5,240, within-person). Results indicated that on days when individuals experienced elevated relationship uncertainty, they engaged in less relationship talk than they typically did. Additionally, we found that individuals with more volatility (intraindividual variability) and inertia (day-to-day carryover) in relationship uncertainty throughout the month enacted less relationship talk on average. Finally, results indicated that attachment insecurity predicted person-specific month-long processes consistent with RTT.

Using the Boundary Element Method for airfoil trailing edge noise prediction

This study investigates two aspects of the prediction of trailing edge noise from an airfoil using the Boundary Element Method to solve the scattering phenomenon. The first one consists in defining the surface pressure spectrum near the trailing edge as a source for the turbulent boundary layer-trailing edge interaction, using either an empirical model or one-dimensional turbulent boundary layer theory. The second aspect focuses on the capability of the Boundary Element Method to handle general boundary conditions. It is applied to predict the noise emitted by a porous trailing edge. This work is conducted in the context of aerodynamic noise from wind turbine blades, but has a wider range of engineering applications.

Structural and Dynamical Quality Assessment of Gap‐Filled Sea Surface Temperature Products

AbstractPrevious studies that intercompared global Level‐4 (L4) sea surface temperature (SST) analyses were centered on the assessment of the accuracy and bias of SST by comparing them with independent near‐surface Argo profile temperature data. This type of assessment is centered on the absolute value of SST rather than on SST spatial properties (gradients), which is more relevant to the study of oceanographic features (e.g., fronts, eddies, etc.) and ocean dynamics. Here, we use, for the first time, the spectrum of singularity exponents to assess the structural and dynamical quality of different L4 gap‐filled products based on the multifractal theory of turbulence. Singularity exponents represent the geometrical projection of the turbulence cascade, and its singular spectrum can be related to the probability density function of the singularity exponents normalized by the scale. Our results reveal that the different schemes used to produce the L4 SST products generate different singularity spectra, which are then used to identify a potential loss of dynamical information or structural coherence. This new diagnostic constitutes a valuable tool to assess the structural quality of SST products and can support data satellite SST producers efforts to improve the interpolation schemes used to generate gap‐filled SST products.

Self-similarity and the direct (enstrophy) cascade in forced two-dimensional fluid turbulence

In the absence of large-scale coherent structures, a widely used statistical theory of two-dimensional turbulence developed by Kraichnan, Leith, and Batchelor (KLB) predicts a power-law scaling for the energy, $E(k)\propto k^\alpha$ with an integral exponent $\alpha ={-3}$ , in the inertial range associated with the direct cascade. A power-law scaling is also observed in the presence of coherent structures, but the scaling exponent becomes fractal and often differs substantially from the value predicted by the KLB theory. Here we present a dynamical theory that sheds new light on the relationship between the spatial and temporal structure of the large-scale flow and the scaling of small-scale structures representing filamentary vorticity. Specifically, we find hyperbolic regions of the large-scale flow to play a key role in the flux of enstrophy between scales. Small-scale vorticity in these regions can be described by dynamically self-similar solutions of the Euler equation, which explains the power-law scaling. Furthermore, we find that correlations between different hyperbolic regions are responsible for the emergence of fractal scaling exponents.

A longitudinal test of relational turbulence theory and serial arguments in romantic relationships

Abstract Relational turbulence theory (RTT) suggests that people perceive their romantic relationships as turbulent when they experience interactions that manifest the deleterious effects of relational uncertainty and altered patterns of interdependence. RTT also positions communication in these episodes as associated with subsequent relational uncertainty and qualities of interdependence. Using three-wave panel data collected at three-week intervals, this study evaluates (a) how communication in serial argument episodes predict relationship parameters (i.e., relational uncertainty and qualities of interdependence), (b) how relationship parameters predict serial argument occurrence, directness, and valence, and (c) whether over time variability in qualities of serial argument communication predict subsequent relational turbulence. Results indicated limited support for reciprocal, between-wave associations between relationship qualities and serial argument communication; however, over time variability in the valence of serial argument communication was associated with higher levels of relational turbulence after 6 weeks. Implications for RTT and research on serial arguments are discussed.

Hidden turbulence in van Gogh's The Starry Night

Turbulent skies have often inspired artists, particularly in the iconic swirls of Vincent van Gogh's The Starry Night. For an extended period, debate has raged over whether the flow pattern in this masterpiece adheres to Kolmogorov's theory of turbulence. In contrast to previous studies that examined only part of this painting, all and only the whirls/eddies in the painting are taken into account in this work, following the Richardson–Kolmogorov's cascade picture of turbulence. Consequently, the luminance's Fourier power spectrum spontaneously exhibits a characteristic −5/3 Kolmogorov-like power-law. This result suggests that van Gogh had a very careful observation of real flows, so that not only the sizes of whirls/eddies in The Starry Night but also their relative distances and intensity follow the physical law that governs turbulent flows. Moreover, a “–1”-like power-law persists in the spectrum below the scales of the smallest whirls, hinting at Batchelor-type scalar turbulence with a high Schmidt number. Our study, thus, unveils the hidden turbulence captured within The Starry Night.

Theory and Modelling of Isotropic Turbulence: From Incompressible through Increasingly Compressible Flows

Homogeneous isotropic turbulence (HIT) has been a useful theoretical concept for more than fifty years of theory, modelling, and calculations. Some exact results are revisited in incompressible HIT, with special emphasis on the 4/5 Kolmogorov law. The finite Reynolds number effect (FRN), which yields corrections to that law, is investigated, using both Kármán–Howarth-type equations and a statistical spectral closure of the Eddy-Damped Quasi-Normal Markovian (EDQNM)-type. This discussion offers an opportunity to give an extended review of such spectral closures, from weak turbulence, as in wave turbulence theory, to a strong one. Extensions of the 4/5 or 4/3 Kolmogorov/Monin laws to anisotropic cases, such as stably stratified and MHD turbulence, are briefly touched on. Before addressing more recent work on compressible isotropic turbulence, the simplest case of quasi-incompressible turbulence subjected to externally imposed isotropic compression or dilatation is presented. Rapid distortion theory is found to be a poor model in this isotropic case, in contrast with its relevance in strongly anisotropic flow cases. Accordingly, a fully nonlinear approach based on a rescaling of all fluctuating variables is used, in order to show its interplay with the linear operator. This opens the discussion on the cases of homogeneous incompressible turbulence, where RDT and nonlinear models are relevant, provided that anisotropy is accounted for. Finally, isotropic compressible flows of increasing complexity are considered. Recent studies using weak turbulence theory, modelling, and DNS are discussed. A final unpublished study involves interactions between the solenoidal mode, inherited from incompressible turbulence, and the acoustic and entropic modes, which are specific to the compressible problem. An approach to acoustic wave turbulence, with resonant triads, is revisited on this occasion.

Personal Recollections on Jack Herring and Developments in Theory of Turbulence, Atmospheric Sciences, and Computational Fluid Dynamics

The majority of articles in this Special Issue illuminate various aspects of Jack Herring’s contributions to the theory of turbulence, atmospheric sciences, and computational fluid dynamics (CFD), be it through his publications, presentations, collaborations, work with colleagues and students, or personal contacts [...]

A Variant of the Local Similarity Theory and Approximations of Vertical Profiles of Turbulent Moments of the Atmospheric Convective Boundary Layer

The approximation of the turbulent moments of the atmospheric convective layer is based on a variant of the local similarity theory using the concepts of the semi-empirical theory of Prandtl turbulence. In the proposed variant of the local similarity theory, the second moment of vertical velocity and the “spectral” Prandtl mixing length are used as basic parameters. This approach allows us to extend Prandtl’s theory to turbulent moments of vertical velocity and buoyancy and additionally offer more than ten new approximations. The comparison of the proposed approximation with other variants of the theory of local similarity is considered. It is shown that the selected basic parameters significantly improve the agreement between the local similarity approximations and experimental data. The approximations are consistent with observations in the turbulent convective layer of the atmosphere, the upper boundary of which nearly corresponds to the lower boundary of the temperature inversion. Analytical approximations of local similarity can find applications in the construction of high-order moment closures in the vortex of resolving numerical turbulence models, as well as in the construction of “mass-flux” parametrization.

Transition from weak turbulence to collapse turbulence regimes in the Majda-McLaughlin-Tabak model.

It is well known that wave collapses can emerge from the focusing one-dimensional (1D) Majda-McLaughlin-Tabak (MMT) model as a result of modulational instability. However, how these wave collapses affect the spectral properties and statistics of the wave field has not been adequately studied. We undertake this task by simulating the forced-dissipated 1D MMT model over a range of forcing amplitudes. Our results show that when the forcing is weak, the spectrum agrees well with the prediction by wave turbulence theory with few collapses in the field. As the forcing strength increases, we see an increase in the occurrence of collapses, together with a transition from a power-law spectrum to an exponentially decaying spectrum. Through a spectral decomposition, we find that the exponential spectrum is dominated by the wave collapse component in the nonintegrable MMT model, which is in analogy to a soliton gas in integrable turbulence.

Solar wind data analysis aided by synthetic modeling: A better understanding of plasma frame variations from temporal data

Context. In situ measurements of the solar wind, a turbulent and anisotropic plasma flow originating at the Sun, are mostly carried out by single spacecraft, resulting in one-dimensional time series. Aims. The conversion of these measurements to the spatial frame of the plasma is a great challenge, but it is required for direct comparison of the measurements with magnetohydrodynamic turbulence theories. Methods. We present a tool kit based on the synthetic modeling of solar wind fluctuations as two-dimensional noise maps with adjustable spectral and power anisotropy that can help with the temporal-spatial conversion of real data. Specifically, by following the spacecraft trajectory through a noise map (relative velocity and angle relative to some mean magnetic field) with properties tuned to mimic those of the solar wind, the likelihood that the temporal data fluctuations represent parallel or perpendicular fluctuations in the plasma frame can be quantified by correlating structure functions of the noise map. Synthetic temporal data can also be generated, which can provide a testing ground for analysis applied to the solar wind data. Results. We demonstrate this tool by investigating Parker Solar Probe’s seventh encounter trajectory and data, and we showcase several possible ways in which it can be used. We find that whether temporal variations in the spacecraft frame come from parallel or perpendicular variations in the plasma frame strongly depends on the spectral and power anisotropy of the measured wind. Conclusions. Data analysis assisted by such underlying synthetic models as presented here could open up new ways to interpret measurements in the future, specifically in the more reliable determination of plasma frame quantities from temporal measurements.

Statistical Dynamics and Subgrid Modelling of Turbulence: From Isotropic to Inhomogeneous

Turbulence is the most important, ubiquitous, and difficult problem of classical physics. Feynman viewed it as essentially unsolved, without a rigorous mathematical basis to describe the statistical dynamics of this most complex of fluid motion. However, the paradigm shift came in 1959, with the formulation of the Eulerian direct interaction approximation (DIA) closure by Kraichnan. It was based on renormalized perturbation theory, like quantum electrodynamics, and is a bare vertex theory that is manifestly realizable. Here, we review some of the subsequent exciting achievements in closure theory and subgrid modelling. We also document in some detail the progress that has been made in extending statistical dynamical turbulence theory to the real world of interactions with mean flows, waves and inhomogeneities such as topography. This includes numerically efficient inhomogeneous closures, like the realizable quasi-diagonal direct interaction approximation (QDIA), and even more efficient Markovian Inhomogeneous Closures (MICs). Recent developments include the formulation and testing of an eddy-damped Markovian anisotropic closure (EDMAC) that is realizable in interactions with transient waves but is as efficient as the eddy-damped quasi-normal Markovian (EDQNM). As a similarly efficient closure, the realizable eddy-damped Markovian inhomogeneous closure (EDMIC) has been developed. Moreover, we present subgrid models that cater for the complex interactions that occur in geophysical flows. Recent progress includes the determination of complete sets of subgrid terms for skilful large-eddy simulations of baroclinic inhomogeneous turbulent atmospheric and oceanic flows interacting with Rossby waves and topography. The success of these inhomogeneous closures has also led to further applications in data assimilation and ensemble prediction and generalization to quantum fields.

Exploring the similarity relationships from the nondimensionalization of atmospheric turbulence

Nondimensionalization, a theoretical approach for establishing interconnections among parameters within a set of equations, has proven to be an effective tool for the analysis of atmospheric turbulence. By applying nondimensionalization to turbulence equations, a concise form of dimensionless turbulence functions can be obtained. This process also yields several dimensionless parameters, defined as combinations of characteristic scales. From the dimensionless tensor B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${B}$$\\end{document} and vector βθ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\varvec{\\beta}}}_{\ heta}$$\\end{document} introduced in this study, the characteristic length scale, zs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${z}^{s}$$\\end{document}, can be defined as an alternative of length scale in similarity theories. Using the data from observational station in Horqin Sandy Land, quantified verifications of similarity relationships are carried out. The dimensionless parameters derived from nondimensionalization is not only in accordance with traditional turbulence theories but also facilitate the derivation of relationships among other dimensionless parameters. This reveals new similarity relationships that supplement the Monin–Obukhov theory. Under conditions of flat terrain and steady motions, the new length scale gives rise to similarity relationships exhibiting “4/3” exponential and near-linear patterns, which are associated with turbulent transport. These results make it possible to obtain the turbulent fluxes directly from the statistics of meteorological elements, even in stable stratifications. Consequently, the method of nondimensionalization can be taken as a reference in parameterization schemes of turbulence and climate models, and is fruitful in prospect of further study on atmospheric turbulence.

Scaling of turbulence-forced capillary waves

We simulate the propagation of capillary waves at the interface between two liquid layers of equal density, viscosity and thickness, which flow in stratified conditions inside a turbulent Poiseuille channel. This setup gives us the possibility to single out the effect of inertia and surface tension – in isolation from gravity – on the propagation of interfacial capillary waves. Numerical simulations, which are based on a combined pseudo spectral/phase field method, are run keeping the shear Reynolds number constant, Reτ=300, and considering four different values of the Weber number: We=0.5, We=1.0, We=1.5 and We=2.0. In the present case, waves are naturally forced by turbulence over a broad range of scales, from the larger ones, whose size is of order of the channel height h, down to the smaller dissipative scales. We show that, by increasing the Weber number, the interface is deformed more vigorously, and is prone to sustain the propagation of waves over a broader range of scales. Upon computation of the power spectra of wave elevation, we show that the behavior of waves agrees with the theoretical predictions given by the weak wave turbulence theory, which predicts a decay Sη(k)∼k−4 in the inertial regime, followed by a steeper decay at large k, in the surface-tension-dominated regime. We clearly show that the transition between the inertial and the surface-tension-dominated regime is pushed towards larger k by increasing the Weber number. In addition, we show that the two-dimensional frequency–wavenumber spectra of the wave elevation – the dispersion relation – is in good agreement with well established theoretical prediction, ω(k)∼k3/2, and that the range of validity of such prediction broadens for increasing Weber number.

A New Perspective on the Scattering Mechanism of S-Band Weather Radar Clear-Air Echoes Based on Communication Models

Clear-air echo studies are usually based on isotropic turbulence theory. But the theory has been considered incomplete by modern turbulence theory. The intermittence of turbulence can reveal obvious shortcomings in the existing studies of clear-air echoes. The mechanism of clear-air echo scattering needs to be supplemented. This paper introduces the troposcatter theory, normally used in over-the-horizon communication, to fill the gap left by Bragg scattering. By treating radar as a self-transmitted and self-received device, the equivalent transmission loss of weather radar is established and compared with the recommendations of the Radiocommunication Sector of the International Telecommunication Union (ITU-R). The results show that the S-band radar transmission loss aligns with ITU-R recommendations. There is also a linear regression relationship between the radar transmission loss and height, which conforms to the troposcatter theory. This means that the theory of troposcatter scattering is a supplement to the theory of Bragg scattering. Tropospheric scattering can be thought of as general Bragg scattering. Meanwhile, based on ITU-R recommendations, this study also provides a new way for the recognition of biological echoes.