Turbulence Theory Research Articles

Singular solutions of nonlinear hydrodynamic equations arising in turbulence theory

Singular solutions of nonlinear hydrodynamic equations arising in turbulence theory

Scour evolution downstream of grade-control structures under unsteady flows: A theoretical analysis

River restoration projects often make use of grade-control structures in order to prevent channel degradation, improve their ecological resources, and regulate floods. The time evolution of localized scour that can occur downstream of grade-control structures represents a crucial component in the design of those projects. This is because erosion processes may undermine the stability of hydraulic structures and compromise their effectiveness. Although localized scour has been widely investigated (mostly via laboratory models), there is significant scatter in results produced by the large number of obtained empirical equations. Consequently, there is a need for more general tools based on theoretical approaches that might overcome the limits of ad-hoc, experimental methods. In particular, a model based on the application of the Phenomenological Theory of Turbulence (PTT) has been recently developed by the last three authors to predict scour-depth evolution under steady flows. Nevertheless, in practical applications, scour phenomena usually occur during floods characterized by variable flow discharges. In this paper, for the first time to the authors’ knowledge, a detailed analysis of the basic assumptions of the PTT-evolution model allowed us to assess their validity for scour processes at different grade-control structures under unsteady (time-dependent) flows. By re-analyzing the dynamics of the scour evolution under unsteady flow conditions, we corroborate its consistency with the steady counterpart and with jet-driven scour problems, evincing that a homothetical evolution of the scour hole occurs during the developed phase. Finally, we provide a confirmation of the general validity of the theoretical approach presented herein. We show that it can be successfully applied to a large range of structure configurations and hydraulic conditions, without any significant modification. This last result is particularly important, since it paves the way to a unique, first-principles-based tool that can be helpful to design hydraulic structures.

Upscale transfer of waves in one-dimensional rotating shallow water

We study the inverse flux of waves in one of the simplest geophysical fluid dynamics models: one-dimensional rotating shallow water equations. Based on direct numerical integration of the governing equations, we find that waves injected at small scales get transferred upscale predominantly via resonant quartic interactions between wave modes. The waves’ upscale transfer is non-local and involves turbulent transfer between disparate scales of the flow. Our analysis reveals that the upscale transfer of waves is extremely intermittent and is a result of localized-in-time bursts in wave action flux. These intermittent events of flux bursts lead to shallower waves’ spectrum and relatively higher amplitude wave fields in physical space. On examining statistics of the flow fields, we find that low-energy high wavenumbers more or less comply with the assumptions used in wave turbulence theory, such as uniformly distributed wave phases and the Gaussian distribution of fields, while non-uniform distribution of wave phases and non-Gaussian statistics dominate at large scales or low wavenumbers that contain a major share of the flow energy. Our findings point out that the one-dimensional rotating shallow water equations, despite being a simple geophysical fluid dynamic model, harbour complex and intricate features associated with the upscale transfer of waves that have not been recognized in the past.

Accurately predicting hypersonic transitional flow on cone via a symmetry approach

A new algebraic transition model is proposed based on a Structural Ensemble Dynamics (SED) theory of wall turbulence, for accurately predicting the hypersonic flow heat transfer on cone. The model defines the eddy viscosity in terms of a two-dimensional multi-regime distribution of a Stress Length (SL) function, and hence is named as SED-SL. This paper presents clear evidence of precise predictions of transition onset location and peak heat flux of a wide range of hypersonic Transitional Boundary Layers (TrBL) around straight cone at zero incidence, to an unprecedented accuracy as validated by over 70 measurements for varying five crucial influential factors (Mach number, temperature ratio, cone half angle, nose Reynolds number and noise level). The results demonstrate the universality of the postulated multi-regime similarity structure, in characterizing not only the spatial non-uniform distribution of the eddy viscosity in hypersonic TrBL on cone, but also the dependence of the transition onset location on the five influential factors. The latter yields a novel correlation formula for transition center Reynolds number which takes similar functional form as the SL function within the symmetry approach. It is concluded that the SED-SL model simulates TrBL around cone with uniformly high accuracy, and then points out to an optimistic alternative way to construct hypersonic transition model.

Boundary layer flow over a bump and the three-dimensional law of the wall

Many turbulence theories in use today are based on two-dimensional equilibrium flows and have limitations when applied to three-dimensional flows. A three-dimensional law of the wall would help to improve simulation fidelity, but while several versions have been proposed, none have been widely accepted. In this study, the three-dimensional attached boundary layer flow over the windward side of the BeVERLI (Benchmark Validation Experiments for RANS/LES Investigations) Hill bump model was measured using near-wall laser Doppler velocimetry in the Virginia Tech Stability Wind Tunnel to study the mean flow and turbulence structure. These mean velocity measurements are compared with the predictions of the proposed three-dimensional (3D) law of the wall of van den Berg [A three-dimensional law of the wall for turbulent shear flows. J Fluid Mech. 1975;70(1):149–160.], which incorporates pressure gradients and inertial effects but assumes alignment of the mean flow gradient and shear-stress angles, and to the sublayer momentum equations, which are exact in the limit of wall-normal . In regions with mild stress/strain misalignment, the van den Berg model compares favourably with the experimental data up to a maximum of , and the sublayer momentum relationship compares favourably with the experimental data in the linear sublayer.

Analysis of the Intranight Variability of BL Lacertae during Its 2020 August Flare

We present an analysis of the BVRI photometry of the blazar BL Lacertae on diverse timescales from 2020 mid-July to mid-September. We have used 11 different optical telescopes around the world and have collected data over 84 observational nights. The observations cover the onset of a new activity phase of BL Lacertae that started in 2020 August (termed as the 2020 August flare by us), and the analysis is focused on the intranight variability. On short-term timescales, (i) flux varied with ∼2.2 mag in the R band, (ii) the spectral index was found to be weakly dependent on the flux (i.e., the variations could be considered mildly chromatic), and (iii) no periodicity was detected. On intranight timescales, BL Lacertae was found to predominantly show bluer-when-brighter chromatism. We also found two cases of significant interband time lags of the order of a few minutes. The duty cycle of the blazar during the 2020 August flare was estimated to be quite high (∼90% or higher). We decomposed the intranight light curves into individual flares and determined their characteristics. On the basis of our analysis and assuming the turbulent jet model, we determined some characteristics of the emitting regions: Doppler factor, magnetic field strength, electron Lorentz factor, and radius. The radii determined were discussed in the framework of the Kolmogorov theory of turbulence. We also estimated the weighted mean structure function slope on intranight timescales, related it to the slope of the power spectral density, and discussed it with regard to the origin of intranight variability.

Fast magneto-acoustic wave turbulence and the Iroshnikov–Kraichnan spectrum

An analytical theory of wave turbulence is developed for pure compressible magnetohydrodynamics in the small $\beta$ limit. In contrast to previous works where the multiple scale method was not mentioned and slow magneto-acoustic waves were included, we present here a theory for fast magneto-acoustic waves for which only an asymptotic closure is possible in three dimensions. We introduce the compressible Elsässer fields (canonical variables) and show their linear relationship with the mass density and the compressible velocity. The kinetic equations of wave turbulence for three-wave interactions are obtained and the detailed conservation is shown for the two invariants, energy and momentum (cross-helicity). An exact stationary solution (Kolmogorov-Zakharov spectrum) exists only for the energy. We find a $k^{-3/2}$ energy spectrum compatible with the Iroshnikov–Kraichnan (IK) phenomenological prediction; this leads to a mass density spectrum with the same scaling. Despite the presence of a relatively strong uniform magnetic field, this turbulence is characterized by an energy spectrum with a power index that is independent of the angular direction; its amplitude, however, shows an angular dependence. We prove the existence of the IK solution using the locality condition, show that the energy flux is positive and hence the cascade direct and find the Kolmogorov constant. This theory offers a plausible explanation for recent observations in the solar wind at small $\beta$ where isotropic spectra with a $-3/2$ power-law index are found and associated with fast magneto-acoustic waves. This theory may also be used to explain the IK spectrum often observed near the Sun. Besides, it provides a rigorous theoretical basis for the well-known phenomenological IK spectrum, which coincides with the Zakharov–Sagdeev spectrum for acoustic wave turbulence.

Propagation of capillary waves in two-layer oil–water turbulent flow

We study the dynamics of capillary waves at the interface of a two-layer stratified turbulent channel flow. We use a combined pseudo-spectral/phase field method to solve for the turbulent flow in the two liquid layers and to track the dynamics of the liquid–liquid interface. The two liquid layers have same thickness and same density, but different viscosity. We vary the viscosity of the upper layer (two different values) to mimic a stratified oil–water flow. This allows us to study the interplay between inertial, viscous and surface tension forces in the absence of gravity. In the present set-up, waves are naturally forced by turbulence over a broad range of scales, from the larger scales, whose size is of order of the system scale, down to the smaller dissipative scales. After an initial transient, we observe the emergence of a stationary capillary wave regime, which we study by means of temporal and spatial spectra. The computed frequency and wavenumber power spectra of wave elevation are in line with previous experimental findings and can be explained in the frame of the weak wave turbulence theory. Finally, we show that the dispersion relation, which gives the frequency ( $\omega$ ) as a function of the wavenumber ( $k$ ), is in good agreement with the well-established theoretical prediction, $\omega (k) \sim k^{3/2}$ .

A Repeated-Measures Study of Relational Turbulence and Transition Processing Communication During the Summer of COVID-19 (Summer 2020)

Drawing upon relational turbulence theory and the experiencing life transitions model, this study examined communication and relationship qualities as married couples experienced work-family changes during the COVID-19 pandemic. Sixty-four American married, different-sex partners affected by job-related changes due to COVID-19 reported their relationship qualities and their own and their spouse’s engagement in transition processing communication weekly for 10 weeks. Hypotheses addressed associations between relationship qualities and communication experiences both averaged across the 10 weeks and indexed by within-week deviations from those averages, controlling for the previous week’s reports. Husbands’ and wives’ own attempts—and also their perceptions of their spouse’s attempts—to increase interaction, promote connection, promote feeling situated, and increase confidence were associated with decreased relationship uncertainty and improved interdependence. Husbands’ self-reported relationship qualities demonstrated the strongest associations with averaged reports of relationship-focused communication, whereas wives’ relational qualities were more strongly associated with week-to-week fluctuations in relationship-focused communication.

Effect of intermittent structures on the spectral index of the magnetic field in the slow solar wind

Abstract. Intermittent structures are ubiquitous in the solar wind turbulence, and they can significantly affect the power spectral index (which reflects the cascading process of the turbulence) of magnetic field fluctuations. However, to date, an analytical relationship between the intermittency level and the magnetic spectral index has not been shown. Here, we present the continuous variation in the magnetic spectral index in the inertial range as a function of the intermittency level. Using the measurements from the Wind spacecraft, we find 42 272 intervals with different levels of intermittency and with a duration of 5–6 min from 46 slow-wind streams between 2005 and 2013. Among them, each of the intermittent intervals is composed of one dominant intermittent structure and background turbulent fluctuations. For each interval, a magnetic spectral index αB is determined for the Fourier spectrum of the magnetic field fluctuations in the inertial range between 0.01 and 0.3 Hz. A parameter Imax, which corresponds to the maximum of the trace of the partial variance increments of the intermittent structure, is introduced as an indicator of the intermittency level. Our statistical result shows that, as Imax increases from 0 to 20, the magnetic spectrum becomes gradually steeper and the magnetic spectral index αB decreases from −1.63 to −2.01. Accordingly, for the first time, an empirical relation is established between αB and Imax: αB=0.4exp⁡(-Imax/5)-2.02. The result will help us to uncover more details about the contributions of the intermittent structures to the magnetic power spectra and, furthermore, about the physical nature of the energy cascade taking place in the solar wind. It will also help to improve turbulence theories that contain intermittent structures.

Turbulence: A Significant Role in Clear-Air Echoes of CINRAD/SA at Night

It is commonly believed that clear-air echoes detected by weather radars are caused by atmobios migration. However, clear-air echoes are sometimes inconsistently related to the activity of living creatures. In some cases, the characteristics of radar products seem to conform to biological scattering, but the movement of echoes cannot be observed. For these reasons, we sought to expand the cause of clear-air echoes from a Chinese Doppler S-band Weather Radar (CINRAD/SA) in Beijing. Some contradictions were discovered in a case which diverged from previous conclusions. It was found that the progression and movement of clear-air echoes do not conform to the rules of biological activities. The frequency distribution of dual-wavelength ratio peaks is 21.5 dB, which is in accordance with Villars–Weisskopf’s turbulence theory. From 1 May to 20 May, the 58% dual-wavelength ratio between the S-band and the X-band was distributed between 18 dB and 24 dB. These results show that more than half of the clear-air echoes of CINRAD/SA at night were caused by turbulence in Beijing. A new model of troposcatter propagation, the reflecting-layers model, was then introduced to explain the radar observations. According to the reflecting-layers model, the echoes’ diurnal variation and reflectivity characteristics are attributed to the effects of turbulent mixing. Excessive turbulent mixing affects the generation of the reflective layer, thereby weakening the echo signal. It is necessary to re-examine the position of turbulence in clear-air echoes.

Energy dissipation rate in the inertial sublayer of turbulent channel flow at large but finite Reτ

A theory of the position dependence of the statistical average of the energy dissipation rate per unit mass ($ϵ$>) in the inertial sublayer of turbulent channel flow is proposed on the basis of the spirit of the Linear Response Theory (LRT) of turbulence. The theory with data consisting of a series of direct numerical simulations up to Re${}_{\ensuremath{\tau}}\ensuremath{\approx}$ 8000 suggests ${\ensuremath{\kappa}}_{ϵ}\ensuremath{\approx}$ 0.44 for the von K\'arm\'an constant of $ϵ$>.

Dispersion of tracer particles by wave turbulence

We have investigated the dispersion of tracer particles by numerically simulating ensemble nonlinear gravity waves on the surface of an infinitely deep fluid. Using the concepts of passive Lagrangian markers and insights from weak wave turbulence theory, we validated fundamental predictions for particle kinematics associated with scaling laws and probability density functions. These results improve the understanding of wave-driven transport phenomena that depart far from equilibrium and the development of the high-fidelity models of transport processes in geophysical systems.

On the dynamical stochastic electrostatic noise fluctuations in Zakharov model

Several of nonlinear plasma and burst electrostatic waves with vastly changing electric fields existed in a widely stochastic ranges in magnetosphere and space. According to the theory of stochastic-growth and density turbulence, it explained that the fluctuations, damping and electrostatic forcing produce in localized space regions. Theoretically, this plasma wave can be described by Schrödinger and Zakharov equations with stochastic term, which is applicable widely in plasma of space applications. Mathematical methods have been applied to produce new stochastic solutions for Zakharov model in subsonic limit with noises in Itô sense. The obtained solutions in the exact solitonic, breather envelopes structure, localized envelopes, dark dissipative blow-up wave, explosive, shocks and shock-like solitons becomes important in investigating electrostatic waves in aurora and solar–Wind. Finally, the effects of noise terms in Itô sense have been discussed and may be applicable to use in bursty waves in cusps regions and solar–wind.

Turbulence via Intermolecular Potential: Viscosity and Transition Range of the Reynolds Number

Turbulence in fluids is an ubiquitous phenomenon, characterized by spontaneous transition of a smooth, laminar flow to rapidly changing, chaotic dynamics. In 1883, Reynolds experimentally demonstrated that, in an initially laminar flow of water, turbulent motions emerge without any measurable external disturbance. To this day, turbulence remains a major unresolved phenomenon in fluid mechanics; in particular, there is a lack of a mathematical model where turbulent dynamics emerge naturally from a laminar flow. Recently, we proposed a new theory of turbulence in gases, according to which turbulent motions are created in an inertial gas flow by the mean field effect of the intermolecular potential. In the current work, we investigate the effect of viscosity in our turbulence model by numerically simulating the air flow at normal conditions in a straight pipe for different values of the Reynolds number. We find that the transition between laminar and turbulent flow in our model occurs, without any deliberate perturbations, as the Reynolds number increases from 2000 to 4000. As the simulated flow becomes turbulent, the decay rate of the time averaged Fourier spectrum of the kinetic energy in our model approaches Kolmogorov’s inverse five-thirds law. Both results are consistent with experiments and observations.

Interference from a partner and anger arousal: a meta-analysis

ABSTRACT A random-effects meta-analysis was conducted to examine proposition 2 of relational turbulence theory, that interference from a partner heightens affective arousal toward the partner. The meta-analysis revealed a positive average correlation between interference from a partner and anger toward that partner (k = 15, N = 6114, r = .419 [95% CI: .376, .459]) with a prediction interval ranging from .264 to .553. Upon uncovering heterogeneity, study moderators were examined to explain variability in study effect sizes including (a) dating versus married samples, (b) average length of romantic relationship, (c) percent of female participants in the study, (d) average age of participants, (e) author research teams, (f) before COVID-19 versus during COVID-19 data collections, and (g) discrete anger measurements versus anger subsumed in negative affect measurements. These study moderators were unable to explain heterogeneity in effect sizes. The computed summary effect and prediction interval support the theoretical logic of proposition 2 from relational turbulence theory.

Semi-theoretical prediction for equilibrium scour depth of a mono-pile supported tidal stream turbine

Local scour around the mono-pile foundation of a tidal stream turbine is an essential issue regarding its structural safety and stability. A semi-theoretical framework for equilibrium scour depth prediction of the mono-pile foundation is proposed with consideration of effects from the rotating rotor. The equivalent mean velocity is obtained by fitting to the accelerated flow beneath the turbine rotor and used to derive the scour depth prediction equation based on the phenomenological theory of turbulence. Both clear-water and live-bed conditions are considered. Results show that the acceleration effect from the turbine rotor turns more pronounced with increasing thrust coefficient and tip clearance. Such acceleration generally leads to an increased scour depth around the mono-pile foundation, reaching the maximum scour depth when the equivalent mean velocity is equal to the critical velocity of sediment entrainment. In addition, a reduction factor of 0.74 is obtained for the bi-directional flow condition. Overall, the proposed prediction framework agrees with the experimental data in the literature with acceptable accuracy, although it could be improved with more data available in the future.

Novel Detection of Atmospheric Turbulence Profile Using Mie-Scattering Lidar Based on Non-Kolmogorov Turbulence Theory.

Turbulence can cause effects such as light intensity fluctuations and phase fluctuations when a laser is transmitted in the atmosphere, which has serious impacts on a number of optical engineering application effects and on climate improvement. Therefore, accurately obtaining real-time turbulence intensity information using lidar-active remote sensing technology is of great significance. In this paper, based on residual turbulent scintillation theory, a Mie-scattering lidar method was developed to detect atmospheric turbulence intensity. By extracting light intensity fluctuation information from a Mie-scattering lidar return signal, the atmospheric refractive index structure constant, Cn2, representing the atmospheric turbulence intensity, could be obtained. Specifically, the scintillation effect on the detection path was analyzed, and the probability density distribution of the light intensity of the Mie-scattering lidar return signal was studied. It was verified that the probability density of logarithmic light intensity basically follows a normal distribution under weak fluctuation conditions. The Cn2 profile based on Kolmogorov turbulence theory was retrieved using a layered, iterative method through the scintillation index. The method for detecting Kolmogorov turbulence intensity was applied to the detection of the non-Kolmogorov turbulence intensity. Through detection using the scintillation index, the corresponding C˜n2 profile could be calculated. The detection of the C˜n2 and Cn2 profiles were compared with the Hufnagel-Valley (HV) night model in the Yinchuan area. The results show that the detection results are consistent with the overall change trend of the model. In general, it is feasible to detect a non-Kolmogorov turbulence profile using Mie-scattering lidar.

Emerging adult romantic relationship narratives during COVID‐19: Understanding relational challenges and opportunities

AbstractThis cross‐sectional study sought to understand how emerging adult college students currently in relationships experienced and dealt with COVID‐19 stress during a key time for romantic relationship development. Using a relational turbulence theory (RTT) framework, we examined romantic relationship narratives in three different groups of students for indicators of uncertainty (break‐up anxiety [BUA]), interference (conflict and negative emotion), facilitation (support), and coping strategies. Of particular interest was whether these indicators would vary as a function of when the stress of the pandemic was assumed to be more compared to less disruptive to relationships (Group 1 Fall 2020 and Group 2 Spring 2021, off campus/unvaccinated; Group 3 Fall 2021, on campus/vaccinated). As expected, students interviewed during periods when pandemic stress was assumed to be higher showed more break‐up anxiety, increased conflict, and angry word use to describe conflict events. However, contrary to expectations, sad emotions and support did not vary as a function of when students were interviewed. Regardless of when students were interviewed, there were several strategies they used to cope with the stresses of COVID‐19. The most frequent strategies students described were using technology/media to stay connected and taking steps to keep themselves and their families healthy.

Exact Solutions of the Generalized ZK and Gardner Equations by Extended GeneralizedG′/G-Expansion Method

In this investigation, the exact solutions of variable coefficients of generalized Zakharov-Kuznetsov (ZK) equation and the Gardner equation are studied with the help of an extended generalizedG′/Gexpansion method. The main objective of this study is to establish the closed-form solutions and dynamics of analytical solutions to the generalized ZK equation and the Gardner equation. The generalized ZK equation and the Gardner equation govern the behavior of nonlinear wave phenomena in the presence of magnetic field in plasma dynamics, turbulence, bottom topography, and quantum field theory. We construct innovative solutions to the models under consideration using various computing tools and a recently developed extended generalizedG′/Gexpansion technique. The extended generalizedG′/Gexpansion technique is a well-defined and simple technique which is based on the initial assumed solutions of the polynomial ofG′/G. The derived solutions for both the equations are the hyperbolic, trigonometric, and rational functions. The obtained solutions have shock/kink waves and multisoliton, which depict the dynamical representations of the acquired solutions through the three-dimensional surface plots and the contour plots.