Anisotropic Turbulence Research Articles

A mathematical model for the interaction of anisotropic turbulence with a rigid leading edge

This paper investigates the effect of anisotropic turbulence on generating leading-edge aerofoil–turbulence interaction noise. Thin aerofoil theory is used to model an aerofoil as a semi-infinite plate, and the scattering of incoming turbulence is solved via the Wiener–Hopf technique. This theoretical solution encapsulates the diffraction problem for gust–aerofoil interaction and is integrated over a wavenumber–frequency spectrum to account for general incoming anisotropic turbulence. We develop a novel axisymmetric wavenumber–frequency model that captures the wall-normal variation in turbulence characteristics, differing from previous approaches. Then, the method of Gaussian decomposition, in which the generalised spectra are approximated through the weighted sum of individual Gaussian eddy models, is applied to fit the turbulence model to experimental data. Comparisons with experimental data show good agreement for a range of anisotropic ratios.

Influence of Alfvén Ion–Cyclotron Waves on the Anisotropy of Solar Wind Turbulence at Ion Kinetic Scales

The power spectra of the magnetic field at ion kinetic scales have been found to be significantly influenced by Alfvén ion–cyclotron (AIC) waves. Here, we study whether and how this influence of the AIC wave depends on the θVB angle (the angle between the local mean magnetic field and the solar wind velocity direction). The wavelet technique is applied to the high time-resolution (11 vectors per second) magnetic field data from WIND spacecraft measurements in a fast solar wind stream associated with an outward magnetic sector. It is found that around the ion kinetic scales (0.52 Hz–1.21 Hz), the power spectrum in the parallel angular bin 0∘<θVB<10∘ has a slope of −4.80±0.15. When we remove the left-handed polarized AIC waves (with normalized reduced magnetic helicity smaller than −0.9) from the fluctuations, the spectral index becomes −4.09±0.11. However, the power spectrum in the perpendicular angular bin 80∘<θVB<90∘ changes very little during the wave-removal process, and its slope remains −3.22±0.07. These results indicate that the influence of the AIC waves on the magnetic spectral index at the ion kinetic scales is indeed dependent on θVB, which is due to the anisotropic distribution of the waves. Apparently, when the waves are removed from the original data, the spectral anisotropy weakens. This result may help us to better understand the physical nature of the spectral anisotropy around the ion scales.

Anisotropy of Self-Correlation Level Contours in Three-Dimensional Magnetohydrodynamic Turbulence

MHD turbulence is considered to be anisotropic owing to the presence of a magnetic field, and its self-correlation anisotropy has been unveiled by solar wind observations. Here, based on numerical results of compressible MHD turbulence with a global mean magnetic field, we explore variations of the normalized self-correlation function’s (NCF) level contours with the scale as well as their evolution. The analyses reveal that the NCF’s level contours tend to elongate in the direction parallel to the mean magnetic field, and the elongation becomes weak with decreasing intervals. These results are consistent with slow solar wind observations. The less anisotropy of the NCF’s level contours with the shorter intervals can be produced by the fact that coherent structures stretch more along the parallel direction at the long intervals than at the short intervals. The analyses also disclose that as the simulation time builds up, the NCF’s level contours change thinner and thinner, and the anisotropy of the NCF’s level contours grows, which can be caused by the break of large coherent structures into small ones. The increased self-correlation anisotropy with time foretells that the self-correlation anisotropy of solar wind turbulence enlarges with the radial distance, which needs to be tested against observations by using Parker Solar Probe (PSP) measurements.

An efficient and low-divergence method for generating inhomogeneous and anisotropic turbulence with arbitrary spectra

This paper proposes a low-divergence method to generate inhomogeneous and anisotropic turbulence. Based on the idea of correlation reconstruction, the proposed method uses the Cholesky decomposition matrix to re-establish turbulence correlation functions. This is in contrast with the time-consuming procedure used in conventional methods to solve eigenvalues and eigenvectors in each location required by coordinate transformation, thereby reducing the computational complexity and improving the efficiency of synthetic turbulence generation. The proposed method is based on the classical spectrum-based method that is widely used to generate homogeneous and isotropic turbulence. By adjusting the generation strategy of specific random vectors, inhomogeneous and anisotropic turbulence with a relatively low divergence level can be obtained in practice, with almost no additional computational burden. Two versions of the proposed method are presented: the inverter and shifter versions. Both the versions are highly efficient, easy to implement, and compatible with high-performance computing. This method is also suitable for providing high-quality initial or boundary conditions for scale-resolving turbulence simulations with large grid numbers (such as direct numerical simulations or large eddy simulations); it can be rapidly implemented using various open-source computational fluid dynamics codes or common commercial software.

Anisotropy of turbulence at the core of the tip and hub vortices shed by a marine propeller

Large-eddy simulation on a grid consisting of 5 billion points was utilized to study the properties of turbulence at the core of the tip and hub vortices shed by a marine propeller across working conditions. Turbulence at the core of the tip vortices was found to be initially isotropic, moving towards a ‘cigar-shaped’ axisymmetric state as instability grows, dominated by turbulent fluctuations of the velocity component directed in the radial direction of the cylindrical reference frame centred at the wake axis. The break-up of the coherence of the tip vortices is instead characterized by turbulence recovering an isotropic state. This process is accelerated by growing load conditions of the propeller. In contrast, during instability of the hub vortex, turbulence at its core develops a ‘pancake-shaped’ axisymmetric state, dominated by the fluctuations of the radial and azimuthal velocities. However, at higher propeller loads turbulence at the core of the hub vortex keeps close to isotropy, thanks to a faster instability. Within both tip and hub vortices the deviations from Boussinesq's hypothesis were found very significant, providing evidence of the unsuitability of conventional turbulence modelling. At the core of the tip vortices they become especially large at their break-up and for increasing load conditions of the propeller, equivalent to more intense structures. In contrast, at the core of the hub vortex they were verified to be decreasing functions of the propeller load.

Performance of HQAM/XQAM Laser Communication System in Anisotropic Non-Kolmogorov Ground–HAP–Satellite Uplink

Satellite laser communication can achieve high-speed, high-precision, and high-security broadband communication without being constrained by the electromagnetic spectrum, which has attracted attention. So, this paper proposes the use of a high-altitude platform (HAP) under anisotropic non-Kolmogorov turbulence to improve the communication performance of the system. Cross quadrature amplitude modulation (XQAM) and hexagon quadrature amplitude modulation (HQAM) are applied to the ground–HAP–satellite (G-H-S) laser communication system. Considering the combined effects of uplink light intensity scintillation, beam wander, and the angle of arrival fluctuation, the G-H-S system’s bit error rate (BER) closure expression is derived under the EW distribution. Simultaneously, the relationship between the G-H-S system’s signal-to-noise ratio (SNR) and BER under different anisotropic factor u values is simulated and compared with the traditional ground–satellite (G-S) system. The results show that the communication performance of the G-H-S system with HQAM modulation is better. In addition, the effects of the zenith angle, receiving aperture, transmitter beam radius, and beam divergence angle on the BER performance of the system are also studied. Finally, the correctness of the analysis results is verified via Monte Carlo simulation. This research will benefit the design and optimization of satellite laser communication systems.

Numerical Investigation of Bed Shear Stress and Roughness Coefficient Distribution in a Sharp Open Channel Bend

The helical flow leads to the redistribution of the bed shear stress and roughness coefficient in open-channel bends, influencing the transport of sediment and riverbed evolution process. To explore the mechanisms underlying this redistribution, a 3D numerical simulation of a 180° sharp bend was performed by solving the Reynolds-averaged Navier–Stokes (RANS) equations using the Reynolds stress equation model (RSM) as the anisotropic turbulence closure approach. The results indicate that the high bed shear stress zone appeared in the inner bank region from the 50° to 110° sections, and the section maximum bed shear stress gradually shifted outwards, owing to the advective momentum transport by the circulation cells. The quantitative analyses of the terms in depth-averaged Navier-Stokes equations indicate that the contributions of the cross-stream circulation and cross-flow to the downstream bed shear stress were of the same order of magnitude, and the overall contribution rate of the cross-stream circulation was 20%. The contribution rates of the cross-flow, cross-stream circulation, turbulence, and pressure gradient to the transverse bed shear stress were approximately 30%, 7%, 3%, and 60%, respectively, indicating that the pressure gradient term arising from the transverse water surface slope played a dominant role. The Chezy resistance coefficient showed an overall decreasing trend along the bend. Therefore, an effective expression considering the streamwise variation along the centreline and transverse variation was successfully established to predict the uneven distribution of the Chezy resistance coefficient.

Spatial coherence length and wave structure function for plane waves transmitted in anisotropic turbulent oceans.

The spatial coherence length and wave phase structure function are two important factors in describing turbulence's effect on light propagation in seawater. This paper derives the wave phase structure function and spatial coherence length of plane waves in moderate to strong turbulent channels by deriving a "modification seawater turbulence power spectrum" and an oceanic-modified Rytov approximation. The evolutions in wave structure function, coherence length with the temperature dissipation rate, energy dissipation rate, anisotropy turbulence factor, signal wavelength, and propagation distance are analyzed by numerical calculation. In the moderate and strong turbulence regions, the phase structure function and spatial coherence length increase and decrease with increasing transmission distance and turbulence strength, respectively, and there is a saturation tendency for both. The fluctuation of seawater salinity has a greater effect on the phase structure function and coherence length than the temperature fluctuation. In addition, the wave structure function decreases with increasing signal wavelength and degree of turbulent anisotropy, but the trend of spatial coherence length is reversed.

Flow patterns and defect dynamics of active nematic liquid crystals under an electric field.

The effects of an electric field on the flow patterns and defect dynamics of two-dimensional active nematic liquid crystals are numerically investigated. We found that field-induced director reorientation causes anisotropic active turbulence characterized by enhanced flow perpendicular to the electric field. The average flow speed and its anisotropy are maximized at an intermediate field strength. Topological defects in the anisotropic active turbulence are localized and show characteristic dynamics with simultaneous creation of two pairs of defects. A laning state characterized by stripe domains with alternating flow directions is found at a larger field strength near the transition to the uniformly aligned state. We obtained periodic oscillations between the laning state and active turbulence, which resembles an experimental observation of active nematics subject to anisotropic friction.

Model and method to predict the turbulent kinetic energy induced by tidal currents, application to the wave-induced turbulence

A prediction model for the turbulent kinetic energy (TKE) induced by tidal-currents is proposed as a function of the barotropic velocity only, along with a robust method evaluating the different parameters involved using Acoustic Doppler Current Profiler (ADCP) measurements from Alderney Race. We find that the model is able to reproduce correctly the TKE profiles with coefficients of correlation on average higher than 0.90 and normalised root-mean-square errors (NRMSE) less than 14%. Different profiles are also tested for the mean velocity, no satisfactory prediction model is found but we are able to have decent estimates of the velocity shear and friction velocity. Two applications are then carried out. First the turbulent budget terms are estimated and discussed. We identify the turbulent production and dissipation of TKE as the most important mechanisms, then we discuss the validity of several theoretical results derived for isotropic turbulence for this application. A strong departure for the estimation of the turbulent dissipation is notably found and explained by the turbulent anisotropy. At last the prediction model for the TKE is used to infer the wave-induced TKE. We show the importance of removing the tidal component, waves can have a strong influence down to mid-depth.

Influence of ground blocking on the acoustic phase variance in a turbulent atmosphere.

Sound propagation through atmospheric turbulence is important in many applications such as localization of low flying aircraft, sonic boom disturbances, and auralization of aircraft during takeoff and landing. This article extends an isotropic turbulence model in the atmospheric boundary layer to account for ground blocking of buoyancy-produced velocity fluctuations. The extended, anisotropic turbulence model is needed to correctly predict the effect of the largest velocity eddies on the statistical characteristics of sound signals. This model and geometrical acoustics are then employed to derive a closed-form expression for the variance of the phase fluctuations of a spherical sound wave for vertical and slanted propagation, without the use of the Markov approximation. A numerical analysis of this expression indicates significant anisotropy of the phase variance due to the buoyancy-produced velocity fluctuations with ground blocking such that it decreases in the vertical direction and increases in the near-horizontal directions. The newly formulated phase variance is compared with data from an outdoor experiment on vertical and slanted sound propagation. By accounting for ground blocking, much better agreement is obtained between the theoretical predictions and experimental data.

Self-nested large-eddy simulations in PALM model system v21.10 for offshore wind prediction under different atmospheric stability conditions

Abstract. Large-eddy simulation (LES) resolves large-scale turbulence directly and parametrizes small-scale turbulence. Resolving micro-scale turbulence, e.g., in wind turbine wakes, requires both a sufficiently small grid spacing and a domain large enough to develop turbulent flow. Refining a grid locally via a nesting interface effectively decreases the required computational time compared to the global grid refinement. However, interpolating the flow between nested grid boundaries introduces another source of uncertainty. Previous studies reviewed nesting effects for a buoyancy-driven flow and observed a secondary circulation in the two-way nested area. Using a nesting interface with a shear-driven flow in LES, therefore, requires additional verification. We use PALM model system 21.10 to simulate a boundary layer in a cascading self-nested domain under neutral, convective, and stable conditions and verify the results based on the wind speed measurements taken at the FINO1 platform in the North Sea. We show that the feedback between parent and child domains in a two-way nested simulation of a non-neutral boundary layer alters the circulation in the nested area, despite spectral characteristics following the reference measurements. Unlike the pure buoyancy-driven flow, a non-neutral shear-driven flow slows down in a two-way nested area and accelerates after exiting the child domain. We also briefly review the nesting effect on the velocity profiles and turbulence anisotropy.

Effect of d-type rib roughness on the turbulent structure of side wall boundary layer for wave-current combined flow

An experimental study has been carried out in a laboratory flume to characterize the turbulence structure and turbulence anisotropy in the boundary layer over smooth and rough side walls for both current alone and wave-current combined flow situations. The rough side wall of the flume comprises a train of circular ribs (diameter, k) attached vertically maintaining uniform spacing p along the streamwise direction. The experiments are performed for smooth surface and rough (ribbed) surfaces with p/k = 2, 3, and 4 to reproduce different cases of d-type rib roughness. The effect of wave-current interaction has been investigated by superposing waves of two different frequencies. Time series data of three velocity components are obtained using Acoustic Doppler Velocimeter. At the near wall region, roughness with higher p/k value enhances the level of turbulent intensity and Reynolds stress significantly. In a channel with smooth side wall, the wave-current combined flow produces lesser turbulence intensity than the current alone flow near the wall. However, for a ribbed wall case, the effect is completely opposite that is, wave-current interacting flow induces higher intensities compared to the reference current alone flow. Substantial decline in the turbulent length scales at the near wall region are observed for ribbed walls, which reveals the strong effect of roughness elements on the turbulent structure. Superposition of wave reduces the length scales even more for both smooth and rough wall cases. As the spacing between two ribs ( p/ k ratio) increases, the energy dissipation rate increases. The analysis of anisotropy invariant map demonstrates a reduction of anisotropy in the vicinity of ribbed wall compared to that for a smooth wall. For wave-current combined flow, the anisotropy invariant data of Reynolds stress tensor varies dramatically within the boundary of map, reflecting significant changes in the state of turbulence.

The Decay of Energy and Scalar Variance in Axisymmetric Turbulence

We review the recent progress in our understanding of the large scales in homogeneous (but anisotropic) turbulence. We focus on turbulence which emerges from Saffman-like initial conditions, in which the vortices possess a finite linear impulse. Such turbulence supports long-range velocity correlations of the form uiu′j=O(r−3), where u and u′ are separated by a distance r, and these long-range interactions dominate the dynamics of large eddies. We show that, for axisymmetric turbulence, the energy and integral scales evolve as u⊥2~u//2~t−6/5 and l⊥~l//~t2/5, where ⊥ and // indicate directions that are perpendicular and parallel to the symmetry axis, respectively. These predictions are consistent with the evidence of direct numerical simulations. Similar results are obtained for the passive scalar variance, where we find that θ2~t−6/5. The primary point of novelty in our discussion of passive scalar decay is that it is based in real (rather than spectral) space, making use of an integral invariant which is a generalization of the isotropic Corrsin integral.

Numerical Study of Shock Wave Boundary Layer Interaction Based on an Anisotropic Turbulence Model

This paper presents numerical simulations of incident oblique shock wave impinging on a flat plate with a turbulent boundary layer using a nonlinear anisotropic turbulence model. Three cases with different incident shock angles under the same free-stream Mach number are studied, including attachment flow, incipient separation flow and outstanding separation flows. Convection terms and diffusion terms are calculated using the third-order Roe’s flux difference splitting scheme and the second-order central difference scheme in the code, respectively. The Runge-Kutta time marching method is employed to solve the governing equations for steady state solutions. Comparisons between the simulated results and experimental data are carried out, which include density contours, pressure distribution, skin friction coefficients distribution and streamline vectors. To capture the essential behavior of this complex anisotropic turbulence separation flow caused by the shock-wave-boundary-layer-interaction (SWBLI), an anisotropic eddy viscosity tensor and a nonlinear Reynolds stress-strain constitutive relationship are used in this model. The anisotropic eddy viscosity tensor is composed of turbulent fluctuation velocity vector and characteristic length vector which are obtained by statistical partial average scheme. With proper modeling of anisotropic eddy viscosity tensor, the numerical results are in good agreement with the experimental data. It is worthy to point it out that the computation accuracy decreases with increasing interaction intensity.

3D numerical investigation of flow behavior in an open channel with uniform and layered vegetation patches under varying submergence conditions

Staggered vegetation patches (central and outer) in an open channel have a significant effect on flow structure and river geomorphology. Although investigation on vegetation patches has been done in the past, however none of the past studies was carried out for varying submergence conditions from outer to central patches or for layered staggered patches with varying submergence. This knowledge gap has been attempted to bridge in this study. In this research work, a numerical technique has been used to analyze the flow behavior in a riverine flow by modeling central and outer vegetation patches as rigid elements in a rectangular channel. The influence of various vegetation patch configurations and submergence conditions were studied on flow characteristics. Computational fluid dynamics (CFD) package FLUENT was used for simulation purposes. Results showed that the flow in the central region of the channel was highly non-uniform, having a longitudinal distribution of velocities with alternating peaks and troughs throughout the central portion of the channel. However, in the outer regions, the velocity pattern acquired sawtooth distribution inside the patch regions (PR) but stabilized in the free regions (FR). Outer FR has 9% less velocities as compared to central FR. Moreover, the inflection point in velocity profiles and the bulge in contour plots was observed which triggered the shear layer at the top of submerged vegetation and at the interface of outer FR and central PR or vice versa. A complex pattern of secondary flow was also observed due to the turbulence anisotropy in the channel and its magnitude ranged from 0.1 to 3.7% of average velocity U. The results also showed that the state of submergence has a notable effect on the flow blockage and percentage of discharge passing through various regions.

Environmental fluid mechanics of minimum energy loss weirs: hydrodynamics and self-aeration at Chinchilla MEL weir during the November–December 2021 flood event

During the recent decades, a number of overflow embankment protection systems were implemented. One design is the Minimum Energy Loss (MEL) weir, developed to pass large flood events with minimum energy loss and low erosion. Several MEL weirs have successfully operated for decades in Australian catchments affected by heavy tropical and sub-tropical rainfalls with very flat gradients. Their historical performances are discussed and confirmed the design capability to pass large floods with small afflux and very small energy loss and little environmental impact including no significant erosion at the abutments. During a major flood, visual and quantitative observations were undertaken at the Chinchilla MEL weir. On the smooth converging chute, the observations showed that the inception of self‐aeration was a three‐dimensional process with a gradual change in free‐surface roughness, from a smooth glassy free-surface to a very-rough choppy surface. An optical technique (OF) was implemented to derive the contour maps of surface velocities based upon video movies taken from a sturdy tripod. The self-aerated flow velocity data were close to the backwater equation in the self‐aerated region, while highlighting regions of high- and low-velocities across the chute. The OF data showed large streamwise surface velocity fluctuations in the aerated flow region, consistent with the broad literature on self-aerated flow measurements. The ratio of transverse to streamwise surface turbulence intensity indicated a strong anisotropy of the free-surface turbulence.

Estimating turbulent energy flux vertical profiles from uncrewed aircraft system measurements: exemplary results for the MOSAiC campaign

Abstract. This study analyzes turbulent energy fluxes in the Arctic atmospheric boundary layer (ABL) using measurements with a small uncrewed aircraft system (sUAS). Turbulent fluxes constitute a major part of the atmospheric energy budget and influence the surface heat balance by distributing energy vertically in the atmosphere. However, only few in situ measurements of the vertical profile of turbulent fluxes in the Arctic ABL exist. The study presents a method to derive turbulent heat fluxes from DataHawk2 sUAS turbulence measurements, based on the flux gradient method with a parameterization of the turbulent exchange coefficient. This parameterization is derived from high-resolution horizontal wind speed measurements in combination with formulations for the turbulent Prandtl number and anisotropy depending on stability. Measurements were taken during the MOSAiC (Multidisciplinary drifting Observatory for the Study of Arctic Climate) expedition in the Arctic sea ice during the melt season of 2020. For three example cases from this campaign, vertical profiles of turbulence parameters and turbulent heat fluxes are presented and compared to balloon-borne, radar, and near-surface measurements. The combination of all measurements draws a consistent picture of ABL conditions and demonstrates the unique potential of the presented method for studying turbulent exchange processes in the vertical ABL profile with sUAS measurements.

Multi-layered turbulence structure of flow under free surface of open channel flow using velocity–vorticity correlation structure method

Turbulence near a free surface is of multi-layered nature and of fundamental significance in understanding vast marine and ocean processes. Direct numerical simulation of an open channel with the free surface modeled by a free-slip boundary is performed. Complex multi-layered turbulence (i.e., the blockage layer, the slip layer, and the Kolmogorov layer) under the free surface are first identified based on predefined anisotropy indices. Later on, statistical features of the anisotropic turbulence inside the three turbulence layers are presented. Most importantly, considering the drastic variation of both velocity and vorticity in near-surface layers, we revisit the complex multi-layered turbulence by the velocity–vorticity correlation structure (VVCS) method [Chen et al., J. Fluid Mech. 742, 291–307 (2014)]. It is found that the VVCS can physically depict evolution of the hairpin vortex inside the channel, including generation in the bottom wall region, expansion in the bulk flow, process of getting flattened in the blockage layer, and then distortion and breakdown in the slip layer and the Kolmogorov layer. Specifically, the correlation structures VVCS11 for streamwise velocity and streamwise vorticity reflect how the hairpin legs are spatially distributed, while the correlation structures VVCS13 for streamwise velocity and spanwise vorticity describe the evolution of the hairpin vortex head. In the end, a cradle-like structure model is proposed based on the VVCS results and the vortex ring collision model is introduced to improve the understanding of the multi-layered turbulence under free surface. Hopefully, this research can be of value in future turbulence modeling work on free surface flows.

Editorial: Celebrating the contributions of Professor Guieppe Marrucci to rheology and non-Newtonian fluid mechanics

Direct numerical simulations (DNS) of generalized Newtonian fluids in a rectangular channel at a friction velocity of 180 are performed using OpenFOAM. The simulations are performed for a Newtonian fluid, a power-law fluid, and a Herschel-Bulkley (HB) fluid. Two-point correlations of the streamwise velocity reveal that a significant increase in the computational domain size is required to avoid statistical correlation due to the shear-thinning and yield stress. The analogous Kolmogorov scale shows that the dissipative structures may be orders of magnitude larger with these two non-Newtonian behaviors. This is demonstrated to be linked to the significant increase in the viscosity. The mean streamwise velocity field integrates to a larger bulk velocity with increased shear-thinning as compared with the Newtonian fluid flow. The addition of the yield stress does the same with respect to the power-law fluid flow. The difference in the velocity profiles is more pronounced in the outer layer. It, however, is observed that the rheology does not play a significant role in the outer layer for the mean streamwise velocity. It is shown that an HB fluid flow does not necessarily show an increase in the fluctuating streamwise velocity with respect to a power-law fluid flow. It is also shown that low power index and high Hedstrom number lead to strong turbulence anisotropy due to the weakening of turbulence. The turbulent kinetic energy (TKE) budget reveals that the studied non-Newtonian behavior mainly affects the inner layer of the flow. The yield stress is also seen to show small variations in the TKE budget in the outer layer. This is attributed to the weakening turbulence as a result of the increased viscosity near the center. It is also demonstrated that the studied HB fluid flow results can be successfully reproduced by under-resolved DNS.