Turbulent Signals Research Articles

Stirring anisotropic turbulence with an active grid

We study spectra and high-order structure functions in anisotropic wind tunnel turbulence, which is generated using an active grid. In the first experiment, we impose homogeneous shear turbulence with a constant gradient of the mean flow and (approximately) homogeneous turbulent fluctuations. We measure mixed structure functions of order 2, 3, 4, 6, and 10 using an array of two-component hotwires. These structure functions, which vanish for isotropic turbulence, display scaling with scaling exponents that highlight intermittency: the return to isotropy at small scales of large fluctuations is much slower than expected on the basis of a simple Kolmogorov-like scaling argument [J. L. Lumley, “Similarity and the turbulent energy spectrum,” Phys. Fluids 10, 855 (1967)]. In the second experiment, we impose anisotropy in otherwise homogeneous turbulence through the time modulation of the active grid. This is done by driving the grid using signals from a turbulence (shell) model, which acts as a convenient turbulent random signal generator. In this way, different statistical properties of different velocity components could be imposed. Similar to the first experiment, our interest is in the return to isotropy of the small-scale turbulent fluctuations, which is quantified using second-order quantities such as spectra and correlation functions. Also, in this case, the strongly anisotropic correlations induced by the forcing at large scales tend to return to isotropy at small, inertial-range scales, but with the imprint of large-scale anisotropy retained.

Deep sea floor observations of typhoon driven enhanced ocean turbulence

The impact of large atmospheric disturbances on deep benthic communities is not well known quantitatively. Observations are scarce but may reveal specific processes leading to turbulent disturbances. Here, we present high-resolution deep-ocean observations to study potential turbulent mixing by a large atmospheric disturbance. We deployed an array of 100-Hz sampling-rate geophysical broadband Ocean Bottom Seismometers (OBSs) on the seafloor. Within the footprint of this array we also deployed an oceanographic 0.5-Hz sampling-rate vertical temperature sensor string covering the water phase between 7 and 207 m above the seafloor at about 3000 m depth off eastern Taiwan between June 2017 and April 2018. In September 2017, all instruments recorded Category 4 cyclone Typhoon Talim’s passage northeast of the array one day ahead of the cyclone’s closest approach when the cyclone’s eye was more than 1,000 km away. For 10 days, a group of near-inertial motions appeared most clearly in temperature. The group contained the largest inertial amplitudes in the ten month time series, and which led to turbulence dissipation rates O(10−7 m2 s−3). The observation reflects the importance of barotropic response to a cyclone and the propagation of inertio-gravity waves in weak density stratification. In addition to internal tides, these waves drove turbulent overturns larger than 200 m that were concurrently recorded by OBSs. The turbulent signals were neither due to seismic activity nor to ocean-surface wave action. Cyclones can generate not only microseisms and earth hums, as well as turbulence in the water column, producing additional ground motions. Quantified turbulence processes may help constrain models on sediment resuspension and its effect on deep-sea benthic life.

Secrecy performance of cognitive underlay hybrid RF/FSO system under pointing errors and link blockage impairments

In this paper, we present the secrecy performance of a cognitive underlay hybrid radio frequency/free space optical (RF/FSO) system, where the licensed spectrum of a primary user (PU) is shared with the secondary user (SU) and an Eavesdropper attempts to intercept the information through the SU RF sub-link. At the SU receiver, the selection combining scheme with heterodyne detection is considered as signal reception. In order to ensure the required quality of service at the PU, the transmit power constraint of SU RF-sub link is assumed to depend only on the interference level arriving at the PU. It is also assumed that all the system RF links follow the Nakagami-m distribution except the Eavesdropper link that is subjected to Rayleigh fading. The SU FSO link is assumed to undergo Gamma–Gamma distribution with pointing error and link blockage. Thus, the cumulative distribution function (CDF) of the SU RF sub-link is obtained and used to derived the system equivalent CDF. By utilizing system equivalent CDF, the closed-form expressions for the secrecy outage probability and average secrecy capacity are derived through the lower bound. In addition, Monte-Carlo simulation is provided to verify the accuracy of the derived expressions. The results demonstrate the influence of atmospheric turbulence, pointing error, link blockage and average signal to noise ratio at the Eavesdropper on the system performance.

Atmospheric turbulence model for direct fire ballistics

Modern system accuracy studies require high fidelity representations of environmental phenomena in order to accurately predict the down range performance of a gun system. One component of the atmosphere that has not been studied in great detail within the ballistic domain is turbulence. The current portrayal of wind leveraged by system analysis efforts ignores this element of atmospheric motion completely and thus its effects on down range dispersion have not been quantified. As a first step in addressing this deficiency, this study develops a methodology for generating synthetic turbulent wind signals along the flight path of a projectile. This goal is accomplished by integrating the work of several authors, developing techniques to fill knowledge gaps, and tailoring the solution to the direct fire domain. The significant contributions of the presented effort include mean flow direction agnostic spectral functions, provisions to account for the non-homogeneity of turbulence parameters along a trajectory, and a higher fidelity signal generation method than was used in previous work. The new information is applied to a sample engagement scenario in order to demonstrate the realization of the given techniques within the small caliber direct fire domain.

Study of Realistic Urban Boundary Layer Turbulence with High-Resolution Large-Eddy Simulation

This study examines the statistical predictability of local wind conditions in a real urban environment under realistic atmospheric boundary layer conditions by means of Large-Eddy Simulation (LES). The computational domain features a highly detailed description of a densely built coastal downtown area, which includes vegetation. A multi-scale nested LES modelling approach is utilized to achieve a setup where a fully developed boundary layer flow, which is also allowed to form and evolve very large-scale turbulent motions, becomes incident with the urban surface. Under these nonideal conditions, the local scale predictability and result sensitivity to central modelling choices are scrutinized via comparative techniques. Joint time–frequency analysis with wavelets is exploited to aid targeted filtering of the problematic large-scale motions, while concepts of information entropy and divergence are exploited to perform a deep probing comparison of local urban canopy turbulence signals. The study demonstrates the utility of wavelet analysis and information theory in urban turbulence research while emphasizing the importance of grid resolution when local scale predictability, particularly close to the pedestrian level, is sought. In densely built urban environments, the level of detail of vegetation drag modelling description is deemed most significant in the immediate vicinity of the trees.

Effects of Optical Turbulence and Density Gradients on Particle Image Velocimetry

Particle image velocimetry (PIV) is a well-established tool to collect high-resolution velocity and turbulence data in the laboratory, in both air and water. Laboratory experiments are often performed under conditions of constant temperature or salinity or in flows with only small gradients of these properties. At larger temperature or salinity variations, the changes in the index of refraction of water or air due to turbulent microstructure can lead to so-called optical turbulence. We observed a marked influence of optical turbulence on particle imaging in PIV. The effect of index of refraction variations on PIV has been described in air for high Mach number flows, but in such cases the distortion is directional. No such effect has previously been reported for conditions of isotropic optical turbulence in water. We investigated the effect of optical turbulence on PIV imaging in a large Rayleigh-Bénard tank for various path lengths and turbulence strengths. The results show that optical turbulence can significantly affect PIV measurements. Depending on the strength of the optical turbulence and path length, the impact can be mitigated in post-processing, which may reduce noise and recover the mean velocity signal, but leads to the loss of the high-frequency turbulence signal.

On the Interpretation of Parker Solar Probe Turbulent Signals

Abstract In this Letter we propose a practical methodology to interpret future Parker Solar Probe (PSP) turbulent time signals even when Taylor’s hypothesis is not valid. By extending Kraichnan’s sweeping model used in hydrodynamics we derive the Eulerian spacetime correlation function in magnetohydrodynamic (MHD) turbulence. It is shown that in MHD, the temporal decorrelation of small-scale fluctuations arises from a combination of hydrodynamic sweeping induced by large-scale fluid velocity and by the Alfvénic propagation along the local magnetic field. The resulting temporal part of the spacetime correlation function is used to determine the field-perpendicular wavenumber range of the turbulent fluctuations that contribute to the power of a given frequency ω of the time signal measured in the spacecraft frame. Our analysis also shows that the shape of frequency power spectrum P sc(ω) of the time signal will follow the same power law of the reduced power spectrum in the plasma frame, where α is the spectral index. The proposed framework for the analysis of PSP time signals entirely relies on two simple dimensionless parameters that can be empirically obtained from PSP measurements, namely, (where V ⊥ is the perpendicular velocity of PSP seen in the plasma frame) and the spectral index α.

Utility of transesophageal echocardiography for intra-operatively assessing pulmonary artery pressure across an isolated ventricular septal defect in children.

The magnitude of pulmonary hypertension (PH) is extremely important with respect to the intra-operative management of children and infants with an isolated ventricular septal defect (VSD). This study aimed to assess the feasibility and accuracy of transesophageal echocardiography for estimating pulmonary arterial systolic pressure (PASP) across isolated VSD. We compared the results of transesophageal echocardiography vs invasive PASP measured simultaneously. This study included 40 patients (age: 6months to 6years; weight: >5kg) who were undergoing elective surgery for isolated VSDs. Flow signals across the VSDs were identified as high velocity turbulent signals in systole via continuous wave Doppler at 0-120° at the mid-esophageal level. Peak velocities were recorded. Radial artery systolic pressures were assessed invasively, and PASPs were obtained after exposing the pulmonary artery intra-operatively. After excluding five patients because of unusable measurements, invasive PASP measurements were obtained in 35 patients (87.5%). There were no significant biases between echocardiographic and catheterization measurements of PASP, with a tight confidence interval measuring, on average, up to 2.6mmHg. However, the±2 standard deviation limits of agreement for mean PASP were -3.8 and 10.6mmHg. PASP measurements via transesophageal echocardiography in cardiac surgical patients under general anesthesia are recommended for use as a screening and monitoring tool for PH in children and infants, but cannot be used as a diagnostic tool.

A Predictive Model for the Streamwise Velocity in the Near‐Neutral Atmospheric Surface Layer

AbstractFor the reconstruction of the streamwise velocity field in the high‐Reynolds‐number (Reτ ≈ Ο(106)) near‐neutral atmospheric surface layer (ASL), a predictive model was proposed in our work. In our model, the streamwise velocity time series at each height are predicted by the superposition of mean velocity, small‐scale turbulent signals, and large‐scale turbulent signals. All the parameters used by our model have clear physical meaning. In addition, the parameters can be directly determined by the experiment that is easy to be achieved in the ASL. The model can predict the streamwise velocity between 0.9 and 30 m in the near‐neutral ASL based only on single‐point‐measured large‐scale streamwise velocity signal. For example, mean velocity, turbulence intensity, and power spectrum of streamwise velocity can be predicted by the new model. Furthermore, the streamwise velocity time series predicted by the new model have a good correlation with the directly measured results, especially for large‐scale structures. It indicates that our model can be used to reconstruct the streamwise velocity field in the logarithmic region of the near‐neutral ASL, and our model is an important complement to the Marusic‐Mathis‐Hutchins model.

Turbulence characteristics in boundary layers over a regular array of cubical roughness

ABSTRACT The paper reports the experimental investigation of free surface turbulent flow under control laboratory flume conditions over a bed artificially roughened by cubes arranged periodically and symmetrically with respect to the channel in a squared configuration. The relative submergence associated with the cube height was 8 which fall into the category of large roughness. Two roughness separations P/k = 5 and 9 were examined at intermediate flow submergence (6 < h/k< 10). Here, k is the cube height and P is the pitch length between consecutive roughness elements. The velocity data were collected at a frequency of 40 Hz using three-dimensional (3D) micro-acoustic Doppler velocimeter (ADV). Mean velocity profiles illustrate the well-known downward shift of the semi-logarithmic portion of the law of the wall. Large variations in the turbulence production and skewness coefficient are found below the roughness top. In this region, the skewness coefficient changes their signs slightly, indicating a changeover of the dominance of the bursting events. Further, to describe the turbulent signal, the probability density function is evaluated which shows the bimodal nature for bottom-normal velocity fluctuation and sharp distribution for stream-wise velocity fluctuations.

Generalized Langevin equation and the linear regression model with memory.

Because the existence of cheap emulators can give rise to potentially drastic computational savings as compared to direct numerical simulations of complex turbulence models, the study of reduced models has good practical relevance, and the guidelines regarding their construction from the true dynamical system model are much in demand. In accordance with this contemporary trend in science and engineering, we provide a new approach for the rigorous derivation of the linear reduced model with memory from the one-dimensional Majda-McLaughlin-Tabak (MMT) prototypical wave turbulence in thermal equilibrium. The basic idea in obtaining the physics-constrained autoregressive model is to perform the discretization in time of the generalized Langevin equation (GLE) governing a single wave profile of the true dynamical system model; the GLE formalized in the Mori-Zwanzig (MZ) projection theory is the exact reduced-order equation and the closed-form rearrangement of the canonical equation of motion for the Hamiltonian system. We study the performance of the resulting linear non-Markovian model in addressing two different manners of uncertainty quantification (UQ) problems: prediction and filtering. In doing so, we perform a comparison analysis against the linear and nonlinear Markovian models describing the MMT system, which are also built upon the GLE under the MZ framework. Finally we discuss an optimal selection of the statistical model in applying the reduced-model approach to the UQ of the turbulent signal.

The Design of Ocean Turbulence Measurement with a Free Fall Vertical Profiler

The newly designed instrument Free Fall Vertical Profiler (FFVP) developed by Ocean University of China (OUC) had been deployed in the Western Pacific in March 08, 2017 and succeed to collect turbulence signals about 350-m-deep water. According to the requirements of turbulence measurement, the mechanical design was developed for turbulence platform to achieve stability and good flow tracking. By analysing the Heading, Pitch and Roll, the results suggested that the platform satisfies the requirements of stability. The power spectrum of the cleaned shear signals using the noise correction algorithm match well with the theoretical Nasmyth spectrum and the rate of turbulence dissipation are approximately 10-8 W/kg. In general, the FFVP was rationally designed and provided a good measurement platform for turbulence observation.

Sparse decomposition of atmospheric turbulence wavefront gradient

Using compressive sensing technology in atmospheric turbulent wavefront detected data compression can greatly reduce the amount of measured data, can effectively reduce the pressure of data transmission and storage, which is good for real-time measurement of turbulent wavefront. However, the wavefront signal is required to be sparse or can be sparsely represented in one transform domain. In this paper, a preliminary study of the sparsity of the atmospheric turbulent wavefront gradient signal is carried out. Based on the statistical characteristics of atmospheric turbulence, the golden section (GS) is used to make the turbulent power spectrum in the frequency domain, and the sparse basis is established to meet the physical characteristics of the turbulent gradient, then the sparsity of the gradient of the turbulent wavefront is clarified. The sparse decomposition of the wavefront gradient is simulated by using the GS sparse base, and the sparse decomposition effect of different sparsity bases on the wavefront gradient is compared. On this basis, using the GS basis as the initialization training dictionary, K singular value decomposition (KSVD) dictionary training is carried out to get the training base (KSVD-GS), and then the sparse representation performance of this training base to the wavefront gradient signal is analyzed. This paper verifies that the wavefront gradient can be sparsely decomposed and build a better sparse basis, and provides the precondition for the application of compressive sensing.

Turbulent characteristics of pulsating flow over hydraulically smooth surface

The mean flow and turbulence characteristics due to the interaction between wave and current is investigated over the hydraulically smooth surface in a wave channel based on measurement of the horizontal, bottom-normal and transverse velocity components in the pulsating fluid using a 3-D micro-acoustic Doppler velocimeter. Pulsations, in the vicinity of the wall are produced by superimposing the surface wave on unidirectional current using plunger type wave-maker. The velocity time series of combined wave–current flow were analysed within the framework of the phase-averaging. Further, to highlight the changes in combined wave–current flow, a comparative study was made between the velocity profiles for current-only flow and those measured for combined wave–current flow. The measured mean stream-wise velocity for waves following a current is reduced towards the free surface when compared with the current-only data. In addition, with an increase in wave height, a further intense reduction of the velocity just below the wave trough occurs. To fully characterize turbulent signals, the joint probability density function of stream-wise and bottom-normal velocity fluctuations is analysed. Spectral analysis was also performed to obtain the oscillation pattern within the flow field that may affect the turbulent properties due to combined wave–current flow.

Interferometric synthetic aperture radar atmospheric correction using a GPS-based iterative tropospheric decomposition model

Atmospheric effects represent one of the major error sources of repeat-pass Interferometric Synthetic Aperture Radar (InSAR), and could mask actual displacements due to tectonic or volcanic deformation. The tropospheric delays vary both vertically and laterally and can be considered as the sum of (i) a vertically stratified component highly correlated with topography and (ii) a turbulent component resulting from turbulent processes in the troposphere varying both in space and time. In this paper, we outline a framework to routinely use pointwise GPS data to reduce tropospheric effects on satellite radar measurements. An Iterative Tropospheric Decomposition (ITD) model is used and further developed to separate tropospheric stratified and turbulent signals and then generate high-resolution correction maps for SAR interferograms. Cross validation is employed to assess the performance of the ITD model and act as an indicator to users of when and where correction is feasible. Tests were carried out to assess the impact of GPS station spacing on the ITD model InSAR correction performance, which provides insights into the trade-off between station spacing and the achievable accuracy. The application of this framework to Sentinel-1A interferograms over the Southern California (USA) and Southern England (UK) regions shows approximately 45–78% of noise reduction even with a sparse (~50–80km station spacing) GPS network and/or with strong and non-random tropospheric turbulence. This is about a 50% greater improvement than previous methods. It is believed that this framework could lead to a generic InSAR atmospheric correction model while incorporating continuous and global tropospheric delay datasets, e.g. numerical weather models.

Accurate signal reconstruction for higher order Lagrangian–Eulerian back-coupling in multiphase turbulence

Multiphase flow simulation serves a vital purpose in applications as diverse as engineering design, natural disaster prediction, and even study of astrophysical phenomena. In these scenarios, it can be very difficult, expensive, or even impossible to fully represent the physical system under consideration. Even still, many such real-world applications can be modeled as a two-phase flow containing both continuous and dispersed phases. Consequentially, the continuous phase is thought of as a fluid and the dispersed phase as particles. The continuous phase is typically treated in the Eulerian frame of reference and represented on a fixed grid, while the dispersed phase is treated in the Lagrangian frame and represented by a sample distribution of Lagrangian particles that approximate a cloud. Coupling between the phases requires interpolation of the continuous phase properties at the locations of the Lagrangian particles. This interpolation step is straightforward and can be performed at higher order accuracy. The reverse process of projecting the Lagrangian particle properties from the sample points to the Eulerian grid is complicated by the time-dependent non-uniform distribution of the Lagrangian particles. In this paper we numerically examine three reconstruction, or projection, methods: (i) direct summation (DS), (ii) least-squares, and (iii) sparse approximation. We choose a continuous representation of the dispersed phase property that is systematically varied from a simple single mode periodic signal to a more complex artificially constructed turbulent signal to see how each method performs in reconstruction. In these experiments, we show that there is a link between the number of dispersed Lagrangian sample points and the number of structured grid points to accurately represent the underlying functional representation to machine accuracy. The least-squares method outperforms the other methods in most cases, while the sparse approximation method is able to capture physically important flow features when under-sampled but at an increased cost. Interestingly, the DS method has been used in the past but in comparison to the other two methods it offers only first order convergence. The above three methods were also compared against standard linear and cubic interpolation techniques from non-uniform Lagrangian points to the Eulerian grid. The performance of these standard interpolation methods were only of the order of DS.

Experimental study on dependence of streak breakdown on disturbance nature

Instability and breakdown of a low-speed streak initially forced by disturbances with a frequency-spectrum similar to that of wall turbulence were examined experimentally. A single low-speed streak was generated using a small piece of screen set normal to a boundary-layer plate, and disturbances made by two loudspeakers were introduced through two small holes. Turbulent velocity fluctuations with various spanwise coherences were used as driving signals to two loudspeakers to see how strongly the streak breakdown depended on the nature of disturbances. Signals were deduced at two spanwise locations with distance Δz+= 24–149 in the buffer region of a turbulent boundary layer by using a pair of hot-wires. The correlation coefficient Ct of the two turbulent signals ranged from −0.11 to 0.56. Both sinuous and varicose modes were excited but the former became dominant downstream in all cases. Development of the disturbance excited was compared to those for anti-symmetric (Ct=−1) and symmetric (Ct=1) excitations with the same turbulent spectrum. The comparison showed that the magnitude of the sinuous instability mode excited was dependent on the initial disturbances, but not strongly for −1≤Ct≤0. That is, in terms of the rms value of velocity fluctuations, the magnitude of the sinuous instability mode for the initial disturbance with Ct∼0 was about 75% of that in the case of anti-symmetric forcing. Even for Ct∼+0.56, the magnitude of the sinuous mode excited was as large as 50% of that in the anti-symmetric case.

Structure and function of the mechanosensory lateral line system in fish and biomimetic

As an adaptation to their environment, fishes and some aquatic amphibians have developed a hydrodynamic receptor system, the lateral line mechanosensory system, which enables them to detect minute water motion, and provide information that can apply for feeding, avoidance, schooling, navigation, and intraspecific communication. This system consists of individual sensory receptors, neuromasts, that either lie within lateral line canals along the body (canal neuromasts, CNM) or occur freestanding on the surface of the skin (superficial neuromasts, SNM). These CNM and SNM act as acceleration (higher frequency response, 60–120 Hz) or velocity receptors (lower frequency response, 20–60 Hz), respectively, which can also provide the signal location and fluidic direction to the central nervous system (CNS). The neuromast consists of discrete clusters of sensory hair cells, support cells and mantle cells. On the apical of hair cell, a bundle of several stereocilia and one kinocilium that protrudes from of the hair cell into a gelatinous cupula. The cupula covers entire neuromast apical and connects the ciliary bundles with surrounding water. Deflection of the cupula induces the directional movement of the hair bundles that results in opening or closing of the transduction channels on the kinocilium, affecting the release of neurotransmitter that conveys the hair cell’s relative excitation to the afferent neuron (in the lateral line ganglion), then, travels to the target areas in the CNS. Between CNM and SNM, there are several morphological differences such as diameter of sensory epithelium, number of hair cells, orientation of axis, and peripheral innervation pattern. They also demonstrate with different biomechanical properties in detecting different types of hydrodynamic signals and play different roles in responding to the various water motion patterns (including micro fluid). All the peripheral information that is integrated at different levels of the CNS centers before guiding the proper behavior of fish. The CNM appears to provide information about fine spatial details that lead to the ability of fish to segregate the fine turbulence signal in near field while towards discrete sources or under rapidly changing water motions surrounding. SNM appears to provide peripheral spatial information of large-scale stimuli, such as rheotaxis to slow motion currents. As a delicate hydrodynamic detector system, a well functioned lateral line system that only consists of two types of peripheral receptor plus their architectural design of arrangement and canal pattern modifications that is far advanced than, up to date, any of the man-made underwater detector in the world. Even more, the system provides a high efficient computing model that is from the peripheral innervation pattern designed for major collection of temporal and spatial hydrodynamic information, the integration of data processing structure, and to the action assignment in the CNS. The mechanosensory lateral line system in fish is starting attracted by research approaches in biomimetic, mathematics, physical and computing sciences, in addition to the biology and neurosciences. The participating of integrative study on the lateral line system, that is not only help developing of new applications but also enhanced biological understand of the system. The biologists have been speculated the function of the CNM, the co-relationship of the canal morphology and fish habits over decades. Until recently, the mathematic model and the device of artificial lateral line receptor established by Klein and Bleckmann (2015), demonstrate that “it is possible to change the transfer function of an artificial line by changing its canal shape, dimensions and pore pattern”. Moreover, their research evident indicate that the lateral line is able to “detect and localize vibrating sources, detect and monitor vortices, localize upstream objects, discriminate between upsteam objects (size and shape), estimate bulk flow velocity in turbulent environment, and measure small amounts of liquid” (Klein, 2016 personal communication). This review paper intends to introduce the fish mechanosensory lateral line system from several aspects that include the studies of morphological characteristics, biological origin of the system, development of the pattern formation, functional properties, behavioral neurophysiology and neural control mechanisms, finally, the biomimetic application and perspectives. We combine both of our knowledge in lateral line and recent research results in the non-canal lateral line system with broad collections of the references in the lateral line studies, especially, the new publications. We hope to provide information to readers across different disciplines, not only for theoretical consideration but also for the application in biomimetic, civic engineering, and the aquatic animal protection /biodiversity program.

Characterisation and improvement of the structure function estimation for application in PSI

Numerical Weather Prediction (NWP) is becoming state of the art for the compensation of atmospheric effects in InSAR and especially in PSI. The structure function (variogram) estimated on the InSAR data and on the NWP data is a required statistical characteristic and very useful in the processing of the data. For example, the structure function (typically the semivariogram) estimated on InSAR data is fundamental for the Kriging of the atmospheric phase screen (APS) based on irregular PSI estimates. The required parameters are the nugget, the range and the slope of the structure function. The practical implementation has shown that the NWP predicted structure function and the InSAR estimated structure function based on the conventional semivariogram equation do not match. Straightforward explanations are wrongly estimated APSs due to an insufficient number of interferograms or hindcasts that fail to capture the turbulent water vapour signal. However, in this paper we explain the effects of noise in interferograms and coarse resolution in the NWP on the conventional structure function estimation resulting in the observed mismatch. In order to avoid the mismatch, an alternative implementation based on wavelets is suggested and demonstrated using real Sentinel-1 data. We show that the wavelet based structure function estimation outperforms the conventional structure function estimation based on a variogram. Application of the proposed structure function alternative based on NWP data are the master selection, the estimation of the effective NWP data resolution, and a statistical consistency check of the estimated InSAR APS. To support the understanding, we demonstrate the effects of noise and resolution using simulated fractals. An application of the proposed wavelet based structure function estimation is the estimation of the effective NWP data resolution, which is demonstrated using the same Sentinel-1 data test case.

Phase relations in a forced turbulent boundary layer: implications for modelling of high Reynolds number wall turbulence.

Phase relations between specific scales in a turbulent boundary layer are studied here by highlighting the associated nonlinear scale interactions in the flow. This is achieved through an experimental technique that allows for targeted forcing of the flow through the use of a dynamic wall perturbation. Two distinct large-scale modes with well-defined spatial and temporal wavenumbers were simultaneously forced in the boundary layer, and the resulting nonlinear response from their direct interactions was isolated from the turbulence signal for the study. This approach advances the traditional studies of large- and small-scale interactions in wall turbulence by focusing on the direct interactions between scales with triadic wavenumber consistency. The results are discussed in the context of modelling high Reynolds number wall turbulence.This article is part of the themed issue 'Toward the development of high-fidelity models of wall turbulence at large Reynolds number'.