Synthetic Velocity Fields Research Articles

Multiscale and anisotropic characterization of images based on complexity: An application to turbulence

This article presents a multiscale, non-linear and directional statistical characterization of images based on the estimation of the skewness, flatness, entropy and distance from Gaussianity of the spatial increments. These increments are characterized by their magnitude and direction; they allow us to characterize the multiscale properties directionally and to explore anisotropy. To describe the evolution of the probability density function of the increments with their magnitude and direction, we use the skewness to probe the symmetry, the entropy to measure the complexity, and both the flatness and distance from Gaussianity to describe the shape. These four quantities allow us to explore the anisotropy of the linear correlations and non-linear dependencies of the field across scales. First, we validate the methodology on two-dimensional synthetic scale-invariant fields with different multiscale properties and anisotropic characteristics. Then, we apply it on two synthetic turbulent velocity fields: a perfectly isotropic and homogeneous one, and a channel flow where boundaries induce inhomogeneity and anisotropy. Our characterization unambiguously detects the anisotropy in the second case, where our quantities report scaling properties that depend on the direction of analysis. Furthermore, we show in both cases that turbulent velocity fluctuations are isotropic, when the mean velocity profile is adequately removed.

Lagrangian Particle Dispersion in a Poor Man’s Magnetohydrodynamic Turbulence Model

Lagrangian dispersion of fluid particle pairs refers to the study of how individual fluid particles disperse and move in a fluid flow, providing insights to understand transport phenomena in various environments, from laminar to turbulent conditions. Here, we explore this phenomenon in synthetic velocity and magnetic fields generated through a reduced-order model of the magnetohydrodynamic equations, which is able to mimic both a laminar and a turbulent environment. In the case of laminar conditions, we find that the average square distance between particle pairs increases linearly with time, implying a dispersion pattern similar to Brownian motion at all time steps. On the other hand, under turbulent conditions, surprisingly enough we observe a Richardson scaling, indicating a super-ballistic dispersion pattern, which aligns with the expected scaling properties for a turbulent environment. Additionally, our study reveals that the magnetic field plays an organizing role. Lastly, we explore a purely hydrodynamic case without magnetic field effects, showing that, even in a turbulent environment, the behavior remains Brownian-like, highlighting the crucial role of the magnetic field in generating the Richardson scaling observed in our model.

A new divergence-free synthetic eddy method for generating homogeneous isotropic turbulence with a prescribed energy spectrum

A new divergence-free synthetic eddy method (SEM) for the generation of homogeneous isotropic turbulence (HIT) with a prescribed energy spectrum is presented. In the proposed method, the divergence-free consistent synthetic velocity field is generated from the curl of a vector potential field following conventional SEM methodology. A carefully designed formula is constructed for the computation of the local vector potential field induced by each synthetic eddy, with which a desired velocity spectrum tensor can be satisfied for the synthetic incompressible turbulence exactly. The proposed method is characterized by a multi-scale feature in which the length scales of synthetic eddies are determined by a dimensionless random variable satisfying a particular probability distribution function (PDF). The mathematical relation, concerning the energy spectrum of the synthetic turbulence, the shape function of the eddies and the PDF of the eddy-scale-related random variable, is derived. This assists in determining either the shape function or the PDF. The performance of the proposed method is examined by synthesizing HIT satisfying a theoretical energy spectrum model and HIT identified in a wind-tunnel experiment.

Spatial outlier detection on discrete GNSS velocity fields using robust Mahalanobis-distance-based unsupervised classification

GNSS velocity field filtering can be identified as a multidimensional unsupervised spatial outlier detection problem. To detect and classify the spatial outliers, we jointly interpret the horizontal and vertical velocity fields with the related standard deviations. We also describe the applied feature engineering process, which represents the underlying problem better than the initial attributes. According to this, we discuss the utilized algorithms and techniques, like the spatial- and non-spatial mapping functions, the k-nearest neighborhood (kNN) technique to retrieve the local environment of each GNSS station, as well as the principal component analysis (PCA) as a dimensionality reduction technique. We also assume that regular velocity field samples containing no outliers come from an approximate multivariate normal distribution (MVN) at the local scale. Regarding this, we evaluate the corresponding sample-wise distance related to model distribution, namely the Mahalanobis distance, with the estimation of the robust covariance matrix derived by the minimum covariant determinant (MCD) algorithm. Subsequently, we introduce the applied binary classification on the values of the derived robust Mahalanobis distances (RMD) which follows the χ2distribution. We also present three cases of artificially generated, pre-labeled synthetic velocity field datasets to perform cross-validation and comparison of the proposed RMD approach to other classification techniques. According to this, we found that k = 12 yields > 95% classification accuracy. While the compared methods have a mean classification accuracy of 96.2–99.8%, the advantage of the RMD approach is that it does not require pre-defined labels to indicate regular and outlier samples. We also demonstrate the proposed RMD based filtering process on a real dataset of the EUREF Permanent Network Densification velocity products. The RMD-based approach has been integrated into the EPN Densification as a quality checking algorithm. According to this, we also introduce a co-developed and regularly updated interactive webpage to disseminate the corresponding results.

Uncertainty of PIV/PTV based Eulerian pressure estimation using velocity uncertainty

This work introduces a method to estimate the uncertainty of the pressure fields reconstructed from particle image velocimetry / particle tracking velocimetry (PIV/PTV) measurements by propagating the instantaneous velocity vector uncertainty through the pressure reconstruction. The uncertainty propagations through the calculation and integration of pressure gradients are modelled as linear transformations. The autocorrelation coefficient was modelled and incorporated in the uncertainty estimation to reproduce the effect of the autocorrelation of velocity errors on the reconstructed pressure’s accuracy. The method was first tested on synthetic velocity fields contaminated with varying levels of artificial noise correlated in space, time, or between components. The error analysis shows that the proposed method could predict the spatiotemporal variations of the pressure errors. The estimated pressure uncertainty also captures the effects of the velocity noise level, the autocorrelation, and the different pressure-gradient integration methods, with more than 80% accuracy in most test cases. The method was applied to an experimental vortex ring flow with planar PIV and a laminar pipe flow with volumetric PTV. The error analysis shows that the obtained pressure uncertainty possessed similar spatial and statistical distributions as the pressure errors. The results also indicate that the performance of the proposed uncertainty estimation method depends on the accuracy of the velocity uncertainty. The proposed uncertainty estimation method exhibits reliability in obtaining the local and instantaneous pressure uncertainty from the PIV/PTV measurements.

Eulerian spatiotemporal correlations in passive scalar turbulence

Using direct numerical simulations we study the spatiotemporal correlations of passive scalars advected both by Navier-Stokes flow and by synthetic velocity fields. Beyond their global behavior, which is either a Gaussian or an exponential decay in time, depending on the carrier flow, we determine with accuracy the prefactors in the exponents, and find a remarkable agreement with recent results from functional renormalization group.

Learning the structure of wind: A data-driven nonlocal turbulence model for the atmospheric boundary layer

We develop a novel data-driven approach to modeling the atmospheric boundary layer. This approach leads to a nonlocal, anisotropic synthetic turbulence model which we refer to as the deep rapid distortion (DRD) model. Our approach relies on an operator regression problem that characterizes the best fitting candidate in a general family of nonlocal covariance kernels parameterized in part by a neural network. This family of covariance kernels is expressed in Fourier space and is obtained from approximate solutions to the Navier–Stokes equations at very high Reynolds numbers. Each member of the family incorporates important physical properties such as mass conservation and a realistic energy cascade. The DRD model can be calibrated with noisy data from field experiments. After calibration, the model can be used to generate synthetic turbulent velocity fields. To this end, we provide a new numerical method based on domain decomposition which delivers scalable, memory-efficient turbulence generation with the DRD model as well as others. We demonstrate the robustness of our approach with both filtered and noisy data coming from the 1968 Air Force Cambridge Research Laboratory Kansas experiments. Using these data, we witness exceptional accuracy with the DRD model, especially when compared to the International Electrotechnical Commission standard.

Triple Hill’s Vortex Synthetic Eddy Method

The generation of initial or inflow synthetic turbulent velocity or scalar fields reproducing statistical characteristics of realistic turbulence is still a challenge. The synthetic eddy method, previously introduced in the context of inflow conditions for large eddy simulations, is based on the assumption that turbulence can be regarded as a superposition of coherent structures. In this paper, a new type of synthetic eddy method is proposed, where the fundamental eddy is constructed by superposing three Hill's vortices, with their axes orthogonal to each other. A distribution of Hill's vortices is used to synthesize an anisotropic turbulent velocity field that satisfies the incompressibility condition and match a given Reynolds stress tensor. The amplitudes of the three vortices that form the fundamental eddy are calculated from known Reynolds stress profiles through a transformation from the physical reference frame to the principal-axis reference frame. In this way, divergence-free anisotropic turbulent velocity fields are obtained that can reproduce a given Reynolds stress tensor. The model was tested on both isotropic and anisotropic turbulent velocity fields, in the framework of grid turbulence decay and turbulent channel flow, respectively. The transition from artificial to realistic turbulence in the proximity to the inflow boundary was found to be small in all test cases that were considered.

LiSBOA (LiDAR Statistical Barnes Objective Analysis) for optimal design of lidar scans and retrieval of wind statistics – Part 1: Theoretical framework

Abstract. A LiDAR Statistical Barnes Objective Analysis (LiSBOA) for the optimal design of lidar scans and retrieval of the velocity statistical moments is proposed. LiSBOA represents an adaptation of the classical Barnes scheme for the statistical analysis of unstructured experimental data in N-dimensional space, and it is a suitable technique for the evaluation over a structured Cartesian grid of the statistics of scalar fields sampled through scanning lidars. LiSBOA is validated and characterized via a Monte Carlo approach applied to a synthetic velocity field. This revisited theoretical framework for the Barnes objective analysis enables the formulation of guidelines for the optimal design of lidar experiments and efficient application of LiSBOA for the postprocessing of lidar measurements. The optimal design of lidar scans is formulated as a two-cost-function optimization problem, including the minimization of the percentage of the measurement volume not sampled with adequate spatial resolution and the minimization of the error on the mean of the velocity field. The optimal design of the lidar scans also guides the selection of the smoothing parameter and the total number of iterations to use for the Barnes scheme. LiSBOA is assessed against a numerical data set generated using the virtual lidar technique applied to the data obtained from a large eddy simulation (LES). The optimal sampling parameters for a scanning Doppler pulsed wind lidar are retrieved through LiSBOA, and then the estimated statistics are compared with those of the original LES data set, showing a maximum error of about 4 % for both mean velocity and turbulence intensity.

A 4D continuous representation of myocardial velocity fields from tissue phase mapping magnetic resonance imaging.

Myocardial velocities carry important diagnostic information in a range of cardiac diseases, and play an important role in diagnosing and grading left ventricular diastolic dysfunction. Tissue Phase Mapping (TPM) Magnetic Resonance Imaging (MRI) enables discrete sampling of the myocardium’s underlying smooth and continuous velocity field. This paper presents a post-processing framework for constructing a spatially and temporally smooth and continuous representation of the myocardium’s velocity field from TPM data. In the proposed scheme, the velocity field is represented through either linear or cubic B-spline basis functions. The framework facilitates both interpolation and noise reducing approximation. As a proof-of-concept, the framework was evaluated using artificially noisy (i.e., synthetic) velocity fields created by adding different levels of noise to an original TPM data. The framework’s ability to restore the original velocity field was investigated using Bland-Altman statistics. Moreover, we calculated myocardial material point trajectories through temporal integration of the original and synthetic fields. The effect of noise reduction on the calculated trajectories was investigated by assessing the distance between the start and end position of material points after one complete cardiac cycle (end point error). We found that the Bland-Altman limits of agreement between the original and the synthetic velocity fields were reduced after application of the framework. Furthermore, the integrated trajectories exhibited consistently lower end point error. These results suggest that the proposed method generates a realistic continuous representation of myocardial velocity fields from noisy and discrete TPM data. Linear B-splines resulted in narrower limits of agreement between the original and synthetic fields, compared to Cubic B-splines. The end point errors were also consistently lower for Linear B-splines than for cubic. Linear B-splines therefore appear to be more suitable for TPM data.

Assessment of pressure field and performance parameter reconstruction from velocity data for ramjet inlet

The current flow measurement devices for ramjet inlet still have many limitations in the supersonic internal flow, such as the qualitative feature of schlieren and the low resolution of pressure tap/probe. In this paper, a non-intrusive measurement scheme for the internal pressure field and performance parameters of ramjet inlet based on particle image velocimetry (PIV) data is established and quantitatively evaluated by using a typical model. This scheme is combined with two important PIV-based pressure reconstruction approaches, namely, the spatial integration in conservative form and the MacCormack method, which are especially suitable for the supersonic flows with shock waves. The feasibility of this scheme is firstly confirmed by using a numerical velocity field, and the turbulence contribution is concluded to be negligible. Then, a simulated PIV experiment is conducted to achieve a synthetic velocity field with various theoretical measurement errors for the comprehensive analyses of the accuracy and robustness of this measurement method. Compared with the traditional equipment, the PIV-based method can obtain the high-resolution velocity and pressure distributions of whole flow field without interference, and the performance parameters are also available along the flow path. The good reconstruction results show the great advantage and attraction of this non-intrusive method in the future wind tunnel test for ramjet inlet.

Vortex detection criteria assessment for PIV data in rotorcraft applications

The aim of the article is to propose a simple engineering method for identifying and characterizing vortical structures within a flow field measured with a classic two-component PIV measurement system. Some of the most popular vortex detection systems are briefly presented. Of these, many fail if spurious vectors are present within the flow field due to poor PIV image quality or due to particle voids in the vortex core. The chosen method is the Γ2-criterion. The investigated method is robust and reliable, because it is based on the velocity field topology without using velocity derivatives sensitive to the spurious vectors. The method is tested on synthetic velocity fields of ideal vortices and on real PIV velocity field of a four-bladed rotor wake. The synthetic velocity fields reproduce single and multiple vortices to investigate the mutual interaction and the effect on the vortex detection criteria. The synthetic velocity fields have different spatial resolution, noise level, and different particle void to perform a parametric assessment. Other vortex identification schemes are applied for comparison. Information on how to use the criterion for different noise conditions, particle void, and presence of multiple vortices is provided.

Using uncertainty to improve pressure field reconstruction from PIV/PTV flow measurements

A novel pressure reconstruction method is proposed to use the uncertainty information to improve the instantaneous pressure fields from velocity fields measured using particle image velocimetry (PIV) or particle tracking velocimetry (PTV). First, the pressure gradient fields are calculated from velocity fields, while the local and instantaneous pressure gradient uncertainty is estimated from the velocity uncertainty using a linear-transformation-based algorithm. The pressure field is then reconstructed by solving an overdetermined linear system which involves the pressure gradients and boundary conditions. This linear system is solved with generalized least squares (GLS) which incorporates the previously estimated variances and covariances of the pressure gradient errors as inverse weights to optimize the reconstructed pressure field. The method was validated with synthetic velocity fields of a 2D pulsatile flow, and the results show significantly improved pressure accuracy. The pressure error reduction by GLS was 50% with 9.6% velocity errors and 250% with 32.1% velocity errors compared to the existing baseline method of solving the pressure Poisson equation (PPE). The GLS was more robust to the velocity errors and provides greater improvement with spatially correlated velocity errors. For experimental validation, the volumetric pressure fields were evaluated from the velocity fields measured using 3D PTV of a laminar pipe flow with a Reynolds number of 630 and a transitional pipe flow with a Reynolds number of 3447. The GLS reduced the median absolute pressure errors by as much as 96% for the laminar pipe flow compared to PPE. The mean pressure drop along the pipe predicted by GLS was in good agreement with the empirical estimation using Darcy–Weisbach equation for the transitional pipe flow.

Tracer Turbulence: The Batchelor–Howells–Townsend Spectrum Revisited

Given a velocity field u(x, t), we consider the evolution of a passive tracer $$\theta $$ governed by $$\partial _t\theta + u\cdot \nabla \theta = \Delta \theta + g$$ with time-independent source g(x). When $$\Vert u\Vert $$ is small in some sense, Batchelor, Howells and Townsend (J Fluid Mech 5:134, 1959) predicted that the tracer spectrum scales as $$|\theta _k|^2\propto |k|^{-4}|u_k|^2$$. In this paper we prove that, for random synthetic two-dimensional incompressible velocity fields u(x, t) with given energy spectra, this scaling does indeed hold probabilistically, asymptotically almost surely for large |k| and small $$\Vert u\Vert $$. We also propose an asymptotic correction factor to the BHT scaling arising from the time-dependence of u.

Unsteady inflow conditions: A variationally based solution to the insurgence of pressure fluctuations

The application of unsteady inflow conditions represents an important aspect when scale resolving turbulence models are adopted in Computational Fluid Dynamics (CFD) analyses. In such context, with reference to Wind Engineering applications, the main concern is often represented by the generation of a synthetic velocity field representative of the turbulent fluctuations impinging on the studied body. Once such synthetic field has been generated, it is applied at the inflow patch. Unfortunately, such operation is not trivial: undesired pressure fluctuations are often generated due to the incompatibility between the applied inflow condition and the flow field found inside the computational domain. In this paper, a procedure able to guarantee a correct application of synthetically generated velocity fields to CFD simulations is described. The procedure, which relies on a variational background, is simple and can be readily implemented in existing codes. Numerical results confirm that, by adopting the proposed corrections, pressure fluctuations are strongly reduced, so confirming the soundness of the proposed approach. An OpenFoam implementation of the method is available for download at https://site.unibo.it/cwe-lamc/en.

4D Flow MRI Pressure Estimation Using Velocity Measurement-Error-Based Weighted Least-Squares.

This work introduces a 4D flow magnetic resonance imaging (MRI) pressure reconstruction method which employs weighted least-squares (WLS) for pressure integration. Pressure gradients are calculated from the velocity fields, and velocity errors are estimated from the velocity divergence for incompressible flow. Pressure gradient errors are estimated by propagating the velocity errors through Navier-Stokes momentum equation. A weight matrix is generated based on the pressure gradient errors, then employed for pressure reconstruction. The pressure reconstruction method was demonstrated and analyzed using synthetic velocity fields as well as Poiseuille flow measured using in vitro 4D flow MRI. Performance of the proposed WLS method was compared to the method of solving the pressure Poisson equation which has been the primary method used in the previous studies. Error analysis indicated that the proposed method is more robust to velocity measurement errors. Improvement on pressure results was found to be more significant for the cases with spatially-varying velocity error level, with reductions in error ranging from 50% to over 200%. Finally, the method was applied to flow in patient-specific cerebral aneurysms. Validation was performed with in vitro flow data collected using Particle Tracking Velocimetry (PTV) and in vivo flow measurement obtained using 4D flow MRI. Pressure calculated by WLS, as opposed to the Poisson equation, was more consistent with the flow structures and showed better agreement between the in vivo and in vitro data. These results suggest the utility of WLS method to obtain reliable pressure field from clinical flow measurement data.

Sensitivity of sonic boom propagation throughout a turbulent atmosphere

Sensitivity of sonic boom propagation throughout a turbulent atmosphere is investigated. Three types of boom of same amplitude but different initial shapes: an ideal N-wave, a measured boom (NASA data for F-18) and a “low” boom (C25D mock-up) with increased rise time are studied. The atmosphere is supposed to be a quiescent and isothermal medium with a superposed synthetic velocity field with homogeneous and isotropic statistical properties satisfying a von Kármán energy spectrum. Using the “random field generation method,” the flow velocity turbulent field is governed by three independent parameters: a random matrix, an intensity parameter and a scale parameter (turbulence integral scale). The flow velocity is then used as a base flow for a in-house software called FLHOWARD designed to compute the propagation of acoustic shock waves in heterogeneous media. In order to reduce the number of simulations compared to a Monte-Carlo approach, the study is performed within the generalized chaos polynomial (gPC) framework. Various convergence tests have been performed to define the optimal discretization and gPC order. Stochastic evolution of selected metrics along a 1 km distance are investigated.

Experimental comparison of PIV-based pressure measurements in supersonic flows with shock waves

This paper evaluates the performance of the novel MacCormack method for particle image velocimetry (PIV)-based pressure reconstruction in real supersonic PIV experiments that include oblique shock waves generated by three different wedges of 10°, 15°, and 20° with a free stream Mach number of 2.91. Besides, the precision of the widely used spatial integration methods and Poisson method for compressible flows is compared with that of the MacCormack method in the supersonic flows. The results show the good performance of the MacCormack method and its significant advantage in complex shock flows. The results are also consistent with the conclusions based on the synthetic velocity fields by (Liu et al 2017 Exp. Fluids 6 1–22), and further confirm the reliability of this method. The non-conservative spatial integration method usually achieves larger pressure distributions behind shock waves than theoretical values while the conservative spatial integration method performs much better but still fails to obtain accurate results in the strong shock flow. The Poisson method does not have obvious optimization compared to the spatial integration method. However, the pressure field reconstructed by the time-marching MacCormack method is constantly in good agreement with theory, which demonstrates the feasibility and advantage of this method in complex supersonic flows with strong shock waves.

Torsional Alfvén waves in a dipolar magnetic field: experiments and simulations

SUMMARY The discovery of torsional Alfvén waves (geostrophic Alfvén waves) in the Earth’s core (Gillet et al. 2010) calls for a better understanding of their properties. We present the first experimental observations of torsional Alfvén waves, performed in the DTS-Ω set-up. In this set-up, 50 L of liquid sodium are confined between an inner sphere (ri = 74 mm) and an outer shell (ro = 210 mm). The inner sphere houses a permanent magnet, imposing a dipolar magnetic field (Bmax = 345 mT). Both the inner sphere and the outer shell can rotate around the vertical axis. Alfvén waves are triggered by a sudden jerk of the inner sphere. We study the propagation of these waves when the fluid is initially at rest, and when it spins at a rotation rate up to 15 Hz. We measure the azimuthal magnetic field of the wave at different radii inside the fluid with magnetometers installed in a sleeve. We also record the electric potential signature on the outer shell at several latitudes. Besides, we probe the associated azimuthal velocity field using ultrasound Doppler velocimetry. With a 15 Hz rotation rate, the dynamical regimes we achieve are characterized by dimensionless numbers in the following ranges: Lundquist number 0.5 < Lu < 12, Lehnert number 0.01 < Le < 0.26, Rossby number Ro ∼ 0.1. We observe that the magnetic signal propagates away from the inner sphere, strongly damped by magnetic diffusion. Rotation affects the magnetic signature in a subtle way. Its effect is more pronounced on the surface electric potentials, which are sensitive to the actual fluid velocity of the wave. The ultrasound Doppler probes provide the first experimental measurement of the fluid velocity of an Alfvén wave. To complement these observations, we ran numerical simulations, using the XSHELLS pseudospectral code with parameters as close as possible to the experimental ones. The synthetic magnetic and electric signals match our measurements. The meridional snapshots of the synthetic azimuthal velocity field reveal the formation of geostrophic cylinders expected for torsional Alfvén waves. We establish scaling laws for the magnetic and kinetic energies of Alfvén waves with and without rotation. In both cases, we find that the magnetic energy EM saturates at a level proportional to $Rm_{\rm jerk}^2$, where Rmjerk = Ujerkro/η is the magnetic Reynolds number built with the maximum azimuthal velocity of the inner sphere during the jerk. The $E_K^{\rm max}/E_M^{\rm max}$ ratio (where $E_K^{\rm max}$ is the maximum kinetic energy), close to 1 for very quick jerks, increases linearly with the jerk duration.

Towards reliable turbulence estimations with phase-detection probes: an adaptive window cross-correlation technique

Air-water flow turbulence was derived from pseudo-instantaneous velocities measured with a dual-tip phase-detection probe. This new technique is proposed based upon adaptive time windows for cross-correlation analysis combined with robust filtering criteria, allowing computation of velocity time series in highly aerated flows. Each velocity estimation corresponded to a small group of bubbles or droplets. Stochastic synthetic velocity fields were generated to assess the limitations and uncertainties related to the proposed analysis. Subsequently, capabilities of the technique were demonstrated through an application to a real two-phase flow on a large-size stepped spillway.