Velocity Gradient Measurements Research Articles

Hierarchical hub-filament structures and gas inflows on galaxy-cloud scales

Abstract We investigated the kinematics and dynamics of gas structures on galaxy-cloud scales in two spiral galaxies NGC5236 (M83) and NGC4321 (M100) using CO (2–1) line. We utilized the FILFINDER algorithm on integrated intensity maps for the identification of filaments in two galaxies. Clear fluctuations in velocity and density were observed along these filaments, enabling the fitting of velocity gradients around intensity peaks. The variations in velocity gradient across different scales suggest a gradual and consistent increase in velocity gradient from large to small scales, indicative of gravitational collapse, something also revealed by the correlation between velocity dispersion and column density of gas structures. Gas structures at different scales in the galaxy may be organized into hierarchical systems through gravitational coupling. All the features of gas kinematics on galaxy-cloud scale are very similar to that on cloud-clump and clump-core scales studied in previous works. Thus, the interstellar medium from galaxy to dense core scales presents multi-scale/hierarchical hub-filament structures. Like dense core as the hub in clump, clump as the hub in molecular cloud, now we verify that cloud or cloud complex can be the hub in spiral galaxies. Although the scaling relations and the measured velocity gradients support the gravitational collapse of gas structures on galaxy-cloud scales, the collapse is much slower than a pure free-fall gravitational collapse.

Feedback from protoclusters does not significantly change the kinematic properties of the embedded dense gas structures

A total of 64 ATOMS sources at different evolutionary stages were selected to investigate the kinematics and dynamics of gas structures under feedback. We identified dense gas structures based on the integrated intensity map of H13CO+ J = 1−0 emission, and then extracted the average spectra of all the structures to investigate their velocity components and gas kinematics. For the scaling relations between the velocity dispersion, σ, the effective radius, R, and the column density, N, of all the structures, σ − N * R always has a stronger correlation compared to σ − N and σ − R. There are significant correlations between velocity dispersion and column density, which may imply that the velocity dispersion originates in gravitational collapse, also revealed by the velocity gradients. The measured velocity gradients for dense gas structures in early-stage sources and late-stage sources are comparable, indicating gravitational collapse through all evolutionary stages. Late-stage sources do not have large-scale hub-filament structures, but the embedded dense gas structures in late-stage sources show similar kinematic modes to those in early- and middle-stage sources. These results may be explained by the multi-scale hub-filament structures in the clouds. We quantitatively estimated the velocity dispersion generated by the outflows, inflows, ionized gas pressure, and radiation pressure, and found that the ionized gas feedback is stronger than other feedback mechanisms. However, although feedback from HII regions is the strongest, it does not significantly affect the physical properties of the embedded dense gas structures. Combined with the conclusions in our previous work on cloud-clump scales, we suggest that although feedback from cloud to core scales will break up the original cloud complex, the substructures of the original complex can be reorganized into new gravitationally governed configurations around new gravitational centers. This process is accompanied by structural destruction and generation, and changes in gravitational centers, but gravitational collapse is always ongoing.

Estimation of Mouse Carotid Arterial Wall Shear Stress Using High-Frequency Ultrasound Imaging.

Wall shear stress (WSS) is a crucial hemodynamic factor that promotes atherosclerosis (plaque) development in arteries; although the relationship between WSS and arterial atherosclerosis has been explored in many animal studies, it is not fully understood. No suitable tool, however, exists for rapidly estimating dynamic WSS in small-animal studies. This study proposes a 40-MHz high-frequency ultrasound (HFUS) imaging system for dynamic WSS estimation based on mouse carotid artery blood flow velocity gradient measurements by vector Doppler imaging (VDI). Aliasing reduces the accuracy of Doppler measurements, which can be prevented by increasing the imaging frame rate. Conventionally, imaging is performed at two tilted angles by alternating between the angles; in the proposed method, the frame rate was doubled by imaging at each tilted angle sequentially and by then temporally aligning the sequences based on pulsatile flow characteristics. Velocity estimation using this method had low errors for both a steady-flow straight-tube and pulsatile flow 60%-stenosis phantom. The method was tested for wild-type (WT) C57BL/6 mice at 16 weeks old and apolipoprotein E knockout (ApoE KO) mice at 16 and 24 weeks old; differences in time-averaged and oscillatory WSS were observed, and histology confirmed that the 24-week ApoE KO mice with the highest oscillatory WSS had the greatest plaque formation. The proposed HFUS WSS imaging method can predict the location and extent of plaque development; thus, this method is useful for small-animal studies investigating the WSS effect on atherosclerotic plaque development.

10-kHz-rate two-color dual-plane stereo-PIV using an optical parametric oscillator

10 kHz-rate dual-plane stereoscopic particle-image velocimetry (DSPIV) was applied in a swirling jet flame measurement simultaneously with hydroxyl planar laser-induced fluorescence (OH-PLIF). A high-speed burst-mode laser system with an optical parametric oscillator were used in double-pulse laser generation of 532 nm for one plane and 607 nm for the second plane. Temporally-resolved particle image velocimetry (PIV) image sequences, nine-component velocity gradient tensor, and simultaneous OH-PLIF images are presented. Flow vorticities and strain rates are also obtained with the measured velocity gradients. Compared with previous polarization-based dual-plane PIV technique, two-color based DSPIV measurement has distinctive advantages in elimination of signal interference from different PIV plane due to polarization leakage and hence improves measurement accuracy.

Derivation of vascular wall shear stress from 1000 fps high-speed angiography (HSA) velocity distributions.

Pathological changes in blood flow lead to altered hemodynamic forces, which are responsible for a number of conditions related to the remodeling and regeneration of the vasculature. More specifically, wall shear stress (WSS) has been shown to be a significant hemodynamic parameter with respect to aneurysm growth and rupture, as well as plaque activation leading to increased risk of stroke. In-vivo measurement of shear stress is difficult due to the stringent requirements on spatial resolution near the wall boundaries, as well as the deviation from the commonly assumed parabolic flow behavior at the wall. In this work, we propose an experimental method of in-vitro WSS calculations from high-temporal resolution velocity distributions, which are derived from 1000 fps high-speed angiography (HSA). The high-spatial and temporal resolution of our HSA detector makes such high-resolution velocity gradient measurements feasible. Presented here is the methodology for calculation of WSS in the imaging plane, as well as initial results for a variety of vascular geometries at physiologically realistic flow rates. Further, the effect of spatial resolution on the gradient calculation is explored using CFD-derived velocity data. Such angiographic-based analysis with HSA has the potential to provide critical hemodynamic feedback in an interventional setting, with the overarching objective of supporting clinical decision-making and improving patient outcomes.

Flow Velocity Gradient Sensing Using a Single Curved Bistable Microbeam

An approach for gas flow velocity measurement at two different closely located points using a simple device incorporating only one thermal sensing element, is introduced and the feasibility to determine different velocities is experimentally demonstrated. An electrostatically actuated initially curved bistable microbeam heated by an electric current and convectively cooled by airflow is switched between two stable locations through the snap-through (ST) and snap-back (SB) buckling mechanisms. The velocities of an air flow at these positions are obtained by measuring the critical ST and SB values of the actuation voltage. In the experiments, our $500~\mu \text{m}$ long, $2~\mu \text{m}$ wide single-crystal Si beam with $\approx \, 2.5\,\mu \text{m}$ nominal initial elevation demonstrated sensitivity of $S_{\text {ST}} \approx 0.25$ V/(m/s) and $S_{\text {SB}} \approx 0.84$ V/(m/s) at the ST and SB points, respectively. In the present device, the distance between the two measurement points is $\approx 10~\mu \text{m}$ . Our experimental results indicate that the suggested approach can be used for the velocity gradient measurements. The sensing principle relying on a single bistable sensing element opens new opportunities for measurement of gas flow velocity and velocity gradients at scales significantly smaller than the state-of-the-art multi hot-wire sensors.

Prevalence and Predictors of Elevated Pulmonary Artery Pressure in Nigerian Children with Sickle Cell Anaemia

Background: Sickle Cell Anaemia (SCA) is the most common inherited disorder in Nigeria. Pulmonary Hypertension (PH) is a known complication of SCA that commences from childhood and progresses as they grow older.
 Aim: To determine the prevalence and predictors of elevated pulmonary artery pressure (PAP) in children with SCA.
 Study Design: This is an analytical cross-sectional study.
 Methods: Three hundred subjects with SCA in steady state and 300 HbAA controls, aged 6 months – 15 years, were enrolled into the study. Children with SCA were recruited by simple random sampling, from the Paediatric Sickle Cell Clinic, Usmanu Danfodiyo University Teaching Hospital, Sokoto. All the children had clinical evaluation, echocardiography, full blood count and reticulocyte count. Pulmonary artery systolic pressure (PASP) and mean pulmonary artery pressure (MPAP) were determined by measurement of tricuspid regurgitant velocity (TRV) and mean gradient (MG) respectively. 
 Results: Seventy-two children with SCA (24.0%) had elevated PAP (defined as TRV ≥ 2.5 m /s) while only 9 (3.0%) had pulmonary hypertension (defined as MPAP ≥ 25 mmHg). Older age (O.R = 1.03, p < 0.001), high systolic blood pressure (O.R = 1.2, p = 0.01), and high reticulocyte count (O.R = 1.33, p = 0.04) were weakly associated with increased likelihood of elevated PAP. Normal pulse oxygen saturation (SPO2 ≥ 95%, O.R = 0.4, p = 0.002) and normal heart sounds on auscultation (O.R = 0.2, p = 0.04) were associated with a reduced likelihood of elevated PAP. Presence of left parasternal heave (O.R = 8.4, p = 0.001) significantly predicted elevated PAP in children with SCA.
 Conclusion: PH occurs as a complication in children with SCA and certain clinical and laboratory parameters do predict this complication in children.

Wall Shear Stress Measurement on Curve Objects with PIV in Connection to Benthic Fauna in Regulated Rivers

The flow characteristics in the vicinity of a set of half-cylinders of different sizes simulating benthic objects were studied experimentally using particle image velocimetry (PIV). The cylinders were mounted on the bottom of an open channel, and the influence of the flow speed on the distribution of the shear stress along the bottom geometry was investigated. Of special interest was how the shear stress changes close to the wall as a function of the flow speed and cylinder arrangement. It was found that the shear stress varies significantly as a function of position. This implies habitat heterogeneity allowing benthic invertebrates with different shear stress tolerance exists when the bottom consists of differently sized stones. It was also shown that direct measurements of near wall velocity gradients are necessary to accurately calculate the wall shear stress for more complex geometries.

Accuracy of tomographic particle image velocimetry data on a turbulent round jet

Tomographic particle image velocimetry (Tomo-PIV) was applied on a turbulent round air jet to quantitatively assess the accuracy of velocity gradients obtained in the self-similar turbulent region. The jet Reynolds number based on the nozzle diameter (d) was Red = 3000. Mean velocity, turbulent intensities, and Reynolds shear stress at the center plane of the jet were measured. In addition, statistical results of Tomo-PIV along the axial direction were assessed by performing a separate set of two-dimensional two-component PIV experiments on a “side view” plane along the jet axis. Moreover, the probability distribution functions of four components of the measured velocity gradients in the axial and radial directions were validated by these “side view” planar PIV data. The root mean square of the velocity divergence values relative to the norm of the velocity gradient tensor was 0.36. Furthermore, the on- and off-diagonal components of the velocity gradients satisfied the axisymmetric isotropy conditions. The divergence error in the data affected only areas with low gradient magnitude. Therefore, turbulent structures in the regions with intense vorticity and dissipation can be closely monitored. On this basis, the joint pdfs of the invariants of the velocity gradient and strain and rotation tensor rates were produced and compared well with those in isotropic turbulence studies.

二次元乱流噴流における速度勾配と変動圧力の結合統計量

Joint statistics between velocity gradient and fluctuating static pressure in a turbulent planar jet are experimentally investigated. The experimental data was acquired by simultaneous measurements of velocity gradient ( ∂u’2/∂x1 and ∂u’1/∂x2, where u’1, u’2, x1 and x2 are fluctuating stereamwise velocity, fluctuating cross-streamwise velocity, streamwise coordinate, and cross-streamwise coordinate, respectively) and fluctuating pressure p′ using two combined probes. One of the combined probe consists of an X-type hot-wire probe and a pressure probe and the other combined probe consists of two I-type hot-wire probes arranged in vertical direction and a pressure probe. In the time-series, it is shown that the events in which the instantaneous Reynolds stress, squared cross-stramwise velocity, and product of velocity gradient and pressure shows significantly large value (so-called “significant events”) occur intermittently with negative pressure fluctuation, this implies that the Reynolds stress transport phenomena are locally active with the large-scale vortex structures observed in a turbulent planar jet. It was also found that the standard deviation of (p’/ρ)(∂u’2/∂x1) conditioned by the negative pressure fluctuation is 21% larger than the unconditioned standard deviation. The cospectra show that the correlation (p’/ρ)(∂u’2/∂x1) greatly contributes the local isotropy of the flow field. On the other hand, we confirmed that ∂u’1/∂x2 is almost unrelated to the pressure fluctuation in the present turbulent planar jet. The results show that there is little contribution on the Reynolds stress transport by the correlation (p’/ρ)(∂u’1/∂x2).

On the accuracy of measurement of turbulent velocity gradient statistics with hot-wire probes

A very high resolution minimal channel flow direct numerical simulation was used to examine, virtually, the ability of various multi-sensor hot-wire probe configurations to measure the statistics of velocity gradient components. Various array and sensor configurations and the spatial resolution of probes with these configurations were studied, building on designs and investigations of various authors. In contrast to our previous studies, which focused on turbulent vorticity, vorticity-velocity correlations, dissipation and production rate, here the measurement accuracy of each component of the velocity vector gradient tensor is analyzed separately. The results of the study show that the virtual experiments compare well with a physical experiment, and that such virtual experiments are a powerful tool to examine the accuracy of velocity gradient measurements. The cross-stream gradients needed to determine the vorticity components can be measured with sufficient accuracy with most of the array and sensor configurations of vorticity probes used so far. A systematic error of some of the gradient measurements can appear due to the array or sensor configurations. None of the examined probe designs can measure, with sufficient accuracy, the streamwise velocity gradients, directly or indirectly, using the continuity equation.

Direct detection and measurement of wall shear stress using a filamentous bio-nanoparticle.

The wall shear stress (WSS) that a moving fluid exerts on a surface affects many processes including those relating to vascular function. WSS plays an important role in normal physiology (e.g. angiogenesis) and affects the microvasculature's primary function of molecular transport. Points of fluctuating WSS show abnormalities in a number of diseases; however, there is no established technique for measuring WSS directly in physiological systems. All current methods rely on estimates obtained from measured velocity gradients in bulk flow data. In this work, we report a nanosensor that can directly measure WSS in microfluidic chambers with sub-micron spatial resolution by using a specific type of virus, the bacteriophage M13, which has been fluorescently labeled and anchored to a surface. It is demonstrated that the nanosensor can be calibrated and adapted for biological tissue, revealing WSS in micro-domains of cells that cannot be calculated accurately from bulk flow measurements. This method lends itself to a platform applicable to many applications in biology and microfluidics.

Initial experience with semi-supine bicycle stress echocardiography in a regional setting

Background: Measurements of peak jet velocity and systolic gradients (PSG) in aortic stenosis (AS) are used in the continuity equation to calculate as well as clinically to classify severity. In 1985, Rozenman and Gotsman demonstrated preservation of the inverse linear relationship between left ventricular ejection time and heart rate (HR) and that the PSG across a stenosis, assuming normal cardiac output, is HR dependent, especially in bradyor tachycardia. We sought to evaluate the potential diagnostic importance ofHR correction in the evaluation of AS severity. Methods: Retrospective database interrogation between 2007 and 2014 detected 93 patients with severe native AS by current guidelines (PSG > 64mmHg) and 103 patients with moderate AS (PSG 36 63mmHg). HRwas determined using semi-quantitative software calculating the aortic R-R interval. Results: In patients with severe AS and HR 70/min (n=39), PSG increased by 11 +/9 [6 to 32] mmHg and 44% were reclassified to severe AS. Conclusion: Using the Rozenman formula, this is the first study to demonstrate the importance of HR correction in a small patient population. A larger prospective study is needed to further elucidate the clinical impact of these findings.

Experimental Characterization of Breakage Rate of Colloidal Aggregates in Axisymmetric Extensional Flow

Aggregates prepared under fully destabilized conditions by the action of Brownian motion were exposed to an extensional flow generated at the entrance of a sudden contraction. Two noninvasive techniques were used to monitor their breakup process [i.e. light scattering and three-dimensional (3D) particle tracking velocimetry (3D-PTV)]. While the first one can be used to measure the size and the morphology of formed fragments after the breakage event, the latter is capable of resolving trajectories of individual aggregates up to the breakage point as well as the trajectories of formed fragments. Furthermore, measured velocity gradients were used to determine the local hydrodynamic conditions at the breakage point. All this information was combined to experimentally determine for the first time the breakage rate of individual aggregates, given in the form of a size reduction rate K(R), as a function of the applied strain rate, as well as the properties of the formed fragments (i.e., the number of formed fragments and the size ratio between the largest fragment and the original aggregate). It was found that K(R) scales with the applied strain rate according to a power law with the slope being dependent on the initial fractal dimension only, while the obtained data indicates a linear dependency of K(R) with the initial aggregate size. Furthermore, the probability distribution function (PDF) of the number of formed fragments and the PDF of the size ratio between the largest fragment and the original aggregate indicate that breakage will result with high probability (75%) in the formation of two to three fragments with a rather asymmetric ratio of sizes of about 0.8. The obtained results are well in agreement with the results from the numerical simulations published in the literature.

Calibration of electrodiffusion probes for turbulent flow measurements

Calibration of electrodiffusion (ED) probes with respect to the wall velocity gradient measurements has been experimentally studied in a fully developed turbulent channel flow over the Reynolds number range 14000–23000. In steady state conditions, direct calibration concerning the mean wall velocity gradient can be provided using simultaneous transient and steady state diffusion limiting current measurements. Indirect calibration of the ED probes regarding the turbulent fluctuation has been undertaken using a spectral analysis performed with probes of different size and geometry. This indirect calibration method has been supported using the measurements of thermo-dependence of molecular diffusivity. In this case, only one ED probe can be used for calibration. Dynamic calibration provides necessary information about critical longitudinal dimension of ED probes. When the longitudinal dimension of the probes exceeds the critical length, the probes becomes sensitive to the normal fluctuation as well. This fact has been confirmed using different orientation of the rectangular ED probe with respect to the flow direction.

Merging of high speed argon plasma jets

Formation of an imploding plasma liner for the plasma liner experiment (PLX) requires individual plasma jets to merge into a quasi-spherical shell of plasma converging on the origin. Understanding dynamics of the merging process requires knowledge of the plasma phenomena involved. We present results from the study of the merging of three plasma jets in three dimensional geometry. The experiments were performed using HyperV Technologies Corp. 1 cm Minirailguns with a preionized argon plasma armature. The vacuum chamber partially reproduces the port geometry of the PLX chamber. Diagnostics include fast imaging, spectroscopy, interferometry, fast pressure probes, B-dot probes, and high speed spatially resolved photodiodes, permitting measurements of plasma density, temperature, velocity, stagnation pressure, magnetic field, and density gradients. These experimental results are compared with simulation results from the LSP 3D hybrid PIC code.

Optimisation of Pulsed Ultrasonic Velocimetry system and transducer technology for industrial applications

Pulsed Ultrasonic Velocimetry, commonly referred to as Ultrasonic Velocity Profiling (UVP) in research and engineering applications, is both a method and a device to measure an instantaneous one-dimensional velocity profile in opaque fluids along a measurement axis by using Doppler echography. Studies have suggested that the accuracy of the measured velocity gradient close to wall interfaces need to be improved. The reason for this is due to distortion caused by cavities situated in front of ultrasonic transducers, measurement volumes overlapping wall interfaces, refraction of the ultrasonic wave as well as sound velocity variations (Doppler angle changes). In order to increase the accuracy of velocity data close to wall interfaces and solve previous problems a specially designed delay line transducer was acoustically characterised and evaluated. Velocity profiles measured using the delay line transducer, were initially distorted due to the effect of finite sample volume characteristics and propagation through the delay line material boundary layers. These negative effects were overcome by measuring physical properties of the ultrasonic beam and implementing a newly developed deconvolution procedure. Furthermore, custom velocity estimation algorithms were developed, which improved the time resolution and penetration depth of the UVP system. The optimised UVP system was evaluated and compared to standard transducers in three different straight pipes (inner diameters of 16, 22.5 and 52.8mm). Velocity data obtained using the optimised UVP system showed significant improvement close to wall interfaces where the velocity gradients are high. The new transducer technology and signal processing techniques reduced previously mentioned problems and are now more suitable for industrial process monitoring and control.

Polynomial element velocimetry (PEV): a technique for continuous in-plane velocity and velocity gradient measurements for low Reynolds number flows

Particle image velocimetry (PIV) selects the maximum of the cross-correlation map to represent the modal displacement, and a wealth of information stored in the cross-correlation is discarded. We introduce a novel method, termed polynomial element velocimetry (PEV), which results in continuous velocity and velocity gradient measurements. PEV utilizes the extra information stored in the cross-correlation to determine continuous velocity measurements with low levels of measurement noise. In contrast to PIV, the continuous nature of velocity measurements facilitates the direct determination of the velocity gradient. The PEV method is applied to two laboratory flows: flow in a channel and flow behind a circular cylinder at Reynolds number, Re = 30, and is shown to greatly reduce the noise in the measurements. In addition, the accuracy of PEV is validated using two computer-generated synthetic flows: parabolic flow in a channel and flow past a square cylinder at Re = 30. In these cases, PEV is shown to reduce the velocity measurement error by up to 45% and the vorticity estimation error by up to 77% when compared to PIV. A key benefit of the PEV method is that it is capable of calculating continuous measures for flow gradient with greatly reduced bias errors. In particular, PEV provides a more accurate measurement of the vorticity near interfaces such as a cylinder wall or channel wall where PIV methods only provide measurement data at half the sampling window size from the wall. Since PEV utilizes the entire shape of the cross-correlation map to determine a local map for the underlying velocity, minimal random error is transmitted to the estimated flow gradient. This feature of the PEV method makes it optimal for flows where flow gradients are well defined and there are insufficient pixels to fully resolve structures in the flow using PIV.

THE SPECTROSCOPIC DIVERSITY OF TYPE Ia SUPERNOVAE

We present 2603 spectra of 462 nearby Type Ia supernovae (SN Ia) obtained during 1993-2008 through the Center for Astrophysics Supernova Program. Most of the spectra were obtained with the FAST spectrograph at the FLWO 1.5m telescope and reduced in a consistent manner, making data set well suited for studies of SN Ia spectroscopic diversity. We study the spectroscopic and photometric properties of SN Ia as a function of spectroscopic class using the classification schemes of Branch et al. and Wang et al. The width-luminosity relation appears to be steeper for SN Ia with broader lines. Based on the evolution of the characteristic Si II 6355 line, we propose improved methods for measuring velocity gradients, revealing a larger range than previously suspected, from ~0 to ~400 km/s/day considering the instantaneous velocity decline rate at maximum light. We find a weaker and less significant correlation between Si II velocity and intrinsic B-V color at maximum light than reported by Foley et al., owing to a more comprehensive treatment of uncertainties and host galaxy dust. We study the extent of nuclear burning and report new detections of C II 6580 in 23 early-time spectra. The frequency of C II detections is not higher in SN Ia with bluer colors or narrower light curves, in conflict with the recent results of Thomas et al. Based on nebular spectra of 27 SN Ia, we find no relation between the FWHM of the iron emission feature at ~4700 A and Dm15(B) after removing the two low-luminosity SN 1986G and SN 1991bg, suggesting that the peak luminosity is not strongly dependent on the kinetic energy of the explosion for most SN Ia. Finally, we confirm the correlation of velocity shifts in some nebular lines with the intrinsic B-V color of SN Ia at maximum light, although several outliers suggest a possible non-monotonic behavior for the largest blueshifts.

Estimate of turbulent dissipation in a decaying grid turbulent flow

The dissipation rate of kinetic energy is a key quantity in turbulent flows. The number of measured components and the resolution of the measurement techniques limit dissipation rate estimates; thus, they are derived by surrogates of dissipation. We examine the validity and accuracy of these estimates by investigating decaying grid turbulence using particle image velocimetry and laser Doppler velocimetry. Dissipation rates are computed and compared via three different methods (i) using the decay rate of turbulent kinetic energy, (ii) direct calculation using measured velocity gradients, and (iii) using second and third order structure functions. Discrepancies have been found between the surrogate methods; specifically, the structure function method requires correction terms. The known factors leading to bias are viscous correction which is significant at low Reynolds numbers and inhomogeneity of the decaying flow. Furthermore, we demonstrate inaccuracy in calculations of the third order structure function which are related to inter-scale dependencies. Test procedures are suggested for decaying and inhomogeneous flows to determine susceptibility to these sources of error.