Power-law Profile Research Articles

Structure of axion miniclusters

The Peak-Patch algorithm is used to identify the densest minicluster seeds in the initial axion density field simulated from string decay. The fate of these dense seeds is found by tracking the subsequent gravitational collapse in cosmological $N$-body simulations. We find that miniclusters at late times are well described by NFW profiles, although for around 80\% of simulated miniclusters a single power-law density profile of $r^{-2.9}$ is an equally good fit due to the unresolved scale radius. Under the assumption that all miniclusters with an unresolved scale radius are described by a power-law plus axion star density profile, we identify a significant number of miniclusters that might be dense enough to give rise to gravitational microlensing if the axion mass is $0.2 \,\mathrm{meV}\lesssim m_a \lesssim 3\,\mathrm{meV}$. Higher resolution simulations resolving the inner structure and axion star formation are necessary to explore this possibility further.

The Luminosity Function of Tidal Disruption Flares for the ZTF-I Survey

The high-cadence survey of the Zwicky Transient Facility (ZTF) has completely dominated the discovery of tidal disruption events (TDEs) in the past few years and resulted in the largest sample of TDEs with optical/UV light curves well sampled around their peaks, providing us an excellent opportunity to construct a peak luminosity function (LF) of tidal disruption flares (TDFs). The new construction is necessary particularly considering that the most updated LF reported in literature has been inferred from only 13 sources from five different surveys. Here we present the optical and blackbody LFs calculated by 33 TDFs discovered in the ZTF-I survey. The optical LF can be described by both a power-law profile , and a Schechter-like function. The blackbody LF can be described by a power-law profile , shallower than the LF made of the previous van Velzen (2018) sample. A possible low-luminosity turnover in the optical LF supports an Eddington-limited emission scenario. The drop of the volumetric rate at high luminosity suggests a rate suppression due to direct captures of the black hole. The total volumetric rate is 1 order of magnitude lower than the previous estimation, which is probably not simply caused by the high fraction postpeak sources (7/13) in the previous sample. Instead, the normalization step during the previous LF construction to reconcile various surveys might adversely amplify the influence of serendipitous discoveries. Therefore, TDFs selected from ongoing and upcoming uniform surveys like ZTF, Vera Rubin Observatory, and the Wide-Field Survey Telescope should yield more accurate LFs.

Probing anisotropies of the Stochastic Gravitational Wave Background with LISA

We investigate the sensitivity of the Laser Interferometer Space Antenna (LISA) to the anisotropies of the Stochastic Gravitational Wave Background (SGWB). We first discussthe main astrophysical and cosmological sources of SGWB which are characterized by anisotropies in the GW energy density, and we build a Signal-to-Noise estimator to quantify the sensitivity of LISA to different multipoles. We then perform a Fisher matrix analysis of the prospects of detectability of anisotropic features with LISA for individual multipoles, focusing on a SGWB with a power-law frequency profile. We compute the noise angular spectrum taking into account the specific scan strategy of the LISA detector. We analyze the case of the kinematic dipole and quadrupole generated by Doppler boosting an isotropic SGWB. We find that β ΩGW ∼ 2 × 10-11 is required to observe a dipolar signal with LISA. The detector response to the quadrupole has a factor ∼ 103 β relative to that of the dipole. The characterization of the anisotropies, both from a theoretical perspective and from a map-making point of view, allows us to extract information that can be used to understand the origin of the SGWB, and to discriminate among distinct superimposed SGWB sources.

Analysis of heat transfer for simultaneously developing temperature profile and fluid flow of power-law fluids in the entrance region of a tube

• Simultaneously developing power-law fluid flow and heat transfer in a tube is addressed. • The model is based on the inlet-filled region concept. • The heat transfer solution is obtained using an integral method. • The model is validated against published results, and asymptotically against the fully developed solution. • Results include Nusselt number profiles and entrance region length for different values of the power-law fluid index and Prandtl number. Developing heat transfer in a tube subjected to a constant heat flux is investigated in the case of simultaneously developing fluid flow of a power law fluid in a circular duct using an integral boundary layer model. The inlet-filled region concept is used to solve the heat transfer problem with the velocity profile determined from a previously published and validated hydrodynamic model. The thermal entrance region is divided into a thermal inlet-hydrodynamic inlet zone, followed by a thermal inlet-hydrodynamic filled zone, and last a thermally filled-hydrodynamic filled zone. The heat transfer results are validated against published power-law Nusselt number profiles, and published thermal boundary layer thickness and local Nusselt number profiles for Newtonian fluids as a special case. Furthermore, the asymptotically reached values are validated against the fully developed values. The model presented is predominantly analytical and requires minor computational work.

Parametric studies and optimal design of the exponents collocation of a segmented acoustic black hole beam

Acoustic black hole (ABH) is a promising way to deal with the flexural vibration of different structures. By utilizing power-law profiles in extremities of structures, ABH could gradually reduce the phase velocity of the propagating flexural waves and realize the capture of waves in the ABH portion. However, it is difficult to manufacture an efficient ABH extremity in practical implementations, and the existence of truncation always leads to greatly reduction of ABH effect at the range of desired frequencies. In this paper, parametric studies of segmented ABH beams have been carried out to investigate how different ABH portions’ exponents affect the vibration attenuation performance of the segmented ABH beam, thus further optimizing the exponents collocation design to get the better ABH effect around the desired frequencies. The detailed meshing scheme has been first studied to guarantee the accuracy of the numerical simulation model. Then, the mobility of different segmented ABH beams with varying exponents collocations are compared to show the influence of power-law exponent changes of the first and second ABH portions on their vibration attenuation performance. Additionally, the energy ratio and reflection coefficient have also been calculated to reveal the underlying physical mechanism. Furthermore, the radial basis function based neural network and simulated annealing method are used to establish the surrogate model and optimize exponents collocation. The optimized segmented ABH beam exhibits more perfect vibration attenuation performance because of possessing lower reflection coefficient. Finally, an experiment has been completed to validate the optimization result.

Testing strong lensing subhalo detection with a cosmological simulation

ABSTRACT Strong gravitational lensing offers a compelling test of the cold dark matter paradigm, as it allows for subhaloes with masses of ∼109 M⊙ and below to be detected. We test commonly used techniques for detecting subhaloes superposed in images of strongly lensed galaxies. For the lens we take a simulated galaxy in a ∼1013 M⊙ halo grown in a high-resolution cosmological hydrodynamical simulation, which we view from two different directions. Though the resolution is high, we note the simulated galaxy still has an artificial core which adds additional complexity to the baryon dominated region. To remove particle noise, we represent the projected galaxy mass distribution by a series of Gaussian profiles which precisely capture the features of the projected galaxy. We first model the lens mass as a (broken) power-law density profile and then search for small haloes. Of the two projections, one has a regular elliptical shape, while the other has distinct deviations from an elliptical shape. For the former, the broken power-law model gives no false positives and correctly recovers the mass of the superposed small halo; however, for the latter we find false positives and the inferred halo mass is overestimated by ∼4–5 times. We then use a more complex model in which the lens mass is decomposed into stellar and dark matter components. In this case, we show that we can capture the simulated galaxy’s complex projected structures and correctly infer the input small halo.

Adhesion of graded elastic materials: A full self-consistent model and its application

A full self-consistent model (FSCM) of the axisymmetric adhesive contact between a rigid punch with an arbitrary surface shape and a power-law graded elastic half-space is developed. The self-consistent equation between the surface gap and the surface interaction (e.g., the Lennard–Jones force law) involves a nonlinear singular integral, posing a great challenge to numerical calculations. By applying the properties of Gauss’s hypergeometric function, the integral singularity is eliminated in the numerical calculation through Riemann–Stieltjes integral. Case studies for power-law punch profiles are performed and the self-consistent equation can be expressed in a dimensionless form with three dimensionless parameters, namely a shape index, a gradient exponent, and a new generalized Tabor number. The FSCM results are obtained by solving the self-consistent equation through the surface central gap control method and Newton–Raphson iterative method. For large generalized Tabor numbers, the force–displacement curves are ‘S-shaped’ and condense to the extended JKR limit in the high-load branch. As the generalized Tabor number decreases, a continuous transition from the extended JKR model to the Bradley model for the adhesion of power-law graded materials is obtained. It is found that the pull-off force of a graded material usually depends on the three dimensionless parameters, but for some cases of the shape index, it is not sensitive to the gradient exponent when the generalized Tabor number is fixed. Asymptotic solutions are derived to predict the unstable jump points, which coincide well with the FSCM predictions. The FSCM is applied to validate the extended Maugis–Dugdale (M–D) model of graded materials and it is found that the accuracy of the original M–D-n-k model using the maximum strength condition to determine the cohesive stress is limited. By introducing the rigid-limit-consistency condition of the pull-off force to determine the cohesive stress, the M–D-n-k model is improved and its predictions show good consistency with the FSCM results.

Prompt cusp formation from the gravitational collapse of peaks in the initial cosmological density field

ABSTRACT I present an analytic model for the early post-collapse evolution of a spherical density peak on the coherence scale of the initial fluctuations in a universe filled with collisionless and pressure-free ‘dust’. On a time-scale which is short compared to the peak’s collapse time t0, its inner regions settle into an equilibrium cusp with a power-law density profile, ρ ∝ r−12/7. Within this cusp, the circular orbit period P at each radius is related to the enclosed mass M by P = t0(M/Mc)2/3 where Mc is a suitably defined characteristic mass for the initial peak. The relaxation mechanism which produces this cusp gives insight into those which are active in high-resolution simulations of first halo formation in cold or warm dark matter universes, and, indeed, a simple argument suggests that the same power-law index γ = −12/7 should describe the prompt cusps formed during the collapse of generic peaks, independent of any symmetry assumption. Further work is needed to investigate the additional factors required to explain the slightly flatter exponent, γ ≈ −1.5, found in high-resolution numerical simulations of peak collapse.

A lensed radio jet at milliarcsecond resolution I: Bayesian comparison of parametric lens models

ABSTRACT We investigate the mass structure of a strong gravitational lens galaxy at z = 0.350, taking advantage of the milliarcsecond (mas) angular resolution of very long baseline interferometric (VLBI) observations. In the first analysis of its kind at this resolution, we jointly infer the lens model parameters and pixellated radio source surface brightness. We consider several lens models of increasing complexity, starting from an elliptical power-law density profile. We extend this model to include angular multipole structures, a separate stellar mass component, additional nearby field galaxies, and/or a generic external potential. We compare these models using their relative Bayesian log-evidence (Bayes factor). We find strong evidence for angular structure in the lens; our best model is comprised of a power-law profile plus multipole perturbations and external potential, with a Bayes factor of +14984 relative to the elliptical power-law model. It is noteworthy that the elliptical power-law mass distribution is a remarkably good fit on its own, with additional model complexity correcting the deflection angles only at the ∼5 mas level. We also consider the effects of added complexity in the lens model on time-delay cosmography and flux-ratio analyses. We find that an overly simplistic power-law ellipsoid lens model can bias the measurement of H0 by ∼3 per cent and mimic flux ratio anomalies of ∼8 per cent. Our results demonstrate the power of high-resolution VLBI observations to provide strong constraints on the inner density profiles of lens galaxies.

Investigation of the imbibition/drainage of two immiscible fluids in capillaries with arbitrary axisymmetric cross-sections: a generalized model

Abstract In this work, we investigate the problem of imbibition/drainage of a fluid in capillaries of arbitrary axisymmetric cross-sections filled initially with another immiscible one. The model predicts the location of the meniscus and its speed along the tube length with time. The two immiscible fluids may assume any density and viscosity contrasts. In addition, the axisymmetric profile of the tube maintains a relatively small angle of tangency to warrant that the axial velocity distribution assumes, approximately, a parabolic profile. The driving forces that may be encountered in this system include the capillary force, pressure force, gravitational force and an opposing viscous force. The orientation of the capillary force can be in the direction of the flow (e.g. during imbibition) or opposite to the flow (e.g. during drainage). Likewise, the gravitational force can be in the direction of the flow or opposite to it. In this work we account for all these possibilities. A differential equation is developed that defines the location of the meniscus with time. A fourth-order-accurate Runge–Kutta scheme has been developed to provide solutions for the different scenarios associated with this system. It is shown that the developed model reduces to those appropriate for straight tubes, which builds confidence in the modelling approach. The effects of changing the tangent along the profile of the tube, which influences the calculation of the radius of curvature of the meniscus, is also considered. Unlike the cases of straight capillary tubes, in tubes with arbitrary symmetric profiles, the friction force depends on the variations of the tube profile. Examples of converging/diverging capillary tubes that follow straight and power law profiles are investigated. In addition, the case of sinusoidal profiles has also been considered.

The rise of the primordial tensor spectrum from an early scalar-tensor epoch

Primordial gravitational waves (PGW) produced during inflation span a large range of frequencies, carrying information on the dynamics of the primordial universe. During an early scalar-tensor dominated epoch, the amplitude of the PGW spectrum can be enhanced over a wide range of frequencies. To study this phenomenon, we focus on a class of scalar-tensor theories, well motivated by high energy theories of dark energy and dark matter, where the scalar is conformally and disformally coupled to matter during the early cosmological evolution. For a conformally dominated epoch, the PGW spectrum has a flat step-like shape. More interestingly, a disformally dominated epoch is characterised by a peaked spectrum with a broken power-law profile, with slopes depending on the scalar-tensor theory considered.We introduce a graphical tool, called broken power-law sensitivity curve, as a convenient visual indicator for understanding whether a given broken power-law profile can be detected by GW experiments. We then analyse the GW spectra for a variety of representative conformal and disformal models, discussing their detectability prospects with the Einstein Telescope (ET), Laser Interferometer Space Antenna (LISA), DECi-hertz Interferometer Gravitational wave Observatory (DECIGO), and Big Bang Observer (BBO).

Inclusive Hyper- to Dilute-Concentrated Suspended Sediment Transport Study Using Modified Rouse Model: Parametrized Power-Linear Coupled Approach Using Machine Learning

The transfer of suspended sediment can range widely from being diluted to being hyper-concentrated, depending on the local flow and ground conditions. Using the Rouse model and the Kundu and Ghoshal (2017) model, it is possible to look at the sediment distribution for a range of hyper-concentrated and diluted flows. According to the Kundu and Ghoshal model, the sediment flow follows a linear profile for the hyper-concentrated flow regime and a power law applies for the dilute concentrated flow regime. This paper describes these models and how the Kundu and Ghoshal parameters (linear-law coefficients and power-law coefficients) are dependent on sediment flow parameters using machine-learning techniques. The machine-learning models used are XGboost Classifier, Linear Regressor (Ridge), Linear Regressor (Bayesian), K Nearest Neighbours, Decision Tree Regressor, and Support Vector Machines (Regressor). The models were implemented on Google Colab and the models have been applied to determine the relationship between every Kundu and Ghoshal parameter with each sediment flow parameter (mean concentration, Rouse number, and size parameter) for both a linear profile and a power-law profile. The models correctly calculated the suspended sediment profile for a range of flow conditions (0.268 mm≤d50≤2.29 mm, 0.00105gmm3≤particle density≤2.65gmm3, 0.197mms≤vs≤96mms, 7.16mms≤u*≤63.3mms, 0.00042≤cˉ≤0.54), including a range of Rouse numbers (0.0076≤P≤23.5). The models showed particularly good accuracy for testing at low and extremely high concentrations for type I to III profiles.

Broadband vibration suppression of rainbow metamaterials with acoustic black hole

Acoustic black hole (ABH) structures have been widely utilized for vibration suppression owing to its tailored indentations according to a power-law profile. Compared with single ABH element, multiple ABHs embedded in periodic lattice configurations exhibit more remarkable attenuation effect within the so-called bandgaps at relatively low frequencies. However, most research is devoted to the analysis of uniform metamaterial embedded with periodic arrangement of ABHs, resulting in the attenuation bandgaps limited to a narrow frequency range. In this research, a rainbow metamaterial beam embedded with graded arrangement of ABHs is proposed to achieve broadband vibration attenuation. Two arrangement types of ABHs in metamaterial beams including linear and sinusoidal profiles are taken into account. The differential quadrature method is used to establish governing equations of infinite beams with periodic ABHs and finite rainbow metamaterials with graded ABHs, respectively. Using the validated mathematical model and genetic algorithm, wave propagation behaviors in infinite metamaterial beams, transmittance characteristics and optimization of finite rainbow metamaterial beams are investigated quantitatively. The results reveal that the generation of broad bandgaps can be explained by the combination of local resonance and Bragg scattering effects for proposed metamaterial beams with periodic asymmetrical ABHs. The corresponding bandgaps are tunable by changing the geometry parameters of ABHs. Moreover, the rainbow metamaterial beams with linear and sinusoidal ABHs own a broader attenuation band and better attenuation effect compared to the periodic ones. Compared with regular graded distributions, disordered arrangement of geometry parameters of neighboring cells leads to further improvement of attenuation strength. This work provides novel perspectives on achieving broadband vibration attenuation by rainbow metamaterial beams with ABHs, and the results could be applied to guide the design of rainbow metamaterial with graded geometry parameters.

Effects of CO-dark Gas on Measurements of Molecular Cloud Stability and the Size–Linewidth Relationship

Stars form within molecular clouds, so characterizing the physical states of molecular clouds is key to understanding the process of star formation. Cloud structure and stability are frequently assessed using metrics including the virial parameter and Larson scaling relationships between cloud radius, velocity dispersion, and surface density. Departures from the typical Galactic relationships between these quantities have been observed in low-metallicity environments. The amount of H2 gas in cloud envelopes without corresponding CO emission is expected to be high under these conditions; therefore, this CO-dark gas could plausibly be responsible for the observed variations in cloud properties. We derive simple corrections that can be applied to empirical clump properties (mass, radius, velocity dispersion, surface density, and virial parameter) to account for CO-dark gas in clumps following power-law and Plummer mass density profiles. We find that CO-dark gas is not likely to be the cause of departures from Larson’s relationships in low-metallicity regions, but that virial parameters may be systematically overestimated. We demonstrate that correcting for CO-dark gas is critical for accurately comparing the dynamical state and evolution of molecular clouds across diverse environments.

Long-term Multiband Near-infrared Variability of the Blazar OJ 287 during 2007–2021

We present the most extensive and well-sampled long-term multiband near-infrared (NIR) temporal and spectral variability study of OJ 287, considered to be the best candidate binary supermassive black hole blazar. These observations were made between 2007 December and 2021 November. The source underwent ∼2–2.5 mag variations in the J, H, and Ks NIR bands. Over these long-term timescales there were no systematic trends in either flux or spectral evolution with time or with the source’s flux states. However, on shorter timescales, there are significant variations in flux and spectra indicative of strong changes during different activity states. The NIR spectral energy distributions show diverse facets at each flux state, from the lowest to the highest. The spectra are, in general, consistent with a power-law spectral profile (within 10%) and many of them indicate minor changes (observationally insignificant) in the shift of the peak. The NIR spectra generally steepen during bright phases. We briefly discuss these behaviors in the context of blazar emission scenarios/mechanisms, OJ 287's well-known traditional behavior, and implications for models of the source central engine invoked for its long-term optical semiperiodic variations.

Supersonic–subsonic transition region in radiative heat flow via self-similar solutions

We study the radiative hydrodynamics flow of radiation-driven heat waves in hot dense plasmas, using approximate self-similar solutions. Specifically, we have focused on the intermediate regime between the pure radiative supersonic flow and the pure subsonic regime. These two regimes were investigated using both exact self-similar solutions and numerical simulations; however, most of the study used numerical simulations, mainly because the radiative heat wave and the shock regions are not self-similar altogether. In a milestone work [Garnier et al., “Self-similar solutions for a nonlinear radiation diffusion equation,” Phys. Plasmas 13, 092703 (2006)], it was found that for a specific power law dependency temperature profile, a unique exact self-similar solution exists that is valid for all physical regimes. In this work, we approximate Garnier's exact solution for a general power-law temperature-dependency, using simple analytical considerations. This approximate solution yields a good agreement compared to numerical simulations for the different thermodynamic profiles within the expected range of validity. In addition, we offer an approximate solution for the energies absorbed in the matter, again, for a general power-law temperature profile. Our approximate self-similar solution for the energy yields very good results compared to exact numerical simulations for both gold and Ta2O5. We also set a comparison of our self-similar solutions with the results of an experiment for radiation temperature measurement in a Hohlraum in low-density foams that is addressed directly to the intermediate regime, yielding a good agreement and similar trends. The different models as well as the numerical simulations are powerful tools to analyze the supersonic–subsonic transition region.

Static Thin Disks with Power-law Density Profiles * * Dedicated to our teacher, Professor Jiří Bičák, on the occasion of his 80th birthday. For O.S., Jiří's interest in thin disks was, 30 yr ago (Bičák et al. ), the first motivation to study the topic.

The task of finding the potential of a thin circular disk with power-law radial density profile is revisited. The result, given in terms of infinite Legendre-type series in the above reference, has now been obtained in closed form thanks to the method of Conway employing Bessel functions. Starting from a closed-form expression for the potential generated by the elementary density term ρ 2l , we cover more generic—finite solid or infinite annular—thin disks using superposition and/or inversion with respect to the rim. We check several specific cases against the series-expansion form by numerical evaluation at particular locations. Finally, we add a method to obtain a closed-form solution for finite annular disks whose density is of “bump” radial shape, as modeled by a suitable combination of several powers of radius. Density and azimuthal pressure of the disks are illustrated on several plots, together with radial profiles of free circular velocity.

Eccentricity evolution in gaseous dynamical friction

ABSTRACT We analyse how drag forces modify the orbits of objects moving through extended gaseous distributions. We consider how hydrodynamic (surface area) drag forces and dynamical friction (gravitational) drag forces drive the evolution of orbital eccentricity. While hydrodynamic drag forces cause eccentric orbits to become more circular, dynamical friction drag can cause orbits to become more eccentric. We develop a semi-analytic model that accurately predicts these changes by comparing the total work and torque applied to the orbit at periapse and apoapse. We use a toy model of a radial power-law density profile, ρ ∝ r−γ, to determine that there is a critical γ = 3 power index, which separates the eccentricity evolution in dynamical friction: orbits become more eccentric for γ < 3 and circularize for γ > 3. We apply these findings to the infall of a Jupiter-like planet into the envelope of its host star. The hydrostatic envelopes of stars are defined by steep density gradients near the limb and shallower gradients in the interior. Under the influence of gaseous dynamical friction, an infalling object’s orbit will first decrease in eccentricity and then increase. The critical separation that delineates these regimes is predicted by the local density slope and is linearly dependent on polytropic index. More broadly, our findings indicate that binary systems may routinely emerge from common envelope phases with non-zero eccentricities that were excited by the dynamical friction forces that drove their orbital tightening.

Quantifying Feedback from Narrow Line Region Outflows in Nearby Active Galaxies. IV. The Effects of Different Density Estimates on the Ionized Gas Masses and Outflow Rates

Active galactic nuclei (AGN) can launch outflows of ionized gas that may influence galaxy evolution, and quantifying their full impact requires spatially resolved measurements of the gas masses, velocities, and radial extents. We previously reported these quantities for the ionized narrow-line region outflows in six low-redshift AGN, where the gas velocities and extents were determined from Hubble Space Telescope long-slit spectroscopy. However, calculating the gas masses required multicomponent photoionization models to account for radial variations in the gas densities, which span ∼6 orders of magnitude. To simplify this method for larger samples with less spectral coverage, we compare these gas masses with those calculated from techniques in the literature. First, we use a recombination equation with three different estimates for the radial density profiles. These include constant densities, those derived from [S ii], and power-law profiles based on constant values of the ionization parameter (U). Second, we use single-component photoionization models with power-law density profiles based on constant U, and allow U to vary with radius based on the [O iii]/Hβ ratios. We find that assuming a constant density of n H = 102 cm−3 overestimates the gas masses for all six outflows, particularly at small radii where the outflow rates peak. The use of [S ii] marginally matches the total gas masses, but also overestimates at small radii. Overall, single-component photoionization models where U varies with radius are able to best match the gas mass and outflow rate profiles when there are insufficient emission lines to construct detailed models.

Measurements on the surface wind pressure characteristics of a thousand-meter scale megatall building by wind tunnel test

This paper investigates the wind pressure distribution of a thousand-meter scale megatall building through a wind tunnel test. One key challenge is that the power-law wind profiles are not applicable due to the height limitation. Therefore, a mesoscale meteorological model WRFv3.4 (Weather Research & Forecasting Model) was used to explore the wind profile covering the height of the thousand-meter scale megatall building. Then the wind profile was reproduced in the wind tunnel representing the actual vertical distribution of wind speed by a passive simulation method, and thereby the wind pressure test was conducted in the wind tunnel. Moreover, to investigate the effect of the inlet wind profile on the wind pressure, a wind tunnel test using the power-law wind profile was carried out for comparison as well. It was revealed that the wind pressure distribution was greatly influenced by the inlet wind profile, which should be investigated with accuracy in the wind tunnel.