Shear Velocity Research Articles

Modified cellulose nanofibrils as enhancers for thin UV-curable matte coatings on digital printing decorative panels

To enhance the surface properties of a thin ultraviolet (UV)-curable matte coating film (less than 20 µm) for the digital printing decorative panels, the aldehyde-containing cellulose nanofibril (CNF) coupled with hyperbranched poly (amido amine) grafting was selected as a reinforcing agent. The viscoelastic properties of the coating and physical and mechanical characteristics of the corresponding decorative panels were evaluated. The results revealed that the incorporation of the modified CNF increased the viscosity of nanocomposite coatings at low shear rates, while at shear velocities exceeding 10 1/s, the viscosity approached that of the pure coating. The modified CNFs dispersed well in coatings and the nanocomposite coating showed excellent stability. Moreover, the incorporation of CNF clearly enhanced the tensile strength of the coating film. Additionally, the modified CNF physically and chemically bonded to the polymer chains within the UV-curable coating. Furthermore, the modified CNF facilitated the optimal distribution of silica in the acrylic polymer when added at 5 wt%. The panels with the modified matte coating exhibited marginally improved clarity compared to that of the original boards. Moreover, the inclusion of the modified CNF resulted in reduced gloss, enhanced hardness, and improved yellowing resistance compared to the original panels, thereby enhancing the quality of the new product.

Numerical investigation on the VIV response of the flexible cylindrical structures subjected to non-linearly varying shear flow

Vortex-induced vibration (VIV) of flexible cylinders is an important phenomenon in ocean structures, which may cause fatigue damage and structural failure of offshore structures. Much literature has extensively explored VIV of flexible cylinders under various flow conditions, such as uniform and linearly varying shear flow. However, real scenarios in most cases involve non-uniform, non-linearly varying shear flow profiles. Hence, it is important to investigate the VIV response of the flexible cylindrical structures subjected to non-linearly varying shear flow profiles. This study employs a wake oscillator model (WOM) that simulates the VIV response of the flexible cylinder under the action of non-linearly varying shear flow velocity profiles. The flexible cylinder subjected to linearly varying sheared flow velocity profiles not only overpredicts but also underpredicts the VIV responses compared to the non-linearly varying sheared flow profile, which represents the real scenario. The VIV mode number for the flexible cylinder subjected to non-linear shear flow was found to have a non-linear relationship with that of linear shear flow. The VIV response of the flexible cylinder subjected to a linearly varying shear flow profile exhibits more travelling wave behaviour compared to non-linear shear flow. The location of the maximum VIV response of the structure having the same aspect ratio (L/D) and shear parameter (β), was different for linear and non-linear flow profiles across the span. The time and frequency domain responses showed a shift in the frequency band. The results obtained from the numerical simulations give better insights into the vibrational behaviour of flexible cylinders in realistic flow environments.

Integrated neural network based simulation of thermo solutal radiative double-diffusive convection of ternary hybrid nanofluid flow in an inclined annulus with thermophoretic particle deposition

Numerical simulation of magnetohydrodynamic radiative double-diffusive convective heat and mass transfer of a ternary hybrid nanofluid with thermophoretic particle deposition in an inclined annulus is studied. Both sides of cylinders are preserved at uniform temperatures, while the remaining sides are thermally insulated. The coupled nonlinear partial differential equations are solved using a finite difference approach. The detailed computational results of heat and mass transfer rate, temperature, concentration and flow fields are presented and discussed for a distinct range of significant physical parameters. The results demonstrated that increasing the angle of the inclination factor increases fluid flow. In light of buoyancy, when the annular cavity is set downward, the enhanced shear velocity is transported upwards, where it reaches its optimal temperature. The higher temperature difference leads to augmented dynamic fluid motion, particularly in the radial direction, as the system seeks to balance the thermal energy through convective transfer. The Brownian motion parameter has abrupt mobility of nanoparticles in liquid, which promotes particle collisions with liquid molecules, yielding kinetic energy. Increased values of thermophoretic parameters augment the thermophoresis force. Thermophoretic particle deposition increases the concentration of nanoparticles in an annulus due to the migration of particles from hot to cold regions under the influence of temperature gradient. The heat and mass transfer characteristics of ternary hybrid nanofluid are forecasted through an artificial neural network-based Levenberg–Marquardt backpropagated algorithm. The built model shows the heat and mass transfer rate root mean squared values through the Levenberg–Marquardt algorithm as one and mean squared error values as 1e-07 and 1e-08 respectively.

Multi-scale processes of the Kelvin-Helmholtz instability at Earth’s magnetopause

The Kelvin-Helmholtz Instability (KHI) is a large scale convective instability which occurs anywhere the velocity shear between two fluids is large, such as Earth’s magnetopause where the fast flowing magnetosheath abuts the relatively stagnant outer magnetosphere. The KHI was initially believed to contribute only to energy and momentum transfer from the solar wind to the magnetosphere, but was eventually shown to support mass transport and plasma heating. Recent advancements in in-situ observational capabilities and high scale computer modeling have once again shifted our understanding of the KHI from a large scale process, to an active environment which connects the global and kinetic scales through a variety of multi-scale processes and phenomena. In this mini-review, we provide an update on the latest findings in Kelvin-Helmholtz (KH) related processes at kinetic scales and the effects of the global environment on KH development.

Interparticle friction behaviors of kaolinite: Insights into macroscale friction from nanoscale

The friction behavior of widespread clay minerals is a major concern in many geo-engineering problems, such as the stability of soft soil foundations and the induction of seismic fault zones. The present work aimed to study the friction behaviors of kaolinite at the particle level using the molecular dynamics method. The effects of normal force (Fn), shear velocity (v), and interfacial water film on nanofriction were discussed. The “stick-slip” phenomenon and periodic evolution of friction forces (Ff) were observed in dry friction and became less pronounced with water lubrication. The dry Ff of kaolinite was found to be insensitive to Fn. However, wet Ff exhibited a linear increase with Fn and then transitioned to a non-linear relationship as slip displacement increases due to the continuous loss of water molecules from the interface during friction. Notably, at high loads (Fn ≥ 30 nN), the peak friction of wet kaolinite showed characteristics similar to dry friction. A velocity-strengthening behavior of kaolinite at high velocities was observed in both dry and wet conditions. The macroscale friction coefficients of kaolinite were predicted from nanofriction data and results showed good agreement with experimental values. This study lays the foundation for bridging micro- and macro-mechanical behaviors, suggesting a new pathway for acquiring precise macroscopic friction of minerals through cross-scale studies.

Determining the debris flow yield strength of weathered soils: a case study of the Miryang debris flow in the Republic of Korea

Debris flow hazards are often interpreted through back-calculated simulation analysis or empirical methods. The mobility of a debris flow is greatly influenced by mechanical and hydrological parameters. The strength parameters play important roles in the debris flow initiation and flow stages. In particular, the rheological parameters of yield strength and plastic viscosity directly affect the debris flow runout distance and velocity. One of the most important parameters to consider when evaluating debris flow hazards is the shear strength. This strength is called the residual shear strength in the failure stage and the yield strength in the post-failure stage. The residual shear strength obtained from ring shear tests can be related to the initiation of mass movements; the yield strength obtained from rheological tests can be related to the mobilization of debris flows. The residual shear stresses obtained from ring shear tests of weathered soils typically range between 10 and 100 kPa and strongly depend on the normal stress and shear velocity. When progressive slope failure (i.e., strain-softening behavior) occurs at a relatively shallow slope depth (e.g., < 1 m), the soil strength ranges from approximately 5–10 kPa. If the liquid limit state (i.e., solid‒liquid transition) is reached, the shear strength of the soil is approximately 2 kPa. Once the soil fails and mixes with ambient water along the slip surface, the yield strength decreases dramatically, resulting in high mobilization. A suggestion on how strength parameters can be applied to estimate debris flow mobility is presented by considering the 2011 Miryang debris flow, which occurred in weathered soil deposits in Miryang city, Republic of Korea. The best approach for debris flow yield strength estimation would be to consider the residual shear strength in the initiation stage, the yield strength in the flow stage, and the reduction in yield strength with the entrainment effect of the flow in the rapid fluidization stage.

Exploring hemodynamic mechanisms and re-intervention strategies for partial false lumen thrombosis in Stanford type B aortic dissection after thoracic aortic endovascular repair

ObjectivesFalse lumen (FL) thrombosis status for Stanford type B aortic dissection (TBAD) after thoracic endovascular aortic repair (TEVAR) is critical for evaluating aortic remodeling and long-term prognosis. This study aimed to monitor the morphology evolution of partial FL thrombosis (PFLT) and its hemodynamic conditions through an innovative approach, providing a re-intervention strategy from both morphologic and hemodynamic perspectives. MethodsThree-dimensional geometries are extracted from a five-year follow-up of CTA images for TBAD after TEVAR. The morphology and hemodynamics of PFLT are comprehensively analyzed based on patient-specific reconstructions and computational fluid dynamics (CFD). The impact of various strategies treating risk factors of PFLT, including proximal entry closure, left renal artery stenting, or accessory renal artery embolism on hemodynamics is assessed. ResultsThe introduced morphologic approaches appropriately reflected the evolution of PFLT. Gradual dilation of FL (surface area from 82.63cm2 to 98.84cm2, volume from 45.12 mL to 63.40 mL, increase in distal tear (from 3.72 cm to 4.32 cm), and fluctuation of thrombosis-blood lumen boundary are observed. For further surgical preparation in the absence of unanimously recognized re-intervention indicators, velocity and wall shear stress distributions reveal different simulated re-interventions have distinctly suppressive effects on hemodynamic conditions within the PFLT, providing valuable insights for further surgical preparation. ConclusionsThe present study demonstrates a re-intervention strategy for PFLT in TBAD patients after TEVAR utilizing morphologic and hemodynamic analyses. Acknowledging the deterioration of PFLT may result in adverse long-term outcomes, this strategy might offer an alternative approach for clinical monitoring and management of related patients.

Three-dimensional joint inversion of surface wave dispersion and gravity data using a petrophysical approach: an application to Los Humeros Geothermal Field

SUMMARY We present a method to jointly invert surface wave dispersion data and gravity measurements for 3-D shear wave velocity and density models. We implemented a petrophysical approach to combine the kernels of both methodologies in a single process. The synthetic experiments show that jointly inverted models recover shear wave velocity and density better than separate inversions. In particular, density models benefit from the good vertical resolution of surface wave dispersion data, while shear velocity models benefit from the good lateral resolution of gravity data. We also proposed two methods to stabilize the solution when using high-grade polynomials. We applied the methodology to the Los Humeros Geothermal area to demonstrate its applicability in a complex geological scenario. Compared with separate inversion, the joint inversion contributes to enhancing key aspects of the geothermal system by (i) delimitating better the geometry of the caldera deposits in the first 0–2.8 km deep by increasing the vertical resolution in density, (ii) delimitating better the lateral borders of low-Vs bodies at different depths interpreted as a part of a complex magmatic chamber system and (iii) estimating the local shear wave velocity–density relationship that conforms to other known relationships for sedimentary and igneous rocks but with some differences that bring us additional information.

Decay of magnetohydrodynamic turbulence in the expanding solar wind: WIND observations

We have studied the decay of turbulence in the solar wind. Fluctuations carried by the expanding wind are naturally damped because of flux conservation, slowing down the development of a turbulent cascade. The latter also damps fluctuations but results in plasma heating. We analyzed time series of the velocity and magnetic field ($ v $ and $ B $, respectively) obtained by the WIND spacecraft at 1 au. Fluctuations were recast in terms of the Elsasser variables, $ z v B rho $, with rho being the average density, and their second- and third-order structure functions were used to evaluate the Politano-Pouquet relation, modified to account for the effect of expansion. We find that expansion plays a major role in the Alfv\'enic stream, those for which $z_+ z_-$. In such a stream, expansion damping and turbulence damping act, respectively, on large and small scales for $z_+$, and also balance each other. Instead, $z_-$ is only subject to a weak turbulent damping because expansion is a negligible loss at large scales and a weak source at inertial range scales. These properties are in qualitative agreement with the observed evolution of energy spectra that is described by a double power law separated by a break that sweeps toward lower frequencies for increasing heliocentric distances. However, the data at 1 au indicate that injection by sweeping is not enough to sustain the turbulent cascade. We derived approximate decay laws of energy with distance that suggest possible solutions for the inconsistency: in our analysis, we either overestimated the cascade of $z_ or missed an additional injection mechanism; for example, velocity shear among streams.

Analysis of Fluid-Structure Interaction Mechanisms for a Native Aortic Valve, Patient-Specific Ozaki Procedure, and a Bioprosthetic Valve.

The Ozaki procedure is a surgical technique which avoids to implant foreign aortic valve prostheses in human heart, using the patient's own pericardium. Although this approach has well-identified benefits, it is still a topic of debate in the cardiac surgical community, which prevents its larger use to treat valve pathologies. This is linked to the actual lack of knowledge regarding the dynamics of tissue deformations and surrounding blood flow for this autograft pericardial valve. So far, there is no numerical study examining the coupling between the blood flow characteristics and the Ozaki leaflets dynamics. To fill this gap, we propose here a comprehensive comparison of various performance criteria between a healthy native valve, its pericardium-based counterpart, and a bioprosthetic solution, this is done using a three-dimensional fluid-structure interaction solver. Our findings reveal similar physiological dynamics between the valves but with the emergence of fluttering for the Ozaki leaflets and higher velocity and wall shear stress for the bioprosthetic heart valve.

Evolution of Turbulent Boundary Conditions on the Surface of Large Barchan Dunes: Anomalies in Aerodynamic Roughness and Shear Velocity, Aeolian Threshold, and the Role of Dune Skewness

AbstractWe recorded aerodynamic roughness and shear velocity along transects on and around mature crescent‐shaped barchan dunes of and height above the horizontal rock‐covered Qatar desert by fitting to the log‐law time‐averaged vertical velocity profiles acquired from triads of ultrasonic anemometers penetrating the inner turbulent boundary layer. Shear velocity first decreased, then recovered as air climbed on the dune, with a local maximum ahead of the crest as predicted by the Jackson and Hunt (1975, https://doi.org/10.1002/qj.49710143015) theory. Unlike flows over gentler bedforms without a slope discontinuity, an anomalous peak of shear velocity also arose on the dune centerline at the brink, which the theory attributed to skewness in the dune transect profile. The onset of aeolian transport produced a log‐law passing through the Bagnold (1941, https://doi.org/10.1007/978‐94‐009‐5682‐7) focal point. It was bracketed by noticeable hysteretic peaks in the correlation between wind speed and entrained sand flux. The dunes' rocky surroundings and topography produced an aerodynamic roughness at odds with the Nikuradse (1933, https://ntrs.nasa.gov/citations/19930093938) data for fully developed turbulent boundary layers. Large‐eddy numerical simulations illustrated the sensitivity of shear velocity to wide changes in aerodynamic roughness from desert floor to dune surface.

Model for plasma transport due to drift waves based on mappings including finite Larmor radius

Abstract Turbulence in plasmas is modeled by fluctuating electric fields that determine particle motion through the E × B drift velocity which can lead to chaotic behavior. When applied to an ensemble of plasma particles in the chaotic regime their transport is studied: the anomalous plasma transport. The fluctuations are modeled by a spectrum of traveling waves with a wide frequency span which converts the equations of motion into an iterative mapping. The statistical properties of transport are derived. When the waves amplitude is small the particle orbits are regular, but as it is increased the behavior becomes increasingly chaotic. The effect of finite Larmor radius and the presence of a background plasma flow is also studied. We show that when a thermal population of particles is considered, the transport becomes non-local, as evidenced by a non-Gaussian particle distribution function (PDF). We have analyzed two different kinds of sheared flows: (1) a monotonic velocity shear and (2) a non-monotonic shear. The presence of transport barriers associated with the sheared velocity is also studied. We also present the application of the same techniques to study the transport of energetic particles that are born with a monoenergetic distribution.

Observations of daytime topside ionospheric irregularities in the afternoon equatorial ionosphere

Previous observations show that the daytime irregularities (the fossil of nighttime irregularities) are mostly located in the low latitude region and at 10–14 LT. In this study, daytime topside ionospheric irregularities were observed along the longitude of ∼ 94oE by Swarms satellite in the afternoon equatorial ionosphere (about 15:00 LT) on 18 June 2020. Simultaneously, the F3 layer were observed in the bottomside ionosphere. Occurrences of F3 layer shows that there are disturbances on the zonal electric field. Multi-satellite observations show that the topside ionospheric irregularities did not occur in the morning equatorial ionosphere. Results indicate that afternoon ionospheric irregularities might be associated with the electrodynamic process of the plasma in the equatorial ionosphere. Simulations show that the combined effects of the disturbed zonal electric field and velocity shears in zonal plasma drift can be favorable to increase the occurrence rate of daytime ionospheric irregularities in the topside equatorial ionosphere.

Numerical model for penetration process of a deformable projectile into ductile metallic target plate considering the interaction of projectile and target

The interaction of projectile and target in a penetration process is key for precise prediction and safety design. However, this interaction is instantaneous and varying with complex physical phenomena, that induces a challenge of panoramically illustrating the penetration process. This work develops a new numerical model that can capture the penetration process of a deformable projectile impacting ductile target plate and meanwhile the interaction of projectile and target is considered. Here, physical mechanisms are explored and accordingly mathematical derivations for theoretical analysis are carried out. The issues of plastic stress wave, contact stress, shear perforation and energy dissipation are involved. Both the deformation of projectile and failure of target plate are addressed which include the upsetting deformation of projectile, pit-opening performance of target plate and perforation of target plate. This model presents the history of the deformation of projectile and target, velocity evolution, penetration resistance and shear perforation with timing. The modelling results show a high-precision prediction by comparing with experimental data of a flat-ended projectile penetrating Weldox 460E steel target plate [54] and other developed models of plugging failure model [13] and plastic wave model [48] for both cases of 16mm- and 20 mm-thickness target plates. This work offers the comprehensive calculation and analysis of penetration process and reveals the insights of the transient phenomena for impact engineering.

A Numerical Approach to Analyzing Shallow Flows over Rough Surfaces

The hydraulic characteristics (such as velocity profiles, near-bed velocity profile, bed shear stress, and resistance coefficients) of shallow flows over rough surfaces were investigated using numerical simulations. A novel method is presented to simulate shallow flows over rough surfaces in a two-dimensional (2D) numerical domain, where the physical numerical domain represents bed topography. Results reveal that the model can accurately predict spatially averaged velocity profiles, turbulence characteristics, shear stresses, and uniform flow depths. The analysis identified two distinct flow regions based on mean and turbulent flow profiles. Results show that the turbulent shear stress profiles provide a more accurate estimation of the bed shear stresses. Resistance coefficients (friction factor or Manning’s roughness coefficient) vary with Froude number and submergence ratio (depth divided by roughness height).

Fine Shallow Structures of Binchuan Basin Inverted From Receiver Functions and Implications for Basin Evolution

AbstractThe understanding of the velocity structure and basement morphology within the basin is crucial for seismic hazard mitigation and the study of basin evolution. To explore the intricate structure of the Binchuan Basin in western Yunnan, China, we deployed a linear dense array in the northern region of the Binchuan Basin, with inter‐station distance ranging from 50 to 100 m. We propose a novel receiver function processing workflow. Initially, we extracted coherent receiver functions based on the dense array, followed by a inversion of basement morphology, S‐wave velocities, and sedimentary layer's average Vp/Vs ratio jointly with frequency‐dependent teleseismic apparent shear velocity and receiver function waveforms. The results show that S‐wave velocity anomalies correspond to the unconsolidated sediments. The thickness of the sedimentary layer ranges from ∼0.5 to ∼0.75 km. The basement morphology suggests the basin is controlled by normal faults. Additionally, intra‐basin faults displayed higher fault displacement to length ratios (∼0.27) than most isolated normal faults (10−3–10−1), which may result from accumulated fault displacement due to interactions between fault segments. These results emphasize the significant role of N–S trending intra‐basin faults in basin evolution, suggesting that micro‐block rotations and transitional movements within the Northwestern Yunnan Rift Zone are primary mechanisms shaping the Binchuan Basin. We further proposed a multi‐stage model for the evolution of the Binchuan Basin. The robustness testing validates that the proposed processing flow is an effective approach to comprehensively image the basin velocity structure and the basement morphology.

3D Shear Velocity Structure of the Caribbean—Northwestern South America Subduction Zone From Ambient Noise and Ballistic Rayleigh Wave Tomography

3D Shear Velocity Structure of the Caribbean—Northwestern South America Subduction Zone From Ambient Noise and Ballistic Rayleigh Wave Tomography

Effects of joint persistence and testing conditions on cyclic shear behavior of en-echelon joints under CNS conditions

Cyclic shear tests on rock joints serve as a practical strategy for understanding the shear behavior of jointed rock masses under seismic conditions. We explored the cyclic shear behavior of en-echelon and how joint persistence and test conditions (initial normal stress, normal stiffness, shear velocity, and cyclic distance) influence it through cyclic shear tests under CNS conditions. The results revealed a through-going shear zone induced by cyclic loads, characterized by abrasive rupture surfaces and brecciated material. Key findings included that increased joint persistence enlarged and smoothened the shear zone, while increased initial normal stress and cyclic distance, and decreased normal stiffness and shear velocity, diminished and roughened the brecciated material. Shear strength decreased across shear cycles, with the most significant reduction in the initial shear cycle. After ten cycles, the shear strength damage factor D varied from 0.785 to 0.909. Shear strength degradation was particularly sensitive to normal stiffness and cyclic distance. Low joint persistence, high initial normal stress, high normal stiffness, slow shear velocity, and large cyclic distance were the most destabilizing combinations. Cyclic loads significantly compressed en-echelon joints, with compressibility highly dependent on normal stress and stiffness. The frictional coefficient initially declined and then increased under a rising cycle number. This work provides crucial insights for understanding and predicting the mechanical response of en-echelon joints under seismic conditions.

Comparison of novel physics-guided machine learning models with empirical equations for predicting longitudinal dispersion coefficient in diverse natural river systems

Accurate calculation of the longitudinal dispersion coefficient (LDC) is crucial for assessing river fate and transport processes. Traditional methods, including experimental, analytical, and empirical equations, each have their challenges, such as complexity and susceptibility to errors. This study introduces and evaluates advanced physics-guided machine learning (ML) models, including hybrid hydrodynamic-particle simulation (HHPS), physics-enhanced ML (PEML), and physics-regularized regression trees (PRRT). Our dataset includes hydraulic and geometric parameters collected from 50 diverse rivers across the US and the UK, encompassing variables such as river width (W), flow depth (H), average flow velocity (U), flow shear velocity (U*), concentration of the solute (μgL) and LDC (ε). Input parameters were dimensioned using the UU∗ and WHratios, while the dimensionless LDC (εHU∗) served as the model output. By embedding core hydrodynamic principles such as continuity, mass and momentum conservation, and Navier-Stokes equations into the ML algorithms, these models achieved high predictive accuracy. Specifically, the HHPS model reached a coefficient of determination of 0.997, and both PEML and PRRT demonstrated Nash-Sutcliffe efficiencies exceeding 0.950, significantly outperforming traditional empirical equations. Additionally, SHapley Additive exPlanations (SHAP) sensitivity analysis revealed the substantial influence of channel width and flow conditions on LDC predictions. Our study highlights the superior performance of physics-guided ML models in providing accurate and reliable tools for river water quality management, demonstrating their substantial improvements over traditional methods and their potential broader applications in environmental studies.

Identification of Kelvin-Helmholtz generated vortices in magnetised fluids

The Kelvin-Helmholtz Instability (KHI), arising from velocity shear across the magnetopause, plays a significant role in the viscous-like transfer of mass, momentum, and energy from the shocked solar wind into the magnetosphere. While the KHI leads to growth of surface waves and vortices, suitable detection methods for these applicable to magnetohydrodynamics (MHD) are currently lacking. A novel method is derived based on the well-established λ-family of hydrodynamic vortex identification techniques, which define a vortex as a local minimum in an adapted pressure field. The J×B Lorentz force is incorporated into this method by using an effective total pressure in MHD, including both magnetic pressure and a pressure-like part of the magnetic tension derived from a Helmholtz decomposition. The λMHD method is shown to comprise of four physical effects: vortical momentum, density gradients, fluid compressibility, and the rotational part of the magnetic tension. A local three-dimensional MHD simulation representative of near-flank magnetopause conditions (plasma β’s 0.5–5 and convective Mach numbers Mf∼0.4) under northward interplanetary magnetic field (IMF) is used to validate λMHD. Analysis shows it correlates well with hydrodynamic vortex definitions, though the level of correlation decreases with vortex evolution. Overall, vortical momentum dominates λMHD at all times. During the linear growth phase, density gradients act to oppose vortex formation. By the highly nonlinear stage, the formation of small-scale structures leads to a rising importance of the magnetic tension. Compressibility was found to be insignificant throughout. Finally, a demonstration of this method adapted to tetrahedral spacecraft observations is performed.