Academic literature on the topic 'Vertical conformal vector field'
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Journal articles on the topic "Vertical conformal vector field"
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Manev, Mancho. "Yamabe Solitons on Conformal Almost-Contact Complex Riemannian Manifolds with Vertical Torse-Forming Vector Field." Axioms 12, no. 1 (January 1, 2023): 44. http://dx.doi.org/10.3390/axioms12010044.
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A Yamabe soliton is considered on an almost-contact complex Riemannian manifold (also known as an almost-contact B-metric manifold), which is obtained by a contact conformal transformation of the Reeb vector field, its dual contact 1-form, the B-metric, and its associated B-metric. A case in which the potential is a torse-forming vector field of constant length on the vertical distribution determined by the Reeb vector field is studied. In this way, manifolds from one of the main classes of the studied manifolds are obtained. The same class contains the conformally equivalent manifolds of cosymplectic manifolds by the usual conformal transformation of the given B-metric. An explicit five-dimensional example of a Lie group is given, which is characterized in relation to the obtained results.
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Siddiqi, Mohd Danish, Ali Hussain Alkhaldi, Meraj Ali Khan, and Aliya Naaz Siddiqui. "Conformal η-Ricci Solitons on Riemannian Submersions under Canonical Variation." Axioms 11, no. 11 (October 27, 2022): 594. http://dx.doi.org/10.3390/axioms11110594.
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This research article endeavors to discuss the attributes of Riemannian submersions under the canonical variation in terms of the conformal η-Ricci soliton and gradient conformal η-Ricci soliton with a potential vector field ζ. Additionally, we estimate the various conditions for which the target manifold of Riemannian submersion under the canonical variation is a conformal η-Ricci soliton with a Killing vector field and a φ(Ric)-vector field. Moreover, we deduce the generalized Liouville equation for Riemannian submersion under the canonical variation satisfying by a last multiplier Ψ of the vertical potential vector field ζ and show that the base manifold of Riemanian submersion under canonical variation is an η Einstein for gradient conformal η-Ricci soliton with a scalar concircular field γ on base manifold. Finally, we illustrate an example of Riemannian submersions between Riemannian manifolds, which verify our results.
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Manev, Mancho. "Almost Riemann Solitons with Vertical Potential on Conformal Cosymplectic Contact Complex Riemannian Manifolds." Symmetry 15, no. 1 (December 30, 2022): 104. http://dx.doi.org/10.3390/sym15010104.
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Almost-Riemann solitons are introduced and studied on an almost contact complex Riemannian manifold, i.e., an almost-contact B-metric manifold, which is obtained from a cosymplectic manifold of the considered type by means of a contact conformal transformation of the Reeb vector field, its dual contact 1-form, the B-metric, and its associated B-metric. The potential of the studied soliton is assumed to be in the vertical distribution, i.e., it is collinear to the Reeb vector field. In this way, manifolds from the four main classes of the studied manifolds are obtained. The curvature properties of the resulting manifolds are derived. An explicit example of dimension five is constructed. The Bochner curvature tensor is used (for a dimension of at least seven) as a conformal invariant to obtain these properties and to construct an explicit example in relation to the obtained results.
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SOLEIMAN, A. "ENERGY β-CONFORMAL CHANGE IN FINSLER GEOMETRY." International Journal of Geometric Methods in Modern Physics 09, no. 04 (May 6, 2012): 1250029. http://dx.doi.org/10.1142/s0219887812500296.
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The present paper deals with an intrinsic generalization of the conformal change and energy β-change on a Finsler manifold (M.L.), namely the energy β-conformal change ([Formula: see text] with [Formula: see text]; [Formula: see text] being a concurrent π-vector field and σ(x) is a function on M). The relation between the two Barthel connections Γ and [Formula: see text], corresponding to this change, is found. This relation, together with the fact that the Cartan and the Barthel connections have the same horizontal and vertical projectors, enable us to study the energy β-conformal change of the fundamental linear connection in Finsler geometry: the Cartan connection, the Berwald connection, the Chern connection and the Hashiguchi connection. Moreover, the change of their curvature tensors is obtained. It should be pointed out that the present work is formulated in a prospective modern coordinate-free form.
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Tannukij, Lunchakorn, and Jae-Hyuk Oh. "Partially massless theory as a quantum gravity candidate." International Journal of Modern Physics A 36, no. 17 (June 2, 2021): 2150122. http://dx.doi.org/10.1142/s0217751x21501220.
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We study partially massless gravity theory (PM gravity theory) and suggest an alternative way to add higher order interaction vertices to the theory. Rather than introducing self-interaction vertices of the gravitational fields to the partially massless gravity action, we consider interactions with matter fields, since it is well known that addition of the self-interaction terms necessarily breaks the [Formula: see text] gauge symmetry that PM gravity theory enjoys. For the coupling with matter fields, we consider two different types of interaction vertices. The first one is given by an interaction Lagrangian density, [Formula: see text], where [Formula: see text] is the PM gravity field and [Formula: see text] is the stress-energy tensor of the matter fields. To retain the [Formula: see text] gauge symmetry, the matter fields also transform accordingly and it turns out that the transform must be nonlocal in this case. The second type of interaction is obtained by employing a gauge covariant derivative with the PM gravity field, where the PM gravity fields play a role of a gauge connection canceling the phase shift of the matter fields. We also study the actions and the equations of motion of the partially massless gravity fields. As expected, it shows 4 unitary degrees of freedom — 2 of them are traceless tensor modes and they are light-like fields and the other 2 are transverse vector modes and their dispersion relation changes as background space–time (de Sitter) evolutes. In the very early time, they are light-like but in the very late time, their velocities become a half of speed of light. The vector mode dispersion relation shows momentum-dependent behavior. In fact, the higher (lower) frequency modes show the faster (slower) velocity. We call this effect “conformal (or de Sitter) prism”. We suggest their quantization, compute Hamiltonians to present their exited quanta and construct their free propagators.
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Khan, Suhail, Amjad Mahmood, and Ahmad T. Ali. "Concircular vector fields for Kantowski–Sachs and Bianchi type-III spacetimes." International Journal of Geometric Methods in Modern Physics 15, no. 08 (June 22, 2018): 1850126. http://dx.doi.org/10.1142/s0219887818501268.
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This paper intends to obtain concircular vector fields (CVFs) of Kantowski–Sachs and Bianch type-III spacetimes. For this purpose, ten conformal Killing equations and their general solution in the form of conformal Killing vector fields (CKVFs) are derived along with their conformal factors. The obtained conformal Killing vector fields are then placed in Hessian equations to obtain the final form of concircular vector fields. The existence of concircular symmetry imposes restrictions on the metric functions. The conditions imposing restrictions on these metric functions are obtained as a set of integrability conditions. It is shown that Kantowski–Sachs and Bianchi type-III spacetimes admit four-, six-, or fifteen-dimensional concircular vector fields. It is established that for Einstein spaces, every conformal Killing vector field is a concircular vector field. Moreover, it is explored that every concircular vector field obtained here is also a conformal Ricci collineation.
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Klaycham, Karun, Chainarong Athisakul, and Somchai Chucheepsakul. "Nonlinear Response of Marine Riser with Large Displacement Excited by Top-End Vessel Motion using Penalty Method." International Journal of Structural Stability and Dynamics 20, no. 04 (March 24, 2020): 2050052. http://dx.doi.org/10.1142/s0219455420500522.
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A marine riser operated in a deep-water field could be substantially affected by large amounts of movement of the floating platform, which is more complicated and very challenging to analyze. This paper presents a mathematical model involving nonlinear dynamic response analysis of a marine riser caused by sways and heave motions at the top end, which are treated as the constraint conditions. The nonlinear equation of motion, arising from the nonlinearity of the ocean current and wave loadings, is derived and written in general matrix form using the finite element method. The excitation caused by platform movement is imposed on the riser system through the time-dependent constrained condition using the penalty method. The advantages of this method are that it is easily implemented on the nonlinear equation of motion and it requires no additional unknown variable, and thus consumes less computational time. By this method, the stiffness matrix and the force vector of the system are then modified, enforcing top-end vessel motion. The dynamic responses are evaluated by using numerical time integration based on Newmark’s method with direct iteration. The effects of the oscillation frequency of top-end vessel sway and heave motions on the nonlinear dynamic characteristics of the riser are investigated. The numerical results reveal that the riser responses to the top-end vessel excitation behave like a periodic motion, which is conformable to the characteristics of vessel movements. The increase in the oscillation frequency of the top-end vessel increases the maximum displacement amplitude for both the horizontal and vertical directions. The directional motion of the vessel also significantly influences the response amplitude of the riser.
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De, Uday Chand, Young Jin Suh, and Sudhakar K. Chaubey. "Conformal vector fields on almost co-Kähler manifolds." Mathematica Slovaca 71, no. 6 (December 1, 2021): 1545–52. http://dx.doi.org/10.1515/ms-2021-0070.
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Abstract In this paper, we characterize almost co-Kähler manifolds with a conformal vector field. It is proven that if an almost co-Kähler manifold has a conformal vector field that is collinear with the Reeb vector field, then the manifold is a K-almost co-Kähler manifold. It is also shown that if a (κ, μ)-almost co-Kähler manifold admits a Killing vector field V, then either the manifold is K-almost co-Kähler or the vector field V is an infinitesimal strict contact transformation, provided that the (1,1) tensor h remains invariant under the Killing vector field.
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MANOFF, S. "CONFORMAL DERIVATIVE AND CONFORMAL TRANSPORTS OVER $\bm{({\bar L}_n,g)}$-SPACES." International Journal of Modern Physics A 15, no. 05 (February 20, 2000): 679–95. http://dx.doi.org/10.1142/s0217751x00000343.
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Transports preserving the angle between two contravariant vector fields but changing their lengths proportional to their own lengths are introduced as "conformal" transports and investigated over [Formula: see text]-spaces. They are more general than the Fermi–Walker transports. In an analogous way as in the case of Fermi–Walker transports a conformal covariant differential operator and its conformal derivative are defined and considered over [Formula: see text]-spaces. Different special types of conformal transports are determined inducing also Fermi–Walker transports for orthogonal vector fields as special cases. Conditions under which the length of a non-null contravariant vector field could swing as a homogeneous harmonic oscillator are established. The results obtained regardless of any concrete field (gravitational) theory could have direct applications in such types of theories.
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Li, Yanlin, Santu Dey, Sampa Pahan, and Akram Ali. "Geometry of conformal η-Ricci solitons and conformal η-Ricci almost solitons on paracontact geometry." Open Mathematics 20, no. 1 (January 1, 2022): 574–89. http://dx.doi.org/10.1515/math-2022-0048.
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Abstract We prove that if an η \eta -Einstein para-Kenmotsu manifold admits a conformal η \eta -Ricci soliton then it is Einstein. Next, we proved that a para-Kenmotsu metric as a conformal η \eta -Ricci soliton is Einstein if its potential vector field V V is infinitesimal paracontact transformation or collinear with the Reeb vector field. Furthermore, we prove that if a para-Kenmotsu manifold admits a gradient conformal η \eta -Ricci almost soliton and the Reeb vector field leaves the scalar curvature invariant then it is Einstein. We also construct an example of para-Kenmotsu manifold that admits conformal η \eta -Ricci soliton and satisfy our results. We also have studied conformal η \eta -Ricci soliton in three-dimensional para-cosymplectic manifolds.
Dissertations / Theses on the topic "Vertical conformal vector field"
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ANSELLI, ANDREA. "PHI-CURVATURES, HARMONIC-EINSTEIN MANIFOLDS AND EINSTEIN-TYPE STRUCTURES." Doctoral thesis, Università degli Studi di Milano, 2020. http://hdl.handle.net/2434/703786.
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The aim of this thesis is to study the geometry of a Riemannian manifold M, with a special structure, called Einstein-type structure, depending on 3 real parameters, a smooth map phi into a target Riemannian manifold N, and a smooth function, called potential function, on M itself. We will occasionally let some of the parameters be smooth functions. The setting generalizes various previously studied situations:, Ricci solitons, almost Ricci-solitons, Ricci-harmonic solitons, quasi-Einstein manifolds and so on. By taking a constant potential function those structures reduces to harmonic-Einstein manifolds, that are a generalization of Einstein manifolds. The main ingredient of our analysis is the study of certain modified curvature tensors on M related to the map phi, called phi-curvatures, obtaining, for instance, their transformation laws under a conformal change of metric, and to develop a series of results for harmonic-Einstein manifolds that parallel those obtained for Einstein manifolds some times ago and also in the very recent literature. Einstein-type structures may be obtained, for some special values of the parameters involved, by a conformal deformation of a harmonic-Einstein manifold or even as the base of a warped product harmonic-Einstein manifold. The latter fact applies not only in the Riemannian but also in the Lorentzian setting and thus some Einstein-type structures are connected with solutions of the Einstein field equations, which are of particular interest in General Relativity. The main result of the thesis is the locally characterization, via a couple of integrability conditions and mild assumptions on the potential function, of Einstein-type structures with vanishing phi-Bach curvature (in the direction of the potential) as a warped product with harmonic-Einstein base and with an open real interval as fibre, extending in a very non trivial way a recent result for Bach flat Ricci solitons. Moreover the map phi depends only on the base of the warped product and not on the fibre . We also consider rigidity, triviality and non-existence results, both in the compact and non-compact cases. This is done via integral formulas and, in the non-compact case, via analytical tools, like the weak maximum principle and the classical results of Obata, Tashiro, Kanai.
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Potter, Harrison D. P. "On Conformal Mappings and Vector Fields." Marietta College Honors Theses / OhioLINK, 2008. http://rave.ohiolink.edu/etdc/view?acc_num=marhonors1210888378.
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Simsir, Muazzez Fatma. "Conformal Vector Fields With Respect To The Sasaki Metric Tensor Field." Phd thesis, METU, 2005. http://etd.lib.metu.edu.tr/upload/12605857/index.pdf.
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On the tangent bundle of a Riemannian manifold the most natural choice of metric tensor field is the Sasaki metric. This immediately brings up the question of infinitesimal symmetries associated with the inherent geometry of the tangent bundle arising from the Sasaki metric. The elucidation of the form and the classification of the Killing vector fields have already been effected by the Japanese school of Riemannian geometry in the sixties. In this thesis we shall take up the conformal vector fields of the Sasaki metric with the help of relatively advanced techniques.
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Pham, Truong Xuan. "Peeling et scattering conforme dans les espaces-temps de la relativité générale." Thesis, Brest, 2017. http://www.theses.fr/2017BRES0034/document.
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Nous étudions l’analyse asymptotique en relativité générale sous deux aspects: le peeling et le scattering (diffusion) conforme. Le peeling est construit pour les champs scalaires linéaire et non-linéaires et pour les champs de Dirac en espace-temps de Kerr (qui est non-stationnaire et à symétrie simplement axiale), généralisant les travaux de L. Mason et J-P. Nicolas (2009, 2012). La méthode des champs de vecteurs (estimations d’énergie géométriques) et la technique de compactification conforme sont développées. Elles nous permettent de formuler les définitions du peeling à tous ordres et d’obtenir les données initiales optimales qui assurent ces comportements. Une théorie de la diffusion conforme pour les équations de champs sans masse de spîn n/2 dans l’espace-temps de Minkowski est construite.En effectuant les compactifications conformes (complète et partielle), l’espace-temps est complété en ajoutant une frontière constituée de deux hypersurfaces isotropes représentant respectivement les points limites passés et futurs des géodésiques de type lumière. Le comportement asymptotique des champs s’obtient en résolvant le problème de Cauchy pour l’équation rééchelonnée et en considérant les traces des solutions sur ces bords. L’inversibilité des opérateurs de trace, qui associent le comportement asymptotique passé ou futur aux données initiales, s’obtient en résolvant le problème de Goursat sur le bord conforme. L’opérateur de diffusion conforme est alors obtenu par composition de l’opérateur de trace futur avec l’inverse de l’opérateur de trace passéThis work explores two aspects of asymptotic analysis in general relativity: peeling and conformal scattering.On the one hand, the peeling is constructed for linear and nonlinear scalar fields as well as Dirac fields on Kerr spacetime, which is non-stationary and merely axially symmetric. This generalizes the work of L. Mason and J-P. Nicolas (2009, 2012). The vector field method (geometric energy estimates) and the conformal technique are developed. They allow us to formulate the definition of the peeling at all orders and to obtain the optimal space of initial data which guarantees these behaviours. On the other hand, a conformal scattering theory for the spin-n/2 zero rest-mass equations on Minkowski spacetime is constructed. Using the conformal compactifications (full and partial), the spacetime is completed with two null hypersurfaces representing respectively the past and future end points of null geodesics. The asymptotic behaviour of fields is then obtained by solving the Cauchy problem for the rescaled equation and considering the traces of the solutions on these hypersurfaces. The invertibility of the trace operators, that to the initial data associate the future or past asymptotic behaviours, is obtained by solving the Goursat problem on the conformal boundary. The conformal scattering operator is then obtained by composing the future trace operator with the inverse of the past trace operator
Books on the topic "Vertical conformal vector field"
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Advanced engineering mathematics. 2nd ed. Upper Saddle River, N.J: Prentice Hall, 1998.
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Advanced engineering mathematics. Englewood Cliffs, N.J: Prentice Hall, 1988.
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Advanced engineering mathematics. Boston, Mass: Pearson Custom Pub., 1998.
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editor, Donagi Ron, Katz Sheldon 1956 editor, Klemm Albrecht 1960 editor, and Morrison, David R., 1955- editor, eds. String-Math 2012: July 16-21, 2012, Universität Bonn, Bonn, Germany. Providence, Rhode Island: American Mathematical Society, 2015.
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Mann, Peter. Linear Algebra. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0037.
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This chapter is key to the understanding of classical mechanics as a geometrical theory. It builds upon earlier chapters on calculus and linear algebra and frames theoretical physics in a new and useful language. Although some degree of mathematical knowledge is required (from the previous chapters), the focus of this chapter is to explain exactly what is going on, rather than give a full working knowledge of the subject. Such an approach is rare in this field, yet is ever so welcome to newcomers who are exposed to this material for the first time! The chapter discusses topology, manifolds, forms, interior products, pullback and pushforward, as well as tangent bundles, cotangent bundles, jet bundles and principle bundles. It also discusses vector fields, integral curves, flow, exterior derivatives and fibre derivatives. In addition, Lie derivatives, Lie brackets, Lie algebra, Lie–Poisson brackets, vertical space, horizontal space, groups and algebroids are explained.
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Mann, Peter. Differential Geometry. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0038.
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This chapter is key to the understanding of classical mechanics as a geometrical theory. It builds upon earlier chapters on calculus and linear algebra and frames theoretical physics in a new and useful language. Although some degree ofmathematical knowledge is required (from the previous chapters), the focus of this chapter is to explain exactlywhat is going on, rather than give a full working knowledge of the subject. Such an approach is rare in this field, yet is ever so welcome to newcomers who are exposed to this material for the first time! The chapter discusses topology, manifolds, forms, interior products, pullback and pushforward, as well as tangent bundles, cotangent bundles, jet bundles and principle bundles. It also discusses vector fields, integral curves, flow, exterior derivatives and fibre derivatives. In addition, Lie derivatives, Lie brackets, Lie algebra, Lie–Poisson brackets, vertical space, horizontal space, groups and algebroids are explained.
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Integrability, Quantization, and Geometry. American Mathematical Society, 2021.
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Ślusarski, Marek. Metody i modele oceny jakości danych przestrzennych. Publishing House of the University of Agriculture in Krakow, 2017. http://dx.doi.org/10.15576/978-83-66602-30-4.
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The quality of data collected in official spatial databases is crucial in making strategic decisions as well as in the implementation of planning and design works. Awareness of the level of the quality of these data is also important for individual users of official spatial data. The author presents methods and models of description and evaluation of the quality of spatial data collected in public registers. Data describing the space in the highest degree of detail, which are collected in three databases: land and buildings registry (EGiB), geodetic registry of the land infrastructure network (GESUT) and in database of topographic objects (BDOT500) were analyzed. The results of the research concerned selected aspects of activities in terms of the spatial data quality. These activities include: the assessment of the accuracy of data collected in official spatial databases; determination of the uncertainty of the area of registry parcels, analysis of the risk of damage to the underground infrastructure network due to the quality of spatial data, construction of the quality model of data collected in official databases and visualization of the phenomenon of uncertainty in spatial data. The evaluation of the accuracy of data collected in official, large-scale spatial databases was based on a representative sample of data. The test sample was a set of deviations of coordinates with three variables dX, dY and Dl – deviations from the X and Y coordinates and the length of the point offset vector of the test sample in relation to its position recognized as a faultless. The compatibility of empirical data accuracy distributions with models (theoretical distributions of random variables) was investigated and also the accuracy of the spatial data has been assessed by means of the methods resistant to the outliers. In the process of determination of the accuracy of spatial data collected in public registers, the author’s solution was used – resistant method of the relative frequency. Weight functions, which modify (to varying degree) the sizes of the vectors Dl – the lengths of the points offset vector of the test sample in relation to their position recognized as a faultless were proposed. From the scope of the uncertainty of estimation of the area of registry parcels the impact of the errors of the geodetic network points was determined (points of reference and of the higher class networks) and the effect of the correlation between the coordinates of the same point on the accuracy of the determined plot area. The scope of the correction was determined (in EGiB database) of the plots area, calculated on the basis of re-measurements, performed using equivalent techniques (in terms of accuracy). The analysis of the risk of damage to the underground infrastructure network due to the low quality of spatial data is another research topic presented in the paper. Three main factors have been identified that influence the value of this risk: incompleteness of spatial data sets and insufficient accuracy of determination of the horizontal and vertical position of underground infrastructure. A method for estimation of the project risk has been developed (quantitative and qualitative) and the author’s risk estimation technique, based on the idea of fuzzy logic was proposed. Maps (2D and 3D) of the risk of damage to the underground infrastructure network were developed in the form of large-scale thematic maps, presenting the design risk in qualitative and quantitative form. The data quality model is a set of rules used to describe the quality of these data sets. The model that has been proposed defines a standardized approach for assessing and reporting the quality of EGiB, GESUT and BDOT500 spatial data bases. Quantitative and qualitative rules (automatic, office and field) of data sets control were defined. The minimum sample size and the number of eligible nonconformities in random samples were determined. The data quality elements were described using the following descriptors: range, measure, result, and type and unit of value. Data quality studies were performed according to the users needs. The values of impact weights were determined by the hierarchical analytical process method (AHP). The harmonization of conceptual models of EGiB, GESUT and BDOT500 databases with BDOT10k database was analysed too. It was found that the downloading and supplying of the information in BDOT10k creation and update processes from the analyzed registers are limited. An effective approach to providing spatial data sets users with information concerning data uncertainty are cartographic visualization techniques. Based on the author’s own experience and research works on the quality of official spatial database data examination, the set of methods for visualization of the uncertainty of data bases EGiB, GESUT and BDOT500 was defined. This set includes visualization techniques designed to present three types of uncertainty: location, attribute values and time. Uncertainty of the position was defined (for surface, line, and point objects) using several (three to five) visual variables. Uncertainty of attribute values and time uncertainty, describing (for example) completeness or timeliness of sets, are presented by means of three graphical variables. The research problems presented in the paper are of cognitive and application importance. They indicate on the possibility of effective evaluation of the quality of spatial data collected in public registers and may be an important element of the expert system.
Book chapters on the topic "Vertical conformal vector field"
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Beauville, Arnaud. "Vector Bundles on Riemann Surfaces and Conformal Field Theory." In Algebraic and Geometric Methods in Mathematical Physics, 145–66. Dordrecht: Springer Netherlands, 1996. http://dx.doi.org/10.1007/978-94-017-0693-3_7.
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Ueno, Kenji. "On conformal field theory." In Vector Bundles in Algebraic Geometry, 283–345. Cambridge University Press, 1995. http://dx.doi.org/10.1017/cbo9780511569319.011.
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Buchbinder, Iosif L., and Ilya L. Shapiro. "Classical fields in curved spacetime." In Introduction to Quantum Field Theory with Applications to Quantum Gravity, 294–313. Oxford University Press, 2021. http://dx.doi.org/10.1093/oso/9780198838319.003.0012.
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This chapter discusses classical fields in an arbitrary Riemann spacetime. General considerations are followed by the formulation of scalar fields with non-minimal coupling. Spontaneous symmetry breaking in curved space is shown to provide the induced gravity action with a cosmological constant. The construction of spinor fields in curved spacetime is based on the notions of group theory from Part I and on the local Lorentz invariance. Massless vector fields (massless vector gauge fields) are described and the interactions between scalar, fermion and gauge fields formulated. A detailed discussion of classical conformal transformations and conformal symmetry for both matter fields and vacuum action is also provided.
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Tu, Loring W. "Fundamental Vector Fields." In Introductory Lectures on Equivariant Cohomology, 87–96. Princeton University Press, 2020. http://dx.doi.org/10.23943/princeton/9780691191751.003.0011.
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This chapter addresses fundamental vector fields. The concept of a connection on a principal bundle is essential in the construction of the Cartan model. To define a connection on a principal bundle, one first needs to define the fundamental vector fields. When a Lie group acts smoothly on a manifold, every element of the Lie algebra of the Lie group generates a vector field on the manifold called a fundamental vector field. On a principal bundle, the fundamental vectors are precisely the vertical tangent vectors. In general, there is a relation between zeros of fundamental vector fields and fixed points of the group action. Unless specified otherwise (such as on a principal bundle), a group action is assumed to be a left action.
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Canarutto, Daniel. "Bundle Prolongations and Connections." In Gauge Field Theory in Natural Geometric Language, 3–24. Oxford University Press, 2020. http://dx.doi.org/10.1093/oso/9780198861492.003.0001.
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A synthetic introduction to the fundamental notions of differential geometry, including tangent, vertical and jet prolongations of fibered manifolds; the Frölicher-Nijenhuis bracket; connections of fibered manifolds and, in particular, linear connections of vector bundles and tangent bundles; the covariant differential of vector-valued forms as a generalisation of the standard covariant derivative.
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Canarutto, Daniel. "Quantum Bundles." In Gauge Field Theory in Natural Geometric Language, 213–24. Oxford University Press, 2020. http://dx.doi.org/10.1093/oso/9780198861492.003.0015.
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A quantum bundle, namely a bundle whose sections are quantum fields, is defined as a classical vector bundle tensorialised by a suitable operator algebra related to a bundle of quantum states. The fundamental differential geometric notions for quantum bundles, including tangent, vertical and jet prolongations, and connections, can be conveniently introduced in terms F-smoothness. A short introduction to anti-fields and Batalin-Vilkovisky algebra naturally fits into this scheme.
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Steward, David R. "Analytic Elements from Complex Functions." In Analytic Element Method, 103–64. Oxford University Press, 2020. http://dx.doi.org/10.1093/oso/9780198856788.003.0003.
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The mathematical functions associated with analytic elements may be formulated using a complex function $\Omega$ of a complex variable ${\zcomplex}$. Complex formulation of analytic elements is introduced in Section 3.1 for exact solutions obtained by embedding point elements that generate divergence, circulation, or velocity within a uniform vector field. Influence functions for analytic elements with circular geometry are obtained using Taylor and Laurent series expansions in Section 3.2, and conformal mapping extends this formulation to analytic elements with the geometry of ellipses (Section 3.3). The Courant's Sewing Theorem is employed in Section 3.4 to develop solutions for interface conditions across straight line segments, and the Joukowsky transformation extends methods to circular arcs and wings (Section 3.5), which satisfy a Kutta condition of non-singular vector field at their trailing edges. Vector fields with spatially distributed divergence and curl are formulated using the complex variable ${\zcomplex}$ with its complex conjugate $\overline{\zcomplex}$ in Section 3.6, and the complex conjugate is further employed in the Kolosov formulas (Section 3.7) to solve force deformation problems for analytic elements with traction or displacement specified boundary conditions.
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Steward, David R. "Analytic Elements from Singular Integral Equations." In Analytic Element Method, 227–84. Oxford University Press, 2020. http://dx.doi.org/10.1093/oso/9780198856788.003.0005.
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Solutions to interface problems may be developed using analytic elements with mathematical solutions to the Laplace equation developed by singular integral equations. This formulation leads to solutions with discontinuities occurring across line segments, where the potential or stream function is discontinuous across double layer elements in Section 5.2, and the normal or tangential component of the vector field is discontinuous across single layer elements in Section 5.3. Examples illustrate a broad range of solutions to interface conditions possible with these elements. Series expansions are used to represent the far-field at larger distances from elements in Section 5.4, which leads to higher-order elements with nearly exact solutions and also provides a simpler representation for contiguous strings of adjacent elements. Such strings of elements are used with polygon elements in 5.5 to solve conditions along the interfaces of heterogeneities, and to provide a common series expansion to represent the far-field for a group of neighboring elements. Methods are extended to analytic elements with curvilinear geometry using conformal mappings (Section 5.6) and to three-dimensional fields in Section 5.7.
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Zaneveld, J. Ronald V. "Optical Closure: from Theory to Measurement." In Ocean Optics. Oxford University Press, 1994. http://dx.doi.org/10.1093/oso/9780195068436.003.0007.
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The intensity and spectrum of the light in the ocean have a major influence on the biological processes. These processes in turn determine the concentrations of much of the suspended and dissolved matter in the ocean, which affect the way in which the light is scattered and absorbed. These relationships can perhaps be most easily illustrated schematically as in Fig. 3-1. At the upper boundary we have the sun and sky radiances and the surface transmission conditions that combine to provide the energy entering through the surface. The ocean itself contains the vertical structure of those optical properties that do not depend on the structure of the light field, but depend only on the properties of the suspended and dissolved materials: the absorption coefficient a(λ,z), the beam attenuation coefficient c(λ,z), and the volume scattering function β(θ,λ,z). These are known as inherent optical properties, because they do not depend on the source radiance field (Preisendorfer, 1976). They are a function only of the suspended and dissolved materials in the water, and of the water itself. How does the vertical structure of the inherent optical properties affect the vertical structure of the radiance field in the ocean itself? This is the problem of radiative transfer in which we try to predict the intensity, direction, and spectrum of the light (spectral radiance) in the ocean, based on a set of given inherent optical properties. Those properties of the light field in the ocean that depend on the radiance are known as the apparent optical properties. Radiance field integrals, such as the vector irradiance, E(λ,z), the scalar irradiance E0(λ,z), and their attenuation coefficients are also apparent optical properties.
Conference papers on the topic "Vertical conformal vector field"
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Heimerl, Joseph, and Moble Benedict. "Flow-Field and Force Measurements on a Cycloidal Rotor Blade in Forward Flight." In Vertical Flight Society 78th Annual Forum & Technology Display. The Vertical Flight Society, 2022. http://dx.doi.org/10.4050/f-0078-2022-17450.
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The paper investigates the unsteady forces and flowfield of a cycloidal rotor blade undergoing forward flight motion through water tunnel experiments. A particle image velocimetry (PIV) system is used in conjunction with an instrumented blade to measure both the two-dimensional flow velocity around the blade and the fluid dynamic forces. The flow-field studies reveal the formation and shedding of strong leading-edge vortices in both the frontal and real halves of the circular blade trajectory, which plays a key role in generating lift as observed from the blade force measurements. Increasing forward speed diminishes the size and strength of these leading-edge vortices due to the reduction in angle of attack, which reflects in the blade forces. With pitch kinematics symmetric between frontal and rear halves of the cycle the blade produced significantly higher forces in the real half compared to the frontal half, which was attributed to the dynamic virtual camber and the differences in relative velocity. The thrust vector was observed to be highly sensitive to both pitch phase offset and spin direction at high advance ratios and required a phase angle around 40-degree for positive propulsive force and lift. At very high advance ratios the blade extracts power from the flow over a large region in the frontal half.
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Deng, Yangbo, Jingming Dong, and Xu Zhen. "Study on Flow Field Characteristics of Low Swirl Injector." In ASME 2014 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/imece2014-37423.
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The flow characteristics of six kinds of LSIs, which are designed by different pore sizes in the center channel screen, are analyzed. The velocity vectors on the spanwise sections and the vertical sections in a channel at atmospheric condition are captured using a Particle Image Velocimetry (PIV) system. The swirl number of the airflow through the LSIs ranges from 0.5 to 0.58, and the inlet velocity is kept at 14m/s. The results show that the swirl number under a threshold can form low swirl flow. The velocity vector distribution of the low swirl flow is a diffuse shape without recirculation, and has the self-similar characteristic. The separation of low speed flow in the center zone and the high speed in the annulus zone generates the unique “W” shape distribution of the through the LSI. With the swirl number increasing, the area of the low vorticity zone decreases, and the vorticity value of the flow in the outer annular zone increases.
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Rooij, Anouk. "Time-Resolved Stereo PIV Measurements of a Cyclorotor in Hover." In Vertical Flight Society 77th Annual Forum & Technology Display. The Vertical Flight Society, 2021. http://dx.doi.org/10.4050/f-0077-2021-16717.
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A cyclorotor is a thrust generating device that can produce thrust in any direction. CycloTech develops cyclorotors which can be used on drones or air taxis. To improve the understanding of a cyclorotor in hover, time-resolved stereo PIV measurements have been performed at an in-house test bench. Part of the flow field on the cyclorotor’s mid-plane has been visualised. In this work the instantaneous velocity vector fields and the spanwise velocity contours were studied for two different RPMs and two maximum pitch amplitudes. In addition, the centers and sizes of the vortices were detected and analysed for a single cycle of the cyclorotor. From the flow fields a very large influence of the pitch amplitude on the results was observed; at large pitch amplitude vortices developed much earlier and more vortices existed. The flow direction inside the cyclorotor also changed. The RPM mainly influences the velocity magnitude outside and inside of the cyclorotor.
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Oliver, Sean M., Dmitro J. Martynowych, Matthew J. Turner, David A. Hopper, Ronald L. Walsworth, and Edlyn V. Levine. "Vector Magnetic Current Imaging of an 8 nm Process Node Chip and 3D Current Distributions Using the Quantum Diamond Microscope." In ISTFA 2021. ASM International, 2021. http://dx.doi.org/10.31399/asm.cp.istfa2021p0096.
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Abstract The adoption of 3D packaging technology necessitates the development of new approaches to failure electronic device analysis. To that end, our team is developing a tool called the quantum diamond microscope (QDM) that leverages an ensemble of nitrogen vacancy (NV) centers in diamond, achieving vector magnetic imaging with a wide field-of-view and high spatial resolution under ambient conditions. Here, we present the QDM measurement of 2D current distributions in an 8-nm flip chip IC and 3D current distributions in a multi-layer PCB. Magnetic field emanations from the C4 bumps in the flip chip dominate the QDM measurements, but these prove to be useful for image registration and can be subtracted to resolve adjacent current traces in the die at the micron scale. Vias in 3D ICs display only Bx and By magnetic fields due to their vertical orientation and are difficult to detect with magnetometers that only measure the Bz component (orthogonal to the IC surface). Using the multi-layer PCB, we show that the QDM’s ability to simultaneously measure Bx, By, and Bz is advantageous for resolving magnetic fields from vias as current passes between layers. We also show how spacing between conducting layers is determined by magnetic field images and how it agrees with the design specifications of the PCB. In our initial efforts to provide further z-depth information for current sources in complex 3D circuits, we show how magnetic field images of individual layers can be subtracted from the magnetic field image of the total structure. This allows for isolation of signal layers and can be used to map embedded current paths via solution of the 2D magnetic inverse. In addition, the paper also discusses the use of neural networks to identify 2D current distributions and its potential for analyzing 3D structures.
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Shiratori, Takahisa, Yuji Tasaka, Yuichi Murai, Kazuya Oyama, Ichiro Kumagai, and Yasushi Takeda. "Flow of a Viscoelastic Fluid Around a Falling Sphere." In ASME-JSME-KSME 2011 Joint Fluids Engineering Conference. ASMEDC, 2011. http://dx.doi.org/10.1115/ajk2011-11003.
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Velocity vector fields around a falling sphere in a 1.0 wt % polyacrylamide (PAA) solution are obtained on a vertical cross section by particle image velocimetry (PIV). PAA solution is known as non-Newtonian fluid, which has shear thinning and viscoelastic property. Strain rate tensor fields and deformation fields are calculated from the velocity vector fields in order to visualize the dynamic behavior of the fluid quantitatively. In velocity vector field, two typical flow regions are observed in the wake of the sphere: approaching flow to the sphere, rising flow called “negative wake” [1]. Results show that the strain rate tensor field gives fluid strain at the approaching flow region and the edge of the negative wake. Furthermore deformation history of one portion of the fluid shows that fluid is strained in the approaching flow region, and the strain rate at the edge of the negative wake represents their recovery to the original status of the fluid in the moving frame.
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Zhao, Bo, Fu-Ping Gao, and Rong-Yu Kang. "Numerical Investigation on Bearing Capacity of a Pipeline on Clayey Soils." In ASME 2010 29th International Conference on Ocean, Offshore and Arctic Engineering. ASMEDC, 2010. http://dx.doi.org/10.1115/omae2010-20200.
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The bearing capacity of a pipeline foundation is crucial for the pipeline stability design. It is usually inappropriate to analyze the bearing capacity for the pipeline with special circular section directly by employing the theory for conventional rectangular strip footings. In this study, the ultimate loads of the pipeline on clayey soils are investigated numerically. A plane-strain finite element model is proposed to simulate the quasi-static process of the pipeline penetrating into the soil, in which the adaptive-grid technique and the ‘contact-pair’ algorithm are employed, and the Drucker-Prager constitutive model is used for modeling the soil plasticity. Based on the proposed numerical model, the development of soil plastic zone and the incremental-displacement vector field beneath the pipeline are examined numerically. It is indicated that, according to the obtained pipeline vertical load-displacement curves, concurrently referring to the plastic strain field and/or the soil incremental-displacement vector field, the shear failure type (e.g., general shear failure, punching shear failure) and the collapse loads can be thereby determined. The present numerical results match well with the analytical solutions of slip-line theory in plasticity mechanics.
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Stoddard, N. R., C. D. Richards, C. T. Crowe, and T. R. Troutt. "Turbulence Modulation in an Initially Quiescent Two-Phase Flow at High Volume Fractions Using Stereoscopic PIV." In ASME/JSME 2003 4th Joint Fluids Summer Engineering Conference. ASMEDC, 2003. http://dx.doi.org/10.1115/fedsm2003-45212.
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Turbulence data for 5 mm solid, spherical particles falling at terminal velocity through an initially quiescent vertical water channel are obtained using a Scheimpflu¨g stereo-PIV system. The system was successfully implemented into the flow facility and optimized to provide 3-D velocity field information for volume fractions at 2.71% and 5.24%. Vector validation played a significant role in reducing the data to obtain turbulence intensities as a function of particle fall time. The system was shown to be capable of measuring in-plane and out-of-plane displacements (velocities) with an error of less than 0.6%, despite blockage effects by the particles and inherent experimental errors. Since similar data in current literature is sparse, a popular length-scale model is provided in support of the data.
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Eskew, Rhea T., Charles F. Stromeyer, and Richard E. Kronauer. "The Constancy of Equiluminant Red-Green Thresholds Examined in Two Color Spaces." In Advances in Color Vision. Washington, D.C.: Optica Publishing Group, 1992. http://dx.doi.org/10.1364/acv.1992.sac2.
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Chromatic detection by the L or long-wavelength cones and M or middle-wavelength cones can be conveniently represented in the diagram shown in Fig. 1. The vertical axis represents the change in the quantal catch of the M cones, produced by a test light modulation, divided by the quantal catch in the M cones produced by all of the steady field components. The horizontal axis represents the analogous quantity for the L cones. Fig. 2 shows the simple model which justifies this representation. The cone classes adapt independently of one another before being combined into the red-green chromatic and luminance mechanisms. The model requires that adaptation take place within each cone pathway, but not necessarily in the cones themselves. Adaptation also occurs at the opponent stage. The linear combinations of cone contrasts shown in the model of Fig. 2 imply that detection contours in cone contrast space should be four straight line segments (only three of which are shown in Fig. 1). Each vector in Fig. 1 represents the threshold for a particular ratio of red to green light modulations, corresponding to a particular ratio of L-to-M cone modulations. The 45° vector represents a test stimulus with the same chromaticity as the adapting field. If there is a detection mechanism which combines cone contrasts linearly and has a response which is independent of luminance modulation, it must have a detection contour which is parallel to the 45° line - that is, it must have a slope of +1.0 — regardless of the adapting chromaticity. No such simple constraint governs the slope of the luminance detection contour. The 45°-225° direction will be here referred to as the “equichromatic” direction.
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Hu, Y. Z., W. W. Chow, and S. U. Koch. "The role of the gain medium in semiconductor laser instabilities." In OSA Annual Meeting. Washington, D.C.: Optica Publishing Group, 1992. http://dx.doi.org/10.1364/oam.1992.thcc2.
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We will show in this paper that the relevant semiconductor gain medium properties giving rise to spatial and temporal instabilities are the differential gain and refractive index, dG/dN and dδn/dN, respectively. The latter is often replaced by the antiguiding or linewidth enhancement factor, α= K0(dδn/dN)/(dG/dN), where K0 is the laser field wave vector in vacuum. A purpose of our paper is to point out the relationships among G,dG/dN, dδn/dN, and α. Owing to these relationships, the common practice of treating these quantities as independent parameters in the modeling of semiconductor lasers is inconsistent. On the other hand, the experimentally obtainable combinations of G, dG/dN, dδn/dN, and αcover a very wide range of situations. This is because one has the choice of bulk and quantum well gain media, and for a quantum well medium, present growth techniques make available a wide range of well widths and strains. This flexibility makes the semiconductor laser an excellent tool for studying laser instabilities. We will use a gain model, which contains the many-body carrier-carrier interactions and the band mixing effects due to quantum confinement and strain, to illustrate the different experimental conditions reachable with a semiconductor laser. As an example, we will present the results of our investigation on the behavior of a vertical cavity surface emitting laser (VCSEL) in the presence of an injected optical field, where we observed relaxation oscillation, sub-harmonic bifurcation, and chaos under certain operating conditions.
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Wang, Xifeng, Kenta Mizushiri, Hiroshi Yokoyama, and Akiyoshi Iida. "Wavenumber-Frequency Spectrum Analysis of Pressure Fields Around an Automobile." In ASME-JSME-KSME 2019 8th Joint Fluids Engineering Conference. American Society of Mechanical Engineers, 2019. http://dx.doi.org/10.1115/ajkfluids2019-4806.
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Abstract In order to evaluate the interior noise caused by the flow around automobiles, it is necessary to clarify the nature of the pressure fluctuations on the surface of vehicle body. The pressure fluctuations around the vehicle which are caused by the fluid motion can be solved by unsteady-compressible Navier-Stokes equation. However, the differences between the scales and intensity of the pressure fluctuations related to the hydrodynamic pressure fluctuation (HPF) of the flow field and the aerodynamic sound (acoustic pressure fluctuation APF) are quite large, these phenomena can be considered separately as two different phenomena. This assumption can help us to understand the contributions of these two components of pressure fluctuations to the structural vibration and interior sound of automobiles. Since both the HPF and the APF are pressure fluctuations, they cannot be separated only by measuring with a single pressure sensor. In this study, we divided these pressure fluctuations by using wavenumber-frequency spectrum analysis. Wind tunnel experiment showed that the HPF and the APF have different wavenumber fields in the wake of a rear-view mirror, and the intensity and wavenumber of the HPF are larger than that of the APF. Flow field was also investigated by using the incompressible flow simulation. As a result of wavenumber-frequency spectrum analysis based on the pressure fields around the vehicle body, the HPF and the APF have different wavenumbers in the case of a boundary layer flow field with no separation such as boundary layer on the vehicle roof. On the other hand, very small wavenumber components of the HPF were observed in the recirculation flow around the rear-view mirror downstream, despite incompressible simulation was done. This is probably due to the flow fields excite the vehicle body in the direction close to the vertical with respect to the vehicle body surface (side shield) in the separated flow region, and the wavenumber vector project on the shield surface apparently become smaller. The wavenumber vector becomes short but the frequency is constant, which leads the speed of pressure propagation apparently increases. In the reverse flow region, even if the uniform flow velocity is smaller than the speed of sound, the HPF may still contribute to vibration and sound generation. At the same time, since the flow velocity is actually slowed in the reverse flow region, large wavenumber components were also observed. Therefore, the wavenumber spectrum was observed in a wide range of the wavelength region. In conclusion, by investigating the wavenumber frequency spectrum, it is possible to estimate the flow field contributing to the interior noise of automobiles.