Academic literature on the topic 'L^infinity Data'
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Journal articles on the topic "L^infinity Data"
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Bamieh, B., M. A. Dahleh, and J. B. Pearson. "Minimization of the L/sup infinity /-induced norm for sampled-data systems." IEEE Transactions on Automatic Control 38, no. 5 (May 1993): 23. http://dx.doi.org/10.1109/9.277236.
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Sychev, M. A. "Characterization of homogeneous scalar variational problems solvable for all boundary data." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 130, no. 3 (June 2000): 611–31. http://dx.doi.org/10.1017/s0308210500000330.
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It is known that the condition ‘either ∂L (F) ≠ Ø or there exist υ1,…,υq ∈ Rnsuch thatF ∈ int co {υ1,…,υq} characterizes solvability of the problem with f(·) = 〈F,·〉.We extend this result to the case of lower semicontinuous integrands L : Rn → R.We also show that validity of this condition for all F ∈ Rn is both a necessary and sufficient requirement for solvability of all minimization problems with sufficiently regular Ω and f. Moreover, the assumptions on Ω and f can be completely dropped if L has sufficiently fast growth at infinity.
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Karpińska, M., M. Wlazło, D. Ramjugernath, P. Naidoo, and U. Domańska. "Assessment of certain ionic liquids for separation of binary mixtures based on gamma infinity data measurements." RSC Advances 7, no. 12 (2017): 7092–107. http://dx.doi.org/10.1039/c6ra25208g.
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Limiting activity coefficients for 64 solutes in [BzMIM][NTf2] and [BzMIM][DCA], the gas–liquid partition coefficients, KL, thermodynamic functions and selectivity for hexane/hex-1-ene, cyclohexane/cyclohexene and ethylbenzene/styrene separation were presented.
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Gajic, Dejan, and Claude Warnick. "Quasinormal Modes in Extremal Reissner–Nordström Spacetimes." Communications in Mathematical Physics 385, no. 3 (June 27, 2021): 1395–498. http://dx.doi.org/10.1007/s00220-021-04137-4.
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AbstractWe present a new framework for characterizing quasinormal modes (QNMs) or resonant states for the wave equation on asymptotically flat spacetimes, applied to the setting of extremal Reissner–Nordström black holes. We show that QNMs can be interpreted as honest eigenfunctions of generators of time translations acting on Hilbert spaces of initial data, corresponding to a suitable time slicing. The main difficulty that is present in the asymptotically flat setting, but is absent in the previously studied asymptotically de Sitter or anti de Sitter sub-extremal black hole spacetimes, is that $$L^2$$ L 2 -based Sobolev spaces are not suitable Hilbert space choices. Instead, we consider Hilbert spaces of functions that are additionally Gevrey regular at infinity and at the event horizon. We introduce $$L^2$$ L 2 -based Gevrey estimates for the wave equation that are intimately connected to the existence of conserved quantities along null infinity and the event horizon. We relate this new framework to the traditional interpretation of quasinormal frequencies as poles of the meromorphic continuation of a resolvent operator and obtain new quantitative results in this setting.
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Arifin, Ajib Setyo, and Tomoaki Ohtsuki. "Capacity of Data Collection in Wireless Sensor Networks Based on Mutual Information and MMSE Estimation." ISRN Sensor Networks 2014 (February 17, 2014): 1–9. http://dx.doi.org/10.1155/2014/389451.
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We investigate the properties of data collection in wireless sensor networks, in terms of both capacity and power allocation strategy. We consider a scenario in which a number of sensors observe a target being estimated at fusion center (FC) using minimum mean-square error (MMSE) estimator. Based on the relationship between mutual information and MMSE (I-MMSE), the capacity of data collection in coherent and orthogonal multiple access channel (MAC) models is derived. Considering power constraint, the capacity is derived under two scenarios: equal power allocation and optimal power allocation of both models. We provide the upper bound of capacity as a benchmark. In particular, we show that the capacity of data collection scales as Θ((1/2)log(1+L)) when the number of sensors L grows to infinity. We show through simulation results that for both coherent and orthogonal MAC models, the capacity of the optimal power is larger than that of the equal power. We also show that the capacity of coherent MAC is larger than that of orthogonal MAC, particularly when the number of sensors L is large and the total power P is fixed.
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Jamali, A., and F. A. Castro. "TOPOLOGICAL 3D ELEVATION DATA INTERPOLATION OF ASTER GDEM BASED ON CONTINUOUS DEFORMATION." ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences XLII-4/W10 (September 12, 2018): 71–77. http://dx.doi.org/10.5194/isprs-archives-xlii-4-w10-71-2018.
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<p><strong>Abstract.</strong> In Geographic Information Science, polynomial methods such as linear estimation and non-polynomial methods including Inverse Distance Weighting and Kriging have been used for elevation data interpolation. In this paper, 3D data interpolation using linear and non-linear homotopy continuation as well as advanced polynomial interpolation methods are researched. Continuous deformations that reconstruct straight lines or algebraic curves between any pair of 3D data are presented. The implemented topological mathematical algorithm for 3D elevation data interpolation is compared to Inverse Distance Weighting and Triangulated Irregular Network (TIN) methods. The presented linear and non-linear mathematical algorithms show better results compared to Inverse Distance Weighting and TIN in terms of Root Mean Square Error and L-infinity.</p>
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LI, DONG, TONG LI, and KUN ZHAO. "ON A HYPERBOLIC–PARABOLIC SYSTEM MODELING CHEMOTAXIS." Mathematical Models and Methods in Applied Sciences 21, no. 08 (August 2011): 1631–50. http://dx.doi.org/10.1142/s0218202511005519.
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We investigate local/global existence, blowup criterion and long-time behavior of classical solutions for a hyperbolic–parabolic system derived from the Keller–Segel model describing chemotaxis. It is shown that local smooth solution blows up if and only if the accumulation of the L∞ norm of the solution reaches infinity within the lifespan. Our blowup criteria are consistent with the chemotaxis phenomenon that the movement of cells (bacteria) is driven by the gradient of the chemical concentration. Furthermore, we study the long-time dynamics when the initial data is sufficiently close to a constant positive steady state. By using a new Fourier method adapted to the linear flow, it is shown that the smooth solution exists for all time and converges exponentially to the constant steady state with a frequency-dependent decay rate as time goes to infinity.
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Chiarello, Felisia Angela, and Paola Goatin. "Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel." ESAIM: Mathematical Modelling and Numerical Analysis 52, no. 1 (January 2018): 163–80. http://dx.doi.org/10.1051/m2an/2017066.
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We prove the well-posedness of entropy weak solutions for a class of scalar conservation laws with non-local flux arising in traffic modeling. We approximate the problem by a Lax-Friedrichs scheme and we provide L∞ and BV estimates for the sequence of approximate solutions. Stability with respect to the initial data is obtained from the entropy condition through the doubling of variable technique. The limit model as the kernel support tends to infinity is also studied.
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Hoang, Luan T., and Thinh T. Kieu. "Interior Estimates for Generalized Forchheimer Flows of Slightly Compressible Fluids." Advanced Nonlinear Studies 17, no. 4 (October 1, 2017): 739–67. http://dx.doi.org/10.1515/ans-2016-6027.
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AbstractThe generalized Forchheimer flows are studied for slightly compressible fluids in porous media with time-dependent Dirichlet boundary data for the pressure. No restrictions are imposed on the degree of the Forchheimer polynomial. We derive, for all time, the interior {L^{\infty}}-estimates for the pressure, its gradient and time derivative, and the interior {L^{2}}-estimates for its Hessian. The De Giorgi and Ladyzhenskaya–Uraltseva iteration techniques are used taking into account the special structures of the equations for both pressure and its gradient. These are combined with the uniform Gronwall-type bounds in establishing the asymptotic estimates when time tends to infinity.
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Sulttan, Mohammed. "Impact of using Infinity-Norm with Initial Radius on Performance and Complexity of SD Algorithm in MIMO systems." INTERNATIONAL JOURNAL OF COMPUTERS & TECHNOLOGY 15, no. 6 (April 20, 2016): 6857–64. http://dx.doi.org/10.24297/ijct.v15i6.1614.
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 In recent years, the iterative decoding techniques have played a role in improving the performance (e.g., bit error rate) and reducing the complexity of various digital communication systems. Techniques of Multiple-Input Multiple-Output (MIMO) are the main technology to enhance and achieve high-speed, high data rates, improved reliability and coverage in wireless communications. The modern wireless communications require a low complexity system for detection, since a high CPU processing involves more energy consumption and thus less flexibility in mobility terms. The sphere decoding (SD) technique proposed to solve this problem, such as an efficient algorithm. The norm-2 or l^2-norm (Euclidean metric) considered as a traditional norm that is used to achieve the tree traversal stage in SD algorithm. This work is divided into two parts; Firstly, we propose to using Infinity-Norm or l^∞-norm instead l^2-norm to decreases the hardware complexity of SD with a loss of performance is negligible, the simulation results show that the proposed l^∞-norm SD needs 14.5% to 5.9% fewer complexities than l^2-norm SD. Secondly, we are investigating the impact of choosing initial radius on the performance and complexity of SD algorithm, we can conclude from the simulation results, that gain a better performance requires increasing in the initial radius of an SD algorithm from d1 (γ =2) to d3 (γ =8), and this mean addition more complexity due to the tradeoff  between performance and complexity.    In recent years, the iterative decoding techniques have played a role in improving the performance (e.g., bit error rate) and reducing the complexity of various digital communication systems. Techniques of Multiple-Input Multiple-Output (MIMO) are the main technology to enhance and achieve high-speed, high data rates, improved reliability and coverage in wireless communications. The modern wireless communications require a low complexity system for detection, since a high CPU processing involves more energy consumption and thus less flexibility in mobility terms. The sphere decoding (SD) technique proposed to solve this problem, such as an efficient algorithm. The norm-2 or -norm (Euclidean metric) considered as a traditional norm that is used to achieve the tree traversal stage in SD algorithm. This work is divided into two parts; Firstly, we propose to using Infinity-Norm or -norm instead -norm to decreases the hardware complexity of SD with a loss of performance is negligible, the simulation results show that the proposed -norm SD needs 14.5% to 5.9% fewer complexities than -norm SD. Secondly, we are investigating the impact of choosing initial radius on the performance and complexity of SD algorithm, we can conclude from the simulation results, that gain a better performance require increasing in the initial radius of an SD algorithm from d1 ( =2) to d3 ( =8), and this mean addition more complexity due to the tradeoff between performance and complexity.
Dissertations / Theses on the topic "L^infinity Data"
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MONTI, FRANCESCA. "Hyperbolic conservation laws: L-infinity data and granular matter." Doctoral thesis, Università degli Studi di Milano-Bicocca, 2009. http://hdl.handle.net/10281/7501.
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In my PhD Thesis I analyzed three problems. The first we studied n × n systems. We considered a particular case of hyperbolic non-linear systems, the
straight-line systems and their perturbations. Both were analized in their most general form i.e. they are
not necessarily conservative.
We proved the existence of global solutions for the related Cauchy problems under the assumption that the
perturbed system was sufficiently near to the straight-line one. The total variation of the initial data must
be bounded, but not necessarily small. Also stability estimates hold.
In an other work we consider a hyperbolic system of conservation laws with genuinely nonlinear characteristic
fields. We extend the classical Glimm-Lax result proving the existence of solutions for L^infinity initial datum,
relaxing the assumptions taken therein on the geometry of the shock–rarefaction curves.
In the last part of the thesis we studied hyperbolic systems that represent the behaviour of granular matter and we have
presented a new model It is a synthesis of the Hadeler–Kuttler and of the Savage–Hutter models. The result
is a 3 × 3 system of balance laws able to describe the deposition–erosion dynamics, as in the former model,
while being compliant with energy dissipation, as in the latter one.
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Lucero, Aldo. "Compressing scientific data with control and minimization of the L-infinity metric under the JPEG 2000 framework." To access this resource online via ProQuest Dissertations and Theses @ UTEP, 2007. http://0-proquest.umi.com.lib.utep.edu/login?COPT=REJTPTU0YmImSU5UPTAmVkVSPTI=&clientId=2515.
Full textBook chapters on the topic "L^infinity Data"
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Zhou, Guoqing, and Qing Wang. "A Hybrid Steepest Descent Method for L-infinity Geometry Problems." In Intelligent Science and Intelligent Data Engineering, 458–65. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-31919-8_59.
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Finney, Mark A., and Torben P. Grumstrup. "Flame Length of Wildland Fires: Effect of Flame Zone Depth." In Advances in Forest Fire Research 2022, 1640–45. Imprensa da Universidade de Coimbra, 2022. http://dx.doi.org/10.14195/978-989-26-2298-9_251.
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Correlations of flame length L (or height) with the intensity of fires (kW/m2 for area sources or kW/m for line sources) have become foundational to fire behavior science (Byram 1959, Thomas 1963, Nelson and Adkins 1986, among others). Empirical data for these correlations have been based on fires fueled by wood cribs, paper strips, natural gas, and spreading fires at laboratory scales. The data and theory outlined by Thomas (1963, 1967) supported the idea that the horizontal dimension D of an infinite line fire could be neglected if L was much greater than D. Here we report on experiments with line-source fires where the energy release and flame zone dimensions were controlled independently (0.2≤L/D≤13.6), finding that increasing D significantly reduced L over the entire range of observations. Our flame length correlation was more similar to that of Thomas’ (1963) than Byram (1959) but the data clearly showed a strong dependency upon D. This finding suggests that fire behavior studies relying on correlations of L and fireline intensity must consider D to have an important effect on physical processes involved in wildfire behavior.
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Lamberti, Edward. "Introduction: Textual Performances in Levinas and Film." In Performing Ethics Through Film Style, 1–26. Edinburgh University Press, 2019. http://dx.doi.org/10.3366/edinburgh/9781474444002.003.0001.
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The introduction begins by exploring how Emmanuel Levinas’s ethical philosophy has been taken up by film theorists to date. Much of this scholarship centres on Levinas’s theories of the Other as found in Totality and Infinity: An Essay on Exteriority (1961), particularly the ‘face’ of the Other, which theorists have discussed in relation to visualre presentation. Levinas developed his ethics, in Otherwise than Being or Beyond Essence (1974), into a performative account of what it feels like to be responsible for the Other. Accordingly, Performing Ethics through Film Style takes a similar approach with film, linking the performativity of Levinas’s writing style with the capacity of films to perform a Levinasian ethics of responsibility for the Other through their styles. The introduction brings in performativity theory – including J. L. Austin’s speech acts, Jacques Derrida’s originary performativity and Judith Butler’s theories of language in the socio-political sphere – to enhance this study of performativity in Levinas and film. And it sets up the subjects of the chapters to follow: the films of the Dardenne brothers, Barbet Schroeder and Paul Schrader. Studying these directors in relation to Levinas shows how films can perform ethics through a wide variety of styles.
Conference papers on the topic "L^infinity Data"
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Anand, Anushka, Leland Wilkinson, and Dang Nhon Tuan. "An L-infinity Norm Visual Classifier." In 2009 Ninth IEEE International Conference on Data Mining (ICDM). IEEE, 2009. http://dx.doi.org/10.1109/icdm.2009.119.
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Lucero, Aldo, Sergio D. Cabrera, and Edward Vidal, Jr. "Residual coding with a JPEG2000 framework for controlling L-infinity error applied to ultraspectral sounder data." In SPIE Optics + Photonics, edited by Roger W. Heymann, Charles C. Wang, and Timothy J. Schmit. SPIE, 2006. http://dx.doi.org/10.1117/12.682052.
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Sarabi, Soroush, and David B. Bogy. "Viscoelastic Effects on Lubricant Depletion and Recovery Under Heat-Assisted Magnetic Recording (HAMR) Conditions." In ASME 2016 Conference on Information Storage and Processing Systems. American Society of Mechanical Engineers, 2016. http://dx.doi.org/10.1115/isps2016-9578.
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Heat Assisted Magnetic Recording (HAMR) is a developing data-storage technology in which a laser delivery system is integrated to the conventional HDD air bearing slider that carries the read and write transducers. The laser beam heats a small spot of around 20nm size on the storage media up to few hundred degrees Celsius [1]. This heating causes several effects on the lubricant such as temperature gradient, thermocapillary shear stress, viscosity drop, and evaporation followed by its depletion. Conventionally [2–8], the disk lubricant is considered as a Newtonian viscous fluid that can be fully described by a viscosity parameter μ. However, in rapid heating and forcing conditions like HAMR, the time dependent nature of the lubricant becomes very important. Measurements [9] show that under some conditions the lubricant behaves like a Maxwell viscoelastic fluid that can be described by two parameters: viscosity μ and Maxwell relaxation time λ. Itoh et al. [10] show that the viscoelastic behavior becomes even more considerable in the case of sub-10nm material confinement. Both the Maxwell relaxation time and viscosity can be functions of temperature and lubricant thickness in case of ultra-thin film lubrication. Karis [9] measured Maxwell relaxation time as a function of temperature for a variety of lubricants, such as Z-dol and Z-tetraol. Fig. 1 shows the results of these measurements for both Z-dol and Z-tetraol. Maxwell relaxation time plays a vital role in determining the behavior of the material under thermal and mechanical loads. In order to have a proper understanding of the effect of Maxwell relaxation time, we non-dimensionalize this parameter by the timescale of the problem to introduce a non-dimensional Deborah number De = λU/L. So, De is a function of HAMR temperature T, disk speed U, laser spot size L, and lubricant type. For purely-viscous materials both the Maxwell relaxation time and De are zero and for purely-elastic materials, both are infinity. So in the case of viscoelasticity, if De ≪ 1 the viscosity mode is dominant, if De ≫ 1 the elasticity is dominant, and if De ≈ 1 the material behaves viscoelastically. Therefore, De is good measure for the viscoelastic behavior of the material. Some attempts have been made to fit the lubrication theory for viscoelastic materials using perturbation methods. However these methods require that the Deborah Number be small enough [11]. Fig. 2 shows the Deborah Number as a function of laser spot size for different lubricant temperatures. Accordingly, at the target of a HAMR laser spot size of L = 20nm, the Deborah number is very large and therefore, the material behaves less viscous and more elastic. Therefore, the traditional methods of lubrication theory cannot describe the lubricant’s behavior in this limit. Consequently, we developed anew direct Finite Element Method (FEM) approach to simulate the behavior of the linear viscoelastic Maxwell fluid lubricants under HAMR conditions.
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Potts, Douglas A., Jonathan R. Binns, Andrew E. Potts, and Hayden Marcollo. "The Effect of Aspect Ratio on the Drag of Bare Cylinders." In ASME 2019 38th International Conference on Ocean, Offshore and Arctic Engineering. American Society of Mechanical Engineers, 2019. http://dx.doi.org/10.1115/omae2019-96431.
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Abstract The drag coefficient for long-slender structures that is typically provided in design guidance has been determined from test specimens of sufficient length that they are unaffected by the aspect ratio (L/D), whereby they are considered to be of “infinite” length. However, it is apparent from the literature that aspect ratio does have a significant non-linear effect at short L/D ratios. DNV provides guidance on the aspect ratio effect on the drag coefficient of a cylinder, for which no experimental source data has been cited. The DNV design guidance has wide usage in the offshore industry and merits critical evaluation. This paper critically reviews the literature and presents the results of a series of tow tank experiments performed by the authors. A series of tow tank tests of a surface-piercing cylinder has been undertaken using a range of aspect ratios as well as testing the effect of various end conditions, where the effects of VIV and ventilation has been deemed insignificant. Correlation of the various data sets of the literature and the experimental test programme provides the basis for developing an alternate design guidance curve for the effect of aspect ratio on the drag coefficient of cylinders.
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Venugopalan, V., J. S. You, and B. J. Tromberg. "An Improved Diffusion Model for Radiative Transport in Turbid Media." In ASME 1998 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 1998. http://dx.doi.org/10.1115/imece1998-0816.
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Abstract The diffusion approximation to the Boltzmann transport equation is commonly used to analyze data obtained from biomedical optical diagnostic techniques. Unfortunately, this approximation has significant limitations which constrains its applicability to highly scattering systems and over length scales much larger than the transport mean free path l*. Using an approach formulated independently by Prahl and Star [11, 13, 14], we present a form of the diffusion approximation which adds a delta function term to both the radiance and phase function expressions. This formulation is presented and solved for steady illumination in infinite media with a collimated source of finite size exhibiting spherical symmetry. The solution is compared to results given by standard diffusion theory and to measurements made in a strongly scattering and strongly absorbing turbid phantom with reduced single scattering albedos a’ of 0.997 and 0.248, respectively. The results show that this approach provides accurate predictions of optical dosimetry in both low and high scattering media and at positions proximal to collimated light sources.
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Stewart, Kelley C., and Pavlos P. Vlachos. "Vortex Ring Formation in Wall-Bounded Domains." In ASME 2010 3rd Joint US-European Fluids Engineering Summer Meeting collocated with 8th International Conference on Nanochannels, Microchannels, and Minichannels. ASMEDC, 2010. http://dx.doi.org/10.1115/fedsm-icnmm2010-31055.
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Vortex ring formation and propagation have been studied extensively in quiescent semi-infinite volumes. However, very little is known about the dynamics of vortex-ring formation in wall-bounded domains where vortex wall interaction will affect both the vortex ring pinch-off and propagation velocity. This study addresses this limitation and studies vortex formation in radially confined domains to analyze the effect of vortex-ring wall interaction on the formation and propagation of the vortex ring. Vortex rings were produced using a pneumatically driven piston cylinder arrangement and were ejected into a long cylindrical tube parallel to the piston cylinder arrangement which defined the confined downstream domain. Two different domains were studied with diameters twice and four times the size of the piston cylinder. A semi-infinite unbounded volume with no downstream cylinder was also investigated for comparison. The piston stroke-to-diameter ratio (L/D0) for the studied vortex rings was varied between 0.75 and 3 with corresponding Reynolds numbers, based on circulation, of approximately 500 to 8,000. Velocity field measurements were performed using planar Time Resolved Digital Particle Image Velocimetry (TRDPIV). The TRDPIV data were processed using an in-house developed cross-correlation PIV algorithm and post processed using Proper Orthogonal Decomposition to remove high frequency noise. The propagation velocity and vorticity were investigated and vortex identification was used to track the changing size, location, and circulation of the vortices. The combination of these parameters was used to investigate the effects of wall interaction on vortex ring formation and propagation.
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Mann, Michael D. "Fatigue Life of Commercial High Pressure Tubing." In ASME 2002 Pressure Vessels and Piping Conference. ASMEDC, 2002. http://dx.doi.org/10.1115/pvp2002-1163.
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Design guidance for high pressure components, has undergone a dramatic change with the release of ASME Section VIII division 3 pressure vessel code. For the first time, a thorough design criteria is available for design of thick wall pressure vessels. The most critical components of a design are safety and reliability. Ultra high-pressure vessels, in most cases, do not have an “infinite” life. The design must therefore be “leak before break” and a design cycle life must be specified. This paper looks at the effects of fatigue on commercial high-pressure tubing under tri-axial fatigue. The tubing investigated is 316 stainless steel 9/16″ and 3/8″ diameter 4100 bar (60,000 psi) tubing. The testing was performed using a tri-axial fatigue machine originally designed by Dr. B. Crossland, Dr. J. L. M. Morrison and Dr. J. S. C. Perry in 1960 and upgraded by the Author. This investigation compares the fatigue life prediction per KD3 in the ASME pressure vessel code Section VIII division 3 and actual test results from the fatigue machine. This verification gives important reliability data for commercial hardware used in high-pressure piping.
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Lia, Carlsen Mathias, and Whitson Curtis Hays. "Numerical RTA Extended to Complex Fracture Systems: Part 2." In SPE Annual Technical Conference and Exhibition. SPE, 2022. http://dx.doi.org/10.2118/210420-ms.
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Abstract This paper is a continuation of the work presented in URTeC 3718584 (Carlsen & Whitson, 2022), and focuses on practical usage of ‘fractional RTA’ theory when applied to both simulated data and field data from the SPE data repository. Most of the theory presented in Part 1 is kept for completeness. An inherent assumption in most industry RTA is equally spaced fractures. However, as shown in several field studies (Raterman 2017, Gale 2018), the distance between individual fractures tends to be unevenly spaced along the wellbore (e.g., "fracture swarms"). In this paper, we extend the original numerical RTA workflow proposed by Bowie and Ewert (2020) to account for uneven fracture spacing. Acuna's (2016, 2020) heterogeneity parameter, delta (δ), is introduced to generalize the linear flow parameter (LFP) to account for complex fracture systems (LFP’ = Akδϕ1-δ = 4nfhxfkδϕ1-δ). For evenly spaced fractures, δ = 0.5, simplifying LFP’ to the familiar LFP = A√k = 4nfhxf√k. For uneven fracture systems, 0 ≤ δ ≤ 0.5. With known (a) well geometry, (b) fluid initialization (PVT and water saturation), (c) relative permeability relations, and (d) bottomhole pressure (BHP) time variation (above and below saturation pressure), three fundamental relationships exist in terms of LFP' and OOIP. Numerical reservoir simulation is used to define these relationships, providing the foundation for numerical RTA, also wells with complex fracture systems. Namely, that wells: (1) with the same value of LFP', the gas, oil and water surface rates will be identical during infinite-acting (IA) behavior; (2) with the same ratio LFP'/OOIP, producing GOR and water cut behavior will be identical for all times, IA and boundary dominated (BD); and (3) with the same values of LFP' and OOIP, rate performance of gas, oil, and water will be identical for all times, IA and BD. These observations lead to an efficient, semi-automated process to perform rigorous RTA, assisted by a symmetry element numerical model. The numerical RTA workflow proposed by Bowie and Ewert solves the inherent problems associated with complex superposition and multiphase flow effects involving time and spatial changes in pressure, compositions and PVT properties, saturations, and complex phase mobilities. This paper extends the approach to complex fracture systems that can be described by the Acuna parameter δ. Numerical RTA workflow decouples multiphase flow data (PVT, initial saturations and relative permeabilities) from well geometry and petrophysical properties (L, xf, h, nf, φ, k, δ), providing a rigorous yet efficient and semi-automated approach to define production performance for many wells. Contributions include a technical framework to perform numerical RTA for unconventional wells, irrespective of fracture spacing. Semi-analytical models, time, and spatial superposition (convolution), pseudopressure and pseudotime transforms are not required.
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Carlsen, Mathias Lia, Braden Bowie, Mohamad Majzoub Dahouk, Stian Mydland, Curtis Hays Whitson, and Ilina Yusra. "Numerical RTA in Tight Unconventionals." In SPE Annual Technical Conference and Exhibition. SPE, 2021. http://dx.doi.org/10.2118/205884-ms.
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Abstract We extend the numerically-assisted RTA workflow proposed by Bowie and Ewert (2020) to (a) all fluid systems and (b) finite conductivity fractures. The simple, fully-penetrating planar fracture model proposed is a useful numerical symmetry element model that provides the basis for the work presented in this paper. Results are given for simulated and field data. The linear flow parameter (LFP) is modified to include porosity (LFPꞌ=LFP√φ). The original (surface) oil in place (OOIP) is generalized to represent both reservoir oil and reservoir gas condensate systems, using a consistent initial total formation volume factor definition (Bti) representing the ratio of a reservoir HCPV containing surface oil in a reservoir oil phase, a reservoir gas phase, or both phases. With known (a) well geometry, (b) fluid initialization (PVT and water saturation), (c) relative permeability relations, and (d) bottomhole pressure (BHP) time variation (above and below saturation pressure), three fundamental relationships exist in terms of LFPꞌ and OOIP. Numerical reservoir simulation is used to define these relationships, providing the foundation for numerical RTA, namely that wells: (1) with the same value of LFPꞌ, the gas, oil and water surface rates will be identical during infinite-acting (IA) behavior; (2) with the same ratio LFPꞌ/OOIP, producing GOR and water cut behavior will be identical for all times, IA and boundary dominated (BD); and (3) with the same values of LFPꞌ and OOIP, rate performance of gas, oil, and water be identical for all times, IA and BD. These observations lead to an efficient, semi-automated process to perform rigorous RTA, assisted by a symmetry element numerical model. The numerical RTA workflow proposed by Bowie and Ewert solves the inherent problems associated with complex superposition and multiphase flow effects involving time and spatial changes in pressure, compositions and PVT properties, saturations, and complex phase mobilities. The numerical RTA workflow decouples multiphase flow data (PVT, initial saturations and relative permeabilities) from well geometry and petrophysical properties (L, xf, h, nf, φ, k), providing a rigorous yet efficient and semi-automated approach to define production performance for many wells. Contributions include a technical framework to perform numerical RTA for unconventional wells, irrespective of fluid type. A suite of key diagnostic plots associated with the workflow is provided, with synthetic and field examples used to illustrate the application of numerical simulation to perform rigorous RTA. Semi-analytical models, time, and spatial superposition (convolution), pseudopressure and pseudotime transforms are not required.