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1
Lam, S. H., e N. Rott. "Eigen-Functions of Linearized Unsteady Boundary Layer Equations". Journal of Fluids Engineering 115, n. 4 (1 dicembre 1993): 597–602. http://dx.doi.org/10.1115/1.2910185.
Abstract (sommario):
The Lam and Rott theory of linearized unsteady boundary layers is revisited, and some new results are obtained. The exact outer eigen-solution for a flat plate found in the original paper is shown to be a special case of the Prandtl-Glauert transposition theorem. The streamwise coordinate-dependent factor of the inner eigen-solutions, first found by M. E. Goldstein for the flat plate, is generalized for arbitrary pressure gradients.
2
Pogorui, Anatoliy A., e Ramón M. Rodríguez-Dagnino. "Goldstein-Kac telegraph equations and random flights in higher dimensions". Applied Mathematics and Computation 361 (novembre 2019): 617–29. http://dx.doi.org/10.1016/j.amc.2019.05.045.
3
FERLINI, VINCENT. "SOLUTIONS TO ZERO-SUM EXPONENT EQUATIONS OVER FINITE CYCLIC GROUPS OF EXPONENT GREATER THAN TWO". International Journal of Algebra and Computation 18, n. 03 (maggio 2008): 423–41. http://dx.doi.org/10.1142/s0218196708004494.
Abstract (sommario):
Lyndon initiated a study of equations over cyclic groups [8] and his approach led him to consider a class of equations of the form ah= (tpat-p)(tqat-q)(tpat-p)-1(tqat-q)-1where t is the variable and the cyclic group is Cn= 〈a|an〉. Work by Ferlini, Goldstein, and Salpukis [3] concentrated, more generally, on equations with the exponent sum of the variable being zero over C2. This paper continues along the path started by Lyndon and shows that a large class of the equations above do have solutions over Cnwhere n ≥ 4. Our approach involves the use of pictures. We also include a result that the equations with h = 1, 2, p = 1, and q = 2 do not have a solution over C3.
4
HILFER, R. "ON FRACTIONAL RELAXATION". Fractals 11, supp01 (febbraio 2003): 251–57. http://dx.doi.org/10.1142/s0218348x03001914.
Abstract (sommario):
Generalized fractional relaxation equations based on generalized Riemann-Liouville derivatives are combined with a simple short time regularization and solved exactly. The solution involves generalized Mittag-Leffler functions. The associated frequency dependent susceptibilities are related to symmetrically broadened Cole-Cole susceptibilities occurring as Johari Goldstein β-relaxation in many glass formers. The generalized susceptibilities exhibit a high frequency wing and strong minimum enhancement.
5
Mallier, Roland. "The nonlinear temporal evolution of a disturbance to a stratified mixing layer". Journal of Fluid Mechanics 291 (25 maggio 1995): 287–97. http://dx.doi.org/10.1017/s0022112095002709.
Abstract (sommario):
Using a nonlinear critical layer analysis, Goldstein & Leib (1988) derived a set of nonlinear evolution equations governing the spatial growth of a two-dimensional instability wave on a homogeneous incompressible tanh y mixing layer. In this study, we extend this analysis to the temporal growth of the García model of an incompressible stratified shear layer. We consider the stage of the evolution in which the growth first becomes nonlinear, with the nonlinearity appearing inside the critical layer. The Reynolds number is assumed to be just large enough so that the unsteady, nonlinear and viscous terms all enter at the same order of magnitude inside the critical layer. The equations are solved numerically for the inviscid case.
6
Galaktionov, V. A., e I. V. Kamotski. "On nonexistence of Baras–Goldstein type for higher-order parabolic equations with singular potentials". Transactions of the American Mathematical Society 362, n. 08 (17 marzo 2010): 4117–36. http://dx.doi.org/10.1090/s0002-9947-10-04855-5.
7
Chalons, Christophe, e Rodolphe Turpault. "High‐order asymptotic‐preserving schemes for linear systems: Application to the Goldstein–Taylor equations". Numerical Methods for Partial Differential Equations 35, n. 4 (21 febbraio 2019): 1538–61. http://dx.doi.org/10.1002/num.22363.
8
El Ibrami, Hassan, e Ahmed Naciri. "Equity Capital-Structure-Based Evaluation Method". International Journal of Accounting and Financial Reporting 2, n. 2 (28 dicembre 2012): 299. http://dx.doi.org/10.5296/ijafr.v2i2.2537.
Abstract (sommario):
Abstract The main purpose of this paper is to theoretically compare three structural models presenting several similarities and using financial statements within the context of real options theory. The models are those suggested by i) Leland (1994); ii) Goldstein, Ju and Leland (2001) and iii) Sarkar and Zapatero (2003). The analysis emphasizes convergence conditions of the three models based on their respective dynamic equations. The results show that the first two models represent special cases of the third one. The paper also presents a new equity and debt valuation method. Keywords: Structural model, Financial statement, Equity, EBIT, Mean reversion, Contingent claim, Convergence.
9
Galaktionov, V. A. "On nonexistence of Baras-Goldstein type without positivity assumptions for singular linear and nonlinear parabolic equations". Proceedings of the Steklov Institute of Mathematics 260, n. 1 (aprile 2008): 123–43. http://dx.doi.org/10.1134/s0081543808010094.
10
Mallier, R., e S. A. Maslowe. "Fully coupled resonant-triad interactions in a free shear layer". Journal of Fluid Mechanics 278 (10 novembre 1994): 101–21. http://dx.doi.org/10.1017/s0022112094003630.
Abstract (sommario):
We report the results of an investigation of the weakly nonlinear evolution of a triad of waves, each slightly amplified on a linear basis, that are superimposed on a tanh y mixing layer. The triad consists of a plane wave and a pair of oblique modes that act as a subharmonic of order 1/2. The oblique modes are inclined at approximately ±60°. to the mean flow direction and because the resonance conditions are satisfied exactly the analysis is entirely self-consistent as an asymptotic theory. The nonlinearity first occurs within the critical layer and the initial interaction is of the parametric resonance type. This produces faster than exponential growth of the oblique waves, behaviour observed recently in the experiments of Corke & Kusek (1993). The critical-layer dynamics lead subsequently to coupled integro-differential equations governing the amplitude evolution and, as first shown in related work by Goldstein & Lee (1992) on boundary layers in an adverse pressure gradient, these equations develop singularities in a finite time.
11
LEE, SANG SOO. "Generalized critical-layer analysis of fully coupled resonant-triad interactions in boundary layers". Journal of Fluid Mechanics 347 (25 settembre 1997): 71–103. http://dx.doi.org/10.1017/s0022112097006617.
Abstract (sommario):
The critical-layer analysis of the nonlinear resonant-triad interaction by Goldstein & Lee (1992) is extended to include viscous effects. A generalized scaling which is valid both for the quasi-equilibrium and non-equilibrium critical-layer analyses in zero- or non-zero-pressure-gradient boundary layers is obtained. A system of partial differential equations which governs the fully coupled non-equilibrium critical-layer dynamics is obtained and it is solved by using a numerical method. Amplitude equations and their viscous limits are also presented. The parametric-resonance growth rate of the non-equilibrium critical-layer solution with finite viscosity is larger than that of the viscous-limit quasi-equilibrium solution. The viscosity delays both the onset of the fully coupled interaction and the ultimate downstream location of the singularity. The difference between the non-equilibrium critical-layer solution and the corresponding quasi-equilibrium critical-layer solution becomes smaller, at least in the parametric resonance region, as the viscosity parameter becomes large. However, the non-equilibrium solution with finite viscosity always ends in a singularity at a finite downstream position unlike the viscous-limit solution.
12
Kolesnik, Alexander D. "Linear combinations of the telegraph random processes driven by partial differential equations". Stochastics and Dynamics 18, n. 04 (agosto 2018): 1850020. http://dx.doi.org/10.1142/s021949371850020x.
Abstract (sommario):
Consider [Formula: see text] independent Goldstein–Kac telegraph processes [Formula: see text] on the real line [Formula: see text]. Each process [Formula: see text] describes a stochastic motion at constant finite speed [Formula: see text] of a particle that, at the initial time instant [Formula: see text], starts from some initial point [Formula: see text] and whose evolution is controlled by a homogeneous Poisson process [Formula: see text] of rate [Formula: see text]. The governing Poisson processes [Formula: see text] are supposed to be independent as well. Consider the linear combination of the processes [Formula: see text] defined by [Formula: see text] where [Formula: see text] are arbitrary real nonzero constant coefficients. We obtain a hyperbolic system of [Formula: see text] first-order partial differential equations for the joint probability densities of the process [Formula: see text] and of the directions of motions at arbitrary time [Formula: see text]. From this system we derive a partial differential equation of order [Formula: see text] for the transition density of [Formula: see text] in the form of a determinant of a block matrix whose elements are the differential operators with constant coefficients. Initial-value problems for the transition densities of the sum and difference [Formula: see text] of two independent telegraph processes with arbitrary parameters, are also posed.
13
Sandham, Neil D., e Adriana M. Salgado. "Nonlinear interaction model of subsonic jet noise". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 366, n. 1876 (16 maggio 2008): 2745–60. http://dx.doi.org/10.1098/rsta.2008.0049.
Abstract (sommario):
Noise generation in a subsonic round jet is studied by a simplified model, in which nonlinear interactions of spatially evolving instability modes lead to the radiation of sound. The spatial mode evolution is computed using linear parabolized stability equations. Nonlinear interactions are found on a mode-by-mode basis and the sound radiation characteristics are determined by solution of the Lilley–Goldstein equation. Since mode interactions are computed explicitly, it is possible to find their relative importance for sound radiation. The method is applied to a single stream jet for which experimental data are available. The model gives Strouhal numbers of 0.45 for the most amplified waves in the jet and 0.19 for the dominant sound radiation. While in near field axisymmetric and the first azimuthal modes are both important, far-field sound is predominantly axisymmetric. These results are in close correspondence with experiment, suggesting that the simplified model is capturing at least some of the important mechanisms of subsonic jet noise.
14
Winters, Kraig B., e Laurence Armi. "Topographic control of stratified flows: upstream jets, blocking and isolating layers". Journal of Fluid Mechanics 753 (16 luglio 2014): 80–103. http://dx.doi.org/10.1017/jfm.2014.363.
Abstract (sommario):
AbstractOptimal solutions to the nonlinear, hydrostatic, Boussinesq equations are developed for steady, density-stratified, topographically controlled flows characterized by blocking and upstream influence. These flows are jet-like upstream of an isolated obstacle and are contained within an asymmetric, thinning stream tube that is accelerated as it passes over the crest. A stagnant, nearly uniform-density isolating layer, surrounded by a bifurcated uppermost streamline, separates the accelerated flow from an uncoupled flow above. The flows are optimal in the sense that, for a given stratification, the solutions maximize the topographic rise above the blocking level required for hydraulic control while minimizing the total energy of the flow. Hydraulic control is defined mathematically by the asymmetry of the accelerated flow as it passes the crest. A subsequent analysis of the Taylor–Goldstein equation shows that these sheared, non-uniformly stratified flows are indeed subcritical upstream, critical at the crest, and supercritical downstream with respect to gravest-mode, long internal waves. The flows obtained are relevant to arrested wedge flows, selective withdrawal, stratified towing experiments, tidal flow over topography and atmospheric flows over mountains.
15
COLONIUS, TIM, SANJIVA K. LELE e PARVIZ MOIN. "Sound generation in a mixing layer". Journal of Fluid Mechanics 330 (10 gennaio 1997): 375–409. http://dx.doi.org/10.1017/s0022112096003928.
Abstract (sommario):
The sound generated by vortex pairing in a two-dimensional compressible mixing layer is investigated. Direct numerical simulations (DNS) of the Navier–Stokes equations are used to compute both the near-field region and a portion of the acoustic field. The acoustic analogy due to Lilley (1974) is also solved with acoustic sources determined from the near-field data of the DNS. It is shown that several commonly made simplifications to the acoustic sources can lead to erroneous predictions for the acoustic field. Predictions based on the quadrupole form of the source terms derived by Goldstein (1976a, 1984) are in excellent agreement with the acoustic field from the DNS. However, despite the low Mach number of the flow, the acoustic far field generated by the vortex pairings cannot be described by considering compact quadrupole sources. The acoustic sources have the form of modulated wave packets and the acoustic far field is described by a superdirective model (Crighton & Huerre 1990). The presence of flow–acoustic interactions in the computed source terms causes the acoustic field predicted by the acoustic analogy to be very sensitive to small changes in the description of the source.
16
AFSAR, M. Z. "Asymptotic properties of the overall sound pressure level of subsonic jet flows using isotropy as a paradigm". Journal of Fluid Mechanics 664 (3 novembre 2010): 510–39. http://dx.doi.org/10.1017/s0022112010003976.
Abstract (sommario):
Measurements of subsonic air jets show that the peak noise usually occurs when observations are made at small angles to the jet axis. In this paper, we develop further understanding of the mathematical properties of this peak noise by analysing the properties of the overall sound pressure level with an acoustic analogy using isotropy as a paradigm for the turbulence. The analogy is based upon the hyperbolic conservation form of the Euler equations derived by Goldstein (Intl J. Aeroacoust., vol. 1, 2002, p. 1). The mean flow and the turbulence properties are defined by a Reynolds-averaged Navier–Stokes calculation, and we use Green's function based upon a parallel mean flow approximation. Our analysis in this paper shows that the jet noise spectrum can, in fact, be thought of as being composed of two terms, one that is significant at large observation angles and a second term that is especially dominant at small observation angles to the jet axis. This second term can account for the experimentally observed peak jet noise (Lush, J. Fluid Mech., vol. 46, 1971, p. 477) and was first identified by Goldstein (J. Fluid Mech., vol. 70, 1975, p. 595). We discuss the low-frequency asymptotic properties of this second term in order to understand its directional behaviour; we show, for example, that the sound power of this term is proportional to the square of the mean velocity gradient. We also show that this small-angle shear term does not exist if the instantaneous Reynolds stress source strength in the momentum equation itself is assumed to be isotropic for any value of time (as was done previously by Morris & Farrasat, AIAA J., vol. 40, 2002, p. 356). However, it will be significant if the auto-covariance of the Reynolds stress source, when integrated over the vector separation, is taken to be isotropic in all of its tensor suffixes. Although the analysis shows that the sound pressure of this small-angle shear term is sensitive to the statistical properties of the turbulence, this work provides a foundation for a mathematical description of the two-source model of jet noise.
17
RAY, PRASUN K., e SANJIVA K. LELE. "Sound generated by instability wave/shock-cell interaction in supersonic jets". Journal of Fluid Mechanics 587 (31 agosto 2007): 173–215. http://dx.doi.org/10.1017/s0022112007007306.
Abstract (sommario):
Broadband shock-associated noise is an important component of the overall noise generated by modern airplanes. In this study, sound generated by the weakly nonlinear interaction between linear instability waves and the shock-cell structure in supersonic jets is investigated numerically in order to gain insight into the broadband shock-noise problem. The model formulation decomposes the overall flow into a mean flow, linear instability waves, the shock-cell structure and shock-noise. The mean flow is obtained by solving RANSequations with a k-ε model. Locally parallel stability equations are solved for the shock structure, and linear parabolized stability equations are solved for the instability waves. Then, source terms representing the instability wave/shock-cell interaction are assembled and the inhomogeneous linearized Euler equations are solved for the shock-noise.Three cases are considered, a cold under-expanded Mj = 1.22 jet, a hot under-expanded Mj = 1.22 jet, and a cold over-expanded Mj = 1.36 jet.Shock-noise computations are used to identify and understand significant trends in peak sound amplitudes and radiation angles. The peak sound radiation angles are explained well with the Mach wave model of Tam & Tanna J. Sound Vib. Vol. 81, 1982, p. 337). The observed reduction of peak sound amplitudes with frequency correlates well with the corresponding reduction of instability wave growth with frequency. However, in order to account for variation of sound amplitude for different azimuthal modes, the radial structure of the instability waves must be considered in additionto streamwise growth. The effect of heating on the Mj = 1.22 jet is shown to enhance the sound radiated due to the axisymmetric instability waves while the other modesare relatively unaffected. Solutions to a Lilley–Goldstein equation show that soundgenerated by ‘thermodynamic’ source terms is small relative to sound from ‘momentum’ sources though heating does increase the relative importance of the thermodynamic source. Furthermore, heating preferentially amplifies sound associated with the axisymmetric modes owing to constructive interference between sound from the momentumand thermodynamic sources. However, higher modes show destructive interference between these two sources and are relatively unaffected by heating.
18
Qiao, Yangyang, e Steinar Evje. "A general cell–fluid Navier–Stokes model with inclusion of chemotaxis". Mathematical Models and Methods in Applied Sciences 30, n. 06 (13 giugno 2020): 1167–215. http://dx.doi.org/10.1142/s0218202520400096.
Abstract (sommario):
The main purpose of this work is to explore a general cell–fluid model which is based on a mixture theory formulation that accounts for the interplay between oxytactically (chemotaxis toward gradient in oxygen) moving bacteria cells in water and the buoyance forces caused by the difference in density between cells and fluid. The model involves two mass balance and two general momentum balance equations, respectively, for the cell and fluid phase, combined with a convection–diffusion–reaction equation for oxygen. In particular, the momentum balance equations include interaction terms which describe the cell–fluid drag force effect. Hence, the model is an extension of the classical Navier–Stokes equation in two different ways: (i) inclusion of two phases (cell and fluid) instead of one; (ii) inclusion of a chemotactic transport mechanism. The model can be understood as a natural generalization of the much studied chemotaxis-Stokes model explored by [I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein, Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA 102 (2005) 2277–2282]. First, we explore the model for parameters in a range which ensures that it lies close to the previously studied chemotaxis-Stokes model (essentially very low cell volume fraction). Main observations are (i) formation of sinking finger-shaped plumes and (ii) convergence to plumes that possibly can be stationary (i.e. persist over time). The general cell–fluid model provides new insight into the role played by the cell–fluid interaction term which controls the competition between gravity segregation and chemotaxis effect on the formation of cell plumes. Second, we explore cases with large cell volume fraction (far beyond the regime captured by the chemotaxis-Stokes model), which gives rise to rich pattern-formation behavior. The general cell–fluid model opens for exploring a hierarchy of different “submodels”. Hence, it seems to be an interesting model for further investigations of various, general cell–fluid spatio-temporal evolution dynamics, both from an experimental and mathematical point of view.
19
Afsar, Mohammed Z., Adrian Sescu e Stewart J. Leib. "Modelling and prediction of the peak-radiated sound in subsonic axisymmetric air jets using acoustic analogy-based asymptotic analysis". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 377, n. 2159 (14 ottobre 2019): 20190073. http://dx.doi.org/10.1098/rsta.2019.0073.
Abstract (sommario):
This paper uses asymptotic analysis within the generalized acoustic analogy formulation (Goldstein 2003 JFM 488 , 315–333. ( doi:10.1017/S0022112003004890 )) to develop a noise prediction model for the peak sound of axisymmetric round jets at subsonic acoustic Mach numbers (Ma). The analogy shows that the exact formula for the acoustic pressure is given by a convolution product of a propagator tensor (determined by the vector Green's function of the adjoint linearized Euler equations for a given jet mean flow) and a generalized source term representing the jet turbulence field. Using a low-frequency/small spread rate asymptotic expansion of the propagator, mean flow non-parallelism enters the lowest order Green's function solution via the streamwise component of the mean flow advection vector in a hyperbolic partial differential equation. We then address the predictive capability of the solution to this partial differential equation when used in the analogy through first-of-its-kind numerical calculations when an experimentally verified model of the turbulence source structure is used together with Reynolds-averaged Navier–Stokes solutions for the jet mean flow. Our noise predictions show a reasonable level of accuracy in the peak noise direction at Ma = 0.9, for Strouhal numbers up to about 0.6, and at Ma = 0.5 using modified source coefficients. Possible reasons for this are discussed. Moreover, the prediction range can be extended beyond unity Strouhal number by using an approximate composite asymptotic formula for the vector Green's function that reduces to the locally parallel flow limit at high frequencies. This article is part of the theme issue ‘Frontiers of aeroacoustics research: theory, computation and experiment’.
20
Mittal, S. "Flow Past Rotating Cylinders: Effect of Eccentricity". Journal of Applied Mechanics 68, n. 4 (29 novembre 2000): 543–52. http://dx.doi.org/10.1115/1.1380679.
Abstract (sommario):
Computational results are presented for flows past a translating and rotating circular cylinder. A stabilized finite element method is utilized to solve the incompressible Navier-Stokes equations in the primitive variables formulation. To validate the formulation and its implementation certain cases, for which the flow visualization and computational results have been reported by other researchers, are computed. Results are presented for Re=5, 200 and 3800 and rotation rate, (ratio of surface speed of cylinder to the freestream speed of flow), of 5. For all these cases the flow reaches a steady state. The values of lift coefficient observed for these flows exceed the limit on the maximum value of lift coefficient suggested by Goldstein based on intuitive arguments by Prandtl. These observations are in line with measurements reported, earlier, by other researchers via laboratory experiments. To investigate the stability of the computed steady-state solution, receptivity studies involving an eccentrically rotating cylinder are carried out. Computations are presented for flow past a rotating cylinder with wobble; the center of rotation of the cylinder does not match its geometric center. These computations are also important from the point of view that in a real situation it is almost certain that the rotating cylinder will be associated with a certain degree of wobble. In such cases the flow is unsteady and reaches a temporally periodic state. However, the mean values of the aerodynamic coefficients and the basic flow structure are still quite comparable to the case without any wobble. In this sense, it is found that the two-dimensional solution is stable to purely two-dimensional disturbances.
21
WU, XUESONG, DIFEI ZHAO e JISHENG LUO. "Excitation of steady and unsteady Görtler vortices by free-stream vortical disturbances". Journal of Fluid Mechanics 682 (8 luglio 2011): 66–100. http://dx.doi.org/10.1017/jfm.2011.224.
Abstract (sommario):
Excitation of Görtler vortices in a boundary layer over a concave wall by free-stream vortical disturbances is studied theoretically and numerically. Attention is focused on disturbances with long streamwise wavelengths, to which the boundary layer is most receptive. The appropriate initial-boundary-value problem describing both the receptivity process and the development of the induced perturbation is formulated for the generic case where the Görtler number GΛ (based on the spanwise wavelength Λ of the disturbance) is of order one. The impact of free-stream disturbances on the boundary layer is accounted for by the far-field boundary condition and the initial condition near the leading edge, both of which turn out to be the same as those given by Leib, Wundrow & Goldstein (J. Fluid Mech., vol. 380, 1999, p. 169) for the flat-plate boundary layer. Numerical solutions show that for a sufficiently small GΛ, the induced perturbation exhibits essentially the same characteristics as streaks occurring in the flat-plate case: it undergoes considerable amplification and then decays. However, when GΛ exceeds a critical value, the induced perturbation exhibits (quasi-) exponential growth. The perturbation acquires the modal shape of Görtler vortices rather quickly, and its growth rate approaches that predicted by local instability theories farther downstream, indicating that Görtler vortices are excited. The amplitude of the Görtler vortices excited is found to decrease as the frequency increases, with steady vortices being dominant. Comprehensive quantitative comparisons with experiments show that the eigenvalue approach predicts the modal shape adequately, but only the initial-value approach can accurately predict the entire evolution of the amplitude. An asymptotic analysis is performed for GΛ ≫ 1 to map out distinct regimes through which a perturbation with a fixed spanwise wavelength evolves. The centrifugal force starts to influence the generation of the pressure when x* ~ ΛRΛG−2/3Λ, where RΛ denotes the Reynolds number based on Λ. The induced pressure leads to full coupling of the momentum equations when x* ~ ΛRΛGΛ−2/5. This is the crucial regime linking the pre-modal and modal phases of the perturbation because the governing equations admit growing asymptotic eigensolutions, which develop into fully fledged Görtler vortices of inviscid nature when x* ~ ΛRΛ. From this position onwards, local eigenvalue formulations are mathematically justified. Görtler vortices continue to amplify and enter the so-called most unstable regime when x* ~ ΛRΛGΛ, and ultimately approach the right-branch regime when x* ~ ΛRΛG2Λ.
22
Badulin, Sergei I., e Victor I. Shrira. "On the irreversibility of internal-wave dynamics due to wave trapping by mean flow inhomogeneities. Part 1. Local analysis". Journal of Fluid Mechanics 251 (giugno 1993): 21–53. http://dx.doi.org/10.1017/s0022112093003325.
Abstract (sommario):
The propagation of guided internal waves on non-uniform large-scale flows of arbitrary geometry is studied within the framework of linear inviscid theory in the WKB-approximation. Our study is based on a set of Hamiltonian ray equations, with the Hamiltonian being determined from the Taylor-Goldstein boundary-value problem for a stratified shear flow. Attention is focused on the fundamental fact that the generic smooth non-uniformities of the large-scale flow result in specific singularities of the Hamiltonian. Interpreting wave packets as particles with momenta equal to their wave vectors moving in a certain force field, one can consider these singularities as infinitely deep potential holes acting quite similarly to the ‘black holes’ of astrophysics. It is shown that the particles fall for infinitely long time, each into its own ‘black hole‘. In terms of a particular wave packet this falling implies infinite growth with time of the wavenumber and the amplitude, as well as wave motion focusing at a certain depth. For internal-wave-field dynamics this provides a robust mechanism of a very specific conservative and moreover Hamiltonian irreversibility.This phenomenon was previously studied for the simplest model of the flow non-uniformity, parallel shear flow (Badulin, Shrira & Tsimring 1985), where the term ‘trapping’ for it was introduced and the basic features were established. In the present paper we study the case of arbitrary flow geometry. Our main conclusion is that although the wave dynamics in the general case is incomparably more complicated, the phenomenon persists and retains its most fundamental features. Qualitatively new features appear as well, namely, the possibility of three-dimensional wave focusing and of ‘non-dispersive’ focusing. In terms of the particle analogy, the latter means that a certain group of particles fall into the same hole.These results indicate a robust tendency of the wave field towards an irreversible transformation into small spatial scales, due to the presence of large-scale flows and towards considerable wave energy concentration in narrow spatial zones.
23
BARAD, MICHAEL F., e OLIVER B. FRINGER. "Simulations of shear instabilities in interfacial gravity waves". Journal of Fluid Mechanics 644 (10 febbraio 2010): 61–95. http://dx.doi.org/10.1017/s0022112009992035.
Abstract (sommario):
An adaptive numerical method is employed to simulate shear instabilities in open-ocean internal solitary-like gravity waves. The method is second-order accurate, employs block-structured adaptive mesh refinement, solves the incompressible Navier–Stokes equations and allows for the simulation of all of the length scales of interest by dynamically tracking important regions with recursively-nested finer grids. Two-dimensional simulations are performed over a range of parameters, which allows us to assess the conditions under which the shear instabilities in the waves occur, including a method to evaluate the critical Richardson number for instability based on the bulk wave parameters. The results show that although the minimum Richardson number is well below the canonical value of 1/4 in all simulations, this value is not a sufficient condition for instability, but instead a much lower Richardson number of 0.1 is required. When the Richardson number falls below 0.1, shear instabilities develop and grow into two-dimensional billows of the Kelvin–Helmholtz type. A linear stability analysis with the Taylor–Goldstein equation indicates that an alternate criterion for instability is given by σiTw > 5, where σi is the growth rate of the instability averaged over Tw, the period in which parcels of fluid are subjected to a Richardson number of less than 1/4. A third criterion for instability requires that Lw/L > 0.86, where Lw is half the length of the region in which the Richardson number falls below 1/4 and L is the solitary wave half-width. An eigendecomposition of the rate-of-strain tensor is performed to show that the pycnocline thickness increases within the wave because of pycnocline-normal strain and not because of diffusion, which has important ramifications for stability. A three-dimensional simulation indicates that the primary instability is two-dimensional and that secondary, three-dimensional instabilities occur thereafter and lead to strong dissipation and mixing.
24
Bensalah, Antoine, Patrick Joly e Jean-Francois Mercier. "Mathematical analysis of Goldstein’s model for time-harmonic acoustics in flows". ESAIM: Mathematical Modelling and Numerical Analysis 56, n. 2 (24 febbraio 2022): 451–83. http://dx.doi.org/10.1051/m2an/2022007.
Abstract (sommario):
Goldstein’s equations have been introduced in 1978 as an alternative model to linearized Euler equations to model acoustic waves in moving fluids. This new model is particularly attractive since it appears as a perturbation of a simple scalar model: the potential model. In this work we propose a mathematical analysis of boundary value problems associated with Goldstein’s equations in the time-harmonic regime.
25
Kang, Irene, Jamie K. Forschmiedt, Michelle M. Loch, William E. Barlow, Danika L. Lew, Julie R. Gralow, Funda Meric-Bernstam et al. "Abstract GS1-04: Patient-reported cognitive impairment in women participating in the RxPONDER trial (SWOG S1007) by menopausal status". Cancer Research 83, n. 5_Supplement (1 marzo 2023): GS1–04—GS1–04. http://dx.doi.org/10.1158/1538-7445.sabcs22-gs1-04.
Abstract (sommario):
Abstract Introduction: Breast cancer treatment is associated with cancer-related cognitive impairment (CRCI). However, the differential effect of endocrine therapy (ET) vs chemotherapy followed by endocrine therapy (CET), including the impact of menopausal status, on CRCI is not well understood. Methods: Participants (pts) with hormone receptor positive, HER2 negative breast cancer with 1-3 positive lymph nodes and an Oncotype DX recurrence score of 0-25 enrolled in the RxPONDER trial were randomly assigned to ET alone versus CET. Until the health-related quality of life (HRQoL) accrual goal was reached, English speaking pts in the US were invited to complete HRQoL questionnaires including the 8-item PROMIS Perceived Cognitive Function Concerns (PCF) Short Form questionnaire shortly after randomization (baseline), as well as 6, 12, and 36 months after baseline. Analysis of measures of anxiety and fatigue is presented separately. Standardized T scores (mean 50; SD 10) for PCF were computed with higher scores indicating less cognitive impairment. The primary endpoint of this exploratory analysis was to compare mean PCF T scores by treatment arm and menopausal status. Separately by menopausal status, a generalized estimating equations (GEE) model was fit to the three timepoints adjusting for baseline to estimate the difference between treatment arms and whether there was a time trend over the three follow-up measures. Results: The HRQoL accrual exceeded the goal of 500 patients, with 74% of pts participating voluntarily until the QOL invitation was removed from the protocol (Dec 1, 2012). A total of 139 pre and 429 postmenopausal pts completed the questionnaires at baseline. T scores were similar between ET and CET arms at baseline [Table 1]. In the ET arm, T scores decreased from baseline to 6 and 12 months but recovered to baseline at 36 months. In the CET arm, T scores decreased from baseline to 6 months and 12 months but did not return to baseline at 36 months. The mean score difference between CET and ET over time was -3.02 (p=0.01) and -2.36 (p=0.003) for pre and postmenopausal pts, respectively. Adjusting for baseline, there was no significant time trend over the three follow-up periods for either premenopausal (p=0.12) or postmenopausal (p=0.49) pts. Dropoff occurred over time with 79%, 76%, 60% of pts at baseline participating at 6, 12, and 36 months. Complete endocrine treatment adherence data are not yet available at each timepoint. Conclusion: Chemoendocrine therapy has a greater negative effect on patient-reported CRCI compared to ET alone in both pre- and post-menopausal pts and it is sustained over 36 months. Interventions to prevent or treat CRCI are needed to improve the long-term quality of life of patients treated with CET. Table 1. Comparisons of mean Cognitive Function score by treatment arm and menopausal status. Citation Format: Irene Kang, Jamie K. Forschmiedt, Michelle M. Loch, William E. Barlow, Danika L. Lew, Julie R. Gralow, Funda Meric-Bernstam, Kathy S. Albain, Daniel F. Hayes, Nancy U. Lin, Edith A. Perez, Lori J. Goldstein, Priya Rastogi, Anne F. Schott, Steven Shak, Priyanka Sharma, Jieling Miao, Debu Tripathy, Lajos Pusztai, Gabriel N. Hortobagyi, Kevin Kalinsky, N. Lynn Henry. Patient-reported cognitive impairment in women participating in the RxPONDER trial (SWOG S1007) by menopausal status [abstract]. In: Proceedings of the 2022 San Antonio Breast Cancer Symposium; 2022 Dec 6-10; San Antonio, TX. Philadelphia (PA): AACR; Cancer Res 2023;83(5 Suppl):Abstract nr GS1-04.
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POLLOCK, M. D. "GOLDSTONE FIELDS IN THE SUPERSTRING THEORY". International Journal of Modern Physics D 17, n. 01 (gennaio 2008): 81–94. http://dx.doi.org/10.1142/s0218271808011882.
Abstract (sommario):
If global supersymmetry is broken by gaugino condensation in the hidden sector of the [Formula: see text] heterotic superstring theory after compactification, then the auxiliary field FB of the modulus B ≡ (B r , B i ) attains a finite value, while that of the dilaton A ≡ (A r , A i ) vanishes, FA = 0, the Goldstone fermion being the modulino [Formula: see text], the spin-½ component of the complex chiral supermultiplet [Formula: see text]. The Goldstone boson of the scale symmetry that is broken when the radius of the internal space is fixed at a constant value is B r , which is determined from the goldstino Lagrangian, compared term by term with the superstring Lagrangian, including higher-derivative gravitational terms [Formula: see text] and [Formula: see text], after linking the space–time curvature to the energy–momentum tensor of the goldstino via the Einstein equations. This non-linear formulation of supersymmetry, due to Volkov and Akulov, is expressed in terms of the goldstino alone, whose Lagrangian contains a negative cosmological constant, which can be cancelled by the super-Higgs mechanism of Deser and Zumino to make the gravitino massive and break supersymmetry at the level [Formula: see text] GeV, while [Formula: see text]. Here, the modulus has been scaled to the Hagedorn value for the heterotic superstring theory, [Formula: see text], and A r , identified with the inverse square of the tree-level gauge coupling, has been scaled to the calculation in the minimal supersymmetric standard model due to Weinberg, that g-2 = 1.39 at the unification mass MX = 2.2 × 1016 GeV, assuming three generations of elementary particles and two Higgs doublets. In the presence of gravitino condensation in the internal space, however, there is an arbitrary additional contribution to the cosmological constant, facilitating reduction of m s to ~ 100 TeV, say, and m3/2 to ~ 1 eV.
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Arnold, Anton, Amit Einav, Beatrice Signorello e Tobias Wöhrer. "Large Time Convergence of the Non-homogeneous Goldstein-Taylor Equation". Journal of Statistical Physics 182, n. 2 (febbraio 2021). http://dx.doi.org/10.1007/s10955-021-02702-8.
Abstract (sommario):
AbstractThe Goldstein-Taylor equations can be thought of as a simplified version of a BGK system, where the velocity variable is constricted to a discrete set of values. It is intimately related to turbulent fluid motion and the telegrapher’s equation. A detailed understanding of the large time behaviour of the solutions to these equations has been mostly achieved in the case where the relaxation function, measuring the intensity of the relaxation towards equally distributed velocity densities, is constant. The goal of the presented work is to provide a general method to tackle the question of convergence to equilibrium when the relaxation function is not constant, and to do so as quantitatively as possible. In contrast to the usual modal decomposition of the equations, which is natural when the relaxation function is constant, we define a new Lyapunov functional of pseudodifferential nature, one that is motivated by the modal analysis in the constant case, that is able to deal with full spatial dependency of the relaxation function. The approach we develop is robust enough that one can apply it to multi-velocity Goldstein-Taylor models, and achieve explicit rates of convergence. The convergence rate we find, however, is not optimal, as we show by comparing our result to those found in [8].
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Espath, Luis. "A continuum framework for phase field with bulk-surface dynamics". Partial Differential Equations and Applications 4, n. 1 (12 dicembre 2022). http://dx.doi.org/10.1007/s42985-022-00218-8.
Abstract (sommario):
AbstractThis continuum mechanical theory aims at detailing the underlying rational mechanics of dynamic boundary conditions proposed by Fischer et al. (Phys Rev Lett 79:893, 1997), Goldstein et al. (Phys D Nonlinear Phenom 240:754–766, 2011), and Knopf et al. (ESAIM Math Model Numer Anal 55:229–282, 2021). As a byproduct, we generalize these theories. These types of dynamic boundary conditions are described by the coupling between the bulk and surface partial differential equations for phase fields. Our point of departure within this continuum framework is the principle of virtual powers postulated on an arbitrary part $$\mathcal {P}$$ P where the boundary $$\partial \mathcal {P}$$ ∂ P may lose smoothness. That is, the normal field may be discontinuous along an edge $$\partial ^2\mathcal {P}$$ ∂ 2 P . However, the edges characterizing the discontinuity of the normal field are considered smooth. Our results may be summarized as follows. We provide a generalized version of the principle of virtual powers for the bulk-surface coupling along with a generalized version of the partwise free-energy imbalance. Next, we derive the explicit form of the surface and edge microtractions along with the field equations for the bulk and surface phase fields. The final set of field equations somewhat resembles the Cahn–Hilliard equation for both the bulk and surface. Moreover, we provide a suitable set of constitutive relations and thermodynamically consistent boundary conditions. In Knopf et al. (2021), a mixed (Robin) type of boundary condition for the chemical potentials is proposed for the model in Fischer et al. (1997), Goldstein et al. (2011). In addition to this boundary condition, we also include this type of mixed boundary condition for the microstructure, that is the phase fields. Lastly, we derive the Lyapunov-decay relations for these mixed type of boundary conditions for both the microstructure and chemical potential.
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Camassa, R., e C. Viotti. "Transient dynamics by continuous-spectrum perturbations in stratified shear flows". Journal of Fluid Mechanics 717 (7 febbraio 2013). http://dx.doi.org/10.1017/jfm.2013.19.
Abstract (sommario):
AbstractThe transient dynamics of the linearized Euler–Boussinesq equations governing parallel stratified shear flows is presented and analysed. Solutions are expressed as integral superpositions of generalized eigenfunctions associated with the continuous-spectrum component of the Taylor–Goldstein linear stability operator, and reveal intrinsic dynamics not captured by its discrete-spectrum counterpart. It is shown how continuous-spectrum perturbations are generally characterized by non-normal energy growth and decay with algebraic asymptotic behaviour in either time or space. This behaviour is captured by explicit long-time/far-field expressions from rigorous asymptotic analysis, and it is illustrated with direct numerical simulations of the whole (non-Boussinesq) stratified Euler system. These results can be helpful in understanding recent numerical observations for parallel and non-parallel perturbed stratified shear flows.
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Pereira, Luiz Mariano, João Nazareno Nonato Quaresma, Jesús Salvador Péerez Guerrero e Renato M. Cotta. "Integral Transform Solution for Natural Convection within Horizontal Concentric Annular Cavities". ASME Journal of Heat and Mass Transfer, 16 marzo 2024, 1–38. http://dx.doi.org/10.1115/1.4065085.
Abstract (sommario):
Abstract Natural convection inside horizontal concentric annular cavities is dealt with through the Generalized Integral Transform Technique (GITT), offering a hybrid numerical-analytical solution of the continuity, Navier-Stokes, and energy equations in cylindrical coordinates. The flow is in steady-state, laminar regime, two-dimensional, buoyancy-induced, and the governing equations are written in the streamfunction-only formulation. Two strategies of integral transformation are adopted to verify the best computational performance, namely, the usual one with eigenvalue problems for both streamfunction and temperature defined in the radial variable, and a novel alternative with eigenvalue problems defined in the azimuthal angular coordinate. First, the eigenfunction expansions convergence behavior is analyzed to critically compare the two integral transform solution strategies. Then, test cases for different aspect ratio and Rayleigh number are validated with experimental data from the classical work of Kuehn and Goldstein. A maximum relative deviation of 5% is found comparing the GITT results for the average Nusselt number against the experimental data, while a 8% maximum relative deviation is found comparing against an empirical correlation by the same authors. It is concluded that the GITT solution with the eigenvalue problem in the angular coordinate yields better convergence rates than the more usual eigenfunction expansion in the radial variable. This is due to the originally homogeneous boundary conditions in the angular direction, which do not require filtering for convergence enhancement, as opposed to the required filter in the radial direction that introduces a source term in the filtered equation for the streamfunction.
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"Singularities encountered in three-dimensional boundary layers under an adverse or favourable pressure gradient". Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences 352, n. 1698 (15 luglio 1995): 45–87. http://dx.doi.org/10.1098/rsta.1995.0058.
Abstract (sommario):
Singularities in solutions of the classical boundary-layer equations are considered, numerically and analytically, in an example of steady hypersonic flow along a flat plate with three-dimensional surface roughness. First, a wide parametric study of the breakdown of symmetry-plane flow is performed for two particular cases of the surface geometry. Emphasis is put on the structural stability of the singularities’ development to local/global variation of the pressure distribution. It is found that, as usual, the solution behaviour under an adverse pressure gradient involves the Goldstein- or marginal-type singularity at a point of zero streamwise skin friction. As the main alternative, typical of configurations with favourable or zero pressure forcing, an inviscid breakdown in the middle of the flow is identified. Similarly to unsteady flows, the main features of the novel singularity include infinitely growing boundary-layer thickness and finite limiting values of the skin-friction components. Subsequent analytical extensions of the singular symmetry-plane solution then suggest two different scenarios for the global boundary-layer behaviour: one implies inviscid breakdown of the flow at some singular line, the other describes the development of a boundary-layer collision at a downstream portion of the symmetry plane. In contrast with previous studies of the collision phenomenon in steady flows, the present theory suggests logarithmic growth of boundary-layer thickness on both sides of the discontinuity. Finally, an example of numerical solution of the full three dimensional boundary layer equations is given. The flow régime chosen corresponds to inviscid breakdown of a centreplane flow under a favourable pressure gradient and development of the discontinuity/collision downstream. The numerical results near the origin of the discontinuity are found to be supportive, producing quantitative agreement with the local analytical description.
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CANTAŞ, Fatih, Serdar ÖZYÖN e Celal YAŞAR. "Runge Kutta Optimization for Fixed Size Multimodal Test Functions". International Scientific and Vocational Studies Journal, 30 dicembre 2022. http://dx.doi.org/10.47897/bilmes.1219033.
Abstract (sommario):
In this study, it is aimed to increase the success of the Runge Kutta (RUN) algorithm, which is used in the solution of many optimization problems in the literature, on fixed-size test functions by changing the parameter values. Optimization can be defined as making a system most efficient at the least possible cost under certain constraints. For this process, many optimization algorithms have been designed in the literature and used to obtain the best solutions for certain problems. The most important parts in solving these problems are modeling the problem correctly, determining the parameters and constraints of the problem, and finally choosing a suitable meta-heuristic algorithm for the solution of the objective function. Not every algorithm is suitable for every problem structure. Therefore, in this study, the suitability of the RUN algorithm for the solution of fixed-size functions will be evaluated. Theoretically, Runge-Kutta methods used in numerical analysis are an important type of the family of closed and open iterative methods for solution approximations of ordinary differential equations. The RUN algorithm is also designed with inspiration from these methods. In order to evaluate the performance of the RUN algorithm on fixed-size functions in the study, 10 fixed-size multimodal test functions (Shekel's Foxholes, Kowalik, Six-Hump Camel-Back, Branin, Goldstein-Price, Hartman3, Hartman6, Shekel5, Shekel7, Shekel10) have been found in the literature before was selected. Solutions for each of the selected functions are obtained by changing the parameter values of the RUN algorithm. The obtained solution values were evaluated by comparing the solutions obtained with Slime Mold Algorithm (SMA) and Hunger Games Search (HGS) algorithms.