Matched asymptotic solutions for turbulent plumes

2012 ◽  
Vol 699 ◽  
pp. 489-499 ◽  
Author(s):  
Fabien Candelier ◽  
Olivier Vauquelin
Keyword(s):  
Asymptotic Expression ◽  
Classical Problem ◽  
Asymptotic Solutions ◽  
Far Field ◽  
Vertical Coordinate ◽  
Outer Solution ◽  
Virtual Origin ◽  
Single Function ◽  
Vertical Evolution ◽  
Made In

AbstractRecent analytical investigations have shown that the vertical evolution of turbulent plumes variables can be derived straightforwardly from the knowledge of a single function $\Gamma (z)$ (called the plume function) which is the solution of a nonlinear differential equation. This article presents matched asymptotic solutions of this equation in the cases corresponding to highly lazy or highly forced plumes. First, it is shown that, far from the source, the asymptotic expression of the plume function can be derived by means of a perturbation method based on a Padé-like approximation. The resulting outer solution is invariant under translation (with respect to the vertical coordinate) so that we are led to the classical problem concerning the location of the plume (asymptotic) virtual origin. In order to determine this virtual origin location as a function of the conditions at the source, the far-field asymptotic solution is matched to an inner expansion of the solution which is valid near the source. Comparisons between these asymptotic solutions and numerical results are finally made in order to test their validity.

scholarly journals The virtual origin as a first-order correction for the far-field description of laminar jets

2002 ◽  
Vol 14 (6) ◽  
pp. 1821-1824 ◽  
Author(s):  
Antonio Revuelta ◽  
Antonio L. Sánchez ◽  
Amable Liñán
Keyword(s):  
Order Correction ◽  
Far Field ◽  
First Order ◽  
Virtual Origin ◽  
Laminar Jets ◽  
First Order Correction

THE INFLUENCE OF A CRITICAL ANGLE ON AN ORIENTATION OF RADIATION OF A SHEAR WAVE THE ANGLE BEAM PROBE

2019 ◽  
pp. 14-25
Author(s):  
V. N. Danilov
Keyword(s):  
Shear Wave ◽  
Critical Angle ◽  
Asymptotic Expression ◽  
Shear Waves ◽  
Far Field ◽  
Directivity Characteristic ◽  
The Third ◽  
Working Frequency

In a far field it is received asymptotic expression of displacement of the shear waves transmitted in the elastic environment by the angle beam probe in view of features of radiation of such waves under a angle of probe, coming nearer to the third critical. At sufficient remoteness from a critical corner this expression passes in received earlier in geometroacustical approximation. The estimations carried out for steel have shown, that for converters with nominal angles of probe 37 – 40 influence of this critical angle causes increase of an angle of registration of a maximum of the signal, observed earlier experimentally. This feature is influenced as distance up to points of registration of a shear wave, and with working frequency of the angle beam probe and its size piezoplate (width of the directivity characteristic).

scholarly journals On one rational integral operator of Fourier – Chebyshev type and approximation of Markov functions

2020 ◽  
pp. 6-27
Author(s):  
Pavel G. Patseika ◽  
Yauheni A. Rouba ◽  
Kanstantin A. Smatrytski
Keyword(s):  
Integral Operator ◽  
Uniform Approximation ◽  
Asymptotic Expression ◽  
Classical Problem ◽  
Rational Functions ◽  
Fixed Number ◽  
Approximation Properties ◽  
Elementary Functions ◽  
Rational Chebyshev ◽  
Rational Integral

The purpose of this paper is to construct an integral rational Fourier operator based on the system of Chebyshev – Markov rational functions and to study its approximation properties on classes of Markov functions. In the introduction the main results of well-known works on approximations of Markov functions are present. Rational approximation of such functions is a well-known classical problem. It was studied by A. A. Gonchar, T. Ganelius, J.-E. Andersson, A. A. Pekarskii, G. Stahl and other authors. In the main part an integral operator of the Fourier – Chebyshev type with respect to the rational Chebyshev – Markov functions, which is a rational function of order no higher than n is introduced, and approximation of Markov functions is studied. If the measure satisfies the following conditions: suppμ = [1, a], a > 1, dμ(t) = ϕ(t)dt and ϕ(t) ἆ (t − 1)α on [1, a] the estimates of pointwise and uniform approximation and the asymptotic expression of the majorant of uniform approximation are established. In the case of a fixed number of geometrically distinct poles in the extended complex plane, values of optimal parameters that provide the highest rate of decreasing of this majorant are found, as well as asymptotically accurate estimates of the best uniform approximation by this method in the case of an even number of geometrically distinct poles of the approximating function. In the final part we present asymptotic estimates of approximation of some elementary functions, which can be presented by Markov functions.

Asymptotic Solutions and Radiation Conditions for Second-Order Diffracted Waves

1989 ◽  
Vol 111 (3) ◽  
pp. 203-207
Author(s):  
H. Huang ◽  
J. Li ◽  
X. Wang
Keyword(s):  
Finite Element ◽  
Circular Cylinder ◽  
Second Order ◽  
Asymptotic Solutions ◽  
Wave Forces ◽  
Far Field ◽  
Two Dimensional ◽  
Diffracted Waves ◽  
Artificial Boundaries ◽  
Radiation Conditions

The far-field asymptotic solutions for the second-order diffracted waves have been developed, both in three and two-dimensional problems. The radiation conditions for the second-order diffracted waves are derived by using the asymptotic solutions. The nonlinear wave forces on a half-circular cylinder on seabed are presented by using finite element methods with the radiation conditions imposed on the artificial boundaries.

Stretching of viscous threads at low Reynolds numbers

2011 ◽  
Vol 683 ◽  
pp. 212-234 ◽  
Author(s):  
Jonathan J. Wylie ◽  
Huaxiong Huang ◽  
Robert M. Miura
Keyword(s):  
Asymptotic Expression ◽  
Classical Problem ◽  
Finite Extension ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Reynolds Numbers ◽  
Initial Surface ◽  
Initial Shape ◽  
Asymptotic Expressions ◽  
Initial Boundary Value

AbstractWe investigate the classical problem of the extension of an axisymmetric viscous thread by a fixed applied force with small initial inertia and small initial surface tension forces. We show that inertia is fundamental in controlling the dynamics of the stretching process. Under a long-wavelength approximation, we derive leading-order asymptotic expressions for the solution of the full initial-boundary value problem for arbitrary initial shape. If inertia is completely neglected, the total extension of the thread tends to infinity as the time of pinching is approached. On the other hand, the solution exhibits pinching with finite extension for any non-zero Reynolds number. The solution also has the property that inertia eventually must become important, and pinching must occur at the pulled end. In particular, pinching cannot occur in the interior as can happen when inertia is neglected. Moreover, we derive an asymptotic expression for the extension.

Diffraction by a wide slit and complementary strip. I

1958 ◽  
Vol 54 (4) ◽  
pp. 479-496 ◽  
Author(s):  
R. F. Millar
Keyword(s):  
Asymptotic Expression ◽  
Normal Incidence ◽  
Present Theory ◽  
Slit Width ◽  
Simple Problem ◽  
Far Field ◽  
Angle Of Incidence ◽  
Current Densities ◽  
Aperture Distribution ◽  
Wide Slit

ABSTRACTIn recent years much attention has been directed towards the asymptotic solution of diffraction problems. In the present work, consideration is given to the relatively simple problem of the diffraction of an E-polarized plane wave by an infinite slit. The solution takes the form of a series in inverse powers of the ratio of slit-width to wavelength of the incident wave, and is based on the solution by successive substitutions of a pair of integral equations. The current densities induced on both halves of the screen are calculated, from which is deduced the electric field in the slit. The far-field is determined from the aperture distribution, and an asymptotic expression is found for the transmission coefficient as a function of the angle of incidence and the ratio of slit-width to wavelength.While most previous work has been confined to near-normal incidence, the present theory is uniformly valid for all angles of incidence.

Wake collapse in a stratified fluid: linear treatment

1972 ◽  
Vol 51 (3) ◽  
pp. 613-618 ◽  
Author(s):  
R. J. Hartman ◽  
H. W. Lewis
Keyword(s):  
Exact Solutions ◽  
Initial Value Problem ◽  
Asymptotic Expression ◽  
Boussinesq Approximation ◽  
Stratified Fluid ◽  
Velocity Fields ◽  
Wave Radiation ◽  
Far Field ◽  
Initial Value ◽  
Detailed Nature

The linear initial-value problem of a partially mixed cylindrical wake in a, uniformly stratified fluid is formulated and exact solutions are given for the density and velocity fields inside and just outside the original cylinder. An asymptotic expression for the far-field internal wave radiation is given and the corresponding solutions for a spherical wake geometry are noted. The treatment emphasizes the inadequancy of the usual linear Boussinesq approximation to describe the detailed nature of similar problems, in particular the fully mixed wake-collapse problem.

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