On the application of mosaic-skeleton approximations of matrices for the acceleration of computations in the vortex method for the three-dimensional Euler equations

We study the possibility of using fast matrix multiplication methods for the approximation of the velocity field when solving the system of differential equations describing the vorticity transport in an ideal incompressible fluid in Lagrangian coordinates. We suggest a numerical scheme that permits effectively using the fast matrix multiplication (the method of mosaic-skeleton approximations). We show that the functions used for the computation of the velocity field and moving grids appearing in the solution of the problem permit one to use the above-mentioned method. We prove the convergence of the resulting numerical solution to the exact solution with regard of the error contributed by the use of the algorithm for approximate fast multiplication of matrices by vectors.     

A pair of stream functions for three-dimensional vortex flows

Introducing a vector potential, that is based on a pair of stream functions, and a velocity potential, antisymmetric equations for the stream functions are derived with the help of a variational principle. It is found that the equations are in a suitable form to investigate flows with helical symmetry, and, for example, to connect upstream axisymmetric flows with downstream helical flows. The special case of a transition from an upstream solid-body vortex to a downstream helical flow is investigated in detail. Furthermore, the stream-function equations are particularly useful to investigate general small-amplitude inertia waves on vortex flows. Time-dependent helical flows that are time-independent in a suitably rotating frame of reference can also be discussed with the proposed method.     

On the existence of mosaic-skeleton approximations for discrete analogues of integral operators

Exterior three-dimensional Dirichlet problems for the Laplace and Helmholtz equations are considered. By applying methods of potential theory, they are reduced to equivalent Fredholm boundary integral equations of the first kind, for which discrete analogues, i.e., systems of linear algebraic equations (SLAEs) are constructed. The existence of mosaic-skeleton approximations for the matrices of the indicated systems is proved. These approximations make it possible to reduce the computational complexity of an iterative solution of the SLAEs. Numerical experiments estimating the capabilities of the proposed approach are described.     

Linearized spin-up in vortex flows

Summary Transient motions of contained laminar vortex flows are investigated by solving a linearized (low Rossby number) spin-up problem for arbitrary primary flow circulation distribution. If the local Ekman number is of the order of one or larger, spin-up occurs solely by viscous diffusion (Rayleigh layers), but if the local Ekman number is small compared with one, spin-up occurs by the action of an Ekman layer induced secondary flow in a time much less than the viscous diffusion time. The unsteady motion is superimposed on a steady state secondary flow which compensates for diffusion of primary flow vorticity.

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Zusammenfassung Zeitabhängige Bewegungen einer laminaren Wirbelströmung in einem Behälter werden untersucht, indem ein (für kleine Rossby-Zahlen) linearisiertes Spin-up-Problem für willkürliche Zirkulationsverteilung der primären Strömung gelöst wird. Wenn die lokale Ekman-Zahl von der Grössenordnung Eins oder grösser wird, so erfolgt der Spin-up ausschliesslich durch viskose Diffusion (Rayleighsche Schichten), doch wenn die lokale Ekman-Zahl verglichen mit Eins klein ist, so erfolgt der Spin-up durch die Wirkung einer durch eine Ekman-Schicht induzierten sekundären Strömung in einer Zeit, die viel kürzer ist als die viskose Diffusionszeit. Die nichtstationäre Bewegung ist einer permanenten Sekundär-Strömung überlagert, welche die Diffusion der primären Strömungsrotation kompensiert.

     

Energy flows in a vortex tube

The energy separation within a vortex tube filled with turbulent compresible fluid is investigated with an order-of-magnitude analysis of the energy equation. The physical processes corresponding to the important terms are: a heat flux due to turbulent mixing of the compressible fluid through radial pressure and temperature gradients, a flux of total energy produced by Archimedean forces, and work fluxes associated with the two most important Reynolds' stresses. All these fluxes will commonly be outwards and will tend to cool the vortex core. Experimental results are used to estimate, the relative magnitudes of the contributions. The Archimedean effect seems to be the least important.

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Zusammenfassung Die Energieverteilung in einem mit turbulenter kompressibler Flüssigkeit gefüllten Wirbelrohr wird durch Berechnung der Gliedergrösse der Energiegleichung untersucht. Die physikalischen Vorgänge, die den wichtigen Gliedern entsprechen, sind folgende: ein Wärmefluss, veranlasst durch turbulente Mischung der kompressible Flüssigkeit infolge der radialen Druck- und Temperaturgradienten; ein Fluss des Gesamtwärmeinhaltes, verursacht durch die Schwimmkraft; und die Arbeitsflüsse, die von den beiden wichtigsten Reynoldsschen scheinbaren Spannungen abhängen. Alle diese Flüsse sind gewöhnlich nach aussen gerichtet und dienen zum Kühlen des Wirbelkerns. Die experimentellen Ergebnisse werden zu einer Abschätzung der Grösse dieser Beiträge gebraucht. Der Schwimmkrafteffekt scheint am wenigsten wichtig.

     

Computation of transitions in Taylor vortex flows

The very reliable numerical methods of [15, 18] were used for the investigation of axisymmetric flows between infinitely long rotating cylinders. Solutions consisting of vortices of different sizes were detected. Varying both the wavelength of the vortices and the Reynolds number, we showed that the solutions are intermediate states between solutions with equally sized vortices (transitions from 2 to 4 and from 2 to 6 vortices).

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Zusammenfassung Die sehr zuverlässigen numerischen Methoden von [15, 18] wurden zur Berechnung von axialsymmetrischen Strömungen zwischen unbeschränkt langen, rotierenden Zylindern benutzt. So wurden Lösungen gefunden, die aus Wirbeln unterschiedlicher Größen bestehen. Durch Variation der Wellenlänge der Wirbel und der Reynoldszahl wurde gezeigt, daß diese Lösungen Übergangszustände sind zwischen verschiedenen Lösungen mit Wirbeln einheitlicher Größe (Übergänge von 2 zu 4 und von 2 zu 6 Wirbeln).

     

Strong approximations of three-dimensional Wiener sausages

We prove that the centered three-dimensional Wiener sausage can be strongly approximated by a one-dimensional Brownian motion running at a suitable time clock. The strong approximation gives all possible laws of iterated logarithm as well as the convergence in law in terms of process for the normalized Wiener sausage. The proof relies on Le Gall [10]șs fine L ²-norm estimates between the Wiener sausage and the Brownian intersection local times. Research supported by the Hungarian National Foundation for Scientific Research, Grants T 037886, T 043037 and K 61052.     

Transient flows in queueing systems via closure approximations

We present a method for obtaining approximations to the distribution of flow times of customers in arbitrary queueing systems. We first propose approximations for uni-variate and multi-variate distributions of non-negative random variables. Then using a closure approximation, we show that the distribution of flow time can be calculated recursively. Computational results for the single server, multi-server and tandem queues are encouraging, with less than 5%average error in the mean flow time in most cases. The average error in the variance of flow times is found to be less than 10% for the more regular distributions.     

Tollmien-Schlichting/vortex interactions in compressible boundary-layer flows

A weakly nonlinear interaction of oblique Tollmien-Schlichtingwaves and longitudinal vortices in compressible, high Reynoldsnumber, boundary-layer flow over a flat plate is consideredfor all ranges of the Mach number. The interaction equationsconsist of equations for the vortex which is indirectly forcedby the waves via a boundary condition, whereas a vortex termappears in the amplitude equation for the wave pressure. Thedownstream solution properties of interaction equations arefound to depend on the sign of an interaction coefficient. Thisparticular type of weakly nonlinear interaction was first proposedby Hall & Smith (1989), who considered incompressible flows;however, there are some errors in their formulation. Correctedresults for the incompressible regime are presented for comparisonwith those calculated for compressible flows. Compressibilityis found to have a significant effect on the interaction properties,principally through its impact on the waves and their governingmechanism, the ‘triple-deck’ structure. It is foundthat, in general, the flow quantities will grow slowly withincreasing downstream coordinate. However, for flows with Machnumber values below 2, there exists a small band of wave obliquenessangles for which the solutions terminate in abrupt, finite-distance‘break-ups’.     

Lagrangian chaos and multiphase processes in vortex flows

We discuss experimental and numerical studies of the effects of Lagrangian chaos (chaotic advection) on the stretching of a drop of an immiscible impurity in a flow. We argue that the standard capillary number used to describe this process is inadequate since it does not account for advection of a drop between regions of the flow with varying velocity gradient. Consequently, we propose a Lagrangian-generalized capillary number C_L number based on finite-time Lyapunov exponents. We present preliminary tests of this formalism for the stretching of a single drop of oil in an oscillating vortex flow, which has been shown previously to exhibit Lagrangian chaos. Probability distribution functions (PDFs) of the stretching of this drop have features that are similar to PDFs of C_L. We also discuss on-going experiments that we have begun on drop stretching in a blinking vortex flow.     

Finite difference approximations for the three-dimensional Laplacian in irregular grids

Two numerical approximations of the three-dimensional Laplacian Operator at a point are each obtained in a form which is suitable for computation at nodes generated in an automatic decomposition of an arbitrary three-dimensional region. The first expression is derived by an extension of a technique previously developed for two-dimensional configurations and is shown to generalise the well known Cartesian form. The second approximation, which assumes a linear dependence of the scalar field on certain nodal values of this field, and for which a variational justification is established, also reduces to the Cartesian form for a particular arrangement of surrounding nodes.

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Zusammenfassung Es werden zwei numerische Näherungen für den dreidimensionalen Laplace-Operator in einem Punkt erhalten; sie eignen sich für Berechnungen in Knotenpunkten, welche durch automatische Zerlegung eines beliebigen dreidimensionalen Feldes entstehen. Der erste Ausdruck wurde durch Verallgemeinerung einer früher entwickelten Methode für zwei Dimensionen gewonnen und führt zur Verallgemeinerung der bekannten kartesischen Form. Die zweite Form der Näherung, bei der eine lineare Abhängigkeit des skalaren Feldes von gewissen Feldwerten im Knotenpunkt angenommen wird (und für die eine Rechtfertigung durch ein Variationsprinzip gegeben wird), reduziert sich ebenfalls auf die kartesische Form bei geeigneter Anordnung der umgebenden Knotenpunkte.

     

On solution of higher approximations to boundary layer flows

Résumé Pour trouver des approximations d'un ordre élevé dans les problèmes de couche limite, l'auteur propose une méthode qui unifie le approches de Meksyn et Görtler. Les valeurs initiales, pour les approximations successives de Görtler, sont evaluées à l'aide de la methode de Meksyn. Toutes les valeurs initiales sont obtaneues par différentiation d'une seule équation. Les résultats obtenus sont au moins équivalent à ceux obtenus par la méthode de Hsu, avec considérablement moins de calculs.     

Finite volume element approximations of nonlocal reactive flows in porous media

In this article, we study finite volume element approximations for two‐dimensional parabolic integro‐differential equations, arising in the modeling of nonlocal reactive flows in porous media. These types of flows are also called NonFickian flows and exhibit mixing length growth. For simplicity, we consider only linear finite volume element methods, although higher‐order volume elements can be considered as well under this framework. It is proved that the finite volume element approximations derived are convergent with optimal order in H¹‐ and L²‐norm and are superconvergent in a discrete H¹‐norm. By examining the relationship between finite volume element and finite element approximations, we prove convergence in L^∞‐ and W^1,∞‐norms. These results are also new for finite volume element methods for elliptic and parabolic equations. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 285–311, 2000     

Symbolic derivation of finite difference approximations for the three-dimensional Poisson equation

A symbolic procedure for deriving various finite difference approximations for the three-dimensional Poisson equation is described. Based on the software package Mathematica, we utilize for the formulation local solutions of the differential equation and obtain the standard second-order scheme (7-point), three fourth-order finite difference schemes (15-point, 19-point, 21-point), and one sixth-order scheme (27-point). The symbolic method is simple and can be used to obtain the finite difference approximations for other partial differential equations. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 593–606, 1998     

Generation of large-scale vortex dislocations in a three-dimensional wake-type flow

Numerical study of three-dimensional evolution of wake-type flow and vortex dislocations is performed by using a compact finite diffenence-Fourier spectral method to solve 3-D incompressible Navier-Stokes equations. A local spanwise nonuniformity in momentum defect is imposed on the incoming wake-type flow. The present numerical results have shown that the flow instability leads to three-dimensional vortex streets, whose frequency, phase as well as the strength vary with the span caused by the local nonuniformity. The vortex dislocations are generated in the nonuniform region and the large-scale chain-like vortex linkage structures in the dislocations are shown. The generation and the characteristics of the vortex dislocations are described in detail.     

The absence of mixing in special flows over rearrangements of segments

Translated from Matematicheskie Zametki, Vol. 55, No. 6, pp. 146–149, June, 1994.     

Macroscopic parameters of three-dimensional flows in free shear turbulence

The initial stage of the onset of turbulence in a three-dimensional compressible inviscid shear flow is studied. An initial deterministic velocity perturbation in the form of one or several Fourier modes leads to the development of a cascade of instabilities, which is numerically simulated. The influence exerted on the formation of the cascade of instabilities and the transition to turbulence by the size of the computational domain, the shear layer width, and the initial conditions is analyzed. It is shown that the mechanism of turbulence onset is essentially three-dimensional. The influence of various flow parameters and initial conditions on the formation of the turbulence cascade is studied numerically.