On self-preserving,variable-density,turbulent free jets

Appropriately defined Gaussian error-functions are shown to represent closed-form solutions of self-preserving, axisymmetric, variable density, free turbulent jets. The turbulent diffusivities of momentum, mass or heat associated with such solutions are found to vary both in the streamwise and radial directions, and are different from each other. An entrainment function for the jet can also be derived from the present analysis. The entrainment coefficient is found to be uniquely related to the turbulent Reynolds number, Re _t , of the jet, which is one of the two parameters in the closed-form solutions. Inverse centreline values of velocity, density, mass fraction and temperature are found to be linearly proportional to the streamwise coordinate as are the spread of velocity and scalar. If the two parameters are determined from jet spread data, the resulting closed-form solutions are in good agreement with measurements.

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Zusammenfassung Es wird gezeigt, daß geeignet definierte Gauss'sche Fehlerfunktionen für ähnliche achssymmetrische Freistrahlen Lösungen in geschlossener Form darstellen. Die mit solchen Lösungen verbundenen Impuls-, Stoff- und Energieaustauschgrößen verändern sich sowohl in Strömungs- als auch in Radialrichtung und sind voneinander verschieden. Auch eine Entrainmentfunktion des Freistrahls ist ableitbar von der vorliegenden Analyse. Es wird gezeigt, daß der Entrainment-koeffizient des Freistrahls einzig von der turbulenten Reynolds-Zahl Re _t , einem der zwei Parameter der Lösung, abhängig ist. Die Inversen der Geschwindigkeits-, Dichte-, Massenverhältnis- und Temperaturwerte auf der Strahlachse steigen in Strömungsrichtung linear an. Wenn die zwei Parameter aus den Versuchswerten der Strahlausbreitung ermittelt werden, ergibt die resultierende analytische Lösung eine gute Übereinstimmung mit den Messungen.

     

On the similarity solutions for a steady MHD equation

In this paper, we investigate the similarity solutions for the steady laminar incompressible boundary layer equations governing the magnetohydrodynamic (MHD) flow near the forward stagnation point of two-dimensional and axisymmetric bodies. This leads to the study of a boundary value problem involving a third order autonomous ordinary differential equation. Our main results are the existence, uniqueness and non-existence for concave or convex solutions.     

Existence of turbulent behavior for non-chaotic two dimensional jets

The classic models of mass transfer for free turbulent jets very often contain neglected terms. This is because these terms are considered so small that their effect can be neglected; however, because the models are usually structurally unstable, these neglected terms, even if they are small, may change the behavior of the system. Furthermore, the existing classic models cannot simulate the far behavior of the fluid, for example, the eddies existing in the jet region of the fluid stream. In this paper, we consider a general implicit small perturbation of the classic model and study the effect of it on the system. We will show that the local bifurcations may imply the existence of eddies for the fluid stream; for example, we will see that, under some conditions, the Hopf bifurcation generates permanent eddies for the system.     

The integral equations for round buoyant jets in stratified flows

The basic integral equations governing the behavior of a round, buoyant momentum jet discharged into an infinite, stratified, turbulent, flowing ambient fluid are derived. The analysis is based on the fundamental partial differential equations for conservation of mass. momentum, concentration and heat energy, transformed by vector operations into a natural coordinate system and integrated in the angular and radial directions using symmetry and similarity assumptions. The common Boussinesq-assumption has not been used. The resulting set of non-linear first-order differential equations differs from previous systems similarly based on integral methods, and which are presently used to model spreading and dilution of chimney gases in the atmosphere or of waste water in the ocean. The differences caused by mathematical errors and methodological inconsistencies in the previous approaches are pointed out and discussed.

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Zusammenfassung In der vorliegenden Arbeit werden die Integralgleichungen zur Berechnung des Ausbreitungsverhaltens runder turbulenter Auftriebsstrahlen in näherungsweise unendlich ausgedehnten, geschichteten und turbulenzbchafteten Querströmungen hergeleitet. Hierzu werden die partiellen Differentialgleichungen der Strömungsmechanik für die Erhaltung von Masse, Impuls. Energie und Fremdstoffbeimengungen in ein problemspezifisches Koordinatensystem transformiert und unter Verwendung der üblichen Symmetrieund Ähnlichkeitsannahmen in azimutaler und radialer Richtung integriert. Die übliche Boussinesq-Vereinfachung wird nicht angewendet. Die sich ergebenden gewöhnlichen Differentialgleichungen erster Ordnung unterscheiden sich signifikant von jenen, die den derzeit gebräuchlichen, ebenfalls auf Integralverfahren basierenden mathematischen Modellen zur Berechnung von Abgasemissionen in die Atmosphäre bzw. Kühl-oder Abwassereinleitungen ins Meer zugrundeliegen. Die die Abweichungen verursachenden Fehler mathematischer und methodischer Art werden aufgezeigt und diskutiert.

     

On similarity solutions for boundary layer flows with prescribed heat flux

This paper is concerned with existence, uniqueness and behaviour of the solutions of the autonomous third‐order non‐linear differential equation f?+(m+2)f f″?(2m+1)f′²=0 on ?⁺ with the boundary conditions f(0)=?γ, f′(∞)=0 and f″(0)=?1. This problem arises when looking for similarity solutions for boundary layer flows with prescribed heat flux. To study solutions we use some direct approach as well as blowing‐up co‐ordinates to obtain a plane dynamical system. Copyright © 2004 John Wiley & Sons, Ltd.     

On the bidirectional vortex and other similarity solutions in spherical coordinates

The bidirectional vortex refers to the bipolar, coaxial swirling motion that can be triggered, for example, in cyclone separators and some liquid rocket engines with tangential aft-end injectors. In this study, we present an exact solution to describe the corresponding bulk motion in spherical coordinates. To do so, we examine both linear and nonlinear solutions of the momentum and vorticity transport equations in spherical coordinates. The assumption will be that of steady, incompressible, inviscid, rotational, and axisymmetric flow. We further relate the vorticity to some power of the stream function. At the outset, three possible types of similarity solutions are shown to fulfill the momentum equation. While the first type is incapable of satisfying the conditions for the bidirectional vortex, it can be used to accommodate other physical settings such as Hill’s vortex. This case is illustrated in the context of inviscid flow over a sphere. The second leads to a closed-form analytical expression that satisfies the boundary conditions for the bidirectional vortex in a straight cylinder. The third type is more general and provides multiple solutions. The spherical bidirectional vortex is derived using separation of variables and the method of variation of parameters. The three-pronged analysis presented here increases our repertoire of general mean flow solutions that rarely appear in spherical geometry. It is hoped that these special forms will permit extending the current approach to other complex fluid motions that are easier to capture using spherical coordinates.     

Large Eddy Simulation of turbulent confined coaxial swirling jets

Large eddy simulations of swirling flow in a coaxial-jet combustor are reported. Two experimental test cases have been chosen from the literature. In both cases the configuration consists of two coaxial jets which enter into an expansion duct with the annular jet being swirled, the inner jet unswirled. The main features of the flow are well predicted in the simulations. The mean velocities and the turbulent fluctuations are compared with the experimental data and show good agreement. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On exact solutions for the semiempirical equation of turbulent diffusion and the second-order closure methods

A few exact solutions are obtained for the one-dimensional semiempirical equation of turbulent diffusion by the second-order closure methods. The data are compared with the solutions of the semiempirical equation closed by the simplest gradient hypothesis.     

Similarity reductions and similarity solutions for the generalized diffusion equation

Herein, the generalized diffusion equation that encompasses the nonlinear diffusion equation with a source term and the Boussinesq equation in hydrology as its particular form and appears in a wide variety of physical and engineering applications has been analyzed via symmetry method that was developed by Steinberg. According to physical situations, in each case, the similarity variables obtained have led us to an ordinary differential equation, and we acquire some new solutions by solving the ODEs. Copyright © 2015 John Wiley & Sons, Ltd     

On symmetrization of jets

Let F = F ^(A,H,t) and F¹ = F^(A1,H1,t1){F^1} = {F^{({A^1},{H^1},{t^1})}} be fiber product preserving bundle functors on the category FM _m of fibred manifolds Y with m-dimensional bases and fibred maps covering local diffeomorphisms. We define a quasi-morphism (A, H, t) → (A ¹, H ¹, t ¹) to be a GL(m)-invariant algebra homomorphism ν: A → A ¹ with t ¹ = ν ∘ t. The main result is that there exists an FM _m -natural transformation FY → F ¹ Y depending on a classical linear connection on the base of Y if and only if there exists a quasi-morphism (A, H, t) → (A ¹, H ¹, t ¹). As applications, we study existence problems of symmetrization (holonomization) of higher order jets and of holonomic prolongation of general connections.     

On a class of similarity solutions of the porous media equation,III

It is shown how existence questions for general multiparameter eigenvalue problems can be treated quite simply using degree theory. The equations to be solved are W_n(λ)x_n = 0 ≠ x_n, n = 1, 2,…, k, where λ ? $\text{R}$^k and each W_n(λ) is a self-adjoint linear operator on a Hilbert space H_n. The W_n, which may be unbounded, depend continuously on λ in a suitable sense. A coercivity condition for large ∥ λ ∥ is used, and is shown to be equivalent, in the “linear” case, to a standard determinantal definiteness condition.     

An extended similarity theory applied to heated flows in complex geometries

In the traditional similarity theory the influence of temperature- and pressure-dependent fluid properties on the flow field and heat transfer is not described by the basic dimensionless parameters, i.e. Prandtl, Reynolds, Rayleigh, . . . number. We present an extended similarity theory that not only takes into account the variable material properties but also can handle small variations in other parameters of the physical model like small changes in the (reference) Prandtl number. The method has general applicability that is suitable for a wide variety of fluid dynamic and heat transfer situations in which variable properties with a strong dependence on temperature and pressure play a significant role. It is especially useful in predicting the behaviour of a certain fluid based on the results for a different one. As an example the Nu?elt number of a lid driven heated cavity is determined with fluid properties being temperature dependent.     

Construction of upper and lower solutions for flow past a non-uniformly heated plate

The Oseen linearization and the modified Oseen linearization are often used in studying fluid mechanical problems, but whether the linearized solution is accurate is usually difficult to assess. For the sample problem of uniform flow past a plate, we use a comparison theorem to show that the Oseen linearization, used in two ways, gives both an upper and a lower solution. Further, we make use of the comparison theorem and the modified Oseen linearization to construct a sharper upper solution valid in the boundary layer. We then go on to consider the case when the plate temperature increases along the plate according to a power law. Upper and lower solutions for the temperature equation are constructed, and bounds on the temperature gradient at the plate are obtained. With a minor modification, similar results for the case when the logarithmic derivative of the plate temperature lies between two power law curves are obtained.     

Isovector fields and similarity solutions of Einstein vacuum equations for rotating fields

The isovector fields (infinitesimal generators of Lie groups) of Einstein vacuum equations for stationary axially symmetric rotating fields, in conventional form, that is a coupled system of nonlinear partial differential equations (PDEs) of second order are derived using the geometric prolongation technique. Some symmetry transformations and similarity (exact) solutions of Einstein vacuum equations are obtained.     

On poset similarity

Let P be a poset, and let A be an element of its strict incidence algebra. Saks (SIAM J. Algebraic Discrete Methods 1 (1980) 211–215; Discrete Math. 59 (1986) 135–166) and Gansner (SIAM J. Algebraic Discrete Methods 2 (1981) 429–440) proved that the kth Dilworth number of P is less than or equal to the dimension of the nullspace of A^k, and that there is some member of the strict incidence algebra of P for which equality is attained (for all k simultaneously). In this paper we focus attention on the question of when equality is attained with the strict zeta matrix, and proceed under a particular random poset model. We provide an invariant depending only on two measures of nonunimodality of the level structure for the poset that, with probability tending to 1 as the smallest level tends to infinity, takes on the same value as the inequality gap between the width of P and the dimension of the nullspace of its strict zeta matrix. In particular, we characterize the level structures for which the width of P is, with probability tending to 1, equal to the dimension of the nullspace of its strict zeta matrix. As a consequence, by the Kleitman–Rothschild Theorem 5, almost all posets in the Uniform random poset model have width equal to the dimension of the nullspace of their zeta matrices. We hope this is a first step toward a complete characterization of when equality holds in Saks’ and Gansner's inequality for the strict zeta matrix and for all k. New to this paper are also the canonical representatives of the poset similarity classes (where two posets are said to be similar if their strict zeta matrices are similar in the matrix-theoretic sense), and these form the setting for our work on Saks’ and Gansner's inequalities. (Also new are two functions that measure the nonunimodality of a sequence of real numbers.)