Initiation of air core in a swirl nozzle using time-independent power-law fluids

Summary Theoretical and experimental analyses have been carried out for determining the injection condition below which the formation of air core does not take place in the course of flow of a time-independent power-law fluid through a swirl nozzle. Analytical solution lends one distinct value of generalized Reynolds number at the inlet to a nozzle below which the air core is not formed. Experiments reveal that there exist two limiting values of such generalized Reynolds number regarding the formation of air core in a nozzle. One value being the upper limit below which steady flow occurs without air core, the other one is the lower limit above which steady flow with fully developed air core persists. In between these two limiting values, there prevails a transition zone through which fully developed air core is set up within the nozzle. For all the nozzles, theoretical results are in fair agreement with the experimental values of upper limit of generalized Reynolds numbers with respect to steady flow without air core. Amongst all the pertinent independent geometrical parameters of a nozzle, the orifice-to-swirl chamber-diameter ratio has the remarkable influence on generalized Reynolds number describing the initiation of air core.Nomenclature D ₁ Swirl chamber diameter - D ₂ Orifice diameter - D _s Diameter of tangential entry ports - E A non-dimensional parameter defined by Eq. (9) - E _R A non-dimensional parameter defined by Eq. (25) - K Flow consistency index - L ₁ Length of the swirl chamber - n Flow behaviour index - P Static pressure inside the nozzle - P _b Back-pressure of the nozzle - Q Volume flow rate - R Radius vector or longitudinal coordinate with respect to spherical coordinate system (Fig. 3) - R ₁ Radius of the swirl chamber - R ₂ Radius of the orifice - Generalized Reynolds number at the inlet to the nozzle - Limiting value of generalized Reynolds number describing initiation of air core - R _z Radius at any section - r Radial distance from the nozzle axis - r _a Air core radius - u Longitudinal component of velocity with respect to spherical coordinate system (Fig. 3) - V _r Radial velocity component - V _z Axial velocity component - V Tangential velocity component - Tangential velocity at inlet to the nozzle - v Component of velocity in the axial plane perpendicular toR (Fig. 3) - w Component of velocity perpendicular to axial plane with respect to the spherical coordinate system (Fig. 3) - z Distance along the nozzle axis from its inlet plane - Half of the spin chamber angle - Boundary layer thickness measured perpendicularly from the nozzle wall - ₂ Boundary layer thickness at the orifice - Angle, which a radius vector makes with the nozzle axis, in spherical coordinate system (Fig. 3) - Density of the fluid - Running coordinate in the azimuthal direction with respect to the cylindrical polar coordinate system as shown in Fig. 3 - Circulation constant With 8 Figures     

Zur Konvergenz der Ableitungen von Interpolationspolynomen

Letp _n denote the polynomial of degreen or less that interpolates a given smooth functionf at the ?eby?ev nodest _j ⁿ =cos(jπ/n), 0≤j≤n, and let ‖·‖ be the maximum norm inC[?1, 1]. It is proved that fork-th derivatives (2≤k≤n) estimates of the following type hold $$\parallel f^{(k)} - p_n^{(k)} \parallel \leqslant c_k n^{k - 1} \inf \{ \parallel f^{(k)} - q\parallel :q \in \Pi _{n - k} \} .$$ In this relationc _k only depends onk andΠ _n?k denotes the space of polynomials up to degreen?k.     

On componentwise error estimates for inverse matrices

LetA be an×n-matrix with the property I–A<1. LetY be an approximation of the inverse ofA. This paper shows how to get a componentwise error estimate forY, that does not require too much numerical effort but generally presents better results than global error estimates do. Although proved by means of interval mathematics, the given error estimate can also be calculated in absence of any implementation of interval arithmetic.

---

Über komponentenweise Fehlerabschätzungen für inverse Matrizen

Zusammenfassung SeiA einen×n-Matrix mit der Eigenschaft I–A<1. SeiY eine Approximation der Inversen vonA. In dieser Arbeit wird gezeigt, wie man eine komponentenweise Fehlerabschätzung fürY erhalten kann, deren Berechnung nicht sehr aufwendig ist, die aber im allgemeinen schärfer ist als globale Fehlerabschätzungen. Obwohl mit intervallmathematischen Mitteln bewiesen, kann die angegebene Fehlerabschätzung auch berechnet werden, wenn keine Intervallarithmetik implementiert ist.

---

---

This research was supported in part by Sonderforschungsbereich 72-Approximation und Optimierung, University of Bonn.     

Explicit,optimal stability functionals and their application to cyclic discretization methods

The present paper contains a stability concept for discretization methods of a certain, very general classM, which is optimal (in the sense of yielding the best general, two-sided error bounds) without being more restrictive than any of the classical stability definitions. The optimal stability functional Ψ_h related to it depends on the linear part of the discretization operator, and has the important property that Ψ_h [δ] may be of orderq+1, i.e. Ψ_h [δ] = O(h ^q+1), even if the local error δ only has orderq, δ = O(h ^q). This result may be used for the construction of methods with maximum order. Its application to linear cyclic methods, for example, furnishes a new approach to the theory of linearM-cyclick-step methods of maximum order.     

Konvergenz von quadratischen Interpolations- und Flächenabgleichssplines

The purpose of this paper is to show that for continuous functions the related quadratic splines converge without any assumption on the spline grid. The points of the interpolatory grid can be chosen between the corresponding points of the spline grid with a division ratio from \(\frac{{\sqrt 2 }}{2}\) to \(1 - \frac{{\sqrt 2 }}{2}\) . In the case of continuously differentiable functions the division ratio can even be taken between 0 and 1; in addition, the order of convergence is increased. For twice differentiable functions the full order of convergence is obtained. Analogous results about the convergence of histo splines are proved.     

Sampling from binomial and poisson distributions: A method with bounded computation times

An accurate acceptance-rejection algorithm is devised and tested. The procedure requires an average of less than 3 uniform deviates whenever the standard deviation of the distribution is at least 4, and this number decreases monotonically to 2.63 as . Variable parameters are permitted, and no subroutines for sampling from other statistical distributions are needed.This research was supported by the Austrian Research Council (Fonds zur Förderung der wissenschaftlichen Forschung).     

Optimal interval enclosing of certain sets of matrices,with application to monotone enclosing of square roots

In this paper, two kinds of partial ordering for symmetric matrices are related to each other, namely, the natural partial ordering ≤ generated by the coneK of elementwise nonnegative matrices, and the definite partial ordering \( \leqslant \cdot\) generated by the coneK _D of nonnegative definite matrices. The main result of this paper shows how a matrix interval in the sense of the definite partial ordering can be enclosed between optimal bounds with respect to the natural partial ordering. By means of this result, it is possible to compute a numerically practicable inclusion based on the natural partial ordering from a given inclusion of some matrix with the definite partial ordering. In this way, an always and moreover quadratically convergent method of elementwise enclosing the square root of a positive definite, symmetric matrix can be constructed.     

Numerical solution of a class of quasilinear hyperbolic equations by reduction to the wave equation

An iterative method of solving quasilinear hyperbolic equations of the type $$ - [p_1 (u_x ,x,t)]_x + [p_2 (u_t ,x,t)]_t = f(x,t)$$ in the domain (0, 1)×(0,T) is proposed. For each given initial-boundary-value problem of this type with boundary conditions of the first kind (second kind), a conjugate problem of the same type that has boundary conditions of the second kind (first kind) is defined. From the relations connecting the solutions of a pair of conjugate problems, a series of wave equations is created. The method proposed consists in calculating the solutions of the wave equations of this series. Theoretical proof of the convergence of the solutions of the wave equations to the solutions of the conjugate quasilinear problems is left as an open question. However, numerical results are presented to demonstrate that, under favorable circumstances, the solutions of the wave equations do converge to the solutions of the conjugate quasilinear problems.     

Unsteady mixed convection from an isothermal circular cylinder

Summary Combined unsteady convection from an isothermal horizontal cylinder in a stream flowing vertically upwards has been investigated. Numerical solutions of the unsteady boundary-layer equations have been obtained at any station along the cylinder using the series truncation method. Solutions which are valid near the front and near stagnation points have been obtained using standard finite-difference methods. A series solution in powers of time has been obtained with which the numerical solutions has been checked.

---

Unstetige, gemischte Konvektion um einen isothermen Kreiszylinder

Zusammenfassung Es wird eine kombinierte, unstetige Konvektion eines isothermen, horizontalen Zylinders in einer vertikal nach oben gerichteten Strömung untersucht. Numerische Lösungen der unstetigen Grenzschichtgleichungen werden an jeder Stelle längs des Zylinders durch die Verwendung der Reihenabbruchsmethode erhalten.Nahe der Vorderseite und nahe bei den Staupunkten gültige Lösungen werden durch Verwendung üblicher Methoden der finiten Differenzen erhalten. Mit Hilfe einer Reihenlösung in Potenzen der Zeit wird das numerische Ergebnis überprüft.

---

Notation a radius of the cylinder - g acceleration of gravity - G _r Grashof number =g|T|a ³/v ² - Q heat transfer - R _e Reynolds number =U ₀ a/v - T temperature of fluid in the boundary layer - T ₀ temperature of the ambient fluid - T ₁ temperature of the cylinder - T temperature difference=T ₁–T ₀ - t time - U ₀ free stream - x co-ordinate measuring distance round the cylinder - y co-ordinate measuring distance normal to the cylinder - G _r /R _e ² - coefficient of expansion - coefficient of kinematic viscosity - _w skin friction With 7 Figures     

Computing turning points of curves implicitly defined by nonlinear equations depending on a parameter

Let the space curveL be defined implicitly by the (n, n+1) nonlinear systemH(u)=0. A new direct Newton-like method for computing turning points ofL is described that requires per step only the evaluation of one Jacobian and 5 function values ofH. Moreover, a linear system of dimensionn+1 with 4 different right hand sides has to be solved per step. Under suitable conditions the method is shown to converge locally withQ-order two if a certain discretization stepsize is appropriately chosen. Two numerical examples confirm the theoretical results.