Acoustic theory of the many-bladed contra-rotating propeller: analysis of the effects of blade sweep on wake interaction noise

2019 ◽  
Vol 868 ◽  
pp. 385-427 ◽  
Author(s):  
M. J. Kingan ◽  
A. B. Parry
Keyword(s):  
Special Functions ◽  
Higher Order ◽  
Stationary Points ◽  
Asymptotic Approach ◽  
Azimuthal Mode ◽  
Special Cases ◽  
Wake Interaction ◽  
Radiated Sound ◽  
Interior Points ◽  
Mode Order

An analytical model is presented for the wake interaction tones produced by a contra-rotating propeller. We re-cast the usual far-field radiation formulae as a double integral over a nominal propeller source annulus. Assuming that the number of blades on both propellers is large, we evaluate the integral asymptotically in terms of its leading-order contributions from interior stationary or boundary critical points which represent the specific locations on the propeller annulus that dominate the sound radiation. The asymptotic approach is powerful producing results in the form of one-line algebraic formulae that contain no integrals or special functions yet remain accurate. The asymptotics show that sweep is not necessarily beneficial and can cause the blade design to become critical for particular tones and directions in terms of a continuum of interior points distributed along a line on the propeller source annulus producing a higher-order result and thus an enhanced radiated sound field. The paper also shows how the interior points are completely consistent with the sub- or super-critical gust response of a swept blade. Tones with low and zero azimuthal mode order are treated as special cases and the asymptotics show that, as the mode order reduces, the radiated sound becomes concentrated around the flight axis where even higher-order solutions are possible, including rings and annuli of stationary points around the propeller annulus. Full numerical calculations confirm the accuracy of the asymptotic approach.

scholarly journals Brownian Swarm Dynamics and Burgers’ Equation with Higher Order Dispersion

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 57
Author(s):  
Max-Olivier Hongler
Keyword(s):  
Burgers Equation ◽  
Higher Order ◽  
Underlying Structure ◽  
Swarm Dynamics ◽  
Special Cases ◽  
Kink Waves ◽  
Systematic Derivation ◽  
Fifth Order ◽  
Higher Order Dispersion ◽  
Order Dispersion

The concept of ranked order probability distribution unveils natural probabilistic interpretations for the kink waves (and hence the solitons) solving higher order dispersive Burgers’ type PDEs. Thanks to this underlying structure, it is possible to propose a systematic derivation of exact solutions for PDEs with a quadratic nonlinearity of the Burgers’ type but with arbitrary dispersive orders. As illustrations, we revisit the dissipative Kotrweg de Vries, Kuramoto-Sivashinski, and Kawahara equations (involving third, fourth, and fifth order dispersion dynamics), which in this context appear to be nothing but the simplest special cases of this infinitely rich class of nonlinear evolutions.

Propeller Wake Sheet Roll-up Modeling in Three Dimensions

1997 ◽  
Vol 41 (01) ◽  
pp. 81-92
Author(s):  
Sangwoo Pyo ◽  
Spyros A. Kinnas
Keyword(s):  
Three Dimensional ◽  
Vortex Sheet ◽  
Experimental Information ◽  
Higher Order ◽  
Three Dimensions ◽  
Tip Vortex ◽  
Propeller Wake ◽  
Wake Interaction ◽  
Radial Circulation ◽  
Wake Sheet

An algorithm for predicting the complete three-dimensional vortex sheet roll-up is developed. A higher order panel method, which combines a hyperboloidal panel geometry with a bi-quadratic dipole distribution, is used in order to accurately model the highly rolled-up regions. For given radial circulation distributions, the predicted wake shapes are shown to be convergent and consistent with those predicted from other methods. Then, a previously developed flow-adapted grid and the three-dimensional wake sheet roll-up algorithm are combined in order to estimate the propeller loading/trailing wake interaction. The complete wake geometry is determined by the method without the need of any experimental information on the shape of the wake. Predicted forces and tip vortex trajectories are shown to agree well with those measured in experiments.

scholarly journals Self-Inductance of the Circular Coils of the Rectangular Cross-Section with the Radial and Azimuthal Current Densities

Physics ◽  
2020 ◽  
Vol 2 (3) ◽  
pp. 352-367
Author(s):  
Slobodan Babic ◽  
Cevdet Akyel
Keyword(s):  
Cross Sections ◽  
Special Functions ◽  
Rectangular Cross Section ◽  
The Self ◽  
Thin Wall ◽  
Elementary Functions ◽  
Special Cases ◽  
Current Densities ◽  
New Formula ◽  
Azimuthal Current

In this paper, we give new formulas for calculating the self-inductance for circular coils of the rectangular cross-sections with the radial and the azimuthal current densities. These formulas are given by the single integration of the elementary functions which are integrable on the interval of the integration. From these new expressions, we can obtain the special cases for the self-inductance of the thin-disk pancake and the thin-wall solenoids that confirm the validity of this approach. For the asymptotic cases, the new formula for the self-inductance of the thin-wall solenoid is obtained for the first time in the literature. In this paper, we do not use special functions such as the elliptical integrals of the first, second and third kind, nor Struve and Bessel functions because that is very tedious work. The results of this work are compared with already different known methods and all results are in excellent agreement. We consider this approach novel because of its simplicity in the self-inductance calculation of the previously-mentioned configurations.

Iterative combinations for Srivastava–Gupta operators

2020 ◽  
pp. 2150108
Author(s):  
Prerna Maheshwari Sharma
Keyword(s):  
Rate Of Convergence ◽  
Modulus Of Continuity ◽  
Simultaneous Approximation ◽  
Higher Order ◽  
Positive Operators ◽  
Integral Type ◽  
Linear Positive Operators ◽  
General Family ◽  
Special Cases

In the year 2003, Srivastava–Gupta proposed a general family of linear positive operators, having some well-known operators as special cases. They investigated and established the rate of convergence of these operators for bounded variations. In the last decade for modified form of Srivastava–Gupta operators, several other generalizations also have been discussed. In this paper, we discuss the generalized modified Srivastava–Gupta operators considered in [H. M. Srivastava and V. Gupta, A certain family of summation-integral type operators, Math. Comput. Modelling 37(12–13) (2003) 1307–1315], by using iterative combinations in ordinary and simultaneous approximation. We may have better approximation in higher order of modulus of continuity for these operators.

Low-Order Modeling to Investigate Clusters of ITA Modes in Annular Combustors

2020 ◽  
Author(s):  
Guillaume Jean Jacques Fournier ◽  
Matthias Haeringer ◽  
Camilo Silva ◽  
Wolfgang Polifke
Keyword(s):  
Reflection Coefficient ◽  
Analytical Model ◽  
Underlying Mechanism ◽  
Riemann Invariants ◽  
Azimuthal Mode ◽  
Annular Combustor ◽  
1D Model ◽  
Phasor Analysis ◽  
Peculiar Behavior ◽  
Mode Order

Abstract The intrinsic thermoacoustic (ITA) feedbackloop constitutes a coupling between flow, flame and acoustics that does not involve the natural acoustic modes of the system. One recent study showed that ITA modes in annular combustors come in significant number and with the peculiar behavior of clusters, i.e. several modes with close frequencies. In the present work an analytical model of a typical annular combustor is derived via Riemann invariants and Bloch theory. The resulting formulation describes the full annular system as a longitudinal combustor with an outlet reflection coefficient that depends on frequency and the azimuthal mode order. The model explains the underlying mechanism of the clustering phenomena and the structure of the clusters associated with ITA modes of different azimuthal orders. In addition, a phasor analysis is proposed, which enclose the conditions for which the 1D model remains valid when describing the thermoacoustic behavior of an annular combustor.

scholarly journals Theoretical framework for higher-order quantum theory

2019 ◽  
Vol 475 (2225) ◽  
pp. 20180706 ◽  
Author(s):  
Alessandro Bisio ◽  
Paolo Perinotti
Keyword(s):  
Quantum Theory ◽  
Convex Sets ◽  
Mathematical Structure ◽  
Higher Order ◽  
Equivalence Relations ◽  
Full Generality ◽  
Special Cases ◽  
Different Types ◽  
Causal Structures

Higher-order quantum theory is an extension of quantum theory where one introduces transformations whose input and output are transformations, thus generalizing the notion of channels and quantum operations. The generalization then goes recursively, with the construction of a full hierarchy of maps of increasingly higher order. The analysis of special cases already showed that higher-order quantum functions exhibit features that cannot be tracked down to the usual circuits, such as indefinite causal structures, providing provable advantages over circuital maps. The present treatment provides a general framework where this kind of analysis can be carried out in full generality. The hierarchy of higher-order quantum maps is introduced axiomatically with a formulation based on the language of types of transformations. Complete positivity of higher-order maps is derived from the general admissibility conditions instead of being postulated as in previous approaches. The recursive characterization of convex sets of maps of a given type is used to prove equivalence relations between different types. The axioms of the framework do not refer to the specific mathematical structure of quantum theory, and can therefore be exported in the context of any operational probabilistic theory.

scholarly journals Some Trapezium-Like Inequalities Involving Functions Having Strongly n-Polynomial Preinvexity Property of Higher Order

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Muhammad Uzair Awan ◽  
Sadia Talib ◽  
Muhammad Aslam Noor ◽  
Yu-Ming Chu ◽  
Khalida Inayat Noor
Keyword(s):  
Fractional Calculus ◽  
Higher Order ◽  
New Class ◽  
Special Cases ◽  
Preinvex Functions ◽  
Class Of Functions

The main objective of this paper is to introduce a new class of preinvex functions which is called as n-polynomial preinvex functions of a higher order. As applications of this class of functions, we discuss several new variants of trapezium-like inequalities. In order to obtain the main results of the paper, we use the concepts and techniques of k-fractional calculus. We also discuss some special cases of the obtained results which show that the main results of the paper are quite unifying one.

On the Rôle of the Group O4 of Local Complex Orthogonal Transformations in a Nonlinear Theory of Elementary Particles

1965 ◽  
Vol 20 (5) ◽  
pp. 649-655 ◽  
Author(s):  
G. Braunss
Keyword(s):  
Conservation Laws ◽  
Affine Connection ◽  
Higher Order ◽  
Nonlinear Electrodynamics ◽  
Spin Representation ◽  
Local Conservation ◽  
Β Decay ◽  
Local Complex ◽  
Special Cases ◽  
Special Complex

It is supposed that there exists a system O′ (intrinsic system) in which the field equation for a spin ½ representation has the simple form γµ ∂Ψ′/∂ϰμ′=0. This system is related to the physical system (in which all measurements are performed) by an affine connection which is induced by a certain group of local transformations. The investigation given here deals with the group of local four-dimensional complex orthogonal transformations. Subjecting ψ' to such a transformation Ω one gets with ψ' (x′) = Ω (x) ψ (x) the following equation γλ ∂Ψ/∂xλ+γλ Ω-1 ∂Ω/∂xλ ψ=0. The interaction term splits up into a vector and a pseudovector part: γλ Ω-1 ∂Ω/∂xλ ≡ γλ Vλ+γλ γ5 Ρλ. The special cases of real local orthogonal (LORENTZ-) transformations (ξλμ= - ξμλ; ξkl real, ξ4l imaginary; ψ → χ) and special complex local orthogonal transformations (ηλμ=- ημλ; ηkl imaginary, η4l real; ψ → φ) are first separately considered. It is required that Vλ and Pλ are to be built up from the fundamental covariants of the field. In order that certain conservation laws hold at least approximately, the following assumptions are made:Im{Vk}=±k2 ɸ̅γkφ, Re {V4}=±k2γ4φ, Im {Pk} = ± l2 χ̅ γk γ5 χ, Re {P4}= ±l2χ̅γ4γ5χ together with the symmetry conditions for the transformation parameters, ξλ[μυ] ≡ 0, η〈λμ,υ〉 ≡ 0, which can be fulfilled by setting, for example, ξλμ,υ = π[λπμ,υ],ηλμ = ϑ[λ,μ]. The remaining parts of Vλ and Ρλ, which are determined by these relations, are of higher order and can be assumed to describe weaker interactions. Neglecting these terms one obtains the following set of equations:(a) γλ ∂χ/∂xλ±k2γλ(ɸ̅γλφ) χ±l2γλγ5(χ̅γλγ5χ) χ≈0(b) γλ ∂χ/∂xλ±k2γλ(ɸ̅γλφ) φ±l2γλγ5(χ̅γλγ5χ) φ≈0Since the pseudovector coupling possesses a greater symmetry, it is assumed that χ represents the baryon and φ the lepton states. Within the approximation, which holds with (a) and (b), it follows the conservation of χ̅γλχand ɸ̅γλφ resp. (conservation of electric charge) and χ̅γλγ5χ and ɸ̅ γ·λγ5φ resp. (conservation of baryonic and leptonic charge resp.). These conservation laws are exact only if the mentioned terms of higher order are neglected; this is equivalent to a strict “local” conservation as can be shown. As to the isospin it is proposed to replace one of its components by a bounded state, i. e. a mixture of χ- and φ-states which would lead in the case of the neutron for example to the components of the /?β-decay. Due to the relations ± k2 ɸ̅γλφ = ¼ηλρ,ρ +O(η2) and ηλu = ϑ[λμ], and in agreement with the reality conditions, it is possible to connect the parameters ϑλ with the electromagnetic field Aλ by setting ϑλ= 8 iAλ. Taking into consideration terms of higher order this would lead to a type of nonlinear electrodynamics.

scholarly journals A family of Finch and Skea relativistic stars

2017 ◽  
Vol 26 (03) ◽  
pp. 1750014 ◽  
Author(s):  
S. D. Maharaj ◽  
D. Kileba Matondo ◽  
P. Mafa Takisa
Keyword(s):  
Bessel Functions ◽  
Special Functions ◽  
Graphical Analysis ◽  
System Of Differential Equations ◽  
Elementary Functions ◽  
Relativistic Stars ◽  
Spacetime Geometry ◽  
Special Cases ◽  
Barotropic Equation ◽  
Charged Model

Several new families of exact solution to the Einstein–Maxwell system of differential equations are found for anisotropic charged matter. The spacetime geometry is that of Finch and Skea which satisfies all criteria for physical acceptability. The exact solutions can be expressed in terms of elementary functions, Bessel functions and modified Bessel functions. When a parameter is restricted to be an integer then the special functions reduce to simple elementary functions. The uncharged model of Finch and Skea [R. Finch and J. E. F. Skea, Class. Quantum Grav. 6 (1989) 467.] and the charged model of Hansraj and Maharaj [S. Hansraj and S. D. Maharaj, Int. J. Mod. Phys. D 15 (2006) 1311.] are regained as special cases. The solutions found admit a barotropic equation of state. A graphical analysis indicates that the matter and electric quantities are well behaved.

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