This study treats the ducted propeller with finite blade number at zero angle of attack in a uniform, incompressible, inviscid flow. The approximation* of a lightly loaded propeller and of thin airfoil theory are made. In the absence of thickness efiects, appropriate vortex distributions represent the blades, the shroud and their respective shed vo:rtices. By means of Fourier analysis of the velocity field in piopeller fixed coordinatcs, the problem for an arbitrary, radial blade circulation distribution is reduced to a form similar to the ring wing integral equations of J. Weissinger. The kernels are not the same for the two cases except for the zeroth harmonic. The equation for this harmonic, which corresponds to a generalized actuator disk formulation, is Ldentical to that of an equivalent, axisymmetric ring wing. The effect of blade number, blade circulation profile and strength, propeller advance ratio and tip clearance, and location of the propeller plane are found and may be evaluated over a range of parameters from tables provided. A numerical example is given. i-i 16 1.6 Formulation of Basic Equation for-y CHAPTER TWO-Derivation of Shroud Vortex Distribution 2.1 Periodic Expansion of Shroud Vortex Distribution 2.2 Reduction of Shroud Contributions 2. 2.3 Decomposition of Governing Equation for-y 28 2.1I Splitting of Governing Complex Equation 2.5 Discussion of Coupled Equations 36 2.6 Shroud and Propeller Loading CHAPTER THREE-Solutions for Duct Leading 3.I Fundamental Solution 43 3.2 Effect of Parameters or the Fundamental Load 47 3.3 Limiting Case of Infinite Advan, 1,1atio 3,14 Infinite Blade uNumber 64 iii CHAPTER THREE 3.5 General Solution for Higber Harmonics 3.6 Example REFERENCES iv Am. .,. amplitudes of sine term of Fourier harmonic 3 A. ... amplitudes of :osne term of Fourier harmonic CU. ... amplitudes of complex Fourier harmonic D distance between vortex element and field point FY, ... functions of complex kernel of intermediate form of governing equations Gn(2) Riegels function 1h, ,ld vector influence functions for axial, radial and circumferential vortex elements respectively IA* ... integ'zl a of chozdwise vortex distribution and decoupled kernel imaginary part of complex function J propeller advance ratio, U/ p Kkernel of final governing equations N blade number P "fcrcin% function" of higher propeller harmc ics Qn_(W) Le%/ndre function of second kind and half order R radius of the shroud camber surfacm in the propeller plane used as reference length real part of complex function R p propeller semi-diameter Sn (Qn+l/2+"n-3/2) Tn (on+1I2-on-312) U uniform free stream V c shroud chord gmN'hmN functions dependent on higher harmonics of shed vorticity 46,j_,k unit vectors in the Cartesian system (xyz) 1-,0i_ unit vectors in the cylindrical system (xr,O) i 4j dummy summation index £blade index number m rank of Fourier harmonic n order of Legendre function p static pressure qfluid velocity t time variable x,r,e cylindrical coordinates fi: ed in propeller x,,rs s cylindrical coordinates of n.t ...
This report presents a tabulation of Legendre functions of second kind and half integral order, nh(z) ' for orders from through 10 , with an accuracof one unit variation in the fifth digit. The argument;, are given by z a (1+y2) with Ay = 0.01 and their range extends from 1.0001 to the terminal argument at which the function is equal to or less than 0.00001. Also included is a tabulation of onh(z) for the same range of order and argument where the argument is given by z-(1+y) with Ay = 0.1. A general description of the comp.tational program and program flow charts is included. iii TABLE OF CONT1ITS INTRODUCTION CHAPTER ONE: MATHEMATICAL RELATIONSHIPS 5 1.1 General Properties of the Legendre Function 1.2 Series Representation for Large Values of the Argument 1.3 Series Representation for Small Values of the Argument 1.4 Recurrence Formulas 9 1.5 Relationship Between 0'.L(z) and Riegels Function 12 CHAPTER TWO: COMPUTATIONAL PROCEDURES 14 2.1 General Discussion 14 2.2 Computation of Small Arguments, 1.0001 z 1.09000 14 2.3 Computation of Moderate Arguments, 1.0900 _< z _< 5.0000 2.4 Computation of Large Arguments, 5.0000 < < 17 2.5 Accuracy Checks 2.6 Sub-routine Selection in the Master Program 19 CHAPTER THREE: FLOW CHARTS 20 CHAPTER FOUR: TABULATIONS 34 REFERENCES 83 iv anp bnp coefficients in the expansion for Onh C complex contour c! integration G n(k2) Riegel* function i (-I) n,p real integer numbers Pn(z) Logandre function of first kind Qn(Z) Legendre function of second kind t dummy variable of integration u a solution of the Legandre equation z argument of Legendre function 6 nO Kronecker delta N GqGawnn "Inction v A CONPUTATIONAL PROGAM AND EXTENDED TABULATION OF LEGENDRE FUNCTIONS OF SECOND KIND AND LhLF ORDER
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.