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This report gives a summary of the work done in Russia from 1911 to 1914, by Professor Joukowski and his pupils. This summary will show that these men were the true originators of the theory, which combines the theory of the wing element and of the slipstream.
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Values of the circulation function have been obtained for dual-rotating propellers. Numerical values are given for four, eight, and twelve-blade dual-rotating propellers and for advance ratios from 2 to about 6. In addition, the circulation function has been determine for single-rotating propellers for the higher values of the advance ratio. The mass coefficient, another quantity of significance in propeller theory, has been introduced.
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Development of a theory for determining the aerodynamic forces for unsteady, compressible subsonic flow on a propeller in forward flight. The acceleration potential method is used in developing the basic downwash integral equation which governs the flow. This integral equation is solved by the doublet-lattice method, which consists of placing 'load' lines at certain locations on the chord and satisfying the downwash condition at other selected positions. The examples presented include the spanwise and chordwise loading on a rotating propeller for incompressible flow, an example of compressible flow calculations, and, finally, a calculation illustrating the loss of aerodynamic damping of a propeller blade due to the passage of the blade over its own wake.
A theory for predicting the noise field of a propeller with a realistic blade geometry is presented. The theory, which utilizes a large blade count approximation, provides an efficient formula for predicting the radiation of sound from all three sources of propeller noise. Comparisons with full numerical integration indicate that the noise levels predicted by this formula are quite accurate. Calculations based on this method also show that the radiation from the Lighthill quadrupole source is rather substantial when compared with thickness and loading noise for high speed propellers. A preliminary application of the theory to the problem of the sensitivity of the peak noise levels generated by a supersonic propeller to the variations in its tip helical Mach number has produced a trend that is in qualitative agreement with the experimental observations.
A theory for predicting the noise field of supersonic propellers with realistic blade geometries is presented. The theory, which utilizes a large-blade-count approximation, provides an efficient formula for predicting the radiation of sound from all three sources of propeller noise. Comparisons with a full numerical integration indicate that the levels predicted by this formula are quite accurate. Calculations also show that, for high speed propellers, the noise radiated by the Lighthill quadrupole source is rather substantial when compared with the noise radiated by the blade thickness and loading sources. Results from a preliminary application of the theory indicate that the peak noise level generated by a supersonic propeller initially increases with increasing tip helical Mach number, but is eventually reaches a plateau and does not increase further. The predicted trend shows qualitative agreement with the experimental observations.
This report presents a calculation of the torsional frequencies of revolving bars with allowance for the air forces. Calculation of the flexural or bonding frequencies of revolving straight or tapered bars in terms of the angular velocity of revolution. Calculation on the basis of Rayleigh's principle of variation. There is also a discussion of error estimation and the accuracy of results. The author then provides an application of the theory to screw propellers for airplanes and the discusses the liability of propellers to damage through vibrations due to lack of uniform loading.
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A technical method is given for calculating the axial interference velocity of a propeller. The method involves the use of certain weight functions p, q, and f. Numerical values for the weight functions are given for two-blade, three-blade, and six-blade propellers.
As the conditions of the ultimate wake are concerned both theoretically and practically, the magnitude of the slipstream contraction has been calculated. It will be noted that the contraction in a representative case is of the order of only 1 percent of the propeller diameter. In consequence, all calculations need involve only first-order effects. Curves and tables are given for the contraction coefficient of two-blade and four-blade propellers for various values of the advance ratio; the contraction coefficient is defined as the contraction in the diameter of the wake helix in terms of the wake diameter at infinity. The contour lines of the wake helix are also shown at four values of the advance ratio in comparison with the contour lines for an infinite number of blades.
Simple and exact expressions are given for the efficiency of single and dual rotating propellers with ideal circulation distribution as given by the Goldstein functions for single-rotating propellers and by the new functions for dual-rotating propellers from part I of the present series. The efficiency is shown to depend primarily on a defined load factor and, to a very small extent, on an axial loss factor. Tables and charts are included for practical use of the results.
An analysis was made to determine the applicability of existing propeller theory and the theory of oscillating airfoils to the problem of determining the magnitude of the forces on propellers in pitch or yaw. Strip calculations using compressible airfoil characteristics were first made as though steady-state conditions existed successively at several blade positions of the propeller blades during one revolution. A theory of oscillating airfoils in pulsating incompressible potential flow was then considered from which it was possible to determine factors which would modify the steady-state forces.