414 



NA TURE 



[Februray 4, 19C9 



bearing on ihe matter. Mr. Alexander, Sir Hiram Maxim, 

 and several other engineers have made researches on the 

 subject of air-propellers. 



Theory, 



Resistance of Smjaces and Solids.^ — A certain resist- 

 ance is experienced when any body is moved through the 

 air, depending on the form of the body and the relative 

 speed. If the air is abruptly parted the sudden alteration 

 or its relative momentum causes a thrust on the body ; 

 its friction against the body produces further resistance, 

 and the partial vacuum at the rear (due to the air not 

 immediately returning) causes still more resistance. The 

 air enters this rear space in a series of whirls or eddies, 

 the kinetic energy of which must be supplied by the moving 

 body. Hence we must consider the front form, the sur- 

 face, and the rear form of the moving body. .-Ml the 

 effects are, at the speeds commonly occurring, nearly pro- 

 portional to the square of the speed. 



If a thin but rigid plane be moved perpendicularly to 

 itself with a speed of V feet per second, it will be sub- 

 jected to a dynamic resistance and also to a negative 

 pressure due to the whirling behind. The skin resistance 

 will be negligible except when the dimensions are very 

 great. The dynamic resistance depends on the quantity 

 of air affected, which again varies with the area, so that 

 we mav write 



P = X. , + i\SV- 



where P is 

 square feet, 

 are constant 



})■- 



(I) 



the total pressure in lb., S is the area in 

 \' the speed in feet per sec, and k and 11 

 < ; h is the mass of a cubic foot of air divided 

 by 2, and is equal to about 00012 at normal temperature 

 and pressure ; 11 is the ratio of the dynamic to the negative 

 pressure, and is generally rather more than 2, so that 



^■( I +-) varies according to different experimenters from 



0-0013 'o 0-0017. Langley's value 00017 'S frequently 

 used, so that we have 



P = /C'SV- (where /• = 0-0017) (2) 



If the plane be turned so that It make an angle y with 

 the direction of motion, the dynamic action Is no longer 

 symmetrical, skin friction becomes Important, and 

 negative pressure decreases. Many rules have been given 

 for this case, but except for very small (say less than 2°) 

 and very large angles (more than 40°) the following rule 

 will serve : — 



P = 2/t'SV2sin7 (3) 



.As the surface becomes nearly coincident with the direc- 

 tion of motion P decreases, but there is a certain residual 

 resistance due to edge dynamic action and skin friction. 

 Lanchester makes this approximately 



F='^2ll .' (,) 



20 

 where F is the total resistance In lb. 



(The author is responsible for this formula.) 



This means that the coefficient of skin friction Is up- 

 wards of 5 per cent, of the coefficient of resistance. There 

 is some difference of opinion as to this, but the value will 

 serve." 



Curved surfaces experience analogous resistances when 

 inclined so as to present a definite convexity or concavity 

 forwards, the coefficient being rather larger. If such 

 surfaces have their chords In the direction of motion, they 

 will be subject to skin friction, and will also experience 

 an upward or downward thrust according as the convexity 

 is beneath or above, provided that the curvature Is easy 

 so that the air may stream Into the concavity. Surfaces 

 laterally great experience more thrust than those the major 

 dimensions of which are In the direction of motion, the 

 ratio of thrust per unit area varying about 30 per cent, 

 above and below that on a square surface. 



The resistance of air to solids in motion Is similar to 

 that of water, but In the decreased ratio of the density 

 of air to water (about i : Soo). 



1 See Lamb's " Hydrodynamics," Lanchester's " Aerodynamic*," also an 

 article by the author on the " Stream Line Theory in Relation to Aero- 

 dynamics," in Acronniitics, August, igo"!. 



2 See Baden-Powell's " Practical .^^erc dynamics," Langlev's " E.vperiments 

 n Aerodynamics,' and the author's botk, "The Problem oi Flight." 



NO. 2049, VOL. 79] 



Centre of Pressure. 



The dynamic resistance is not symmetrical, the resultant 

 pressure being ahead of the centre of area. More in- 

 formation is required as to this displacement. For planes 

 inclined at an angle y to the direction of motion, the 

 following rule, given by Joessel and .Vvanzini, is much 

 used ; — 



A = o-3(l-sln7)L (5) 



where A is the distance in feet from the centre of area to 

 the centre of pressure, and L the length in feet of the 

 plane In the direction of motion. 



Turnbull (Phys. Review, xxiv., March, 1907) contests 

 this rule, and states that his experlrnents indicate that 

 when 7 is less than 18°, A simply varies with y, so that 

 when 7 = 0, A = o. For surfaces having a convex under- 

 side or concavity In front and convexity at the rear (both 

 on the underside), he gets a law similar to, but in excess 

 of, Joessel 's. He maintains that these two types of surface 

 only are stable. 



As this quantity enters into all the stability formula?, 

 further experiment is urgently required.' 



Energy Required for Flight (Aerofilanc). 

 Since the normal pressure varies as the area of the plane 

 and the square of the speed, the component of this in 

 the direction of motion will similarly vary. Thus if the 



-Equilibi 





.\croplane Running Hori- 



thrust is in the direction of motion we have R the resist- 

 ance of the plane in lb. 



R=Psin 7 = 2/!-SV=sin-^7 (6) 



and if a further resistance CV" be allowed (where C is the 

 projected area In square feet of the car at right angles 

 to the direction of motion) for the car and framework, we 

 have 



H = (R-t-CV=)V 



= (2/C-Ssin=7-fC)V'' (7) 



where H is in foot-pounds per second. 



Hence the power required appears to vary as the cube 

 of the velocity. 7, however, is not necessarily constant. 



Fig. 2.— Equilibrium of Forces in .\eroplanes : Aeroplane Ascending. 



so that we may diminish the power by decreasing 7, 

 always remembering that C is invariable. The limiting 

 value of 7 Is determined by the weight, for the vertical 

 thrust must never be less than the weight. If the direc- 

 tion of motion Is horizontal, then we have 



W = Pcos7 = 2-('SV'-sin 7COS7 . ... (8) 

 where W is the weight in lb., so that V being known, 

 7 can be computed, or vice versA. It will follow from 

 this that if a certain starting value for 7 is assumed, the 

 value V, found from equation (8), will be the lowest soar- 

 ing speed, i.e. the starting speed required. 



1 See Turnbull's paper, also Kummer, " Berlin Akademie Abhandlungen," 

 1P7S-6 ; Joessel, " Ginie Maritime," 1870; " Langley, "Experiments in 

 Aercdynamics " ; Moedebeck's " Pocket-Book," 



