# Category Aerodynamics for Engineering Students

## The hovercraft

In conventional winged aircraft lift, associated with circulation round the wings, is used to balance the weight, for helicopters the ‘wings’ rotate but the lift generation is the same. A radically different principle is used for sustaining of the hovercraft. In machines of this type, a more or less static region of air, at slightly more than atmospheric pressure, is formed and maintained below the craft. The difference between the pressure of the air on the lower side and the atmospheric pressure on the upper side produces a force tending to lift the craft. The trapped mass of air under the craft is formed by the effect of an annular jet of air, directed inwards and downwards from near the periphery of the underside. The downwards ejection of the annular jet produces an upwards reaction on the craft, tending to lift it. In steady hovering, the weight is balanced by the jet thrust and the force due to the cushion of air below the craft. The difference between the flight of hovercraft and normal jet-lift machines lies in the air cushion effect which amplifies the vertical force available, permitting the direct jet thrust to be only a small fraction of the weight of the craft. The cushion effect requires that the hovering height/diameter ratio of the craft be small, e. g. 1/50, and this imposes a severe limitation on the altitude attainable by the hovercraft.

Consider the simplified system of Fig. 9.10, showing a hovercraft with a circular planform of radius r, hovering a height h above a flat, rigid horizontal surface. An annular jet of radius r, thickness t, velocity V and density p is ejected at an angle в to the horizontal surface. The jet is directed inwards but, in a steady, equilibrium state, must turn to flow outwards as shown. If it did not, there would be a continuous increase of mass within the region C, which is impossible. Note that such an increase of mass will occur for a short time immediately after starting, while the air cushion is being built up. The curvature of the path of the air jet shows that it possesses a centripetal acceleration and this is produced by a difference between the pressure pc within the air cushion and the atmos­pheric pressure pq. Consider a short peripheral length 6s of the annular jet and assume:

(i) that the pressure pc is constant over the depth h of the air cushion

(ii) that the speed V of the annular jet is unchanged throughout the motion.

Then the rate of mass flow within the element of peripheral length 6s is pVt fa kgs-1. This mass has an initial momentum parallel to the rigid surface (or ground) of pVt6sVcos 0 = pV^t cos 0 6s inwards.

After turning to flow radially outwards, the air has a momentum parallel to the ground of pVt6sV = pV2t 6s outwards. Therefore there is a rate of change of momen­tum parallel to the ground of pV2t( + cos 0) 6s. This rate of change of momentum is due to the pressure difference (pc — po) and must, indeed, be equal to the force exerted on the jet by this pressure difference, parallel to the ground, which is (pc — po)h 6s. Thus

(Pc — po)h6s = pV2t(l + cos0) 6s

or

(Pc~Po)=^Y^( 1 + COS0) (9.73)

Thus the lift Lc due to the cushion of air on a circular body of radius r is

7i■or2V2t

Lc = nr2(pc – po) =—- —— (l+cos0) (9.74)

n

The direct lift due to the downwards ejection of the jet is

Lj = pVt 2-кг F sin 0 = Іттгр V2t sin 0 (9.75)

and thus the total lift is

 L = 7rrpF2* |2sin0 4-

(l+cos0)} (9.76)

If the craft were remote from any horizontal surface such as the ground or sea, so that the air cushion has negligible effect, the lift would be due only to the direct jet  thrust, with the maximum value L-ia — 2nrpV2t when # = 90°. Thus the lift amplifica­tion factor, L/Lj0, is

Differentiation with respect to # shows that this has a maximum value when „ 2 h tan# = —

Since machines of this type are intended to operate under conditions such that h is very small compared to r, it follows that the maximum amplification is achieved when # is close to zero, i. e. the jet is directed radially inwards. Then with the approximations sin # = 0, cos # = 1: L__h

r  and

It will be noted that the direct jet lift is now, in fact, negligible.

The power supplied is equal to the kinetic energy contained in the jet per unit time which is і 2nr pVt V2 = 7гг p V3t

Denoting this by P, combining Eqns (9.80) and (9.81), and setting lift L equal to the weight W, leads to

P__Vh W~ 2r

as the minimum power necessary for sustentation, while, if в Ф 0,

P _ Vh W r(l + cos#)

ignoring a term involving sin#. Thus if Vis small, and if h is small compared to r, it becomes possible to lift the craft with a comparatively small power.

The foregoing analysis applies to hovering flight and has, in addition, involved a number of simplifying assumptions. The first is the assumption of a level, rigid surface below the machine. This is reasonably accurate for operation over land but is not justified over water, when a depression will be formed in the water below the craft. It must be remembered that the weight of the craft will be reacted by a pressure distributed over the surface below the machine, and this will lead to deformation of a non-rigid surface.

Another assumption is that the pressure pc is constant throughout the air cushion. In fact, mixing between the annular jet and the air cushion will produce eddies leading to non-uniformity of the pressure within the cushion. The mixing referred to above, together with friction between the air jet and the ground (or water) will lead to a loss of kinetic energy and speed of the air jet, whereas it was assumed that the speed of the jet remained constant throughout the motion. These effects produce only small corrections to the results of the analysis above.

If the power available is greater than is necessary to sustain the craft at the selected height h, the excess may be used either to raise the machine to a greater height, or to propel the craft forwards.

Exercises

1 If an aircraft of wing area S and drag coefficient Сд is flying at a speed of V in air of density p and if its single airscrew, of disc area A, produces a thrust equal to the aircraft drag, show that the speed in the slipstream Vs, is, on the basis of Froude’s momentum theory

y.= y{i+lcD

2 A cooling fan is required to produce a stream of air, 0.5 m in diameter, with a speed of 3 m s-1 when operating in a region of otherwise stationary air of standard density. Assuming the stream of air to be the fully developed slipstream behind an ideal actuator disc, and ignoring mixing between the jet and the surrounding air, estimate the fan diameter and the power input required. (Answer: 0.707 m diameter; 3.24 W)

3 Repeat Example 9.2 in the text for the case where the two airscrews absorb equal powers, and finding (i) the thrust of the second airscrew as a percentage of the thrust of the first, (ii) the efficiency of the second and (iii) the efficiency of the combination.

4 Calculate the flight speed at which the airscrew of Example 9.3 of the text will produce a thrust of 7500 N, and the power absorbed, at the same rotational speed.

5 At 1.5 m radius, the thrust and torque gradings on each blade of a 3-bladed airscrew revolving at 1200 rpm at a flight speed of 90 m s-1 TAS at an altitude where er = 0.725 are 300 Nm-1 and 1800 N mm-‘respectively. If the blade angle is 28°, find the blade section absolute incidence. Ignore compressibility. (Answer. 1°48’) (CU)

6 At 1.25 m radius on a 3-bladed airscrew, the aerofoil section has the following characteristics:

solidity = 0.1; 0 = 29°7′; a = 4°7′; CL = 0.49; L/D = 50

Allowing for both axial and rotational interference find the local efficiency of the element. (Answer: 0.885) (CU)

7 The thrust and torque gradings at 1.22 m radius on each blade of a 2-bladed airscrew are 2120Nm-1 and 778 Nmm-1 respectively. Find the speed of rotation (in rads-1) of the airstream immediately behind the disc at 1.22m radius.

8 A 4-bladed airscrew is required to propel an aircraft at 125ms-1 at sea level, the rotational speed being 1200 rpm. The blade element at 1.25 m radius has an absolute incidence of 6° and the thrust grading is 2800 Nm’1 per blade. Assuming a reason­able value for the sectional lift curve slope, calculate the blade chord at 1.25 m radius. Neglect rotational interference, sectional drag and compressibility.

9 A 3-bladed airscrew is driven at 1560 rpm at a flight speed of 110 m s-1 at sea level. At 1.25 m radius the local efficiency is estimated to be 87%, while the lift/drag ratio of the blade section is 57.3. Calculate the local thrust grading, ignoring rotational interference.

10 Using simple momentum theory develop an expression for the thrust of a pro­

peller in terms of its disc area, the air density and the axial velocities of the air a long way ahead, and in the plane, of the propeller disc. A helicopter has an engine developing 600kW and a rotor of 16m diameter with a disc loading of 170Nm-2. When ascending vertically with constant speed at low altitude, the product of the lift and the axial velocity of die air through the rotor disc is 53% of the power available. Estimate the velocity of ascent. (.Answer. 110 m min-1) (U of L) Quite often ‘dimension’ appears in the form ‘a dimension of 8 metres’ and thus means a specified length. This meaning of the word is thus closely related to the engineer’s ‘unit’, and implies linear extension only. Another common example of its use is in ‘three-dimensional geometry’, implying three linear extensions in different directions. References in later chapters to two-dimensional flow, for example, illustrate this. The meaning above must not be confused with either of these uses.

* Some authorities express temperature in terms of length and time. This introduces complications that are briefly considered in Section 1.2.8.

 Since many valuable texts and papers exist using those units, this book contains, as Appendix 4, a table ot factors Гог converting from the Imperial system to the SI system.

 For example, the Aeronautical Research Committee Current Paper No. 369 which was also published in the Journal of the Royal Aeronautical Society, November 1958.

 Recent reviews are given by M. Gad-el-Hak (1999) The fluid mechanics of microdevices – The Freeman Scholar Lecture. J. Fluids Engineering, 121, 5—33; L. Lofdahl and M. Gad-el-Hak (1999) MEMS applica­tions in turbulence and flow control. Prog, in Aerospace Sciences, 35, 101-203.

 It should be noted that in a general system the fluid would also do work which should be taken into the equation, but it is disregarded here for the particular case of flow in a stream tube.

 In fact, this statement is somewhat of an over-simplification. Technically the turbulence characteristics of the oncoming flow also influence the details of the flow field.

 Continuity equation for two-dimensional flow in polar coordinates (a) Consider a two-dimensional flow field expressed in terms of the cylindrical coordinate system (г, ф, z) where all flow variables are independent of the azimuthal angle ф. For example, the flow over a circular cylinder. If the velocity components (и, v) correspond to the coordinate directions (г, ф) respectively, show that the continuity equation is given by

 Here ro is the radius of the equipotcntial ф = 0 for the isolated source and the isolated sink, but not for the combination.

 J. L. Hess and A. M.O. Smith ‘Calculation of Potential Flow about Arbitrary Bodies’ Prog, in Aero. Set, 8 (1967).

 W. H. Press etal. (1992) Numerical Recipes. The Art of Scientific Computing. 2nd ed. Cambridge Uni­versity Press.

W. Kutta (1902) ‘Lift forces in flowing fluids’ (in German), III. Aeronaut. Mitt., 6, 133.

 see footnote on page 161.

j N. Zhukovsky ‘On the shape of the lifting surfaces of kites’ (in German), Z. Flugtech. Motorluftschiffahrt, 1, 281 (1910) and 3, 81 (1912).

 Z. Flugtech. Motorluftschiffahrt, 9, 111 (1918). f NACA Report, No. 411 (1931).

 N. D. Halsey (1979) Potential flow analysis of multi-element airfoils using conformal mapping, AIAA J., 12, 1281.

*NACA Report, No. 142 (1922).

8 Aeronautical Research Council, Reports and Memoranda No. 910 (1924).



 See R and M, No. 1095, for the complete analysis.

* D. A. Spence, The lift coefficient of a thin, jet flapped wing, Proc. Roy. Soc. A., No. 1212, Dec. 1956. D. A. Spence, The lift of a thin aerofoil with jet augmented flap, Aeronautical Quarterly, Aug. 1958.

 II is suggested that this section be omitted from general study until the reader is familiar with these derivatives and their use.

Lighthill, M. J. (1951) ‘A new approach to thin aerofoil theory’, Aero. Quart., 3, 193. ^ J. Weber (1953) Aeronautical Research Council, Reports & Memoranda No. 2918.

 P. E. Rubbert (1964) Theoretical Characteristics of Arbitrary Wings by a Nonplanar Vortex Lattice Method D6-9244, The Boeing Co.

 J. L. Hess (1972) Calculation of Potential Flow about Arbitrary Three-Dimensional Lifting Bodies Douglas Aircraft Co. Rep. MDC J5679/01.

*A full description is given in J. D. Anderson (1985) Fundamentals of Aerodynamics McGraw-Hill.

 see Bibliography.

* Prandtl, L. (1918), Tragfliigellheorie, Nachr. Ges. Wiss., Gottingen, 107 and 451.

 There is no fully convincing physical explanation for the production of the starting vortex and the generation of the circulation around the aerofoil. Various incomplete explanations will be found in the references quoted in the bibliography. The most usual explanation is based on the large viscous forces associated with the high velocities round the trailing edge, from which it is inferred that circulation cannot be generated, and aerodynamic lift produced, in an inviscid fluid. It may be, however, that local flow acceleration is equally important and that this is sufficiently high to account for the failure of the flow to follow round the sharp trailing edge, without invoking viscosity. Certainly it is now known, from the work of T. Weis-Fogh [Quick estimates of flight fitness in hovering animals, including novel mechanisms for lift production, J. Expl. Biol., 59, 169-230, 1973] and M. J. Lighthill [On the Weis-Fogh mechanism of lift generation, J. Fluid Mech., 60,1-17,1973] on the hovering flight of the small wasp Encarsia formosa, that it is possible to generate circulation and lift in the complete absence of viscosity.

In practical aeronautics, fluid is not inviscid and the complete explanation of this phenomenon must take account of viscosity and the consequent growth of the boundary layer as well as high local velocities as the motion is generated.

 Saffman, P. G. 1992 Vortex Dynamics, Cambridge University Press.

 The leader is flying in a flow regime that has additional vertical flow com­

ponents induced by the following vortices. Upward components appear from the bound vortices агсг, азСз, trailing vortices C2d2, азЬз and downward

L. Deighton (1977) Fighter Jonathan Cape Ltd.

 Pohlhamus, E. C. (1966), ‘A Concept of the Vortex Lift of Sharp-Edge Delta Wings Based on a Leading – Edge-Suction Analogy’, NASA TN D-3767; See also ‘Applying Slender Wing Benefits to Military Aircraft’, AIAA J. Aircraft, 21, 545-559, 1984.

 See B. Hunt (1978) ‘The panel method for subsonic aerodynamic flows: A survey of mathematical for­mulations and numerical models and an outline of the new British Aerospace scheme’, in Computational Fluid Dynamics, ed. by W. Kollmann, Hemisphere Pub. Corp., 100-165; and a review by J. L. Hess (1990) ‘Panel methods in computational fluid dynamics’, Ann. Rev. Fluid Mech, 22, 255-274.

 D. N. Holder el at., ARCR and M, 2782, 1953.

 Th. Meyer, Ober zweidimensionale Bewegungsvorgange in einem Gas das mit Oberschallgeschwindigkeit strdmt, 1908.

t See, for example, E. L. Houghton and A. E. Brock, Tables for the Compressible Flow of Dry Air, 3rd Edn, Edward Arnold, 1975.

 J. Ackeret, Z. Flugtech Motorluftschiff, pp. 72-4, 1925. Trans. D. M. Milner in NAC & TM, 317.

T From suitable tables, e. g. E. L. Houghton and A. E. Brock, Tables for the Compressible Flow of Dry Air, 3rd Edn, Edward Arnold, 1975.

 A. Feni, Experimental results with aerofoils tested in the high-speed tunnel at Guidomia, Atti Guidornia, No. 17, September 1939.

 e. g. E. L. Houghton and A. E. Brock, Tables for the Compressible Flow of Dry Air, 3rd Edn., Edward. Arnold, 1975.

 A. Jameson, ‘Full-Potential, Euler and Navier-Stokes Schemes’, in Applied Computational Aerodynamics, Vol. 125 of Prog, in Astronautics and Aeronautics (ed. By P. A. Henne), 39-88 (1990), AIAA: New York.

 This chapter is concerned mainly with incompressible flows. However, the general arguments developed are also applicable to compressible flows.

P. R. Owen and L. Klanl’er, RAE Reports Aero., 2508, 1953.

* Reynolds number here is defined as V^Djw where Vx is the free stream velocity, v is the kinematic viscosity in the free stream and D is the cylinder diameter.

1.8 x 10-5

 W. Tollmien (1929) Uber die Entstehung der Turbulenz. I. Mill. Nachr. Ges. Wiss. Gottingen, Math. Phy. s. Klas. se. pp. 21-44.

f H. Schlichling (1933) Zur Entstehung der Turbulenz bei der Plattenstromung. Z. angew. Math. Mech.. 13, 171-174.

G. B. Schubauer and H. K. Skramstadt (1948) Laminar boundary layer oscillations and transition on a flat plate. NACA Rep., 909.

 J. Boussinesq (1872) Essai sur la theorie des eaux courantes. Memoires Acad, des Science, Vol. 23, No. 1, Paris.

* L. Prandtl (1942) Bemerkungen zur Theorie der freien Turbulenz, ZAMM, 22, 241-243.

1 H. Reichardt (1942) Gesetzmassigkeiten der freien Turbulenz, VDI-Forschmgsheft, 414, 1st Ed., Berlin.

 L. Prandtl (1925) Bericht fiber Untersuchunger zur ausgebildeten Turbulenz, ZAMM, 5, 136-139.

Th. von Karman (1930) Mechanische Ahnlichkeit und Turbulenz, Nachrichten der Akademie der Wissenschaften Gd’ttingen, Math.-Phys. Klasse, p. 58.

tG. I. Barenblatt and V. M. Prostokishin (1993) Scaling laws for fully-developed turbulent shear flows, J. Fluid Mech., 248, 513-529.

 P. W. Carpenter (1997) The right sort of roughness, Nature, 388, 713-714.

 S. J. Kline, W. J. Reynolds, F. A. Schraub and P. W. Runstadler (1967) The structure of turbulent boundary layers, J. Fluid Mech., 30, 741-773.

K. Narahari Rao, R. Narasimha and M. A. Badri Narayanan (1971) The ‘bursting’ phenomenon in a turbulent boundary layer, J. Fluid Mech., 48, 339.

1 Blackwelder, R. F. and Haritonidis, J. H. (1983) Scaling of the bursting frequency in turbulent boundary layers, J. Fluid Mech., 132. 87-103.

* R. L. Panton (ed.) (1997) Self-Sustaining Mechanisms of Wall Turbulence. Computational Mechanics Publications, Southampton.

 For a general description of MEMS technology see Maluf, N. (2000) An Introduction to Microelectro­mechanical Systems Engineering. Artech House; Boston/London. And for a description of their potential use for flow control see Gad-el-Hak, M. (2000) Flow Control. Passive. Active, and Reactive Flow Manage­ment, Cambridge University Press.

*The concept of the synthetic jet was introduced by Smith, B. L. and Glezer, A. (1998) The formation and evolution of synthetic jets, Phys. Fluids, 10, 2281-2297. For its use for flow control, see: Crook, A., Sadri. A. M. and Wood, N. J. (1999) The development and implementation of synthetic jets for the control of separated flow, AIAA Paper 99-3176′, and Amitay, M. elal. (2001) Aerodynamics flow control over an unconventional airfoil using synthetic jet actuators, AIAA Journal. 39, 361-370. A study of the use of these actuators for boundary-layer control based on numerical simulation is described by Lockerby. D. A., Carpenter, P. W. and Davies, C. (2002) Numerical simulation of the interaction of MEMS actuators and boundary layers, AIAA Journal, 39, 67-73.

 B. Thwaites (1949) ‘Approximate calculation of the laminar boundary layer’, Aero. Quart., 1, 245.

1 M. R. Head (1958) ‘Entrainment in the turbulent boundary layers’, Aero. Res. Council, Rep. & Mem., 3152.

 See N. A. Jaffe, T. T. Okamura and A. M.O. Smith (1970) ‘Determination of spatial amplification factors and their application to predicting transition’, AIAA J., 8, 301-308.

f A. R. Wazzan, C. Gazley and A. M.O. Smith (1981) ‘H-Rx method for predicting transition’, AIAA J., 19, 810-812.

 For example, see Fersiger, J. H. (1998) Numerical Methods for Engineering Application, 2nd Ed., Wiley; Fersiger, J. H. and Peric, M. (1999) Computational Methods for Fluid Dynamics, 2nd Ed., Springer; Schli – chting, H. and Gersten, K. (2000) Boundary Layer Theory, Chap. 23, 8th Ed., McGraw-Hill.

TA guide on how to use commercial CFD is given by Shaw, C. T. (1992) Using Computational Fluid Mechanics, Prentice Hall.

 Keller, H. B. (1978) Numerical methods in boundary-layer theory, Annual Review FluidMech., 10,417-433.

 Spalart, P. R. and Watmuff, J. H. (1993) Direct simulation of a turbulent boundary layer up to Re = 1410, J. Fluid Mech., 249, 337-371; Moin, P. and Mahesh, K. (1998) Direct numerical simulation: A tool in turbulence research, Annual Review Fluid Mech., 30, 539-578; Friedrich, R. etal. (2001) Direct numerical simulation of incompressible turbulent flows, Computers & Fluids, 30, 555-579.

1 Agarwal, R. (1999) Computational fluid dynamics of whole-body aircraft, Annual Review Fluid Mech., 31, 125-170.

Cebeci, T. and Smith, A. M.O. (1974) Analysis of Turbulent Boundary Layers, Academic Press.

^ Van Driest, E. R. (1956) On the turbulent flow near a wall, J. Aeronautical Sciences, 23, 1007-1011.

1 Clauser, F. H. (1956) The turbulent boundary layer, Adv. in Applied Mech., 4, 1-51.

5Corrsin, S. and Kistler, A. L. (1954) The free-stream boundaries of turbulent flows, NACA Tech. Note 3133. KlebannoiT, P. S. (1954) Characteristics of influence in a boundary layer with zero pressure gradient, NACA Tech. Note 3178 and NACA Rep.1247.

 Smagorinsky, J. (1963) General circulation experiments with the primitive equations: 1. The basic equa­tions, Mon. Weather Rev., 91, 99-164.

1 Tucker, P. G. (2001) Computation of Unsteady Internal Flows. Kluwer Academic Publishers, Norwell, MA, U. S.A.

 A. M.O. Smith (1975) ‘High-Lift Aerodynamics’, J. Aircraft, 12, 501-530. Many of the topics discussed in Sections 8.1 and 8.2 are covered in greater depth by Smith.

B. S. Stratford (1959) The prediction of separation of the turbulent boundary layer. J. Fluid Mech., 5, 1-16.

 R. H. Liebeck (1973) A class of aerofoils designed for high, lift in incompressible flow. J. of Aircraft, 10, 610-617.

fP. B.S. Lissaman (1983) ‘Low-Reynolds-number airfoils’, Annual Review of Fluid Mechanics, 15,223-239.

The most complete account is given by A. D. Young (1953) ‘The aerodynamic characteristics of flaps’. Aero. Res. Council, Rep. & Mem. No. 2622.

 L. Prandtl and O. G. Tietjens Applied Hydro – and Aeromechanics, Dover, New York, p. 227.

* A. Nakayama, H.-P. Kreplin and H. L. Morgan (1990) ‘Experimental investigation of flowfield about a multielement airfoil’, AlAA J., 26. 14-21.

* Many of the ideas described in the following passages are due to A. M.O. Smith (1975) ibid.

The information for this section comes from two main sources, namely, R. G. Dominy (1992) ‘Aerodynamics of Grand Prix Cars’, Proc. I. Mech. E„ Part D: J. of Automobile Engineering, 206, 267-274; and P. G. Wright (1982) ‘The influence of aerodynamics on the design of Formula One racing cars’, Int. J. of Vehicle Design, 3(4), 383-397.

 R. H. Liebeck (1978) ‘Design of subsonic airfoils for high lift’, AIAA J. of Aircraft, 15(9), 547-561.

 R. H. Liebeck (1978) ‘Design of subsonic airfoils for high lift’, AIA A J. of Aircraft, 15(9), 547-561.

Ї D. Jeffrey, X. Zhang and D. W. Hurst (2000) ‘Aerodynamics of Gurney flaps on a single-element high-lift wing’, AIAA J. of Aircraft, 37(2), 295-301; D. Jeffrey, X. Zhang and D. W. Hurst (2001) ‘Some aspects of the aerodynamics of Gurney flaps on a double-element wing’, Trans, of ASME, J. of Fluids Engineering, 123, 99-104.

The information on helicopter aerodynamics used here is based on an article by R. W. Prouty, The Gurney Flap, Part 2’ in the March 2000 issue of Rotor & Wing (http://www. aviationtoday. com/reports/ rotorwing/).

 The account given here is based on a more detailed treatment by D. W. Bechert, M. Bruse, W. Hage and R. Meyer (1997) ‘Biological surfaces and their technological application – Laboratory and flight experi­ments on drag reduction and separation control’, AIAA Paper 97-1960.

f W. Liebe (1975) ‘Der Auftrieb am Tragflugel: Enstehung and Zusammenbruch’, Aerokurier, Heft 12, 1520-1523.

5 B. Malzbender (1984) ‘Projekte der FV Aachen, Erfolge im Motor – und Segelflug’, Aerokurier, Heft 1, 4.

 A more complete recent account is to be found in M. Gad-el-Hak (2000) Flow Control: Passive, Active and Reactive Flow Management, Cambridge University Press.

*L. Prandtl (1904) ‘Uber Fliissigkeitsbewegung bei sehr kleiner Reibung’, in Proc. 3rd Int. Math. Mech., 5, 484-491, Heidelberg, Germany.

 For a recent review on the aerodynamics of the Coanda effect, see P. W. Carpenter and P. N. Green (1997) The aeroacoustics and aerodynamics of high-speed Coanda devices’, J. Sound & Vibration, 208(5), 777-801.

 See R. W. Prouty’s articles on ‘Aerodynamics’ in Aviation Today: Rotor & Wing May, June and July, 2000 (http://www. aviationtoday. com/reports/rotorwing/).

 See the recent review by D. Greenblatt and I. Wygnanski (2000) ‘The control of flow separation by periodic excitation’, Prog, in Aerospace Sciences, 36, 487-545.

* A. Seifert and L. G. Pack (1999) ‘Oscillatory control of separation at high Reynolds number’, AIAA J., 37(9), 1062-1071.

 For example, see J. P. Johnston and M. Nishi (1990) ‘Vortex generator jets – means for flow separation control’, AlAA J., 28(6), 989-994; see also the recent reviews by Greenblatt and Wygnanski (2000) refer­enced in Section 8.4.2 and Gad-el-Hak (2000) referenced at the beginning of Section 8.4., and J. C. Magill and K. R. McManus (2001) ‘Exploring the feasibility of pulsed jet separation control for aircraft config­urations’, AlAA J. of Aircraft, 38(1), 48-56.

 M. Gad-el-Hak (2000) Flow Control: Passive, Active and Reactive Flow Management, Cambridge Uni­versity Press.

+ Reviews of many aspects of this subject are to be found in Viscous Drag Reduction in Boundary Layers, edited by D. M. Bushnell and J. N. Hefner, AIAA: Washington, D. C. (1990).

* H. Holstein (1940) ‘Messungen zur Laminarhaltung der Grenzschicht an einem Fliigel’, Lilienlhal Bericht, S10, 17-27; J. Ackeret, M. Ras, and W. Pfenninger (1941) ‘Verhinderung des Turbulentwerdens einer Grenzschicht durch Absaugung’, Naturwissenschaflen, 29, 622-623; and M. Ras and J. Ackeret (1941) ‘Uber Verhinderung der Grenzschicht-Turbulenz durch Absaugung’, Helv. Phys. Acta, 14, 323.

 A. L. Braslow (2000) Laminar-Flow Control, NASA web-based publication on http://www/dfrc. nasa. gov/ History/Publications/LFC.

* J. Gray (1936) ‘Studies in animal locomotion. VI The propulsive powers of the dolphin’, J. Experimental Biology, 13, 192-199.

* M. O. Kramer (1957, I960) ‘Boundary layer stabilization by distributed damping’, J. Aeronautical Sci­ences, 24, 459; and J. American Society of Naval Engineers, 74, 341-348.

 T. B. Benjamin (1960) ‘Effects of a flexible boundary on hydrodynamic stability’, J. Fluid Mech., 9, 513-532. ‘

* P. W. Carpenter and A. D. Garrad (1985) ‘The hydrodynamic stability of flows over Kramer-type com­pliant surfaces. Pt. 1. Tollmien-Schlichting instabilities’, J. Fluid Mech., 155, 465-510.

 M. Gaster (1987) ‘Is the dolphin a red herring?’, Proc. of IUTAM Symp. on Turbulence Management and Relaminarisation, edited by H. W. Liepmann and R. Narasimha, Springer, New York, pp. 285-204. See also A. D. Lucey and P. W. Carpenter (1995) ‘Boundary layer instability over compliant walls: comparison between theory and experiment’, Physics of Fluids, 7(11), 2355-2363.

* Detailed reviews of recent progress can be found in: P. W. Carpenter (1990) ‘Status of transition delay using compliant walls’, Viscous Drag Reduction in Boundary Layers, edited by D. M. Bushnell and J. N. Hefner, Vol. 123, Progress in Astronautics and Aeronautics, AIAA, Washington, D. C., pp. 79-113; M. Gad-el-Hak (2000) Flow control, Cambridge University Press; and P. W. Carpenter, A. D. Lucey and C. Davies (2001) ‘Progress on the use of compliant walls for laminar-flow control’, AIAA J. of Aircraft, 38(3), 504-512.

1 See D. W. Bechert, M. Bruse, W. Hage and R. Meyer (1997) ‘Biological surfaces and their technological application – laboratory and flight experiments and separation control’, AIAA Paper 97-1960.

8 MJ. Walsh and L. M. Weinstein (1978) ‘Drag and heat transfer on surfaces with longitudinal fins’, AIAA Paper No. 78-1161.

 See W.-E. Reif and A. Dinkelacker (1982) ‘Hydrodynamics of the squamation in fast swimming sharks’, Neues Jahrbuch fur Geologie und Paleontologie, 164,184-187; and D. W. Bechert, G. Hoppe and W.-E. Reif (1985) ‘On the drag reduction of the shark skin’, AIAA Paper No. 85-0546.

 See A. Filippone (1999-2001) ‘Wing-tip devices’ (http://aerodyn. org/Drag/tip_devices. html).

 R. T. Whitcomb (1956) ‘A study of the zero-lift drag-rise characteristics of wing-body combinations near the speed of sound’, NACA Rep. 1273.

 -"*—)2tfr

 Some authors denote mass flow by m in rocketry, using the mass discharged (per second, understood) as the parameter.

 The power supplied to the jet will also contain a term relating to the increase in potential (pressure) energy, since the jet static pressure will be slightly greater than atmospheric. Since the jet pressure will be approximately equal to pc, which is, typically, about 750 N m“2 above atmospheric, the increase in pressure energy will be very small and has been neglected in this simplified analysis.

## The free motion of a rocket-propelled body

Imagine a rocket-propelled body moving in a region where aerodynamic drag and lift and gravitational force may be neglected, i. e. in space remote from any planets, etc. At time t let the mass of the body plus unburnt fuel be M, and the speed of the body relative to some axes be V. Let the fuel be consumed at a rate of m, the resultant gas being ejected at a speed of v relative to the body. Further, let the total rearwards momentum of the rocket exhaust, produced from the instant of firing to time t, be /relative to the axes. Then, at time t, the total forward momentum is

H{ = MV-I (9.64)

At time (/ + 6t) the mass of the body plus unbumt fuel is (M – m6t) and its speed is (V + 6V), while a mass of fuel m6t has been ejected rearwards with a mean speed, relative to the axes, of (v — V — j6V). The total forward momentum is then

tf2 = {M – m8t){V + 8V) – rh6t{v – V-^6V)-I

Now, by the conservation of momentum of a closed system:

#, =#2

i. e.

MV – I =MV + M6V – mV6t – mStSV – mv6t + mV6t + ^mStSV-I

which reduces to

M6V — – mStSV — mvSt = 0 2

Dividing by 6t and taking the limit as 6t —> 0, this becomes

Note that this equation can be derived directly from Newton’s second law, force = mass x acceleration, but it is not always immediately clear how to apply this law to bodies of variable mass. The fundamental appeal to momentum made above removes any doubts as to the legitimacy of such an application. Equation (9.65) may now be rearranged as

dV _ m

Table 9.1

 t (min) M (1000 kg) Acceleration (m s 2) f (ms!) x (km) 0 11 4.55 0 0 0.5 10 5.00 143 2.18 1 9 5.55 300 9.05 1.5 8 6.25 478 19.8 2 7 7.15 679 37.6 2.5 6 8.33 919 61.4 3 5 10.0 1180 92.0 3.5 4 12.5 1520 133 4 3 16.7 1950 185 4.5 2 25.0 2560 256 5 1 50.0 3600 342 5.5 1 0 3600 450

Substituting the above values into the appropriate equations leads to the final results given in Table 9.1. The reader should plot the curves defined by the values in Table 9.1. It should be noted that, in the 5 minutes of burning time, the missile travels only 342 kilometres but, at the end of this time, it is travelling at 3600 m s_ 1 or 13 000 km h-1. Another point to be noted is the rapid increase in acceleration towards the end of the burning time, consequent on the rapid percentage decrease of total mass. In Table 9.1, the results are given also for the first half-minute after all-burnt.

## The rocket motor

As noted on page 527 the rocket motor is the only current example of aeronautical interest in Class II of propulsive systems. Since it does not work by accelerating atmospheric air, it cannot be treated by Froude’s momentum theory. It is unique among current aircraft power plants in that it can operate independently of air from the atmosphere. The consequences of this are:

(i) it can operate in a rarefied atmosphere, or an atmosphere of inert gas

(ii) its maximum speed is not limited by the thermal barrier set up by the high ram – compression of the air in all air-breathing engines.

In a rocket, some form of chemical is converted in the combustion chamber into gas at high temperature and pressure, which is then exhausted at supersonic speed through a nozzle. Suppose a rocket to be travelling at a speed of V, and let the gas leave the nozzle with a speed of v relative to the rocket. Let the rate of mass flow of gas be m This gas is produced by the consumption, at the same rate, of the chemicals in the rocket fuel tanks (or solid charge). Whilst in the tanks the mean m of fuel has a forward momentum of mV. After discharge from the nozzle the gas has a rearward momentum of m(v — V). Thus the rate of increase of rearward momentum of the fuel/gas is m(v — V) — (—mV) — mv

and this rate of change of momentum is equal to the thrust on the rocket. Thus the thrust depends only on the rate of fuel consumption and the velocity of discharge relative to the rocket. The thrust does not depend on the speed of the rocket itself. In particular, the possibility exists that the speed of the rocket V can exceed the speed of the gas relative to both the rocket, v, and relative to the axes of reference, v — V.

When in the form of fuel in the rocket, the mass m of the fuel has a kinetic energy of mV2. After discharge it has a kinetic energy of m(v — V)2. Thus the rate of change of kinetic energy is ^ = ^[(v – V)2 – V2] = т(^ – 2vV)

the units being Watts.

Useful work is done at the rate TV, where T = mv is the thrust. Thus the propulsive efficiency of the rocket is

rate of useful work rate of increase of KE of fuel 2vV 2

v2 — 2vV (v/V) -2

Now suppose v/V = 4. Then

7/p = = 1 or 100%

If v/V < 4, i. e. V > v/4, the propulsive efficiency exceeds 100%. This derivation of the efficiency, while theoretically sound, is not normally accepted, since the engineer is unaccustomed to efficiencies in excess of 100%. Accordingly an alternative measure of the efficiency is used. In this the energy input is taken to be the energy liberated in the jet, plus the initial kinetic energy of the fuel while in the tanks. The total energy input is then

giving for the efficiency 2(v/V)

{v/V)2 + 1

By differentiating with respect to v/V, this is seen to be a maximum when v/V = 1, the propulsive efficiency then being 100%. Thus the definition of efficiency leads to a maximum efficiency of 100% when the speed of the rocket equals the speed of the exhaust gas relative to the rocket, i. e. when the exhaust is at rest relative to an observer past whom the rocket has the speed V.  If the speed of the rocket V is small compared with the exhaust speed v, as is the case for most aircraft applications, V2 may be ignored compared with v2 giving

## Translational helicopter flight

It is assumed that the effect of the actuator disc used to approximate the rotor is to add incremental velocities vy and z^, vertically and horizontally respectively, at the disc. It is further assumed, in accordance with the simple axial momentum theory of Section 9.1, that in the slipstream well behind the disc these incremental velocities increase to 2vy and 2respectively. The resultant speed through the disc is denoted by U and the resultant speed in the fully developed slipstream by U. Then, by considering vertical momentum:

 W = pAU(2uv) = 2 pAUvy (9.52) Also, from the vector addition of velocities: U2 — {V + Vh)2 + (z’v)2 (9.53) where V is the speed of horizontal flight. By consideration of horizontal momentum ^pV2ACD = 2pAUvh (9.54) where Co is the drag coefficient of the fuselage, etc., based on the rotor area A. Power input = rate of increase of KE, i. e. P=X-pAU{U2-V2) (9.55) and from vector addition of velocities: U2 = {V + 2uh)1 + {2uy)2 (9.56) The most useful solution of the five equations Eqn (9.52) to Eqn (9.56) inclusive is obtained by eliminating U, Vh and zv W Vy~ 2pAU (9.52a) pV2ACo Co y2 h 2 pAU 4U (9.54a)

Then, from Eqn (9.53):

U2 = Vі + 2Vvh + vl + v2 Substituting for vy and z^, and multiplying by U2 gives

(9.57)

a quartic equation for U in terms of given quantities. Since, from Eqn (9.56), ■ 4Vi>h + 4i>l + 4i>l

Then

 4С°У’+-6<*и Uyip0J     P=X-pAU{U-V2) = X-pAU[4Vvb+4vl

which, with the value of U calculated from Eqn (9.57) and the given quantities, may be used to calculate the power required.

Example 9.6 A helicopter weighs 24 000 N and has a single rotor of 15 m diameter. Using momentum theory, estimate the power required for level flight at a speed of 15ms-1 at sea level. The drag coefficient, based on the rotor area, is 0.006.

A =t(15)2 = 176.7m2 4

, W 24000 ,

/de =~r = —r= ,36Nm

Aa 176.7 x 1

Idc 136 2

= о—- = 55-6 m s

2po 2 x 1.226

With the above values, and with V = 15ms_1, Eqn (9.57) is t/4 – 225U2 – l – U(0.006)(3375) = (55.6)2 +

2 16

l. e.

U4 – 225U2- 10.1256/ = 3091  This quartic equation in U may be solved by any of the standard methods (e. g. Newton – Raphson), the solution being U = 15.45ms-1 to four significant figures. Then

This is the power required if the rotor behaves as an ideal actuator disc. A practical rotor would require considerably more power than this.

## Vertical climbing flight

The problem of vertical climbing flight is identical to that studied in Section 9.1, with the thrust equal to the helicopter weight plus the air resistance of the fuselage etc., to the vertical motion, and with the oncoming stream speed V equal to the rate of climb of the helicopter.

9.5.2 Slow, powered, descending flight

In this case, the air approaches the rotor from below and has its momentum decreased on passing through the disc. The associated loss of kinetic energy of the air appears as a power input to the ideal actuator, which therefore acts as a windmill. A real rotor will, however, still require to be driven by the engine, unless the rate of descent is large. This case, for the ideal actuator disc, may be treated by the methods of Section 9.1 with the appropriate changes in sign, i. e. V positive, Vs < Vo < V, p > pi and the thrust T = – W.

## The momentum theory applied to the helicopter rotor

In most, but not all, states of helicopter flight the effect of the rotor may be approxi­mated by replacing it by an ideal actuator disc to which the simple momentum theory applies. More specifically, momentum theory may be used for translational, i. e. forward, sideways or rearwards, flight, climb, slow descent under power and hovering.

9.5.1 The actuator disc in hovering flight

In steady hovering flight the speed of the oncoming stream well ahead of (i. e. above) the disc is zero, while the thrust equals the helicopter weight, ignoring any downward force arising from the downflow from the rotor acting on the fuselage, etc. If the weight is W, the rotor area A, and using the normal notation of the momentum theory, with p as the air density

W = pA V0( Vs – V) = pA V0 Vs (9.44)

since V = 0. Vs is the slipstream velocity and Vq the velocity at the disc.

The general momentum theory shows that

V0=^(VS+V) (Eqn(9.8)) — – К in this case 2

or

Vs = 2V0

 which, substituted in Eqn (9.44), gives W = 2pAVl i. e.

 (9.46) (9.47) (9.48)

 V0 = y/WflpA Defining the effective disc loading, !&, as /de = W/A(7 where и is the relative density of the atmosphere, then W W 1 a 1 2pA Aa 2 p 2po de po being sea-level standard density. Then

 (9.49)

 Vo — ykc/lpo = l-pVoVlA = 2pAVl

 (9.50)

 Substituting for Vo from Eqn (9.47) leads to

 ‘ w3/2 jw3

 P = 2pA

 (9.51a)

 2pA

 2pA)  (9.51b)

This is the power that must be supplied to the ideal actuator disc. A real rotor would require a considerably greater power input.

## The performance of a blade element

Consider an element, of length 6r and chord c, at radius r of an airscrew blade. This element has a speed in the plane of rotation of Qr. The flow is itself rotating in the same plane and sense at Ш, and thus the speed of the element relative to the air in

this plane is flr(l — b). If the airscrew is advancing at a speed of F the velocity through the disc is F(1 + a), a being the inflow at the radius r. Note that in this theory it is not necessary for a and b to be constant over the disc. Then the total velocity of the flow relative to the blade is Fr as shown in Fig. 9.9.

If the line CC’ represents the zero-lift line of the blade section then в is, by definition, the geometric helix angle of the element, related to the geometric pitch, and a is the absolute angle of incidence of the section. The element will therefore experience lift and drag forces, respectively perpendicular and parallel to the relative velocity Fr, appropriate to the absolute incidence a. The values of Cl and Co will be those for a two-dimensional aerofoil of the appropriate section at absolute incidence a, since three-dimensional effects have been allowed for in the rotational interference term, MI. This lift and drag may be resolved into components of thrust and ‘torque-force’ as in Fig. 9.9. Here SL is the lift and 6D is the drag on the element. <57? is the resultant aerodynamic force, making the angle 7 with the lift vector. 6R is resolved into components of thrust 6T and torque force SQ/r, where SQ is the torque required to rotate the element about the airscrew axis. Then

 tan7 = SD/SL = CD/CL (9.24) Fr = F(1 + a)cosec = flr(l — Ь)ъесф (9.25) ST = <57?cos( + 7) (9.26) Щ – = SRsin(

The efficiency of the element, 771, is the ratio, useful power out/power input, i. e. VST V cos (<ft + 7)

^1 Si SQ Sir sin(</> + 7) Now from the triangle of velocities, and Eqn (9.28):

whence, by Eqn (9.29):

 1—6 tan ф 1 + a tan(ф + y)

 (9.30)

 *71  Let the solidity of the annulus, <7, be defined as the ratio of the total area of blade in annulus to the total area of annulus. Then

From Fig. 9.9

ST = SL cos ф – SD sin ф

— BcSr^pV{CLcosф – C£isin^) Therefore   whence, using Eqn (9.32a and b)

= 2nr2a^pV^(C£ sin^ + Co cos ф)

Writing now

 q=CL sin{ф + i) (9.36) leads to — m^aqpV^ total (9.37a) l 9 = Bcr -^pV^q total (9.37b) l, — cr-pV^q per blade (9.37c) The quantities dT/dr and dQ/dr are known as the thrust grading and the torque grading respectively. Consider now the axial momentum of the flow through the annulus. The thrust ST

is equal to the product of the rate of mass flow through the element with the change in the axial velocity, i. e. ST = mSV. Now

m = area of annulus x velocity through annulus x density = (27rr<5r)[F(l + a)p = livrpSr V(l + a)

AV = Vs – V = V(1 + 2a) – V = 2aV

whence

ST = 2TrrpSrV22a(l + a)

giving d T 2 /1 — = 47rprV2a(l + a) Equating Eqn (9.38) and (9.35a) and using also Eqn (9.25), leads to: 4nprV2a(l +a) = nrcrtpV2(l + a)2 cosec2 ф whence a l 2j ■ =-at cosec ф

 (9.38)

 (9.39)

 l + a 4

In the same way, by considering the angular momentum

SQ = mAwr1

where Aw is the change in angular velocity of the air on passing through the airscrew. Then SQ = (2irr8r)pV{ + a)(2bQ)r2
= 47rr3pF6(l + a)Q6r ^ = 4nr3pVb(l + a)Q,8

Now, as derived previously,

^ = 7ГPvqpVl (Eqn (9.37a))

Substituting for FR both expressions of Eqn (9.25), this becomes = m^trplVfi + fl)cosec0][I2r(l — b) sec^

Equating this expression for dQ/dr to that of Eqn (9.41) gives after manipulation

61 / ,

——— = – crq cosec ф sec ф

1 — b 4 = ^aq cosec 2ф

The local efficiency of the blade at the element, r? i, is found as follows.

(SlT

Useful power output = V ST = F—<5r

Power input – 2xn SQ 2-717? Sr

Therefore

F dT/dr ^ 27Г7І dQ/dr

V 2жra pVt 2жп 2тггга jpV^q

V t 2mr q

which is an alternative expression to Eqn (9.30).

With the expressions given above, dT/dr and dQ/dr may be evaluated at several radii of an airscrew blade given the blade geometry and section characteristics, the forward and rotational speeds, and the air density. Then, by plotting dT/dr and

dQjdr against the radius r and measuring the areas under the curves, the total thrust and torque per blade and for the whole airscrew may be estimated. In the design of a blade this is the usual first step. With the thrust and torque gradings known, the deflection and twist of the blade under load can be calculated. This furnishes new values of 9 along the blade, and the process is repeated with these new values of 9. The iteration may be repeated until the desired accuracy is attained.

A further point to be noted is that portions of the blade towards the tip may attain appreciable Mach numbers, large enough for the effects of compressibility to become important. The principal effect of compressibility in this connection is its effect on the lift-curve slope of the aerofoil section. Provided the Mach number of the relative flow does not exceed about 0.75, the effect on the lift-curve slope may be approximated by the Prandtl-Glauert correction (see Section 6.8.2). This correction states that, if the lift curve slope at zero Mach number, i. e. in incompressible flow, is ao the lift-curve slope at a subsonic Mach number M is ам where

ao

VI – M2

Provided the Mach number does not exceed about 0.75 as stated above, the effect of compressibility on the section drag is very small. If the Mach number of any part of the blade exceeds the value given above, although the exact value depends on the profile and thickness/chord ratio of the blade section, that part of the blade loses lift while its drag rises sharply, leading to a very marked loss in overall efficiency and increase in noise.

Example 9.5 At 1.25m radius on a 4-bladed airscrew of 3.5m diameter the local chord of each of the blades is 250 mm and the geometric pitch is 4.4 m. The lift-curve slope of the blade section in incompressible flow is 0.1 per degree, and the lift/drag ratio may, as an approxima­tion, be taken to be constant at 50. Estimate the thrust and torque gradings and the local efficiency in flight at 4600m (cr = 0.629, temperature = —14.7°С), at a flight speed of 67ms-1 TAS and a rotational speed of 1500rpm.

The solution of this problem is essentially a process of successive approximation to the values of a and b.

Be 4×0.25

solidity a = -— = ——— —r-r = 0.1273

2 2-кг 27ГХІ.25

1500 rpm = 25rps = n  tan7 = — whence 7=1.15°

Suitable values for initial guesses for a and b are a = 0.1, b = 0.02. Then

итф = 0.3418^ = 0.383 и. Уо

ф = 20.93°, a = 29.3 – 20.93 = 8.37° Kr V(l +a) cosec ф V(l + a) _ 67 x 1.1 sin<£ 0.357

M = ^ = 0-635, Vl-M2 = 0.773

= = °’1295 Per degree

da 0.773

Since a is the absolute incidence, i. e. the incidence from zero lift:

CL = a^ = 0.1295 x 8.37 = 1.083 da

Then    q= Cz, sin(0 + 7) = 1.083 sin(20.93 + 1.15)° =0.408
t = Cz, cos(0 + 7) = 1.083 cos 22.08° = 1.004

giving

, 0.0384 „

b = – = 0.0371

1.0384

a 1 2j 0.1274x 1.004

—— = – at cosecІф = -—_ — = 0.2515

1 + a 4 Y 4 x 0.357 x 0.357

giving 0.2515 л a= ,-=0.336

Thus the assumed values a = 0.1 and b = 0.02 lead to the better approximations a = 0.336 and b = 0.0371, and a further iteration may be made using these values of a and b. A rather quicker approach to the final values of a and b may be made by using, as the initial values for an iteration, the arithmetic mean of the input and output values of the previous iteration. Thus, in the present example, the values for the next iteration would be a = 0.218 and b = 0.0286. The use of the arithmetic mean is particularly convenient when giving instructions to computers (whether human or electronic).

The iteration process is continued until agreement to the desired accuracy is obtained between the assumed and derived values of a and b. The results of the iterations were:

a = 0.1950 b = 0.0296

to four significant figures. With these values for a and b substituted in the appropriate equations, the following results are obtained:

ф = 22°48′ a = 6°28′

KR = 207 ms-‘ M = 0.640

giving

= i;pVlcl = 3167Nm 1 per blade dr 2 K

and

= ^pV^crq = 1758 N mm 1 per blade

Thus the thrust grading for the whole airscrew is 12 670Nm_1 and the torque grading is 7032 N mm-1.

The local efficiency is

– = 0.768 or 76.8%

2nnr q

This theory permits direct calculation of the performance of an airscrew and the design of an airscrew to achieve a given performance.

9.4.1 The vortex system of an airscrew

An airscrew blade is a form of lifting aerofoil, and as such may be replaced by a hypothetical bound vortex. In addition, a trailing vortex is shed from the tip of each blade. Since the tip traces out a helix as the airscrew advances and rotates, the trailing vortex will itself be of helical form. A two-bladed airscrew may therefore be con­sidered to be replaced by the vortex system of Fig. 9.8. Photographs have been taken of aircraft taking off in humid air that show very clearly the helical trailing vortices behind the airscrew.

 Real trailing helical vortices Fig. 9.8 Simplified vortex system for a two-bladed airscrew

Rotational interference The slipstream behind an airscrew is found to be rotating, in the same sense as the blades, about the airscrew axis. This rotation is due in part to the circulation round the blades (the hypothetical bound vortex) and the remainder is induced by the helical trailing vortices. Consider three planes: plane (i) immediately ahead of the airscrew blades; plane (ii), the plane of the airscrew blades; and plane (iii) immediately behind the blades. Ahead of the airscrew, in plane (i) the angular velocity of the flow is zero. Thus in this plane the effects of the bound and trailing vortices exactly cancel each other. In plane (ii) the angular velocity of the flow is due entirely to the trailing vortices, since the bound vortices cannot produce an angular velocity in their own plane. In plane (iii) the angular velocity due to the bound vortices is equal in magnitude and opposite in sense to that in plane (i), and the effects of the trailing and bound vortices are now additive.

Let the angular velocity of the airscrew blades be Г2, the angular velocity of the flow in the plane of the blades be bQ, and the angular velocity induced by the bound vortices in planes ahead of and behind the disc be ±/3f2. This assumes that these planes are equidistant from the airscrew disc. It is also assumed that the distance between these planes is small so that the effect of the trailing vortices at the three planes is practically constant. Then, ahead of the airscrew (plane (i)):

(b – p)£l = 0

i. e.

b = i3

Behind the airscrew (plane (iii)), if u) is the angular velocity of the flow

uj = (b + /3)f2 = 2bfl

Thus the angular velocity of the flow behind the airscrew is twice the angular velocity in the plane of the airscrew. The similarity between this result and that for the axial velocity in the simple momentum theory should be noted.

## Experimental mean pitch

The experimental mean pitch is defined as the advance per revolution when the airscrew is producing zero net thrust. It is thus a suitable parameter for experimental measurement on an existing airscrew. Like the geometric pitch, it has a definite value for any given airscrew, provided the conditions of test approximate reasonably well to practical flight conditions.

9.3.2 Effect of geometric pitch on airscrew performance

Consider two airscrews differing only in the helix angles of the blades and let the blade sections at, say, 70% radius be as drawn in Fig. 9.5. That of Fig. 9.5a has a fine pitch, whereas that of Fig. 9.5b has a coarse pitch. When the aircraft is at rest, e. g. at the start of the take-off run, the air velocity relative to the blade section is the resultant Vr of the velocity due to rotation, 2ттг, and the inflow velocity, V-m. The blade section of the fine-pitch airscrew is seen to be working at a reasonable incidence, the lift 8L will be large, and the drag 8D will be small. Thus the thrust ST will be large and the torque SQ small and the airscrew is working efficiently. The section of the coarse-pitch airscrew, on the other hand, is stalled and therefore gives little lift and much drag. Thus the thrust is small and the torque large, and the airscrew is inefficient. At high flight speeds the situation is much changed, as shown in Fig. 9.5c, d. Here the section of the coarse – pitch airscrew is working efficiently, whereas the fine-pitch airscrew is now giving a negative thrust, a situation that might arise in a steep dive. Thus an airscrew that has  Fig. 9.5 Effect of geometric pitch on airscrew performance

a pitch suitable for low-speed flight and take-off is liable to have a poor performance at high forward speeds, and vice versa. This was the one factor that limited aircraft performance in the early days of powered flight.

A great advance was achieved consequent on the development of the two-pitch airscrew. This is an airscrew in which each blade may be rotated bodily, and set in either of two positions at will. One position gives a fine pitch for take-off and climb, whereas the other gives a coarse pitch for cruising and high-speed flight. Consider Fig. 9.6 which shows typical variations of efficiency tj with J for (a) a fine-pitch and (b) a coarse-pitch airscrew.

For low advance ratios, corresponding to take-off and low-speed flight, the fine pitch is obviously better whereas for higher speeds the coarse pitch is preferable. If the pitch may be varied at will between these two values the overall performance Fig. 9.6 Efficiency for a two-pitch airscrew Fig. 9.7 Efficiency for a constant-speed airscrew

attainable is as given by the hatched line, which is clearly better than that attainable from either pitch separately.

Subsequent research led to the development of the constant-speed airscrew in which the blade pitch is infinitely variable between predetermined limits. A mech­anism in the airscrew hub varies the pitch to keep the engine speed constant, per­mitting the engine to work at its most efficient speed. The pitch variations also result in the airscrew working close to its maximum efficiency at all times. Figure 9.7 shows the variation of efficiency with J for a number of the possible settings. Since the blade pitch may take any value between the curves drawn, the airscrew efficiency varies with J as shown by the dashed curve, which is the envelope of all the separate 77, J curves. The requirement that the airscrew shall be always working at its optimum efficiency while absorbing the power produced by the engine at the predetermined constant speed calls for very skilful design in matching the airscrew with the engine.

The constant-speed airscrew, in turn, led to the provision of feathering and reverse – thrust facilities. In feathering, the geometric pitch is made so large that the blade sections are almost parallel to the direction of flight. This is used to reduce drag and to prevent the airscrew turning the engine (windmilling) in the event of engine failure. For reverse thrust, the geometric pitch is made negative, enabling the airscrew to give a negative thrust to supplement the brakes during the landing ground run, and also to assist in manoeuvring the aircraft on the ground.

## Airscrew pitch

By analogy with screw threads, the pitch of an airscrew is the advance per revolution. This definition, as it stands, is of little use for airscrews. Consider two extreme cases. If the airscrew is turning at, say, 2000 rpm while the aircraft is stationary, the advance per revolution is zero. If, on the other hand, the aircraft is gliding with the engine stopped the advance per revolution is infinite. Thus the pitch of an airscrew can take any value and is therefore useless as a term describing the airscrew. To overcome this difficulty two more definite measures of airscrew pitch are accepted.

 / Fig. 9.4

9.3.1 Geometric pitch

Consider the blade section shown in Fig. 9.4, at radius r from the airscrew axis. The broken line is the zero-lift line of the section, i. e. the direction relative to the section of the undisturbed stream when the section gives no lift. Then the geometric pitch of the element is 2nr tan в. This is the pitch of a screw of radius r and helix angle (90 – в) degrees. This geometric pitch is frequently constant for all sections of a given air­screw. In some cases, however, the geometric pitch varies from section to section of the blade. In such cases, the geometric pitch of that section at 70% of the airscrew radius is taken, and called the geometric mean pitch.

The geometric pitch is seen to depend solely on the geometry of the blades. It is thus a definite length for a given airscrew, and does not depend on the precise conditions of operation at any instant, although many airscrews are mechanically variable in pitch (see Section 9.3.3).