# Category Aerodynamics for Engineering Students

## Activity factor

The activity factor is a measure of the power-absorbing capacity of the airscrew, which, for optimum performance, must be accurately matched to the power pro­duced by the engine.

Consider an airscrew of diameter D rotating at n with zero forward speed, and consider in particular an element of the blade at a radius of r, the chord of the element being c. The airscrew will, in general, produce a thrust and therefore there will be a finite speed of flow through the disc. Let this inflow be ignored, however. Then the motion and forces on the element are as shown in Fig. 9.3. 1

 2.1ГГП

Fig. 9.3

and therefore the torque associated with the element is

SQ = 2ir1pCj}nl(cri)Sr

It is further assumed that Co is constant for all blade sections. This will not normally be true, since much of the blade will be stalled. However, within the accuracy required by the concept of activity factor, this assumption is acceptable. Then the total torque required to drive an airscrew with В blades is

Thus the power absorbed by the airscrew under static conditions is approximately

In a practical airscrew the blade roots are usually shielded by a spinner, and the lower limit of the integral is, by convention, changed from zero (the root) to 0.1 D. Thus

Defining the activity factor (AF) as

Further work on the topic of airscrew coefficients is most conveniently done by means of examples.

Example 9.3 An airscrew of 3.4 m diameter has the following characteristics:

Calculate the forward speed at which it will absorb 750 kW at 1250 rpm at 3660 m (a = 0.693) and the thrust under these conditions. Compare the efficiency of the airscrew with that of the ideal actuator disc of the same area, giving the same thrust under the same conditions.

Power = 2m Q

Therefore

„ 750000×60 „„Л1кТ

torque g= 27rxl250 = 5730 Nm

= 20.83 rps n1 = 435 (rps)2

Therefore

Plotting the given values of kg and 77 against J shows that, for kg = 0.0368, J = 1.39 and 77 = 0.848. Now J = V/nD, and therefore

V = JnD = 1.39 x 20.83 x 3.4 = 98.4ms_I

Since the efficiency is 0.848 (or 84.8%), the thrust power is

750 x 0.848 = 635 kW

Therefore the thrust is

Power 635000 T = г = = 6460 N

whence

a = 0.0417

Thus the ideal efficiency is

Thus the efficiency of the practical airscrew is (0.848/0.958) of that of the ideal actuator disc. Therefore the relative efficiency of the practical airscrew is 0.885, or 88.5%.

Example 9.4 An aeroplane is powered by a single engine with speed-power characteristic:

The fixed-pitch airscrew of 3.05m diameter has the following characteristics:

 J 0.4 0.42 0.44 0.46 0.48 0.5 кт 0.118 0.115 0.112 0.109 0.106 0.103 kQ 0.0157 0.0154 0.015 0.0145 0.0139 0.0132

and is directly coupled to the engine crankshaft. What will be the airscrew thrust and efficiency during the initial climb at sea level, when the aircraft speed is 45ms~’?

Preliminary calculations required are:

Q = kepn2D5 = 324.2 kQn2

after using the appropriate values for p and D.

J = V/nD = 14.75/n

The power required to drive the airscrew, Pr, is

Pr = 2 miQ

With these expressions, the following table may be calculated:

 rpm 1800 1900 2000 2100 Fa(kW) 1072 1113 1156 1189 n (rps) 30.00 31.67 33.33 35.00 /72(rps)2 900 1003 1115 1225 J 0.492 0.465 0.442 0.421 kQ 0.013 52 0.01436 0.01494 0.015 38 G(Nm) 3950 4675 5405 6100 P r(kW) 745 930 1132 1340

In this table, P3 is the brake power available from the engine, as given in the data, whereas the values of kQ for the calculated values of J are read from a graph.

A graph is now plotted of P3 and PT against rpm, the intersection of the two curves giving the equilibrium condition. This is found to be at a rotational speed of 2010 rpm, i. e. n = 33.5 rps. For this value of n, J = 0.440 giving kT = 0.112 and kQ = 0.0150. Then

T = 0.112 x 1.226 x (33.5)2 x (3.05)4 = 13 330N

and

As a check on the correctness and accuracy of this result, note that

thrust power = TV = 13 300 x 45 = 599 кW

At 2010 rpm the engine produces 1158kW (from engine data), and therefore the efficiency is 599 x 100/1158 = 51.6%, which is in satisfactory agreement with the earlier result.

## Torque coefficient

The torque Q is a force multiplied by a length, and it follows that a rational expression for the torque is

Q = kQpn2Ds (9.19)

kQ being the torque coefficient which, like kj, depends on the airscrew design and on Re, M and J.

9.2.2 Efficiency

The power supplied to an airscrew is P;n where

Pin = 2imQ

whereas the useful power output is Pout where

Pout = TV

Therefore, the airscrew efficiency, jj, is given by

TV kTpn2D4V

2nnQ к<2рп2052пп 1 kj V 1 kj 2-k kQ nD 2-k kQ

9.2.3 Power coefficient

The power required to drive an airscrew is

P = 2’KnQ = 2nn(kQpn2D5) = 2irkQpn3D5 The power coefficient, CP, is then defined by

P Cppn2D5 i. e.

p

CP =

Cp.

## Airscrew coefficients

The performance of an airscrew may be determined by model tests. As is the case with all model tests it is necessary to find some way of relating these to the full-scale performance, and dimensional analysis is used for this purpose. This leads to a number of coefficients, analogous to the lift and drag coefficients of a body. These coefficients also serve as a very convenient way of presenting airscrew performance data, which may be calculated by blade-element theory (Section 9.4), for use in aircraft design.

9.2.1 Thrust coefficient

Consider an airscrew of diameter D revolving at n revolutions per second, driven by a torque Q, and giving a thrust of T. The characteristics of the fluid are defined by its density, p, its kinematic viscosity, v, and its modulus of bulk elasticity, K. The forward speed of the airscrew is V. It is then assumed that

T = h{D, n,p, v, K, V)

= CDanbpcvdKeVf (9.15)

Then, putting this in dimensional form,

[MLT~2] = [(L)fl(T)_*(ML_3)c(L2T~1)‘/(ML_IT_2)‘?(LT-1/’]

Separating this into the three fundamental equations gives

(M) 1 =c + e

(L) 1 = a-3c + 2d-e+f

(T) 2 = b + d + 2e+f

Solving these three equations for a, b

a = 4-2e-2d-f b = 2-d-2e-f c = 1-е

Substituting these in Eqn (9.15) gives

(9.16)

Consider the three factors within the square brackets.

(i) v/D[76] n; the product Dn is a multiple of the rotational component of the blade tip speed, and thus the complete factor is of the form v/(length x velocity), and is therefore of the form of the reciprocal of a Reynolds number. Thus ensuring equality of Reynolds numbers as between model and full scale will take care of this term.

(ii) KjpLP-n2-, K/p = a2, where a is the speed of sound in the fluid. As noted above, Dn is related to the blade tipspeed and therefore the complete factor is related to (speed of sound/velocity)2, i. e. it is related to the tip Mach number. Therefore care in matching the tip Mach number in model test and full-scale flight will allow for this factor.

(iii) V/nD; V is the forward speed of the airscrew, and therefore V/n is the distance advanced per revolution. Then VjnD is this advance per revolution expressed as a multiple of the airscrew diameter, and is known as the advance ratio, denoted by J.

Thus Eqn (9.16) may be written as

T = Cpn2D4h(Re, M, J) (9.17)

The constant C and the function h(Re, M, J) are usually collected together, and denoted by кт, the thrust coefficient. Thus, finally.

T = kTpn2D4 (9.18)

кт being a dimensionless quantity dependent on the airscrew design, and on Re, M and J. This dependence may be found experimentally, or by the blade-element theory.

## Froude’s momentum theory of propulsion

This theory applies to propulsive systems of Class I. In this class, work is done on air from the atmosphere and its energy increased. This increase in energy is used to increase the rearwards momentum of the air, the reaction to which appears as a thrust on the engine or airscrew.

The theory is based on the concept of the ideal actuator disc or pure energy supplier. This is an infinitely thin disc of area S’ which offers no resistance to air

 Fig. 9.1 The ideal actuator disc, and flow in the slipstream

passing through it. Air passing through the disc receives energy in the form of pressure energy from the disc, the energy being added uniformly over the whole area of the disc. It is assumed that the velocity of the air through the disc is constant over the whole area and that all the energy supplied to the disc is transferred to the air.

Consider the system shown in Fig. 9.1. This represents an actuator disc at rest in a fluid that, a long way ahead of the disc, is moving uniformly with a speed of V and has a pressure of po. The outer curved lines represent the streamlines that separate the fluid which passes through the disc from that which does not. As the fluid between these streamlines approaches the disc it accelerates to a speed Vo, its pressure decreasing to p. At the disc, the pressure is increased to p2 but continuity prohibits a sudden change in speed. Behind the disc the air expands and accelerates until, well behind the disc, its pressure has returned to po, when its speed is Vs. The flow between the bounding streamlines behind the disc is known as the slipstream.

In unit time:

mass of fluid passing through disc = pSV0 (9.1)

Increase of rearward momentum of this mass of fluid

= pSV0(Vs-V) (9.2)

and this is the thrust on the disc. Thus

T = pSV0(Vs-V) (9.3)

The thrust can also be calculated from the pressures on the two sides of the disc as

T = S{p2-pi) (9.4)

The flow is seen to be divided into two regions 1 and 2, and Bernoulli’s equation may be applied within each of these regions. Since the fluid receives energy at the disc Bernoulli’s equation may not be applied through the disc. Then

Po+pV2=Pi+pVl (9.5)

and

From Eqns (9.5) and (9.6)

(p*+pvi) – {pi+pvi) = {p*+pvi) – {р»+ру1)

i. e.

P2~Pi=^p(V2- V2)

Substituting this in Eqn (9.4) and equating the result to Eqn (9.3), i. e. equating the two expressions for the thrust:

X-pS{Vl-V2)=pSV, o(Fs-F)

and dividing this by pS(Vs – V) gives

Vo = {V, + V) (9.8)

showing that the velocity through the disc is the arithmetic mean of the velocities well upstream, and in the fully developed slipstream. Further, if the velocity through the disc Vo is written as

Vo = F(1 + a) (9.9)

it follows from Eqn (9.8) that

Vs + V = 2V0 = 2V(1 +a)

whence

Fs = F(l + 2 a) (9.10)

The quantity a is termed the inflow factor.

Now unit mass of the fluid upstream of the disc has kinetic energy of j V2 and a pressure energy appropriate to the pressure po, whereas the same mass well behind the disc has, after passing through the disc, kinetic energy of V2 and pressure energy appropriate to the pressure po. Thus unit mass of the fluid receives an energy increase ofl(Fs2 — V2) on passing through the disc. Thus the rate of increase of energy of the fluid in the system, dE/dt, is given by

f=rsv^(v:-r*)

= PSVo(V2-V2)

This rate of increase of energy of the fluid is, in fact, the power supplied to the actuator disc.

If it is now imagined that the disc is moving from right to left at speed V into initially stationary fluid, useful work is done at the rate TV. Thus the efficiency of the disc as a propulsive system is

TV

4~pSVo(V?-Vi)

 k(V* + V)

 {V* + V) 2 [1 + {V,/V) v_ V0 1 (1+e)

Of particular interest is Eqn (9.12a). This shows that, for a given flight speed V, the efficiency decreases with increasing Vs. Now the thrust is obtained by accelerating a mass of air. Consider two extreme cases. In the first, a large mass of air is affected, i. e. the diameter of the disc is large. Then the required increase in speed of the air is small, so Va/V differs little from unity, and the efficiency is relatively high. In the second case, a disc of small diameter affects a small mass of air, requiring a large increase in speed to give the same thrust. Thus Vs/V is large, leading to a low efficiency. Therefore to achieve a given thrust at a high efficiency it is necessary to use the largest practicable actuator disc.

An airscrew does, in fact, affect a relatively large mass of air, and therefore has a high propulsive efficiency. A simple turbo-jet or ram-jet, on the other hand, is closer to the second extreme considered above, and consequently has a poor propulsive efficiency. However, at high forward speeds compressibility causes a marked reduc­tion in the efficiency of a practical airscrew, when the advantage shifts to the jet engine. It was to improve the propulsive efficiency of the turbo-jet engine that the by­pass or turbo-fan type of engine was introduced. In this form of engine only part of the air taken is fully compressed and passed through the combustion chambers and turbines. The remainder is slightly compressed and ducted round the combustion chambers. It is then exhausted at a relatively low speed, producing thrust at a fairly high propulsive efficiency. The air that passed through the combustion chambers is
ejected at high speed, producing thrust at a comparatively low efficiency. The overall propulsive efficiency is thus slightly greater than that of a simple turbo-jet engine giving the same thrust. The turbo-prop engine is, in effect, an extreme form of by-pass engine in which nearly all the thrust is obtained at high efficiency.

Another very useful equation in this theory may be obtained by expressing Eqn (9.3) in a different form. Since

V0 = V(1 + a) and Fs = V(1 + 2d)

T = pSV0(Vs – V) = pSV( 1 + a)[V( 1 + 2a) – V = 2pSV2a{ + a)

Example 9.1 An airscrew is required to produce a thrust of 4000 N at a flight speed of 120ms-1 at sea level. If the diameter is 2.5m, estimate the minimum power that must be supplied, on the basis of Froude’s theory.

T = 2pSV1a{+a)

i. e.

Now T = 4000N, V = 120ms 1 and S = 4.90m2. Thus

whence

a = 0.0227

Then the ideal efficiency is

Пі 1.0227

Useful power = TV = 480 000 W

Therefore minimum power supplied, P, is given by

P = 480000 x 1.0227 = 491 kW

The actual power required by a practical airscrew would probably be about 15% greater than this, i. e. about 560 kW.

Example 9.2 A pair of airscrews are placed in tandem (Fig. 9.2), at a streamwise spacing sufficient to eliminate mutual interference. The rear airscrew is of such a diameter that it just fills the slipstream of the front airscrew. Using the simple momentum theory calculate: (i) the efficiency of the combination and (ii) the efficiency of the rear airscrew, if the front airscrew has a Froude efficiency of 90%, and if both airscrews deliver the same thrust. (U of L)

For the front airscrew, тц = 0.90 = Therefore

 Fig. 9.2 Actuator discs in tandem and

Vs=V(l+2a)=^V

The thrust of the front airscrew is

T = pSyVo(K – F) = Psl (j vj f) = ^pSy V2

The second airscrew is working entirely in the slipstream of the first. Therefore the speed of the approaching flow is F3, i. e. Y F. The thrust is

T = p52Tj(F’ – V) = pS2V’0(V3 – Vs)

Now, by continuity:

pS2V0 =PSlV0

and also the thrusts from the two airscrews are equal. Therefore

T = pSy V0(VS – V) = p52F'(F’ – V) = pSy F0(F’ – V3)

whence

Fs-F=F’-F5

i. e.

f; = 2F3-F=(^-i)f = HK Then, if the rate of mass flow through the discs is m:

/13 11 2

thrust of rear airscrew = m(Vs — V,) = m (——- — J V = – mF

The useful power given by the second airscrew is TV, not TV3, and therefore:

2

useful power from 2nd airscrew = – mV2

Kinetic energy added per second by the second airscrew, which is the power supplied by (and to) the second disc, is

Thus the efficiency of the rear components is

imV2

-r = 0.75 or 75% fjmV2

TV

Power input to front airscrew =

TV

0.75

Therefore

Therefore

## Propellers and propulsion

Preamble

Propulsive systems using atmospheric air include propellers, turbo-jets, ramjets, helicopters and hovercraft. Those which are independent of the atmosphere (if any) through which they move include rocket motors. In every case mentioned above the propulsive force is obtained by increasing the momentum of the working gas in the direction opposite to that of the force, assisted in the case of the hovercraft by a cushioning effect. A simple momentum theory of propulsion is applied to airscrews and rotors that permits performance criteria to be derived. A blade-element theory is also described. For the rocket motor and rocket-propelled body a similar momentum treatment is used. The hovercraft is briefly treated separately.

The forward propulsive force, or thrust, in aeronautics is invariably obtained by increasing the rearward momentum of a quantity of gas. Aircraft propulsion systems may be divided into two classes:

(I) those systems where the gas worked on is wholly or principally atmospheric air; (II) other propulsive systems, in which the gas does not contain atmospheric air in any appreciable quantity.

Class I includes turbo-jets, ram-jets and all systems using airscrews or helicopter rotors. It also includes ornithopters (and, in nature, birds, flying insects, etc.). The only example of the Class II currently used in aviation is the rocket motors.

## Reduction of wave drag

Aspects of this have been covered in the discussion of swept wings in Section 5.7 and of supercritical aerofoils in Sections 7.9 and 8.2. In the latter case it was found that keeping the pressure uniform over the upper wing surface minimized the shock strength, thereby reducing wave drag. A somewhat similar principle holds for the whole wing-body combination of a transonic aircraft. This was encapsulated in the area rule formulated in 1952 by Richard Whitcomb[75] and his team at NACA Langley. It was known that as the wing-body configuration passed through the speed of sound, the conventional straight fuselage, shown in Fig. 8.41a, experienced a sharp rise in wave drag. Whitcomb’s team showed that this rise in drag could be consider­ably reduced if the fuselage was waisted, as shown in Fig. 8.41b, in such a way as to keep the total cross-sectional area of the wing-body combination as uniform as possible. Waisted fuselages of this type became common features of aircraft designed for transonic operation.

The area rule was first applied to a production aircraft in the case of the Convair F-102A, the USAF’s first supersonic interceptor. Emergency application of the area rule became necessary owing to a serious problem that was revealed during the flight tests of the prototype aircraft, the YF-102. Its transonic drag was found to exceed the thrust produced by the most powerful engine then available. This threatened to

 Fig. 8.41 Application of the area rule for minimizing wave drag

jeopardize the whole programme because a supersonic flight speed was an essential USAF specification. The area rule was used to guide a major revised design of the fuselage. This reduced the drag sufficiently for supersonic Mach numbers to be achieved.

## Reduction of form drag

Form drag is kept to a minimum by avoiding flow separation and in this respect has already been discussed in the previous sections. Streamlining is vitally important for reducing form drag. It is worth noting that at high Reynolds numbers a circular cylinder has roughly the same overall drag as a classic streamlined aerofoil with a chord length equal to 100 cylinder radii. Form drag is overwhelmingly the main contribution to the overall drag for bluff bodies like the cylinder, whereas in the case of streamlined bodies skin-friction drag is predominant, form drag being less than ten per cent of the overall drag. For bluff bodies even minimal streamlining can be very effective.

8.4 Reduction of induced drag

Aspects of this topic have already been discussed in Chapter 5. There it was shown that, in accordance with the classic wing theory, induced drag falls as the aspect ratio of the wing is increased. It was also shown that, for a given aspect ratio, elliptic­shaped wings (strictly, wings with elliptic wing loading) have the lowest induced drag. Over the past 25 years the winglet has been developed as a device for reducing induced drag without increasing the aspect ratio. A typical example is depicted in Figs 8.37a and 8.40. Winglets of this and other types have been fitted to many different civil aircraft ranging from business jets to very large airliners.

The physical principle behind the winglet is illustrated in Figs. 8.37b and 8.37c. On all subsonic wings there is a tendency for a secondary flow to develop from the high – pressure region below the wing round the wing-tip to the relatively low-pressure region on the upper surface (Fig. 8.37b). This is part of the process of forming the trailing vortices. If a winglet of the appropriate design and orientation is fitted to the wing-tip, the secondary flow causes the winglet to be at an effective angle of incidence, giving rise to lift and drag components Lw and Dw relative to the winglet, as shown in Fig. 8.37c. Both Lw and Dw have components in the direction of flight. Lw provides a component to counter the aircraft drag, while Dw provides one that augments the aircraft drag. For a well-designed winglet the contribution of LK predominates, resulting in a net reduction in overall drag, or a thrust, equal to AT (Fig. 8.37c). For example, data available for the Boeing 747—400 indicate that

 Fig. 8.37 Using winglets to reduce induced drag

winglets reduce drag by about 2.5% corresponding to a weight saving of 9.5 tons at take-off.[74]

The winglet shown in Fig. 8.37a has a sharp angle where it joins the main wing. This creates the sort of corner flow seen at wing-body junctions. Over the rear part of the wing the boundary layer in this junction is subject to an adverse streamwise pressure gradient from both the main wing and the winglet. This tends to intensify the effect of the adverse pressure gradient leading to a risk of flow separation and increased drag. This can be avoided by the use of blended winglets (Fig. 8.38a) or a winglet that is shifted downstream (Fig. 8.38b). Variants of both these designs are very common. The pressure distributions over the upper surface of the main wing close to the wing-tip are plotted in Fig. 8.39 for all three types of winglet and for the unmodified wing. The winglet with the sharp comer has a distribution with a narrow suction peak close to the leading edge that is followed by a steep adverse pressure gradient. This type of pressure distribution favours early laminar-turbulent transition and also risks flow separation. In contrast, the other two designs, especially the

Fig. 8.38 Alternative winglet designs, (a) The blended winglet; (b) The winglet shifted downstream.

downstream-shifted winglet, have much more benign pressure distributions. Calcula­tions using the panel method indicate that all three winglet types lead to a similar reduction in induced drag.* This suggests that the two winglet designs shown in Fig. 8.38 are to be preferred to the one with a sharp comer.

‘Winglets. a close look’ (http://beadecl. ea. bs. dlr. de/airfoils/wmgltl. htm).

 Fig. 8.40 A view of the Airbus A340 showing the winglets attached to the wing-tips. These devices are used in order to reduce induced drag. See Fig. 8.37, page 523. In the foreground is the wing of the Airbus A320-200 fitted with another wing-tip device known as a wing-tip fence. {The photograph was provided by Gert Wunderlich.)

## Riblets

A moderately effective way of reducing turbulent skin friction involves surface modification in the form of riblets. These may take many forms, but essentially consist of minute streamwise ridges and valleys. One possible configuration is depicted in Fig. 8.36b. Similar triangular-shaped riblets are available in the form of polymeric film from the 3M Company. The optimum, non-dimensional, spanwise spacing between the riblets is given in wall units (see Section 7.10.5) by

j+ = sy/rw/v = 10 to 20

This corresponds to an actual spacing of 25 to 75 jim for flight conditions. (Note that the thickness of a human hair is approximately 70 pm). The 3M riblet film has been flight tested on an in-service Airbus A300-600 and on other aircraft. It is currently being used on regular commercial flights of the Airbus A340-300 aircraft by Cathay Pacific. The reduction in skin-friction drag observed was of the order of 5 to 8%. Skin-friction drag accounts for about 50% of the total drag for the Airbus A340-300 (a rather higher proportion than for many other types of airliner). Probably only about 70% of the surface of the aircraft is available to be covered with riblets leading to about 3% reduction in total drag.* This is fairly modest but represents a worth­while savings in fuel and increase in payload. Riblets have also been used on Olympic-class rowing shells in the United States and on the hull of the Stars and Stripes, the winner of the 1987 America’s Cup yacht race.

The basic concept behind the riblets had many origins, but it was probably the work at NASA Langley® in the United States that led to the present developments.

 Fig. 8.36 The effect of riblets on the near-wall structures in a turbulent boundary layer

The concept was also discovered independently in Germany through the study of the hydrodynamics of riblet-like formations on shark scales.[73] The non-dimensional riblet spacings found on shark scales lie in the range 8 < < 18, i. e. almost identical

to the range of values given above for optimum drag reduction in the experiments of NASA and others on man-made riblets.

Given that the surface area is increased by a factor of 1.5 to 2.0, the actual reduction in mean surface shear stress achieved with riblets is some 12-16%. How do riblets produce a reduction in skin-friction drag? At first sight it is astonishing that such minute modifications to the surface should have such a large effect. The

phenomenon is also in conflict with the classic view in aerodynamics and hydro­dynamics that surface roughness should lead to a drag increase. A plausible explanation for the effect of riblets is that they interfere with the development of the near-wall structures in the turbulent boundary that are mainly responsible for generating the wall shear stress. (See Section 7.10.8) These structures can be thought of as ‘hairpin’ vortices that form near the wall, as depicted in Fig. 8.36a. As these vortices grow and develop in time they reach a point where the head of the vortex is violently ejected away from the wall. Simultaneously the contra-rotating, streamwise- oriented, legs of the vortex move closer together, thereby inducing a powerful down – wash of high-momentum fluid between the vortex legs. This sequence of events is often termed a ‘near-wall burst’. It is thought that riblets act to impede the close approach of the vortex legs, thereby weakening the bursting process.

## Compliant walls: artificial dolphin skins

It is widely thought that some dolphin species possess an extraordinary laminar-flow capability. Certainly mankind has long admired the swimming skills of these fleet creatures. Scientific interest in dolphin hydrodynamics dates back at least as far as 1936 when Gray^ published his analysis of dolphin energetics. It is widely accepted that species like the bottle-nosed dolphin (Tursiops truncatus) can maintain a sustained swimming speed of up to 9 m/s. Gray followed the usual practice of marine engineers in modelling the dolphin’s body as a one-sided flat plate of length 2 m. The corresponding value of Reynolds number based on overall body length was about 20 x 106. Even in a very-low-noise flow environment the Reynolds number, Rext, for transition from laminar to turbulent flow does not exceed 2 to 3 x 106 for flow over a flat plate. Accordingly, Gray assumed that if conventional hydrodynamics were involved, the flow would be mostly turbulent and the dolphin would experience a large drag force. So large, in fact, that at 9 m/s its muscles would have to dehver about seven times more power per unit mass than any other mammalian muscle. This led him and others to argue that the dolphin must be capable of maintaining laminar flow by some extraordinary means. This hypothesis has come to be known as Gray’s Paradox.

Little in detail was known about laminar-turbulent transition in 1936 and Gray would have been unaware of the effects of the streamwise pressure gradient along the boundary layer (see Section 7.9). We now know that transition is delayed in favourable pressure gradients and promoted in adverse ones. Thus, for the dolphin, the transition point would be expected to occur near the point of minimum pressure. For Tursiops truncatus this occurs about half way along the body corresponding to Rext = 10 x 10°. When this is taken into account the estimated drag is very much less and the required power output from the muscles only exceeds the mammalian norm by no more than a factor of two. There is also some recent evidence that dolphin muscle is capable of a higher output. So on re-examination of Gray’s paradox there is now much less of an anomaly to explain. Nevertheless, the dolphin may still find advantage in a laminar-flow capability. More­over, there is ample evidence which will be briefly reviewed below, that the use of properly designed, passive, artificial dolphin skins, i. e. compliant walls, can maintain laminar flow at much higher Reynolds numbers than found for rigid surfaces.

In the late 1950s, Max Kramer,* a German aeronautical engineer working in the United States, carried out a careful study of the dolphin epidermis and designed

compliant coatings closely based on what he considered to be its key properties. Figure 8.34 shows his compliant coatings and test model. Certainly, his coatings bore a considerable resemblance to dolphin skin, particularly with respect to dimensions (see also Fig. 8.35). They were manufactured from soft natural rubber and he mimicked the effects of the fatty, more hydrated tissue, by introducing a layer of highly viscous silicone oil into the voids created by the short stubs. He achieved drag reductions of up to 60% for his best compliant coating compared with the rigid-walled control in sea-water at a maximum speed of 18 m/s. Three grades of rubber and various silicone oils with a range of viscosities were tested to obtain the largest drag reduction. The optimum viscosity was found to be about 200 times that of water.

Although no evidence existed beyond the drag reduction, Kramer believed that his compliant coatings acted as a form of laminar-flow control. His idea was that they reduced or suppressed the growth of the small-amplitude Tollmien-Schlichting waves, thereby postponing transition to a much higher Reynolds number or even eliminating it entirely. He believed that the fatty tissue in the upper dermal layer of

 Fig. 8.34 Kramer’s compliant coating and model. All dimensions are in mm. (a) Cross-section; (b) Cut through stubs; (c) Model: shaded regions were coated Source; Based on Fig. 1 of P. W. Carpenter, C. Davies and A. D. Lucey (2000) ‘Hydrodynamics and compliant walls: Does the dolphin have a secret?’, Current Science, 79(6), 758-765

 Fig. 8.35 Structure of dolphin skin, (a) Cross-section; (b) Cut through the dermal papillae at AA’; (c) Front view. Key: a, cutaneous ridges or microscales; b, dermal papillae; c, dermal ridge; d, upper epidermal layer; e, fatty tissue Source: Based on Fig. 1 of P. W. Carpenter etal. (see Fig. 8.34)

the dolphin skin and, by analogy the silicone oil in his coatings, acted as damping to suppress the growth of the waves. This must have seemed eminently reasonable at the time. Surprisingly, however, the early theoretical work by Benjamin,[71] while showing that wall compliance can indeed suppress the growth of Tollmien-Schlichting waves, also showed that wall damping in itself promoted wave growth (i. e. the waves grew faster for a high level of damping than for a low level). This led to considerable scepticism about Kramer’s claims. But the early theories, including that of Benjamin, were rather general in nature and made no attempt to model Kramer’s coatings theoretically. A detailed theoretical assessment of the laminar-flow capabilities of his coatings was carried out much later by Carpenter and Garrad* who modelled the coatings as elastic plates supported on spring foundations with the effects of visco-elastic damping and the viscous damping fluid included. Their results broadly

confirm that the Kramer coatings were capable of substantially reducing the growth of Tollmien-Schlichting waves.

Experimental confirmation for the stabilizing effects of wall compliance on Tollmien – Schlichting waves was provided by Gaster[72] who found close agreement between the measured growth and the predictions of the theory. Subsequently many authors have used versions of this theory to show how suitably designed compliant walls can achieve a fivefold or greater increase in the transitional Reynolds number, Rexl, as compared with the corresponding rigid surfaces. Although, compliant walls have yet to be used for laminar-flow control, there is little doubt that they have the potential for this in certain marine applications. In principle, they could also be used in aeronautical applications. But, in practice, owing to the need to match the inertias of the air and the wall, the wall structure would have to be unpractically light and flimsy.*

## Laminar flow control by boundary-layer suction

Distributed suction acts in two main ways to suppress laminar-turbulent transition. First, it reduces the boundary-layer thickness. Recall from Section 7.9 that for a fixed pressure gradient a critical Reynolds number based on boundary-layer thickness must be reached before transition is possible. Second, it creates a much fuller velocity profile within the boundary layer, somewhat similar to the effect of a favourable pressure gradient. This makes the boundary layer much more stable with respect to the growth of small disturbances (e. g. Tollmien-Schlichting waves). In effect, this also greatly increases the critical Reynolds number. The earliest work on laminar-flow control (LFC) including the use of suction was carried out in Germany and Switzer­land during the late 1930s in wind-tunnels.* The first flight tests were carried out in the United States in 1941 using a B-18 bomber fitted with a wing glove. The maximum flight speed available and the chord of the wing glove limited the transi­tional Reynolds number achieved to a lower value than that obtained in wind-tunnel tests.

Research on suction-type LFC continued up to the 1960s in Great Britain and the United States. This included several flight tests using wing gloves on aircraft like the F-94 and Vampire. In such tests full-chord laminar flow was maintained on the wing’s upper surface at Reynolds numbers up to ЗО x 106. To achieve this transition delay exceptionally well-made smooth wings were required. Even very small surface roughness, due to insect impact, for example, caused wedges of turbulent flow to form behind each individual roughness element. Further flight tests in the United States and Great Britain (the latter used a vertically mounted test wing on a Lan­caster bomber) revealed that it was much more difficult to maintain laminar flow over swept wings. This was because swept leading edges bring into play more powerful routes to transition than the amplification of Tollmien-Schlichting waves. First of all, turbulence propagates along the leading edge from the wing roots, this is termed leading-edge contamination. Secondly, completely different and more powerful

disturbances form in the boundary layer over the leading-edge region of swept wings. These are called cross-flow vortices.

Owing to the practical difficulties and to the relatively low price of aviation fuel, LFC research was discontinued at the end of the 1960s. More recently, with the growing awareness of the environmental requirements for fuel economy and limiting engine emissions, it has been revived. LFC is really the only technology currently available with the potential for very substantial improvement to fuel economy. For transport aircraft, the reduction in fuel burnt could exceed 30%. Recent technical advances have also helped to overcome some of the practical difficulties. The princi­pal such advances are:

(i) Krueger (Fig. 8.33) flaps at the leading edge that increase lift and act to protect the leading-edge region from insect impact during take-off and climb-out;

(ii) Improved manufacturing techniques, such as laser drilling and electron-beam technology, that permit the leading edges to be smooth perforated titanium skins;

(iii) The use of hybrid LFC.

The application of the first two innovations is illustrated in Fig. 8.33. Perforated skins give distributed suction which is more effective than the use of discrete suction slots. Hybrid LFC would be particularly useful for swept-back wings because it is not possible to maintain laminar flow over them by means of natural LFC alone. This depends on shaping the wing section in order to postpone the onset of an adverse pressure gradient to as far aft as possible. Tollmien-Schlichting waves can be sup­pressed in this way, but not the more powerful transition mechanisms of leading-edge contamination and cross-flow vortices found in the leading-edge region on swept wings. With hybrid LFC, suction is used only in the leading-edge region in order to suppress the cross-flow vortices and leading-edge contamination. Over the remainder of the wing where amplification of Tollmien-Schlichting waves is the main route to

 Electron-beam perforated Fig. 8.33 Leading-edge arrangement for 1983-1987 flight tests conducted on a JetStar aircraft at NASA Dryden Flight Research Center. Important features were: (1) Suction on upper surface only; (2) Suction through electron-beam-perforated skin; (3) Leading-edge shield extended for insect protection; (4) De-icer insert on shield for ice protection; (5) Supplementary spray nozzles for protection from insects and ice Source: Based on Fig. 12 of Braslow (2000) ibid

transition, wing-profile shaping can be used to reduce the effects of an adverse pressure gradient. In practice, it is easier to achieve this for the upper surface only. Owing to the higher flow speeds there, the upper surface produces most of the skin – friction drag. Hybrid LFC wings were extensively and successfully flight tested by Boeing on a modified 757 airliner during the early 1990s. Although, LFC based on the use of boundary-layer suction has yet to be used in any operational aircraft, the technology in the form of the less risky hybrid LFC has been established as practically realizable. In this way, based on proven current technology, a 10 to 20% improvement in fuel consumption could be achieved for moderate-sized subsonic commercial aircraft.

A detailed account of LFC technology and its history is given by Braslow.[70]