## 951 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 since V 0. Vs is the slipstream velocity and Vq the velocity at the disc. The general momentum theory shows that which, substituted...

## 1

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 (2nr6r) V(l + a) p 2irrp6r V(l + a) AV Vs - V V(l +2a) V 2 aV Equating Eqn (9.38) and (9.35a) and using also Eqn (9.25), leads to 47rprV2a(l...

## 942 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 fir. The flow is itself rotating in the same plane and sense at b l, and thus the speed of the element relative to the air in 1 b tan < j> 1 + a tan(0 + 7) Let the solidity of the annulus, < r, be defined as the ratio of the total area of blade in annulus to the total area of annulus. Then where B is the number of blades. Now

## 941 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...

## 7101 Reynolds averaging and turbulent stress

Turbulent flow is a complex motion that is fundamentally three-dimensional and highly unsteady. Figure 7.34a depicts a typical variation of a flow variable, , such as velocity or pressure, with time at a fixed point in a turbulent flow. The usual approach in engineering, originating with Reynolds*, is to take a time average. Thus the instantaneous velocity is given by where the time average is denoted by ( ) and ( )' denotes the fluctuation (or deviation from the time average). The strict...

## 777 Conditions at transition

It is usually assumed for boundary-layer calculations that the transition from laminar to turbulent flow within the boundary layer occurs instantaneously. This is obviously not exactly true, but observations of the transition process do indicate that the transition region (streamwise distance) is fairly small, so that as a first approximation the assumption is reasonably justified. An abrupt change in momentum thickness at the transition point would imply that d0 dx is infinite. The simplified...

## 7116 The ke method A typical twoequation method

Probably the most widely used method for calculating turbulent flows is the k e model which is incorporated into most commercial CFD software. It was independently developed at Los Alamos* and at Imperial College London.* Baldwin, B.S. and Lomax, H. (1978) Thin layer approximation and algebraic model for separated turbulent flows. AIAA Paper 78-257. * Harlow, F.H. and Nakayama, P.I. (1968) Transport of turbulence energy decay rate. Univ. of California, Los Alamos Science Lab. Rep. LA-3854. *...

## 10227

Therefore minimum power supplied, P, is given by 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...

## 7121 The momentum integral expression for the drag of a twodimensional body

Consider a two-dimensional control volume fixed in space (see Fig. 7.48) of unit width, with two faces (planes 0 and 2) perpendicular to the free stream, far ahead of and far behind the body respectively, the other two lying parallel to the undisturbed flow direction, and situated respectively far above and far below the body. For any stream tube (of vertical height Sy) that is contained within the wake at the downstream boundary, the mass flow per unit time is pu2Sy2 and the velocity reduction...

## 724 Laminar and turbulent flows

Closer experimental study of boundary-layer flows discloses that, like flows in pipes, there are two different regimes which can exist laminar flow and turbulent flow. In laminar flow, the layers of fluid slide smoothly over one another and there is little interchange of fluid mass between adjacent layers. The shearing tractions that develop due to the velocity gradients are thus due entirely to the viscosity of the fluid, i.e. the momentum exchanges between adjacent layers are on a molecular...

## 77 Approximate methods for a boundary layer on a flat plate with zero pressure gradient

In this section, the momentum integral equation (7.59) will be solved to give approximate expressions for the skin-friction drag and for the variation of 6, 6*, 6 and Cf along a flat plate with laminar, turbulent and mixed laminar turbulent boundary layers. This may seem a rather artificial and restrictive case to study in depth. It should be noted, however, that these results can be used to provide rough, but reasonable, estimates for any streamlined body. The equivalent flat plate for a...

## 07485

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...

## Info

By substituting A< j> , Eqn (6.121), y rj, Eqn (6.131), but from Eqn (6.122) To preserve the identity, A y'l MJ, and the transformed potential y X - MIq < j> , as previously shown in Eqn (6.127). The horizontal flow perturbations, pressure coefficients and lift coefficients follow as before. Glauert explained the latter transformation in physical terms by appealing to the fact that the flow at infinity in both the original compressible plane and the transformed, ideal or Laplace plane...

## 29 Properties of the Navier Stokes equations

At first sight the Navier-Stokes equations, especially the three-dimensional version, Eqns (2.95), may appear rather formidable. It is important to recall that they are nothing more than the application of Newton's second law of motion to fluid flow. For example, the left-hand side of Eqn (2.95a) represents the total rate of change of the x component of momentum per unit volume. Indeed it is often written as Dt Where Dt dt + Ud-x + Vd-y + Wd-z is called the total or material derivative. It...

## 66 Mach waves

The point source moves through a distance ut in the time the wave moves through the greater distance at. Once again the waves signalling the pressure disturbance will move through the whole region of fluid, ahead of and behind the moving source, (c) If the steady speed of the source is increased beyond that of the acoustic speed the individual sound waves (at any one instant) are seen in Fig. 6.11c to be eccentric intersecting circles with their centres on the...

## 64 Onedimensional flow plane normal shock waves

In the previous section the behaviour of gas when acting as a transmitter of waves of infinitesimal amplitude was considered and the waves were shown to travel at an (acoustic) speed of a s dpjdp relative to the gas, while the gas properties of pressure, density etc. varied in a continuous manner. If a disturbance of large amplitude, e.g. a rapid pressure rise, is set up there are almost immediate physical limitations to its continuous propagation. The accelerations of individual particles...

## 40 400 16 000

It can be seen that this will become unity, the incompressible value, at M 0. This is the practical meaning of the incompressibility assumption, i.e. that any velocity changes are small compared with the speed of sound in the fluid. The result given in Eqn (2.32) is the correct one, that applies at all Mach numbers less than unity. At supersonic speeds, shock waves may be formed in which case the physics of the flow are completely altered. Table 2.1 shows the variation of CPo with Mach number....

## A computational routine in Fortran

In order to see how the calculation of the influence coefficients works in practice, a computational routine written in standard FORTRAN 77 is given below, with a description of each step. SUBROUTINE INFLU XC, YC, AN, AT, NHAT, THAT, N, NM) On exit XC and YC are column matrices of length N containing the co-ordinates of the collocation points AN and AT are the N*N influence coefficient matrices and NHAT and THAT are the N*2 matrices containing the co-ordinates of the unit normal and tangent...

## 776 Drag coefficient for a flat plate with wholly turbulent boundary layer

The local friction coefficient Q may now be expressed in terms of x by substituting from Eqn (7.81) in Eqn (7.80). Thus Cf 0.0468 v ua - 77 (7-85) Tw - Jf'L 0-029 75 50 1 5 (7 g6) The total surface friction force and drag coefficient for a wholly turbulent boundary layer on a flat plate follow as

## M

It is shown above that a slight pressure change in supersonic flow is propagated along an oblique wave inclined at p, to the flow direction. The pressure difference is across, or normal to, the wave and the gas velocity will alter, as a consequence, in its component perpendicular to the wave front. If the downstream pressure is less, the flow velocity component normal to the wave increases across the wave so that the resultant downstream flow is inclined at a greater angle to the wave front,...

## UXUen78

One might question the assumption that two terms are the same order of magnitude. But, the slope of the streamlines in the boundary layer is equal to v u by definition and will also be given approximately by 6 L, so Eqn (7.8) is evidently correct. We will now use Eqns (7.3)-(7.6) and (7.8) to estimate the orders of magnitude of the terms in the Navier-Stokes equations (2.92a,b). We will assume steady flow, ignore the body-force terms, and divide throughout by p (noting that the kinematic...

## 10 1276

It is often convenient to regard the effects of compressibility as negligible if the flow speed nowhere exceeds about 100 m s-1. However, it must be remembered that this is an entirely arbitrary limit. Compressibility applies at all flow speeds and, therefore, ignoring it always introduces an error. It is thus necessary to consider, for each problem, whether the error can be tolerated or not. In the following examples use will be made of the equation (1.6d) for the speed of sound that can also...

## Gxl

The maximum speed of a rocket in free space will be reached when all the fuel is burnt, i.e. at the instant the motor ceases to produce thrust. Let the mass with all fuel burnt be My. Then, from Eqn (9.66) where R is the mass ratio M0 My. Note that if the mass ratio exceeds e 2.718 , the base of natural logarithms, the speed of the rocket will exceed the speed of ejection of the exhaust relative to the rocket. Now if the distance travelled from the instant of firing is x in time t vt In Mo v...

## 26 The momentum equation

The momentum equation for two- or three-dimensional flow embodies the application of Newton's second law of motion (mass times acceleration force, or rale of change of momentum force) to an infinitesimal control volume in a fluid flow (see Fig. 2.8). It lakes the form of a set of partial differential equations. Physically it states that the rate of increase in momentum within the control volume plus the net rate at which momentum flows out of the control volume equals the force acting on the...

## 7t

And again the centre of pressure moves as the lift or incidence is changed. Now, from Section 1.5.4, and comparing Eqns (4.44) and (4.45) gives This shows that, theoretically, the pitching moment about the quarter chord point for a thin aerofoil is constant, depending on the camber parameters only, and the quarter chord point is therefore the aerodynamic centre. It is apparent from this analysis that no matter what the camber-line shape, only the first three terms of the cosine series...

## Mdf

If the filter function is chosen appropriately this has the effect of 'averaging' over the sub-grid scales. Two common choices of filter function The choice made for the size of A or Aa in Eqns (7.158) or (7.159) determines the sub-grid scale. Filtering the Navier-Stokes equations gives -5- + s i + i2 + s w 3 - - s +ZA7zM,- I 1,2,3 at ox i 0x2 0x3 poxj where m, 112 and 3 denote u, v and w, and xi, and X3 denote x, y and z and where uiuj Hfij + Ufa + u'j i + u' j (7.162) * Deardorff, J.W. (1970)...

## 13

Where and aE are often termed effective turbulence Prandtl numbers. Further modification of Eqns (7.154 and 7.155) is required to deal with relatively low Reynolds numbers. See Wilcox (1993) for details of this and the choice of wall boundary conditions. The k e model is intended for computational calculations of general turbulent flows. It is questionable whether it performs any better than, or even as well as, the zero-equation models described in Section 7.11.5 for boundary layers. But it...

## Te

Fig. 1.12 Centre of pressure position for Example 1.5 Attempts have been made to rationalize the definitions and terminology associated with drag*. On the whole the new terms have not been widely adopted. Here we will use the widely accepted traditional terms and indicate alternatives in parentheses. This is formally defined as the force corresponding to the rate of decrease in momentum in the direction of the undisturbed external flow around the body, this decrease being calculated between...

## 32

For the theoretical estimation of rt and ( , of the complete aircraft, the contribu tions of the tailplane must be added. These are given here for completeness. It has been shown that quite general camber lines may be used in the theory satisfactorily and reasonable predictions of the aerofoil characteristics obtained. The reverse problem may be of more interest to the aerofoil designer who wishes to obtain the camber-line shape to produce certain desirable characteristics. The general design...

## 712 Estimation of profile drag from velocity profile in wake

At the trailing edge of a body immersed in a fluid flow, there will exist the boundary layers from the surfaces on either side. These boundary layers will join up and move downstream in the form of a wake of retarded velocity. The velocity profile will change with distance downstream, the wake cross-section increasing in size as the magnitude of its mean velocity defect, relative to free stream, decreases. At a sufficient distance downstream, the streamlines will all be parallel and the static...

## J

It may now be argued that very close to the wall, in the viscous sublayer (u 0), the velocity u will not depend on the overall size of the pipe, i.e. that u f(a). If this is so, then it immediately follows that rw, which is p(du dy)w, cannot depend on the pipe diameter and therefore the term aK7 4)-(i 4) jn jjqn (7.77) must be unity in order not to affect the expression for rw. For this to be so, 7n 4 -1 4 0 which immediately gives n j. Substituting this back into Eqn (7.74) gives u Um (y...

## Vrv

As the reference velocity that is subsequently used to render the velocity in the near-wall region non-dimensional. Integrate Eqn (7.115) and divide by V* to obtain the non-dimensional velocity profile in the fully turbulent region, and also re-write (7.114) to obtain the same in the viscous sub-layer. Thus where C and C2 are constants of integration to be determined by comparison with experimental data and rj or y+ yV, u is the dimensionless distance from the wall the length + u Vt is usually...

## Q

Now Cf is the local surface shear stress coefficient at the base of the boundary layer, and at the wake centre, where the two boundary layers join, there is no relative velocity and therefore no shearing traction. Thus, for each half of the wake, Q is zero and Eqn (7.171) becomes It is clear from this that if the mainstream velocity outside the wake is constant, then dUe dx 0 and the right-hand side becomes zero, i.e. the momentum thickness of the wake is constant. This would be expected from...

## T

Consider this by the reverse argument. Look again at Fig. 4.3b. By definition the velocity potential of C relative to A (( ca) must be equal to the velocity potential of C relative to B (< > cb) in a potential flow. The integration continued around ACB gives This is for a potential flow only. Thus, if T is finite the definition of the velocity potential breaks down and the curve ACB must contain a region of rotational flow. If the flow is not potential then Eqn (ii) in Section 3.2 must give...

## 954 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 u , 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 2vv and respectively. The resultant speed through the disc is denoted by U and the resultant speed in the fully developed slipstream by U ....

## H

Where (' ) is used to denote a time derivative. The instantaneous rate of rotation of a fluid element is given by (a 3) 2 - see above. This corresponds to a fundamental property of fluid flow called the vorticity that, using Eqn (2.71), in two-dimensional flow is defined as In three-dimensional flow vorticity is a vector given by _ , . _ fdw dv du dw dv du It can be seen that the three components of vorticity are twice the instantaneous rates of rotation of the fluid element about the three...

## 921 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 MLT 2 (L)(T)*(ML-3)f(L2T V(ML-'T V T1 Separating this into the three fundamental equations gives T h(D,n,p, v,K, V) CDanbpcvdKeVf Then, putting this in dimensional form, Solving...

## 7131 Nearnormal shock interaction with laminar boundary layer

There appear to be three general possibilities when a near-normal shock interacts with a laminar boundary layer. With a relatively weak shock, corresponding to an upstream Mach number just greater than unity, the diffused pressure rise may simply cause a gradual thickening of the boundary layer ahead of the shock with no transition and no separation. The gradual thickening causes a family of weak compression waves to develop ahead of the main shock (these are required to produce the supersonic...

## 7107 Distribution of Reynolds stresses and turbulent kinetic energy across the boundary layer

Figure 7.38 plots the variation of Reynolds shear stress and kinetic energy (per unit mass), k (i 2 + va + w 2) 2 across the boundary layer. What is immediately striking is how comparatively high the levels are in the near-wall region. The Reynolds shear stress reaches a maximum at about y+ 100 while the turbulence kinetic energy appears to reach its maximum not far above the edge of the viscous sub-layer. Figure 7.39 plots the distributions of the so-called turbulence intensities of the...

## Xac

Fig. 1.11 Determination of the centre of pressure position Fig. 1.11 Determination of the centre of pressure position Again making the approximations that cos a 1 and CD sin a can be ignored, the Eqn (1.54), above, becomes At first sight this would suggest that kcp is always less than xac c. However, CVAC is almost invariably negative, so that in fact cp is numerically greater than jcac c and the centre of pressure is behind the aerodynamic centre. Example 1.5 For the aerofoil section of...

## 11 Units and dimensions

A study in any science must include measurement and calculation, which presupposes an agreed system of units in terms of which quantities can be measured and expressed. There is one system that has come to be accepted for most branches of science and engineering, and for aerodynamics in particular, in most parts of the world. That system is the Syst me International d'Unit s, commonly abbreviated to SI units, and it is used throughout this book, except in a very few places as specially noted....

## O

Showing that for large values of Re (recall that turbulence is a phenomenon that only occurs at large Reynolds numbers) the viscous shear stress will be negligible compared with the Reynolds shear stress. Boussinesq* drew an analogy between viscous and Reynolds shear stresses by introducing the concept of the eddy viscosity Ej'. r i cf. -pwV PSTg r y( W) Boussinesq, himself, merely assumed that eddy viscosity was constant everywhere in the flow field, like molecular viscosity but very much...

## 96 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...

## 91 Froudes 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 deal actuator disc, and flow in the...

## Ttt

Fig. 6.21 Wave reflection from an open boundary Fig. 6.21 Wave reflection from an open boundary deflects the flow towards the wall where the compressive reflected wave from the wall (P2Q2) is required to bring the flow back parallel to the wall and in so doing increases its pressure to pi (greater than p). The requirement of the reflection of P2Q2 in the open boundary is thus expansive wavelet Q2P3 which brings the pressure back to the ambient value p again. And so the cycle repeats itself. The...

## P

And has the dimensions L2T-1 and the units m2 s-1. It may be regarded as a measure of the relative magnitudes of viscosity and inertia of the fluid and has the practical advantage, in calculations, of replacing two values representing fj, and p by a single value. 1.2.7 Speed of sound and bulk elasticity The bulk elasticity is a measure of how much a fluid (or solid) will be compressed by the application of external pressure. If a certain small volume, V, of fluid is subjected to a rise in...

## 10

Fig. 3.39 Calculation of pressure coefficient for NACA 0024 aerofoil What are the advantages of the panel method compared to other numerical methods such as finite differences and finite elements Both of the latter are field methods that require that the whole of the flow field be discretized. The panel method, on the other hand, only requires the discretization of the body surface - the boundary of the flow field. The dimensions of the solution are thereby reduced by one compared to the field...

## Eyk

Cl oo (a < *o) - e from Eqn (5.57) r oc cax (a - ao) - e -r a three-dimensional lift slope Fig. 5.35 Three different wing planforms with the same elliptic chord distribution Fig. 5.35 Three different wing planforms with the same elliptic chord distribution for an elliptic chord distribution, so that on substituting in Eqn (5.63) and rearranging This equation gives the lift-curve slope a for a given aspect ratio (AR) in terms of the two-dimensional slope of the aerofoil section used in the...

## 000000

H - 0.34375(1 +0.5cos0)(1 - O.O5455cos0) Hot 0.032 995(1 +0.5 cos 0)(1 -0.054 55 cos 0)(1 + 0.363 64 cos 0) where a is now in radians. For convenience Eqn (5.60) is rearranged to fia sin0 Ai sin0(sin0 + n) + A3 sin30(sin0 + 3 i) + ssin50(sin0 + 5 i) + A7 sin 70(sin 9 + 7 i) and since the distribution is symmetrical the odd coefficients only will appear. Four coefficients will be evaluated and because of symmetry it is only necessary to take values of between 0 and tt 2, i.e. tt 8, tt 4, 3tt 8,...

## Lwb

Fig. 5.33 (a) Elliptic distribution gives constant downwash and minimum drag, (b) Non-elliptic distribution gives varying downwash. (c) Equivalent variation for comparison purposes and since J twfi(z) 0 in Eqn (5.53) D (b) oc 5' j Comparing Eqns (5.54) and (5.55) DV(b) Dv it)+-m and since fi(z) is an explicit function of z, since (f,(z))2 is always positive whatever the sign of fi(z). Hence DV(b) is always greater than Dy ). 5.6 Determination of the load distribution on a given wing This is the...

## L

pV2sL J ' TnA sin9 A sinn9 dQ This can be demonstrated by multiplying out the first three (say) odd harmonics, thus 1 (A sin0 + 3 3 sin 39 + sin sin9 + A3 sin39 + A5 sin9)d9 I A sin2 9 + 3Al sin2 9 + SA sin2 9 + A i A3 sin 9 sin 30 and Jo other like terms which are products of different multiples of 0 d9 On carrying out the integration from 0 to -k all terms other than the squared terms vanish leaving 1 J (Aj sin2 9 + 3A sin2 39 + 5A sin2 50 H----)d0

## Kof

Fig. 5.29 Circulation superimposed on forward wind velocity and downwash to give lift and vortex drag (induced drag) respectively the end-effects are more dominant. It seems therefore that a wing that is large in the spanwise dimension, i.e. large aspect ratio, is a better wing - nearer the ideal - than a short span wing of the same section. It would thus appear that a wing of large aspect ratio will have better aerodynamic characteristics than one of the same section with a lower aspect ratio....

## JC

The induced drag per unit span (dv), or the induced drag grading, again by the Kutta-Zhukovsky theorem is and by similar integration over the span This expression for Dv shows conclusively that if w is zero all along the span then Dv is zero also. Clearly, if there is no trailing vorticity then there will be no induced drag. This condition arises when a wing is working under two-dimensional conditions, or if all sections are producing zero lift. As a consequence of the trailing vortex system,...

## A

Fig. 5.23 Modelling the lifting effect by a distribution of horseshoe vortex elements Fig. 5.24 Equivalence between distributions of (a) horseshoe and (b) L-shaped vortices when calculating the induced velocity. These problems can be overcome by recom-bining the elements in the way depicted in Fig. 5.24. Here it is recognized that partial cancellation occurs for two elemental horseshoe vortices occupying adjacent span-wise positions, z and z + 6z. Accordingly, the horseshoe-vortex element can...

## 4

Where x (z) denotes the leading edge of the wing. In general, Eqn (5.23) is fairly cumbersome and nowadays modern computational techniques like the panel method (see Section 5.8) are used. In the special case of Fig. 5.20 Modelling the displacement effect by a distribution of sources Fig. 5.20 Modelling the displacement effect by a distribution of sources wings having high aspect ratio, intuition would suggest that the flow over most of the wing behaves as if it were two-dimensional. Plainly...

## B

To find the velocity at P due to the length AB the sum of induced velocities due to all such elements is required. Before integrating, however, all the variables must be quoted in terms of a single variable. A convenient variable is (see Fig. 5.10) and the limits of the integration are since < j> passes through zero when integrating from A to B. sin 0 cos < p, r2 h2 sec2 < p dr d(h tan < j> ) h sec2 < > d < j> The integration of Eqn (5.8) is thus - rCOS diA sin - p + sin - a...

## 778 Mixed boundary layer flow on a flat plate with zero pressure gradient

Figure 7.26 indicates the symbols employed to denote the various physical dimensions used. At the leading edge, a laminar layer will begin to develop, thickening with distance downstream, until transition to turbulence occurs at some Reynolds number Ret Uxxt v. At transition the thickness increases suddenly from < 5l, in the laminar layer to & rt in the turbulent layer, and the latter then continues to grow as if it had started from some point on the surface distant xjt ahead of...

## Xtq

Consider the three factors within the square brackets. (i) v lPn 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) KlplP-n2-, K p a2, where a is the speed of sound in the fluid. As noted above, Dn is related to the blade...

## 925 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,...

## Joid

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 J 1.06 1.19 1.34 1.44 kQ 0.0410 0.0400 0.0378 0.0355 7 0.76 0.80 0.84 0.86 J 0.40 0.42 0.44 0.46 0.48 0.50 kT 0.118 0.115 0.112 0.109 0.106 0.103 kQ 0.0157 0.0154 0.0150 0.0145 0.0139 0.0132 and is directly coupled to the engine crankshaft. What will be the airscrew thrust and...

## 1340

In this table, P3 is the brake power available from the engine, as given in the data, whereas the values of Icq 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. 33.5 rps. For this value of , J 0.440 giving kT 0.112 and kQ 0.0150. Then r 0.112 x 1.226 x (33.5) x (3.05)4 13 330N

## 93 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...

## 931Geometric 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 2irr tan . This is the pitch of a screw of radius r and helix angle (90 - 6) degrees. This geometric pitch is frequently constant for all sections of a given airscrew. In some cases, however, the geometric pitch varies...

## 87 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, ft was also shown that, for a given aspect ratio, elliptic-shaped 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...

## 852 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...

## 851 Laminar flow control by boundarylayer 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...

## 85 Reduction of skinfriction drag

Four main types of drag are found in aerodynamics - see Section 1.5.5 - namely skin-friction drag, form drag, induced drag, and wave drag. The methods in use for * For example, see J.P. Johnston and M. Nishi (1990) 'Vorlex 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)...

## 843 Other methods of separation control

Passive flow control through the generation of streamwise vortices is frequently used on aircraft and other applications. Some of the devices commonly in use are depicted * 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. Seifert and L.G. Pack (1999) 'Oscillatory control of separation at high Reynolds number', AIAA J., 37(9), 1062-1071. in Fig. 8.32. Figure 8.32a shows a row of...

## 841 Boundarylayer suction

The basic principle was demonstrated experimentally in Prandtl's paper that introduced the boundary-layer concept to the world.* He showed that the suction through a slot could be used to prevent flow separation from the surface of a cylinder. The basic principle is illustrated in Fig. 8.22. The layer of low-energy ('tired') air near the surface approaching the separation point is removed through a suction slot. * A more complete recent account is to be found in M. Gad-el-Hak (2000) Flow...

## 84 Boundary layer control for the prevention of separation

Many of the widely used techniques have already been described in Section 8.3. But there are various other methods of flow-separation control that are used on aircraft and in other engineering applications. These are described here.* Some of the devices used are active, i.e. they require the expenditure of additional power from the propulsion units others are passive and require no additional power. As a general rule, however, the passive devices usually lead to increased drag at cruise when...

## 833Offthesurface recovery

What happens with a typical multi-element aerofoil, as shown in Figs 8.9 and 8.13, is that the boundary layer develops in the adverse pressure gradient of the slat, Fig. 8.12 Effect of a vane (modelled by a vortex) on the velocity distribution over the main wing Fig. 8.12 Effect of a vane (modelled by a vortex) on the velocity distribution over the main wing reaches the trailing edge in an unseparated state, and then leaves the trailing edge forming a wake. The slat wake continues to develop in...

## 832The vane effect

In a similar way the effect of the downstream element (e.g. the vane) on the immediate upstream element (e.g. the main aerofoil) can also be modelled approximately by placing a vortex near the trailing edge of the latter. This effect is illustrated in Fig. 8.12. This time the vane (vortex) near the trailing edge induces a velocity over the main aerofoil surface that leads to a rise in velocity on both upper and lower surfaces. In the case of the upper surface this is beneficial because it...

## 831 The slat effect

To appreciate qualitatively the effect of the upstream element (e.g. the slat) on the immediate downstream element (e.g. the main aerofoil) the former can be modelled by a vortex. The effect is illustrated in Fig. 8.11. When one considers the component of the velocity induced by the vortex in the direction of the local tangent to the aerofoil contour in the vicinity of the leading edge (see inset in Fig. 8.11), it can be seen that the slat (vortex) acts to reduce the velocity along the edge of...

## Aerodynamics In Civil Engineering

Fig. 8.4 Upper-wing-surface pressure distributions with laminar rooftop Fig. 8.4 Upper-wing-surface pressure distributions with laminar rooftop other hand, only modest maximum suction pressures are permissible before sonic conditions are reached. In this case, therefore, the pressure distribution is very flat. An example of the practical application of these ideas for low flight speeds is illustrated schematically in Fig. 8.5. This shows a Liebeck* aerofoil. This sort of aerofoil was used as a...

## 24 Twodimensional flow

Consider flow in two dimensions only. The flow is the same as that between two planes set parallel and a little distance apart. The fluid can then flow in any direction between and parallel to the planes but not at right angles to them. This means that in the subsequent mathematics there are only two space variables, x and y in Cartesian (or rectangular) coordinates or r and 9 in polar coordinates. For convenience, a unit length of the flow field is assumed in the z direction perpendicular to x...

## 0228

When the component of the free-stream velocity perpendicular to the leading edge is greater than the local speed of sound the wing is said to have a supersonic leading edge. In this case, as illustrated in Fig. 6.56, there is two-dimensional supersonic flow over much of the wing. This flow can be calculated using supersonic aerofoil theory. For the rectangular wing shown in Fig. 6.56 the presence of a wing-tip can only be communicated within the Mach cone apex which is located at the wing-tip....

## 35 Computational panel methods

In Section 3.3.7, it was shown how the two-dimensional potential flow around an oval-shaped contour, the Rankine oval, could be generated by the superposition of a source and sink on the ,v axis and a uniform flow. An analogous three-dimensional flow can also be generated around a Rankine body - see Section 3.4.4 above - by using a point source and sink. Thus it can be demonstrated that the potential flow around certain bodies can be modelled by placing sources and sinks in the interior of the...

## 842 Control by tangential blowing

Since flow separation is due to the complete loss of kinetic energy in the boundary layer immediately adjacent to the wall, another method of preventing it is to re-energize the 'tired' air by blowing a thin, high-speed jet into it. This method is often used with trailing-edge flaps (Fig. 8.25). To obtain reasonable results with this (a) Normal coanda flow (b) Jet break-away Fig. 8.26 The Coanda effect - the flow of a jet around a circular cylinder Source Based on Fig. 1 of P.W. Carpenter and...

## 7133 Shockwaveboundarylayer interaction in supersonic flow

One of the main differences between subsonic and supersonic flows, as far as boundary-layer behaviour is concerned, is that the pressure gradient along the flow is of opposite sign with respect to cross-sectional area change. Thus in a converging supersonic flow the pressure rises and in a diverging flow the pressure falls in the stream direction (see Section 6.2). As a result the pressure gradient at a convex corner is negative and the boundary layer will generally negotiate the corner without...

## A solution of integrals of the type of Glauerts integral

o cos 0 cos In Chapters 4 and 5 much use is made of the integral This may be proved, by contour integration, as follows. In the complex plane, integrate the function with respect to z round the circle of unit radius centred at the origin. On this circle z eltf and therefore z2 - 2z cos 0i + 1 7_lr tm - 2eie cos + 1 which, cancelling eifl from numerator and denominator, putting e'e cos 0 + i sin and using De Moivre's theorem, reduces to * This section may be omitted at a first reading. The...

## Aircraft Downwash

5.2.3 Variation of velocity in vortex flow To confirm how the velocity outside a vortex core varies with distance from the centre consider an element in a thin shell of air (Fig. 5.12). Here, flow conditions depend only on the distance from the centre and are constant all round the vortex at any given radius. The small element, which subtends the angle 86 at the centre, is circulating round the centre in steady motion under the influence of the force due to the radial pressure gradient....

## 933Effect 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, 2irnr, and the inflow velocity, Vm. The blade section of the fine-pitch airscrew is seen to be working at...

## 836 Gurney flaps

As well as being a great racing-car driver, Dan Gurney is also well-known for his technical innovations. His most widely emulated innovation is probably the now-obligatory practice of winning drivers spraying their supporters with champagne from vigorously shaken bottles. But it is for the Gurney flap that he is known in aerodynamics. This is a deceptively simple device consisting merely of a small plate fixed to and perpendicular to the trailing edge of a wing. It can be seen attached to the...

## Twodimensional wing theory

Here the basic fluid mechanics outlined previously is applied to the analysis of the flow about a lifting wing section. It is explained that potential flow theories of themselves offer little further scope for this problem unless modified to simulate certain effects of real flows. The result is a powerful but elementary aerofoil theory capable of wide exploitation. This is derived in the general form and applied to a number of discrete aeronautical situations, including the flapped aerofoil and...

## Finite wing theory

Whatever the operating requirements of an aeroplane may be in terms of speed endurance, pay-load and so on, a critical stage in its eventual operation is in the low-speed flight regime, and this must be accommodated in the overall design process. The fact that low-speed flight was the classic flight regime has meant that over the years a vast array of empirical data has been accumulated from flight and other tests, and a range of theories and hypotheses set up to explain and extend these...

## 82 Maximizing lift for singleelement aerofoils

This section addresses the question of how to choose the pressure distribution, particularly that on the upper wing surface, to maximize the lift. Even when a completely satisfactory answer is found to this rather difficult question, it still remains to determine the appropriate shape the aerofoil should assume in order to produce the specified pressure distribution. This second step in the process is the so called inverse problem of aerofoil design. It is very much more demanding than the...

## 741 Separation bubbles

On many aerofoils with relatively large upper-surface curvatures, high local curvature over the forward part of the chord may initiate a laminar separation when the aerofoil is at quite a moderate angle of incidence Fig. 7.14 . Small disturbances grow much more readily and at low Reynolds numbers in separated, as compared to attached, boundary layers. Consequently, the separated laminar boundary layer may well undergo transition to turbulence with characteristic rapid thickening. This rapid...

## 837 Movable flaps artificial bird feathers

This concept is illustrated in Fig. 8.20. Superficially it appears similar to the Gurney flap. However, the mode of operation is quite different. And, in any case, for positive high lift the Gurney flap would be attached to the trailing edge pointing downwards. The basic idea here is that at high angles of attack when flow separation starts to occur near the trailing edge, the associated reversed flow causes the movable flap to be raised. This then acts as a barrier to the further migration of...

## 835Use of multielement aerofoils on racing cars

In the 1960s and early 1970s several catastrophic accidents occurred in which racing cars became airborne. In some cases aerodynamic interference from nearby competing vehicles was undoubtedly a factor. Nevertheless, these accidents are a grim reminder of what can happen to a racing car if insufficient aerodynamic downforce is generated. Modern Grand Prix cars generate their prodigious aerodynamic downforces from two main sources, namely 'ground effect' and inverted wings. Under current...