Category Aerodynamics for Engineering Students

If the boundary layer that develops on the surface of a flat plate held edgeways on to the free stream is studied, it is found that, in general, a laminar boundary layer starts to

develop from the leading edge. This laminar boundary layer grows in thickness, in accordance with the argument of Section 7.2, from zero at the leading edge to some point on the surface where a rapid transition to turbulence occurs (see Fig. 7.29). This transition is accompanied by a corresponding rapid thickening of the layer. Beyond this transition region, the turbulent boundary layer then continues to thicken steadily as it proceeds towards the trailing edge. Because of the greater shear stresses within the turbulent boundary layer its thickness is greater than for a laminar one. But, away from the immediate vicinity of the transition region, the actual rate of growth along the plate is lower for turbulent boundary layers than for laminar ones. At the trading edge the boundary layer joins with the one from the other surface to form a wake of retarded velocity which also tends to thicken slowly as it flows away downstream (see Fig. 7.5).

On a flat plate, the laminar profile has a constant shape at each point along the surface, although of course the thickness changes, so that one non-dimensional relationship for м = f{y) is sufficient (see Section 7.3.4). A similar argument applies to a reasonable approximation to the turbulent layer. This constancy of profile shape means that flat-plate boundary-layer studies enjoy a major simplification and much work has been undertaken to study them both theoretically and experimentally.

However, in most aerodynamic problems, the surface is usually that of a stream­line form such as a wing or fuselage. The major difference, affecting the boundary – layer flow in these cases, is that the mainstream velocity and hence the pressure in a streamwise direction is no longer constant. The effect of a pressure gradient along the flow can be discussed purely qualitatively initially in order to ascertain how the boundary layer is likely to react.

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 scale only.

In turbulent flow considerable seemingly random motion exists, in the form of velocity fluctuations both along the mean direction of flow and perpendicular to it. As a result of the latter there are appreciable transports of mass between adjacent layers. Owing to these fluctuations the velocity profile varies with time. However, a time-averaged, or mean, velocity profile can be defined. As there is a mean velocity gradient in the flow, there will be corresponding interchanges of streamwise momen­tum between the adjacent layers that will result in shearing stresses between them. These shearing stresses may well be of much greater magnitude than those that develop as the result of purely viscous action, and the velocity profile shape in a turbulent boundary layer is very largely controlled by these Reynolds stresses (see Section 7.9), as they are termed.

As a consequence of the essential differences between laminar and turbulent flow shearing stresses, the velocity profiles that exist in the two types of layer are also different. Figure 7.4 shows a typical laminar-layer profile and a typical turbulent – layer profile plotted to the same non-dimensional scale. These profiles are typical of those on a flat plate where there is no streamwise pressure gradient.

In the laminar boundary layer, energy from the mainstream is transmitted towards the slower-moving fluid near the surface through the medium of viscosity alone and only a relatively small penetration results. Consequently, an appreciable proportion of the boundary-layer flow has a considerably reduced velocity. Throughout the

Fig. 7.4 boundary layer, the shearing stress r is given by r = fi(du/dy) and the wall shearing stress is thus tw = fj(du/dy)y-0 = /i(<9u/<9y)w(say).

In the turbulent boundary layer, as has already been noted, large Reynolds stresses are set up owing to mass interchanges in a direction perpendicular to the surface, so that energy from the mainstream may easily penetrate to fluid layers quite close to the surface. This results in the turbulent boundary away from the immediate influ­ence of the wall having a velocity that is not much less than that of the mainstream. However, in layers that are very close to the surface (at this stage of the discussion considered smooth) the velocity fluctuations perpendicular to the wall are evidently damped out, so that in a very limited region immediately adjacent to the surface, the flow approximates to purely viscous flow.

In this viscous sublayer the shearing action becomes, once again, purely viscous and the velocity falls very sharply, and almost linearly, within it, to zero at the surface. Since, at the surface, the wall shearing stress now depends on viscosity only, i. e. rw = fi(du/dy)w, it will be clear that the surface friction stress under a turbulent layer will be far greater than that under a laminar layer of the same thickness, since (du/dy)w is much greater. It should be noted, however, that the viscous shear-stress relation is only employed in the viscous sublayer very close to the surface and not throughout the turbulent boundary layer.

It is clear, from the preceding discussion, that the viscous shearing stress at the surface, and thus the surface friction stress, depends only on the slope of the velocity profile at the surface, whatever the boundary-layer type, so that accurate estimation of the profile, in either case, will enable correct predictions of skin-friction drag to be made.

In order to make the idea of a boundary layer realistic, an arbitrary decision must be made as to its extent and the usual convention is that the boundary layer extends to a distance 6 from the surface such that the velocity и at that distance is 99% of the local mainstream velocity Ue that would exist at the surface in the absence of the boundary layer. Thus 6 is the physical thickness of the boundary layer so far as it needs to be considered and when defined specifically as above it is usually designated the 99%, or general, thickness. Further thickness definitions are given in Section 7.3.2.

7.2.1 Non-dimensional profile

In order to compare boundary-layer profiles of different thickness, it is convenient to express the profile shape non-dimensionally. This may be done by writing U = u/Ue and у = yjS so that the profile shape is given by u=f(y). Over the range у — 0 to у — 6, the velocity parameter й varies from 0 to 0.99. For convenience when using и values as integration limits, negligible error is introduced by using n = 1.0 at the outer boundary, and considerable arithmetical simplification is achieved. The vel­ocity profile is then plotted as in Fig. 7.3b.

Consider again a small element of fluid (Fig. 7.2) of unit depth normal to the flow plane, having a unit length in the direction of motion and a thickness by normal to the flow direction. The shearing stress on the lower face AB will be r = /r(du/dy) while that on the upper face CD will be т + (дт/ду)ёу, in the directions shown, assuming и to increase with y. Thus the resultant shearing force in the x-direction will be [t + (дт/ду)ёу] – t = (дт/ду)ёу (since the area parallel to the x-direction is unity) but r = fi(du/dy) so that the net shear force on the element = ^(сРи/ду^ёу. Unless fi be zero, it follows that d2u/dy2 cannot be infinite and therefore the rate of change of the velocity gradient in the boundary layer must also be continuous.

Also shown in Fig. 7.2 are the streamwise pressure forces acting on the fluid element. It can be seen that the net pressure force is -(dp/dx)8x. Actually, owing to the very small total thickness of the boundary layer, the pressure hardly varies at all normal to the surface. Consequently, the net transverse pressure force is zero to a very good approximation and Fig. 7.2 contains all the significant fluid forces. The

effects of streamwise pressure change are discussed in Section 7.2.6 below. At this stage it is assumed that dp/dx = 0.

If the velocity и is plotted against the distance у it is now clear that a smooth curve of the general form shown in Fig. 7.3a must develop (see also Fig. 7.11). Note that at the surface the curve is not tangential to the и axis as this would imply an infinite gradient du/dy, and therefore an infinite shearing stress, at the surface. It is also evident that as the shearing gradient decreases, the retarding action decreases, so that

Fig. 7.3

at some distance from the surface, when dujdy becomes very small, the shear stress becomes negligible, although theoretically a small gradient must exist out to у = oo.

For the flow around a body with a sharp leading edge, the boundary layer on any surface will grow from zero thickness at the leading edge of the body. For a typical aerofoil shape, with a bluff nose, boundary layers will develop on top and bottom surfaces from the front stagnation point, but will not have zero thickness there (see Section 2.10.3).

On proceeding downstream along a surface, large shearing gradients and stresses will develop adjacent to the surface because of the relatively large velocities in the mainstream and the condition of no slip at the surface. This shearing action is greatest at the body surface and retards the layers of fluid immediately adjacent to the surface. These layers, since they are now moving more slowly than those above them, will then influence the latter and so retard them. In this way, as the fluid near the surface passes downstream, the retarding action penetrates farther and farther away from the surface and the boundary layer of retarded or ‘tired’ fluid grows in thickness.

Preamble

This chapter introduces the concept of the boundary layer, describes the flow phenomena involved, and explains how the Navier Stokes equations can be simplified for the analysis of boundary-layer flows. Certain useful solutions to the boundary-layer equations are described. There are two phenomena, governed by viscous effects and the behaviour of the boundary layer, that are vitally important for engineering applications of aerodynamics. These are How separation and transition from laminar to turbulent flow. A section is devoted to each in turn. The momentum-integral form of the boundary-layer equations is derived. Its use for obtaining approximate solutions for laminar, turbulent, and mixed laminar-turbulent boundary layers is explored in detail. Its application for estimating profile drag is also described. These approximate techniques are illustrated with examples chosen to show how to estimate the aerodynamic characteristics, such as drag, that depend on the behaviour of the boundary layer. Computational methods for obtaining numerical solutions to the boundary-layer equations are presented and reviewed. Some of the computational methods are explained in detail. A substantial section is devoted to the flow physics of turbulent boundary layers with illustrations from aeronautical applications. Computational methods for turbulent boundary layers and other flows are also reviewed. As viscosity is the key physical property governing the behaviour of boundary layers and related phenomena, a quantitative treatment of compressible effects is omitted. But the chapter closes with a detailed qualitative description of the influence of compressible effects on boundary layers, particularly their interaction with shock waves.

7.1 Introduction

In the other chapters of this book, the effects of viscosity, which is an inherent property of any real fluid, have, in the main, been ignored. At first sight, it would seem to be a waste of time to study inviscid fluid flow when all practical fluid

Effects of viscosity negligible in regions not In close proximity to the body

problems involve viscous action. The purpose behind this study by engineers dates back to the beginning of the previous century (1904) when Prandtl conceived the idea of the boundary layer.

In order to appreciate this concept, consider the flow of a fluid past a body of reasonably slender form (Fig. 7.1). In aerodynamics, almost invariably, the fluid viscosity is relatively small (i. e. the Reynolds number is high); so that, unless the transverse velocity gradients are appreciable, the shearing stresses developed [given by Newton’s equation r = ц(ди/ду) (see, for example, Section 1.2.6 and Eqn (2.86))] will be very small. Studies of flows, such as that indicated in Fig. 7.1, show that the transverse velocity gradients are usually negligibly small throughout the flow field except for thin layers of fluid immediately adjacent to the solid boundaries. Within these boundary layers, however, large shearing velocities are produced with conse­quent shearing stresses of appreciable magnitude.

Consideration of the intermolecular forces between solids and fluids leads to the assumption that at the boundary between a solid and a fluid (other than a rarefied gas) there is a condition of no slip. In other words, the relative velocity of the fluid tangential to the surface is everywhere zero. Since the mainstream velocity at a small distance from the surface may be considerable, it is evident that appreciable shearing velocity gradients may exist within this boundary region.

Prandtl pointed out that these boundary layers were usually very thin, provided that the body was of streamline form, at a moderate angle of incidence to the flow and that the flow Reynolds number was sufficiently large; so that, as a first approximation, their presence might be ignored in order to estimate the pressure field produced about the body. For aerofoil shapes, this pressure field is, in fact, only slightly modified by the boundary-layer flow, since almost the entire lifting force is produced by normal pressures at the aerofoil surface, it is possible to develop theories for the evaluation of the lift force by consideration of the flow field outside the boundary layers, where the flow is essentially invisdd in behaviour. Herein lies the importance of the inviscid flow methods considered previously. As has been noted in Section 4.1, however, no drag force, other than induced drag, ever results from these theories. The drag force is mainly due to shearing stresses at the body surface (see Section 1.5.5) and it is in the estimation of these that the study of boundary-layer behaviour is essential.

The enormous simplification in the study of the whole problem, which follows from Prandtl’s boundary-layer concept, is that the equations of viscous motion need

be considered only in the limited regions of the boundary layers, where appreciable simplifying assumptions can reasonably be made. This was the major single impetus to the rapid advance in aerodynamic theory that took place in the first half of the twentieth century. However, in spite of this simplification, the prediction of boundary – layer behaviour is by no means simple. Modern methods of computational fluid dynamics provide powerful tools for predicting boundary-layer behaviour. However, these methods are only accessible to specialists; it still remains essential to study boundary layers in a more fundamental way to gain insight into their behaviour and influence on the flow field as a whole. To begin with, we will consider the general physical behaviour of boundary layers.

Computational methods for compressible flows, particularly transonic flow over wings, have been the subject of a very considerable research effort over the past three decades. Substantial progress has been made, although much still remains to be done. A discussion of these methods is beyond the scope of the present book, save to note that for the linearized compressible potential flow Eqn (6.118) panel methods (see Sections 3.5, 4.10 and 5.8) have been developed for both subsonic and supersonic flow. These can be used to obtain approximate numerical solutions in cases with exceedingly complex geometries. A review of the computational methods developed for the full inviscid and viscous equations of motion is given by Jameson.[34]

Exercises

1 A convergent-divergent duct has a maximum diameter of 150 mm and a pitot – static tube is placed in the throat of the duct. Neglecting the effect of the pitot-static tube on the flow, estimate the throat diameter under the following conditions:

(i) air at the maximum section is of standard pressure and density, pressure differ­ence across the pitot-static tube =127 mm water;

(ii) pressure and temperature in the maximum section are 101 300 N m-2 and 100 °С respectively, pressure difference across pitot-static tube = 127 mm mercury.

(Answer: (i) 123 mm; (ii) 66.5 mm)

2 In the wing-flow method of transonic research an aeroplane dives at a Mach

number of 0.87 at a height where the pressure and temperature are 46 500 N m-2 and -24.6 °С respectively. At the position of the model the pressure coefficient is —0.5. Calculate the speed, Mach number, 0.7p M2, and the kinematic viscosity of the flow past the model. _

(Answer: 344m s"1; M = 1.133; 0.1 pM2 = 30 800N nr2; v = 2.64 x 10-3m2s )

3 What would be the indicated air speed and the true air speed of the aeroplane in Exercise 2, assuming the air-speed indicator to be calibrated on the assumption of incompressible flow in standard conditions, and to have no instrument errors?

(Answer: TAS = 274m s"1; IAS = 219m s"1)

4 On the basis of Bernoulli’s equation, discuss the assumption that the compressi­bility of air may be neglected for low subsonic speeds.

A symmetric aerofoil at zero lift has a maximum velocity which is 10% greater than die free-stream velocity. This maximum increases at the rate of 7% of the free – stream velocity for each degree of incidence. What is the free-stream velocity at which compressibility effects begins to become important (i. e. the error in pressure coefficient exceeds 2%) on the aerofoil surface when the incidence is 5°?

(Answer: Approximately 70 m s_1) (U of L)

5 A closed-return type of wind-tunnel of large contraction ratio has air at standard conditions of temperature and pressure in the settling chamber upstream of the contraction to the working section. Assuming isentropic compressible flow in the tunnel estimate the speed in the working section where the Mach number is 0.75. Take the ratio of specific heats for air as 7 = 1.4. (Answer: 242 m s_1) (U of L)

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. The same consideration would apply to any inboard three-dimensional effects, such as the ‘kink’ at the centre-line of a swept-back wing.

The opposite case is when the component of free-stream velocity perpendicular to the leading edge is less than the local speed of sound and the term subsonic leading edge is used. A typical example is the swept-back wing shown in Fig. 6.57. In this case the Mach cone generated by the leading edge of the centre section subtends the whole wing. This implies that the leading edge of the outboard portions of the wing influences the oncoming flow just as for subsonic flow. Wings having finite thickness and incidence actually generate a shock cone, rather than a Mach cone, as shown in

Fig. 6.57 A wing with a subsonic leading edge

Fig. 6.58

Fig. 6.58. Additional shocks are generated by other points on the leading edge and the associated shock angles will tend to increase because each successive shock wave leads to a reduction in the Mach number. These shock waves progressively decelerate the flow, so that at some section, such as AA’, the flow approaching the leading edge will be subsonic. Thus subsonic wing sections would be used over most of the wing.

Wings with subsonic leading edges have lower wave drag than those with super­sonic ones. Consequently highly swept wings, e. g. slender deltas, are the preferred configuration at supersonic speeds. On the other hand swept wings with supersonic leading edges tend to have a greater wave drag than straight wings.

The shock-expansion approximation

The supersonic linearized theory has the advantage of giving relatively simple for­mulae for the aerodynamic characteristics of aerofoils. However, as shown below in Example 6.13 the exact pressure distribution can be readily found for a double-wedge aerofoil. Hence the coefficients of lift and drag can be obtained.

Example 6.13 Consider a symmetrical double-wedge aerofoil at zero incidence, similar in shape to that in Fig. 6.44 above, except that the semi-wedge angle eo = 10°. Sketch the wave pattern for Mx = 2.0, calculate the Mach number and pressure on each face of the aerofoil, and hence determine Co – Compare the results with those obtained using the linear theory. Assume the free-stream stagnation pressure, po<» = 1 bar.

The wave pattern is sketched in Fig. 6.54a. The flow properties in the various regions can be obtained using isentropic flow and oblique shock tables.[33] In region 1 M — MB0 = 2.0 and pooo = 1 bar. From the isentropic flow’ tables po/pі = 7.83 leading to p = 0.1277 bar. In region 2 the oblique shock-wave tables give pijp = 1.7084 (leading to p2 = 0.2182 bar), M2 = 1.6395 and shock angle = 39.33°. Therefore

^ Pi — Рос (pi/px)-1 (Pi/P^)- 1

*~p2 — ] — ] — і

2 l’oc 2 7 (А» /Трсо) vx —

_ (0.2182/0.1277)-! _

0.5 x 1.4 x 22

Fig. 6.54

(Using the linear theory, Eqn (6.145) gives

2s 2 x (10s/180)

Сгі – ~ vp-i ■o m)

In order to continue the calculation into region 3 it is first necessary to determine the Prandtl-Meyer angle and stagnation pressure in region 2. These can be obtained as follows using the isentropic flow tables: ркіірі = 4.516 givingрю = 4.516 x 0.2182 = 0.9853 bar; and Mach angle, pa = 37.57° and Prandtl-Meyer angle, V2 = 16.01°.

Between regions 2 and 3 the flow expands isentropically through 20° so V3 = V2 + 20° = 36.01°. From the isentropic flow tables this value of V3 corresponds to М3 = 2.374, /із = 24.9° and

/>оз//>з = 14.03. Since the expansion ръ = 0.9853/14.03 = 0.0702 bar. Thus

There is an oblique shock wave between regions 3 and 4. The obUque shock tables give рлІРз — 1.823 andM4 = 1.976 giving/>4 = 1.823 x 0.0702 = 0.128 bar and a shock angle of 33.5°. The drag per unit span acting on the aerofoil is given by resolving the pressure forces, so that

D = 2(p2 — рз) x. x sin(10°)

so

Cd = (Cp2 – Срз) tan(10°) = 0.0703 (Using the linear theory, Eqn (6.151) with a = 0 gives

It can be seen from the calculations above that, although the linear theory does not approx­imate the value of Cp very accurately, it does yield an accurate estimate of Cd.

When Mao = 1.3 it can be seen from the obUque shock tables that the maximum compres­sion angle is less than 10°. This imphes that in this case the flow can only negotiate the leading edge by being compressed through a shock wave that stands off from the leading edge and is normal to the flow where it intersects the extension of the chord line. This leads to a region of subsonic flow being formed between the stand-off shock wave and the leading edge. The corresponding flow pattern is sketched in Fig. 6.54b.

A similar procedure to that in Example 6.13 can be followed for aerofoils with curved profiles. In this case, though, the procedure becomes approximate because it ignores the effect of the Mach waves reflected from the bow shock wave – see Fig. 6.55. The so-called shock-expansion approximation is made clearer by the example given below.

Example 6.14 Consider a biconvex aerofoil at zero incidence in supersonic flow at = 2, similar in shape to that shown in Fig. 6.46 above so that, as before, the shape of the upper surface is given by

у = хєо (l — “) giving local flow angle 0(= a) = arc tan e0 ^1 —

Fig. 6.55

Calculate the pressure and Mach number along the surface as functions of x/c for the case of єо = 0.2. Compare with the results obtained with linear theory. Take the freestream stagnation pressure to be 1 bar.

Region 1 as in Example 6.13, i. e. Mi = 2.0, рої = 1 bar and p = 0.1277 bar At x = 0 в — arctan(0.2) = 11.31°. Hence initially the flow is turned by the bow shock through an angle of 11.31°, so using the oblique shock tables gives Р2ІР1 = 1.827 and М2 = 1.59. Thus p2 = 1.827 x 0.1277 = 0.233 bar. From the isentropic flow tables it is found that М2 = 1.59 corresponds to Р02ІР2 = 4.193 giving рог = 0.977 bar.

Thereafter the pressures and Mach numbers around the surface can be obtained using the isentropic flow tables as shown in the table below.

Example 6.12 Using Ackeret’s theory obtain expressions for the lift and drag coefficients of the cambered double-wedge aerofoil shown in Fig. 6.52. Hence show that the minimum lift-drag ratio for the uncambered double-wedge aerofoil is y/2 times that for a cambered one with h = tjl. Sketch the flow patterns and pressure distributions around both aerofoils at the incidence for (LjD)max. (U of L)