Category Aerodynamics for Engineering Students

The local friction coefficient Q may now be expressed in terms of x by substituting from Eqn (7.81) in Eqn (7.80). Thus

These expressions are shown plotted in Fig. 7.25 (upper curve). It should be clearly understood that these last two coefficients refer to the case of a flat plate for which the boundary layer is turbulent over the entire streamwise length.

In practice, for Reynolds numbers (Re) up to at least 3 x 105, the boundary layer will be entirely laminar. If the Reynolds number is increased further (by increasing the flow speed) transition to turbulence in the boundary layer may be initiated (depending on free-stream and surface conditions) at the trailing edge, the transition point moving forward with increasing Re (such that Rex at transition remains approximately constant at a specific value, Reu say). However large the value of Re there will inevitably be a short length of boundary layer near the leading edge that will remain laminar to as far back on the plate as the point corresponding to Rex = Ret. Thus, for a large range of practical Reynolds numbers, the boundary – layer flow on the plate will be partly laminar and partly turbulent. The next stage is to investigate the conditions at transition in order to evaluate the overall drag coeffi­cient for the plate with mixed boundary layers.

Note that

dв _ 26{L)

where в(Ь) is the value of the momentum thickness at x = L. Thus using Eqn (7.70) in Eqn (7.71) gives

CF – 1.293/Jte1/2 and

CDv = 2.586/Jte1/2 (7.72)

These expressions are shown plotted in Fig. 7.25 (lower curve).

Example 7.3 A flat plate of 0.6 m chord at zero incidence in a uniform airstream of 45 m s-1. Estimate (i) the displacement thickness at the trailing edge, and (ii) the overall drag coefficient of the plate.

At the trailing edge, x = 0.6 m and

1.85 x 106

Therefore, using Eqn (7.69),

7.7.1 Turbulent velocity profile

A commonly employed, turbulent-boundary-layer profile is the seventh-root profile, which was proposed by Prandtl on the basis of friction-loss experiments with turbulent flow in circular pipes correlated by Blasius. The latter investigated experi­mental results on the resistance to flow and proposed the following empirical rela­tionships between the local skin friction coefficient at the walls, Cf(=rw/|p£/ ) and the Reynolds number of the flow Re (based on the average flow velocity U in the pipe and the diameter D). Blasius proposed the relationship

This expression gives reasonably good agreement with experiment for values of Re up to about 2.5 x 105.

Assuming that the velocity profile in the pipe may be written in the form

where и is the velocity at distance у from the wall and a = pipe radius Djl, it remains to determine the value of n. From Eqn (7.74), writing Um = CU, where C is a constant to be determined: i. e.

(7.75)

Substituting for Cf in the expression above for surface friction stress at the wall,

"І-”2 °-0791у1/4 lpU2 = 0.03955pU7/4(^y/4 (7.76)

From Eqn (7.75)

-7/4_«7/4 faln/A ~ СУ4

so that Eqn (7.76) becomes

i. e.

It may now be argued that very close to the wall, in the viscous sublayer (и Ф 0), the velocity и will not depend on the overall size of the pipe, i. e. that и Ф f(a). If this is so, then it immediately follows that rw, which is p(du/dy)v,, cannot depend on the pipe diameter and therefore the term aKVn/4)—(i/4)] jn pqn (7.77) must be unity in order not to affect the expression for rw. For this to be so, 7л/4 -1/4 = 0 which immediately gives n = j. Substituting this back into Eqn (7.74) gives u/Um = (yja)1/7. This expres­sion thus relates the velocity и at distance у from the surface to the centre-line velocity Um at distance a from the surface. Assuming that this will hold for very large pipes, it may be argued that the flow at a section along a flat, two-dimensional plate is similar to that along a small peripheral length of pipe, so that replacing a by 6 will give the profile for the free boundary layer on the flat plate. Thus

This is Prandtl’s seventh-root law and is found to give surprisingly good overall agreement with practice for moderate Reynolds numbers (Rex < 107). It does, how­ever, break down at the wall where the profile is tangential to the surface and gives an infinite value of (дй/дуХr In order to find the wall shear stress, Eqn (7.77) must be used. The constant Cmay be evaluated by equating expressions for the total volume flow through the pipe, i. e. (using Eqns (7.75) and (7.78)),

Га – fa /v 1/7 49 _

naiU = 2n urdr = 2nUC (-) (a — y)dy = — irUCa1

Jo Jo ‘a’ 60

giving C = — = 1.224. Substituting for C and n in Eqn (7.77) then gives

rw = 0.0234/эи7/4

that, on substituting for и from Eqn (7.78), gives

Finally, since

for a free boundary layer:

Using Eqns (7.78) and (7.80) in the momentum integral equation enables the growth of the turbulent boundary layer on a flat plate to be investigated.

The seventh-root profile with the above thickness quantities indicated is plotted in Fig. 7.24.

Example 7.4 A wind-tunnel working section is to be designed to work with no streamwise pressure gradient when running empty at an airspeed of 60ms-1. The working section is 3.6 m long and has a rectangular cross-section which is 1.2 m wide by 0.9 m high. An approximate allowance for boundary-layer growth is to be made by allowing the side walls of the working section to diverge shghtly. It is to be assumed that, at the upstream end of the working section, the turbulent boundary layer is equivalent to one that has grown from zero thickness over a length of 2.5 m; the wall divergence is to be determined on the assumption that the net area of flow is correct at the entry and exit sections of the working section. What must be the width between the walls at the exit section if the width at the entry section is exactly 1.2 m?

For the seventh-root profile:

At entry, x = 2.5 m. Therefore

Uxx 60 x 2.5 v = 14.6 x 10-6 RelJ5 = 25.2

„ 0.0479 x 2.5

=—— ^r-r—– = 0.00475m

25.2

At exit, x = 6.1 m. Therefore

60 x 6.1

14.6 x 10-6 RelJs = 30.2

^ = 00479,x 6л = ooo968m

Thus <5* increases by (0.00968 — 0.00475) = 0.00493 m. This increase in displacement thick­ness occurs on all four walls, i. e. total displacement area at exit (relative to entry) = 0.00493 x 2(1.2 + 0.9) = 0.0207m2.

The allowance is to be made on the two side walls only so that the displacement area on side walls = 2 x 0.9 x Д” = 1.8Д*т2, where Д* is the exit displacement per wall. Therefore

Д*=^ = 0.0115т 1.0

This is the displacement for each wall, so that the total width between side walls at the exit section = 1.2 + 2 x 0.0115 = 1.223 m.

The rate of increase of the boundary-layer thickness 6 may be found by integrating Eqn (7.67), after setting Л = 0 in Eqns (7.64b and c) and substituting for / and Q. Thus Eqn (7.67) becomes

d<5 Cf 140 /і dx~21~ 13 pU^d

Therefore

whence

<52 _ 140 px

T = Upt^

The integration constant is zero if x is measured from the fore stagnation point where (5 = 0, i. e.

6 = 4.64jc/{Rex)in

The other thickness quantities may now be evaluated using Eqns (7.64a, b) with Л = 0. Thus

6* = 0.375(5 = 1.74л /{Rex)l/1 6 = 0.139(5 = QM6x/{Rex)1/2

For the flat plate dp/dx = 0 and Ue = Ux = const, so that dUe/dx = 0. Accordingly, the momentum integral equation (7.59) reduces to the simple form

(7.66)

Now the shape factor / = 6/6 is simply a numerical quantity which depends only on the shape of the velocity profile.

So Eqn (7.66) may then be expressed in the alternative form

(7.67)

where / has been assumed to be independent of x. Equations (7.66) and (7.67) are forms of the simple momentum integral equation.

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 8, S*, в 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 specific streamlined body would have the same surface area and total streamwise length as the body. In this way reasonable estimates can be obtained, especially for the skin-friction drag, provided that the transition point is correctly located using the guidelines given at the end of Section 7.9.

As explained in the previous subsection an approximate expression is required for the velocity profile in order to use the momentum integral equation. A reasonably accurate approximation can be obtained by using a cubic polynomial in the form:

m(= u/Ue) = a + by + су1 + dy3 (7.60)

where у — y/6. In order to evaluate the coefficients a, b, c and d four conditions are required, two at у = 0 and two at у = 1. Two of these conditions are readily avail­able, namely

її = 0 at у = 0 (7.61a)

m = 1 at у = 1 (7.61b)

In real boundary-layer velocity profiles – see Fig. 7.10 – the velocity varies smoothly to reach Ue; there is no kink at the edge of the boundary layer. Accordingly, it follows that the velocity gradient is zero at у = S giving a third condition, namely

— 0 at y= 1 (7.61c)

dy

To obtain the fourth and final condition it is necessary to return to the boundary – layer equation (7.14). At the wall у = 0, и = v = 0, so both terms on the left-hand side are zero at у = 0. Thus, noting that r = p. du/dy, the required condition is given by

Since у = y8 and p + pU2 = const, the above equation can be rearranged to read

(7.62a) (7.62b) (7.62c) (7.62d)

a – b + c + d■ b – I- 2c – I – 3d : 2c-

Equations (7.62b, c, d) can be readily solved for b, c and d to give the following approximate velocity profile

The parameter Л in Eqn (7.63) is often called the Pohlhausen parameter. It determines the effect of an external pressure gradient on the shape of the velocity profile. Л > 0 and <0 correspond respectively to favourable and unfavourable pressure gradients. For Л = —6 the wall shear stress rw = 0 and for more negative values of Л flow reversal at the wall develops. Thus Л = —6 corresponds to boundary-layer separation. Velocity profiles corresponding to various values of Л are plotted in Fig. 7.6. In this figure, the flat-plate profile corresponds to Л = 0; Л = 6 for the favourable pressure gradient; Л = —4 for the mild adverse pressure gradient; Л = —6 for the strong adverse pressure gradient; and Л = —9 for the reversed-flow profile.

For the flat-plate case Л = 0, the approximate velocity profile of Eqn (7.63) is compared with two other approximate profiles in Fig. 7.22. The velocity profile labelled Blasius is the accurate solution of the differential equations of motion given in Section 7.3.4 and Fig. 7.10.

The various quantities introduced in Sections 7.3.2 and 7.3.3 can readily be evaluated using the approximate velocity profile (7.63). For example, if Eqn (7.63) is substituted in turn into Eqns (7.16), (7.17) and (7.19), with use of Eqn (7.20), the following are obtained.

24^223

, (3 Л Л /1 Л

(4 + 8) + 6 + S 16

3_ Л.

8 48

Dimensionless velocity parallel to surface, u[-u/U^ Fig. 7.22 Laminar velocity profile

The quantities /1 and / depend only on the shape of the velocity profile, and for this reason they are usually known as shape parameters. If the more accurate differential form of the boundary-layer equations were to be used, rather than the momentum – integral equation with approximate velocity profiles, the boundary-layer thickness 8 would become a rather less precise quantity. For this reason, it is more common to use the shape parameter H = 8*j0. Frequently, H is referred to simply as the shape parameter.

For the numerical methods discussed in Section 7.11 below for use in the general case with an external pressure gradient, it is preferable to use a some­what more accurate quartic polynomial as the approximate velocity profile. This is particularly important for predicting the transition point. This quartic velocity profile is derived in a very similar way to that given above, the main differences are the addition of another term, ev4, on the right-hand side of Eqn (7.60), and the need for an additional condition at the edge of the boundary layer. This latter requires that

This has the effect of making the velocity profile even smoother at the edge of the boundary layer, and thereby improves the approximation. The resulting quartic velocity profile takes the Form

u = 2y – 2y3 + v4 + – (у – 3у1 + 3 v3 – у4)

Using this velocity profile and following similar procedures to those outlined above leads to the following expressions:

Note that it follows from Eqn (7.64c’) that with the quartic velocity profile the separation point where rw = 0 now corresponds to Л = —12.

The accurate evaluation of most of the quantities defined above in Sections 7.3.2 and 7.3.3 requires the numerical solution of the differential equations of motion. This will be discussed in Section 7.11. Here an integral form of the equations of motion is derived that allows practical solutions to be found fairly readily for certain engineer­ing problems.

The required momentum integral equation is derived by considering mass and momentum balances on a thin slice of boundary layer of length 6x. This slice is illustrated in Fig. 7.21. Remember that in general, quantities vary with x, i. e. along the surface; so it follows from elementary differential calculus that the value of a quantity / say, on CD (where the distance from the origin is. v + 6x) is related to its value on AB (where the distance from the origin is x) in the following way:

f(x + 6x) ~f{x) + (7-54)

First the conservation of mass for an elemental slice of boundary layer will be considered, see Fig. 7.21b. Since the density is assumed to be constant the mass flow balance for slice ABCD states, in words, that

Volumetric flow rate into the slice across AB = Volumetric flow rate out across CD

+ Volumetric flow rate out across AD + Volumetric flow rate out across BC

The last item in the volumetric flow balance allows for the possibility of flow due to suction passing through a porous wall. In the usual case of an impermeable wall Vs — 0. Expressed mathematically this equation becomes

Note that Eqn (7.54) has been used, Q,- replacing/where

Qi =

Cancelling common factors, rearranging Eqn (7.55) and taking the limit 6x —> d. v leads to an expression for the perpendicular velocity component at the edge of the boundary layer, i. e.

The definition of displacement thickness, Eqn (7.16), is now introduced to give

Ч8*

(c)

y – Momentum balance

(d) Forces in ^-direction (e) Momentum fluxes in x – direction

The у-momentum balance for the slice ABCD of boundary layer is now consid­ered. This is illustrated in Fig. 7.21c. In this case, noting that the у-component of momentum can be carried by the flow across side AD only and that the only force in the у direction is pressure,* the momentum theorem states that the

The force of gravity is usually ignored in aerodynamics.

Rate at which ^-component The net pressure force in the у direction of momentum crosses AD = acting on the slice ABCD

Or in mathematical terms

pV*6x = (/>w – pi)6x

Thus cancelling the common factor 6x leads to

Pv-Pi =pVl

It can readily be shown from this result that the net pressure difference across the boundary layer is negligible, i. e. pw ~ pi, as it should be according to boundary-layer theory. For simplicity the case of the boundary layer along an impermeable flat plate when Ue = C/oc(=const.) and Vs = 0 is considered, so that from Eqn (7.56)

d<5* , /d<5*2

Ve _ U°° dJ ^ Pw “Pl ~ pU°° Vd^J

It must be remembered, however, that the boundary layer is very thin compared with the length of the plate thus dS*/dx – c 1, so that its square is negligibly small. This argument can be readily extended to the more general case where Ue varies along the edge of the boundary layer. Thus it can be demonstrated that the assumption of a thin boundary layer implies that the pressure does not vary appreciably across the boundary layer. This is one of the major features of boundary-layer theory (see Section 7.3.1). It also implies that within the boundary layer the pressure p is a function of x only.

Finally, the x-momentum balance for the slice ABCD is considered. This case is more complex since there are both pressure and surface friction forces to be con­sidered, and furthermore the x component of momentum may be carried across AB, CD and AD. The forces involved are depicted in Fig. 7.21d while the momentum fluxes are shown in Fig. 7.21e. In this case, the momentum theorem states that

Rate at which / Rate at which / Net pressure force

momentum leaves ] — I momentum enters I = I in x direction acting across CD and AD/ V across AB / on ABCD

out across CD

(7.57)

on AB

where

Mi,

After cancelling common factors, taking the limit 8x —► dx and simplification Eqn (7.57) becomes

dp _ TT dUe

dx P e dx

After substituting Eqn (7.56) for Ve, introducing the definition, Eqn (7.17), of momentum thickness and using the result given above, Eqn (7.58) reduces to

This is the momentum integral equation first derived by von Karman. Since no assumption is made at this stage about the relationship between rw and the velocity gradient at the wall, the momentum integral equation applies equally well to laminar or turbulent flow.

When suitable forms are chosen for the velocity profile the momentum integral equation can be solved to provide the variations of 6, 8*, в and Cf along the surface. A suitable approximate form for the velocity profile in the laminar boundary layer is derived in Section 7.6.1. In order to solve Eqn (7.59) in the turbulent case additional semi-empirical relationships must be introduced. In the simple case of the flat plate the solution to Eqn (7.59) can be found in closed form, as shown in Section 7.7. In the general case with a non-zero pressure gradient it is necessary to resort to computa­tional methods to solve Eqn (7.59). Such methods are discussed in Section 7.11.

The art of the seam bowler in cricket is also explainable with reference to boundary – layer transition and separation. The bowling technique is to align the seam at a small angle to the flight path (see Fig. 7.20). This is done by spinning the ball about an axis perpendicular to the plane of the seam, and using the gyroscopic inertia to stabilize this seam position during the trajectory. On the side of the front stagnation point where the boundary layer passes over the seam, it is induced to become turbulent before reaching the point of laminar separation. On this side, the boundary layer remains attached to a greater angle from the fore stagnation point than it does on the other side where no seam is present to trip the boundary layer. The flow past the ball thus becomes asymmetric with a larger area of low pressure on the turbulent side, producing a lateral force tending to move the ball in a direction normal to its flight path. The range of flight speeds over which this phenomenon can be used corresponds to those of the medium to medium-fast pace bowler. The diameter of a cricket ball is between 71 and 72.5 mm. In air, the critical speed for a smooth ball would be about 75 m s_I. However, in practice it is found that transition to turbulence for the seam-free side occurs at speeds in the region of 30 to 35ms’1, because of inaccuracies in the spherical shape and minor surface irregularities. The critical speed for a rough ball with early transition (Re ю 105) is about 20ms_I and below this speed the flow asymmetry tends to disappear because laminar separation occurs before the transition, even on the seam side.

Thus within the speed range 20 to about 30 m s_1, very approximately, the ball may be made to swing by the skilful bowler. The very fast bowler will produce a flight speed in excess of the upper critical and no swing will be possible. A bowler may make the ball swing late by bowling at a speed just too high for the asymmetric condition to exist, so that as the ball loses speed in flight the asymmetry will develop later in the trajectory. It is obvious that considerable skill and experience is required to know at just what speed the delivery must be made to do this.

It will also be realized that the surface condition, apart from the seam, will affect the possibility of swinging the ball, e. g. a new, smooth-surfaced ball will tend to

maintain laminar layers up to separation even on the seam side, while a badly worn ball will tend to induce turbulence on the side remote from the seam. The slightly worn ball is best, especially if one side can be kept reasonably polished to help maintain flow on that side only.

In the early days of the sport, golf balls were made with a smooth surface. It was soon realized, however, that when the surface became worn the ball travelled farther when driven, and subsequently golf balls were manufactured with a dimpled surface to simulate the worn surface. The reason for the increase in driven distance with the rough surface is as follows.

The diameter of a golf ball is about 42 mm, which gives a critical velocity in air, for a smooth ball, of just over 135ms-1 (corresponding to Re = 3.85 x 105). This is much higher than the average flight speed of a driven ball. In practice, the critical speed would be somewhat lower than this owing to imperfections in manufacture, but it would still be higher than the usual flight speed. With a rough surface, promoting early transition, the critical Reynolds number may be as low as 105, giving a critical speed for a golf ball of about 35 ms-1, which is well below the flight speed. Thus, with the roughened surface, the ball travels at above the critical drag speed during its flight and so experiences a smaller decelerating force throughout, with consequent increase in range.

The effect of free-stream turbulence on the Reynolds number at which the critical drag decrease occurs was widely used many years ago to ascertain the turbulence level in the airstream of a wind-tunnel working section. In this application, a smooth sphere is mounted in the working section and its drag, for a range of tunnel speeds, is read off on the drag balance. The speed, and hence the Reynolds number, at which the drag suddenly decreases is recorded. Experiments in air of virtually zero small-scale

turbulence have indicated that the highest critical sphere Reynolds number attainable is 385 000. A turbulence factor, for the tunnel under test, is then defined as the ratio of 385 000 to the critical Reynolds number of the test tunnel.

A major difficulty in this application is the necessity for extreme accuracy in the manufacture of the sphere, as small variations from the true spherical shape can cause appreciable differences in the behaviour at the critical stage. As a result, this technique for turbulence measurement is not now in favour, and more recent methods, such as hot-wire anemometry, took its place some time ago.