Category Aerodynamics for Engineering Students

Although it is not valid in the outer part of the boundary layer, Eqn (7.117) can be used to obtain the following more accurate semi-empirical formulae for the local skin-friction coefficient and the corresponding drag coefficient for turbulent bound­ary layers over flat plates.

cf =r^- = (2 log10 Rex – 0.65)-2’3 jPUSo

_ Df 0.455

m-PUlaBL (log10 Ref)25*

where В and L are the breadth and length of the flat plate. The Prandtl-Schlichting formula (7.122) is more accurate than Eqn (7.88) when Rel > Ю7.

Effects of wall roughness

Turbulent boundary layers, especially at high Reynolds numbers, are very sensitive to wall roughness. This is because any roughness element that protrudes through the viscous sub-layer will modify the law of the wall. The effect of wall roughness on the boundary layer depends on the size, shape and spacing of the elements. To bring a semblance of order Nikuradze matched each ‘type’ of roughness against an equivalent sand-grain roughness having roughness of height, ks. Three regimes of wall roughness, corresponding to the three regions of the near-wall region, can be defined as follows:

Hydraulically smooth If ks Vt/v < 5 the roughness elements lie wholly within the viscous sublayer, the roughness therefore has no effect on the velocity profile or on the value of skin friction or drag.

Completely rough If ks V*jv > 50 the roughness elements protrude into the region of fully developed turbulence. This has the effect of displacing the logarithmic profile downwards, i. e. reducing the value of Cj in Eqn (7.117). In such cases the local skin-friction and drag coefficients are independent of Reynolds number and are given by

Transitional roughness If 5 < ksV, ju < 50 the effect of roughness is more complex and the local skin-friction and drag coefficients depend both on Reynolds number and relative roughness, ks/6.

The relative roughness plainly varies along the surface. But the viscous sub-layer increases slowly and, although its maximum thickness is located at the trailing edge, the trailing-edge value is representative of most of the rest of the surface. The degree of roughness that is considered admissible in engineering practice is one for which the surface remains hydraulically smooth throughout, i. e. the roughness elements remain within the viscous sub-layer all the way to the trailing edge. Thus

In the case of a flat plate it is found that Eqn (7.125) is approximately equivalent to

kadm ОСOC(7.126) * oo лЄь

Thus for plates of similar length the admissible roughness diminishes with increasing Rbl – In the case of ships’ hulls admissible roughness ranges from 7 pm (large fast ships) to 20 pm (small slow ships); such values are utterly impossible to achieve in practice, and it is always neccessary to allow for a considerable increase in drag due to roughness. For aircraft admissible roughness ranges from 10 pm to 25 pm and that is just about attainable in practice. Model aircraft and compressor blades require the same order of admissible roughness and hydraulically smooth surfaces can be obtained without undue difficulty. At the other extreme there are steam-turbine blades that combine a small chord (L) with a fairly high Reynolds number (5 x 106) owing to the high velocities involved and to the comparatively high pres­sures. In this cases admissible roughness values are consequently very small, ranging from 0.2 pm to 2 pm. This degree of smoothness can barely be achieved on newly manufactured blades and certainly the admissible roughness would be exceeded after a period of operation owing to corrosion and the formation of scaling.

The description of the aerodynamic effects of surface roughness given above has been in terms of equivalent sand-grain roughness. It is important to remember that the aerodynamic effects of a particular type of roughness may differ greatly from that of sand-grain roughness of the same size. It is even possible (see Section 8.5.3) for special forms of wall ‘roughness’, such as riblets, to lead to a reduction in drag.[43]

As the wall is aproached it has a damping effect on the turbulence, so that very close to the wall the viscous shear stress greatly exceeds the Reynolds shear stress. This region right next to the wall where viscous effects dominate is usually known as the viscous sub-layer. Beyond the viscous sub-layer is a transition or buffer layer
where the viscous and Reynolds shear stresses are roughly equal in magnitude. This region blends into the fully turbulent region where the Reynolds shear stress is very much larger than the viscous shear stress. It is in this fully turbulent near­wall region that the mixing-length theory can be used. The outer part of the boundary layer is more like a free shear layer and there the Reynolds shear stress is given by Eqn (7.111).

A major assumption is that the fully turbulent layer begins at a height above the wall of у «: <5, so that

dr

T = Tw + -7-у – I– ^ Tw

d У

Near the wall in the viscous sub-layer the turbulence is almost completely damped, so only molecular viscosity is important, thus

r = u^- = rw therefore m = —y (7.114)

dу V

In the fully turbulent region the Reynolds shear stress is much greater than the viscous shear stress, so:

r = —pu’v’ = rw

So if Eqn (7.112) is used and it is assumed that £m oc y, then

ф2“7? implyi”8 <7’115)

where we have introduced the friction velocity:

К = fbjp (7.116)

as the reference velocity that is subsequently used to render the velocity in the near­wall 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

Fully turbulent flow:

|- = ад(^) + с2 (7.117)

Viscous sub-layer:

h _yK V. і/

where Ci 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 known as the wall unit.

Figure 7.37 compares (7.117) and (7.118) with experimental data for a turbulent boundary layer and we can thereby deduce that

, yV„ du ——

5 < -—- < 50 u— puV

v dy

Fully developed turbulence:

yV[42] „ du – j-j

—- >50 ц—С-ріїм1

v dy

The constants C and C% can be determined from comparison with the experi­mental data so that (7.117) becomes:

Logarithmic velocity profile:

— = 2.54 in у *

Ci is often written as 1/k where к = 0.41 is known as the von Karman constant* because he was the first to derive the logarithmic velocity profile. Equation (7.117) is often known as the Law of the wall. It applies equally well to the near-wall region of turbulent pipe and channel flows for which better agreement with experimental data is found for slightly different values of the constants. It is worth noting that it is not essential to evoke Prandtl’s mixing-length theory to derive the law of the wall. The logarithmic form of the velocity profile can also be derived purely by means of dimensional analysis.*

The outer boundary layer

The outer part of the boundary layer that extends for 70 or 80% of the total thickness is unaffected by the direct effect of the wall. It can be seen in Fig. 7.37 that the velocity profile deviates considerably from the logarithmic form in this outer part of the boundary layer. In many respects it is analogous to a free shear layer, especially a wake. It is sometimes referred to as the defect layer or wake region. Here inertial effects dominate and viscous effects are negligible, so the appropriate reference velocity and length scales to use for non-dimensionalization are Ue (the streamwise flow speed at the boundary-layer edge) and 8 (the boundary-layer thickness) or some similar length scale. Thus the so-called outer variables are:

Equation (7.111) is not a good approximation in the region of the turbulent bound­ary layer or pipe flow near the wall. The eddy viscosity varies with distance from the wall in this region. A commonly used approach in this near-wall region is based on

Fig. 7.36

Prandtl’s mixing-length theory.[41] This approach to modelling turbulence is loosely based on the kinematic theory of gases. A brief account is given below and illustrated in Fig. 7.36.

Imagine a blob of fluid is transported upward by a fluctuating turbulent velocity v’ through an average distance im – the mixing length – (analogous to the mean free path in molecular dynamics). In the new position, assuming the streamwise velocity of the blob remains unchanged at the value in its original position, the fluctuation in velocity can be thought to be generated by the difference in the blob’s velocity and that of its new surroundings. Thus

Term (i) is the mean flow speed in the new environment. In writing the term in this form it is assumed that im <C 8, so that, in effect, it is the first two terms in a Taylor’s series expansion.

Term (ii) is the mean velocity of blob.

If it is also assumed that V ~ (du/dy)£m, then

(7.112)

Term (iii) is written with an absolute value sign so that the Reynolds stress changes sign with дй/ду, just as the viscous shear stress would.

Away from the immediate influence of the wall which has a damping effect on the turbulent fluctuations, the Reynolds shear stress can be expected to be very much
greater than the viscous shear stress. This can be seen by comparing rough order – of-magnitude estimates of the Reynolds shear stress and the viscous shear stress, i. e.

-Г7 e da

p uV c. f. ti­dy

Assume that u’v’ ~ CU^ (where C is a constant), then

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[40] drew an analogy between viscous and Reynolds shear stresses by introducing the concept of the eddy viscosity Єт-

r — Q —

T = /i^ cf. – piSV = рєтщ: £T^>v{=p/p) (7.109)

Boussinesq, himself, merely assumed that eddy viscosity was constant everywhere in the flow field, like molecular viscosity but very much larger. Until comparatively recently, his approach was still widely used by oceanographers for modelling turbu­lent flows. In fact, though, a constant eddy viscosity is a very poor approximation for wall shear flows like boundary layers and pipe flows. For simple turbulent free shear layers, such as the mixing layer and jet (see Fig. 7.35), and wake it is a reasonable assumption to assume that the eddy viscosity varies in the streamwise direction but not across a particular cross section. Thus, using simple dimensional analysis Prandtl^ and Reichardt* proposed that

єг= к x AU x 6 (7.110)

const. Velocity difference across shear layer shear-layer width

к is often called the exchange coefficient and it varies somewhat from one type of flow to another. Equation (7.110) gives excellent results and can be used to determine the variation of the overall flow characteristics in the streamwise direction (see Example 7.9).

The outer 80% or so of the turbulent boundary layer is largely free from the effects of the wall. In this respect it is quite similar to a free turbulent shear layer. In this

Uj Inviscid Jet boundary

V

Mixing-layer region

(b) Real turbulent jet

Fig. 7.35 An ideal inviscid jet compared with a real turbulent jet near the nozzle exit

outer region it is commonly assumed, following Laufer (1954), that the eddy viscosity can be determined by a version of Eqn (7.110) whereby

єт = nUe6*

Example 7.9 The spreading rate of a mixing layer

Figure 7.35 shows the mixing layer in the intial region of a jet. To a good approximation the external mean pressure field for a free shear layer is atmospheric and therefore constant. Furthermore, the Reynolds shear stress is very much larger than the viscous stress, so that, after substituting Eqns (7.109) and (7.110), the turbulent boundary-layer equation (7.108b) becomes

The only length scale is the mixing-layer width, S(x), which increases with x, so dimensional arguments suggest that the velocity profile does not change shape when expressed in terms of dimensionless y, i. e.

This is known as making a similarity assumption. The assumed form of the velocity profile implies that

_yjdA

Sdx)

dq/dx where F'(rf) = dF/drj.

Integrate Eqn (7.108a) to get

v =

so

6 f

v=Uj — G{rf) where G — I rjF'(rj)drj

The derivatives with respect to у are given by

дй=дп*й=Ц1

dy dydV S w

cPu _ dr] d /<9w dy2 dydr) dy)

The results given above are substituted into the reduced boundary-layer equation to obtain, after removing common factors,

— = const. or a OC X dx

Setting the term, depending on 77, with F" as numerator, equal to a constant leads to a differential equation for F that could be solved to give the velocity profile. In fact, it is easy to derive a good approximation to the velocity profile, so this is a less valuable result.

When a turbulent (or laminar) flow is characterized by only one length scale – as in the present case – the term self-similarity is commonly used and solutions found this way are called similarity solutions. Similar methods can be used to determine the overall flow characteristics of other turbulent free shear layers.

For the applications considered here, namely two-dimensional boundary layers (more generally, two-dimensional shear layers), only one of the Reynolds stresses is significant, namely the Reynolds shear stress, —pu’v’. Thus for two-dimensional turbulent boundary layers the time-averaged boundary-layer equations (c. f. Eqns 7.7 and 7.14), can be written in the form

dv

+ ^~ = 0
dy

_du _du dp df

U dx^V dy dx dy

The chief difficulty of turbulence is that there is no way of determining the Reynolds stresses from first principles, apart from solving the unsteady three-dimensional Navier-Stokes equations. It is necessary to formulate semi-empirical approaches for modelling the Reynolds shear stress before one can begin the process of solving Eqns (7.108a, b).

The momentum integral form of the boundary-layer equations derived in Section 7.6.1 is equally applicable to laminar or turbulent boundary layers, providing it is recognized that the time-averaged velocity should be used in the definition of momentum and displacement thicknesses. This is the basis of the approximate methods described in Section 7.7 that are based on assuming a l/7th. power velocity profile and using semi-empirical formulae for the local skin-friction coefficient.

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 mathematical definition of the time average is

(7.100)

where to is the time at which measurement is notionally begun. For practical meas­urements T is merely taken as suitably large rather than infinite. The basic approach is often known as Reynolds averaging.

f

V

f

(a)

(b)

Fig. 7.34 * Reynolds, O. (1895) ‘ On the dynamical theory of incompressible viscous fluids and the determination of the criterion’, Philosophical Transactions of the Royal Society of London, Series A, 186, 123.

We will now use the Reynolds averaging approach on the continuity equation (2.94) and x-momentum Navier-Stokes equation (2.95a). When Eqn (7.99) with и for / and similar expressions for v and w are substituted into Eqn (2.94) we obtain

Taking a time average of a fluctuation gives zero by definition, so taking a time average of Eqn (7.101) gives

We now take a time average of each term, noting that although the time average of a fluctuation is zero by definition (see Fig. 7.34b), the time average of a product of fluctuations is not, in general, equal to zero (e. g. plainly u’u’ = ua > 0, see Fig. 7.34b). Let us also assume that the turbulent boundary-layer flow is two-dimensional when time-averaged, so that no time-averaged quantities vary with z and w = 0. Thus if we take the time average of each term of Eqn (7.104), it simplifies to

dp (d2u ~d~x + tlM

The term marked with * can be written as

=0 from Eqn (7.103)

<9uV

h

=0 no variation with z

So that Eqn (7.105) becomes

( du _du dp (daxx daxy .

where we have written

– du ~n ’ axy = ^-f^v

This notation makes it evident that when the turbulent flow is time-averaged — pua and – pu’V take on the character of a direct and shear stress respectively. For this reason, the quantities are known as Reynolds stresses or turbulent stresses. In fully turbulent flows, the Reynolds stresses are usually very much greater than the viscous stresses. If the time-averaging procedure is applied to the full three-dimensional Navier-Stokes equations (2.95), a Reynolds stress tensor is generated with the form

(7.107)

It can be seen that, in general, there are nine components of the Reynolds stress comprising six distinct quantities.

It was mentioned in Section 7.2.5 above that transition from laminar to turbulent flow usually occurs at some point along the surface. This process is exceedingly complex and remains an active area of research. Owing to the very rapid changes in both space and time the simulation of transition is, arguably, the most challenging problem in computational fluid dynamics. Despite the formidable difficulties how­ever, considerable progress has been made and transition can now be reliably predicted in simple engineering applications. The theoretical treatment of transition is beyond the scope of the present work. Nevertheless, a physical understanding of transition is vital for many engineering applications of aerodynamics, and accord­ingly a brief account of the underlying physics of transition in a boundary layer on a flat plate is given below.

Transition occurs because of the growth of small disturbances in the boundary layer. In many respects, the boundary layer can be regarded as a complex nonlinear oscillator that under certain circumstances has an initially linear wave-like response to external stimuli (or inputs). This is illustrated schematically in Fig. 7.28. In free flight or in high-quality wind-tunnel experiments several stages in the process can be discerned. The first stage is the conversion of external stimuli or disturbances into low-amplitude waves. The external disturbances may arise from a variety of different sources, e. g. free-stream turbulence, sound waves, surface roughness and vibration. The conversion process is still not well understood. One of the main difficulties is that the wave-length of a typical external disturbance is invariably very much larger than that of the wave-like response of the boundary layer. Once the low-amplitude wave is generated it will propagate downstream in the boundary layer and, depending on the local conditions, grow or decay. If the wave-like disturbance grows it will eventually develop into turbulent flow.

While their amplitude remains small the waves are predominantly two-dimen­sional (see Figs 7.28 and 7.29). This phase of transition is well understood and was first explained theoretically by Tollmien[38] with later extensions by Schlichtingt and many others. For this reason the growing waves in the early so-called linear phase of transition are known as Tollmien-Schlichting waves. This linear phase extends for some 80% of the total transition region. The more advanced engineering predictions

are, in fact, based on modern versions of Tollmien’s linear theory. The theory is linear because it assumes the wave amplitudes are so small that their products can be neglected. In the later nonlinear stages of transition the disturbances become increas­ingly three-dimensional and develop very rapidly. In other words as the amplitude of the disturbance increases the response of the boundary layer becomes more and more complex.

This view of transition originated with Prandtl* and his research team at Gottingen, Germany, which included Tollmien and Schlichting. Earlier theories, based on neglecting viscosity, seemed to suggest that small disturbances could not grow in the boundary layer. One effect of viscosity was well known. Its so-called dissipative action in removing energy from a disturbance, thereby causing it to decay. Prandtl realized that, in addition to its dissipative effect, viscosity also played a subtle but essential role in promoting the growth of wave-like disturbances by causing energy to be transferred to the disturbance. His explanation is illustrated in Fig. 7.30. Consider a small-amplitude wave passing through a small element of fluid within the boundary

L. Prandtl (1921) Bermerkungen iiber die Enstehung der Turbulenz, Z. angew. Math. Mech., 1, 431-436.

layer, as shown in Fig. 7.30a. The instantaneous velocity components of the wave are (г/, V) in the (jt, y) directions, u’ and V are very much smaller than u, the velocity in the boundary layer in the absence of the wave. The instantaneous rate of increase in kinetic energy within the small element is given by the difference between the rates at which kinetic energy leaves the top of the element and enters the bottom, i. e.

-pu’v’^- + higher order terms

dy

In the absence of viscosity vi and V are exactly 90 degrees out of phase and the average of their product over a wave period, denoted by u’V, is zero, see Fig. 7.30b. However, as realized by Prandtl, the effects of viscosity are to increase the phase difference between u’ and V to slightly more than 90 degrees. Consequently, as shown in Fig. 7.30c, mV is now negative, resulting in a net energy transfer to the disturbance. The quantity — puV is, in fact, the Reynolds stress referred to earlier in Section 7.2.4. Accordingly, the energy transfer process is usually referred to as energy production by the Reynolds stress. This mechanism is active throughout the transition process and, in fact, plays a key role in sustaining the fully turbulent flow (see Section 7.10).

Tollmien was able to verify Prandtl’s hypothesis theoretically, thereby laying the foundations of the modern theory for transition. It was some time, however, before the ideas of the Gottingen group were accepted by the aeronautical community. In part this was because experimental corroboration was lacking. No sign of Tollmien- Schlichting waves could at first be found in experiments on natural transition. Schubauer and Skramstadt[39] did succeed in seeing them but realized that in order to study such waves systematically they would have to be created artificially in a controlled manner. So they placed a vibrating ribbon having a controlled frequency, cu, within the boundary layer to act as a wave-maker, rather than relying on natural sources of disturbance. Their results are illustrated schematically in Fig. 7.31. They found that for high ribbon frequencies, see Case (a), the waves always decayed. For intermediate frequencies (Case (b)) the waves were attenuated just downstream of the ribbon, then at a greater distance downstream they began to grow, and finally at still greater distances downstream decay resumed. For low frequencies the waves grew until their amplitude was sufficiently large for the nonlinear effects, alluded to above, to set in, with complete transition to turbulence occurring shortly afterwards. Thus, as shown in Fig. 7.31, Schubauer and Skramstadt were able to map out a curve of non­dimensional frequency versus Rex{= U^xjv) separating the disturbance frequencies that will grow at a given position along the plate from those that decay. When disturbances grow the boundary-layer flow is said to be unstable to small disturbances, conversely when they decay it is said to be stable, and when the disturbances neither grow nor decay it is in a state of neutral stability. Thus the curve shown in Fig. 7.31 is known as the neutral-stability boundary or curve. Inside the neutral-stability curve, production of energy by the Reynolds stress exceeds viscous dissipation, and vice versa outside. Note that a critical Reynolds number Rec and critical frequency ujc exist. The Tollmien-Schlichting waves cannot grow at Reynolds numbers below Rec or at frequencies above cuc. However, since the disturbances leading to transition to turbu­lence are considerably lower than the critical frequency, the transitional Reynolds number is generally considerably greater than Rec.

The shape of the neutral-stability curve obtained by Schubauer and Skramstadt agreed well with Tollmien’s theory, especially at the lower frequencies of interest for

Boundary-layer edge _________

(a)

тшшщтшшїшяшшшг/.

Wall

(b)

фтт/Ш/ІЩ№

жттттштжттх

transition. Moreover Schubauer and Skramstadt were also able to measure the growth rates of the waves and these too agreed well with Tollmien and Schlichting’s theoretical calculations. Publication of Schubauer and Skramstadt’s results finally led to the Gottingen ‘small disturbance’ theory of transition becoming generally accepted.

It was mentioned above that Tollmien-Schlichting waves could not be easily observed in experiments on natural transition. This is because the natural sources of disturbance tend to generate wave packets in an almost random fashion in time and space. Thus at any given instant there is a great deal of ‘noise’, tending to obscure the wave-like response of the boundary layer, and also disturbances having a wide range of frequencies are continually being generated. In contrast, the Tollmien – Schlichting theory is based on disturbances with a single frequency. Nevertheless, providing the initial level of the disturbances is low, what seems to happen is that the boundary layer responds preferentially, so that waves of a certain frequency grow most rapidly and are primarily responsible for transition. These most rapidly grow­ing waves are those predicted by the modern versions of the Tollmien-Schlichting theory, thereby allowing the theory to predict, approximately at least, the onset of natural transition.

It has been explained above that provided the initial level of the external distur­bances is low, as in typical free-flight conditions, there is a considerable difference between the critical and transitional Reynolds number. In fact, the latter is about 3 x 106 whereas Rec ~ 3 x 105. However, if the initial level of the disturbances rises, for example because of increased free-stream turbulence or surface roughness, the

downstream distance required for the disturbance amplitude to grow sufficiently for nonlinear effects to set in becomes shorter. Therefore, the transitional Reynolds number is reduced to a value closer to Rec. In fact, for high-disturbance environ­ments, such as those encountered in turbomachinery, the linear phase of transition is by-passed completely and laminar flow breaks down very abruptly into fully developed turbulence.

The Tollmien-Schlichting theory can also predict very successfully how transition will be affected by an external pressure gradient. The neutral-stability boundaries for the flat plate and for typical adverse and favourable pressure gradients are plotted schematically in Fig. 7.32. In accordance with the theoretical treatment Reg is used as the abscissa in place of Rex. However, since the boundary layer grows with passage downstream Reg can still be regarded as a measure of distance along the surface. From Fig. 7.32 it can be readily seen that for adverse pressure gradients not only is (Reg)c smaller than for a flat plate, but a much wider band of disturbance frequencies are unstable and will grow. When it is recalled that the boundary-layer thickness also grows more rapidly in an adverse pressure gradient, thereby reaching a given critical value of Reg sooner, it can readily be seen that transition is promoted under these circumstances. Exactly the converse is found for the favourable pressure gradient. This circumstance allows rough and ready predictions to be made for the transition

point on bodies and wings, especially in the case of the more classic streamlined shapes. These guidelines may be summarized as follows:

(i) If 105 < ReL < 107 (where ReL = V^Ljv is based on the total length or chord of the body or wing) then transition will occur very shortly downstream of the point of minimum pressure. For aerofoils at zero incidence or for streamlined bodies of revolution, the point of minimum pressure often, but not invariably, coincides with the point of maximum thickness.

(ii) If for an aerofoil ReL is kept constant increasing the angle of incidence advances the point of minimum pressure towards the leading edge on the upper surface, causing transition to move forward. The opposite occurs on the lower surface.

(iii) At constant incidence an increase in ReL tends to advance transition.

(iv) For ReL > 107 the transition point may slightly precede the point of minimum pressure.

The effects of external pressure gradient on transition also explain how it may be postponed by designing aerofoils with points of minimum pressure further aft. A typical modern aerofoil of this type is shown in Fig. 7.33. The problem with this type of aerofoil is that, although the onset of the adverse pressure gradient is postponed, it tends to be correspondingly more severe, thereby giving rise to bound­ary-layer separation. This necessitates the use of boundary-layer suction aft of the point of minimum pressure in order to prevent separation and to maintain laminar flow. See Section 7.4 and 8.4.1 below.

7.10 The physics of turbulent boundary layers

In this section, a brief account is given of the physics of turbulent boundary layers. This is still very much a developing subject and an active research topic. But some classic empirical knowledge, results and methods have stood the test of time and are worth describing in a general textbook on aerodynamics. Moreover, turbulent flows are so important for engineering applications that some understanding of the rele­vant flow physics is essential for predicting and controlling flows.

For the general solution of the momentum integral equation it is necessary to resort to computational methods, as described in Section 7.11. It is possible, however, in certain cases with external pressure gradients to find engineering solutions using the momentum integral equation without resorting to a computer. Two examples are given here. One involves the use of suction to control the boundary layer. The other concerns determining the boundary-layer properties at the leading-edge stagnation point of an aerofoil. For such applications Eqn (7.59) can be written in the alter­native form with H = 6*/в:

When, in addition, there is no pressure gradient and no suction, this further reduces to the simple momentum integral equation previously obtained (Section 7.7.1, Eqn (7.66)), i. e. Cr = 2(d0/dx).

Example 7.7 A two-dimensional divergent duct has a total included angle, between the plane diverging walls, of 20°. In order to prevent separation from these walls and also to maintain a laminar boundary-layer flow, it is proposed to construct them of porous material so that suction may be applied to them. At entry to the diffuser duct, where the flow velocity is 48 ms"1 the section is square with a side length of 0.3 m and the laminar boundary layers have a general thickness (<5) of 3 mm. If the boundary-layer thickness is to be maintained constant at this value, obtain an expression in terms of x for the value of the suction vel­ocity required, along the diverging walls. It may be assumed that for the diverging walls the laminar velocity profile remains constant and is given approximately by ii= 1.65 j7 _ 4.30/ + 3.65)’.

i. e.

Equation (7.16) gives

Then

dU,

Finally

initial boundary-layer thickness it can be assumed that the flow in the vicinity of the stagnation point is similar to that approaching a flat plate oriented perpendicularly to the free-stream, as shown in Fig. 7.27b. For this flow Ue = cx (where c is a constant) and the boundary-layer thickness does not change with x. In the example given below the momentum integral equation will be used to estimate the initial boundary-layer thickness for the flow depicted in Fig. 7.27b. An exact solution to the Navier-Stokes equations can be found for this stagnation-point flow (see Section 2.10.3). Here the momentum integral equation is used to obtain an approximate solution.

Example 7.8 Use the momentum integral equation (7.59) and the results (7.64a’, b’, d) to obtain expressions for <5, <5*, 9 and Q. It may be assumed that the boundary-layer thickness does not vary with x and that Ue = cx.

= —c = const, v

Hence 9 = const, also and Eqn (7.59) becomes

After rearrangement this equation simplifies to

It is known that A lies somewhere between 0 and 12 so it is relatively easy to solve this equation by trial and error to obtain

Using Eqns (7.64a’, b’, c’) then gives

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 <$l, 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 transition, this distance being given by the relationship

0.383лгт,

for the seventh-root profile.

The total skin-friction force coefficient Cf for one side of the plate of length L may be found by adding the skin-friction force per unit width for the laminar boundary layer of length xt to that for the turbulent boundary layer of length (L — xt), and

dividing by where L is here the wetted surface area per unit width. Working

in terms of Reu the transition position is given by

= —Ret

U OC

The laminar boundary-layer momentum thickness at transition is then obtained from Eqn (7.70):

The corresponding turbulent boundary-layer momentum thickness at transition then follows directly from Eqn (7.83):

The equivalent length of turbulent layer (xT,) to give this thickness is obtained from setting ви = вТі – using Eqn (7.93) and (7.94) this gives

/ 1/2 / 1/5

leading to

4/5 0-646 (

Kj’ 0.037 V

*Tt = 35.5 -^Re*/8

U OO

Now, on a flat plate with no pressure gradient, the momentum thickness at transition is a measure of the momentum defect produced in the laminar boundary layer between the leading edge and the transition position by the surface friction stresses only. As it is also being assumed here that the momentum thickness through transi­tion is constant, it is clear that the actual surface friction force under the laminar boundary layer of length xt must be the same as the force that would exist under a turbulent boundary layer of length jcTt. It then follows that the total skin-friction force for the whole plate may be found simply by calculating the skin-friction force under a turbulent boundary layer acting over a length from the point at a distance jcTi ahead of transition, to the trailing edge. Reference to Fig. 7.26 shows that the total effective length of turbulent boundary layer is, therefore, L — jct + *Tt – Now, from Eqn (7.21),

where Cf is given from Eqn (7.85) as

Thus

Now, Cf = F/^pU^L, where L is the total chordwise length of the plate, so that

i. e.

(7.96)

This result could have been obtained, alternatively, by direct substitution of the appropriate value of Re in Eqn (7.87), making the necessary correction for effective chord length (see Example 7.5).

The expression enables the curve of either Cf or CDf, for the flat plate, to be plotted against plate Reynolds number Re = (U^Ljv) for a known value of the transition Reynolds number Rex. Two such curves for extreme values of Ret of 3 x 105 and 3 x 106 are plotted in Fig. 7.25.

It should be noted that Eqn (7.96) is not applicable for values of Re less than Ret, when Eqns (7.71) and (7.72) should be used. For large values of Re, greater than about 108, the appropriate all-turbulent expressions should be used. However,

Eqns (7.85) and (7.88) become inaccurate for Re > 107. At higher Reynolds numbers the semi-empirical expressions due to Prandtl and Schlichting should be used, i. e.

For the lower transition Reynolds number of 3 x 105 the corresponding value of Re, above which the all-turbulent expressions are reasonably accurate, is 107.

Example 7.5 (1) Develop an expression for the drag coefficient of a flat plate of chord c and infinite span at zero incidence in a uniform stream of air, when transition occurs at a distance pc from the leading edge. Assume the following relationships for laminar and turbulent boundary layer velocity profiles, respectively:

(2) On a thin two-dimensional aerofoil of 1.8 m chord in an airstream of 45 m s_1, estimate the required position of transition to give a drag per metre span that is 4.5 N less than that for transition at the leading edge.

(1) Refer to Fig. 7.26 for notation.

From Eqn (7.95), setting xt = pc

,, = 3

Equation (7.88) gives the drag coefficient for an all-turbulent boundary layer as Cor = 0.1488/ReV. For the mixed boundary layer, the drag is obtained as for an all-turbulent layer of length [x-г, + (1 — p)c. The corresponding drag coefficient (defined with reference to length [хт, + (1 —p)c]) is then obtained directly from the all-turbulent expression where Re is based on the same length [xr, + (1 — p)c]. To relate the coefficient to the whole plate length c then requires that the quantity obtained should now be factored by the ratio

This form of expression (as an alternative to Eqn (7.96)) is convenient for enabling a quick approximation to skin-friction drag to be obtained when the position of transition is likely to be fixed, rather than the transition Reynolds number, e. g. by position of maximum thickness, although strictly the profile shapes will not be unchanged with length under these conditions and neither will Ue over the length.

(2) With transition at the leading edge:

0. 1488

C^=W = °[37]

The corresponding aerofoil drag is then Z)f = 0.00667 x 0.6125 x (45)2 x 1.8 = 14.88 N. With transition at pc, D¥ = 14.86 — 4.5 = 10.36 N, i. e.

CV = x 0.00667 = 0.00465

Using this value in (i), with Res’s = 16480, gives 0.1488

55.8 x 105

5.84 x 10y/8 – 55.8 x 105/> = f55**^5′) ‘ – 55.8 x 105 = (35.6 – 55.

V 0.1488 )

55.8p – 5.84p5/8 = 20.2

The solution to this (by successive approximation) is p = 0.423, i. e.

pc = 0.423 x 1.8 = 0.671 m behind leading edge

Example 7.6 A light aircraft has a tapered wing with root and tip chord-lengths of 2.2 m and

1.8 m respectively and a wingspan of 16 m. Estimate the skin-friction drag of the wing when the aircraft is travelling at 55 m/s. On the upper surface the point of minimum pressure is located at 0.375 chord-length from the leading edge. The dynamic viscosity and density of air may be taken as 1.8 x 10_5kgs/m and 1.2kg/m3 respectively.

The average wing chord is given by c = 0.5(2.2 + 1.8) = 2.0 m, so the wing is taken to be equivalent to a flat plate measuring 2.0 m x 16 m. The overall Reynolds number based on average chord is given by

1.2x55x2.0

= 7.33 x 106

Since this is below 107 the guidelines at the end of Section 7.9 suggest that the transition point will be very shortly after the point of minimum pressure, so ~ 0.375 x 2.0 = 0.75 m; also Eqn (7.96) may be used.

Rex = 0.375 x Re = 2.75 x 106

So Eqn (7.96) gives

CF = ^.°74^ {7.33 x 106 – 2.75 x 106 + 35.5(2.75 x 106)5/8}4/5 = 0.0023

f. j j X 1 и

Therefore the skin-friction drag of the upper surface is given by

D = ^-pU^csCf = 0.5 x 1.2 x 552 x 2.0 x 16 x 0.0023 = 133.8N

Finally, assuming that the drag of the lower surface is similar, the estimate for the total skin – friction drag for the wing is 2 x 133.8 ~ 270N.

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 momentum integral equation (7.66) shows that this in turn implies that the local skin-friction coefficient Cf would be infinite. This is plainly unacceptable on physical grounds, so it follows that the momentum thickness will remain constant across the transition position. Thus

ви = 0r, (7.89)

where the suffices L and T refer to laminar and turbulent boundary layer flows respectively and t indicates that these are particular values at transition. Thus

ви (^Jq «(1 – tf)dj^ = 6r, Qf m(1 – M)dj^

The integration being performed in each case using the appropriate laminar or turbulent profile. The ratio of the turbulent to the laminar boundary-layer thick­nesses is then given directly by

(7.90)

Using the values of / previously evaluated for the cubic and seventh-root profiles (Eqns (ii), Sections 7.6.1 and 7.7.3):

This indicates that on a flat plate the boundary layer increases in thickness by about 40% at transition.

It is then assumed that the turbulent layer, downstream of transition, will grow as if it had started from zero thickness at some point ahead of transition and developed along the surface so that its thickness reached the value <5rt at the transition position.