Category When Is A Flow Compressible?

Turbulent Boundary Layers

The one uncontroversial fact about turbulence is that it is the most complicated kind of fluid motion.

Peter Bradshaw Imperial College of Science and Technology, London 1978

Turbulence was, and still is, one of the great unsolved mysteries of science, and it intrigued some of the best scientific minds of the day. Arnold Sommerfeld, the noted German theoretical physicist of the 1920s, once told me, for instance, that before he died he would like to understand two phenomena—quantum mechanics and turbulence. Sommerfeld died in 1924. I believe he was somewhat nearer to an understanding of the quantum, the discovery that led to modern physics, but no closer to the meaning of turbulence.

Theodore von Karman, 1967

19.1 Introduction

The subject of turbulent flow is deep, extensively studied, but at the time of writing still imprecise. The basic nature of turbulence, and therefore our ability to predict its characteristics, is still an unsolved problem in classical physics. Many books have been written on turbulent flows, and many people have spent their professional lives working on the subject. As a result, it is presumptuous for us to try to carry out a thorough discussion of turbulent boundary layers in this chapter. Instead, the purpose of this chapter is simply to provide a contrast with our study of laminar boundary layers in Chapter 18. Here, we will only be able to provide a flavor of turbulent

boundary layers, but this is all that is necessary in the present book. Turbulence is a subject that we leave for you to study more extensively as a subject on its own.

Before proceeding further, return to Section 15.2 and review the basic discussion of the nature of turbulence that is given there. In the present chapter, we will pick up where Section 15.2 leaves off.

Also, we note that no pure theory of turbulent flow exists. Every analysis of turbulent flows requires some type of empirical data in order to obtain a practical answer. As we examine the calculation of turbulent boundary layers in the following sections, the impact of this statement will become blatantly obvious. Finally, because this chapter is short, there is no need for a roadmap to act as a guide.

Derivation of the Linearized Supersonic Pressure Coefficient Formula

For the case of supersonic flow, let us write Equation (11.18) as

л^-4=о

Эх2 dy2

where Л = J— 1. A solution to this equation is the functional relation

Ф — f(x-ky) [12.2]

We can demonstrate this by substituting Equation (12.2) into Equation (12.1) as follows. The partial derivative of Equation (12.2) with respect to x can be written as

З ф. d(x — ky)

= / (x — ky)——- —

dx dx

In Equation (12.3), the prime denotes differentiation of / with respect to its argument, x — ky. Differentiating Equation (12.3) again with respect to x, we obtain

d-±= f" dx2 1

Similarly,

Substituting Equations (12.4) and (12.6) into (12.1), we obtain the identity

Xі f" – Xі f" = 0

Hence, Equation (12.2) is indeed a solution of Equation (12.1).

Examine Equation (12.2) closely. This solution is not very specific, because / can be any function of x — Xy. However, Equation (12.2) tells us something specific about the flow, namely, that ф is constant along lines of л – Xy = constant. The slope of these lines is obtained from

x — Xy — const

Hence, ± = ! =_______ ‘___

dx x !Mlc -1

From Equation (9.31) and the accompanying Figure 9.25, we know that

tan/r = , : [12.8]

where p. is the Mach angle. Therefore, comparing Equations (12.7) and (12.8), we see that a line along which ф is constant is a Mach line. This result is sketched in Figure 12.1, which shows supersonic flow over a surface with a small hump in the middle, where 9 is the angle of the surface relative to the horizontal. According to Equations (12.1) to (12.8), all disturbances created at the wall (represented by the perturbation potential ф) propagate unchanged away from the wall along Mach waves. All the Mach waves have the same slope, namely, dy/dx — (M^. — 1)~1/2. Note that the Mach waves slope downstream above the wall. Hence, any disturbance at the wall cannot propagate upstream; its effect is limited to the region of the flow downstream of the Mach wave emanating from the point of the disturbance. This is a further substantiation of the major difference between subsonic and supersonic flows mentioned in previous chapters, namely, disturbances propagate everywhere throughout a subsonic flow, whereas they cannot propagate upstream in a steady supersonic flow.

Keep in mind that the above results, as well as the picture in Figure 12.1, pertain to linearized supersonic flow [because Equation (12.1) is a linear equation]. Hence, these results assume small perturbations; that is, the hump in Figure 12.1 is small,

and thus в is small. Of course, we know from Chapter 9 that in reality a shock wave will be induced by the forward part of the hump, and an expansion wave will emanate from the rearward part of the hump. These are waves of finite strength and are not a part of linearized theory. Linearized theory is approximate; one of the consequences of this approximation is that waves of finite strength (shock and expansion waves) are not admitted.

The above results allow us to obtain a simple expression for the pressure coeffi­cient in supersonic flow, as follows. From Equation (12.3),

and from Equation (12.5),

~ 9<^ і f’

v = — = – A./
dy

Eliminating /’ from Equations (12.9) and (12.10), we obtain

Figure 1 2.2 Variation of the linearized

pressure coefficient with Mach number (schematic).

portion. This is denoted by the (+) and (—) signs in front of and behind the hump shown in Figure 12.1. This is also somewhat consistent with our discussions in Chapter 9; in the real flow over the hump, a shock wave forms above the front portion where the flow is being turned into itself, and hence p > whereas an expansion wave occurs over the remainder of the hump, and the pressure decreases. Think about the picture shown in Figure 12.1; the pressure is higher on the front section of the hump, and lower on the rear section. As a result, a drag force exists on the hump. This drag is called wave drag and is a characteristic of supersonic flows. Wave drag was discussed in Section 9.7 in conjunction with shock-expansion theory applied to supersonic airfoils. It is interesting that linearized supersonic theory also predicts a finite wave drag, although shock waves themselves are not treated in such linearized theory.

Examining Equation (12.15), we note that Cp oc (Af£, — l)-l/2; hence, for su­personic flow, Cp decreases as M0c increases. This is in direct contrast with subsonic flow, where Equation (11.51) shows that Cp cx (1 — M^,)^1/2; hence, for subsonic flow, Cp increases as M^ increases. These trends are illustrated in Figure 12.2. Note that both results predict Cp —>■ oo as M —> 1 from either side. However, keep in mind that neither Equation (12.15) nor (11.51) is valid in the transonic range around Mach 1.

Recovery Factor

As a corollary to the above case for the adiabatic wall, we take this opportunity to define the recovery factor—a useful engineering parameter in the analysis of aerody­namic heating. The total enthalpy of the flow at the upper plate (which represents the

upper boundary on a viscous shear layer) is, by definition,

И2

ho = he + ~

(The significance and definition of total enthalpy are discussed in Section 7.5.) Com­pare Equation (16.45), which is a general definition, with Equation (16.39), repeated below, which is for the special case of Couette flow:

haw = he + Pr^ [16.36]

Note that haw is different from ho, the difference provided by the value of Pr as it appears in Equation (16.39). We now generalize Equation (16.39) to a form which holds for any viscous flow, as follows:

Similarly, Equation (16.40) can be generalized to

[16.46b]

In Equations (16.46a and b), r is defined as the recovery factor. It is the factor that tells us how close the adiabatic wall enthalpy is to the total enthalpy at the upper boundary of the viscous flow. If r = 1, then haw = ho – An alternate expression for the recovery factor can be obtained by combining Equations (16.46) and (16.45) as follows. From Equation (16.46),

haw he

Г = «2/2

From Equation (16.45),

и2

~f=ho~he

Inserting Equation (16.48) into (16.47), we have

where To is the total temperature. Equation (16.49) is frequently used as an alternate definition of the recovery factor.

In the special case of Couette flow, by comparing Equation (16.39) or (16.40) with Equation (16.46a) or (16.46b), we find that

[16.50]

For Couette flow, the recovery factor is simply the Prandtl number. Note that, if Pr < 1, then haw < h0; conversely, if Pr > 1, then haw > h0.

In more general viscous flow cases, the recovery factor is not simply the Prandtl number; however, in general, for incompressible viscous flows, we will find that the recovery factor is some function of Pr. Hence, the Prandtl number is playing its role as an important viscous flow parameter. As expected from Section 15.6, for a compressible viscous flow, the recovery factor is a function of Pr along with the Mach number and the ratio of specific heats.

The Issue of Accuracy for the Prediction of Skin Friction Drag

The aerodynamic drag on a body is the sum of pressure drag and skin friction drag. For attached flows, the prediction of pressure drag is obtained from inviscid flow analyses such as those presented in Parts 2 and 3 of this book. For separated flows, various approximate theories for pressure drag have been advanced over the last century, but today the only viable and general method of the analysis of pressure drag for such flows is a complete numerical Navier-Stokes solution.

The prediction of skin friction on the surface of a body in an attached flow is nicely accomplished by means of a boundary-layer solution coupled with an inviscid flow analyses to define the flow conditions at the edge of the boundary layer. Such an approach is well-developed, and the calculations can be rapidly carried out on

local computer workstations. Therefore, the use of boundary-layer solutions for skin friction and aerodynamic heating is the preferred engineering approach. However, as mentioned above, if regions of flow separation are present, this approach cannot be used. In its place, a full Navier-Stokes solution can be used to obtain local skin friction and heat transfer, but these Navier-Stokes solutions are still not in the category of “quick engineering calculations.”

Zoom view of protuberance grid along the bottom surface of the airfoil.

This leads us to the question of the accuracy of CFD Navier-Stokes solutions for skin friction drag and heat transfer. There are three aspects that tend to diminish the accuracy of such solutions for the prediction of tw and qw (or alternately, c/ and ChY

1. The need to have a very closely spaced grid in the vicinity of the wall in order to obtain an accurate numerical value of (du/dy)w and (ЗT/dy)w, from which rw and qw are obtained.

2. The uncertainty in the accuracy of turbulence models when a turbulent flow is being calculated.

3. The lack of ability of most turbulent models to predict transition from laminar to turbulent flow.

Computed velocity vector field around and downstream of the protuberance.

In spite of all the advances made in CFD to the present, and all the work that has gone into turbulence modeling, at the time of writing the ability of Navier-Stokes

solutions to predict skin friction in a turbulent flow seems to be no better than about 20 percent accuracy, on the average. A recent study by Lombardi et al. (Reference 92) has made this clear. They calculated the skin friction drag on an NACA 0012 airfoil at zero angle of attack in a low-speed flow using both a standard boundary-layer code and a state-of-the-art Navier-Stokes solver with three different state-of-the-art turbulence models. The results for friction drag from the boundary-layer code had been validated with experiment, and were considered the baseline for accuracy. The boundary-layer code also had a prediction for transition that was considered reliable. Some typical results reported in Reference 92 for the integrated friction drag coefficient C/ are as follows, where NS represents Navier-Stokes solver and with the turbulence model in parenthesis. The calculations were all for Re = 3 x 106.

Cf X 103

NS (Standard к — є)

7.486

NS (RNG к-є)

6.272

NS (Reynolds stress)

6.792

Boundary Layer Solution

5.340

Clearly, the accuracy of the various Navier-Stokes calculations ranged from 18 percent to 40 percent.

More insight can be gained from the spatial distribution of the local skin friction coefficient Cf along the surface of the airfoil, as shown in Figure 20.15. Again the three different Navier-Stokes calculations are compared with the results from the boundary layer code. All the Navier-Stokes calculations greatly overestimated the peak in c/ just downstream of the leading edge, and slightly underestimated c/ near the trailing edge.

For a completely different reason not having to do with our discussion of accuracy, but for purposes of showing and contrasting the physically different distribution of Cf along a flat plate compared with that along the surface of the airfoil, we show Figure 20.16. Here the heavy curve is the variation of с/ with distance from the leading edge for a flat plate; the monotonic decrease is expected from our previous discussions of flat plate boundary layers. In contrast, for the airfoil Cf rapidly increases from a value of zero at the stagnation point to a peak value shortly downstream of the leading edge. This rapid increase is due to the rapidly increasing velocity as the flow external to the boundary layer rapidly expands around the leading edge. Beyond the peak, c/ then monotonically decreases in the same qualitative manner as for a flat plate. It is simply interesting to note these different variations for c/ over an airfoil compared to that for a flat plate, especially since we devoted so much attention to flat plates in the previous chapters.

20.5 Summary

With this, we end our discussion of viscous flow. The purpose of all of Part 4 has been to introduce you to the basic aspects of viscous flow. The subject is so vast that it demands a book in itself—many of which have been written (see, e. g., References 41 through 45). Here, we have presented only enough material to give you a flavor for some of the basic ideas and results. This is a subject of great importance in aerody­namics, and if you wish to expand your knowledge and expertise of aerodynamics in general, we encourage you to read further on the subject.

We are also out of our allotted space for this book. Therefore, we hope that you have enjoyed and benefited from our presentation of the fundamentals of aerodynam­ics. However, before closing the cover, it might be useful to return once again to Figure 1.38, which is the block diagram categorizing the different general types of aerodynamic flows. Recall the curious, uninitiated thoughts you might have had when you first examined this figure during your study of Chapter 1, and compare these with the informed and mature thoughts that you now have—honed by the aerodynamic knowledge packed into the intervening pages. Hopefully, each block in Figure 1.38 has substantially more meaning for you now than when we first started. If this is true, then my efforts as an author have not gone in vain.

[1] у –

1 H—- ~r(Mj — 1)

Y + 1 .

From Equation (8.68), we see that the entropy change S2 — s 1 across the shock is a function of Mi only. The second law dictates that

S2 — S >0

In Equation (8.68), if Mi = l, s2 = v,, and if Mi > 1, then, v2 — .? 1 > 0, both of which

[2] = Voo tan в

[3] vx

Detached Shock Wave in Front of a Blunt Body

The curved bow shock which stands in front of a blunt body in a supersonic flow is sketched in Figure 8.1. We are now in a position to better understand the properties of this bow shock, as follows.

The flow in Figure 8.1 is sketched in more detail in Figure 9.21. Here, the shock wave stands a distance 8 in front of the nose of the blunt body; 8 is defined as the shock detachment distance. At point a, the shock wave is normal to the upstream flow; hence, point a corresponds to a normal shock wave. Away from point a, the shock wave gradually becomes curved and weaker, eventually evolving into a Mach wave at large distances from the body (illustrated by point e in Figure 9.21).

Figure 9.31 Flow over a supersonic blunt body.

A curved bow shock wave is one of the instances in nature when you can observe all possible oblique shock solutions at once for a given freestream Mach number M|. This takes place between points a and e. To see this more clearly, consider the в-Р-М diagram sketched in Figure 9.22 in conjunction with Figure 9.21. In Figure 9.22, point a corresponds to the normal shock, and point e corresponds to the Mach wave. Slightly above the centerline, at point b in Figure 9.21, the shock is oblique but pertains to the strong shock-wave solution in Figure 9.22. The flow is deflected slightly upward behind the shock at point b. As we move further along the shock, the wave angle becomes more oblique, and the flow deflection increases until we encounter point c. Point c on the bow shock corresponds to the maximum deflection angle shown in Figure 9.22. Above point c, from c to e, all points on the shock correspond to the weak shock solution. Slightly above point c, at point c’, the flow behind the shock becomes sonic. From a to c the flow is subsonic behind the bow shock; from c’ to e, it is supersonic. Hence, the flow field between the curved bow shock and the blunt body is a mixed region of both subsonic and supersonic flow. The dividing line between the subsonic and supersonic regions is called the sonic line, shown as the dashed line in Figure 9.21.

The shape of the detached shock wave, its detachment distance <5, and the com­plete flow field between the shock and the body depend on M and the size and shape

Figure 9.22 9-fi-hA diagram for the sketch shown in Figure 9.21.

of the body. The solution of this flow field is not trivial. Indeed, the supersonic blunt – body problem was a major focus for supersonic aerodynamicists during the 1950s and 1960s, spurred by the need to understand the high-speed flow over blunt-nosed missiles and reentry bodies. Indeed, it was not until the late 1960s that truly suffi­cient numerical techniques became available for satisfactory engineering solutions of supersonic blunt-body flows. These modem techniques are discussed in Chapter 13.

Elements of Hypersonic Flow

Almost everyone has their own definition of the term hypersonic. If we were to conduct something like a public opinion poll among those present, and asked everyone to name a Mach number above which the flow of a gas should properly be described as hypersonic there would be a majority of answers round about 5 or 6, but it would be quite possible for someone to advocate, and defend, numbers as small as 3, or as high as 12.

P. L. Roe, comment made in a lecture at the von Karman Institute, Belgium January 1970

14.1 Introduction

The history of aviation has always been driven by the philosophy of “faster and higher,” starting with the Wright brothers’ sea level flights at 35 mi/h in 1903, and progressing exponentially to the manned space flight missions of the 1960s and 1970s. The current altitude and speed records for manned flight are the moon and 36,000 ft/s—more than 36 times the speed of sound—set by the Apollo lunar capsule in 1969. Although most of the flight of the Apollo took place in space, outside the earth’s atmosphere, one of its most critical aspects was reentry into the atmosphere after completion of the lunar mission. The aerodynamic phenomena associated with very high-speed flight, such as encountered during atmospheric reentry, are classified as hypersonic aerodynamics—the subject of this chapter. In addition to reentry vehicles, both manned and unmanned, there are other hypersonic applications on the horizon, such as ramjet-powered hypersonic missiles now under consideration by the military and the concept of a hypersonic transport, the basic technology of which is now being studied by NASA. Therefore, although hypersonic aerodynamics is at one extreme

end of the whole flight spectrum (see Section 1.10), it is important enough to justify one small chapter in our presentation of the fundamentals of aerodynamics.

This chapter is short; its purpose is simply to introduce some basic considerations of hypersonic flow. Therefore, we have no need for a chapter road map or a summary at the end. Also, before progressing further, return to Chapter 1 and review the short discussion on hypersonic flow given in Section 1.10. For an in-depth study of hypersonic flow, see the author’s book listed as Reference 55.

The Boundary-Layer Equations

For the remainder of this chapter, we consider two-dimensional, steady flow. The nondimensionalized form of the x-momentum equation (one of the Navier-Stokes equations) was developed in Section 15.6 and was given by Equation (15.29);

Let us now reduce Equation (15.29) to an approximate form which holds reasonably well within a boundary layer.

Consider the boundary layer along a flat plate of length c as sketched in Fig­ure 17.7. The basic assumption of boundary-layer theory is that a boundary layer is

very thin in comparison with the scale of the body; that is,

Consider the continuity equation for a steady, two-dimensional flow,

d(pu) d(pv) __ dx dy

In terms of the nondimensional variables defined in Section 15.6, Equation (17.16) becomes

d(p’u’) Э(рУ) dx1 dy’

Because u’ varies from 0 at the wall to 1 at the edge of the boundary layer, let us say that u’ is of the order of magnitude equal to 1, symbolized by 0(1). Similarly, p’ = 0(1). Also, since x varies from 0 to c, x’ = 0(1). However, since у varies from 0 to <5, where 8 <£ c, then y’ is of the smaller order of magnitude, denoted by у’ = О(8/с). Without loss of generality, we can assume that c is a unit length. Therefore, y’ = 0(8). Putting these orders of magnitude in Equation (17.17), we have

!в + Ш=0 [17.18]

0(1) 0(8)

Hence, from Equation (17.18), clearly v’ must be of an order of magnitude equal to 8; that is, v’ = 0(8). Now examine the order of magnitude of the terms in Equation (15.29). We have

Let us now introduce another assumption of boundary-layer theory, namely, the Reynolds number is large, indeed, large enough such that

Then, Equation (17.19) becomes

From Equation (17.26), we see that dp’/dy’ = 0(8) or smaller, assuming that yM^ = 0(1). Since 8 is very small, this implies that dp’/dy’ is very small. There­fore, from the у-momentum equation specialized to a boundary layer, we have

Equation (17.26a) is important; it states that at a given x station, the pressure is constant through the boundary layer in a direction normal to the surface. This implies that the pressure distribution at the outer edge of the boundary layer is impressed directly to the surface without change. Hence, throughout the boundary layer, p — P(x) = pe (x).

It is interesting to note that if is very large, as in the case of large hypersonic Mach numbers, then from Equation (17.26) dp’/dy’ does not have to be small. For example, if Mqo were large enough such that 1/yM^ = 0(8), then dp’/dy’ could be as large as 0(1), and Equation (17.26) would still be satisfied. Thus, for very large hypersonic Mach numbers, the assumption that p is constant in the normal direction through a boundary layer is not always valid.

Consider the general energy equation given by Equation (15.26). The nondi­mensional form of this equation for two-dimensional, steady flow is given in Equa­tion (15.33). Inserting e = h — p/p into this equation, subtracting the momentum equation multiplied by velocity, and performing an order-of-magnitude analysis sim­ilar to those above, we can obtain the boundary-layer energy equation as

The details are left to you.

In summary, by making the combined assumptions of 8 <SC c and Re > 1/82, the complete Navier-Stokes equations derived in Chapter 15 can be reduced to simpler forms which apply to a boundary layer. These boundary-layer equations are

[17.30]

[17.31]

Note that, as in the case of the Navier-Stokes equations, the boundary-layer equations are nonlinear. However, the boundary-layer equations are simpler, and therefore are more readily solved. Also, since p = p,. (x), the pressure gradient expressed as dp/dx in Equations (17.23) and (17.27) is reexpressed as dpe/dx in Equations (17.29) and

(17.31) . In the above equations, the unknowns are u, v, p, and h p is known from p = pe (x), and д and к are properties of the fluid which vary with temperature. To complete the system, we have

and h=cpT [17.33]

Hence, Equations (17.28), (17.29), and (17.31) to (17.33) are five equations for the five unknowns, и, n, p, T, and h.

The boundary conditions for the above equations are as follows:

At the wall: у = 0 и = 0 v = О T — Tw

At the boundary-layer edge: у —»• oo и —*■ ue T —*■ Te

Note that since the boundary-layer thickness is not known a priori, the boundary con­dition at the edge of the boundary layer is given at large y, essentially у approaching infinity.

Prandtl-Glauert Compressibility orrection

The aerodynamic theory for incompressible flow over thin airfoils at small angles of attack was presented in Chapter 4. For aircraft of the period 1903-1940, such theory

was adequate for predicting airfoil properties. However, with the rapid evolution of high-power reciprocating engines spurred by World war II, the velocities of military fighter planes began to push close to 450 mi/h. Then, with the advent of the first operational jet-propelled airplanes in 1944 (the German Me 262), flight velocities took a sudden spurt into the 550 mi/h range and faster. As a result, the incompressible flow theory of Chapter 4 was no longer applicable to such aircraft; rather, high-speed airfoil theory had to deal with compressible flow. Because a vast bulk of data and experience had been collected over the years in low-speed aerodynamics, and because there was no desire to totally discard such data, the natural approach to high-speed subsonic aerodynamics was to search for methods that would allow relatively simple corrections to existing incompressible flow results which would approximately take into account the effects of compressibility. Such methods are called compressibility corrections. The first, and most widely known of these corrections is the Prandtl – Glauert compressibility correction, to be derived in this section. The Prandtl-Glauert method is based on the linearized perturbation velocity potential equation given by Equation (11.18). Therefore, it is limited to thin airfoils at small angles of attack. Moreover, it is purely a subsonic theory and begins to give inappropriate results at values of Moo = 0.7 and above.

Consider the subsonic, compressible, inviscid flow over the airfoil sketched in Figure 11.3. The shape of the airfoil is given by у = f(x). Assume that the airfoil is thin and that the angle of attack is small; in such a case, the flow is reasonably approximated by Equation (11.18). Define

P2 – і – мі

so that Equation (11.18) can be written as

2д2ф д2ф

Let us transform the independent variables x and у into a new space, £ and ij, such that

у

Ко

%=x [11.36a]

П = РУ [11.36b]

Moreover, in this transformed space, consider a new velocity potential ф such that

</>(£, rj) = Рф(х, у) [11.36c]

To recast Equation (11.35) in terms of the transformed variables, recall the chain rule of partial differentiation; that is,

Зф 3ф 3§ 3ф dt]

дх 3£ dx dr] dx

dip d0 3§ 30 dr]

dy Э§ 3у dr] dy

From Equations (11.36a and b), we have

Differentiating Equation (11.41) with respect to x (again using the chain rule), we obtain

З20 1 d2ф

3×2 “ W

Differentiating Equation (11.42) with respect to y, we hnd that the result is

Substitute Equations (11.43) and (11.44) into (11.35):

Examine Equation (11.45)—it should look familiar. Indeed, Equation (11.45) is Laplace’s equation. Recall from Chapter 3 that Laplace’s equation is the governing relation for incompressible flow. Hence, starting with a subsonic compressible flow in physical (x, у) space where the flow is represented by ф(х, у) obtained from Equation (11.35), we have related this flow to an incompressible flow in transformed (£, і)) space, where the flow is represented by ф(%, і) ‘) obtained from Equation (11.45). The relation between ф and ф is given by Equation (11.36c).

Consider again the shape of the airfoil given in physical space by у = f(x). The shape of the airfoil in the transformed space is expressed as r) — q(^). Let us compare the two shapes. First, apply the approximate boundary condition, Equation

(11.34) , in physical space, noting that df/dx = tan в. We obtain

df _дф _ 1 дф _ дф 00 dx ду p ду 3 г)

Similarly, apply the flow-tangency condition in transformed space, which from Equa­tion (11.34) is

dq_ _ дф

°°d$ ~ dr)

Examine Equations (11.46) and (11.47) closely. Note that the right-hand sides of these two equations are identical. Thus, from the left-hand sides, we obtain

df_ _dq_

dx </£

Equation (11.48) implies that the shape of the airfoil in the transformed space is the same as in the physical space. Hence, the above transformation relates the compress­ible flow over an airfoil in (x, y) space to the incompressible flow in (£, >)) space over the same airfoil.

The above theory leads to an immensely practical result, as follows. Recall Equa­tion (11.32) for the linearized pressure coefficient. Inserting the above transformation into Equation (11.32), we obtain

2Й 2 3ф 2 1 3ф 2 1 3ф ^ ^ 49]

Question: What is the significance of 3$/3£ in Equation (11.49)? Recall that ф is the perturbation velocity potential for an incompressible flow in transformed space. Hence, from the definition of velocity potential, дф/ді; = й, where й is a perturbation

velocity for the incompressible flow. Hence, Equation (11.49) can be written as

From Equation (11.32), the expression f—2u/V00) is simply the linearized pressure coefficient for the incompressible flow. Denote this incompressible pressure coeffi­cient by Орд. Hence, Equation (11.50) gives

or recalling that f = ,/F— M^, we have

[11.51]

Equation (11.51) is called the Prandtl-Glauert rule; it states that, if we know the incompressible pressure distribution over an airfoil, then the compressible pressure distribution over the same airfoil can be obtained from Equation (11.51). Therefore, Equation (11.51) is truly a compressibility correction to incompressible data.

Consider the lift and moment coefficients for the airfoil. For an inviscid flow, the aerodynamic lift and moment on a body are simply integrals of the pressure distribution over the body, as described in Section 1.5. (If this is somewhat foggy in your mind, review Section 1.5 before progressing further.) In turn, the lift and moment coefficients are obtained from the integral of the pressure coefficient via Equations (1.15) to (1.19). Since Equation (11.51) relates the compressible and incompressible pressure coefficients, the same relation must therefore hold for lift and moment coefficients:

The Prandtl-Glauert rule, embodied in Equations (11.51) to (11.53), was histor­ically the first compressibility correction to be obtained. As early as 1922, Prandtl was using this result in his lectures at Gottingen, although without written proof. The derivation of Equations (11.51) to (11.53) was first formally published by the British aerodynamicist, Hermann Glauert, in 1928. Hence, the rule is named after both men. The Prandtl-Glauert rule was used exclusively until 1939, when an improved com­pressibility correction was developed. Because of their simplicity, Equations (11.51) to (11.53) are still used today for initial estimates of compressibility effects.

Recall that the results of Chapters 3 and 4 proved that inviscid, incompressible flow over a closed, two-dimensional body theoretically produces zero drag—the well – known d’Alembert’s paradox. Does the same paradox hold for inviscid, subsonic,

compressible flow? The answer can be obtained by again noting that the only source of drag is the integral of the pressure distribution. If this integral is zero for an incompressible flow, and since the compressible pressure coefficient differs from the incompressible pressure coefficient by only a constant scale factor, p, then the integral must also be zero for a compressible flow. Hence, d’Alembert’s paradox also prevails for inviscid, subsonic, compressible flow. However, as soon as the freestream Mach number is high enough to produce locally supersonic flow on the body surface with attendant shock waves, as shown in Figure 137b, then a positive wave drag is produced, and d’Alembert’s paradox no longer prevails.

Example 11.1 | At a given point on the surface of an airfoil, the pressure coefficient is —0.3 at very low speeds. If the freestream Mach number is 0.6, calculate Cp at this point.

Solution

From Equation (11.51),

^ _ Cp, o _ ~~0-3

p ~~ Vl – M2 ~ y/ – (0.6)2

Example 1 1.2 | From Chapter 4, the theoretical lift coefficient for a thin, symmetric airfoil in an incompressible

flow is с/ = 2ла. Calculate the lift coefficient for Mx = 0.7.

Solution

From Equation (11.52),

C/,o 2na

C‘ ~ У1 – Ml ~ Vl – (0.7)2

Note: The effect of compressibility at Mach 0.7 is to increase the lift slope by the ratio 8.8/27Г = 1.4, or by 40 percent.

The Navier-Stokes Equations

In Chapter 2, Newton’s second law was applied to obtain the fluid-flow momentum equation in both integral and differential forms. In particular, recall Equations (2.13a to c), where the influence of viscous forces was expressed simply by the generic terms (•Tudviscous, (dd'(viscousi and (d%)viSCous – The purpose of this section is to obtain the analogous forms of Equations (2.13a to c) where the viscous forces are expressed explicitly in terms of the appropriate flow-field variables. The resulting equations are called the Navier-Stokes equations—probably the most pivotal equations in all of theoretical fluid dynamics.

In Section 2.3, we discussed the philosophy behind the derivation of the governing equations, namely, certain physical principles are applied to a suitable model of the fluid flow. Moreover, we saw that such a model could be either a finite control volume (moving or fixed in space) or an infinitesimally small element (moving or fixed in space). In Chapter 2, we chose the fixed, finite control volume for our model and obtained integral forms of the continuity, momentum, and energy equations directly from this model. Then, indirectly, we went on to extract partial differential equations from the integral forms. Before progressing further, it would be wise for you to review these matters from Chapter 2.

For the sake of variety, let us not use the fixed, finite control volume employed in Chapter 2; rather, in this section, let us adopt an infinitesimally small moving fluid element of fixed mass as our model of the flow, as sketched in Figure 15.11. To this model let us apply Newton’s second law in the form F = ma. Moreover, for the time being consider only the jc component of Newton’s second law:

Fx = max [15.14]

In Equation (15.14), Fx is the sum of all the body and surface forces acting on the fluid element in the x direction. Let us ignore body forces; hence, the net force acting on the element in Figure 15.11 is simply due to the pressure and viscous stress distributions over the surface of the element. For example, on face abed, the only force in the jc direction is that due to shear stress, rvv dx dz. Face efgh is a

Figure 1 5.1 1 Infinitesimally small, moving fluid element. Only the forces in the x direction are shown.

distance dy above face abed; hence, the shear force in the x direction on face efgh is [xyx + (дтух/ду) dy] dx dz. Note the directions of the shear stress on faces abed and efgh; on the bottom face, zyx is to the left (the negative x direction), whereas on the top face, zyx + (dzyx/dy) dy is to the right (the positive x direction). These directions are due to the convention that positive increases in all three components of velocity, u, v, and w, occur in the positive directions of the axes. For example, in Figure 15.11, и increases in the positive у direction. Therefore, concentrating on face efgh, и is higher just above the face than on the face; this causes a “tugging” action which tries to pull the fluid element in the positive x direction (to the right) as shown in Figure 15.11. In turn, concentrating on face abed, и is lower just beneath the face than on the face; this causes a retarding or dragging action on the fluid element, which acts in the negative x direction (to the left), as shown in Figure 15.11. The directions of all the other viscous stresses shown in Figure 15.11, including zxx, can be justified in a like fashion. Specifically, on face degh, zzx acts in the negative x direction, whereas on face abfe, zzx + (dzzx/8z)dz acts in the positive x direction. On face adhe, which is perpendicular to the x axis, the only forces in the x direction are the pressure force p dy dz, which always acts in the direction into the fluid element, and zxx dy dz, which is in the negative x direction. In Figure 15.11, the reason why zxx on face adhe is to the left hinges on the convention mentioned earlier for the direction of increasing velocity. Here, by convention, a positive increase in и takes place in the positive x direction. Hence, the value of и just to the left of face adhe is smaller than the value of и on the face itself. As a result, the viscous action of the normal stress

acts as a “suction” on face adhe: that is, there is a dragging action toward the left that wants to retard the motion of the fluid element. In contrast, on face bcgf, the pressure force [p + (dp/dx) dx] dy dz presses inward on the fluid element (in the negative x direction), and because the value of и just to the right of face bcgf is larger than the value of и on the face, there is a “suction” due to the viscous normal stress which tries to pull the element to the right (in the positive x direction) with a force equal to [txx + (dzxx/dx)dx]dydz.

Return to Equation (15.14). Examining Figure 15.11 in light of our previous discussion, we can write for the net force in the л direction acting on the fluid element:

і dP
p-p+-dx

Equation (15.15) represents the left-hand side of Equation (15.14). Considering the right-hand side of Equation (15.14), recall that the mass of the fluid element is fixed and is equal to

m = p dx dy dz

Also, recall that the acceleration of the fluid element is the time rate of change of its velocity. Hence, the component of acceleration in the x direction, denoted by ax, is simply the time rate of change of n; since we are following a moving fluid element, this time rate of change is given by the substantial derivative (see Section 2.9 for a review of the meaning of substantial derivative). Thus,

Du _ .

ax = — [15.17]

Dt

Combining Equations (15.14) to (15.17), we obtain

[15.18a]

which is the x component of the momentum equation for a viscous flow. In a similar fashion, the у and z components can be obtained as

Dw dp drxz dzyz drzz

Dt dz dx dy dz

Equations (15.18a to c) are the momentum equations in the x, y, and z directions, respectively. They are scalar equations and are called the Navier-Stokes equations in

honor of two men—the Frenchman M. Navier and the Englishman G. Stokes—who independently obtained the equations in the first half of the nineteenth century.

With the expressions for rxy = xyx, xyi = xzy, rzx = rxz, rxx, xyy, and xzz from Equations (15.5) to (15.10), the Navier-Stokes equations, Equations (15.18a to c), can be written as

dи Эи Эи Зи dp Э / Эи

РЧ7 + риТ + pv^~ + pwV~ = UV • V + 2/х —

dt dx dy dz dx dx dx J

Equations (15.19a to c) represent the complete Navier-Stokes equations for an un­steady, compressible, three-dimensional viscous flow. To analyze incompressible vis­cous flow, Equations (15.19a to c) and the continuity equation [say, Equation (2.52)] are sufficient. However, for a compressible flow, we need an additional equation, namely, the energy equation to be discussed in the next section.

In the above form, the Navier-Stokes equations are suitable for the analysis of laminar flow. For a turbulent flow, the flow variables in Equations (15.19a to c) can be assumed to be time-mean values over the turbulent fluctuations, and p can be replaced by p + є, as discussed in Section 15.3. For more details, see References 42 and 43.

Results for Turbulent Boundary Layers on a Flat Plate

In this section, we discuss a few results for the turbulent boundary layer on a flat plate, both incompressible and compressible, simply to provide a basis of comparison with the laminar results described in the previous section. For considerably more detail on the subject of turbulent boundary layers, consult References 42 to 44.

For incompressible flow over a flat plate, the boundary-layer thickness is given approximately by

Note from Equation (19.1) that the turbulent boundary-layer thickness varies approx­imately as RejTl/5 in contrast to Re“I/2 for a laminar boundary layer. Also, turbulent values of <5 grow more rapidly with distance along the surface; 8 <x x4/5 for a turbulent flow in contrast to 8 oc x1 /2 for a laminar flow. With regard to skin friction drag, for incompressible turbulent flow over a flat plate, we have

Note that for turbulent flow, C f varies as Re~1//5 in comparison with the Re’1/2 vari­ation for laminar flow. Hence, Equation (19.2) yields larger friction drag coefficients for turbulent flow in comparison with Equation (18.22) for laminar flow.

The effects of compressibility on Equation (19.2) are shown in Figure 19.1, where C f is plotted versus ReTO with Мто as a parameter. The turbulent flow results are shown toward the right of Figure 19.1, at the higher values of Reynolds numbers where turbulent conditions are expected to occur, and laminar flow results are shown toward the left of the figure, at lower values of Reynolds numbers. This type of figure—friction drag coefficient for both laminar and turbulent flow as a function of Re on a log-log plot—is a classic picture, and it allows a ready contrast of the two types of flow. From this figure, we can see that, for the same Re^, turbulent skin

friction is higher than laminar; also, the slopes of the turbulent curves are smaller than the slopes of the laminar curves—a graphic comparison of the Re,/? variation in contrast to the laminar Re 1/2 variation. Note that the effect of increasing Мж is to reduce Cf at constant Re and that this effect is stronger on the turbulent flow results. Indeed, C/ for the turbulent results decreases by almost an order of magnitude (at the higher values of Re,*,) when M0c is increased from 0 to 10. For the laminar flow, the decrease in Су as Мж is increased though the same Mach number range is far less pronounced.