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