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 NavierStokes solution.
The prediction of skin friction on the surface of a body in an attached flow is nicely accomplished by means of a boundarylayer solution coupled with an inviscid flow analyses to define the flow conditions at the edge of the boundary layer. Such an approach is welldeveloped, and the calculations can be rapidly carried out on
local computer workstations. Therefore, the use of boundarylayer 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 NavierStokes solution can be used to obtain local skin friction and heat transfer, but these NavierStokes 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 NavierStokes 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 NavierStokes
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 lowspeed flow using both a standard boundarylayer code and a stateoftheart NavierStokes solver with three different stateoftheart turbulence models. The results for friction drag from the boundarylayer code had been validated with experiment, and were considered the baseline for accuracy. The boundarylayer 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 NavierStokes 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 кє)

6.272

NS (Reynolds stress)

6.792

Boundary Layer Solution

5.340

Clearly, the accuracy of the various NavierStokes 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 NavierStokes calculations are compared with the results from the boundary layer code. All the NavierStokes 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