CFD Applications: Transonic Airfoils and Wings
The analysis of subsonic compressible flow over airfoils discussed in this chapter, resulting in classic compressibility corrections such as the Prandtl-Glauert mle (Section 11.4), fits into the category of “closed-form” theory as discussed in Section 2.17.1. Although this theory is elegant and useful, it is restricted to:
1. Thin airfoils at small angles of attack
2. Subsonic numbers that do not approach too close to one, that is, Mach numbers typically below 0.7
3. Inviscid, irrotational flow
However, modern subsonic transports (Boeing 747, 111, etc.) cruise at freestream Mach numbers on the order of 0.85, and high-performance military combat airplanes spend time at high subsonic speeds near Mach one. These airplanes are in the transonic flight regime, as discussed in Section 1.10.4 and noted in Figure 1.37. The closed – form theory discussed in this chapter does not apply in this flight regime. The only approach that allows the accurate calculation of airfoil and wing characteristics at transonic speeds is to use computational fluid dynamics; the basic philosophy of CFD is discussed in Section 2.17.2, which should be reviewed before you progress further.
The need to calculate accurately the transonic flow over airfoils and wings was one of the two areas that drove advances in CFD in the early days of its development, the other area being hypersonic flow. The growing importance of high-speed jet civil transports during the 1960s and 1970s made the accurate calculation of transonic flow imperative, and CFD was (and still is) the only way of making such calculations. In this section we will give only the flavor of such calculations; see Chapter 14 of Reference 21 for more details, as well as the modern aerodynamic literature for the latest developments.
Beginning in the 1960s, transonic CFD calculations historically evolved through four distinct steps, as follows:
1. The earliest calculations numerically solved the nonlinear small-perturbation potential equation for transonic flow, obtained from Equation (11.6) by dropping all terms on the right-hand side except for the leading term, which is not small near M„о = 1. This yields
(l – Ml;)— + ~ = Ml
^ 8x 8v
which in terms of the perturbation velocity potential is
Equation (11.69) is the transonic small perturbation potential equation; it is nonlinear due to the term on the right-hand side, which involves a product of derivatives of the dependent variable <j>. This necessitated a numerical CFD solution. However, the results were limited to the assumptions embodied in this equation, namely, small perturbations and hence thin airfoils at small angles of attack.
2. The next step was numerical solutions of the full potential equation, Equation
(11.12) . This allowed applications to airfoils of any shape at any angle of attack. However, the flow was still assumed to be isentropic, and even though shock waves appeared in the results, the properties of these shocks were not always accurately predicted.
3. As CFD algorithms became more sophisticated, numerical solutions of the Euler equations (the full continuity, momentum, and energy equations for inviscid flow, such as Equations (7.40), (7.42), and (7.44)) were obtained. The advantage of these Euler solutions was that shock waves were properly treated. However,
none of the approaches discussed in steps 1-3 accounted for the effects of viscous flow, the importance of which in transonic flows soon became more appreciated because of the interaction of the shock wave with the boundary layer. This interaction, with the attendant flow separation is dominant in the prediction of drag.
4. This led to the CFD solution of the viscous flow equations (the Navier-Stokes equations, such as Equations (2.43), (2.61), and (2.87) with the viscous terms included) for transonic flow. The Navier-Stokes equations are developed in detail in Chapter 15. Such CFD solutions of the Navier-Stokes equations are currently the state of the art in transonic flow calculations. These solutions contain all of the realistic physics of such flows, with the exception that some type of turbulence model must be included to deal with turbulent boundary layers, and such turbulent models are frequently the Archilles heel of these calculations.
An example of a CFD calculation for the transonic flow over an NACA 0012 airfoil at 2° angle of attack with = 0.8 is shown in Figure 11.21. The contour lines shown here are lines of constant Mach number, and the bunching of these lines together clearly shows the nearly normal shock wave occurring on the top surface. In reference to our calculation in Example 11.3 showing that the critical Mach number for the NACA 0012 airfoil at zero angle of attack is 0.74, and the experimental confirmation of this shown in Figure 11.1 Ofo, clearly the flow over the same airfoil shown in Figure 11.21 is well beyond the critical Mach number. Indeed, the boundary layer downstream of the shock wave in Figure 11.21 is separated, and the airfoil is squarely in the drag-divergence region. The CFD calculations predict this separated flow because a version of the Navier-Stokes equations (called the thin shear layer approximation) is being numerically solved, taking into account the viscous flow effects. The results shown in Figure 11.21 are from the work of Nakahashi and Deiwert at the NASA Ames Research Center (Reference 74); these results are a graphic illustration of the power of CFD applied to transonic flow. For details on these types of CFD calculations, see the definitive books by Hirsch (Reference 75).
Today, CFD is an integral part of modem transonic airfoil and wing design. A recent example of how CFD is combined with modem optimization design techniques for the design of complete wings for transonic aircraft is shown in Figures 11.22 and 11.23, taken from the survey paper by Jameson (Reference 76). On the left side of Figure 11.22a the airfoil shape distribution along the semispan of a baseline, initial wing shape at Мх = 0.83 is given, with the computed pressure coefficient distributions shown at the right. The abrupt drop in Cp in these distributions is due to a relatively strong shock wave along the wing. After repeated iterations, the optimized design at the same = 0.83 is shown in Figure 11.22b. Again, the new airfoil shape distribution is shown on the left, and the Cp distribution is given on the right. The new, optimized wing design shown in Figure 11.22b is virtually shock free, as indicated by the smooth Cp distributions, with a consequent reduction in drag of 7.6 percent. The optimization shown in Figure 11.22 was subject to the constraint of keeping the wing thickness the same. Another but similar case of wing design optimization is shown in Figure 11.23. Flere, the final optimized wing planform shape is shown for Mqq = 0.86, with the final computed pressure contour lines shown on
Figure 1 1.21 Mach number contours in the transonic flow over an NACA 001 2 airfoil at Mtx; = 0.8 and at 2° angle of attack. (Source: Nakahasi and Deiwert, Reference 74.1
the planform. Straddling the wing planform on both the left and right of Figure 11.23 are the pressure coefficient plots at six spanwise stations. The dashed curves show the Cp variations for the initial baseline wing, with the tell-tale oscillations indicating a shock wave, whereas the solid curves are the final Cp variations for the optimized wing, showing smoother variations that are almost shock-free. At the time of writing, the results shown in Figures 11.22 and 11.23 are reflective of the best combination of multidisciplinary design optimization using CFD for transonic wings. For more details on this and other design applications, see the special issue of the Journal of Aircraft, vol. 36, no. 1, Jan./Feb. 1999, devoted to aspects of multidisciplinary design optimization.
Figure I 1.22 The use of CFD for optimized transonic wing design. Moo = 0.83. (a) Baseline wing with a shock wave, (b) Optimized wing, virtually shock free. Source: Jameson, Reference 76.