Computational Aerodynamics: Solvers and Shape Optimization

Since the development of the IAS computer built from 1942 to 1951 at the Institute of Advanced Studies under the supervision of John von Neumann, computer engineers’ push to make the hardware more practical and efficient has been driven and challenged by a handful of applied fields. While it is widely recognized that Aerodynamics played a key role in the development of modern scientific computing, it is instructive to understand why. We argue that the trigon formed by a compelling technological problem, the availability of useful mathematical models of increasing complexity, and the relentless pace of improvement in computing platforms imposed by Moore’s law, is responsible for the amazing advances in computational aerodynamics of the past 50 yr. Moreover, the particular nature of the aerospace industry imposes on computational aerodynamics stringent requirements on both the accuracy and robustness of the computations, which provided the need and the impetus for the development of advanced numerical techniques. This set of circumstances has brought and maintains computational aerodynamics at the forefront of modern scientific computing. In the early dawn of aviation, empiricism dominated aerodynamic design. Airfoil shapes were selected based on observation of nature (e.g., lilienthal) or good physical insight and new designs evolved following a build-test-modify process. By the late 1930 s, a deeper theoretical understanding of airfoil performance at subsonic speeds had been gained primarily thanks to Prandtl’s and Glauert’s pioneering work. This approach culminated in the development of the NACA 6 series of airfoils, which was obtained by hand calculations using the Theodorsen method for conformal mapping. Nevertheless, wind tunnel testing remained the main tool for aerodynamic analysis and design. The quest for supersonic flight, initiated in the late 1930’s in Germany and Italy, had moved after World War II to the USA and the USSR, and the onset of the cold war exacerbated the technological competition between the two superpowers. The Mig 19, the first massproduced true supersonic fighter (Mach 1.35), entered production in 1955, and was faster than the F-100 Super Saber, which was only capable of Mach 1.05 in level flight. By the end of the decade the speed had topped Mach 2.00 with the F104 (1958) and the Mig21 (1959). Again, advances were made principally by superior physical insight confirmed by wind tunnel tests. The discovery of the area-rule by Richard T. Whitcomb, his development of supercritical airfoils, and later of winglets are among the most notable examples of this traditional build-test-modify approach. This process was expensive, and in the 1960 s cost escalated with the complexity of newer projects. For example, more that 20,000 h of wind tunnel testing were needed for the development of the F111 or the Boeing 747. The need of gaining air-superiority together with the explosive growth of civil air traffic consolidated the strategic importance of aeronautical sciences in general and aerodynamics in particular. By 1960, it began to be apparent that digital computers had improved to the point of making it possible to attempt their use for the calculation of the aerodynamic characteristics—at least of isolated aircraft components—by solving a suitable mathematical model. The conservation of mass, momentum and energy for a viscous Newtonian fluid, which are generally referred to as the Navier–Stokes equation, govern the dynamics of any flow under the assumption that the fluid is a continuum; they have been known for approximately 150 yr, but solution of this nonlinear set of partial-differential equations is still daunting. Fortunately, since efficient flight can be achieved only by establishing highly coherent flows, useful predictions can be made with simplified mathematical models. In particular, since the Reynolds’ number of a typical aircraft is of the order of 10, aerodynamic forces such as lift, Induced drag and in the case of transonic or supersonic flight wave-drag, can be computed by using inviscid flow models. It was precisely the availability of a hierarchy of models of increased complexity and fidelity, which yield useful prediction at different stage of a design, that allowed computational aerodynamics to develop in-step with Moore’s law. The 1960s were dominated by the development of boundary integral methods (panel methods) based on the solution of a linear-potential equation both for purely subsonic or supersonic flow, for arbitrarily complex geometry [1–3]. The late 1960s and early 1970s have witnessed the emergence of computational fluid dynamics for more general industrial problems. In this wider arena, Spalding’s group at Imperial College led the way. The split treatment of the pressure terms, which culminated in the development of the SIMPLE method [5], was born from unmatched physical insight as much as it was grounded in the mathematical properties of the equations. The development of advanced numerical methods coupled with the path-breaking advances in turbulence modeling made in the same period by Launder and Spalding [6] enabled the practical use of CFD in an industrial setting. In aeronautics, prediction at transonic speeds were needed. The importance of the transonic flight regimes is twofold. To a first approximation, cruising efficiency is proportional to ML/D, the product of the Mach number M with the lift L to drag D ratio (aerodynamic efficiency). Since the aerodynamic efficiency is insensitive to the velocity, as long as shock waves are not present,