Direct simulation of low-density flow over airfoils

The low-density aerodynamics about airfoils in the transition region is considered using a direct simulation Monte Carlo method. Numerical results are presented for two airfoils (the NACA-0009 and a 9% thick, circular- arc airfoil) traveling at Mach 4 and 5, at an angle of attack of 1.25 deg and altitudes 56 and 62 miles above sea level. The flow, having a Knudsen number range between 0.47-1.15 and a Reynolds number range between 15-114, departs considerably from the continuum theory. Results indicate that both lift and drag are very much penalized in the transition region. In contrast with previous work on slightly rarefied flow with smaller Knudsen numbers (slip flow regime), the effect of rarefaction becomes much more dominant than that due to viscosity. URRENT interest in high-altitude flight has prompted new exploration of low-density aerodynamics. Unfor- tunately, the theoretical models and computational capabil- ities developed to date have proven insufficient to fully char- acterize these flows. The difficulty has to do with the high degree of rarefaction of the flow. The low-density flows of interest occur in a region between the continuum flow region and the molecular flow region. In this "transition region," the concept of transport coefficients (which are the basis for the Navier-Stokes equations) becomes invalid. Thus, the tra- ditional Navier-Stokes analysis ceases to give accurate results. The objective of this research is to investigate the flow physics of high-altitude flight and to develop a new computational scheme for assessing vehicle performance in this important flow region. In order to achieve sufficient lift for level flight, a high- altitude vehicle must fly at high speed. A typical flight en- velope would include freestream Mach numbers between 3 and 15, at an altitude of 30-60 miles above sea level. This flight regime, although considerably beyond the capability of existing aircraft, lies within the flight envelope of the National Aerospace Plane (X-30). The aerodynamic challenge is to achieve sufficiently high-lift coefficients, at the prescribed al- titudes but at a relatively low Mach number, so that certain hypersonic flow problems (such as heat transfer, chemical reactions, and surface catalytic activities) can be circum- vented. In contrast with continuum flow, transitional flow requires a higher-order approximation to the Boltzmann equation. Nu- merical solution to the Boltzmann equation consists of 1) evaluation of the collision integral; and 2) integration of the differential equation. Although the integral form of the col- lision term causes much of the mathematical difficulty in solv- ing the Boltzmann equation, the use of the velocity space coordinates as independent variables in partial differential equations requires large amounts of computer time and stor- age. The advert of high-speed computers has spurred the development of numerical solutions to the Boltzmann equa- tion for several basic, one-dimensional cases.1