Aerothermal analysis of supersonic flow around an airfoil in transitional rarefied conditions

Abstract

When the physical length scale of a fluid mechanical system becomes comparable to the molecular mean free path (the average distance traveled by each molecule of gas between successive intermolecular collisions), the system could not able to attain the flow and thermodynamic equilibrium conditions. Such cases are known as rarefied conditions. In general, these rarefied conditions occur when the gas flow is at very low density (low pressure) environment and/or involves very small length scales such as- flight of high-speed vehicles at very high altitude in the earth’s atmosphere, micro- and nanoelectromechanical systems (MEMS/NEMS), flow around spacecraft thrusters, and so on. In rarefied gas flows, the discrete particle effects (representing atoms and molecules of gas) become dominant and a sharp gradient of the macroscopic quantities such as velocity, temperature, pressure and so on are observed. As a result, the well-established continuum descriptions of gas flows that take into account the molecular behavior in an aggregate sense (bulk) collapse at rarefied conditions. Therefore, the mathematical modeling of such gas flows with Navier-Stokes (NS) equations becomes inaccurate. In these cases, the kinetic theory of gas i.e. the Boltzmann equation is to be considered to predict the rarefied flow phenomena which take into account the particle collisions and the post-collision molecular velocity distributions. The degree of rarefaction is defined by Knudsen number (Kn) which is defined as the ratio between the molecular mean free path and the characteristics length of the flow system. The continuum hypothesis is valid for very small values of Kn (Kn < 0.01); whereas the hypothesis breaks down at slip regime (0.01 < Kn < 0.1) and transitional regime (0.1 < Kn < 10). In this present work, supersonic flow over an airfoil in the transitional rarefied conditions has been studied. The aerothermal characteristics of the system have modeled and computationally solved using Direct Simulation Monte Carlo (DSMC) approach. The computations have been performed with ‘dsmcFOAM’ solver which is available in the open-source CFD software platform ‘OpenFOAM.’ ‘dsmcFOAM’ is a validated and widely used DSMC solver that is dedicated to solving the Boltzmann equation stochastically for the rarefied gas flow problems. A classical series of symmetric airfoil NACA 00XX has been considered. DSMC computation has been performed for three airfoils; namely- NACA 0012 (12% thick), NACA 0010 (10% thick) and NACA 0007 (10% thick) airfoils at 0o angle of attack (AOA). Supersonic flows with freestream Mach number in the range of M∞ = 2 – 4 and transitional rarefied conditions in the range of Kn = 0.5 - 5 are considered. The results have been validated with the available relevant experimental data as well as with the solution data from other DSMC computations. DSMC results showed that the shock waves which are obvious in the supersonic flow field around the airfoil vanishes in the transitional rarefied conditions and a series of diffusive waves are observed. These diffusive waves deviate the free stream flow properties from the airfoil leading edge. The surface-particles interactions which result the shear stress shows decreasing behavior with the increase of Kn. However, this stress was found to be increased with the increase of free-stream Mach number, M∞ for a particular degree of rarefaction (Kn). The cumulative effects of shear stress, pressure and compressibility are considered to determine the aerodynamic drag force. It is found that the aerodynamic drag increases with the increase of Kn at a fixed Mach number. On the other hand, for a constant Kn, a linear decrement of drag is found while Mach number increases. The non-equilibrium effects, i.e. velocity slip and temperature jump at the airfoil surfaces are identified by the present DSMC approach. Whenever the degree of rarefication (Kn) increases, the velocity increases at the vicinity of the airfoil surface which results in significant-velocity slip. At supersonic speeds, gas molecules accumulate at the leading edge of the airfoil and the macroscopic properties such as density, pressure and temperature increase. Moreover, at these speeds, a large number of molecules collide at the leading edge and their kinetic energy is transported into force action and heating. As the Mach number increases, the particle density at the airfoil leading-edge increases. However, the particle density decreases over the airfoil surface. The Kn number has effects on particle interactions with airfoil surfaces. At Kn = 5.0, the lowest amount of particle density linearly distributed over the airfoil surface is observed. At lower Kn number, gas molecules possess a larger amount of kinetic and internal energy which are responsible to affect the flow temperatures. Conversely, the molecules at higher Kn number have negligible kinetic and internal energy. The heat load, i.e. surface heat flux is higher for the slower particles within the viscous regime of the flow filed. Therefore, at lower Kn number, the airfoil surface encounters a higher heat load than a higher Kn number. Mach number has an adverse effect on heat flux. The higher Mach number provides higher heat flux at all the rarefied conditions studied here. Additionally, the thickness effects of the airfoil in the transitional rarefied condition have been investigated based on their aerodynamic performance and surface heat load. Consequently, the airfoil with lower thickness encounters optimum aerodynamic drag and surface heat flux.