Comprehensive mesh study for a Direct Numerical Simulation of the transonic flow at Rec = 500,000 around a NACA 0012 airfoil

For the method of a Direct Numerical Simulation (DNS), a mesh study of the transonic flow around the well known NACA 0012 airfoil at a moderate Reynolds number of Rec=5·105 is presented. The three-dimensional Navier-Stokes equations for an unsteady, compressible flow are discretized in a generalized curvilinear coordinate system. The spatial derivatives of first-order are approximated by a fifth-order WENO scheme, the second-order derivatives by a sixth-order central scheme and the time derivatives by a fourth-order Runge-Kutta scheme. The focus of the investigation is on the demonstration of the applicability of DNS for simulating an airfoil flow at a moderate Reynolds number and to study upstream running pressure waves around the airfoil. At this Reynolds number, for a full resolution of all turbulent length scales, theoretically estimated numbers of mesh points are far away from realizable mesh sizes. In the present study seven three-dimensional meshes are compared where each of the two largest meshes consists of one billion mesh points (4096 × 512 × 512 and 8192 × 512 × 256). This mesh size is close to the practical limit of recent simulations since the numerical effort is about 16 · 106 core-hours on a supercomputer for one simulation. The other meshes are gradually coarsened resulting in only four million mesh points for the coarsest mesh. The mesh study is performed by the comparison of aerodynamical and turbulent quantities. On the one hand the main flow features are studied, which are mostly determined by large flow scales. Pressure waves are studied for all meshes, which are generated at the trailing edge, moving upstream. These pressure waves are analyzed in the vicinity of the airfoil. Acoustic phenomena in the far field are not studied. For the present study, a mesh with 67 million mesh points (M3) was sufficient to resolve the main flow features and flow phenomena caused by the pressure waves in the vicinity of the airfoil. On the other hand the turbulent intensities are compared, which are influenced by the smallest turbulent scales. The analysis of the wall units show that even the finest mesh spacings are slightly too large to fulfil the requirements of a fully-resolved DNS. In this context, the energy spectrum of the turbulent kinetic energy is useful to evaluate the quality of the turbulent boundary layer.