In this paper, the trailing edge noise generated by a 2D airfoil around the critical angle of attack for vortex shedding is numerically investigated using an in-house code with high accuracy and efficiency. In the present method, a fourth-order upwind compact finite-difference scheme with dispersion relation preserving (DRP) property is applied for the convection terms, and a fourth-order Runge-Kutta scheme is used for temporal discretization. The reflection of sound on the boundary is suppressed with Navier-Stokes characteristics boundary condition (NSCBC). To improve computational efficiency, a novel parallel computing strategy for the high-order compact schemes is employed. Thus, direct numerical simulation (DNS) can be realized for the flows of low Reynolds number (Re), while implicit large eddy simulation (ILES) would be carried for the flows of high Reynolds number. The present numerical method is validated by comparing the lift coefficient, drag coefficient and Strouhal number (St) to the previous publications. Based on the high accuracy and high-fidelity method, the flow field and sound field of a two-dimensional NACA0012 airfoil around critical angle of attack (AoA) at Re = 1000 are simultaneously solved. The results indicate that sound source is dipole centered at the surface of the airfoil at vortex shedding frequency, and is dipole, quadrupole or more complex sources located at the wake close to the trailing edge at higher order frequencies. These findings will help to improve understanding about the generation and propagation mechanisms of trailing edge noises at low Reynolds number.

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In this paper, based on the boundary approximation approach for parallelization of the compact difference schemes, a novel strategy for the sub-domain boundary approximation schemes is proposed to maintain consistent accuracy and dispersion with the compact scheme in the interior points. In this strategy, not only the order of accuracy of the sub-domain boundary scheme is the same as the interior scheme, but the coefficient of the first truncation error term is also equal to that of the internal scheme. Furthermore, to realize the consistent dispersion performance for a class of high order upwind compact schemes, which usually include two expressions, we modify the opposite expression to be the sub-domain boundary scheme. As an example of application, the present strategy is applied to a fourth-order upwind compact scheme, and its accuracy is verified by a numerical test. The resolution and efficiency of the newly proposed parallel method are examined by four numerical examples, including propagation of a wave-packet, convection of isentropic vortex, Rayleigh–Taylor instability problems, and propagation of Gauss pulse. The results obtained demonstrate that the present strategy for compact difference schemes has the feasibility to solve the flow problems with high accuracy, resolution and efficiency in parallel computation.

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In this paper, a high‐order compact finite difference algorithm is established for the stream function‐velocity formulation of the two‐dimensional steady incompressible Navier‐Stokes equations in general curvilinear coordinates. Different from the previous work, not only the stream function and its first‐order partial derivatives but also the second‐order mixed partial derivative is treated as unknown variable in this work. Numerical examples, including a test problem with an analytical solution, three types of lid‐driven cavity flow problems with unusual shapes and steady flow past a circular cylinder as well as an elliptic cylinder with angle of attack, are solved numerically by the newly proposed scheme. For two types of the lid‐driven trapezoidal cavity flow, we provide the detailed data using the fine grid sizes, which can be considered the benchmark solutions. The results obtained prove that the present numerical method has the ability to solve the incompressible flow for complex geometry in engineering applications, especially by using a nonorthogonal coordinate transformation, with high accuracy.@en