ALBERT

Beschreibung: A mapped Chebyshev pseudospectral method is extended to solve three-dimensional unsteady flow problems. As the classical Chebyshev spectral approach can lead to numerical instabilities due to ill conditioning of the spectral matrix, the Chebyshev points are evenly redistributed over the domain by an inverse sine mapping function. The mapped Chebyshev pseudospectral method can be used as an alternative time-spectral approach that uses a Chebyshev collocation operator to approximate the time derivative terms in the unsteady flow governing equations, and the method can make general applications to both nonperiodic and periodic problems. In this study, the mapped Chebyshev pseudospectral method is employed to solve three-dimensional periodic problem to verify the spectral accuracy and computational efficiency with those of the Fourier pseudospectral method and the time-accurate method. The results show a good agreement with both of the Fourier pseudospectral method and the time-accurate method. The flow solutions also demonstrate a good agreement with the experimental data. Similar to the Fourier pseudospectral method, the mapped Chebyshev pseudospectral method approximates the unsteady flow solutions with a precise accuracy at a considerably effective computational cost compared to the conventional time-accurate method.

Beschreibung: Unsteady helicopter rotor flows are solved by a Chebyshev pseudospectral method with overset mesh topology which employs Chebyshev polynomials for solution approximation and a Chebyshev collocation operator to represent the time derivative term of the unsteady flow governing equations. Spatial derivative terms of the flux Jacobians are discretized implicitly while the Chebyshev spectral derivative term is treated in explicit form. Unlike the Fourier spectral method, collocation points of standard Chebyshev polynomials are not evenly distributed and heavily clustered near the extremities of the time interval, which makes the spectral derivative matrix ill-conditioned and deteriorates the stability and convergence of the flow solution. A conformal mapping of an arcsin function is applied to redistribute those points more evenly and thus to improve the numerical stability of the linear system. A parameter study on the condition number of the spectral derivative matrix with respect to the control parameters of the mapping function is also carried out. For the validation of the proposed method, both periodic and nonperiodic unsteady flow problems were solved with two-dimensional problems: an oscillating airfoil with a fixed frequency and a plunging airfoil with constant plunging speed without considering gravitational force. Computation results of the Chebyshev pseudospectral method showed excellent agreements with those of the time-marching computation. Subsequently, helicopter rotor flows in hovering and nonlifting forward flight are solved. Moving boundaries of the rotating rotor blades are efficiently managed by the overset mesh topology. As a set of subgrids are constructed only one time at the beginning of the solution procedure corresponding to the mapped Chebyshev collocation points, computation time for mesh interpolation of hole-cutting between background and near-body grids becomes drastically reduced when compared to the time-marching computation method where subgrid movement and the hole-cutting need to be carried out at each physical time step. The number of the collocation points was varied to investigate the sensitivity of the solution accuracy, computation time, and memory. Computation results are compared with those from the time-marching computation, the Fourier spectral method, and wind-tunnel experimental data. Solution accuracy and computational efficiency are concluded to be great with the Chebyshev pseudospectral method. Further applications to unsteady nonperiodic problems will be left for future work.