An Extension of the Time-Spectral Method to Overset Solvers

Abstract

Relative motion in the Cartesian or overset framework causes certain spatial nodes to move in and out of the physical domain as they are dynamically blanked by moving solid bodies. This poses a problem for the conventional Time-Spectral approach, which expands the solution at every spatial node into a Fourier series spanning the period of motion. The proposed extension to the Time-Spectral method treats unblanked nodes in the conventional manner but expands the solution at dynamically blanked nodes in a basis of barycentric rational polynomials spanning partitions of contiguously defined temporal intervals. Rational polynomials avoid Runge's phenomenon on the equidistant time samples of these sub-periodic intervals. Fourier-and rational polynomial-based di↵erenti-ation operators are used in tandem to provide a consistent hybrid Time-Spectral overset scheme capable of handling relative motion. The hybrid scheme is tested with a linear model problem and implemented within NASA's OVERFLOW Reynolds-averaged NavierStokes (RANS) solver. The hybrid Time-Spectral solver is then applied to inviscid and turbulent RANS cases of plunging and pitching airfoils and compared to time-accurate and experimental data. A limiter was applied in the turbulent case to avoid undershoots in the undamped turbulent eddy viscosity while maintaining accuracy. The hybrid scheme matches the performance of the conventional Time-Spectral method and converges to the time-accurate results with increased temporal resolution.

Recent advances in linear barycentric rational interpolation

“…Our examples should also have made clear that there basically is no limit to the number of nodes that can be accommodated. As a final remark, we wish to emphasize the simple Lagrange representation j f j (v) j (x) with the (v) j of (8.1): it allows for its straightforward use as an ansatz in the solution of functional equations and constitutes one further reason why we are confident that these interpolants will have a bright future in the solution of various problems in numerical analysis and scientific computing (see the very recent example in [35]). …”

Time-Spectral Rotorcraft Simulations on Overset Grids

“…Relative motion of overset grids produces spatial nodes that lack complete time-histories, obviating the ability to uniquely represent the aperiodic solutions with a Fourier series. To circumvent this result, the approach presented by the authors in [16] expands the solution at a given dynamically blanked node within intervals of consecutively unblanked time samples onto a global basis spanning the same sub-periodic interval. Spatial nodes may have multiple associated temporal intervals so solutions are expanded with independent bases within each partition.…”

“…Fully periodic intervals are still expanded and differentiated in the Fourier basis, resulting in a hybrid approach that employs the optimal basis available. Readers are referred to [16,30] for further detail.…”

“…However, careful implementation limits the storage requirements. Details of memory scaling for the OVERFLOW Time-Spectral solver are outlined in [16] and suggest that a large number of temporal modes can be applied without risk of exhausting the memory budget on suitably parallelized calculations.…”

Development of a High-Fidelity Time-Dependent Aero-Structural Capability for Analysis and Design

“…This has been achieved by analytically deriving the timespectral coefficients for the second derivative term, in addition to the coefficients typically used for the first time derivative term in traditional CFD applications. 31 The time-spectral solver makes use of an approximate factorization scheme [32][33][34] where the spatial factor is solved using the same direct or iterative solvers described previously. The in-house structural model has been tested on various representative configurations including various brick element solid models, the CRM shell-based wing box model, and a shell-based wind turbine structural model, as depicted in Figure 2.…”