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.
In response to the stagnation of computer microprocessor speeds over the past decade, the design emphasis of novel supercomputing architectures has focused primarily on increasing the overall number of available cores and reducing communication bottlenecks. Typically, flow solvers have been able to achieve parallel efficiency using domain decomposition, but this approach has the natural limitation that saturation will manifest itself on a finite number of cores at which point parallel speedup stalls and eventually deteriorates. In order to improve parallel scalability we seek to leverage the existing knowledge base on spatial decomposition, while attempting to exploit additional parallelism in the temporal dimension. Specifically, we explore the case of time periodic flows using Parallel-in-Time (PinT) variants of the Time-Spectral (TS) method. A framework employing a Pythonbased infrastructure is described including a standalone library that can be coupled to existing flow solvers in order to facilitate PinT calculations. A model problem of a periodic density pulse is used to examine the different discretization options. Implications for application to wind turbines and rotors are addressed.
The Time-Spectral method is derived as a Fourier collocation scheme and applied to NASA's overset Reynolds-averaged Navier-Stokes (RANS) solver OVERFLOW. The paper outlines the Time-Spectral OVERFLOW implementation. Successful low-speed laminar plunging NACA 0012 airfoil simulations demonstrate the capability of the Time-Spectral method to resolve the highly-vortical wakes typical of more expensive three-dimensional rotorcraft configurations. Dealiasing, in the form of spectral vanishing viscosity (SVV), facilitates the convergence of Time-Spectral calculations of high-frequency flows. Finally, simulations of the isolated V-22 Osprey tiltrotor for both hover and forward (edgewise) flight validate the three-dimensional Time-Spectral OVERFLOW implementation. The Time-Spectral hover simulation matches the time-accurate calculation using a single harmonic. Significantly more temporal modes and SVV are required to accurately compute the forward flight case because of its more active, high-frequency wake.
Overset and Cartesian solvers have traditionally been employed to efficiently resolve unsteady flows with conventional time-marching methods. Temporal pseudospectral schemes have demonstrated the ability to dramatically reduce the computational effort required to simulate the important subclass of time-periodic phenomena. Incorporation of the Time-Spectral method within these approaches is desirable, but direct application is infeasible. Relative motion in an overset framework introduces the dynamic blanking of spatial nodes which move interior to the impermeable boundaries of solid bodies; the solution at these nodes is therefore undefined over specific intervals of time. This proves problematic for the the conventional Time-Spectral approach, as it expands the temporal variation at every node as an infinitely-supported Fourier series. An extension of the Time-Spectral method is outlined that treats the solution at dynamically-blanked nodes in an alternative manner. The standard Fourier differentiation operator and differentiation operators derived from barycentric rational interpolants are applied in tandem providing a hybrid Time-Spectral scheme capable of consistently resolving relative motion. The proposed scheme is applied to the cases of high-amplitude and high-frequency plunging airfoils, and the results compared against corresponding simulations obtained with a traditional time marching scheme. The results demonstrate that the hybrid scheme mirrors the performance of the conventional Time-Spectral method, and monotonically converges to corresponding high-resolution, time-accurate simulations with increasing temporal modes.
Micro air vehicle design is an emerging area in aeronautics requiring accurate solutions of flapping wing flight in combination with numerical optimization. The highly unsteady and complex nature of the flow fields associated with flapping wings can translate to massive computational costs. A large-scale optimization over a significant design space can result in hundreds to thousands of objective evaluations making it critical to ensure maximal efficiency while maintaining the requisite level of accuracy. Full-scale optimization can be used for both the trajectory and geometry of the wings but this work focuses on a reduced two-dimensional problem with fixed geometry. A series of analyses are performed to isolate the minimum spatial and temporal requirements to accurately and efficiently model a twodimensional flapping rigid airfoil subjected to sinusoidal pitching and plunging motion. NASA's highly parallelized overset flow solver OVERFLOW is used for all of the computations on a rigid NACA 0012 airfoil. Grid-spacing and domain size parameters are explored using spatial analysis. The physical time-step with implicit sub-iterations in the context of OVERFLOW's dual time-stepping scheme is explored using temporal analysis. Finally, a method for evaluating periodic convergence in the force and moment time-history signals is discussed. Nomenclature f Flapping frequency, Hz U ref Reference velocity k Reduced frequency, 2πf c U ref
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