Adaptive Harmonic Balance Solutions to Euler's Equation

Maple, Raymond C.; King, Paul I.; Oxley, Mark E.

doi:10.2514/2.7316

Cited by 13 publications

(8 citation statements)

References 8 publications

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“…if a node alternates between blanked and unblanked at every consecutive time instance, the solution will be considered constant in time at each of the nodes and therefore the time derivative will be approximated as zero universally. This may prove problematic, but perhaps such situations would require higher temporal resolution to resolve the high-frequency motion, suggesting the incorporation of temporal adaptation [26,27]…”

Section: Dynamic Hole Cutting and The Hybrid Time-spectral Methodsmentioning

confidence: 99%

Time-Spectral Method for Overlapping Meshes

2017

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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.

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Section: Dynamic Hole Cutting and The Hybrid Time-spectral Methodsmentioning

confidence: 99%

Time-Spectral Method for Overlapping Meshes

2017

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“…With the goal of improving efficiency, future work includes parallelization in the temporal dimension [24] and a zone-based frequency adaptation strategy to more optimally allocate the temporal degrees of freedom to more dynamic regions of the flow. Frequency adaptation in the context of temporal Fourier schemes has been developed [46,47,48,49,50,51,52], however refinement on a point-by-point basis is not practical in the context of overset grids. Rather, the adaptive mesh refinement model can be employed in time so that every grid has an associated number of time-instances that can be refined as the simulation advances.…”

Section: Summary and Future Workmentioning

confidence: 99%

Time-Spectral Rotorcraft Simulations on Overset Grids

Leffell

Murman

Pulliam

2014

32nd AIAA Applied Aerodynamics Conference

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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.

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“…The derivation of the NLFD method starts with equation (9) and the assumption thatw = Ωw and R can be represented by separate Fourier series:…”

Section: B Temporal Discretizationmentioning

confidence: 99%

“…In the case of real problems, the Fourier coefficients for negative wave numbers are simply the complex conjugates of the coefficients for the positive wave numbers and can be eliminated in computation. The Fourier representations are then substituted into the semi-discrete form of the governing equations as described in equation (9) to yield,…”

Section: B Temporal Discretizationmentioning

confidence: 99%

“…The primary concept of the pNLFD approach is to augment the Fourier series in each control volume independent of the others based on the level of the spectral energy of the highest mode. An adaptive non-linear frequency domain approach was initially proposed by Maple et al [8][9][10] The authors demonstrated their approach on a quasi-one-dimensional flow and showed a significant improvement in memory usage. 8 The extension of the adaptive concept to 2D viscous flows was performed by the present authors.…”

Section: Introductionmentioning

confidence: 99%

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An Implicit Adaptive Non-Linear Frequency Domain Method (pNLFD) for Viscous Periodic Steady State Flows on Deformable Grids

Mohamadi

Nadarajah

2011

49th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition

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In the present study, an implicit and adaptive Nonlinear Frequency Domain method (pNLFD) has been implemented to the Navier-Stokes equations on deformable grids. Although the computational time for periodic flows is drastically reduced by using the NLFD approach over classical time marching schemes, implementing the pNLFD concept leads to an even faster numerical algorithm. Besides that, the need for a large amount of memory, which is the main disadvantage of the NLFD method, is resolved in the present pNLFD approach. Moreover, the concept of dynamic or moving/deformable grid, which is a need in many problems dealing with periodic flows, is extended to the pNLFD method. Finally, in order to accelerate the convergence, the nonlinear LU-SGS technique which is an implicit time marching method, is implemented. In the LU-SGS technique the cells are treated locally, hence its implementation is quite suitable for the pNLFD method, where different cells have different number of modes and therefore has to be treated individually. Results are presented for 2D stationary, oscillating and pitching cylinders and are compared with previous numerical results as well as experimental data.

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Adaptive Harmonic Balance Solutions to Euler's Equation

Cited by 13 publications

References 8 publications

Time-Spectral Method for Overlapping Meshes

Time-Spectral Method for Overlapping Meshes

Time-Spectral Rotorcraft Simulations on Overset Grids

An Implicit Adaptive Non-Linear Frequency Domain Method (pNLFD) for Viscous Periodic Steady State Flows on Deformable Grids

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