Adaptive Harmonic Balance Solutions to Euler's Equation

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

doi:10.2514/6.2002-2953

Cited by 4 publications

(8 citation statements)

References 7 publications

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“…(14a) forn n 1 ðx; sÞ subject to the initial conditionn n 1 ðx; 0Þ ¼n nðx; 0Þ, the initial condition for Eq. (13). The approximate solution is then obtained by solving Eq.…”

Section: Split-domain Harmonic Balance Form Of the Governing Equationsmentioning

confidence: 99%

“…Numerical stability considerations require a reduced time step when solving Eq. (13) for large N [9,12]. To improve the stability characteristics of the numerical solution, Eq.…”

Section: Split-domain Harmonic Balance Form Of the Governing Equationsmentioning

confidence: 99%

“…To achieve shorter computation times for this class of problem, CFD techniques have been developed that take advantage of the time-periodic nature of the flow. These include the time-linearization technique [1][2][3][4], the time-averaged nonlinear harmonic technique [5][6][7], and the harmonic balance technique [8][9][10][11][12][13].…”

Section: Introductionmentioning

confidence: 99%

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Adaptive harmonic balance method for nonlinear time-periodic flows

Maple

King

Orkwis

et al. 2004

Journal of Computational Physics

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“…(14a) forn n 1 ðx; sÞ subject to the initial conditionn n 1 ðx; 0Þ ¼n nðx; 0Þ, the initial condition for Eq. (13). The approximate solution is then obtained by solving Eq.…”

Section: Split-domain Harmonic Balance Form Of the Governing Equationsmentioning

confidence: 99%

“…Numerical stability considerations require a reduced time step when solving Eq. (13) for large N [9,12]. To improve the stability characteristics of the numerical solution, Eq.…”

Section: Split-domain Harmonic Balance Form Of the Governing Equationsmentioning

confidence: 99%

See 1 more Smart Citation

Adaptive harmonic balance method for nonlinear time-periodic flows

Maple

King

Orkwis

et al. 2004

Journal of Computational Physics

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“…A pitching airfoil allows for simulations with both rigid-and relative-body motion so that the temporal discretization can be isolated and compared directly. The airfoil pitches about its quarter chord with incidence, ↵ (t), defined as ↵ (t) = 0.016 + 2.51 sin (kt) (14) with reduced frequency k = 0.1627 radians per non-dimensional time unit and free stream Mach number M 1 = 0.755. A hierarchy of Time-Spectral simulations are computed using an increasing number of modes (N 2 {3, 5, 9, 17, 33}) to investigate the performance of the scheme with increased temporal resolution and ensure that spurious higher-frequency modes are not generated and propagated into the solution.…”

Section: Transonic Agard 702 Pitching Airfoilmentioning

confidence: 99%

“…Additionally, adjoint-based optimization techniques [10,11] can be applied to provide the ability to perform design optimization without resorting to costly unsteady adjoint methods. Frequency-adaptive methods [12,13,14] can o↵er even greater e ciency by refining the the number of modes resolved at each grid point to the frequency content in its solution.…”

Section: Introductionmentioning

confidence: 99%

An Extension of the Time-Spectral Method to Overset Solvers

Leffell

Murman

Pulliam

2013

51st AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition

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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.

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Prediction of dynamic derivatives for air-breathing hypersonic vehicle using a harmonic balance method

Chai

Liu

Yang

et al. 2019

Proceedings of the Institution of Mechanical Engineers, Part G:

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The harmonic balance method is an efficient frequency-domain approach for computing periodically unsteady flows. Applying harmonic balance method to the rapid calculation of forced periodic motions, an efficient prediction method for dynamic derivatives is established based on the Etkin unsteady aerodynamic model. The method is firstly validated by a standard model of the hypersonic missile hyper ballistic shape, and the harmonic balance method results show great consistency with both time-domain results and experimental data, consolidating the efficiency of our numerical algorithm. The method is subsequently applied for an air-breathing hypersonic vehicle to investigate the capability of the harmonic balance method in modeling unsteady flows around complex geometric configurations. Comparisons with the results of the time-domain method demonstrate that the harmonic balance method is capable of resolving such complicated flowfield with both accuracy and efficiency. To be specific, for the air-breathing hypersonic vehicle, the maximum difference of the dynamic derivatives calculated by the harmonic balance method and the time-domain method is only 6.56%, and the harmonic balance method shows at least an order of magnitude efficiency over the general time-domain method in the current study.

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Adaptive Harmonic Balance Solutions to Euler's Equation

Cited by 4 publications

References 7 publications

Adaptive harmonic balance method for nonlinear time-periodic flows

Adaptive harmonic balance method for nonlinear time-periodic flows

An Extension of the Time-Spectral Method to Overset Solvers

Prediction of dynamic derivatives for air-breathing hypersonic vehicle using a harmonic balance method

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