# zbMATH — the first resource for mathematics

Frequency-wavenumber spectral analysis of spatio-temporal flows. (English) Zbl 1404.76007
Summary: We propose a fully spatio-temporal approach for identifying spatially varying modes of oscillation in fluid dynamics simulation output by means of multitaper frequency-wavenumber spectral analysis. One-dimensional spectrum estimation has proven to be a valuable tool in the analysis of turbulence data applied spatially to determine the rate of energy transport between spatial scales, or temporally to determine frequencies of oscillatory flows. It also allows for the quantitative comparison of flow characteristics between two scenarios using a standard basis. It has the limitation, however, that it neglects coupling between spatial and temporal structures. Two-dimensional frequency-wavenumber spectral analysis allows one to decompose waveforms into standing or travelling variety. The extended higher-dimensional multitaper method proposed here is shown to have improved statistical properties over conventional non-parametric spectral estimators, and is accompanied by confidence intervals which estimate their uncertainty. Multitaper frequency-wavenumber analysis is applied to a canonical benchmark problem, namely, a direct numerical simulation of von Kármán vortex shedding off a square wall-mounted cylinder with two inflow scenarios with matching momentum-thickness Reynolds numbers $$\mathrm{Re}_\theta\approx 1000$$ at the obstacle. Frequency-wavenumber analysis of a two-dimensional section of these data reveals that although both the laminar and turbulent inflow scenarios show a turbulent $$-5/3$$ cascade in wavenumber $$(\nu)$$ and frequency $$(f)$$, the flow characteristics differ in that there is a significantly more prominent discrete harmonic oscillation near $$(f,\nu)=(0.2,0.21)$$ in wavenumber and frequency in the laminar inflow scenario than the turbulent scenario. This frequency-wavenumber pair corresponds to a travelling wave with velocity near one near the centre path of the vortex street.
##### MSC:
 76A02 Foundations of fluid mechanics 76D17 Viscous vortex flows
##### Keywords:
mathematical foundations; vortex flows; vortex streets
sapa
Full Text:
##### References:
 [1] Berkooz, G.; Holmes, P.; Lumley, J. L., The proper orthogonal decomposition in the analysis of turbulent flows, Annu. Rev. Fluid Mech., 25, 539-575, (1993) [2] Cressie, N., Transformations and the jackknife, J. Roy. Statist. Soc. B, 43, 177-182, (1981) · Zbl 0467.62035 [3] Cressie, N.; Wikle, C. K., Statistics for Spatio-Temporal Data, (2011), Wiley · Zbl 1273.62017 [4] Efron, B.; Stein, C., The jackknife estimate of variance, Ann. Stat., 9, 586-596, (1981) · Zbl 0481.62035 [5] Grünbaum, F. A., Toeplitz matrices commuting with tridiagonal matrices, Linear Algebr. Applics., 40, 25-36, (1981) · Zbl 0477.15005 [6] Hanssen, A., Multidimensional multitaper spectral estimation, Signal Process., 58, 327-332, (1997) · Zbl 1005.94515 [7] Hayashi, Y., Space – time spectral analysis of rotary vector series, J. Atmos. Sci., 36, 757-766, (1979) [8] Holmes, P.; Lumley, J. L.; Berkooz, G., Turbulence, Coherent Structures, Dynamical Systems and Symmetry, (1998), Cambridge University Press · Zbl 0923.76002 [9] Kirby, J. F., Estimation of the effective elastic thickness of the lithosphere using inverse spectral methods: the state of the art, Tectonophysics, 631, 87-116, (2014) [10] Mallows, C. L., Linear processes are nearly Gaussian, J. Appl. Probab., 4, 313-329, (1967) · Zbl 0155.25601 [11] Percival, D. B.; Walden, A. T., Spectral Analysis for Physical Applications: Multitaper and Conventional Univariate Techniques, (1993), Cambridge University Press · Zbl 0796.62077 [12] Pratt, R. W., The interpretation of space – time spectral quantities, J. Atmos. Sci., 33, 6, 1060-1066, (1976) [13] Rowley, C. W., Model reduction for fluids, using balanced proper orthogonal decomposition, Intl J. Bifurcation Chaos, 15, 997-1013, (2005) · Zbl 1140.76443 [14] Rowley, C. W.; Mezić, I.; Bagheri, S.; Schlatter, P.; Henningson, D. S., Spectral analysis of nonlinear flows, J. Fluid Mech., 641, 115-127, (2009) · Zbl 1183.76833 [15] Schmidt, O. T.; Towne, A.; Colonius, T.; Cavalieri, A. V. G.; Jordan, P.; Brès, G. A., Wavepackets and trapped acoustic modes in a turbulent jet: coherent structure eduction and global stability, J. Fluid Mech., 825, 1153-1181, (2017) · Zbl 1374.76074 [16] Simons, F. J.; Wang, D. V., Spatiospectral concentration in the Cartesian plane, Intl J. Geomath., 2, 1, 1-36, (2011) · Zbl 1226.42017 [17] Slepian, D., Prolate spheroidal wave functions, Fourier analysis and uncertainty IV, Bell System Tech. J., 43, 3009-3057, (1964) · Zbl 0184.08604 [18] Slepian, D., Prolate spheroidal wave functions, Fourier analysis, and uncertainty V: the discrete case, Bell System Tech. J., 57, 5, 1371-1429, (1978) · Zbl 0378.33006 [19] Von Storch, H.; Zwiers, F. W., Statistical Analysis in Climate Research, (1999), Cambridge University Press [20] Thomson, D. J., Spectrum estimation and harmonic analysis, Proc. IEEE, 70, 9, 1055-1096, (1982) [21] Thomson, D. J., Quadratic-inverse spectrum estimates: applications to paleoclimatology, Phil. Trans. R. Soc. Lond. A, 332, 539-597, (1990) · Zbl 0714.62110 [22] Thomson, D. J.1994An overview of multiple-window and quadratic-inverse spectrum estimation methods. Proc. ICASSPVI, 185-194; invited plenary lecture. [23] Thomson, D. J. & Chave, A. D.1991Jackknifed error estimates for spectra, coherences, and transfer functions. In Advances in Spectrum Analysis and Array Processing (ed. Haykin, S.), vol. 1, chap. 2, pp. 58-113. Prentice-Hall. [24] Vinuesa, R.; Schlatter, P.; Malm, J.; Mavriplis, C.; Henningson, D. S., Direct numerical simulation of the flow around a wall-mounted square cylinder under various inflow conditions, J. Turbul., 16, 555-587, (2015) [25] Wilks, D. S., Statistical Methods in the Atmospheric Sciences, (2011), Academic Press
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.