## Frequency–wavenumber spectral analysis of spatio-temporal flows

## Abstract

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$$Re_{\unicode[STIX]{x1D703}}\approx 1000$$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$$cascade in wavenumber ($$\unicode[STIX]{x1D708}$$) and frequency ($$f$$), the flow characteristics differ in that there is a significantly more prominent discrete harmonic oscillation near$$(f,\unicode[STIX]{x1D708})=(0.2,0.21)$in wavenumber and frequency in the laminar inflow scenario than the turbulent scenario. Here, this frequency–wavenumber pair corresponds to a travelling wave with velocity near one near the centre path of the vortex street.

- Authors:

- Argonne National Lab. (ANL), Lemont, IL (United States)
- Argonne National Lab. (ANL), Lemont, IL (United States); Univ. of Chicago, Chicago, IL (United States)

- Publication Date:

- Research Org.:
- Argonne National Lab. (ANL), Argonne, IL (United States)

- Sponsoring Org.:
- National Science Foundation (NSF); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)

- OSTI Identifier:
- 1461555

- Grant/Contract Number:
- AC02-06CH11357

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of Fluid Mechanics

- Additional Journal Information:
- Journal Volume: 848; Journal ID: ISSN 0022-1120

- Publisher:
- Cambridge University Press

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; Coherent Structures; Direct Numerical Simulation; Fourier Analysis; Power Spectrum; Wall-Mounted Square Cylinder; von Karman Vortex Street

### Citation Formats

```
Geoga, Christopher J., Haley, Charlotte L., Siegel, Andrew R., and Anitescu, Mihai. Frequency–wavenumber spectral analysis of spatio-temporal flows. United States: N. p., 2018.
Web. doi:10.1017/jfm.2018.366.
```

```
Geoga, Christopher J., Haley, Charlotte L., Siegel, Andrew R., & Anitescu, Mihai. Frequency–wavenumber spectral analysis of spatio-temporal flows. United States. doi:10.1017/jfm.2018.366.
```

```
Geoga, Christopher J., Haley, Charlotte L., Siegel, Andrew R., and Anitescu, Mihai. Fri .
"Frequency–wavenumber spectral analysis of spatio-temporal flows". United States. doi:10.1017/jfm.2018.366. https://www.osti.gov/servlets/purl/1461555.
```

```
@article{osti_1461555,
```

title = {Frequency–wavenumber spectral analysis of spatio-temporal flows},

author = {Geoga, Christopher J. and Haley, Charlotte L. and Siegel, Andrew R. and Anitescu, Mihai},

abstractNote = {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$Re_{\unicode[STIX]{x1D703}}\approx 1000$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 ($\unicode[STIX]{x1D708}$) and frequency ($f$), the flow characteristics differ in that there is a significantly more prominent discrete harmonic oscillation near$(f,\unicode[STIX]{x1D708})=(0.2,0.21)$in wavenumber and frequency in the laminar inflow scenario than the turbulent scenario. Here, this frequency–wavenumber pair corresponds to a travelling wave with velocity near one near the centre path of the vortex street.},

doi = {10.1017/jfm.2018.366},

journal = {Journal of Fluid Mechanics},

number = ,

volume = 848,

place = {United States},

year = {2018},

month = {6}

}