Experimental investigation of the turbulent cascade development by injection of single large-scale Fourier modes
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RESEARCH ARTICLE
Experimental investigation of the turbulent cascade development by injection of single large‑scale Fourier modes Margherita Dotti1,2 · Rasmus K. Schlander1,3 · Preben Buchhave4 · Clara M. Velte1 Received: 27 April 2020 / Revised: 4 July 2020 / Accepted: 21 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The current work presents an experimental investigation of the dynamic interactions between flow scales caused by repeated actions of the nonlinear term of the Navier–Stokes equation. Injecting a narrow band oscillation, representing a single Fourier mode, into a round jet flow allows the measurement of the downstream generation and development of higher harmonic spectral components and to measure when these components are eventually absorbed into fully developed turbulence. Furthermore, the dynamic evolution of the measured power spectra observed corresponds well to the measured cascaded delays reported by others. Closely matching spectral development and cascade delays have also been derived directly from a one-dimensional solution of the Navier–Stokes equation described in a companion paper. The results in the current work provide vital information about how initial conditions influence the development of the shape of the spectrum and about the extent of the timescales in the triad interaction process, which should be of significance to turbulence modelers.
Financial support from the Poul Due Jensen Foundation (Grundfos Foundation) for this research is gratefully acknowledged. Grant Number 2018-039.
1
Department of Mechanical Engineering, Technical University of Denmark, Nils Koppels Allé, Bldg. 403, 2800 Kgs. Lyngby, Denmark
2
Rasmus K. Schlander [email protected]
Present Address: Department of Chemical and Biochemical Engineering, Technical University of Denmark, Søltofts Plads 228A, 2800 Kgs. Lyngby, Denmark
3
Preben Buchhave [email protected]
Present Address: Department of Aeronautics, Imperial College, London SW7 2AZ, UK
4
Intarsia Optics, Sønderskovvej 3, 3460 Birkerød, Denmark
* Clara M. Velte [email protected] Margherita Dotti [email protected]
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Vol.:(0123456789)
214
Page 2 of 13
Experiments in Fluids
(2020) 61:214
Graphic abstract
1 Introduction Most of the twentieth century knowledge of turbulence is founded on the equilibrium gas dynamics analogy to small and intermediate scales of turbulence, as put forth by Kolmogorov (1991). Indeed, almost all turbulence theories and models are in one way or another based on the Kolmogorov theory of turbulence (which the authors will hereafter refer to as the K41 theory). In particular, many approaches rely on the existence of a continuous exchange of turbulent energy from small to large wavenumbers, the Richardson cascade, where predominantly local interactions between scales are assumed to occur (L’vov and Procaccia 1997; Richardson and Lynch 2007; Kraichnan 1959). By taking the Fourier transform of the non-linear advection term in the Navier–Stokes equation, one
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