Dynamic Triad Interactions and Evolving Turbulence -- Part 1: 4D Modal Interactions
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arXiv
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| Natura: | Preprint |
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2019
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| _version_ | 1866911302582009856 |
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| author | Buchhave, Preben Velte, Clara M. |
| author_facet | Buchhave, Preben Velte, Clara M. |
| contents | We investigate the effect of a four-dimensional Fourier transform on the formulation of the Navier-Stokes equation in Fourier space and the way the energy is transferred between Fourier components. Since time in a sampled high intensity turbulence must be considered a stochastic variable in the energy exchange between scales, we refer to these dynamic triad interactions as modal interactions, rather than the commonly referred to triad interactions in the classical 3-dimensional analysis. The inclusion of time as a parameter broadens the phase match condition from the classical one, $Δ\bm{k} \cdot \bm{r} = \left [ \bm{k} - (\bm{k}_1 + \bm{k}_2 ) \right ] \cdot \bm{r}$, to the more general formulation that also includes temporal frequencies: $Δ\bm{k} \cdot \bm{r} - Δωt = \left [ \bm{k} - (\bm{k}_1 + \bm{k}_2 ) \right ] \cdot \bm{r} - \left [ ω- \left (ω_1 + ω_2 \right ) \right ] t$. This renders possible the occurrence of `delayed' and `advanced' interactions. The observation that mismatches in the wavevector triadic interactions may be compensated by a corresponding mismatch in the frequencies supports the empirically deduced delayed interactions reported in [Josserand \textit{et al.}, \textit{J. Stat. Phys.} (2017)]. These results explain the occurrence and inherent time development of the so-called Richardson cascade and also how finite temporal overlap of wave components can result in significant non-local interactions and consequently non-equilibrium turbulence, e.g., fractal grid generated turbulence. The consequences of including time as a parameter in practical experiments or simulations in terms of limited resolution, domain size etc. are treated in the companion paper (Part 2) of the present work. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1906_04756 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Dynamic Triad Interactions and Evolving Turbulence -- Part 1: 4D Modal Interactions Buchhave, Preben Velte, Clara M. Fluid Dynamics We investigate the effect of a four-dimensional Fourier transform on the formulation of the Navier-Stokes equation in Fourier space and the way the energy is transferred between Fourier components. Since time in a sampled high intensity turbulence must be considered a stochastic variable in the energy exchange between scales, we refer to these dynamic triad interactions as modal interactions, rather than the commonly referred to triad interactions in the classical 3-dimensional analysis. The inclusion of time as a parameter broadens the phase match condition from the classical one, $Δ\bm{k} \cdot \bm{r} = \left [ \bm{k} - (\bm{k}_1 + \bm{k}_2 ) \right ] \cdot \bm{r}$, to the more general formulation that also includes temporal frequencies: $Δ\bm{k} \cdot \bm{r} - Δωt = \left [ \bm{k} - (\bm{k}_1 + \bm{k}_2 ) \right ] \cdot \bm{r} - \left [ ω- \left (ω_1 + ω_2 \right ) \right ] t$. This renders possible the occurrence of `delayed' and `advanced' interactions. The observation that mismatches in the wavevector triadic interactions may be compensated by a corresponding mismatch in the frequencies supports the empirically deduced delayed interactions reported in [Josserand \textit{et al.}, \textit{J. Stat. Phys.} (2017)]. These results explain the occurrence and inherent time development of the so-called Richardson cascade and also how finite temporal overlap of wave components can result in significant non-local interactions and consequently non-equilibrium turbulence, e.g., fractal grid generated turbulence. The consequences of including time as a parameter in practical experiments or simulations in terms of limited resolution, domain size etc. are treated in the companion paper (Part 2) of the present work. |
| title | Dynamic Triad Interactions and Evolving Turbulence -- Part 1: 4D Modal Interactions |
| topic | Fluid Dynamics |
| url | https://arxiv.org/abs/1906.04756 |