A spatio-temporal random synthetic turbulent velocity field: The underlying Gaussian structure

Abstract

We develop, simulate and extend an initial proposition by Chaves et al. concerning a random incompressible vector field able to reproduce key ingredients of three-dimensional turbulence in both space and time. In this article, we focus on the important underlying Gaussian framework. Presently, the statistical spatial structure of this velocity field is consistent with a divergence-free fractional Gaussian vector field that encodes all known properties of homogeneous and isotropic fluid turbulence at a given finite Reynolds number, up to second-order statistics. The temporal structure of the velocity field is introduced through a stochastic evolution of the respective Fourier modes. In the simplest picture, Fourier modes evolve according to an Ornstein-Uhlenbeck process, where the characteristic time scale depends on the wave-vector amplitude. For consistency with direct numerical simulations (DNSs) of the Navier-Stokes equations, this time scale is inversely proportional to the wave vector amplitude. As a consequence, the characteristic velocity that governs the eddies is independent of their size and is related to the velocity standard deviation, which is consistent with some features of the so-called sweeping effect. To ensure differentiability in time while respecting the Markovian nature of the evolution, we use the methodology developed by Viggiano et al. to propose a fully consistent stochastic picture. We finally derive analytically all statistical quantities in a continuous setup and develop precise and efficient numerical schemes of the corresponding periodic framework. Both exact predictions and numerical estimations of the model are compared to DNSs provided by the Johns Hopkins database.

Figures (4)

• Figure 1: Comparison of the instantaneous and statistical spatial structure of a DNS velocity field $\bu^{\text{\tiny{DNS}},\nu}(t,\bx)$ and of a realization of the model which coincides with a fractional Gaussian vector field at a given instant of time in the statistically stationary regime. (a) Instantaneous spatial profile of the norm of the DNS velocity field $|\bu^{\text{\tiny{DNS}},\nu}(t,\bx)|$ in the plane $y=0$, at the initial time of the DNS dataset. (b) Same as (a) for the model, we used the same colorbar in both representations. (c) Estimation of the longitudinal PSDs $E_\nu^{\text{\tiny{E,long}}}(k)$\ref{['eq:DefPSD1dLong']} based on the variance of the Fourier modes of the one-dimensional discrete Fourier transform. Statistics (DNS in orange and model in blue) are estimated by averaging both in space and time. We superimpose with a black dashed line the inviscid limit of the functional form provided in \ref{['eq:ModelPSD1dLong']}, corresponding to the power-law $D_2|\bk|^{-5/3}$. (d) Statistical estimation of the second-order longitudinal structure function $S_{2,\nu}^{\text{\tiny{long}}}(\ell)$\ref{['eq:DefS2FiniteVisc']}, same colors as (c). We superimpose with dashed lines the inviscid predictions in the three ranges of scales of interest: (i) at large scales of order $L_{\text{\tiny{tot}}}$, $S_2^{\text{\tiny{long}}}(\ell)$ reaches the plateau $\frac{2}{3}\sigma^2$, where $\sigma^2$ is related to the free parameter $D_2$ according to \ref{['eq:RelD2VelVariance']}, (ii) in the intermediate inertial range of scales, we represent the prediction made in \ref{['eq:DefS2Holder13LongWithCk']}, and (iii) by the smooth behavior $\propto \ell^2$ in the dissipative range, as it is predicted in \ref{['eq:TaylorS2Long']}.
• Figure 2: Temporal correlation structure of the longitudinal velocity Fourier modes \ref{['eq:Def1dLongFourierModes']}. (a) Longitudinal correlation function $C_{\text{\tiny{long}}}^{\nu}(\tau,k_n)$\ref{['eq:DefTimeCorrLongFM']} of the DNS velocity field $\widehat{u}_{\text{\tiny{long}}}^{\text{\tiny{DNS}},\nu}(t,k_n)$, for $k_n L_{\text{\tiny{tot}}}=42,\ 75,\ 107,\ 141,\ 173,\ 205$. The characteristic large time scale $T_E^{\text{\tiny{DNS}}} = 1.99$ is defined in the https://turbulence.idies.jhu.edu/docs/isotropic/README-isotropic.pdf of the DNS. Inset: Re-scaled correlation functions in semi-log scale for the same wave numbers $k_n$, with $D_3$ provided in the text and the theoretical gaussian expression in solid line by GorBal21. (b) Similar representation of the longitudinal correlation function $C_{\text{\tiny{long}}}^{(4),\nu}(\tau,k_n)$\ref{['eq:DefTimeCorrLongFM']} as in (a) but for the Gaussian model $\widehat{u}_{\text{\tiny{long}}}^{(4),\nu}(t,k_n)$. Solid lines represent theoretical predictions while dot-lines are numerical estimations for the same wavenumbers $k_n$ as (a). The characteristic large time scale $T_E = 1.43$ is defined by $T_E := \sqrt{3}\frac{L_{\text{\tiny{int}}}}{\sigma^\nu}$ with $L_{\text{\tiny{int}}}$ being the integral time scale defined in \ref{['eq:RelatLtoLint']} Inset: Re-scaled correlation functions in semi-log scale for the same wave numbers $k_n$, with the same $D_3$ as for DNS and provided in the text. (c) Numerical estimation of the characteristic time scale $T_{k_n}$ of the time correlation functions displayed in (a) and (b). We superimpose the numerical estimation of $T_{k_n}$ with a dashed black line given by the large $k$ asymptotic $(D_3 k)^{-1}$ of the theoretical expression \ref{['eq:DefTkBeta']}, with relevant free parameters provided in the text. Inset: re-scaled estimated $T_{k_n}$ by $(D_3 k)^{-1}$, the asymptotical behavior of $T_k$ at large $k$.
• Figure 3: Direct comparison of the temporal structure of the DNS and modeled velocity fields, and estimation of PSDs and second-order structure functions. (a) Spatio-temporal representation of the DNS velocity field $|\bu^{\text{\tiny{DNS}},\nu}(t,\bx)|$ along the spatial line $\bx\in([-\pi,\pi],0,0)$ and across time $t\in[0,5027\Delta t]$ where the time stepping $\Delta t$ is provided by the Hopkins database. We use the same characteristic large time scale $T_E^{\text{\tiny{DNS}}}$ as in Fig. \ref{['Fig:TempCorr3D']} to adimensionalize time. (b) Similar representation as in panel (a) but for the model $|\bu^{(4),\nu}(t,\bx)|$ with a characteristic large time scale $T_E$ of the order of $T_E^{\text{\tiny{DNS}}}$. We use the same colorbar for panels (a) and (b). (c) Estimation of the temporal power spectral density $E_\nu^{\text{\tiny{T}}}(\omega)$\ref{['eq:DefTimeSpectrum']} obtained as the variance of the temporal Fourier modes (see text). We use orange for DNS and blue for the model. The dashed black line corresponds to the power-law given in \ref{['eq:TimeSpectrumFNOmegaInfty']} without any fitting parameters. (d) Estimation of the second-order temporal structure function $S_2^{\text{\tiny{T}},\nu}(\tau)$ (given by \ref{['eq:TemporalS2']} in the inviscid limit), same colors as in (c). Dashed black lines corresponds to two times the variance at large time lags, to \ref{['eq:TemporalS2AnyHEquiv1FN']} in the inertial range, and to \ref{['eq:TemporalS2AnyHViscFN']} in the dissipative range.
• Figure 4: One dimensional temporal Fourier mode . (a) 1D Fourier mode for the set of Fourier modes $k_n L_{\text{\tiny{tot}}} =7,\ 15,\ 31,\ 63,\ 127,\ 255$. Solid line curves represent the theoretical predictions $F^{(N)}\lp {\tau}/{T_k}\rp$\ref{['eq:FNTAU']} entering in \ref{['eq:1DTempCorr']} for the aforementioned Fourier modes while dots are numerical estimations. This simulation has been lead with the same parameters as the 3D one except for the number of layer $N$, here equal to $N=8$. Time is rescaled by $T_k$ in the inset showing that all curves collapse onto a single nearly Gaussian decreasing function. (b) Pointwise convergence of the 1D Fourier mode onto a Gaussian for a single Fourier mode $k_n L_\text{\tiny{tot}} = 15$ as a function of $\tau/T_k$ when increasing the number of layer $N$. Solid line are the theoretical predictions $F^{(N)}$\ref{['eq:FNTAU']} for $N = 1,\ 2,\ 4,\ 8$, their pointwise limit $F^{(\infty)}$\ref{['eq:FInfiniteNTAU']} in pink and dots are numerical estimations for the same number of layers $N$.