Reexamining the framework for intermittency in Lagrangian stochastic models for turbulent flows: A way to an original and versatile numerical approach

The characterization of intermittency in turbulence has its roots in the refined similarity hypotheses of Kolmogorov, and if no proper definition is to be found in the literature, statistical properties of intermittency were studied and models were developed in an attempt to reproduce it. The first contribution of this work is to propose a requirement list to be satisfied by models designed within the Lagrangian framework. Multifractal stochastic processes are a natural choice to retrieve multifractal properties of the dissipation. Among them, we investigate the Gaussian multiplicative chaos formalism, which requires the construction of a log-correlated stochastic process $X_t$. The fractional Gaussian noise of Hurst parameter $H=0$ is of great interest because it leads to a log correlation for the logarithm of the process. Inspired by the approximation of fractional Brownian motion by an infinite weighted sum of correlated Ornstein-Uhlenbeck processes, our second contribution is to propose a stochastic model: $X_t={\int }_0^{\infty }{Y}_t^{x}k\left(x\right)dx$, where $Y_t^x$ is an Ornstein-Uhlenbeck process with speed of mean reversion $x$ and $k$ is a kernel. A regularization of $k\left(x\right)$ is required to ensure stationarity, finite variance, and logarithmic autocorrelation. A variety of regularizations are conceivable, and we show that they lead to the aforementioned multifractal models. To simulate the process, we eventually design a new approach relying on a limited number of modes for approximating the integral through a quadrature $X_t^N={\sum }_{i=1}^N{\omega }_i{Y}_t^{x_i}$, using a conventional quadrature method. This method can retrieve the expected behavior with only one mode per decade, making this strategy versatile and computationally attractive for simulating such processes, while remaining within the proposed framework for a proper description of intermittency.

Figure 1

Temporal evolution of the pseudodissipation along three particle trajectories obtained from the DNS dataset of [24].

Figure 2

Probability density function of the normalized variable $ln\phi$ compared to the Gaussian distribution (dashed black line). The $y$ axis is on a logarithmic scale.

Figure 3

Autocorrelation of $X_t^T$ (red line) compared with expected logarithmic behavior (black dotted line). The $x$ axis is on a logarithmic scale.

Figure 4

Comparison of autocorrelation of processes of Pope and Chen [7] $X_t^{\text{OU}}$, Schmitt [38] $X_t^S$, and Pereira et al. [22] $X_t^P$ with logarithmic behavior. Processes are rescaled for comparable variance of $ln(T_L/{\tau }_{\eta })$ and the $x$ axis is on a logarithmic scale.

Figure 5

Five correlated Ornstein-Uhlenbeck processes, driven by the same Wiener increments. Plots are shifted up for a better visualization and the color shades are darker with increasing characteristic times $x^{-1}$.

Figure 6

Possible regularizations of the kernel $k\left(x\right)$ in a log-log scale.

Figure 7

The kernel behavior $k\left(x\right)\sim x^{-1/2}$ in the dashed line is regularized with Heaviside cutting functions in dark blue and compared to its quadrature representation in light yellow. The $x$ and $y$ scales are logarithmic.

Figure 8

Comparison of the autocorrelations of the analytical process (dotted black line) and the discrete one for a finite number of modes. The inertial range covers 5 decades. The $x$ axis is on a logarithmic scale and the dashed line represents the expected logarithmic behavior.

Figure 9

Error in log-log scale between the $L_2$ norm of the discrete process $X_t^N$ and the analytical process $X_t^{\infty }$. The dashed line represents a slope of $-2$.

Figure 10

Autocorrelation of $X_t$ for different inertial ranges in log-log scale, compared with logarithmic behavior in dashed lines. The number of points chosen in the quadrature $N$ corresponds to the number of decades covered by the inertial range $[{\tau }_{\eta },T_L]$.

Figure 11

The yellow line represents the functions $B$ (on the left), $C$ (in the middle), and $D$ (on the right) plotted for three examples of $\theta$. The dotted blue line is $[g_{\tau }\left(r^2\right)-g_{T_L}\left(r^2\right)]$ and the dashed red line is $[g_{{\tau }_{\eta }/{tan}^2\theta }\left(r^2\right)-g_{T_L/{tan}^2\theta }\left(r^2\right)]$.