Numerical study of Lagrangian velocity structure functions using acceleration statistics and a spatial-temporal perspective

TL;DR

This study investigates the debated inertial-range scaling of the Lagrangian second-order velocity structure function in forced isotropic turbulence using direct numerical simulation across $R_\lambda\in[140,1300]$. By expressing $D_2^L(\tau)$ in terms of the acceleration autocorrelation and by decomposing the Lagrangian increment into convective and local components, the authors reveal that the expected plateau is not clearly observed due to finite time spans and strong particle displacements. They find weak Reynolds-number growth of the peak value $C_0^*$ (approximately $C_0^*\propto R_\lambda^{0.20}$) and a stronger Reynolds dependence for the acceleration variance $a_0\propto R_\lambda^{0.25}$, while the incomplete cancellation between $\mathbf{v}_C$ and $\mathbf{v}_L$ contributes to intermittency. The results emphasize the roles of limited time-scale range and large-scale particle transport in shaping Lagrangian statistics and offer a spatial-temporal framework for interpreting tracer-level turbulence measurements and high-$R_\lambda$ intermittency.

Abstract

A fundamental relation in Lagrangian Kolmogorov theory is concerned with inertial range scaling of the second-order velocity structure function over intermediate time lags at sufficiently high Reynolds numbers. However, the scaling is not well observed, and it is uncertain whether the scaling constant ($C_0$) truly approaches a constant value at asymptotic Reynolds numbers. In this paper, direct numerical simulation of forced isotropic turbulence at Taylor-scale Reynolds numbers between 140 and 1300 is used to help advance understanding in this subject. Uncertainties arising from modest simulation time spans are addressed by expressing the velocity structure function in terms of the acceleration autocorrelation, which suggests that $C_0$ may be sensitive to intermittency effects, leading to a sustained, although weak, Reynolds number dependence. The Lagrangian velocity increment is examined from a spatial-temporal perspective, as a combination of convective (spatial) and local (temporal) contributions, which are subject to a strong but incomplete mutual cancellation dependent on Reynolds number and time lag. The convective increment is strongly influenced by the particle displacement, which is driven by large-scale dynamics and can thus grow into inertial range dimensions in space within just a few Kolmogorov time scales, without fully satisfying classical Lagrangian inertial-range requirements. An overall conclusion in this work is that both the limited range of time scales (narrower than for length scales) and the effects of particle displacements have significant roles in the observed behavior of the second-order Lagrangian velocity structure function.

Figures (10)

• Figure 1: (a) Normalized second-order Lagrangian structure function at ${ R_\lambda }$ from 140 to 1300 per Table I (red, green, blue, black, magenta respectively) (b) peak of the curve in (a), denoted by $C_0^*$ (squares) and $a_0$ (circles). Dashed gray lines indicate power laws fits for comparison: ${ R_\lambda }^{0.2}$ (for $C_0^*$) and ${ R_\lambda }^{0.25}$ (for $a_0$).
• Figure 2: Results from $12288^3$ simulation at the highest ${ R_\lambda }$ (1300) (a) acceleration autocorrelation and (c) same as (a) but multiplied by time lag; structure function from (b) a direct calculation or (d) recovered from the quantities $\rho_a(\tau)$ and $\langle a^2\rangle$. Lines in red and blue are for data processed up to about $8~{\tau_\eta}$ and $10~{\tau_\eta}$ respectively. (They nearly coincide except in portions of (c).)
• Figure 3: (a) Evolution of $\rho(\mathbf{v}_L,\mathbf{v}_C)$ in time at ${ R_\lambda }~140, 390, 650, 1000, 1300$ (increasing in the direction of the arrow), (b) and (c) PDFs of $\theta$, and $\cos(\theta)$ respectively, at ${ R_\lambda }~390$ and time lags $\tau/{\tau_\eta}\approx 1, 2, 4, 8, 16, 32$, (in the order red, green, blue, black, magenta, cyan).
• Figure 4: Conditional PDFs of $v_C^{'}$ given $v_L^{'}$ (left column) and $v_L^{'}$ given $v_C^{'}$ (right column), with primes denoting normalization by unconditional root-mean-square (r.m.s.), using data at ${ R_\lambda }$ 390 on a $1536^3$ grid. Each frame consists of 5 lines, with the conditioning variable at -4, -2, 0, 2, 4 times the r.m.s values (red, green, blue, black, magenta) from the mean. Top, middle, and bottom rows show results at $\tau/{\tau_\eta} \approx 0.1$, $4$ and $100$, respectively.
• Figure 5: Same as top row Fig. \ref{['fig:cpdf_re390']}, but for data at ${ R_\lambda }~1300$, on a $12288^3$ grid.