Multiscale velocity correlations in turbulence and Burgers turbulence: Fusion rules, Markov processes in scale, and multifractal predictions

We compare different approaches towards an effective description of multiscale velocity field correlations in turbulence. Predictions made by the operator-product expansion, the so-called fusion rules, are placed in juxtaposition to an approach that interprets the turbulent energy cascade in terms of a Markov process of velocity increments in scale. We explicitly show that the fusion rules are a direct consequence of the Markov property provided that the structure functions exhibit scaling in the inertial range. Furthermore, the limit case of joint velocity gradient and velocity increment statistics is discussed and put into the context of the notion of dissipative anomaly. We generalize a prediction made by the multifractal model derived by Benzi et al. [R. Benzi et al., Phys. Rev. Lett. 80, 3244 (1998)] to correlations among inertial range velocity increment and velocity gradients of any order. We show that for the case of squared velocity gradients such a relation can be derived from first principles in the case of Burgers equations. Our results are benchmarked by intensive direct numerical simulations of Burgers turbulence.

Figure 1

(a) Schematic depiction of a shock in Burgers turbulence. The local energy dissipation rate is peaked at the center of the shock $\varepsilon \left(x\right)$. Depending on the strength of the shock, the velocity field at $v(x-r)$ and at $v(x+r)$ possesses the symmetry $v(x-r)=-v(x+r)$, which leads to Burgers scaling (55). (b) In the case of cusplike structures, $\varepsilon \left(x\right)$ is still peaked in the center of the cusp. The symmetry of the cusps, however, leads to the vanishing of the dissipation anomaly in Eq. (52).

Figure 2

Parametric space of the exponents of Eq. (62) and the corresponding fusion-rule prediction. Here I–III correspond to the three regions that emerge from the bifractal Burgers scaling (58).

Figure 3

Examination of the inertial-inertial fusion rules (63) for runs 1–3 via the quantity $F_{p,q}(r,R)$ defined in Eq. (62) as a function of $r/R$ for different values of $p$ and $q$ and for different Reynolds numbers: (a) $p=2$ and $q=4$ (region I), (b) $p=q=0.4$ (region II), and (c) $p=0.6$ and $q=2$ (region III). The black line corresponds to the fusion-rule prediction in Burgers turbulence summarized in the three regions I–III of Fig. 2. The inset depicts $F_{p,q}(r,R)$ divided by this prediction.

Figure 4

Parametric space of the exponents of Eq. (64) and the MF prediction for the viscous-inertial fusion rules. Here I–III correspond to the three regions that emerge from the bifractal Burgers scaling (58).

Figure 5

Examination of the viscous-inertial fusion rules prediction (65) via the quantity $D_{p,q}(\mathrm{Re},R)$ defined in Eq. (64) as a function of $R/\eta$ for different values of $p$ and $q$ and for different Reynolds numbers. The insets depict the results divided by the corresponding viscous-inertial fusion rules prediction of regions I–III of Fig. 4. (a) $p=2$ and $q=4$ (region I). The quantity $\nu D_{2,4}(\mathrm{Re},R)$ is shown here. It is independent of $R$, and the collapse of the data within error bars supports the suggestion of a generalized dissipative anomaly discussed in Sec. 2 and Eq. (9). (b) $p=q=0.4$ (region II). (c) $p=0.6$ and $q=2$ (region III).

Figure 6

(a) Moments $M_p$ from Eq. (67) as a function of the Taylor-Reynolds number in Burgers turbulence. The low-Reynolds-number regime exhibits Gaussian statistics (dashed black lines), whereas the high-Reynolds-number regime agrees well with the multifractal prediction (17) and the result from [31, 32, 33] (solid black line) for Burgers turbulence (68). The lines correspond to flat regions of the logarithmic derivative of the moments $\chi \left(n\right)$ [see (b)]. (b) Logarithmic derivative of the moments (69). The straight black lines correspond to the theoretical predictions (68), $\chi \left(4\right)=2$, $\chi \left(6\right)=4$, and $\chi \left(8\right)=6$.