Degrees of freedom and the dynamics of fully developed turbulence

While the degrees of freedom needed to represent the dynamics of high Reynolds number turbulence are extremely large, we show using well-resolved direct numerical simulations that one can capture essential physics with only a fraction of modes obeying the Navier-Stokes equations; the other modes can be modeled with very simple dynamics. This result suggests that the attractor for the dynamics of fully developed turbulence is robust to modeling errors and the strongly nonlinear dynamics may reside on fewer degrees of freedom than traditionally thought. The proposed approach is validated in terms of dissipation rate, skewness of velocity gradients, the energy and transfer spectrum, and structure functions. The mixed-dynamics model explored here, which may open different venues for turbulence modeling, may also be applicable to a broader set of physical phenomena governed by nonlinear complex dynamics with a wide range of scales.

Figure 1

Normalized dissipation rate $C_{\epsilon }=\langle \epsilon \rangle L/u^{\prime 3}$. Lines correspond to U (dashed) and V (solid) distributions for different values of $P_r$ in different colors. Symbols are for other DNS studies.

Figure 2

Skewness of velocity gradients. Lines correspond to U (dashed) and V (solid) distributions for different values of $P_r$ in different colors. Symbols except ($◊$) are for other DNS studies. The ($◊$) symbol corresponds to experimental data and are taken from Ref. [20].

Figure 3

Flatness of velocity gradients. Lines are for U (dashed) and V (solid) distributions. Symbols except ($◊$) are for other DNS studies. The ($◊$) symbol corresponds to experimental data and are taken from Ref. [19].

Figure 4

Compensated energy spectrum [left panels, (a)–(c)] and transfer spectrum [right panels, (d)–(f)] vs $k\eta$ for different Reynolds numbers (indicated). Lines correspond to DNS (red solid), U distribution (blue dashed), and V distribution (blue solid).

Figure 5

Normalized longitudinal structure functions of order 2 (left), 3 (middle), and 4 (right) vs $r/\eta$ for different Reynolds numbers (indicated). Lines correspond to DNS (red solid), U distribution (blue dashed), and V distribution (blue solid). The horizontal solid black line on the middle panels corresponds to 4/5 for reference.