Geometric constraints on energy transfer in the turbulent cascade

The energy cascade is the most significant feature that separates turbulence from other unsteady flows, and results from the behavior of the nonlinear term in the Navier-Stokes equations. The mathematical form of this term, however, places constraints on exactly how it can act. Here we consider the action of the nonlinear term in physical space rather than in Fourier space, where the energy transfer between scales can be interpreted as a mechanical process where some scales do work on others. This formulation reveals the fundamental role played by geometry, as work can only be done when the eigenframes of the turbulent stress and strain rate are appropriately aligned. By comparing a direct numerical simulation of the Navier-Stokes equations, an ensemble of random solenoidal vector fields, and a random sampling of uniform eigenframe alignments, we show that this geometric alignment plays a much stronger role in determining the flux between scales than do the magnitudes of the stress and strain rate. We also show that when the alignment is effectively two dimensional, even when embedded in a three-dimensional flow, the energy flux is typically inverse, suggesting that the inverse cascade in two-dimensional turbulence may have a kinematic origin. Our results point to some potentially fruitful directions for turbulence modeling.

Figure 1

Sketch of the eigenframes of the stress and strain rate and the $ZXZ$ Euler angles used to move between the two. Transforming from the eigenbasis of the stress to the strain rate corresponds to a rotation by $\varphi$ around ${\widehat{\mathit{\lambda }}}_3$, a rotation by $\theta$ about the rotated ${\widehat{\mathit{\lambda }}}_1$, and finally a rotation by $\psi$ about the rotated ${\widehat{\mathit{\lambda }}}_3$.

Figure 2

Mean efficiencies $\langle \mathrm{\Gamma }\rangle$ as a function of scale $r/\eta$ for combinations of eigenvalues and Euler angles taken from different data sets. The line styles distinguish the data sets from which the angles were drawn. Data with blue solid lines correspond to angles taken from the DNS, with red dashed lines to angles taken from the random vector fields, and with black dash-dotted lines to uniformly sampled random angles. The symbols distinguish the data sets from which the eigenvalues were drawn. Triangles correspond to eigenvalues taken from the DNS, circles to eigenvalues taken from the random vector fields, and crosses to uniformly sampled random eigenvalues.

Figure 3

Probability density functions of $\mathrm{\Gamma }$ for (a) $r/\eta =8$, (b) $r/\eta =40$, and (c) $r/\eta =200$. All eigenvalues were taken from the DNS. As in Fig. 2, the line style shows the data set from which the angles were taken. Blue solid lines correspond to the DNS, red dashed lines to the random vector fields, and black dash-dotted lines to uniformly sampled random angles.

Figure 4

Marginal PDFs of the Euler angles $\varphi , \psi$, and $\theta$ for (a)–(c) the DNS and (d)–(f) the random vector fields. Data are shown for $r/\eta =8$, 12, 16, 20, 40, 80, 120, 160, and 200, as distinguished by color. The dash-dotted horizontal lines show the corresponding PDFs for the uniformly distributed case.

Figure 5

Mean efficiency as a function of scale for combinations of eigenvalues and angles taken from different data sets, as in Fig. 2, but for the case of 2D configurations. As in Fig. 2, the line styles show the data sets from which the angles were taken: Solid lines correspond to (projected) angles from the DNS, dashed lines to the random vector fields, and black dash-dotted lines to uniformly sampled random angles. Symbols show the data sets from which the eigenvalues were taken. Triangles correspond to eigenvalues taken from the DNS, circles to eigenvalues taken from the random vector fields, and crosses to uniformly sampled random eigenvalues.