Complex network approach to turbulent velocity gradient dynamics: High- and low-probability Lagrangian paths

Understanding the dynamics of the turbulent velocity gradient tensor (VGT) is essential to gain insights into the Navier-Stokes equations and improve small-scale turbulence modeling. However, characterizing the VGT dynamics conditional on all its relevant invariants in a continuous fashion is extremely difficult. Hence, in this paper, we represent the Lagrangian dynamics using a network where each node represents a unique flow state in the space of the invariants of the VGT. The sign and relative magnitude ranking of the invariants resulting from a Schur decomposition of the VGT determines the discrete flow state vector. Our analysis reveals intriguing features of the resulting network dynamics, such as a grouping of the influential nodes where the eigenvalues of the VGT are real, in the proximity of the right Vieillefosse tail. We relate our complex network approach to the well-established VGT discretization based on the sign of its principal invariants, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>Q</a:mi></a:math> and <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>R</b:mi></b:math>, and its discriminant, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi mathvariant="normal">Δ</c:mi></c:math> into six regions representing very different physical and topological states. We study the shortest paths on the network based on the six regions to which their starting and ending nodes belong. This analysis shows that the typically clockwise path on the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mrow><e:mi>Q</e:mi><e:mtext>−</e:mtext><e:mi>R</e:mi></e:mrow></e:math> diagram arises in a varying manner: sometimes region transitions are based on multiple pathways of more equal probability, while others are focused on specific nodes that are pivotal in these interregion dynamics. Comparisons to an enhanced Gaussian closure model show that the model captures the out-degree distribution for the nodes well. However, it uses a greater number of nodes to transition from real to complex eigenvalues and is far less efficient at generating clockwise motions that are counter to the restricted Euler dynamics.