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An Objective $Q$-Criterion

Tiemo Pedergnana, Florian Kogelbauer

TL;DR

This work addresses the non-objectivity of classical Eulerian vortex criteria in unsteady flows by formulating a spatio-temporal variational principle that minimizes a modified time derivative of the strain-rate tensor. The authors derive an optimal spin tensor $\bm{T}$ and an objective co-moving velocity field $\bm{v}_{\text{s}} = \bm{v} - \bm{T}[\bm{x}-\overline{\bm{x}}] - \\overline{\bm{v}}$, and define the objective $Q$-criterion $Q_{\text{s}} = \frac{1}{2}(\|\bm{W}-\text{skew}[\bm{T}]\|^2 - \|\bm{S}-\text{symm}[\bm{T}]\|^2)$. This framework resolves all previously known pathologies where Eulerian or earlier objective diagnostics produced false positives/negatives, and is demonstrated on analytical and 3D wake data, showing frame-invariance and improved alignment with Lagrangian particle motion. The approach is computationally efficient and readily applicable to complex 3D flows, with potential extensions to objective vorticity, potential vorticity, and helicity analyses for transport in unsteady flows.

Abstract

Classical Eulerian vortex criteria, such as the $Q$-, $Δ$-, $λ_2$-, $λ_{ci}$-, and Okubo--Weiss criterion, depend on frame-dependent quantities and therefore fail to provide objective, observer-independent diagnostics in unsteady frames. In this work, we address this longstanding limitation by introducing an objective variant of the $Q$-criterion, derived from a spatio-temporal variational principle that minimizes a modified time-derivative of the strain-rate tensor. Although several objective variants of the $Q$-criterion have been proposed before, none of these has successfully reconciled Eulerian diagnostics with the underlying Lagrangian motion of fluid particles, even in simple analytical solutions of the Navier--Stokes equations. Here, we present the first vortex criterion that consistently resolves all known pathological examples leading to false positives and false negatives in Eulerian vortex criteria applied to unsteady flows. The results establish a unified and objective framework for Eulerian vortex detection, opening new directions for the analysis and control of unsteady flow phenomena.

An Objective $Q$-Criterion

TL;DR

This work addresses the non-objectivity of classical Eulerian vortex criteria in unsteady flows by formulating a spatio-temporal variational principle that minimizes a modified time derivative of the strain-rate tensor. The authors derive an optimal spin tensor and an objective co-moving velocity field , and define the objective -criterion . This framework resolves all previously known pathologies where Eulerian or earlier objective diagnostics produced false positives/negatives, and is demonstrated on analytical and 3D wake data, showing frame-invariance and improved alignment with Lagrangian particle motion. The approach is computationally efficient and readily applicable to complex 3D flows, with potential extensions to objective vorticity, potential vorticity, and helicity analyses for transport in unsteady flows.

Abstract

Classical Eulerian vortex criteria, such as the -, -, -, -, and Okubo--Weiss criterion, depend on frame-dependent quantities and therefore fail to provide objective, observer-independent diagnostics in unsteady frames. In this work, we address this longstanding limitation by introducing an objective variant of the -criterion, derived from a spatio-temporal variational principle that minimizes a modified time-derivative of the strain-rate tensor. Although several objective variants of the -criterion have been proposed before, none of these has successfully reconciled Eulerian diagnostics with the underlying Lagrangian motion of fluid particles, even in simple analytical solutions of the Navier--Stokes equations. Here, we present the first vortex criterion that consistently resolves all known pathological examples leading to false positives and false negatives in Eulerian vortex criteria applied to unsteady flows. The results establish a unified and objective framework for Eulerian vortex detection, opening new directions for the analysis and control of unsteady flow phenomena.

Paper Structure

This paper contains 15 sections, 39 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Strain-Rate Unsteadiness Minimization
  4. Examples
  5. Spatially linear unsteady velocity field
  6. Separation and reattachment flow
  7. Cubic unsteady Navier--Stokes flow with hidden mixing region
  8. Unsteady three-dimensional flow in a wind wake
  9. Discussion
  10. Conclusion
  11. First variation of $\mathcal{S}$
  12. Transformation properties of $\bm{T}$
  13. Transformation property of ${\bm{M}}$
  14. Transformation property of $\overline{\bm{M}^{T}\operatorname{vec}[\partial_t \bm{S}]}$
  15. Transformation property of $\operatorname{vec}[\bm{T}]$

Figures (2)

  • Figure 1: Vortex detection in analytical flow fields. a The objective $Q_\text{s}$-criterion predicts the correct fluid particle motion for the linear family of flow fields \ref{['Linear field']} as a function of its parameters. See Fig. 4 of KogelbauerPedergnana2025 for corresponding predictions of the classical $Q$-criterion and its other objective analogues. b,c Analysis of a two-dimensional flow defined by the streamfunction \ref{['streamfunction']}. b Instantaneous streamlines of the velocity field at time $t=\pi/4$. c Instantaneous streamlines of the objective field $\bm{v}_\text{s}$ defined by \ref{['streamfunction2']} at the same time instant correctly identify the flow topology of the fluid particle motion. Brighter values correspond to larger values of the streamfunction. The zero isocontour, indicating the separation line of the flow, is colored in red. d,e Analysis of the spatially nonlinear unsteady Navier--Stokes field \ref{['unsteady 2d NS']}. d Instantaneous streamlines of the velocity field $\bm{v}$, shown here for $t_0=\pi/4$, indicate a vortical flow. Streamlines at different times show analogous flow patterns. e The streamlines of the corresponding objective field $\bm{v}_\text{s}$ correctly indicate a mixing region near the origin.
  • Figure 2: Vortex detection in the wind wake behind the research vessel Tangaroa. a, b Comparison of the $Q$- and $Q_\text{s}$-criteria and c$Q$-criterion computed in a rotating frame described by \ref{['Tangaroa observer change']}, all in in the plane $y=0.005$. The black curves show the zero isocontours. The small insets show the position of the larger insets and all criteria in a global view of the wind wake. d, e, f Zero isocontours of the three vortex criteria. The out-of-plane axes are scaled for visualization purposes. The transparent plane shows the location where the iscontours in a, b, c were extracted.