An Objective Measure of Unsteadiness

TL;DR

The paper addresses the frame-dependence of the unsteady component of velocity and its impact on vortex diagnostics. It introduces deformation unsteadiness $[rac{ ext{∂v}}{ ext{∂t}}]_{ ext{d}}$, derived by optimally subtracting bulk rigid-body motion via a variational principle, yielding an objective measure of local flow change. An explicit optimal unsteadiness frame characterized by $m{ ext{ω}}_{ ext{US}}=m{ extΘ}_v^{-1}ar{m{v}_{ ext{d}} imes rac{ ext{∂}m{v}_{ ext{d}}}{ ext{∂}t}}$ makes $[rac{ ext{∂v}}{ ext{∂t}}]_{ ext{d}}$ objective, and an objective $Q$-criterion, $Q_{ ext{US}}$, is defined using this frame, improving vortex detection in unsteady flows. The approach is demonstrated on analytical and Navier–Stokes-based flows in 2D/3D and on simulated data, showing that deformation unsteadiness reveals features obscured by frame changes and that $Q_{ ext{US}}$ can better identify coherent structures than traditional criteria. The work provides a fast, Eulerian, frame-invariant diagnostic with potential for broad application in turbulence analysis and flow visualization.

Abstract

Unsteadiness lies at the heart of turbulent fluid dynamics, eddy formation and instabilities in flows thus making it central to both understanding and controlling fluid systems. In this work, we present an objective measure for the unsteadiness of a time-dependent velocity field, the deformation unsteadiness, derived from a spatio-temporal variational principle, allowing for a frame-independent assessment of the unsteadiness of a given flow field. Additionally, as an application of our main result, we define an objective analogue of the classic $Q$-criterion based on extremizers of unsteadiness minimization. We apply our results to several examples of analytical flows as well as simulated flow data sets in two and three dimensions. In particular, we apply our newly derived vortex criterion to several explicit, time-dependent solutions of the Navier--Stokes equation and compare the results to existing vortex criteria. We give a physical interpretation of the deformation unsteadiness and discuss future research directions.

Figures (7)

• Figure 1: Sketch of multiple co-moving observers (UAVs) measuring a velocity field $\bm{v}$ of a coherent structure (tornado). Due to the frame-dependence of $\bm{v}$, each observer's measurements of the velocity field will generally be different from all others', and also different from the velocity field measured in the rest frame of the earth. These disagreements in the velocity field carry over to the partial time derivative $\partial_t \bm{v}$, the vorticity $\bm{\nabla}\times\bm{v}$, and other quantities derived from $\bm{v}$.
• Figure 2: Isocontours of the norms of (a) the partial time derivative and (b) the deformation unsteadiness for the two-dimensional unsteady flow field described by the stream function \ref{['streamfunction']}. The coexistence of two diametrically opposed flow cells is revealed only by the objective deformation unsteadiness, but not by the non-objective partial time derivative. In inset (b), the zero contour of the deformation unsteadiness is colored in red.
• Figure 3: (a) Streamlines of the spatially quadratic, unsteady Navier--Stokes field \ref{['Quadratic field']} at time $t=0.5$. (b) Streamlines at $t=0.5$ of the auxiliary velocity field $\bm{v}_{\text{d},\text{US}}$ based on the results of the extremizer $\Omega_\text{US}$ of the spatio-temporal variational principle introduced in Section \ref{['unsteadiness minimization']}. The fluid particle motion induced by this velocity field is elliptical, yet the streamlines of the velocity field are hyperbolic. In contrast, the streamlines of the auxiliary field $\bm{v}_{\text{US}}$ are elliptical, correctly pronouncing the vortical nature of the flow.
• Figure 4: Predictions of the $Q$-criterion and its objective analogues for the unsteady Navier--Stokes flow given by Eq. \ref{['Linear field']} as a function of $\omega$ and $C$. Predictions at the discrete, green points are compared in Table \ref{['Table 1']}.
• Figure 5: Unsteadiness analysis of a a simulated flow across a step with obstacles. (a), (b) Norm of the partial time derivative in the resting $\bm{x}$-frame. (c), (d) Norm of the partial time derivative in the $\bm{y}$-frame, defined by a time-dependent observer change $\bm{x}(t)=\bm{y}(t)+\bm{b}_1(t)$, where $\bm{b}_1(t)$ is defined in \ref{['observer changes']}. (e), (f) Norm of the deformation unsteadiness in the $\bm{y}$-frame.

Theorems & Definitions (2)

• Remark 1
• Remark 2