Abstract
Reference frame optimization is a generic framework to calculate a spatially varying observer field that views an unsteady fluid flow in a reference frame that is as-steady-as-possible. In this paper, we show that the optimized vector field is objective, i.e., it is independent of the initial Euclidean transformation of the observer. To check objectivity, the optimized velocity vectors and the coordinates in which they are defined must both be connected by an Euclidean transformation. In this paper, we show that a recent publication applied this definition incorrectly, falsely concluding that reference frame optimizations are not objective. Furthermore, we prove the objectivity of the variational formulation of the reference frame optimization that was recently proposed and discuss how the variational formulation relates to recent local and global optimization approaches to unsteadiness minimization.
Highlights
In fluid mechanics, an important property of vortex detectors is whether their corresponding vortex criteria are objective, i.e., indifferent to the reference frame in which they are computed
We show that the optimized vector field is objective, i.e., it is independent of the initial Euclidean transformation of the observer
We prove the objectivity of the variational formulation of the reference frame optimization that was recently proposed and discuss how the variational formulation relates to recent local and global optimization approaches to unsteadiness minimization
Summary
An important property of vortex detectors is whether their corresponding vortex criteria are objective, i.e., indifferent to the reference frame in which they are computed. This is a major drawback of nonobjective methods that often corresponds to the fact that the detected features lack a clear physical meaning or cannot occur physically at all, as pointed out by many authors, from the early work of Haller until a recent analysis.2 With this motivation in mind, a variety of vortex criteria have been designed to be objective by definition, i.e., the associated method can directly be proven to be indifferent to the motion of the input reference frame, and all observers agree on the result of the evaluated criteria. Its immediate physical meaning is only that if a method is objective, different physical observers come to the same conclusions, for example regarding the location of a vortex This is true for all generic “objectivization” approaches by Gu€nther et al., Hadwiger et al., Baeza Rojo and Gu€nther, Gu€nther and Theisel, Gu€nther and Theisel, and Rautek et al.. We focus purely on objectivization with the standard meaning of objectivity and show that the corresponding mathematical proof given by Haller is incorrect and that such an objectivization is possible
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