The inviscid Euler limit as a critical boundary for moment-based aerodynamic system identification

TL;DR

This paper shows that inviscid Euler equations have a unique power-law decay affecting system identification, leading to diverging moments and no intrinsic memory time scale, unlike viscous flows.

Contribution

It introduces a diagnostic for analyzing moment convergence and demonstrates the critical boundary nature of the inviscid limit for aerodynamic system modeling.

Findings

Inviscid Euler impulse response decays as t^{-3/2} causing divergence in second moments.

A new temporal-moment diagnostic, ν_t(T), quantifies memory growth with observation window T.

Numerical simulations confirm the √ln T scaling and the role of dissipation as a regularizer.

Abstract

Finite-dimensional state-space representations of unsteady aerodynamics implicitly assume a system with fading memory. However, the impulse response of the two-dimensional inviscid (Euler) equations is characterized by an asymptotic $t^{-3/2}$ power-law decay due to the persistence of shed vorticity. The present work demonstrates that this decay rate constitutes a critical boundary for moment convergence: the second temporal moment diverges logarithmically, causing the characteristic memory time to grow as $\sqrt{\ln T}$ with the observation window $T$. As a result, no window-independent characteristic time scale exists, and finite-dimensional models fitted to inviscid data effectively parameterize the observation horizon rather than intrinsic flow physics. To quantify this behavior, a temporal-moment diagnostic, $\nu_t(T)$, is introduced based on the ratio of the second and zeroth windowed moments of the impulse response kernel. Exponential models exhibit stable memory time plateaus, as their sufficiently fast decay ensures convergence of the moment diagnostic. Compressible Euler simulation results confirm the predicted $\sqrt{\ln T}$ scaling at intermediate times, while numerical dissipation inherent to the discretization acts as an artificial regularizer that enforces convergence at late times. These results establish the two-dimensional inviscid limit as a critical boundary for moment-based system identification, where the absence of a dissipative mechanism prevents the definition of a window-independent characteristic memory time.