Curve fitting 3-D experimental turbulent flows with the poor man's Navier--Stokes equation

The 3-D poor man's Navier--Stokes (PMNS) equation is a discrete dynamical system (DDS) whose solutions retain much of the dynamical behavior of the partial differential equations from which it is derived, and yet is very easily executed---far faster than real time. We briefly outline derivation of this DDS and then discuss a general procedure for curve fitting DDSs to chaotic experimental data. This technique (first introduced by McDonough {\it et al., Appl.\ Comp.\ Math.\ }1998 and later used by Yang {\it et al., AIAA J.\ }2003 in a 2-D Navier--Stokes setting) employs a least-squares method to generate a global (long-time) fit of chaotic data that produces details of experimental time series in a manner more appropriate for representing fluid turbulence (including sensitivity to initial conditions) than often used short-time extrapolation techniques can. We apply this least-squares approach to three-component velocity measurements in grid turbulence described by Bailey and Tavoularis, {\it J.\ Fluid Mech.\ }2008, and demonstrate that the PMNS equation can reproduce the structure of all three experimental velocity components. We present comparisons of time series, energy spectra, and other typical turbulence statistics, {\it e.g.}, flatness and skewness. A possible application of such curve fits would be to real-time control of physical turbulent fluid flows.