Application of the Poor Man′s Navier‐Stokes Equations to Real‐Time Control of Fluid Flow

Abstract

Control of fluid flow is an important, underutilized process possessing potential benefits ranging from avoidance of separation and stall on aircraft wings to reduction of friction in oil and gas pipelines to mitigation of noise from wind turbines. But the Navier-Stokes (N.-S.) equations, whose solutions describe such flows, consist of a system of time-dependent, multidimensional, nonlinear partial differential equations (PDEs) which cannot be solved in real time using current computing hardware. The poor man's Navier-Stokes (PMNS) equations comprise a discrete dynamical system that is algebraic—hence, easily (and rapidly) solved—and yet which retains many (possibly all) of the temporal behaviors of the PDE N.-S. system at specific spatial locations. Herein, we outline derivation of these equations and discuss their basic properties. We consider application of these equations to the control problem by adding a control force. We examine the range of behaviors that can be achieved by changing this control force and, in particular, consider controllability of this (nonlinear) systemvianumerical experiments. Moreover, we observe that the derivation leading to the PMNS equations is very general and may be applied to a wide variety of problems governed by PDEs and (possibly) time-delay ordinary differential equations such as, for example, models of machining processes.

Numerical Investigation of the “Poor Man’s Navier–Stokes Equations” with Darcy and Forchheimer Terms

“…A discussion regarding the connection of such DDSs with the physics of turbulence is presented by McDonough et al [2003], along with results from anisotropic cases. Polly [2011] and Polly and McDonough [2012] extended the system to 3-D, with more complicated structure and more bifurcation parameters, and discovered that the system behavior is at least slightly different from that of the 2-D system. It has been shown by McDonough [2009] that the 3-D system can lead to scaling for the turbulence kinetic energy spectrum in accord with Kolmogorov's K41 theory (see, e.g.…”