An efficient method of solving the Navier-Stokes equations for flow control

Abstract

A new method of solving the Navier-Stokes equations e ciently by reducing their number of modes is proposed in the present paper. It is based on the Karhunen-Lo eve decomposition which is a technique of obtaining empirical eigenfunctions from the experimental or numerical data of a system. Employing these empirical eigenfunctions as basis functions of a Galerkin procedure, one can a priori limit the function space considered to the smallest linear subspace that is su cient to describe the observed phenomena, and consequently reduce the Navier-Stokes equation deÿned on a complicated geometry to a set of ordinary di erential equations with a minimum degree of freedom. The present algorithm is well suited for the problems of ow control or optimization, where one has to compute the ow ÿeld repeatedly using the Navier-Stokes equation but one can also estimate the approximate solution space of the ow ÿeld based on the range of control variables. The low-dimensional dynamic model of viscous uid ow derived by the present method is shown to produce accurate ow ÿelds at a drastically reduced computational cost when compared with the ÿnite di erence solution of the Navier-Stokes equation. ?

Review of Automated Design and Optimization of MEMS

“…An approach used for highly nonlinear system over a wide range of operating conditions was proposed by Rewienski et al [23,24], where the algorithm presented has been demonstrated to be effective and accurate for highly nonlinear systems, but its performance still depends on a few parameters, which need to be adjusted more or less arbitrarily for a given application example. The reduced-order models generated by the Karhunen-Loeve/Galerkin approach [25,26], on the other hand, are based on the basis functions extracted from an ensemble of the snapshots of the physical fields (e.g., pressure distribution or temperature distribution) under certain actuation conditions, and have been proved to be effective and accurate for macromodeling nonlinear systems too. However, this approach requires expensive coupled-domain FEM/FDM runs to provide enough snapshot data for extracting basis functions.…”

Recursive solution of inverse natural convection problems by means of mode reduction

“…Since the empirical eigenfunctions, which constitute the space for the low dimensional dynamic model, are expressed linearly in terms of velocity and temperature fields (Park and Cho 1996;Park and Lee 1998), the velocity and temperature fields must be prepared such that they fully characterize the system dynamics. We assume that the heat flux function qðx; tÞ can be expressed as:…”

“…As explained in Park and Cho (1996) and Park and Lee (1998), the empirical eigenfunctions of the KarhunenLoève decompositions are obtained by solving the following integral equations. where uðxÞ and uðxÞ are the velocity empirical eigenfunction and the temperature empirical eigenfunction, respectively, and K ðvÞ and K ðTÞ are the corresponding twopoint correlation functions defined as:…”

“…The only way of circumventing this difficulty is to devise a reduced order model or a low dimensional dynamic model that simulates the system faithfully. An appropriate technique for this purpose is the Karhunen-Loève Galerkin procedure (Park and Cho 1996;Park and Lee 1998). The Karhunen-Loève Galerkin procedure is a type of Galerkin method that employs as basis functions the empirical eigenfunctions of the KarhunenLoève decomposition (Loève 1977).…”

“…But recent works (Park and Cho 1996;Park and Lee 1998) have extended the applicability of the Karhunen-Loève decomposition to the analyses of nonstationary, nonhomogeneous deterministic as well as stochastic fields, and allowed the derivation of rigorous reduced order models that simulate the given systems almost exactly. This extension of the original Karhunen-Loève decomposition is called the KarhunenLoève Galerkin procedure (Park and Cho 1996;Park and Lee 1998), and can be adopted in the inverse analysis. Through the Karhunen-Loève Galerkin procedure, one can a priori limit the function space considered to the smallest linear subspace that is sufficient to describe the observed phenomena and consequently reduce the governing partial differential equations to a minimal set of ordinary differential equations.…”