A practical guide for using proper orthogonal decomposition in engine research

Abstract

Proper orthogonal decomposition has been utilized for well over a decade to study turbulence and cyclic variation of flow and combustion properties in internal combustion engines. In addition, proper orthogonal decomposition is useful to quantitatively compare multi-cycle in-cylinder measurements with numerical simulations (large-eddy simulations). However, the application can be daunting, and physical interpretation of proper orthogonal decomposition can be ambiguous. In this paper, the mathematical procedure of proper orthogonal decomposition is described conceptually, and a compact MATLAB® code is provided. However, the major purpose is to empirically illustrate the properties of the proper orthogonal decomposition analysis and to propose practical procedures for application to internal combustion engine flows. Two measured velocity data sets from a motored internal combustion engine are employed, one a highly directed flow (each cycle resembles the ensemble average), and the other an undirected flow (no cycle resembles the average). These data are used to illustrate the degree to which proper orthogonal decomposition can quantitatively distinguish between internal combustion engine flows with these two extreme flow properties. In each flow, proper orthogonal decomposition mode 1 is an excellent estimate of ensemble average, and this study illustrates how it is thus possible to unambiguously quantify the cyclic variability of Reynolds-averaged Navier–Stokes ensemble average and turbulence. In addition, this study demonstrates the benefits of comparing two different samples of cycles using a common proper orthogonal decomposition mode set derived by combining the two samples, the effect of spatial resolution, and a method to evaluate the number of snapshots required to achieve convergence.

Evaluating Large-Eddy Simulation (LES) and High-Speed Particle Image Velocimetry (PIV) with Phase-Invariant Proper Orthogonal Decomposition (POD)

“…Thus, as in previous work [7,8], the mass-specific kinetic energy contribution of mode 1 in each snapshot is a good measure of its RANS ensemble average content, and the mass-specific kinetic energy contribution of modes 2 through M in each snapshot is a good measure of its the RANS turbulence content. The mass-specific kinetic energy contribution of a particular mode m to a snapshot k (ke m,k ) is calculated from the POD coefficient c k m as follows:…”

“…To demonstrate the attributes of this technique, phase-dependent POD was used to analyze snapshots at 76°ATDCE from 60 consecutive cycles from PIV and LES, respectively. Previous studies [7,8] have shown that the first mode Measured and computed ensemble-average (every second vector shown) and RMS velocity distributions.…”

“…Cosadia et al [6] found that determining a cut-off mode was not possible, as the cumulative energy fractions of the POD of the instantaneous velocity fields converged very slowly. Chen et al [7,8] proposed that, if the first mode is a good estimate of the mean flow, the POD coefficients of the first mode may be used to estimate the mass-specific kinetic energy contribution of the mean flow to any velocity field used in the decomposition, thus estimating how closely a particular velocity field compares to the mean flow. Similarly, the mass-specific kinetic energy contribution of all higher modes estimates the RANS turbulence in a specific velocity field.…”

Identification of Large-Scale Structure Fluctuations in IC Engines using POD-Based Conditional Averaging

“…Here, we focus on the snapshot POD method [6,7,24]. A detailed description of the underlying theory and the application for experimental and simulation data can be found in [21,24,25].…”

“…Furthermore, the remaining fluctuations are expected not to be associated with a variation of a coherent motion but rather being smaller scale turbulent fluctuations. To achieve this, our method combines POD, which has been used previously for the identification and investigation of velocity field fluctuations based both on experimental and numerical data [3][4][5][6][7][8][9], and conditional averaging [10].…”