Development and Application of Bivariate 2D-EMD for the Analysis of Instantaneous Flow Structures and Cycle-to-Cycle Variations of In-cylinder Flow

Abstract

The bivariate two dimensional empirical mode decomposition (Bivariate 2D-EMD) is extended to estimate the turbulent fluctuations and to identify cycle-to-cycle variations (CCV) of in-cylinder flow. The Bivariate 2D-EMD is an adaptive approach that is not restricted by statistical convergence criterion, hence it can be used for analyzing the nonlinear and non-stationary phenomena. The methodology is applied to a high-speed PIV dataset that measures the velocity field within the tumble symmetry plane of an optically accessible engine. The instantaneous velocity field is decomposed into a finite number of 2D spatial modes. Based on energy considerations, the in-cylinder flow large-scale organized motion is separated from turbulent fluctuations. This study is focused on the second half of the compression stroke. For most of the cycles, the maximum of turbulent fluctuations is located between 50 and 30 crank angle degrees before top dead center (TDC). In regards to the phase-averaged velocity field, the contribution of CCV to the fluctuating kinetic energy is approximately 55% near TDC.

Appendix

A synthetic Lamb–Oseen vortex, $\text {U}_{i}(x,y)$ that mimics large-scale organized motion is perturbed by an experimental HIT velocity field, $\text {P}(x,y)$ within $127 \times 127$ data points with spatial resolution of 0.16 mm (Galmiche et al. 2014), shown in Fig. 22a and b, respectively. The HIT flow has a longitudinal integral length-scale of 3.2 mm and the scale of vortex is approximately five times larger. The primary perturbed velocity field in Fig. 22c is interpolated on $254 \times 254$ data points with spatial resolution of 0.08 mm, as presented in Fig. 22d.

Fig. 22

a Large-scale synthetic flow, b experimental HIT velocity field as a perturbation, c, d perturbed velocity field within $127 \times 127$ and $254 \times 254$ data points, respectively. Every four vectors are displayed, the unit is $\hbox {m.s}^{-1}$

The primary and interpolated velocity fields are decomposed by Bivariate 2D-EMD. The modes corresponding to the horizontal velocity components, U are shown in Fig. 23. As one can see the interpolated velocity field has one mode more than the primary field, however the last modes of both fields are quite similar. Also the 5-th mode of the primary perturbed velocity field corresponds to the 6-th mode of the interpolated one and so on up to the first mode of the primary field that is distributed on the first and second modes of the interpolated velocity field.

Fig. 23

Horizontal velocity field in the first row and the corresponding modes in following. a Primary, b interpolated velocity fields, the unit is $\hbox {m.s}^{-1}$

Figure 24 presents the energy content of each mode for two decomposed velocity fields in logarithmic scale. For both cases, a significant increase in the energy of last mode is observed. This mode is considered as the organized motion (${\mathbf {U}}_{\mathbf {Lf}}$) and the sum of the other modes represents the HIT velocity field (${\mathbf {U}}_{\mathbf {Hf}}$). These flow fields are illustrated in the first and second row of Fig. 25 for the primary and interpolated velocity field, respectively. By observation, there is a good agreement between the flow structures of two Hf velocity fields as well as two Lf ones. Moreover the relative mean square error between mean kinetic energy of the Hf velocity fields is 6.7% and that of the Lf velocity fields is 0.3%.

As a conclusion, the interpolation of the velocity field on the grid points with higher spatial resolution affects slightly the feature of the low order modes that contain highest spatial frequency. Indeed the first mode of the primary perturbed velocity field, that can be considered as a measurement error or incoherent noise, is distributed on the first and second modes of the interpolated velocity field. However, it has no influence on the higher order modes, in particular the last one that represents the flow large-scale organized motion i.e., Lf part of the flow.

Fig. 24

Energy content of each mode for two decomposed perturbed velocity fields

Fig. 25

a, b Flow Lf and Hf velocity field, respectively correspond to the primary perturbed velocity field. Every four vectors are displayed, c, d flow Lf and Hf velocity field, respectively correspond to the interpolated perturbed velocity field. Every eight vectors are displayed, the unit is $\hbox {m.s}^{-1}$