Quantitative Imaging in Proper Orthogonal Decomposition of Flow Past a Delta Wing

Snapshot proper orthogonal decomposition is used to characterize the unsteady e ow past a delta wing, focusing on the structure of the leading-edge vortices. The decomposition is applied to two-dimensional velocity e elds obtained via a laser-scanning version of high-image-density particle image velocimetry to extract the most coherent structures in the e ow. Data are analyzed for cases involving both a stationary wing and a wing undergoing harmonic oscillations in roll. In each case, the velocity e elds are reconstructed as a truncated series expansion in terms of the computed eigenfunctions from which the corresponding contours of constant vorticity and sectional streamline patterns are calculated. Comparison with the original data demonstrates that the analysis provides an accurate representation of the velocity e elds while eliminating extraneous small-scale features. Inclusion of as few astwo eigenfunctionsin thereconstruction series reproduces the largest-scalefeaturesoftheleading-edge vortices, whereas the inclusion of half of the total number of eigenfunctions produces a reconstructed e eld that captures the majority of the e ow features. By appropriately combining the spatial and temporal components of the proper orthogonal decomposition analysis, one obtains the dynamical structures that evolve in both space and time, such as the e uctuations in the location of vortex breakdown. Nomenclature ak = modal amplitudes for velocity b = span C = correlation matrix co = centerline chord dI = interrogation window size E = mean total e uctuation energy M = total number of snapshots N = number of modes retained in Eq. (4) T = period of oscillation t = time U1 = freestream velocity u = x component of velocity V = velocity vector v = y component of velocity xvb = location of vortex breakdown, measured along wing centerline from apex x; y = coordinates ®k = kth eigenvector of the correlation matrix 1t = time interval between snapshots · = reduced frequency, .ob/=2U1T ¸k = kth eigenvalue of the correlation matrix 8 = roll angle Ak = kth empirical velocity eigenfunction N = time-averaged quantities 0 = e uctuation quantities