Data recovery and reconstruction methods for unsteady flow fields with spatio-temporal missing data are studied based on proper orthogonal decomposition (POD) and on Kriging interpolation. It is found that for sufficient temporal resolution, POD-based methods outperform Kriging interpolation. However, for insufficient temporal resolution, large spatial gappiness or for flow fields with black zones, Kriging interpolation is more effective. The comparison is performed based on randomly generated laminar and turbulent flow fields obtained from simulations of uniform flow past a circular cylinder.

Proper orthogonal decomposition (the Karhunen-Loève expansion) is applied to convective flows in a tall differentially heated cavity. Empirical spatial eigenfunctions are computed from a multicellular solution at supercritical conditions beyond the first Hopf bifurcation. No assumption of periodicity is made, and the computed velocity and temperature eigenfunctions are found to be centro-symmetric. A lowdimensional model for the dynamical behaviour is then constructed using Galerkin projection. The reduced model successfully predicts the first Hopf bifurcation of the multicellular flow. Results determined from the low-order model are found to be in qualitative agreement with known properties of the full system even at conditions far from criticality.

Methods for data reconstruction and spatial enhancement of experimental data for a transitional boundary layer with laminar separation bubble are investigated. Particularly, proper orthogonal decomposition (POD) is applied to direct numerical simulation (DNS) data to extract the DNS-based POD modes, which are projected onto the experimental data (via a least-squares procedure) in order to obtain model coefficients. These model coefficients are then used to reconstruct, “interpolate,” and smooth the experimental data based on the DNS modes. In addition, in order to compare and assess the effectiveness of the present DNS-based procedure, Kriging interpolation is performed on the experimental (as well as numerical) data. These procedures are applied to time periodic (experimental) instantaneous spanwise vorticity (ωz) at a constant spanwise location. We have demonstrated that particle-image-velocimetry (PIV)-based POD modes can be smoothed by Kriging interpolation, thus a noise-free reconstruction of PIV data can be achieved. It is also found that for very low resolution experimental data, DNS-based interpolation is superior over Kriging interpolation. On the other hand, Kriging interpolation based on the Gaussian correlation model works very well for sufficiently high resolution experimental data. The correlation parameter can be used to control the degree of smoothness in the data reconstruction. Both procedures effectively eliminate the unwanted noise in the experimental data. One important difference between the two procedures is that, with quite some confidence, the DNS-based procedure can also be used for “extrapolation” since the model coefficients do not depend on spatial variation. In fact, we show that near-wall spanwise vorticity, which is not available from experimental data, can be recovered faithfully. Moreover, the enhancement (interpolation and smoothing) of full three-dimensional PIV data has been performed by Kriging interpolation employing a Gaussian correlation model.

Reduced dynamical models are derived for transitional flow and heat transfer in a periodically grooved channel. The full governing partial differential equations are solved by a spectral element method. Spontaneously oscillatory solutions are computed for Reynolds number Reу300 and proper orthogonal decomposition is used to extract the empirical eigenfunctions at Reϭ430, 750, 1050, and Prϭ0.71. In each case, the organized spatio-temporal structures of the thermofluid system are identified, and their dependence on Reynolds number is discussed. Low-dimensional models are obtained for Reϭ430, 750, and 1050 using the computed empirical eigenfunctions as basis functions and applying Galerkin's method. At least four eigenmodes for each field variable are required to predict stable, self-sustained oscillations of correct amplitude at ''design'' conditions. Retaining more than six eigenmodes may reduce the accuracy of the low-order models due to noise introduced by the low-energy high order eigenmodes. The low-order models successfully describe the dynamical characteristics of the flow for Re close to the design conditions. Far from the design conditions, the reduced models predict quasi-periodic or period-doubling routes to chaos as Re is increased. The case Prϭ7.1 is briefly discussed.

Kriging method for data reconstruction and spatial enhancement of stereo-particle image velocimetry ͑S-PIV͒ data for a transitional boundary layer with a laminar separation bubble is investigated. Particularly, the effect of various variogram models and their parameters are studied in detail. In addition, we show that missing data clusters, or black zones, which often occur in PIV measurements, can be estimated using kriging provided the data are well correlated. An important issue in PIV measurements is that built-in PIV data processing software might have problems to detect or to correct spurious erroneous vectors called "outliers." It is shown that these outliers can be eliminated or greatly alleviated using kriging. 2 isosurface and stream traces show that noisy vortical structures are eliminated but the main coherent structures are well preserved and smoothed, thus procedures for the detection and tracking of vortex core lines can be effectively applied on kriged data. Analytical test data for a more quantitative evaluation of the performance of kriging are given in the Appendices.