RECONSTRUCTION OF TURBULENT DATA WITH GAPPY POD METHOD

Gappy POD is a method of data reconstruction based on the proper orthogonal decomposition (POD). We study the applicability of gappy POD to the reconstruction of fluid turbulence configurations and focus mainly on two factors. The first factor is the complexity of the data, which mostly depends on the number of POD modes with non-zero eigenvalues. The second factor is the area and the geometry of the gap. By taking these factors into account, we reformulate the gappy POD reconstruction and derive a formula to compute the reconstruction error. Rotating turbulence data is used as a case study of gappy POD reconstruction, where the reconstruction error can be separated into two parts. The first contribution to the reconstruction error is from the truncation error during the POD expansion and it is amplified by the smallest eigenvalue of the matrix, which consists of POD modes at known indexes. This error mainly depends on the flow complexity, e.g. for flow of moderate complexity, this error decreases with the increase in number of POD modes employed during the reconstruction process. For flow of large complexity, a small POD truncation error can be detrimental and contribute signification to the reconstruction error. Therefore, all POD modes should be considered when utilizing Gappy POD reconstruction to eliminate the truncation error, especially for the turbulent flow field. The second part of the reconstruction error appears when the matrix composed of POD modes at the known points is not of full column rank. This part of error depends on the area and the geometry of the gap. The gap area determines the amount of the lost information. For the same gap area, the gap geometry determines the correlation of the lost information. Gappy POD reconstruction works well when both the amount and the correlation of the lost information are small.