An eigen-ensemble algorithm for localization of scalar sources in turbulent environments

Identifying the location and intensity of a passive scalar source from remote measurements in a turbulent flow is a challenging inverse problem. We introduce a new algorithm that can be realized using simulation data or physical measurements, and that makes use of prior tests and new sensor data to provide accurate predictions of the source parameters. We start with a set of trial sources to estimate the right and left singular vectors of the impulse-response system, namely the eigen-sources and corresponding eigen-measurements. Once we have new sensor data, we project them onto the eigen-measurements and use the results to synthesize the true source in terms of a superposition of the eigen-sources, appropriately weighted by the corresponding singular values. Since the eigen-sources converge for long time horizons, and are related to statistical properties of the scalar fields, the difficulty of identifying the sources from noisy measurements can be related to the statistics of the scalar field. Results that demonstrate the accuracy of the source reconstruction and the associated uncertainties will be discussed in the context of canonical turbulent channel flow at $Re_{\tau} = 180$.