POD-BASED ESTIMATION OF THE FLOW FIELD FROM FREE-SURFACE VELOCITY IN THE BACKWARD-FACING STEP

With a view to “nowcasting” of depth-scale flow fields in rivers, estuaries, and the near-coast, this work develops empirical “measurement models” that correlate velocity at a water surface to subsurface flow fields. Free-surface flow over a backstep, as simulated by LES, is chosen as a test case. Proper Orthogonal Decomposition (POD) is first applied to both subsurface and surface velocity fields from a “training data set”, whence we develop regression models between the vectors of surface and subsurface POD coefficients. These models are then applied to estimate the subsurface flow field outside the training data based on virtual “measurements” of streamwise velocity along a streamwise line at the water surface. Good predictive capability is obtained in a region some 20 step heights downstream of the step that is characterized by strong upwellings. INTRODUCTION This paper applies measurement models developed by Mokhasi et al. [1], who were motivated by “micronowcasting” and back-casting of chemical release in a city, and the “multi-time” extensions reported by Nguyen et al. [2]. “Nowcasting” referred originally to detailed, short-term weather forecasting, e.g. to predict the development of thunderstorms. The mathematics are the same as a “traditional” forecasting system, and typically include a nonlinear Kalman Filter (KF) that continually updates the “current estimate” for the vector of variables that specify the state of the system (atmosphere, ocean, etc.) in the chosen dynamical model. Since such physical systems are chaotic, monitoring data must be continually fed back to keep the current estimate close to reality; in the KF this data is compared with the “current state” via a “measurement model”. In the scenario of [1], data from sensors distributed through a cityscape would feed back on a KF designed to track the turbulent flow; then, based on the estimated flow field, chemical concentration measurements would be “back-tracked” to estimate release location(s). As a prototype problem, they considered unidirectional flow past a surface-mounted cube. The measurement models proposed in [1] develop an idea from the fluids engineering literature known as “stochastic estimation”, in which one estimates a flow field, via its POD coefficients, based on a limited number of measurements, typically pressure at one or more surface points. The dominant example has been Linear Stochastic Estimation (LSE). For example, Taylor and Glauser [3] built a low dimensional pressure sensing and control system via POD and LSE for a flow between a backward facing ramp and an adjustable flap; Durgesh and Naughton [4] applied a multitime delay LSE-POD that used pressure to estimate POD coefficients of an unsteady near wake flow. We are motivated by nowcasting of environmental flows with a free surface, notably estuaries and rivers. Existing systems (e.g. the NYHOPS network, [5]) are typically based on sparse point measurements, and cannot resolve horizontal variations much finer than the river width. For the future, we expect that High Frequency radar and/or imaging methods will be able to monitor surface velocity over a certain area. Indeed, highly-resolved data on surface velocity were obtained as early as 1967 by large-format aerial photography when rising flood waters generated surface bubbles as “PIV tracers”, and the patterns of surface divergence even allowed Kinoshita [6] to infer the instantaneous spatial structure of subsurface bedforms. With this justification we investigate here the use of data measured over a portion of free surface as input for stochastic estimation. In the context of stochastic estimation, the introduction of a free surface was the key novelty in our recent communication [7]. That work considered simulated flow over a sinusoidal bed, with a free surface modeled as a stress-free lid. To simplify analysis, we confined attention to the flow velocity in three vertical-streamwise planes, and built POD-based correlations for the subsurface flow based on “measured” surface velocity. In this paper, we also use information on free-surface velocity as input for estimation models. However, we instead study the flow past a backstep as a prototype for a subaqueous dune. Such “bedforms” are important because of their form drag, and because of their key role in sediment dynamics. In addition, the current work applies the correct dynamical conditions at the free surface. The LES of the backward-facing step flow with a free-surface is followed by a brief mathematical introduction of POD. Next, single-time and multi-time POD-based estimations are described and the performances of the POD-based estimations are discussed.