Static pressure is a scalar magnitude that expresses the force per unit area exerted by a fluid at rest. As such, it constitutes one of the two mechanisms through which fluid flows generate forces on bodies. Moreover, static pressure is not only relevant in the definition of surf
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Static pressure is a scalar magnitude that expresses the force per unit area exerted by a fluid at rest. As such, it constitutes one of the two mechanisms through which fluid flows generate forces on bodies. Moreover, static pressure is not only relevant in the definition of surface loads, as it plays a key role in a number of fields, such as turbulence research due to its impact on the amplification or damping of turbulent flow instabilities, or medical research, provided its key paper on cardiovascular disease.
Accordingly, different approaches exist that allow to obtain pressure information in Fluid Dynamics applications. Among these, simplified analytical models, Computational Fluid Dynamics and experimental measurements stand out given their extensive use. While each of these comes with its own advantages and limitations, the latter typically offers the advantage of being conducted with real flows, hence providing a reliable source of information if proper similarity parameters and set-up are achieved.
While there exist different techniques to measure pressure experimentally, among which pressure tapping and pressure sensitive paint stand out, these present major limitations, such as the limited spatial resolution that can be achieved without intrusion effects or the challenges encountered during calibration, respectively. Consequently, more recent methods have been developed that allow to reconstruct pressure from velocity fields obtained via Particle Image Velocimetry. An instance of such algorithms is the Poisson Solver, which is based on the application of the incompressible relation to the momentum conservation equations, yielding a boundary-value problem for pressure. Nonetheless, while this approach benefits from the instantaneous and simultaneous nature of PIV measurements, it presents its own challenges, among which the propagation of noise from the velocity field into the reconstructed pressure field stands out.
More recently, with the proliferation of Machine Learning, the number of applications in Fluid Mechanics has grown. In particular, an approach that stands out are Physics-Informed Neural Networks, which optimize Deep NN models minimizing a loss function with contributions from labeled flow data variables and residuals from physical equations, thus learning the flow field variables. While diverse use cases have been reported for PINNs, such as the generation of reduced order models or the direct simulation of flows, especial emphasis has been placed in research on their ability to infer unsteady or mean pressure fields from velocity measurements, via the application of the Navier-Stokes equations.
Even if research has shown PINNs offer key advantages with respect to traditional pressure reconstruction methods, such as robustness to Gaussian noise or lack of discretization errors, analysis of the research available highlights key areas that require further exploration in the establishment of PINNs as a reliable alternative to traditional methods. Specifically, the study of PINNs performance with real experimental data and its comparison with solvers as the Poisson against direct experimental measurements is of paramount relevance, provided that the vast majority of publications concern the use of artificial experimental data from CFD simulations.
In accordance, a PINN framework has been developed and its accuracy in the reconstruction of surface pressure has been tested using time-averaged data from both CFD simulations and experimental tests of the two-dimensional flow around a cylinder. Particularly, comparison of the PINNs and the Poisson surface pressure reconstructions with pressure tap data showed superior performance of the former, with respective MSE reductions of -1% and -21% for the flows around a smooth cylinder and one fitted with zig-zag strips at θ = ±45 º.
Additionally, sensitivity studies to understand the effect of various parameters in the PINN training process has resulted in the identification of various trends. Among these, especial attention is required by the ability of PINNs to add regularization in areas affected by correlated noise such as reflections via the addition of collocation points, where the PDE loss is evaluated. Further noteworthy findings concern the benefit of using physical boundary conditions at solid surfaces in the form of the no-slip, no-penetration and no-fluctuations constraints. In this study, it has been proven that these allow not only to bypass non-physical pressure fluctuations that derive from spatially-correlated noise, but also to reduce surface pressure reconstruction error when data gaps exist close to the surface, achieving reductions of up to -92% for a radial data gap of 75% of the cylinder radius from the cylinder surface. Finally, it has been shown that, provided that the dataset contains points that allow to define a reference pressure value, the provision of sparse pressure tap data to the NN during training results in local, rather than generalized, error reductions.
In conclusion, the sensitivity studies carried out on the smooth cylinder dataset have resulted in pressure MSE reductions as substantial as -50\% with respect to the Poisson solver when both are compared with static pressure tap data, supporting the establishment of PINNs as an alternative method to conventional pressure reconstruction algorithms as the Poisson solver, despite the time penalty that these can represent given the instantaneous nature of the latter. On the qualitative side, it has been proven that PINNs can provide a flexible framework to embed prior knowledge of the solution, such as the positive nature of normal Reynolds Stress components or boundary conditions. Along the same lines, it is shown throughout the thesis how PINNs can be used to perform debugging steps that allow to identify sources of error.