Hyperbolic model of fluid flow under Bayesian paradigm

Ferdinand Evert Uilhoorn

Zeszyt 73

Theoretical and numerical modeling of fluid transients in pipeline systems is a challenging field of research. The governing flow equations constitute a system of nonlinear hyperbolic partial dierential equations enforcing the conservation laws for mass, momentum and energy. The application of these mathematical models might be limited due to the absence of complete knowledge about the physical phenomena and uncertainties. Knowledge about the initial and boundary conditions is usually obtained from measurements. The presence of noise and inaccuracies in these measurements, as well as inexactness of the fluid flow model and approximations for solving the full mathematical system, can lead to predictions that significantly dier from reality. 
Our interest is to deal with the problem of extracting information about states, or parameters, or both, of the system in real time given noisy measurements. We investigated the performance of dierent nonlinear data assimilation methods within the Bayesian framework applied to a quasilinear nonhomogeneous hyperbolic system of partial diserential equations of first order describing fluid transients. These methods merge sparse data into numerical models to optimize predictions and reduce uncertainties in the modeled state variables. The performance of the extended Kalman filter, unscented Kalman filter and two particle filters, namely sequential importance resampling and its variant, the auxiliary particle filter, were investigated.
Numerical experiments were conducted for an isothermal and nonisothermal flow field. The isothermal fluid flow model in mathematical conservationlaw form was solved with the two-step Lax–Wendro scheme and a semidiscrete finite volume scheme using flux limiters. The latter high-resolution technique was applied to estimate flow transients using the extended Kalman filter while allowing for solutions that contain discontinuities, such as shock waves. The nonisothermal flow model in nonconservative form was solved with the method of lines using a classical five-point, fourth-order finite dierence approximation.
The semidiscrete approximations were integrated with a multistage explicit Runge–Kutta scheme.
With respect to estimation accuracy, robustness and computation time of the Bayesian algorithms, we discussed the impact of inverse crime, ensemble size and resampling algorithm in the particle filter, spatial and temporal resolution of sensor readings, noise statistics and gradient steepness in the mass flow boundary conditions. Simulations were conducted for fluids in dense liquid or gaseous phase. 
In general, we can conclude that in most of the situations the Bayesian approach is successful in estimating fluid transients. Taking into account the computation time of the unscented Kalman filter, robustness issues of the particle filters and numerical eciency of computing the Jacobian matrix, the extended Kalman filter would be a better choice for real-time state estimation.

Keywords:  Bayesian filtering, data assimilation, particle filters, Kalman filters, fluid flow modeling, hyperbolic partial dierential equations

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