An Artificial Compressibility Method for 1D Simulation of Open-Channel and Pressurized-Pipe Flow

Abstract

Piping systems (e.g., storm sewers) that transition between free-surface flow and surcharged flow are challenging to model in one-dimensional (1D) networks as the continuity equation changes from hyperbolic to elliptic as the water surface reaches the pipe ceiling. Previous network models are known to have poor mass conservation or unpredictable convergence behavior at such transitions. To address this problem, a new algorithm is developed for simulating unsteady 1D flow in closed conduits with both free-surface and surcharged flow. The shallow-water (hydrostatic) approximation is used as the governing equations. The artificial compressibility (AC) method is implemented as a dual-time-stepping discretization for a finite-volume solver with timescale interpolation used for face reconstruction. A new formulation for the AC celerity parameter is proposed such that the AC celerity matches the equivalent gravity wave speed for the local hydraulic head—which has some similarities to the classic Preissmann Slot used to approximate pressurized flow in conduits. The new approach allows the AC celerity to be set locally by the flow (i.e., non-uniform in space) and removes it as a free parameter of the AC solution method. The derivation of the AC method provides for only a minor change in the form of the solution equations when a computational element switches from free-surface to surcharged. The new solver is tested for both unsteady free-surface (supercritical, subcritical) and surcharged flow transitions in a circular pipe and is implemented in an open-source Python code available under the name “PipeAC.” The results are compared to laboratory experiments that include rapid flow changes due to opening/closing of gates. Results show that the new algorithm is satisfactory for 1D representation of unsteady transition behavior with two caveats: (i) sufficient grid resolution must be applied, and (ii) the shallow-water equation approximations (hydrostatic, single-fluid) limit the accuracy of the solution with regards to the celerity of the turbulent unsteady bore that propagates upstream. This research might benefit any piping network model that must smoothly handle unsteady transitions from free surface to surcharged flow.

Artificial Compressibility with Riemann Solvers: Convergence of Limiters on Unstructured Meshes

“…In this paper, we focus on the method of artificial compressibility, which scales better than matrix inversion as it requires less communication between processors. Indeed, although artificial compressibility is not new, it is currently very relevant due to its suitability for implementation on massively parallel architectures [12,13,14].…”

“…PIMPLE, a combination of SIM-PLE (semi-implicit method for pressure-linked equations) [10] and PISO (pressure-implicit with splitting of operators) [11], is a predictor-corrector method that relies on matrix inversion. While such methods require fewer total iterations than artificial compressibility and therefore would be more efficient in serial [12], artificial compressibility is easier and more efficient to parallelise [12,13,14] and can lead to almost linear speed-up [15]. This is because solving local wave propagation problems requires less communication between processors than matrix inversion [12].…”

“…While such methods require fewer total iterations than artificial compressibility and therefore would be more efficient in serial [12], artificial compressibility is easier and more efficient to parallelise [12,13,14] and can lead to almost linear speed-up [15]. This is because solving local wave propagation problems requires less communication between processors than matrix inversion [12]. Therefore, it is worthwhile to depart from the convention of using PIMPLE in OpenFOAM ® and investigate implementing artificial compressibility instead because of its scaling potential on the massively parallel architectures of today.…”

“…Even though the new solver does not use much of the numerics in OpenFOAM ® , it is useful to develop the solver in OpenFOAM ® because the lowlevel structure does not have to be built from scratch. A downside is that the form of parallelisation in OpenFOAM ® , domain decomposition, is not optimal for artificial compressibility as the waves propagate locally [12]. However, we focus on the numerics in this paper, leaving the implementation of efficient parallelisation to future work.…”