Analytical ‘steady-state’-based derivation and clarification of the courant-friedrichs-lewy condition for pipe flow
Zdzisław Kowalczuk, Marek S. Tatara
Abstract
This article addresses the problem of choosing the optimal discretization grid for emulating fluid flow through a pipeline. The aggregated basic flow model is linearized near the operating point obtained from the steady state analytic solution of the differential equations under consideration. Based on this model, the relationship between the Courant number (μ) and the stability margin is examined. The numerically set coefficient μopt, ensuring the maximum margin of stability, is analyzed in terms of the physical and technological parameters of the flow. As a result of this analysis, a specific formula is obtained based on parameters describing the mechanics (geometry and physics) of the flow through the pipeline, which leads to the optimal value of the Courant number, separately for smooth and rough pumping conditions. A more detailed analysis of the distribution of the optimal μ coefficient in relation to the parameters of the pipeline flow mechanics shows four cases to consider when determining the coefficient μopt. Surprisingly, in three cases, the CFL condition is insufficient, which is expressed in the form of the proposed procedure for choosing the optimal value of μ. The final dichotomous model is derived from the Monte Carlo simulation results in which the effect of each parameter on the optimal Courant number is estimated and consolidated. Taking into account the recognized general laws of physics and using numerical methods and mathematical analysis, simple and useful analytical relationships describing the flow process are obtained. In addition, computer simulations are performed to verify the correctness of the proposed procedure, as well as a number of other considerations related to the modeling of fluid flow in transport pipelines.