Author(s): Naila Matin; Hasan Zobeyer; Abul B. M. Baki

Linked Author(s): Naila Matin

Keywords: Dvection; Higher order schemes; Flux limiters; Accuracy; Numerical diffusion

Abstract: In Computational Fluid Dynamics (CFD), perfect modeling of the convection-dominated flow still remains a challenge. A great many numerical schemes have been developed in the last few decades for better approximation of the convective terms in CFD. This paper deals with a comprehensive review of 30 such well-recognized higher-order upwinding methods and an assessment of their performance in terms of accuracy and diffusivity. Flux limiters of different forms were imposed on many of these schemes, which resulted in total variation diminishing (TVD) solutions. The schemes were applied to unsteady, one-dimensional pure advection at constant velocity scenario for three different initial profiles (a triangular, a rectangular, and a half-circle distribution). Each case was simulated for 450 seconds and for two different courant numbers, Cr = 0.5 and 0.9, at identical conditions to ensure comparable results. The performance of the schemes was assessed against the analytical solution with the help of two parameters: the normalized root mean square error (NRMSE) and the normalized variance (NV) index, to quantify the deviation from the exact solution and to reflect the scheme’s compressive or diffusive nature, respectively. First-order upwinding method was also tested alongside - which provided a good base condition for visual comparison, being one of the most stable differencing methods. Adaptive-stencil-expansion scheme, fourth and fifth-order upwinding, SUPERBEE, QUICKEST, MUSCL were some of the methods that yielded considerably better results. Accuracy of the results greatly increased with higher courant numbers. The approximation to the exact solution was much closer for the triangular profile, where the slope was more uniform than the rectangular and the half-circle distributions. Moreover, the use of flux limiters was found to improve the stability of several schemes noticeably. NRMSE value for the triangular distribution and for Cr = 0.9, was calculated to be in the range of 1.93 to 2.89 for the top six performing schemes, suggesting reasonably good accuracy, and the NV indices were within ±0.5 for these six schemes, indicating that these methods were neither too diffusive nor compressive. However, for the same scenario, NRMSE and NV index values for the first-order upwinding method were 10.09 and -10.37, respectively- which was much less accurate and more diffusive compared to the top six higher-order schemes. The results of this study offer some valuable observations regarding the performance of the schemes, and the findings are likely to help with the selection process of better, more accurate, and stable methods for simulating the advection terms. As a continuation of the current study, the best performing six schemes will be further tested for accuracy and computational efficiency in a coupled depth-averaged (DA) and Reynolds averaged Navier-Stokes (RANS) 3D model developed for open channel flow.

DOI: https://doi.org/10.3850/IAHR-39WC2521711920221277

Year: 2022