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This study presents an innovative implicit–explicit time-stepping algorithm based on a first-order temporal accuracy method, addressing challenges in simulating all-regimes of fluid flows. The algorithm's primary focus is on mitigating stiffness inherent in the density-based “Roe” method, pivotal in finite volume approaches employing unstructured meshes. The objective is to comprehensively evaluate the method's efficiency and robustness, contrasting it with the explicit fourth-order Runge–Kutta method. This evaluation encompasses simulations across a broad spectrum of Mach numbers, including scenarios of incompressible and compressible flow. The scenarios investigated include the Sod Riemann problem to simulate compressible Euler equations, revealing the algorithm's versatility, and the low Mach number Riemann problem to analyze system stiffness in incompressible flow. Additionally, Navier–Stokes equations are employed to study viscous and unsteady flow patterns around stationary cylinders. The study scrutinizes two time-stepping algorithms, emphasizing accuracy, stability, and computational efficiency. The results demonstrate the implicit–explicit Runge–Kutta algorithm's superior accuracy in predicting flow discontinuities in compressible flow. This advantage arises from the semi-implicit nature of the equations, reducing numerical errors. The algorithm significantly enhances accuracy and stability for low Mach number Riemann problems, addressing increasing stiffness as Mach numbers decrease. Notably, the algorithm optimizes computational efficiency for both low Mach number Riemann problems and viscous flows around cylinders, reducing computational costs by 38%–68%. The investigation extends to a two dimensional hypersonic inviscid flow over cylinder and double Mach reflection case, showcasing the method's proficiency in capturing complex and hypersonic flow behavior. Overall, this research advances the understanding of time discretization techniques in computational fluid dynamics, offering an effective approach for handling a wide range of Mach numbers while improving accuracy and efficiency.
This study presents an innovative implicit–explicit time-stepping algorithm based on a first-order temporal accuracy method, addressing challenges in simulating all-regimes of fluid flows. The algorithm's primary focus is on mitigating stiffness inherent in the density-based “Roe” method, pivotal in finite volume approaches employing unstructured meshes. The objective is to comprehensively evaluate the method's efficiency and robustness, contrasting it with the explicit fourth-order Runge–Kutta method. This evaluation encompasses simulations across a broad spectrum of Mach numbers, including scenarios of incompressible and compressible flow. The scenarios investigated include the Sod Riemann problem to simulate compressible Euler equations, revealing the algorithm's versatility, and the low Mach number Riemann problem to analyze system stiffness in incompressible flow. Additionally, Navier–Stokes equations are employed to study viscous and unsteady flow patterns around stationary cylinders. The study scrutinizes two time-stepping algorithms, emphasizing accuracy, stability, and computational efficiency. The results demonstrate the implicit–explicit Runge–Kutta algorithm's superior accuracy in predicting flow discontinuities in compressible flow. This advantage arises from the semi-implicit nature of the equations, reducing numerical errors. The algorithm significantly enhances accuracy and stability for low Mach number Riemann problems, addressing increasing stiffness as Mach numbers decrease. Notably, the algorithm optimizes computational efficiency for both low Mach number Riemann problems and viscous flows around cylinders, reducing computational costs by 38%–68%. The investigation extends to a two dimensional hypersonic inviscid flow over cylinder and double Mach reflection case, showcasing the method's proficiency in capturing complex and hypersonic flow behavior. Overall, this research advances the understanding of time discretization techniques in computational fluid dynamics, offering an effective approach for handling a wide range of Mach numbers while improving accuracy and efficiency.
This paper introduces a novel platform that integrates three preconditioning matrices based on conservative variables: the Turkel, Choi–Merkle and Onur matrices. The platform aims to compare these matrices in terms of accuracy and robustness by investigating their performance in solving three distinct and challenging high-gradient laminar flow problems: (i) Bi-Plane NACA0012 airfoil, (ii) lid-driven flow in a square cavity and (iii) flow in a planar T-junction. These problems serve as new and challenging test cases to accurately determine the abilities of the preconditioning matrices. The preconditioning matrices are applied to evaluate the numerical solutions, and their performance in complex flow fields is assessed in terms of accuracy and efficiency. By solving these flow problems, the effectiveness of the preconditioning matrices is thoroughly analyzed. By integrating these preconditioning matrices into a single platform, this paper significantly contributes to the field. The approach enables a direct and meaningful comparison of the performance of the Turkel, Choi–Merkle and Onur matrices in solving these new and challenging laminar flow problems. While all three matrices demonstrate comparable accuracy in predicting flow characteristics like pressure coefficients, Turkel’s method shows superior convergence acceleration across the almost test cases due to alpha parameter and modifications of the momentum equations. In the external flow case, Turkel converges 22–53% faster than the other matrices by adjusting momentum terms with an alpha parameter. For the internal lid-driven cavity case, Turkel again accelerates convergence up to 38% over Choi–Merkle as Reynolds and Mach numbers increase. However, Onur’s method stalls at high Reynolds/Mach numbers. At low values of Reynolds and Mach numbers, Onur reduces computational cost by 17%. Finally, for the T-junction case, Turkel and Choi–Merkle perform almost identically, decreasing CPU time by 55% vs Onur, which lacks convergence robustness. While the preconditioning matrices have similar accuracy, Turkel offers the best convergence improvement by accounting for momentum effects. Onur works well at low Reynolds/Mach numbers but shows limitations at higher values. The selection of the optimal preconditioning matrix should be based on whether momentum or viscosity dominates in the flow field, as well as the intensity of the viscosity gradient rate.
The accurate computation of different turbulent statistics poses different requirements on numerical methods. In this paper, we investigate the capabilities of two representative numerical schemes in predicting mean velocities, Reynolds stress and budget of turbulent kinetic energy (TKE) in low Mach number flows. With concerns on numerical order of accuracy, dissipation and dispersion properties, a high-order upwind scheme with relatively good dispersion and dissipation and a second-order non-dissipative central scheme with perfect dissipation but poor dispersion are adopted for this comparative study. By carrying out a series of numerical simulations including Taylor-Green vortex, turbulent channel flow at $$Re_{\tau }=180$$ R e τ = 180 and turbulent flow over periodic hill at $$Re_{b}=10595$$ R e b = 10595 , it can be obtained that although the high-order upwind scheme lacks perfection of dissipation in high wave number range, it still demonstrates superior predictive capability compared with the second-order non-dissipative central scheme, especially with relatively coarse grids. Finally, by taking the high-order upwind scheme as a suitable selection for turbulence simulation, the turbulent flow over a 30P30N multi-element airfoil is investigated as an application study. After briefly comparing the simulated profiles and spectrum with reference experimental results as validation, the budget of TKE is analyzed to locate the dominant flow structures and regions. It is found that the production and dissipation terms behave in a “monopole” pattern in the locations with strong shears and wakes. Whereas the advection and diffusion terms show an “inward” pattern and an “outward” pattern, which indicate the spatial transport of TKE between the center of the shear layer and nearby locations.
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