Abstract
We compare the performance of line Gauss–Seidel (LGS), point Gauss–Seidel (PGS), and alternating direction implicit (ADI) linear solvers used in the artificial compressibility method for the numerical solution of the three-dimensional incompressible Navier–Stokes equations. Spatial discretization is carried out using a fifth-order WENO scheme for the convective terms and a second-order central difference scheme for the viscous terms. A comparison is made by simulating the spherical Couette flow problem, with only the inner sphere rotating and the outer one fixed. OpenMP is used for numerical computation in parallel for the three schemes. First, we compare the numerical efficiency of the solvers by computing 0-vortex flow for a medium-gap σ=R2−R1/R1=0.25. Second, we make a residual comparison for a steady-state flow calculation based on the CFL number and artificial compressibility factor. Finally, we compare the three solvers for unsteady flow computations based on the artificial compressibility factor. The results show that the LGS solver is more reliable than the PGS and the ADI solvers. To show the accuracy of the LGS scheme, we compute different flow modes for an intermediate-gap clearance ratio σ=R2−R1/R1=0.14. The computed results have good agreement with the existing numerical results.
Highlights
The approximate solutions of three-dimensional incompressible Navier–Stokes equations (3D-INSEs) are required in several engineering applications such as in the ship industry and problems in geosciences and hydraulic machinery.1,2 Many researchers have solved these equations analytically for flow problems,3–6 but complicated flow problems require numerical methods
It indicates that the line Gauss–Seidel (LGS) and point Gauss–Seidel (PGS) schemes are accurate enough for a range of β and the reduction in the residual by 2–3 orders of magnitude is sufficiently accurate for unsteady flow simulations
Yavorskaya et al.35 developed an empirical formula Rec = 41.3(1 + σ)σ−3/2 for the narrow, intermediate, and some medium-gap ranges 0.08 ≤ σ ≤ 0.25. He developed an experimental formula for the maximum number of Taylor vortex (TV) pairs i on each side of the hemisphere as i = 0.21σ−4/3
Summary
The approximate solutions of three-dimensional incompressible Navier–Stokes equations (3D-INSEs) are required in several engineering applications such as in the ship industry and problems in geosciences and hydraulic machinery. Many researchers have solved these equations analytically for flow problems, but complicated flow problems require numerical methods. The artificial compressibility method (ACM) does not need to solve the pressure Poisson equation. When the pseudo-time derivative converges to zero, the continuity equation is satisfied The advantage of this method is that it can be applied with many numerical schemes that were initially developed for compressible flow problems in fluid dynamics. Many researchers in the past have investigated medium-gap clearance ratios, but there are few studies available in the literature for the intermediate-gap clearance ratio σ = 0.14.38,48,50 The authors in Ref. 48 obtained supercritical spiral Taylor-vortex flow (TVF) by rotating the inner sphere and fixing the outer sphere for different Reynolds numbers. In Ref. 38, the author used the artificial compressibility method along with an ADI scheme to find different flow transitions for the intermediate-gap σ = 0.14 in detail.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
