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doi: 10.1002/fld.1650170702pmid: N/A
Relaxation‐based multigrid solvers for the steady incompressible Navier–Stokes equations are examined to determine their computational speed and robustness. Four relaxation methods were used as smoothers in a common tailored multigrid procedure. The resulting solvers were applied to three two‐dimensional flow problems, over a range of Reynolds numbers, on both uniform and highly stretched grids. In all cases the L2 norm of the velocity changes is reduced to 10−6 in a few 10's of fine‐grid sweeps. The results of the study are used to draw conciusions on the strengths and weaknesses of the individual relaxation methods as well as those of the overall multigrid procedure when used as a solver on highly stretched grids.
doi: 10.1002/fld.1650170703pmid: N/A
The unsteady flow over an oscillatory NACA0012 aerofoil has been simulated by the calculation with Euler equations. The equations are discretized by an implicit Euler in time, and a second‐order space‐accurate TVD scheme based on flux vector splitting with van Leer's limiter. Modified eigenvalues are proposed to overcome the slope discontinuities of split eigenvalues at Mach = 0·0 and ± 1·0, and to generate a bow shock in front of the aerofoil. A moving grid system around the aerofoil is generated by Sorenson's boundary fitted co‐ordinates for each time step. The calculations have been done for two angles of attack θ = 5·0° sin (ωt) and θ = 3·0° + 3·0° sin (ωt) for the free‐stream Mach numbers 2·0 and 3·0. The results show that pressure and Mach cells flow along characteristic lines. To examine unsteady effects, the responses of wall pressure and normal force coefficients are analysed by a Fourier series expansion.
doi: 10.1002/fld.1650170704pmid: N/A
The advent of vector and massively parallel computers offers researchers the possibility of enormous gains in execution time for scientific and engineering programs. From the numerical point of view, such programs are frequently based on the inversion of sparse, diagonally banded matrices. Conventional scalar solvers often perform poorly on vector machines due to short effective vector lengths, and thus appropriate methods must be chosen for use with vector machines. In this paper a number of commonly used solvers are tested for the Navier–Stokes equations, in both scalar and vector form, on two vector architecture machines. A new method is presented which performs well in both vector and scalar form on a range of vector architectures.
doi: 10.1002/fld.1650170705pmid: N/A
The flow of viscoelastic fluids through a porous channel with one impermeable wall is computed. The flow is characterized by a boundary value problem in which the order of the differential equation exceeds the number of boundary conditions. Three solutions are developed: (i) an exact numerical solution, (ii) a perturbation solution for small R, the cross‐flow Reynold's number and (iii) an asymptotic solution for large R. The results from exact numerical integration reveal that the solutions for a non‐Newtonian fluid are possible only up to a critical value of the viscoelastic fluid parameter, which decreases with an increase in R. It is further demonstrated that the perturbation solution gives acceptable results only if the viscoelastic fluid parameter is also small. Two more related problems are considered: fluid dynamics of a long porous slider, and injection of fluid through one side of a long vertical porous channel. For both the problems, exact numerical and other solutions are derived and appropriate conclusions drawn.
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