A wavenumber parallel computational code for the numerical integration of the Navier-Stokes equations

A parallel computational code for the numerical integration of the Navier–Stokes equations has been developed. The system of partial differential equations describing the non-steady flow of a viscous incompressible fluid in three dimensions is considered and applied to the channel flow problem. A mixed spectral-finite difference technique for the numerical integration of the governing equations is devised: Fourier decomposition in both streamwise and spanwise directions and finite differences in the direction orthogonal to the solid walls are used, while a semi-implicit procedure of Runge–Kutta and Crank–Nicolson type is utilised for the advancement in time. A wavenumber parallelism is implemented for the execution of the calculations. Within each time step of integration, the computations are executed in two distinct phases, each phase corresponding to a different way of decomposing the computational domain, vertically and horizontally, respectively; in both phases of the whole calculation process, each portion of the computing domain is handled by a different CPU on a Convex SPP 1200/XA parallel computing system. Results are presented in terms of performance of the calculation procedure with the use of 2,4,6 and 8 processors respectively and are compared with the single-processor performance. Also the accuracy of the parallel algorithm has been tested, by analysing the evolution in time of small amplitude disturbances of the mean flow; a satisfactory agreement with the theoretical solution given by the hydrodynamic stability theory is found, provided that a given number of grid points in the y direction are present.