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A pointwise simultaneous solution algorithm based on dual time stepping was developed by De Wilde et al. (2002). With increasing grid aspect ratios, the efficiency of the point method quickly drops. Most realistic flow cases, however, require high grid aspect ratio grids, with the highest grid spacing in the streamwise direction. In this direction, the stiffness is efficiently removed by applying preconditioning (Weiss and Smith, 1995). In the direction perpendicular to the stream wise direction, stiffness remains because of the viscous and the acoustic terms. To resolve this problem, a line method is presented. All nodes in a plane perpendicular to the stream wise direction, a so-called line, are solved simultaneously. This allows a fully implicit treatment of the fluxes in the line, removing the stiffness in the line wise directions. Calculations with different grid aspect ratios are presented to investigate the convergence behavior of the line method. The line method presented is particularly suited for parallelization. At each pseudo-time step, the lines (typically hundreds) can be solved independently of each other. The Message Passing Interface (MPI) standard (Snir et al., 1996) is used. The communication between the processors can be easily reduced by solving a block of lines per processor. The communication is then limited to information regarding only the outer lines of the block. In common practice, the number of lines is much higher than the number of processors available. In this region of the lines/processor space, the reduction of the calculation time is linear with the number of processors that is used.Copyright © 2004 by ASME

Transient 3D simulation of the gas-solid flow in large diameter (> 0.3 m) risers is performed using a new density based solution algorithm. Unlike the conventional pressure based algorithm used so far for riser simulation, the density based solution method uses the pre-conditioning of time derivatives, does not have the internal pressure and velocity correction loop and hence provides a much faster convergence speed. The two phase flow is highly oscillatory with many small scale high frequency (10Hz) fluctuations and a few dominating low frequency (0.02 Hz) oscillations. The transient simulation of a case belonging to a fast fluidized regime with Geldart B particles, demonstrated the density inversion phenomena, experimentally observed before. Cluster formation is simulated on a real time scale for a dilute phase riser having solid fraction < 0.0007, although the time averaged flow fields still resemble a core-annular flow structure.Copyright © 2002 by ASME