Finite-volume scheme QUICK

P 61] The numerical simulations were based on the solution of the incompressible Navier-Stokes equation and a convection-diffusion equation for a concentration field by means of the finite-volume method [152], The Einstein convention of summation over repeated indices was used. For pressure-velocity coupling, the SIMPLEC algorithm and for discretization of the species concentration equation the QUICK differencing scheme were applied. Hybrid and the central differencing schemes referred to velocities and pressure, respectively (commercial flow solvers CFX4 and CFX5). 

Hayase, T, Humphrey, J.A.C. and Grief, R. (1992), A constantly formulated QUICK scheme for fast and stable convergence using finite volume iterative calculation procedures, J. Computat. Phys., 98, 108-118. 

The snapshot approach for gas-liquid flows was implemented using a commercial CFD code, FLUENT (Fluent Inc., USA). User-defined subroutines were used for this purpose. Half of the vessel was considered as a solution domain. The solution domain and details of the finite volume grid used was similar to those used for singlephase flows discussed earlier (however, the number of cells in the 6 direction were half of that used in single-phase simulations). A QUICK discretization scheme with SUPERBEE limiter function was used to integrate all the equations (Fluent User Guide, 1997). Simulations were carried out for three values of dimensionless gas flow rates (Qc/ND ), 0.01, 0.02 and 0.03. 

Alternative higher-order methods are available in the literature, and most of them share the salient characteristics of the spatial discretization schemes used in finite-volume codes (such as, for example, QUICK and MUSCL). Some of them are based on polynomial reconstructions with some minor modifications in order to ensure positivity of the NDF, as explained in Laurent (2006), whereas others are based on the maximization of Shannon entropy (see, for example, Massot et at. (2010)). 

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