Finite-volume scheme upwind

The finite volume methods have been used to discretised the partial differential equations of the model using the Simple method for pressure-velocity coupling and the second order upwind scheme to interpolate the variables on the surface of the control volume. The segregated solution algorithm was selected. The Reynolds stress turbulence model was used in this model due to the anisotropic nature of the turbulence in cyclones. Standard fluent wall functions were applied and high order discretisation schemes were also used. 

To illustrate the principles of the finite volume method, as a first approach, the implicit upwind differencing scheme is used for a multi-dimensional problem. Although the upwind differencing scheme is very diffusive, this scheme is frequently recommended on the grounds of its stability as the preferred method for treatment of convection terms in multiphase flow and determines the basis for the implementation of many higher order upwinding schemes. 

Based on the finite volume method, the control equation can be converted to a numerical method for solving algebraic equations. Convection of equation use second-order upwind difference during the discrete process, the solver is based on the pressure, the pressure-velocity coupling adopt the SIMPLE algorithm, pressure interpolation scheme use PRESTO Format. 

The model equations were solved numerically by using the commercial software FLUENT 6.2 with finite volume method. The SIMPLEC algorithm was used to solve the pressure-velocity coupling problem in the momentum equations. The second-order upwind spatial discretization scheme was employed for all differential equations. 

All governing equations are all solved using a finite volume discretization, see [7]. All vectors quantities, e.g. position vector, velocity and moment of momentum, are expressed in Cartesian coordinates. Non-staggered variable arrangement is used to define dependent variables all physical quantities are stored and computed at cell centers. An interpolation practice of second order accuracy is adopted to calculate the physical quantities at cell-face center [8]. The deferred correction approach [9] is used to compute the convection term appearing in the governing equations by blending the upwind difference and the centi difference scheme. 

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