Finite-volume scheme second order

For non-Newtonian fluids the viscosity p is fitted to flow curves of experimental data. The models for this fit are discussed in the next chapter. The energy equation is also implemented in the code and can be used for temperature-dependent problems, but it is not needed for the simulation of fluid dynamic problems like jet breakup due to the uncoupling of the density in the incompressible formulation. The finite volume scheme uses the Marker and Cell (MAC) method to discretize the computational domain in space. The convective and diffusive terms are discretized with second-order accuracy and the fluxes are calculated with a Godunov-type scheme. 

The coupled set of flow and solid equations (Eqs. (3.1)-(3.8), (3.11)) were solved simultaneously. A finite volume scheme was adopted for the spatial discretization of the flow equations and solution was obtained with a SIMPLER method for the pressure-velocity field [8]. In the case of transient simulation, the 1-D transient solid energy equation was solved with a second order accurate, fully implicit scheme by using a quadratic backward time discretization [9]. The coupled flow and solid phases were solved iteratively and convergence was achieved at each time step when the solid temperamre did not vary at any position along the wall by more than 10 K. 

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. 

The finite approximations to be used in the discretization process have to be selected. In a finite difference method, approximations for the derivatives at the grid points have to be selected. In a finite volume method, one has to select the methods of approximating surface and volume integrals. In a weighted residual method, one has to select appropriate trail - and weighting functions. A compromise between simplicity, ease of implementation, accuracy and computational efficiency has to be made. For the low order finite difference- and finite volume methods, at least second order discretization schemes (both in time and space) are recommended. For the WRMs, high order approximations are normally employed. 

The thermodynamics of the I-N phase transition has been extensively investigated for resolving the issue concerning the order of the transition. Following the Ehrenfest scheme, a phase transition is classified into a first-order transition or a second-order one, depending upon the observation of finite discontinuities in the first or the second derivatives of the relevant thermodynamic potential at the transition point. An experimental assessment of the order of the I-N transition has turned out to be not a simple task because of the presence of only small discontinuities in enthalpy and specific volume. It follows from high-resolution measurements that I-N transition is weakly first order in nature [85]. 

The computational approach is based on a colocated, finite-volume, energy-con-serving numerical scheme on unstructured grids [10] and solves the low-Mach number, variable density gas-phase flow equations. Numerical solution of the governing equations of continuum phase and droplet phase are staggered in time to maintain time-centered, second-order advection of the fluid equations. Denoting the time level by a superscript index, the velocities are located at time level f and... 

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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