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ABSTRACT
In this study, the implementation of boundary conditions for the Navier–Stokes and the energy equations, including the pressure and pressure correction equations, are presented in the context of finite volume formulation on cell-centered, colocated unstructured grids. The implementation of boundary conditions is formulated in terms of the contribution of boundary face of a cell to the coefficients of the discretized equation for either Dirichlet- or Neumann-type boundary conditions. Open boundaries through which the flow is not fully developed are also considered. In this case, a data reconstruction method is proposed for finding the boundary values of the variables at the correction stage. The validity of implementations is checked by comparing the results with some well-known benchmark problems.
Nomenclature
| A, A | = | area in scalar or vector form |
| C | = | cp for energy or 1 for momentum equations |
| C | = | coefficient in vector form defined in appendix |
| cp | = | constant pressure specific heat |
| d | = | distance from a cell center |
| e | = | unit vector |
| f | = | body forces per unit volume |
| fr | = | interpolation factor |
| g | = | gradient |
| H | = | height |
| I | = | unit tensor |
| J, J | = | flux in scalar or vector form |
| k | = | thermal conductivity |
| L | = | length |
| = | mass flow rate | |
| = | pseudomass flow rate | |
| n | = | unit vector in normal direction |
| nb(P) | = | neighbors of cell P |
| nf(P) | = | number of faces of cell P |
| p | = | pressure |
| p′ | = | pressure correction |
| S, S | = | source term of the discretized equation in scalar or vector form |
| s, s | = | source term of the differential equation in scalar or vector form |
| t | = | time |
| u, v, w | = | velocity components in x, y, z-directions |
| V | = | volume |
| = | volume flow rate | |
| = | pseudovolume flow rate | |
| v | = | velocity vector |
| x, y, z | = | Cartesian coordinates |
| α | = | relaxation parameter |
| β | = | coefficient of thermal expansion |
| ϕ | = | dependent variable |
| Γ | = | diffusion coefficient |
| μ | = | fluid viscosity |
| ρ | = | fluid density |
| Subscripts | = | |
| b | = | boundary face |
| f | = | face |
| N | = | neighbor node |
| P | = | central node |
| t | = | tangential |
| Superscripts | = | |
| cd | = | cross derivative term |
| conv | = | convection |
| dif | = | diffusion |
| H | = | higher order convection terms |
| l | = | current time |
| n | = | current iteration |
| T | = | transpose |
| U | = | upwind |
Nomenclature
| A, A | = | area in scalar or vector form |
| C | = | cp for energy or 1 for momentum equations |
| C | = | coefficient in vector form defined in appendix |
| cp | = | constant pressure specific heat |
| d | = | distance from a cell center |
| e | = | unit vector |
| f | = | body forces per unit volume |
| fr | = | interpolation factor |
| g | = | gradient |
| H | = | height |
| I | = | unit tensor |
| J, J | = | flux in scalar or vector form |
| k | = | thermal conductivity |
| L | = | length |
| = | mass flow rate | |
| = | pseudomass flow rate | |
| n | = | unit vector in normal direction |
| nb(P) | = | neighbors of cell P |
| nf(P) | = | number of faces of cell P |
| p | = | pressure |
| p′ | = | pressure correction |
| S, S | = | source term of the discretized equation in scalar or vector form |
| s, s | = | source term of the differential equation in scalar or vector form |
| t | = | time |
| u, v, w | = | velocity components in x, y, z-directions |
| V | = | volume |
| = | volume flow rate | |
| = | pseudovolume flow rate | |
| v | = | velocity vector |
| x, y, z | = | Cartesian coordinates |
| α | = | relaxation parameter |
| β | = | coefficient of thermal expansion |
| ϕ | = | dependent variable |
| Γ | = | diffusion coefficient |
| μ | = | fluid viscosity |
| ρ | = | fluid density |
| Subscripts | = | |
| b | = | boundary face |
| f | = | face |
| N | = | neighbor node |
| P | = | central node |
| t | = | tangential |
| Superscripts | = | |
| cd | = | cross derivative term |
| conv | = | convection |
| dif | = | diffusion |
| H | = | higher order convection terms |
| l | = | current time |
| n | = | current iteration |
| T | = | transpose |
| U | = | upwind |