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ABSTRACT
This paper presents a novel numerical model for incompressible flows on unstructured hybrid grids by combining the pressure-implicit with splitting of operator (PISO) algorithm and volume-integrated average and point value-based multimoment (VPM) method. Implementing the spatial discretization of VPM to the PISO solution procedure results in a novel formulation that is unconditionally stable and superior in numerical accuracy and robustness in comparison with the conventional finite volume method. The present VPM/PISO formulation provides a numerical framework of great practical significance that well balances the numerical accuracy and algorithmic complexity. Numerical verifications demonstrate that the present model can significantly improve numerical accuracy. Moreover, the numerical dissipation is effectively suppressed, which shows a great potential for simulations of high-Reynolds number flows.
Nomenclature
| aP, anb | = | diagnal and off-diagonal coefficients matrix for discrete convective and diffusion term |
| A, B | = | swichting parameters in VPM reconstruction function of different element shape |
| E | = | formula of L1 or L2 error |
| I | = | number of grid cells |
| J,K | = | number of faces and vertices in each cell |
| L | = | number of cells sharing each vertex |
| n | = | unit normal vector at surface |
| p | = | pressure |
| r | = | distance vector in space |
| Re | = | Reynolds number |
| Res | = | residual of momentum and pressure equation |
| Sb | = | explicit source term of discretized momentum equation |
| t | = | time |
| TEC | = | interpolation operator of TEC formula |
| u | = | velocity vector |
| x, y, z | = | Cartesian coordinate |
| α | = | parameters for switching BDF1 and BDF2 |
| Γ | = | surface element |
| θ | = | vertice element |
| μ | = | kinematic viscosity |
| ξ, η, ζ | = | local coordinate |
| ρ | = | density |
| ϕ | = | prognostic variables |
| Φ | = | VPM reconstruction polynomial |
| ψ | = | basis function of VPM reconstruction |
| ω | = | weight function of Riemann solver |
| Ω | = | cell element |
| Subscripts and superscript | = | |
| i | = | index of cell |
| ic | = | centroid of Ωi |
| ij, ik | = | vertices and boundary segament of Ωi |
| m | = | intermediate level of iterative loop |
| n | = | time level |
| * | = | intermediate values in algorithm |
Nomenclature
| aP, anb | = | diagnal and off-diagonal coefficients matrix for discrete convective and diffusion term |
| A, B | = | swichting parameters in VPM reconstruction function of different element shape |
| E | = | formula of L1 or L2 error |
| I | = | number of grid cells |
| J,K | = | number of faces and vertices in each cell |
| L | = | number of cells sharing each vertex |
| n | = | unit normal vector at surface |
| p | = | pressure |
| r | = | distance vector in space |
| Re | = | Reynolds number |
| Res | = | residual of momentum and pressure equation |
| Sb | = | explicit source term of discretized momentum equation |
| t | = | time |
| TEC | = | interpolation operator of TEC formula |
| u | = | velocity vector |
| x, y, z | = | Cartesian coordinate |
| α | = | parameters for switching BDF1 and BDF2 |
| Γ | = | surface element |
| θ | = | vertice element |
| μ | = | kinematic viscosity |
| ξ, η, ζ | = | local coordinate |
| ρ | = | density |
| ϕ | = | prognostic variables |
| Φ | = | VPM reconstruction polynomial |
| ψ | = | basis function of VPM reconstruction |
| ω | = | weight function of Riemann solver |
| Ω | = | cell element |
| Subscripts and superscript | = | |
| i | = | index of cell |
| ic | = | centroid of Ωi |
| ij, ik | = | vertices and boundary segament of Ωi |
| m | = | intermediate level of iterative loop |
| n | = | time level |
| * | = | intermediate values in algorithm |
Acknowledgment
This work was supported in part by JSPS KAKENHI Grant Numbers 15H03916 and 15J09915.