http://orcid.org/0000-0002-6526-2728
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ABSTRACT
This paper develops a new tri-quadratic non-inverted skew upwind scheme (NISUS) for additional refinement of nodal integration points in numerical advection–diffusion of scalar transport. Using a control-volume-based finite element method, the performance of the eight-noded hexahedral formulation is compared with tri-quadratic hexahedral elements (27-noded hexahedral). As an extension of the NISUS formulation developed with eight-noded hexahedral elements, the new 27-noded hexahedral version uses isoparametric shape functions and integration point interpolation. The proposed method is applied to three cases of advection–diffusion of heat transfer and energy transport, including radial heat flow in a rotating hollow sphere, advection–diffusion in a cubical cavity, and combined advection/diffusion in an inlet/outlet tank. Performance improvement of the two versions of NISUS in terms of speed, accuracy, and stability are presented as a comparative assessment for the design of energy conversion systems.
Nomenclature
| A | = | local coefficient matrix |
| cp | = | specific heat capacity (J/kg°C) |
| C | = | convective length scale |
| D | = | diffusive length scale |
| e* | = | error at the center of the sphere |
| ip | = | integration point |
| J | = | Jacobian |
| k | = | thermal conductivity (W/mK) |
| = | total mass flow rate into SCV-i | |
| Ni | = | shape function for node i |
| Nup,i | = | upwind influence function for node i |
| Pe | = | Peclet number |
| qd, qc | = | diffusive, convective fluxes |
| Q | = | element control surface flux |
| ŝ | = | streamline coordinate |
| e,n,t | = | local coordinates in the x-, y-, z-directions, respectively |
| S | = | volumetric heat generation |
| [S] | = | source term matrix |
| SCV | = | subcontrol volume |
| Ŝ | = | volumetric heat generation |
| t | = | time (s) |
| u,v,w | = | fluid velocity components in (x,y,z) (m/s) |
| ṽ | = | average velocity (m/s) |
| v | = | fluid velocity vector (m/s) |
| V | = | finite control volume (m3) |
| x,y,z | = | global Cartesian coordinate (m) |
| [β] | = | transient-term coefficient matrix |
| Γ | = | diffusion coefficient |
| θ | = | non-b-dimensional temperature value |
| ρ | = | fluid density (kg/m3) |
| ṁip | = | mass flow rate at the integration |
| φ | = | scalar-dependent variable |
| ω | = | angular velocity of the sphere about the x-axis (rad/s) |
| Subscripts | = | |
| αip, bip, cip, dip | = | boundary integration point associated with a, b, c, and d, respectively |
| b | = | boundary surface |
| i | = | nodal point |
| n | = | normal to the boundary surface |
| up | = | upstream point |
| Superscripts | = | |
| c | = | convective |
| d | = | diffusive |
| ip | = | integration point |
| n | = | previous time level |
| n + 1 | = | current time level |
Nomenclature
| A | = | local coefficient matrix |
| cp | = | specific heat capacity (J/kg°C) |
| C | = | convective length scale |
| D | = | diffusive length scale |
| e* | = | error at the center of the sphere |
| ip | = | integration point |
| J | = | Jacobian |
| k | = | thermal conductivity (W/mK) |
| = | total mass flow rate into SCV-i | |
| Ni | = | shape function for node i |
| Nup,i | = | upwind influence function for node i |
| Pe | = | Peclet number |
| qd, qc | = | diffusive, convective fluxes |
| Q | = | element control surface flux |
| ŝ | = | streamline coordinate |
| e,n,t | = | local coordinates in the x-, y-, z-directions, respectively |
| S | = | volumetric heat generation |
| [S] | = | source term matrix |
| SCV | = | subcontrol volume |
| Ŝ | = | volumetric heat generation |
| t | = | time (s) |
| u,v,w | = | fluid velocity components in (x,y,z) (m/s) |
| ṽ | = | average velocity (m/s) |
| v | = | fluid velocity vector (m/s) |
| V | = | finite control volume (m3) |
| x,y,z | = | global Cartesian coordinate (m) |
| [β] | = | transient-term coefficient matrix |
| Γ | = | diffusion coefficient |
| θ | = | non-b-dimensional temperature value |
| ρ | = | fluid density (kg/m3) |
| ṁip | = | mass flow rate at the integration |
| φ | = | scalar-dependent variable |
| ω | = | angular velocity of the sphere about the x-axis (rad/s) |
| Subscripts | = | |
| αip, bip, cip, dip | = | boundary integration point associated with a, b, c, and d, respectively |
| b | = | boundary surface |
| i | = | nodal point |
| n | = | normal to the boundary surface |
| up | = | upstream point |
| Superscripts | = | |
| c | = | convective |
| d | = | diffusive |
| ip | = | integration point |
| n | = | previous time level |
| n + 1 | = | current time level |