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ABSTRACT
The purpose of this paper is to introduce a new physical-based computational approach for the solution of convection heat transfer problems on co-located non-orthogonal grids in the context of an element-based finite volume method. The approach has already been presented in the context of two-dimensional incompressible flow problems without heat transfer. It has been shown that the pressure–velocity coupling on co-located grids can be correctly modeled via the so-called method of proper closure equations (MPCE). Here, MPCE is extended to the numerical simulation of natural, forced, and mixed convection heat transfer problems. It is shown that the couplings between pressure, velocity, and temperature can be conveniently handled on co-located grids by resorting again to the modified forms of the governing equations, i.e., the proper closure equations. The set of discrete equations is solved in a fully coupled manner in this study. Here, in part I of the paper, only the basic methodology is described; in part II, the results of application of the method to some test problems are presented.
Nomenclature
| A | = | area of control surface, matrix of coefficients |
| = | influence coefficients (for example see Eq. (39)) | |
| B | = | matrix of coefficients |
| bU, bV, bP, bθ | = | vectors of known values |
| = | influence coefficients (for example see Eq. (49)) | |
| dU, dV, dP, dθ | = | vectors of known values |
| F | = | flow term |
| g | = | gravitational acceleration |
| = | ||
| = | Grash of number | |
| = | influence coefficients (for example see Eq. (45)) | |
| k | = | linearization parameter |
| L | = | length |
| M | = | dimensionless volumetric flow rate |
| N | = | bi-linear shape functions |
| P | = | dimensionless pressure |
| p | = | pressure |
| = | Peclet number | |
| = | Prandtl number | |
| = | Rayleigh number | |
| = | Reynolds number | |
| Res | = | Residual |
| = | Richardson number | |
| 𝒮, S, s | = | source terms |
| s, t | = | local coordinates |
| T | = | Temperature |
| U, V | = | dimensionless velocity components |
| u, v | = | velocity components |
| = | velocity vector | |
| X, Y | = | dimensionless Cartesian coordinates |
| x, y | = | Cartesian coordinates |
| ∀ | = | volume |
| ∇ | = | Gradient |
| Greek symbols | = | |
| α | = | thermal diffusivity |
| β | = | thermal expansion coefficient |
| ω | = | relaxation parameter |
| θ | = | dimensionless temperature |
| ρ | = | density |
| λ1, λ2, λ3 | = | dimensionless coefficients in the governing equations |
| ν | = | kinematic viscosity |
| ϕ | = | a general scalar variable |
| η | = | aspect ratio |
| Subscripts | = | |
| 0 | = | reference value |
| d | = | down |
| i, j, m, n, … | = | dummy indices |
| ip | = | integration point |
| l | = | left |
| max | = | Maximum |
| r | = | Right |
| u | = | Up |
| Superscripts | = | |
| n | = | iteration level |
| VCU, … | = | used and explained in Eq. (47) and (48) |
| VD | = | used and explained in (43) and (B1) |
Nomenclature
| A | = | area of control surface, matrix of coefficients |
| = | influence coefficients (for example see Eq. (39)) | |
| B | = | matrix of coefficients |
| bU, bV, bP, bθ | = | vectors of known values |
| = | influence coefficients (for example see Eq. (49)) | |
| dU, dV, dP, dθ | = | vectors of known values |
| F | = | flow term |
| g | = | gravitational acceleration |
| = | ||
| = | Grash of number | |
| = | influence coefficients (for example see Eq. (45)) | |
| k | = | linearization parameter |
| L | = | length |
| M | = | dimensionless volumetric flow rate |
| N | = | bi-linear shape functions |
| P | = | dimensionless pressure |
| p | = | pressure |
| = | Peclet number | |
| = | Prandtl number | |
| = | Rayleigh number | |
| = | Reynolds number | |
| Res | = | Residual |
| = | Richardson number | |
| 𝒮, S, s | = | source terms |
| s, t | = | local coordinates |
| T | = | Temperature |
| U, V | = | dimensionless velocity components |
| u, v | = | velocity components |
| = | velocity vector | |
| X, Y | = | dimensionless Cartesian coordinates |
| x, y | = | Cartesian coordinates |
| ∀ | = | volume |
| ∇ | = | Gradient |
| Greek symbols | = | |
| α | = | thermal diffusivity |
| β | = | thermal expansion coefficient |
| ω | = | relaxation parameter |
| θ | = | dimensionless temperature |
| ρ | = | density |
| λ1, λ2, λ3 | = | dimensionless coefficients in the governing equations |
| ν | = | kinematic viscosity |
| ϕ | = | a general scalar variable |
| η | = | aspect ratio |
| Subscripts | = | |
| 0 | = | reference value |
| d | = | down |
| i, j, m, n, … | = | dummy indices |
| ip | = | integration point |
| l | = | left |
| max | = | Maximum |
| r | = | Right |
| u | = | Up |
| Superscripts | = | |
| n | = | iteration level |
| VCU, … | = | used and explained in Eq. (47) and (48) |
| VD | = | used and explained in (43) and (B1) |