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ABSTRACT
The paper deals with the formulation of a variety of boundary conditions for incompressible and compressible flows in the context of the segregated pressure-based unstructured finite volume method. The focus is on the derivation and the implementation of these boundary conditions and their relation to the various physical boundaries and geometric constraints. While a variety of boundary conditions apply at any of the physical boundaries (inlets, outlets, and walls), geometric constraints define the type of boundary condition to be used. The emphasis is on relating the mathematical derivation of the boundary conditions to the algebraic equations defined at each centroid of the boundary elements and their coefficients. All derived boundary conditions are validated through a set of test cases with comparison of computed results to available numerical and/or experimental data.
Nomenclature
| = | coefficients in discretized momentum equations | |
| = | coefficients in discretized equations | |
| = | source term in the discretized momentum equation | |
| = | source term in the discretized equations | |
| cp | = | specific heat at constant pressure |
| Cρ | = | variable equal to 1/RT |
| dCF | = | vector joining the grid points C and F |
| dCF | = | magnitude of dCF |
| D | = | tensor operator |
| D | = | scalar defined by Eq. (17) |
| e | = | unit vector |
| E | = | Distance vector in the direction of dCF |
| E | = | magnitude of E |
| fb | = | body force per unit volume |
| Fb | = | force exerted by wall on fluid |
| I | = | identity matrix |
| k | = | thermal conductivity |
| ℓ | = | chord length |
| M | = | Mach number |
| = | mass flow rate | |
| p | = | pressure |
| p′ | = | pressure correction |
| = | heat generation per unit volume | |
| R | = | gas constant |
| S | = | surface vector |
| S | = | magnitude of S |
| t | = | time |
| T | = | temperature |
| T | = | vector equal to S − E |
| u, v, w | = | velocity components in x, y, and z direction, respectively |
| v | = | velocity vector |
| V | = | cell volume |
| Greek Symbols | = | |
| φ | = | general variable |
| σ⊥ | = | normal stress |
| μ | = | dynamic viscosity |
| γ | = | ratio of specific heats |
| ρ | = | fluid density |
| τ | = | deviatoric stress tensor |
| Ψ, Φ | = | dissipation terms in energy equation |
| Subscripts | = | |
| b | = | refers to boundary |
| C | = | refers to main grid point |
| f | = | refers to control volume face |
| F | = | refers to the F grid point |
| nb | = | refers to values at the faces obtained by interpolation between C and its neighbors |
| NB | = | refers to the neighbors of the C grid point |
| wall | = | refers to wall |
| x,y,z | = | refer to x,y, and z component, respectively |
| 0 | = | refers to stagnation condition |
| □ | = | component of a vector parallel to a surface |
| ⊥ | = | component of a vector normal to a surface |
| Superscripts | = | |
| p | = | refers to pressure |
| T | = | refers to temperature |
| T | = | refers to the transpose of a vector |
| u, v, w | = | refers to the u, v, and w-velocity component, respectively |
| n | = | refers to value at the previous iteration |
| = | refers to an interpolated value | |
| * | = | refers to an updated value during an iteration |
| ○ | = | refers to an old time value |
Nomenclature
| = | coefficients in discretized momentum equations | |
| = | coefficients in discretized equations | |
| = | source term in the discretized momentum equation | |
| = | source term in the discretized equations | |
| cp | = | specific heat at constant pressure |
| Cρ | = | variable equal to 1/RT |
| dCF | = | vector joining the grid points C and F |
| dCF | = | magnitude of dCF |
| D | = | tensor operator |
| D | = | scalar defined by Eq. (17) |
| e | = | unit vector |
| E | = | Distance vector in the direction of dCF |
| E | = | magnitude of E |
| fb | = | body force per unit volume |
| Fb | = | force exerted by wall on fluid |
| I | = | identity matrix |
| k | = | thermal conductivity |
| ℓ | = | chord length |
| M | = | Mach number |
| = | mass flow rate | |
| p | = | pressure |
| p′ | = | pressure correction |
| = | heat generation per unit volume | |
| R | = | gas constant |
| S | = | surface vector |
| S | = | magnitude of S |
| t | = | time |
| T | = | temperature |
| T | = | vector equal to S − E |
| u, v, w | = | velocity components in x, y, and z direction, respectively |
| v | = | velocity vector |
| V | = | cell volume |
| Greek Symbols | = | |
| φ | = | general variable |
| σ⊥ | = | normal stress |
| μ | = | dynamic viscosity |
| γ | = | ratio of specific heats |
| ρ | = | fluid density |
| τ | = | deviatoric stress tensor |
| Ψ, Φ | = | dissipation terms in energy equation |
| Subscripts | = | |
| b | = | refers to boundary |
| C | = | refers to main grid point |
| f | = | refers to control volume face |
| F | = | refers to the F grid point |
| nb | = | refers to values at the faces obtained by interpolation between C and its neighbors |
| NB | = | refers to the neighbors of the C grid point |
| wall | = | refers to wall |
| x,y,z | = | refer to x,y, and z component, respectively |
| 0 | = | refers to stagnation condition |
| □ | = | component of a vector parallel to a surface |
| ⊥ | = | component of a vector normal to a surface |
| Superscripts | = | |
| p | = | refers to pressure |
| T | = | refers to temperature |
| T | = | refers to the transpose of a vector |
| u, v, w | = | refers to the u, v, and w-velocity component, respectively |
| n | = | refers to value at the previous iteration |
| = | refers to an interpolated value | |
| * | = | refers to an updated value during an iteration |
| ○ | = | refers to an old time value |