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ABSTRACT
In this article, a recently developed pressure-based, fully coupled solver capable of predicting fluid flow at all speeds is extended to deal with turbulent flows in a rotating frame of reference and emphasizing turbomachinery applications. The pressure–velocity coupling at the heart of the Navier-Stokes equations is resolved by deriving a pressure equation in a similar fashion to a segregated SIMPLE algorithm but with implicit treatment of the velocity and pressure fields. The resulting system of coupled equations is solved using an algebraic multigrid solver. The above numerical procedures have been implemented within OpenFOAM®, which is an open-source code framework capable of dealing with industrial-scale flow problems. The OpenFOAM-based coupled solver is validated using experimental and numerical data available from reference literature test cases as well as with a segregated solver based on the SIMPLE algorithm. This is done in addition to evaluating its performance by solving an industrial problem. In comparison with the segregated solver, the coupled solver results indicate substantial reduction in computational cost with increased robustness.
Nomenclature
| = | coefficients in the discretized equations | |
| A | = | convection-diffusion type matrix |
| = | source term in the discretized equations | |
| = | source term in the discretized momentum equation | |
| B | = | discrete gradient matrix |
| C | = | main grid point |
| dCF | = | vector joining the grid points C and F |
| D | = | tensor operator |
| F | = | refers to neighbor of the C grid point |
| g | = | geometric interpolation factor |
| H | = | total enthalpy |
| k | = | thermal conductivity |
| = | mass flow rate | |
| N | = | total number of elements |
| p | = | pressure |
| Q | = | general source term |
| r | = | radial coordinate |
| = | normalized radial coordinate | |
| res | = | residuals |
| R | = | gas constant |
| RMS | = | root-mean-square residual |
| S | = | surface vector |
| T | = | temperature |
| u, v, w | = | velocity components in x, y, and z directions, respectively |
| u | = | velocity vector |
| V | = | cell volume |
| θ | = | flow angle |
| μ | = | dynamic viscosity |
| ρ | = | fluid density |
| τ | = | deviatoric stress tensor |
| φ | = | general variable |
| Subscripts | = | |
| C | = | refers to main grid point |
| eff | = | refers to effective viscosity or thermal conductivity |
| f | = | refers to control-volume face |
| F | = | refers to the F grid point |
| l | = | refers to laminar flow |
| 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 |
| o | = | refers to total conditions |
| ref | = | refers to reference conditions |
| t | = | refers to turbulent flow |
| Superscripts | = | |
| n | = | refers to value at the previous iteration |
| p | = | refers to pressure |
| u | = | refers to the u-velocity component |
| v | = | refers to the v-velocity component |
| w | = | refers to the w-velocity component |
| = | refers to an interpolated value |
Nomenclature
| = | coefficients in the discretized equations | |
| A | = | convection-diffusion type matrix |
| = | source term in the discretized equations | |
| = | source term in the discretized momentum equation | |
| B | = | discrete gradient matrix |
| C | = | main grid point |
| dCF | = | vector joining the grid points C and F |
| D | = | tensor operator |
| F | = | refers to neighbor of the C grid point |
| g | = | geometric interpolation factor |
| H | = | total enthalpy |
| k | = | thermal conductivity |
| = | mass flow rate | |
| N | = | total number of elements |
| p | = | pressure |
| Q | = | general source term |
| r | = | radial coordinate |
| = | normalized radial coordinate | |
| res | = | residuals |
| R | = | gas constant |
| RMS | = | root-mean-square residual |
| S | = | surface vector |
| T | = | temperature |
| u, v, w | = | velocity components in x, y, and z directions, respectively |
| u | = | velocity vector |
| V | = | cell volume |
| θ | = | flow angle |
| μ | = | dynamic viscosity |
| ρ | = | fluid density |
| τ | = | deviatoric stress tensor |
| φ | = | general variable |
| Subscripts | = | |
| C | = | refers to main grid point |
| eff | = | refers to effective viscosity or thermal conductivity |
| f | = | refers to control-volume face |
| F | = | refers to the F grid point |
| l | = | refers to laminar flow |
| 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 |
| o | = | refers to total conditions |
| ref | = | refers to reference conditions |
| t | = | refers to turbulent flow |
| Superscripts | = | |
| n | = | refers to value at the previous iteration |
| p | = | refers to pressure |
| u | = | refers to the u-velocity component |
| v | = | refers to the v-velocity component |
| w | = | refers to the w-velocity component |
| = | refers to an interpolated value |