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ABSTRACT
The following study introduces an algorithm for numerically solving a coupled system of differential equations that discretize to produce a banded pentadiagonal matrix with tridiagonal submatrices on either side. The article then proceeds to illustrate a step-by-step procedure to solve a two-equation system in a coupled manner. A generalization of this method to multiequation system follows. The proposed algorithm is highly efficient as it fully exploits the banded nature of the matrix. Finally, the solution of the two-equation k − ω and the four-equation turbulence models in plane channel flows by the current method serves as a demonstration exercise and wraps up the paper.
Nomenclature
| ϵ | = | dissipation rate for the turbulent kinetic energy |
 | = | matrix elements for the ith node for an m-equation system |
 | = | right-hand-side vector for the ith node for an m-equation system |
| 𝔣 | = | elliptic relaxation function |
| = | matrices for the ith node for the recursive relationships for an m-equation system | |
| ν | = | kinematic viscosity |
| νt | = | tubulent viscosity |
| ω | = | inverse time-scale for turbulence |
| = | scalar denoting normal Reynolds stress perpendicular to the wall | |
| ϕ | = | general unknown variable |
| ψj | = | jth unknown variable in an m-equation system |
| ρ | = | fluid density |
| τw | = | shear stress at the wall |
| = | co-efficients in the linear equations for a two-equation system | |
| H | = | channel half-height |
| h | = | computational cell height |
| k | = | turbulent kinetic energy |
| L | = | length scale |
| li | = | elements of the right-hand-side vector for a two-equation system |
| m | = | number of variables |
| n | = | number of computational nodes |
| P | = | mean pressure |
| = | vectors needed for back-substitution for the two-equation system | |
| Reτ | = | Reynolds number |
| T | = | time-scale |
| U | = | mean streamwise velocity |
| uτ | = | friction velocity |
| x | = | spatial coordinate in the direction of mean flow |
| y | = | spatial coordinate in the direction normal to the wall |
| p(i) | = | vector for the ith node for the recursive relationships for an m-equation system |
Nomenclature
| ϵ | = | dissipation rate for the turbulent kinetic energy |
| = | matrix elements for the ith node for an m-equation system |
| = | right-hand-side vector for the ith node for an m-equation system |
| 𝔣 | = | elliptic relaxation function |
| = | matrices for the ith node for the recursive relationships for an m-equation system | |
| ν | = | kinematic viscosity |
| νt | = | tubulent viscosity |
| ω | = | inverse time-scale for turbulence |
| = | scalar denoting normal Reynolds stress perpendicular to the wall | |
| ϕ | = | general unknown variable |
| ψj | = | jth unknown variable in an m-equation system |
| ρ | = | fluid density |
| τw | = | shear stress at the wall |
| = | co-efficients in the linear equations for a two-equation system | |
| H | = | channel half-height |
| h | = | computational cell height |
| k | = | turbulent kinetic energy |
| L | = | length scale |
| li | = | elements of the right-hand-side vector for a two-equation system |
| m | = | number of variables |
| n | = | number of computational nodes |
| P | = | mean pressure |
| = | vectors needed for back-substitution for the two-equation system | |
| Reτ | = | Reynolds number |
| T | = | time-scale |
| U | = | mean streamwise velocity |
| uτ | = | friction velocity |
| x | = | spatial coordinate in the direction of mean flow |
| y | = | spatial coordinate in the direction normal to the wall |
| p(i) | = | vector for the ith node for the recursive relationships for an m-equation system |