The default coefficients of the Reynolds-averaged Navier– Stokes turbulence model often yield inaccurate results. To improve predictive capability, a modified \(k-\varepsilon \) model with variable coefficients is considered. In this model, the coefficients are optimized for a range of Reynolds numbers from 5100 to 86000 based on an incompressible, axisymmetric round jet. The model is implemented in OpenFOAM. The main advantage of the proposed model is its dependence on the Reynolds number parameter at the nozzle exit. It has been shown to enhance the precision of incompressible jets in both the near and far fields. Furthermore, the model performs well in other benchmark cases from the ERCOFTAC database, such as a mixing layer, flow over a two-dimensional hill, and a backward-facing step. These results demonstrate the model’s robustness and suitability for practical engineering applications.