A new two-equation turbulence model is described. It combines an algebraic, non-linear expression of the Reynolds stresses in terms of strain rate and vorticity tensor components, with a modified transport equation for the dissipation rate. Thanks to the cubic law for the Reynolds stresses, the influence on turbulence from streamline curvature is accounted for, while the increase in computational costs is small. The classical transport equation for the dissipation rate is altered, in order to bring more physics into this equation. As a result, more realistic values for the turbulence quantities are obtained. A new low-Reynolds source term has been introduced and a model parameter is written in terms of dimensionless strain rate and vorticity. The resulting model is firstly applied to the inert turbulent flow over a backward-facing step, demonstrating the quality of the turbulence model. Next, application to an inertly mixing round jet reveals that the spreading rate of the mixture fraction is correctly predicted. Afterwards, a piloted-jet diffusion flame is considered. Finally, inert and reacting flows in a bluff-body burner are addressed. It is illustrated for both reacting test cases that the turbulence model is important with respect to the flame structure. It is more important than the chemistry model for the chosen test cases. Results are compared to what is obtained by linear turbulence models. For the reacting test cases, the conserved scalar approach with pre-assumed beta -probability density function (PDF) is used.