Unconditionally positive implicit procedure for two-equation turbulence models: Application to k–ω turbulence models

A new general implicit procedure that guarantees the positivity of turbulence model equations dependent variables is presented. The implicit procedure is based on designing the implicit Jacobian to be an M-matrix. A unified approach for the treatment of the convection, diffusion and source terms is introduced and employed. An appropriate design of the M-matrix guarantees the positivity of the turbulence equation dependent variables for any time step, without the use of any clipping. The procedure is employed and demonstrated using a k–ω turbulence model. An efficient construction of the Jacobian matrix is proposed, resulting in diagonal matrices. The model is tested by simulating three test cases of two- and three-dimensional complex flow fields. Results from the numerical simulations show that the new procedure exhibits very good convergence characteristics, is extremely robust, and always preserves the positivity of the turbulence variables.