Numerical simulation is performed for stagnating turbulent flows of impinging and countercurrent jets by the Reynolds stress model(RSM). Results are compared with those of the ${kappa}-{varepsilon}$ model and available data to assess the flow characteristics and turbulence modes. Three variants of the RSM tested are those of Gibson and Launder(GL), Craft and Launder(GL-CL) and Speziale, Sarkar and Gatski(SSG). As well known, the ${kappa}-{varepsilon}$ model overestimates turbulent kinetic energy near the wall significantly. Although the RSM is superior to the ${kappa}-{varepsilon}$ model, it shows considerable difference according to how the redistributive pressure-strain term is modeled. Results of the RSM for countercurrent jets are improved with the modified coefficients for the dissipation rate, $C_{{varepsilon}1};and;C_{{varepsilon}2}$ suggested by Champion and Libby. The performance of the three variants of the RSM model for stagnating flows are assessed.