I am computing natural convection applying the Boussinesq approximation. So, I am solving the Navier-Stokes equations with a buoyant term and a convection-diffusion equation for the temperature. If the environmental temperature, that is considered in the buoyant term (T-Tinf) of the momentum equation, is not zero, the nonlinear solver diverges, otherwise everything works fine. Mathemetically spoken, the system of coupled nonlinear PDEs is not invariant against a shift in temperature. Analytically every constant added to the temperature should cancel, if respected in the bouyant term. I should mention that I use a stabilized Navier-Stokes discretization. Has someone an explanation for this phenomena?

If buoyant term (T-Tinf) of the momentum eq. vanishes, temperature is decoupled from Momentum and Continuity eqs. In this case, you can verify whether the solver for temerature Eq. is correct. Since it overflow, there is something wrong with your T equation solver.