There is considerable effort to solve the problem of well-posedness, regularity, and global existence
for the Navier-Stokes equations. While progress has been made for some particular cases, this
Millennium problem remains unsolved in general. These issues also apply to the transport equation
or equations for modeling the effects of turbulence. Without this mathematical foundation, one
cannot know a priori if there exist a solution or possibly multiple solutions. Currently, scientists and
engineers generally rely only upon numerical demonstrations to determine if there exist a realistic
eddy viscosity from turbulence models, which primarily depends on solving transport equations.
In this presentation, the mathematical and discrete issues concerning a representative two-equation
turbulence model are examined. The challenges for convergence in steady state problems as well
as unsteady problems, when considering a dual time-stepping algorithm, are discussed. Although
the focus will be on a two-equation model, some representative convergence behaviors of a full
Reynolds Stress Model (RSM) are also presented. Results are shown for three different airfoil cases
at different flow conditions, which includes transonic flow conditions. Various issues in verification
of turbulence models are also briefly considered. A perspective on requirements for an effective
and efficient numerical algorithm to solve the transport equations is also examined. In particular,
the following issues are considered: (1) stiffness of the governing transport equations, (2) boundary
conditions, (3) positivity and realizability, (3) boundary conditions, (4) strong solution algorithms,
(5) linear and nonlinear stability, (6) analysis concerning behavior of the transport equations.
