For a discrete adjoint method, is the adjoint solver should be similar to the N-S flow solver? May I use the explicit multigrid method for N-S flow solver and implicit method for adjoint solver? Thanks in advance.

Depends on what your goals are. If you are after optimization then yes, it is fine to combine the two provided that everything else is the same, i.e. the same turbulence model, same mesh, etc. Most importantly this is fine if you are looking into steady state problems, just make sure that explicit solver has converged fully, otherwise gradients are inaccurate especially around stationary points of functionals.

If your goals are error analysis, superconvergent computation of various functionals, and similar fun stuff, then no, it is not OK to mix the two formulations. You must have the same error accumulation and propagation in order to correctly account for them. Also, for transient analysis it is important to use exactly the same discretizations going forward and backward since you want to make sure that the system is taking the same path in both directions. If different discretization and time marching techniques are used in two directions then directional derivatives computed at initial time using adjoint solver do not represent correctly the sensitivities with respect to terminal conditions.

Good luck.

Thanks

Hi: What is the "adjoint"