Aerodynamic sensitivity analysis is performed for the Navier-Stokes equatio
ns, coupled with two-equation turbulence. models using a discrete adjoint m
ethod and a direct differentiation method, respectively. Like the mean flow
equations, the turbulence model equations are also hand differentiated to
calculate accurately the sensitivity derivatives of how quantities with res
pect to design variables in turbulent viscous hows. Both the direct differe
ntiation code and the adjoint variable code adopt the same time integration
scheme with the Row solver to solve the differentiated equations efficient
ly. The sensitivity codes are then compared with the flow solver in terms o
f solution accuracy, computing time, and computer memory requirements. The
sensitivity derivatives obtained from the sensitivity codes with different
turbulence models are compared with each other. Using two-equation turbulen
ce models, it is observed that a usual assumption of constant turbulent edd
y viscosity in adjoint methods may lead to inaccurate results in a case of
turbulent hows involving strong shocks, The capability of the present sensi
tivity codes to treat complex geometry is successfully demonstrated by anal
yzing the Rows over multielement airfoils on chimera overlaid grid systems.