Sensitivity Analysis for the Navier-Stokes Equations with Two-Equation Turbulence Models

Abstract

Aerodynamic sensitivity analysis is performed for the Navier– Stokes equations, coupled with two-equation turbulence models using a discrete adjoint method and a direct differentiation method, respectively. Like the mean ow equations, the turbulence model equations are also hand differentiated to calculate accurately the sensitivity derivatives of ow quantities with respect to design variables in turbulent viscous ows. Both the direct differentiation code and the adjoint variable code adopt the same time integration scheme with the ow solver to solve the differentiated equations ef ciently. The sensitivity codes are then compared with the ow solver in terms of 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 turbulence models, it is observed that a usual assumption of constant turbulent eddy viscosity in adjoint methods may lead to inaccurate results in a case of turbulent ows involving strong shocks. The capability of the present sensitivity codes to treat complex geometry is successfully demonstrated by analyzing the ows over multielement airfoils on chimera overlaid grid systems.