Based on the optimal control theory and Navier-Stokes equations, aerodynamic design of airfoil with multi-constraint conditions, such as aerodynamic and geometric constraint conditions, is studied. According to a given problem, the corresponding adjoint equations, boundary conditions and final cost function formulation are derived in the computational space. In order to achieve the requirements of the numerical solution, final formulations in the physical space is also achieved. Numerical methods are developed effectively. By integrating the aspects, such as the flow analysis, the solution of adjoint equations, gradient solution, optimal arithmetic and grid generation etc., an aerodynamic design method involving drag reduction is successfully developed. Testing results show that the method has outstanding merits in the above aspects. It is effective and feasible for aerodynamic design with a large number of design variables. Computational time consumption is less than the conventional aerodynamic design method.

It is interesting for designer how to reduce drag of transonic wing which meets design lift requirement. Based on adjoint equation theory and using Euler equations, an optimization design method of transonic wing with drag reduction and constant lift coefficient is developed. According to the cost function given, three dimensional adjoint equations and their boundary conditions in physical space are dreived, and a numerical method is developed to solve the three dimensional adjoint equations. The sensitivity derivation of objective function with respect to design variables can be obtained by using grid perturbation method, and the quasi-Newtonian algorithms are used. Some numerical tests are made for the drag reduction with constant lift coefficient. The test results show that shock strength on wing surface will become much weaker than that of initial wing, thus drag may be reduced effectively.