EU Regional School - Mittal Seminar
Prof. Mittal, Ph.D. - Aerodynamic Shape Optimization Using Adjoint-Based Methods
Indian Institute for Technology Kanpur
A method for aerodynamic shape optimization using adjoint variables is developed and implemented. A stabilized finite element method is used to solve the governing equations. The validation of the formulation and its implementation is carried out via steady flow past an elliptical bump whose eccentricity is used as a design parameter. Results for, both, optimal design and inverse problems are presented. Using different initial guesses, multiple optimal shapes are obtained. A multi-objective function with additional constraints on the volume and the drag coefficient of the bump is utilized. It is seen that as more constraints are added to the objective function the design space is constrained and the multiple optimal shapes become progressively similar to each other. Next, the shape of an airfoil is optimized for various different objectives and for various values of Reynolds number. Very interesting shapes are discovered at low Reynolds numbers. The non-monotonic behavior of the objective functions with respect to the design variables is demonstrated. The method is extended to design airfoils for a range of Reynolds number and angles of attack. Next, the approach for optimizing shapes that are associated with unsteady flows is developed. The objective function is typically based on time-averaged aerodynamic coefficients. Interesting shapes are obtained, especially when the objective is to produce high performance airfoils. The method is utilized to obtain high performance airfoils for Re=1000 and 10,000 using relatively large number of design variables. Beyond a certain number of control points the optimization leads to a spontaneous appearance of corrugations on the upper surface of the airfoil. The corrugations are responsible for generation of small vortices that add to suction on the upper surface of the airfoil and lead to enhanced lift. Preliminary results will be presented for optimization for finite wings.
Figure: Time-averaged pressure field for the optimal airfoils for desired values of the time-averaged lift coefficient. The Reynolds number, based on chord length, is 1000 and the angle of attack is 4 degree.