An aerodynamic inverse design method for viscous flow over airfoils is presented. In this approach, pressure distribution on the airfoil surfaces are prescribed as design target and the airfoil geometry is modified so as to reach the desired shape. In the design method, the walls are assumed to be moving with a virtual wall velocity that would balance the current momentum flux with the target pressure distribution; this virtual wall velocity drives the airfoil geometry to the shape that would produce the target pressure distribution where it would asymptotically vanish. This method was extended to address multi-point design and multi-element airfoils, and to use the pressure distribution of the airfoil suction surface as design variable. The approach is consistent with the viscous flow assumption and is incorporated into the governing equations which are expressed in an Arbitrary Lagrangian-Eulerian form, and are solved in a time accurate fashion. A cell-vertex finite volume scheme of the Jameson type is used for spacial discretization and time integration is performed by dual time stepping. Baldwin-Lomax turbulence model is used for turbulence closure. The validation of this approach is carried out for NACA 4-digit and 5-digit airfoils and RAE 2822 airfoil in transonic flow regime. The redesign cases include NACA 5-digit and 4-digit airfoils, the latter design experiencing large separation, RAE 2822 airfoil in transonic regime, multi-element airfoil and a dual-point design case.
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