A continuous second‐order sensitivity equation method for time‐dependent incompressible laminar flows

This paper presents a general formulation of the continuous sensitivity equation method (SEM) for computing first‐ and second‐order sensitivities of time‐dependent, incompressible laminar flows. The formulation accounts for complex parameter dependence and is suitable for a wide range of problems. The SEM formulation is verified on a problem with a closed‐form solution. Systematic grid convergence studies confirm the theoretical rates of convergence in both space and time. The methodology is then applied to uniform flow around a circular cylinder. The flow starts with a symmetrical solution and transitions to the traditional Von Karman street (alternate vortex shedding). Sensitivities are used to demonstrate fast evaluation of nearby flows. The accuracy of nearby flows is much improved when second‐order sensitivities are used. The sensitivity of the Strouhal number with respect to the Reynolds number agrees well with the computed and experimental St – Re relationship.