A Transonic Potential Solver with an Embedded Wake Approach using Multivalued Finite Elements

Abstract

A potential transonic solver with an embedded wake is presented. The flow outside of attached boundary layers of streamlined bodies flying at high Reynold numbers can be assumed to be irrotational and isentropic. This assumption reduces the Navier-Stokes equations to a single scalar nonlinear partial differential equation, namely the full-potential equation (FPE). The FPE expresses the conservation of mass in terms of the velocity potential. In this work, the FPE is discretized using a standard Galerkin finite element method, and the nonlinear system of equations stemming from the discretization is solved using Newton's method. An artificial compressibility method is used to stabilize the problem in supersonic flow regions. This method prevents the Jacobian from becoming singular and allows capturing shock waves. To include the viscosity effects in the lift generation, a model for the trailing wake needs to be introduced. In the presented method, the wake is modeled as a straight surface in the free-stream direction. This assumption is relaxed allowing mass flux across the wake. To capture the discontinuity in the velocity potential across the wake, a multivalued element method is employed. This implicit description of the wake within the mesh presents an effective approach to perform fluidstructure interaction computations and apply aeroelastic optimization methods, where the position of the wake changes during consecutive iterations. The solver is implemented in Kratos Multi-Physics and verified for different angles of attack and free-stream conditions. Since the pressure does not change in the transverse direction of the boundary layer, the FPE yields accurate lift, induced drag, and moment computations.