Dr. Roy Charles Swanson, NASA Langley Research Center

Title: On boundary value problems for RANS equations and two-equation turbulence models – Talk 2

Abstract:

Currently, in engineering computations for high Reynolds number turbulent flows, turbulence modeling continues to be the most frequently used approach to represent the effects of turbulence. Such models generally rely on solving either one or two transport equations along with the Reynolds-Averaged Navier-Stokes (RANS) equations. The solution of the boundary-value problem of any system of partial differential equations (PDEs) requires the complete delineation of the equations and the boundary conditions, including any special restrictions and conditions. In the literature, such a description is often incomplete, neglecting important details related to the boundary conditions and possible restrictive conditions, such as how to ensure satisfying prescribed values of the dependent variables of the transport equations in the far field of a finite domain. In these two lectures, which build on each other, we consider the following topics:

Talk 2:

In this work, we address the issue of a properly defined boundary-value problem, what we call a well-defined problem. To obtain a unique numerical solution for the well-defined problem, well-posedness is required. The basic requirements of a well-posed problem are reviewed, and the current status of proving well-posedness, including the relevance of the boundary data and a bounded solution, is briefly discussed. This sets the stage for the focus on well-posedness as it pertains to considering the RANS equations and the transport equations for modeling the effects of turbulence as a weakly coupled system. Emphasis is placed on the equations of the turbulence model, and how turbulence modeling can be interpreted as a parameter identification problem, specificaly an inverse
problem. A compelling argument (although not a proof) for ill-posedness is made for both direct and inverse problems. This argument is based on using surface pressure and skin-friction data that produces very different results for the distribution of the turbulent viscosity. Numerical examples for the RANS equations and two-equation turbulence models are presented that confirm the theoretical findings.

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