On Boundary-Value Problems for RANS Equations and Two-Equation Turbulence Models
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On Boundary-Value Problems for RANS Equations and Two-Equation Turbulence Models Stefan Langer1 · R. C. Swanson2 Received: 13 March 2020 / Revised: 16 June 2020 / Accepted: 21 September 2020 / Published online: 12 October 2020 © The Author(s) 2020
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 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 this article, we discuss the possible influence of boundary values, as well as near-field and far-field behavior, on the solution of the RANS equations coupled with transport equations for turbulence modeling. In so doing, we defne the concept of a welldefined boundary-value problem. Additionally, a three-dimensional, rather than a simpler one-dimensional analysis is performed to analyze the near-wall and far-field behavior of the turbulence model variables. This allows an assessment of the decay rate of these variables required to realize the boundary conditions in the far field. This paper also addresses the impact of various transformations of two-equation models (e.g., the model of Wilcox) to remove the singular behavior of the dissipation rate (ω) at the surface boundary. Finally, the issue of well-posedness regarding the governing equations is considered. A compelling argument (although not a proof) for ill-posedness is made for both direct and inverse problems. Keywords Navier–Stokes equations · Two-equation turbulence models · Boundary-value problems
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Stefan Langer [email protected]
1
German Aerospace Center, Member of the Helmholtz Association, Institute of Aerodynamics and Flow Technology, Lilienthalplatz 7, 38108 Braunschweig, Germany
2
NASA Langley Research Center, Computational AeroSciences Branch, Hampton, VA 23681, USA
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Journal of Scientific Computing (2020) 85:20
1 Introduction In this paper we consider the Reynolds-averaged Navier–Stokes (RANS) equations. The RANS equations are based upon Reynolds expansions for the dependent variables and time (Reynolds) averaging. In practice, this definition is extended to include applying a combination of Reynolds expansions (for the pressure and density) and mass weighted (Favre) expansions for the other dependent flow variables (see for example the textbooks of Wilcox and Pope [1,2]). In so doing, the time-averaged equations can
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