Turbulence Modelling

Simple, standard Boussinesq-viscosity models have proven insufficient to correctly predict complex flow situations. Their inability to render the fundamental physics of turbulence, such as curvature-driven shear-stress variation and secondary motion, is a well-known fact. Thus, attention is drawn to more reliable approaches towards the modelling of turbulence. Taking into account that RSTM have not yet reached the state of industrial maturity, and furthermore considering that no difference in priority between accuracy and efficiency is made in the design process, the attention is focussed on improved one- and two-equation practices. A variety of models was either developed or adopted from literature, implemented in the MEGAFLOW software and thouroughly validated. Six different approaches have been considered, and will be briefly described. While the first two models represent linear two-equation models which are expanded for an enhanced range of validity, the second two models are non-linear Explicit Algebraic Stress Models (EASM), which are characterized by a mathematically rigorous derivation from RSTM. Finally, two one-equation models are included. The classical Wilcox $k$-$\omega$ formulation [3] serves as a baseline reference.

The Menter SST (Shear Stress Transport) $k$-$\omega$ model [4] was designed to remedy two major flaws incorporated in the Wilcox approach, viz. the free-stream dependency and the unsatisfactory predictive performance in adverse-pressure-gradient flows. The former is adressed by blending from $k$-$\omega$ in the inner region of the boundary layer to $k$- $\varepsilon$ in the outer region and free shear flows (BSL model). The latter is tackled by sensitizing the eddy viscosity to the transport of the shear stress magnitude by incorporating the Bradshaw hypothesis (SST modification). The Menter model has gained significant popularity in the aeronautical community and can be regarded as one of the standard approaches today [5]. It can be used with either both BSL and SST (usually referred to as the full Menter SST model) or with SST only (labelled as Wilcox+SST).

The LLR (Local Linear Realizable) $k$-$\omega$ model [6] is a local linear two-parameter model derived from realizability and non-equilibrium turbulence constraints. The coefficients of the stress-strain relation and the turbulence-transport equations are all functions of the non-dimensional invariants of the mean strain and vorticity rates. The approach thus tries to accomplish consistent stress-strain distributions not only in plane shear flow, but also in more general flow situations.

The RQEVM (Realizable Quadratic Explicit Algebraic Stress Model) [7] stems from an explicit solution to the second-moment closure in the limit of equilibrium turbulence. This approach can be regarded as a generalized (non-linear) two-parameter model, which retains the predictive benefits of the second-moment closure methodology, while numerical advantages of the Boussinesq-viscosity concept are conserved. Additional key features of the modelling practice are topography-independent low-Re formulations, obedience of the realizability principle, consistency to the hydrodynamic stability theory and an approximately self-consistent representation of non-equilibrium turbulence. The current formulation is cast in terms of the Wilcox $k$-$\omega$ background model. Besides the full non-linear model, a linear truncation of the non-linear constitutive relation named LEA (Linearized Explicit Algebraic Stress Model) $k$-$\omega$ is available.

The EARSM (Explicit Algebraic Reynolds Stress Model) [8] by Wallin is another EASM derived similarly to the RQEVM. However, contrary to the latter, it is based on a fully self-consistent formulation. Following a suggestion by Wallin [9], the Kok $k$-$\omega$ approach [10] is used as the background model. Additionally, in an early stage of the project, a linear truncation of this model based on Wilcox $k$-$\omega$, thus designated as L-EARSM + Wilcox $k$-$\omega$, was investigated.

Due to their cost-effectiveness, one-equation models enjoy a wide popularity in practical application-oriented methods. Especially the Spalart-Allmaras (SA) model [11], which solves a transport equation for a modified eddy viscosity, has, besides the SST approach, become a standard model in aeronautical applications [5]. The model was assembled in a bottom-up manner, mainly by arguments of empiricism and dimensional analysis, also incorporating the Bradshaw hypothesis.

Finally, the SALSA (Strain-Adaptive Linear Spalart-Allmaras) model [12] is included, which is based on the original SA formulation. However, it offers an enhanced range of validity with respect to non-equilibrium flows. Unlike in standard one-equation approaches, which inherently contain the assumption of local equilibrium of production and destruction of turbulent energy, the reconstruction of the production-to-destruction ratio from mixing-length hypothesis elements via a sensitisation of the production term coefficient to variable strain rates allows for a more realistic representation of non-equilibrium states.