### 12.1.1. Scale-Adaptive Simulation (SAS)

In principle, all RANS models can be solved in unsteady mode (URANS). Experience shows, however, that classical URANS models do not provide any spectral content, even if the grid and time step resolution are sufficient for that purpose. It has long been argued that this behavior is a natural outcome of the RANS averaging procedure (typically time averaging), which eliminates all turbulence content from the velocity field. By that argument, it has been concluded that URANS can work only in situations of a "separation of scales," where only time variations that are of much lower frequency than turbulence are resolved. An example would be the flow over a slowly oscillating airfoil, where the turbulence is modeled entirely by the RANS model and only the slow super-imposed motion is resolved in time. A borderline case for this scenario is the flow over bluff bodies, like a cylinder in crossflow. For such flows, the URANS simulation provides unsteady solutions even without an independent external forcing. The frequency of the resulting vortex shedding is not necessarily much lower than the frequencies of the largest turbulent scales. This scenario is depicted in Figure 12.1, which shows that URANS models (in this case SST) produce a single mode vortex shedding even at a relatively high number of . The vortex stream extends far into the cylinder wake, maintaining a single frequency. This result is in contradiction to experimental observations of a broadband turbulence spectrum.

As shown in a series of publications (for example, Menter and Egorov, 2010 [18], Egorov et al., 2010 [6]), a class of RANS models can be derived based on a theoretical concept dating back to Rotta (see Rotta, 1972 [24]). These models perform like standard RANS models in steady flows, but enable the formation of a broadband turbulence spectrum for certain types of unstable flows (for the types of flows, see Section 12.2). Such models are termed Scale-Adaptive Simulation (SAS) models. This scenario is illustrated by Figure 12.2, which shows the same simulation as in Figure 12.1 but with the SAS-SST model. The behavior seen in Figure 12.1 is therefore not inherent to all RANS models, but only to those derived in a special fashion.

Figure 12.1: URANS computations of a flow past a circular cylinder (SST model)

Figure 12.2: SAS simulation of flow past a circular cylinder (SAS-SST model)

The SAS concept is described in much detail in the cited references and will not be repeated here. However, the basic model formulation must be provided for a discussion of the model’s characteristics. The difference between standard RANS and SAS models lies in the treatment of the scale-defining equation (typically -, -, or -equation). In classic RANS models, the scale equation is modeled based on an analogy with the -equation using simple dimensional arguments. The scale equation of SAS models is based on an exact transport equation for the turbulence length scale as proposed by Rotta. This method was re-visited by Menter and Egorov, 2010 [18] and avoids some limitations of the original Rotta model. As a result of this re-formulation, it was shown that the second derivative of the velocity field must be included in the source terms of the scale equation. The original SAS model (Menter and Egorov, 2010 [18]) was formulated as a two-equation model, with the variable for the scale equation:

| (12–1)

| (12–2)

| (12–3)

The main new term is the one including the von Karman length scale , which does not appear in any standard RANS model. The second velocity derivative allows the model to adjust its length scale to those structures already resolved in the flow. This functionality is not present in standard RANS models. This leads to the behavior shown in Figure 12.2, which agrees more closely with the experimental observations for such flows.

The term can be transformed and implemented into any other scale-defining equation resulting in SAS capabilities as in the case of the SAS-SST model. For the SAS-SST model, the additional term in the -equation resulting from the transformation has been designed to have no (or at least minimal) effect on the SST model’s RANS performance for wall boundary layers. It can have a moderate effect on free shear flows (Davidson, 2006 [4]).

The SAS model will remain in steady RANS mode for wall bounded flows, and can switch to SRS mode in flows with large and unstable separation zones (see Section 12.2).

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