As mentioned in the previous section, regions close to the walls suppose an additional problem for RANS and LES in terms of modelling, DNS not requiring any modelling. RANS has been well-adapted for this different modelling with more or less success as stated in the previous section. Equivalently, LES relies on a certain amount of modelling to close the equations to be solved as shown in App. B. Thus, unresolved terms rely upon these models to account for missing physics. Adequate models may be found in cases where turbulence is isotropic, but these need usually to be adapted for strongly sheared flows such as those encountered near walls. Even for sub-grid scale models that are designed for wall-resolved situations, like the WALE model in App. B.3, these require additional hypotheses if meshes aren’t sufficiently fine. This implies that any unresolved physics will either be wrongly modelled (Smagorinsky model) or neglected (WALE, SIGMA) if no other condition is added. There are thus two options; either define a mesh sufficiently fine to account for the smallest scales or to use an additional model. Simulations done in Chap. 6 are done in a wall-resolved context but the most widely used alternatives in industry are nevertheless presented.
Two main great families of modelling exist in the context of LES, • Wall-modelled LES
Modelling results from the need to determine the interaction between the smaller and the larger scales of the flow which are filtered and resolved respectively. This applies to the whole domain, however, it is of great importance in the near-wall region as the boundary layer physics dominate this area. The problem is that subgrid-scale models by themselves are not able to account for the shear stresses at the wall if y+, a non-dimensional distance
parameter, is larger than a certain value O (10). On the other hand, from the universal velocity distribution law in the presence of walls it is possible to provide a relation to account for the wall influence. Taking this law and the associated velocity distribution, it is possible to estimate the shear stress. This has been a historical problem initially approached by Schumann and Deardorff [163, 47] that has led to many laws and their variants. The common feature to all laws is that they require information of the velocity field outside of what is known as the viscous sublayer (see Chap. 5). The mesh must be
2.2 Near-wall turbulence modelling adapted so the position at which the information is extracted has a physical meaning as it is crucial to the correct behaviour of the wall law. The wall-modelled LES approach is thus based on predicting the shear stress (from a purely aerodynamical point of view) using a velocity value given at a normal distance from the wall to surpass the need for a high resolution in the proximity of the wall.
• Wall-resolved LES
In this approach, the objective is to capture the dynamics of the locally most energetic scales. This imposes that the mesh cell size decreases in the near-wall region to account for the smaller scales present in the flow. The problem associated to this reduction is the increase in cost of the simulation. The estimation of the cost associated to this type of simulation is linked to the value of the Reynolds number and scales as Re13/7 following
Choi and Moin [31]. This implies that the cost is near to that of a DNS as the value of y+, should not be higher than 5 for channel simulations [140] and even lower when
curvature effects are present. In this low value y+ region, contributions from the subgrid-
scale models should be well adapted and an adequate SGS model (cf. the P1 property in App. B) has to be used.
It is known that different levels of refinement and discretization lead to different predictions and the only way to correctly assess this influence is by comparing different grids. This is especially critical if a wall-resolved approach is chosen, which increases the number of degrees of freedom, but more importantly, limits the time step to be used. This observation has led some authors to qualify such predictions as Quasi-Direct Numerical Simulations (QDNS) [171]. It is of note to underline the fact that in a wall model context, the effort required to obtain a good near-wall model is especially critical for blade flows characterized by an adverse pressure gradient. Most industrial-like application of LES relies on the use of laws-of-the-wall [189]. The main issue in this approach is the existence of pressure gradients which are not taken into account in classical approaches [148] or more recently [115] for example. The importance of taking into account the local effects of tangential pressure gradient was first noted by Wang and Moin [190] and good agreement has been found in papers such as Duprat et al. [60] and Maheu et al. [114]. In all approaches however, there is still the issue of the prediction of the transition position. The law-of-the-wall is designed to behave properly in turbulent boundary layers, whereas this might not be the case on the whole blade surface being then necessary to deactivate the model. Only the wall-resolved approach provides a sufficient degree of confidence today for such near wall flow predictions around blades where a transitioning flow is expected. Indeed, the solution provided by a well designed wall-resolved LES allows to capture all the effects provided an adequate local grid resolution. The main difficulty is hence not so much the physics in the boundary layer but the capability to capture effects such as strain or curvature [172] as well as rely on adequate SGS models.
Although the subject of this discussion stresses the importance of the turbulence formalism, it is important to note and this is especially evidenced in the context of LES, grid resolution is of foremost importance. The cell grid size parameter determines the filter size and in the case
of wall bounded flows qualifies the prediction as a wall-resolved simulation. Great care on grid generation is hence needed but different numerical solutions call for different code capabilities on this specific matter as detailed in the next section.