A multiscale approach to hybrid RANS/LES wall modeling within a high‐order discontinuous Galerkin scheme using function enrichment

Abstract

Summary We present a novel approach to hybrid Reynolds‐averaged Navier‐Stokes (RANS)/ large eddy simulation (LES) wall modeling based on function enrichment, which overcomes the common problem of the RANS‐LES transition and enables coarse meshes near the boundary. While the concept of function enrichment as an efficient discretization technique for turbulent boundary layers has been proposed in an earlier article by Krank & Wall (A new approach to wall modeling in LES of incompressible flow via function enrichment. J Comput Phys. 2016;316:94‐116), the contribution of this work is a rigorous derivation of a new multiscale turbulence modeling approach and a corresponding discontinuous Galerkin discretization scheme. In the near‐wall area, the Navier‐Stokes equations are explicitly solved for an LES and a RANS component in one single equation. This is done by providing the Galerkin method with an independent set of shape functions for each of these two methods; the standard high‐order polynomial basis resolves turbulent eddies, where the mesh is sufficiently fine and the enrichment automatically computes the ensemble‐averaged flow if the LES mesh is too coarse. As a result of the derivation, the RANS model is applied solely to the RANS degrees of freedom, which effectively prevents the typical issue of a log‐layer mismatch in attached boundary layers. As the full Navier‐Stokes equations are solved in the boundary layer, spatial refinement gradually yields wall‐resolved LES with exact boundary conditions. Numerical tests show the outstanding characteristics of the wall model regarding grid independence, superiority compared to equilibrium wall models in separated flows, and achieve a speed‐up by two orders of magnitude compared to wall‐resolved LES.

Wall modeling via function enrichment: Extension to detached-eddy simulation

“…However, an alternative hybrid RANS/LES turbulence modeling approach can be constructed based on the enrichment, which a priori circumvents these problems and the associated log-layer mismatch. This turbulence model is described in a follow-up paper [12].…”

“…A strong adverse pressure gradient and flow separation from the curved boundary are challenging for many statistical modeling approaches, but DDES yielded very good agreement with a reference LES in that study. Also, all previous publications on wall modeling via function enrichment [1,2,12] used this benchmark example, and very promising results were obtained if a turbulence resolving approach was used. Reference data for this flow is provided by DNS at the lower Reynolds number [19] (available for download at [24]) and water-channel experiments [25] at the higher Reynolds number.…”

“…Near separation or reattachment locations, it may happen that the wall shear stress becomes zero, which renders the function space linear dependent. However, considering that the first off-wall point is located very close to the wall in terms of y + at these locations, a simple and consistent solution is to temporally "switch off" the enrichment in the respective cells, see [2] for details and [12] for an evaluation of the method in WMLES in the context of another turbulence modeling approach. If wall shear stress becomes larger at these locations at a later instance, the enrichment is "switched on" again.…”

“…In the framework of the present enrichment methodology, it is possible to construct an alternative hybrid RANS/LES turbulence model, which does not show a log-layer mismatch by definition. We have recently developed such an approach, which is the topic of a subsequent publication [12]. The RMS velocities and the Reynolds shear stress are also presented in Figure 3 up to Re τ = 5,200 and compared with the DNS data.…”

ExaDG: High-Order Discontinuous Galerkin for the Exa-Scale

“…We have applied this solver framework to conduct direct numerical simulations of turbulent channel flow in [45], the first direct numerical simulation of the turbulent flow over a periodic hill at Re ≈ 10 4 in [46], and to large-eddy simulation of the FDA benchmark nozzle problem in [25]. Furthermore, we have developed multiscale wall modeling approaches that allow to use the proposed highly efficient schemes also for industrial cases with even higher Reynolds numbers than what is feasible for wall-resolved large eddy simulation [47].…”