On locally embedded two-scale solution for wall-bounded turbulent flows

Abstract

Recent findings on wall-bounded turbulence have prompted a new impetus for modelling development to capture and resolve the Reynolds-number-dependent influence of outer flow on near-wall turbulence in terms of the ‘foot-printing’ of the large-scale coherent structures and the scale-interaction associated ‘modulation’. We develop a two-scale method to couple a locally embedded near-wall fine-mesh direct numerical simulation (DNS) block with a global coarser mesh domain. The influence of the large-scale structures on the local fine-mesh block is captured by a scale-dependent coarse–fine domain interface treatment. The coarse-mesh resolved disturbances are directly exchanged across the interface, while only the fine-mesh resolved fluctuations around the coarse-mesh resolved variables are subject to periodic conditions in the streamwise and spanwise directions. The global near-wall coarse-mesh region outside the local fine-mesh block is governed by the augmented flow governing equations with forcing source terms generated by upscaling the space–time-averaged fine-mesh solution. The validity and effectiveness of the method are examined for canonical incompressible channel flows at several Reynolds numbers. The mean statistics and energy spectra are in good agreement with the corresponding full DNS data. The results clearly illustrate the ‘foot-printing’ and ‘modulation’ in the local fine-mesh block. Noteworthy also is that neither spectral-gap nor scale-separation is assumed, and a smooth overlap between the global-domain and the local-domain energy spectra is observed. It is shown that the mesh-count scaling with Reynolds number is potentially reduced from $O(R{e^2})$ for the conventional fully wall-resolved large-eddy simulation (LES) to $O(Re)$ for the present locally embedded two-scale LES.

Two-scale solution for tripped turbulent boundary layer

“…In relation to the common limit of the previous MFU methods, the distinctive scale-dependent interface treatment in the original two-scale framework (He 2018) becomes particularly relevant. As illustrated by Chen & He (2022) for a canonical channel flow, the scale-dependent interface treatment enables the large scales to directly pass through the interface, so that the 'foot-printing' can now be captured in the local near-wall fine-mesh block. The well-resolved local fine-mesh domain subject to well-captured 'footprints' then provides a much more suitable base on which the source terms can be generated to correct the global under-resolved coarse-mesh domain.…”

“…However, the near-trip vortex shedding process, responsible for generating these large-scale disturbances, needs to be resolved locally with the fine-mesh resolution. As such, the wall-normal distance of the frontal fine-mesh block for tripping should be higher than the low bound of the log-region, which is used to determine the fine-mesh block height in a fully turbulent regime, as shown by Chen & He (2022). This requirement can be easily met however if we take a value of the log region low-bound at a downstream position (thus a more developed and thicker boundary layer).…”

“…A distinctive alternative approach the authors adopt is the two-scale methodology (He 2018; Chen & He 2022). The method is to couple a locally embedded near-wall fine-mesh DNS block (or a small number of blocks) with a global coarser mesh domain.…”

“…The source terms are generated by upscaling the target space–time-averaged solution within the locally embedded fine-mesh block. The two-scale methodology has been implemented, verified and demonstrated in the canonical channel flow by Chen & He (2022) where the mesh-count scaling with Reynolds number is potentially reduced from for the full wall-resolved LES to for the two-scale solutions.…”

Linear and Nonlinear Dimensionality Reduction from Fluid Mechanics to Machine Learning

“…Among the manifold learning techniques, the most popular ones are arguably the Locally Linear Embedding (LLE) Ghojogh et al (2020b); Roweis and Saul (2000); Saul and Roweis (2001) and ISOMAPs (Ghojogh et al, 2020a;Tenenbaum et al, 2000). These are now entering the fluid dynamics comunity as alternatives to the POD for reduced order modeling (Chen and He, 2022) and for finding compressed representation of fluid flows (Ehlert et al, 2019;Farzamnik et al, 2022;Tauro et al, 2014).…”