Reclassification of the logarithmic-layer-mismatch phenomenon in wall-modeled large-eddy simulations

The wall-modeled large-eddy simulation (WMLES) is recognized as a high-fidelity tool for simulating high Reynolds-number wall-bounded turbulent flows. However, the logarithmic-layer mismatch (LLM) phenomenon, where the mean streamwise velocity profile deviates from the logarithmic law, has plagued the WMLES community for many years. Understanding the causes of the LLM phenomenon is essential for the development of the wall model with the aim of alleviating or eliminating LLM. Based on the initial definition of LLM, it can be classified into positive (or negative) LLM, where the mean velocity profile is higher (or lower) than the logarithmic law. In this review, we further refine the classification based on the velocity gradient in the transition region by introducing GI-LLM (gradient-increasing LLM) and GD-LLM (gradient-decreasing LLM). To the best of the author’s knowledge, only the pGI-LLM (positive-GI LLM) has been observed in the hybrid RANS/LES method. As for the wall-shear-stress model, there exist three types of LLM in literature, i.e., pGI-LLM, pGD-LLM (positive-GD LLM), and nGI-LLM (negative-GI LLM).We revisit and discuss the explanations and remedies of different types of LLM observed in the previous studies. We then propose that the LLM can be broadly classified into GI-LLM and GD-LLM, according to their underlying mechanisms. Influenced by “high-dissipation” models or numerical schemes, the near-wall flow field tends to exhibit characteristics of “high dissipation, weak fluctuations, and low Reynolds number”. As the flow transitions to the LES region, where turbulence becomes more active, the mean strain (or mean velocity gradient) increases and thus results in GI-LLM. In contrast, GD-LLM is associated with “low-dissipation” models or numerical schemes, which cannot effectively suppress or dissipate the unphysical fluctuations induced by the numerical and/or modeling errors in the near-wall region. As a result, the near-wall flow field exhibits “low dissipation, strong fluctuations and high Reynolds number” characteristics, and the mean strain (or mean velocity gradient) decreases as the flow transitions to the LES region. Additionally, it is found that in the same WMLES cases, slip boundary conditions can enhance near-wall fluctuations, thereby contributing to the GD-LLM phenomenon. It is important to emphasize that the previous explanations and remedies of LLM remain valid, but focus on different contributing factors. The formation of LLM is sensitive to various elements, including numerical schemes, wall models, subgrid-scale stress models, and even grid configurations. This highlights the importance of establishing general constraints for wall models. To address these challenges, we introduce the total-shear-stress-conserved (TSSC) constraint, which ensures the “correct dissipation” for near-wall flows and effectively eliminates the LLM phenomenon. Furthermore, we demonstrate the critical role of the convection term (equivalent to the resolved Reynolds shear stress in fully developed turbulent channel flow) in enforcing the TSSC constraint.