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Reynolds Equation is the most commonly used governing equation in turbulence. However, its application in wall-bounded turbulent shear flows may involve a defect. In general, Reynolds averaging should be ensemble averaging and Reynolds stresses are supposed to express all the actions of turbulence on the mean field. In statistically steady three-dimensional flows, Reynolds stresses are usually defined as correlations of temporal velocity fluctuations so that they cannot contain the influences of steady components of streamwise vortices. This is believed to be one of the reasons why many closure models in RANS meet problems in flows where streamwise vortices play significant roles. In this paper, Spatial-Temporal (S-T) averaged Reynolds stresses were defined, which separates the turbulence actions caused by temporal or spatial velocity fluctuations. DNS data for a fully developed channel flow were then used to check balancing of equations. Comparison showed that the balancing errors in the S-T averaged Reynolds equations were obviously smaller than those in the temporal averaged one, in particular, in the near wall region where the streamwise vortices located. Thus, a combination of traditional model with a supplemental model expressing influences of streamwise vortices might be a way out to improve the turbulence modeling.

This study addresses the vivid internal flow structure variations through horizontal double-layered vegetation (HDLV) under subcritical flow conditions for an inland tsunami. The computational domain was built in ANSYS Workbench, while post-processing and simulation were performed using the computational fluid dynamics (CFD) tool FLUENT with the three-dimensional (3D) Reynolds stress model (RSM). Two alternative arrangements of HDLV were considered, namely Configuration 1 (short submergent layer $(Ls)+\text{tall}$ emergent layer (Lt)) and Configuration 2 (tall emergent layer $(Lt)+\text{short}$ submergent layer (Ls)) along with varying flow depths. Strong inflections in velocity and Reynolds stress profiles were observed at the interface near the top of Ls, Whereas, these profiles were almost constant from bed to the top of vegetations inside Lt. A shear layer zone was formed above the top of Ls, which extended to the downstream region in Configuration 2 while it was restricted by Lt in Configuration 1. The normal Reynolds stresses at the bed were significantly greater within Ls in Configuration 2 than inside Lt in Configuration 1. Hence, Configuration 1 was performed relatively better than Configuration 2 in terms of reducing velocity within the vegetation, while Configuration 2 played a key role in attenuating the increased velocities and confining the shear layer above the short submergent layer.