# Direct Numerical Simulation of Turbulent Boundary Layer over Cubical Roughness Elements
## 1 Introduction  
The rapid urbanization following the Industrial Revolution has led to increased population density and the proliferation of apartments and high-rise buildings within cities [1]. This expansion significantly releases pollutants from human activities into the atmosphere, leading to severe atmospheric pollution when not expelled promptly [2]. At turbulent/non-turbulent interfaces of the boundary layers, an active exchange of energy and substances occurs [3]. Additionally, high-rise buildings introduce concerns of building-induced wind, posing threats to human life and property. Assessing the impact of building-induced wind has become a critical aspect of architectural design. Urban authorities now require studies on how this wind influences pedestrians in large construction projects [4]. The refore, it is essential to accurately scrutinize the complex and swirling flows around buildings and their interactions. Rough-wall turbulence is ubiquitous in both nature environments and industrial fields. Recently, there has been a significant focus on the study of wall-bounded turbulence over rough surfaces [5, 6]. Extensive studies on rough-wall turbulence have been conducted experimentally and numerically on 2D rods and 3D cubes [7, 8, 9, 10, 11, 12, 13], hemispheres [14, 15], LEGG\({}^{\circledR}\) bricks [16, 17], randomly distributed roughness [18, 19], and real urban terrain [20, 21]. In urban regions, the canopy layer includes bluff bodies representing buildings, typically one-tenth the size of the boundary layer thickness (\(\delta\)). The characteristics ofrough-wall flows can be determined based on the height (\(k\)), arrangement, and spacing of roughness elements [22; 23; 24]. A major parameter is the plane density (\(\lambda_{P}\)), defined as the plan area of elements per unit area of the urban array. [PERSON] and [PERSON] [24] identified flow regimes in urban regions based on urban surface density, including \(\lambda_{P}\). To concentrate on flows around elements and their interactions, 3D cubic roughness elements are utilized with \(\lambda_{P}\) = 0.25 in a staggered arrangement. Several studies of turbulent flows over a staggered cube array with \(\lambda_{P}\) = 0.25 to describe urban-like terrain have been conducted. [PERSON] et al. [25] employed hot-wire anemometry and laser Doppler anemometry to measure the velocity of urban canopy layers with \(k/\delta\) = 0.135 at \(Re_{\theta}\) = 12,000. Here, \(Re_{\theta}\) is the momentum thickness defined as \(U_{\infty}\theta/\
u\), where \(U_{\infty}\) is the free-stream velocity, \(\theta\) is the momentum thickness, and \(\
u\) is the kinematic viscosity. [PERSON] et al. [25] reported the dominant scales of turbulence in the roughness sublayer of urban flow are of the same order as the height of elements. Additionally, they observed two energetic scales around the top of elements, with small scales superimposed onto larger scales, supported by the results using particle image velocimetry (PIV) [26]. [PERSON] et al. [27] conducted experiments with hot-wire anemometry to measure the velocity of the turbulent boundary layer (TBL) at \(Re_{\tau}\) = 5250 with the same configuration as [PERSON] et al. [25] with \(k/\delta\) = 0.085. Here, \(Re_{\tau}\) is the friction Reynolds number as \(u_{\tau}\delta/\
u\), where \(u_{\tau}\) is the friction velocity. [PERSON] et al. [28] conducted experiments on TBLs with cubic roughnesses of \(k/\delta\) = 0.044 at \(Re_{\tau}\) = 32,400 and \(k/\delta\) = 0.045 at \(Re_{\tau}\) = 49,900 using hot-wire measurements. [PERSON] et al. [29; 30] performed experiments at the same facilities as [PERSON] et al. [28] using stereoscopic PIV at two wall-parallel planes. A spectral analysis revealed that large scales influence the flow within the roughness sublayer, and these scales depend on the arrangement of cubes. Experimental results can enhance the understanding of high-Reynolds-number flows, but there are limits to the relatively small field of view and low-resolved data near the wall due to roughness elements.