# Direct Numerical Simulation of Turbulent Boundary Layer over Cubical Roughness Elements
## 1 Introduction
[PERSON] [31] performed a large-eddy simulation of turbulent channel flows over the staggered arrays with \(\lambda_{P}\) = 0.25 of cubes with \(k/\delta\) = 0.167. Wall-normal profiles of Reynolds shear stress were reported near the wall under elements, and low-speed streaks and vortical structures were visualized in instantaneous flow fields. To fully resolve all turbulent scales, [PERSON] et al. [32; 33] conducted the direct numerical simulation (DNS) of turbulent channel flows at \(Re_{\tau}\) = 500 over the staggered cubes with \(\lambda_{P}\) = 0.25 and \(k/\delta\) = 0.133. [PERSON] et al. [33] observed vortical structures around low-momentum regions in the logarithmic layer, reminiscent of the hairpin vortex model [34]. However, the sizes of the computational domains are not sufficiently large to resolve \(\delta\)-scale turbulence structures [35]. While the free-slip boundary condition was applied to the top boundary of channel flows, there are limitations in mimicking the real urban boundary layer due to the periodicity in the streamwise direction and in fully resolving the flow around individual elements with uniform grids in the wall-normal direction [33]. The refore, the DNS of TBL is essential to fully resolve turbulent flows under the roughness sublayer and large scales. The re are DNS studies that conduct TBLs over 3D cubic roughness elements with \(\lambda_{P}\) = 0.25 [36; 37]. The height of elements is approximately 0.06\(\delta\), equivalent to 26\(\sim\)30 wall units based on \(Re_{\tau}\) = 434\(-\)488. [PERSON] et al. [36] and [PERSON] et al. [37] concentrated on examing the effects of streamwise and spanwise spacing of elements on the properties of TBLs. Considering the average thickness (=600 m) of urban boundary layers [38], DNS studies of TBLs for higher \(k/\delta\) and \(Re_{\tau}\) are required to understand turbulent flows around buildings. The objective of the present study was to explore the energy transport near roughness elements in rough-wall turbulence. To this end, DNS was conducted on a TBL (\(Re_{\tau}\) = 816) over a staggered array of 3D cubical roughness elements with \(k/\delta\) = 0.121. For comparison, DNS data from a zero-pressure-gradient (ZPG) TBL with a smooth wall at a similar \(Re_{\tau}\) were employed. This paper is organized into four sections. Section 2 describes the numerical procedure for DNS. Section 3.1 analyzes the streamwise variations of turbulence statistics following a roughness step change. Turbulence statistics are compared with the results of other experiments and DNS studies for validation in Section 3.2. In Section 3.3, turbulence statistics are conditionally averaged, and spectral analysis is conducted. Section 3.4 employs quadrant analysis to investigate Reynolds shear stress and the contributions of each quadrant to flows near the roughness elements. Additionally, Section 3.5 calculates the transport equation of Reynolds stress and conditionally averages each budget term to examine the energy transfer near the elements. Finally, Section 4 concludes the paper with a summary of the present results.