Investigation of wall pressures and surface flow patterns on a wall-mounted square cylinder using very high-resolution Cartesian mesh

Journal of Wind Engineering & Industrial Aerodynamics 188 (2019) 1–18

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Investigation of wall pressures and surface flow patterns on a wall-mounted square cylinder using very high-resolution Cartesian mesh Yong Cao *, Tetsuro Tamura, Hidenori Kawai Department of Architecture and Building Engineering, Tokyo Institute of Technology, Yokohama, Japan

A R T I C L E I N F O

A B S T R A C T

Keywords: Wall pressure Surface flow pattern Surface-mounted square cylinder Critical point Building cube method Immersed boundary method

Surface flow patterns on a wall-mounted square cylinder and their relationship with pressure are investigated numerically using a fine Cartesian grid. A combination of the building cube and immersed boundary methods and results, including mean pressure, root mean square (r.m.s) pressure, and flow fields are validated qualitatively and quantitatively through experiments. At an attack angle (α) of 0
, it is found that the mean pressure has a distinct vertical classification, even for short cylinders (aspect ratio of 3 or 4). The side-wall pressures are divided into three categories from bottom to top: “junction region,” “2D-like region,” and “free-end region.” Their borders correspond to two notable saddle points in the side-wall streamlines. The mean base pressures can be classified into “lower region” and “near-wake vortex influence region” in the vertical direction. The base-pressure tendency is similar to the change in wake formation lengths. The border is near the upper end of the attachment line in back-wall streamlines. At α ¼ 15
, the mean pressures on the flow-reattachment wall are classified using the curvature of the attachment line, wherein strong two-dimensional effects arise locally in the 2D-like region. Additionally, the similarity between 2D-like regions and infinite cylinders is discussed.

1. Introduction Surface-mounted square cylinders with finite height are commonly encountered in building engineering. Their aerodynamic characteristics possess salient three-dimensional effects along the vertical direction. The aerodynamics of a finite square cylinder is strongly dependent on the following factors: (i) the attack angle (α); (ii) the aspect ratio of the prism (AR ¼ H/D, where H is the height of the cylinder and D is the width); (iii) Reynolds number (Re ¼ UD/ν, where U is the characteristic velocity and ν is the kinematic viscosity); (iv) the relative thickness of the boundary layer on the ground (δ/H or δ/D). In this study, AR ¼ 3 and 4 and δ/D  20 in order to imitate a typical high-rise building completely immersed in an atmospheric boundary layer. The Reynolds number was set to o(104). This study tested cases where α ¼ 0
and α ¼ 15
. Most previous studies focused on the former angle (e.g., Bourgeois et al., 2011; Wang and Zhou, 2009; Kawai et al., 2012; Sumner et al., 2017). The latter angle approaches the critical or glancing angle, where the mean drag force coefficient of the cylinder is minimized, the mean lift magnitude is maximized, and the Strouhal number (St) is maximized. The effects of incidence angles were investigated experimentally by Sakamoto (1985), McClean and Sumner (2014), Unnikrishnan et al. (2017), and Sohankar et al. (2018).

In the past, studies on square cylinders with finite height primarily focused on (i) measurement and study of cylinder pressure, (ii) vortical structures and their connection in the near and far wakes, and (iii) nearwall flow patterns (i.e., surface flow patterns). To the best of the authors’ knowledge, the first two (i.e., the pressure characteristics on the cylinder and three-dimensional vortical structures) have attracted most of the research interest in this field and were investigated using either experimental or numerical tools. Sarode et al. (1981), Sakamoto and Oiwake (1984), Sakamoto (1985), and McClean and Sumner (2014) measured the time-averaged, fluctuating pressures and forces acting on wall-mounted square cylinders while considering the influence of the aspect ratio, incidence angle, and boundary-layer thickness. Many researchers, e.g., Sakamoto and Arie (1983), Okuda and Taniike (1993), Wang and Zhou (2009), Bourgeois et al. (2011), Kawai et al. (2012), Sumner et al. (2017), Unnikrishnan et al. (2017), and Sohankar et al. (2018) focused on the near-wake or far-wake flow fields. In particular, Sakamoto and Arie (1983) reported two dominant vortex-shedding patterns behind surface-mounted cylinders: symmetric “arch vortex shedding” and anti-symmetric “Karman vortex shedding” for cylinders with aspect ratios smaller or greater than a critical value, respectively. Wang and Zhou (2009) emphasized the coexistence of arch vortex shedding

* Corresponding author. E-mail address: [email protected] (Y. Cao). https://doi.org/10.1016/j.jweia.2019.02.013 Received 8 September 2018; Received in revised form 24 January 2019; Accepted 20 February 2019 0167-6105/© 2019 Elsevier Ltd. All rights reserved.

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Journal of Wind Engineering & Industrial Aerodynamics 188 (2019) 1–18

is well-suited for massive computation with high parallel efficiency, which is useful for predicting surface-flow patterns with high resolution in this study. On the other hand, treating the wall in a Cartesian grid is an important issue, particularly for complex curved surfaces. Mittal and Iaccarino (2005) summarized the different approaches of the immersed boundary method (IBM). This study used a modified IBM, similar to the well-known ghost cell method, which was proposed and implemented by Onishi et al. (2013, 2014, 2018). Examination and validation of the BCM-IBM method will be based on the relatively simple but typical flow configuration, i.e., high-Re turbulent flow past a wall-mounted square cylinder. It becomes possible to comprehensively compare the results with available experimental data. Numerical validation is also an essential prerequisite for discussing the relationship between near-body flow and surface pressure on the cylinder. To summarize, this study aims to provide a surface flow pattern database for square cylinders with finite height and to clarify the relationship between the pressure on the cylinder and the surface flow pattern using simulations with very high spatial resolution. Specifically, the main goals include: (1) examining and validating numerical methods for use in the next-generation K computer based on a typical building configuration, (2) describing the mean near-wall flow streamlines of the cylinder, (3) investigating the three-dimensional effects of mean pressure along the vertical direction, (4) identifying the relationship between the mean pressure and wall streamlines.

and Karman vortex shedding and suggested that the inclination of the spanwise vortex generated the tip and base vortices. Bourgeois et al. (2011) proposed a vortex-shedding model based on particle image velocimetry (PIV) measurements when a cylinder (AR ¼ 4) was partially immersed in a boundary layer of δ/D ¼ 0.72. The model was composed of an alternating “half-loop shed structure.” In addition, Okuda and Taniike (1993) focused on flow next to the side wall and found the presence of inverted and standing conical vortices. These were associated with severe negative pressures at various attack angles. In contrast, quantitative data on near-wall flow patterns are rarely found, possibly owing to technical difficulties. Nevertheless, surface oil flow visualization was extensively performed. For example, Martinuzzi and Tropea (1993) examined mean near-wall flow on prismatic obstacles, and Sumner (2013) summarized the surface flow above the free-end of a circular cylinder. However, oil flow visualization typically cannot provide the precise locations of critical points (Tian et al., 2004; Depardon et al., 2005). Gravity prevails over shear friction for pigments on lateral side walls, which drags the pigment downward and increases the difficulty in distinguishing critical points. Quantitatively, Castro and Dianat (1983) and Dianat and Castro (1984) used pulsed wall gauges to measure the mean and fluctuating skin friction of the top surfaces of wall-mounted rectangular bodies with long span. They tested the situation where the shear layer reattached or did not reattach near the trailing edge of the body. They argued that this measurement was the first set of quantitative surface flow patterns that unambiguously determined the location of critical points. Depardon et al. (2005) introduced the near-wall PIV technique and provided quantitative skin friction patterns on the walls of a surface-mounted cube. Recently, near-wall PIV visualization was extended to the area above the free end of a finite square cylinder at α ¼ 0
(Sumner et al., 2017). Complete surface flow patterns on square cylinders of finite height are lacking. For this reason, one of the goals of this study is to provide a database of near-wall flows about this body shape, which is often encountered in building engineering. This work is expected to be useful for improving the integrity of building aerodynamics and provide a better understanding of the external flow around a body. The three aspects that make up the major studies on finite-height square cylinder (i.e., pressures on the cylinder, surface-flow patterns, and vortical structures) are indeed interrelated. Once one of them is known, we aim to infer the remaining information to some extent. The desire motivates an attempt to clarify the relationship between the pressure on the cylinder and the surface-flow pattern. The link between the surface-flow pattern and the three-dimensional vortical structure will be further investigated in future studies. The relationship among pressure on the cylinder, surface-flow patterns, and vortical structures are expected to be applied or referred to in wind tunnel studies or in field observation situations where measured data is limited, but one wishes to infer more information. In this study, the term “limiting streamline” or “wall streamline” is used to represent the surface flow pattern, which is defined as the limit position of the streamline when the distance between the streamline and the surface tends to zero. The limiting streamline tends to follow the skin friction line when approaching the surface of a body (Delery, 2013). Numerical simulations with high spatial resolution are regarded as a sensible tool for accurately describing near-wall flow. Fortunately, the development of supercomputers and numerical methods with high parallel efficiency have gradually facilitated simulations with high spatial resolution. The present simulations were performed using K-computer, the national high-performance computer of Japan. The numerical method, under the framework of the building cube method (BCM) proposed by Nakahashi (2003, 2005), was originally designed to realize true high fidelity computational fluid dynamics (CFD) simulations using high performance computers with a high density hierarchical Cartesian mesh. It provides simplicity in all flow simulation stages, i.e., the mesh generation, solution algorithm, and post-processing stages (Nakahashi, 2005). In comparison with conventional unstructured grids, the BCM framework

2. Numerical method and model 2.1. Numerical method The topology independent IBM approach (Onishi et al., 2013, 2014, 2018) was proposed and implemented under the framework of the building cube method (Nakahashi, 2003, 2005). This study serves validation and application purposes of these methods. In the following, BCM and the modified IBM are briefly introduced. BCM divides the computational domain into sub-domains called cubes based on block-structured Cartesian meshes, which are further decomposed into cells. Each cube has the same number of equally-spaced cells. An example based on an inclined square cylinder is shown in Fig. 1. Here, a three-dimensional cube consists of 16  16  16 cells. BCM has the following advantages over the use of generalized curvilinear and unstructured meshes: (1) it allows the use of simple pre-processing methods, flow solvers, and post-processing methods (Nakahashi, 2005); (2) it is suitable for highly parallel computation (Komatsu et al., 2011). The incompressible continuity and Navier-Stokes equations are shown in the dimensionless form in Eqs. (1) and (2). ui and p are the velocity components and the pressure defined at the center of the cell, respectively. Ui is the contravariant velocity component that is perpendicular to the cell face. In particular, the IBM external forcing term fi is expressed in Eq. (3) in order to satisfy unþ1 ¼ V nþ1 , where the RHS is the

Fig. 1. Computational mesh in BCM with an inclined square cylinder as an example. A cube consists of 16  16  16 equally-spaced cells in three dimensions. 2

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Journal of Wind Engineering & Industrial Aerodynamics 188 (2019) 1–18

cell in the solid that has at least one neighbor in the fluid. For instance, the ghost cell is marked with a hollow square, and its value is represented by qGC . The imaginary point is defined by mirroring the ghost-cell point around the intersection point to the fluid region, which is indicated with an open triangle. The value at the imaginary point (qIP ) is interpolated from the surrounding fluid cells (i.e., linear interpolation in one dimension). The sign-inverse velocity (i.e.,  qIP ) is imposed on the corresponding ghost cell in the solid (i.e., qRC ). Extending the one-dimensional process to three dimensions is straightforward, i.e., the horizontal axis in Fig. 2 can represent any of x, y, or z, and each dimension is implemented separately. In the original “ghost cell” method and its modified version in Nakahashi (2011), the line connecting the ghost point and the imaginary point is perpendicular to the original immersed boundary. The present method degenerates geometric data into cell-oriented information and adopts an axis projection technique. Consequently, the adjacent cell-orientated geometric faces are not guaranteed to be connected and continuous (Onishi et al., 2013). This means the present method possibly has stronger dependence on cell resolution. A relatively coarse resolution may possibly lead to a large gap between the neighboring cell-oriented geometric planes. It is difficult to guarantee numerical stability and mass conservation with high accuracy. For these reasons, we suspect that small cells may be required in the current version of the method to provide high accuracy. On the other hand, the simplicity of the wall-boundary treatment and the faster computational speed can meet the very fine resolution requirements.

sum of the convective, diffusion, and pressure gradient terms, and V nþ1 is the desired velocity at step n þ 1. The convective and diffusion terms are spatially discretized using the second-order central-difference scheme, while 5% of the first-order upwind scheme is blended to estimate the convective flux on the cell face. The fractional-step method is used for time marching, as shown in Eqs. (4)–(6). The semi-implicit Crank–Nicolson method is used to treat the convective and diffusion terms. The semi-implicit velocity equation in Eq. (4) and Poisson equation in Eq. (5) are solved using the red/black successive over-relaxation method (SOR) method.

∂ Ui ¼ 0 ∂xi

(1)

∂ui ∂ui ∂p 1 ∂2 ui þ Uj ¼ þ þ fi ∂t ∂xj ∂xi Re ∂xj ∂xj

(2)

nþ1=2

fi

¼ RHSnþ1=2 þ

u*i ¼ uni þ

Δt 2




(3) 

 U nj

∂2 pnþ1 1 ∂u*j ¼ ∂xj ∂xj Δt ∂xj ¼ u*i  Δt unþ1 i

V nþ1  uni i Δt

∂pnþ1 ∂xi

∂u*i 1 ∂2 u*i Δt þ þ 2 ∂xj Re ∂xj ∂xj


 U nj

∂uni 1 ∂2 uni þ ∂xj Re ∂xj ∂xj

 (4)

(5)

(6)

2.2. Numerical model

Methods for representing geometric data and defining no-slip wall boundary conditions can be found in Onishi et al. (2018). The following provides a conceptual introduction. Preliminary processing is first performed by degenerating a computer-aided design (CAD) model into cell-oriented CFD boundary information. This is called a “topology independent boundary representation,” which allows the physical solver to handle “dirty” CAD data (e.g., with gaps/overlaps) without any clean-up. Compared with unstructured-grid solvers, this greatly reduces the workload required to clean up geometric data and define a closed volume, particularly for complicated geometries. The characteristic plane equation containing wall-boundary information is constructed in each cell. The characteristic plane intersects the grid lines (connecting cell centers) at intersection points. Meanwhile, the axis projection technique is introduced, i.e., the search process for the imaginary point is projected to the axial direction. As a result, it becomes possible to mathematically express the boundary condition treatment in one dimension. Fig. 2 shows the procedure in the boundary condition treatment. The red open circle on the projected horizontal axis is the intersection point. Like the “ghost cell” approach in Mittal and Iaccarino (2005), a ghost cell is defined as a

The cases where AR ¼ 3 and AR ¼ 4 were studied here. The Reynolds number Re ¼ UHD/ν  5  104, where UH is the mean inflow velocity at the height z ¼ H. Two angles of attack are tested (i.e., α ¼ 0
and 15
). The ratio of the thickness of the turbulent boundary layer (TBL) to the cylinder width is δ/D ¼ 20.1. The computational domain size is 32D  8D  32D, where the origin of the axis is at the center of the bottom of the cylinder. Although there is no qualitative effect on flow near the cylinder, it is expected that blockage caused by the relatively small lateral dimension (8D) of the computation domain will slightly reduce the growth rate of the far wake compared with the case where the cylinder is placed in a wide domain. Therefore, it is expected that the fluctuation-related quantities will be smaller in the far wake. However, the qualitative and quantitative fields around the cylinder are expected to remain unaffected. The grid system in the symmetry plane is shown in Fig. 3(a). The cut plane at the middle height of the cylinder is shown in Fig. 3(b). The case where H/D ¼ 3 is displayed as an example since the case where H/D ¼ 4 has a similar cell density distribution. Cubes with the same size are placed as carefully as possible in the shear-layer and near-wake regions. Each cube has a halo region that contains information from all neighbors. Data exchange between neighboring cubes takes place via the halo region. When the size (resolution) of two neighboring cubes varies, spatial interpolation (small to large cubes) or extrapolation (large to small cubes) is used, yielding lower-order accuracy near the cube boundary. The minimum cell size is determined from the thickness of the boundary layer on the cylinder (δB ). δB is estimated using the formulas based on the laminar boundary layer on a flat plate. The characteristic length for calculating δB is the distance from the frontal stagnation point to the cylinder edge. The boundary layer is assumed to be laminar because Re0:5D  2:5  104 and is too low to trigger the laminarturbulent transition, even under inflow turbulence effects. According to pffiffiffiffiffiffiffiffiffiffiffiffiffi White (2006), δB  5:5  ð0:5DÞ= Re0:5D ¼ 0:0174D. In this study, the cell size in the wall-normal direction is Δ ¼ 0.00195D, which means that the boundary layer spans about 9 cells. In other directions near the cylinder walls, the cell size is the same as Δ because the elemental cells are in the shape of a cube. This means that approximately 512  512 cells are

Fig. 2. Procedure for determining the ghost-cell values on the projected axis. qIP is the value at the imaginary point, qGC is the value at ghost cell, Δ is the cell size, and d is the distance between the intersection point and the nearest cell center on the fluid side. 3

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Journal of Wind Engineering & Industrial Aerodynamics 188 (2019) 1–18

Fig. 3. Grid system: (a) in the symmetry plane; (b) in the cut plane at the middle of the cylinder when AR ¼ 3 and α ¼ 0
. Note that the domain sizes are normalized by D. The cell sizes are also tabbed in different regions, relative to the minimum cell size Δ just near the immersed boundary.

distributed on a wall-surface area of 1D  1D. This resolution is much finer than most of the numerical simulations or experimental PIV measurements to date and has the advantage of capturing finer flow structures. Δ is also the smallest cell size throughout the computational domain and is considered to be the reference grid resolution used to describe other regions. The region from ground to z/D ¼ 1 has cell size of 8Δ before approaching the cylinder. In total, 460 million and 598 million cells were used when AR ¼ 3 and AR ¼ 4 at α ¼ 0
, respectively, and 1.2 billion cells were used when AR ¼ 4 and α ¼ 15
. The inviscid estimate of the mean energy dissipation rate ε based on Tennekes and Lumley (1972) was used to estimate the Kolmogorov

shown in Fig. 4. Note that the r.m.s velocity includes the vortex-shedding component, thus it tends to overestimate U . L is estimated as the thickness of the shear layer in the shear-layer region (L SL =D  0:05) and the cylinder width in the wake (L wake =D  1). In the shear-layer region, η=D > 3:0  104 (U SL  0:4; L SL  0:05); in the near wake, η=D > 5:7  104 (U nwake  0:3; L wake  1:0); in the intermediate wake, η= D > 4:6  104 (U iwake  0:4; L wake  1:0). The cube resolution (16  16  16 cells) is also indicated by the horizontal and vertical white lines in Fig. 4. The shear-layer region has cell resolution of 2Δ=D ¼ 3:9  103 , the near wake has cell resolution of 2Δ=D ¼ 3:9  103 , and the intermediate wake has cell resolution of 4Δ=D ¼ 7:8  103 . The present cell sizes are nearly the same order of magnitude as η in the regions of particular interest. Most of the kinematic turbulent energy is dissipated in the scales of 10η. Therefore, most previous direct numerical simulations used 10η as the criteria for determining the appropriate grid resolution. To conclude, the present resolution should be sufficient for

length scale η ¼ ðν3 =εÞ1=4 , i.e., ε  U 3 =L , where U and L are the velocity and length scales of large-scale turbulent motion. Accurate estimation of U and L is very difficult, especially for the complex flow considered here. Here, U is crudely estimated as the root mean square pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (r.m.s) velocity (i.e., σ 2u þ σ 2v þ σ 2w ), and σ 2u þ σ 2v þ σ 2w is distributed as

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fig. 4. Distribution of r.m.s velocity (i.e., σ 2u þ σ 2v þ σ 2w ), taking the mid-height plane of the case with AR ¼ 3 and α ¼ 0
as an example. The range of values indicated in the contour indicates the r.m.s velocity values in the corresponding region. The white horizontal and vertical lines are used to indicate cube sizes (one cube has 16  16  16 cells). 4

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Journal of Wind Engineering & Industrial Aerodynamics 188 (2019) 1–18

Fig. 5. Temporal convergence of the statistical computation: (a) mean and r.m.s pressures; (b) mean and r.m.s force coefficients at AR ¼ 3 and α ¼ 0
. Cp ðt * Þ ¼ 1=ðt * 

constructed by Tokyo Polytechnic University (TPU) (http://wind.arch. t-kougei.ac.jp/system/contents/code/tpu). In this study, the Lund method (Lund et al., 1998) is used to generate a fully developed turbulent boundary layer inflow on a smooth ground wall. The sampled data were scaled and input into the main domain with a finite-height square cylinder through the inlet boundary condition. Fig. 6 shows the mean velocity and turbulence intensity profiles at a location very close to the inlet (i.e., x/D ¼ 7.4). The experimental and numerical results are consistent. To ensure that this consistency is maintained before approaching the cylinder, Fig. 7(a) shows the profiles along the streamwise direction, including x/D ¼ 7.4, 5.4, and 3.4. As expected, the profiles remain nearly unchanged before approaching the bluff body with the present numerical grids and methods. The power spectra of the two velocities measured at the height of the cylinder (AR ¼ 4) are plotted in Fig. 7(b)-(c). The integral length scale Lx is included in these figures, where Lx/D  2 at these two stations. The positions of (x/D, y/D, z/D) are (7.4, 0, 4) and (3.4, 0, 4), and the cell size at two positions is approximately 0.03D. The present spectra and Karman spectrum at low and intermediate frequencies are consistent. By comparing Fig. 7(b)-(c), the energy at higher frequencies recovers slightly as the measurement station moves downstream along the streamwise direction.

the implicit large-eddy simulations (because the upwind scheme is included when discretizing the convective term). The time step (Δt * ¼ ΔtU∞ =D, where U∞ is the free stream velocity) is 2  104, resulting in a maximum Courant number of approximately 0.3 at α ¼ 0
and approximately 1.2 at α ¼ 15
. Statistical analysis begins after the instant when the flow becomes statistically stationary. The duration of the statistical average is about 200t * (specifically, t * ¼ 56  256). Fig. 5 shows the temporal variation in the mean pressure, r.m.s pressure, and forces when AR ¼ 3 and α ¼ 0
. The pressures are captured at the center of the rear and side walls. The mean pressure in R t* time is defined as Cp ðt * Þ ¼ 1=ðt *  t *start Þ t * Cp ðt * Þdt * , where t *start is the start

initial time where the statistical average is calculated and Cp is defined in Eq. (7). Fig. 5 shows that the statistical values converged over the present duration. The computations were performed on the K-computer, which is among the highest caliber of supercomputers in the world. One computational node has a SPARC64™ VIIIfx 2 GHz CPU, 128 GF performance, and 16 GB memory. Each node contains eight cores. The network is based on Tofu Interconnect (6D Mesh/Torus). The wall-clock time varies from case to case. The case AR ¼ 3 and α ¼ 0
(460 million cells) used 2881 computational nodes. The wall-clock time was about 7 days for 1,300,000 time steps. 3072 computational nodes were used for the case AR ¼ 4 and α ¼ 15
(1.2 billion cells). Calculations over 1,300,000 time steps required 11 days in total.

3. Numerical validation of BCM-IBM method 3.1. Wall-mounted square cylinder at α ¼ 0


2.3. Inflow condition Time-averaged and r.m.s pressure coefficients (denoted Cp and σ p , respectively) are compared with results of experiments under similar inflow conditions to validate the method presented here. The pressure coefficients Cp are defined in Eq. (7), where p is the pressure on the

The inflow conditions are first examined by comparing the numerical results with experiments, including Katsumura's experiment (Maruyama et al., 2013) and the aerodynamic database of high-rise buildings

Fig. 6. Time-averaged velocity and turbulence intensity profiles at x ¼ 7.4: (a) comparison with the experiment by Katsumura (Maruyama et al., 2013); (b) comparison with the TPU experimental database. 5

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Journal of Wind Engineering & Industrial Aerodynamics 188 (2019) 1–18

Fig. 7. (a) Examination of profile variations along the streamwise direction before approaching the bluff body; (b)–(c) power spectra density at (x/D, y/D, z/D)¼(-7.4, 0, 4) and (3.4, 0, 4), respectively, where the cell size at two positions is approximately 0.03D.

surfaces of the cylinder, pH and UH are the reference pressure and velocity measured near the inlet and at the same height as cylinders, respectively, and ρ is the density of the fluid. Figs. 8–11 compares the present results with the experiment by Katsumura (Maruyama et al., 2013) when α ¼ 0
. The subplots show the distributions at different cylinder heights. The present results agree well with Katsumura's measurement, regardless whether AR ¼ 3 or AR ¼ 4. This means that the flow patterns for AR ¼ 3 and AR ¼ 4 are similar under the same inflow condition. Therefore, the mean flow patterns and surface pressures at α ¼ 0
are analyzed further for AR ¼ 3.

Cp ¼

p  pH 0:5ρU 2H

(7)

Fig. 12 shows a comparison of the mean flow fields with recent PIV measurements (Sumner et al., 2017; Unnikrishnan et al., 2017). It is worth noting that this comparison should be taken as qualitative because the thickness of the present boundary layer (δ/D ¼ 20) is much larger than that in the experiments (δ/D ¼ 1.5). Fig. 12(a) shows the similarity in the streamlines very close to the top wall, where the distance (z') between the streamline plane and the top wall is set equal to the value used in a previous study (Sumner et al., 2017), i.e., z'=D ¼ 0:016. Unfortunately, the near-wall streamlines on the other faces are unavailable in the

Fig. 8. Comparison of time-averaged pressure distributions with the experiment conducted by Katsumura (Maruyama et al., 2013) at α ¼ 0
. Note that AR ¼ 3 here, while AR ¼ 4 in Katsumura's experiment. 6

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Journal of Wind Engineering & Industrial Aerodynamics 188 (2019) 1–18

Fig. 9. Comparison of r.m.s pressure distributions with the experiment conducted by Katsumura (Maruyama et al., 2013) at α ¼ 0
. Note that AR ¼ 3 here, while AR ¼ 4 in Katsumura's experiment.

Fig. 10. Comparison of time-averaged pressure distributions with the experiment conducted by Katsumura (Maruyama et al., 2013) at α ¼ 0
. Note that AR ¼ 4 in the numerical and experimental cases.

experiments. Fig. 12(b) shows the far-wake flow pattern in the y-z plane when x is outside the mean recirculation zone behind the cylinder. The longitudinal vortices (or tip vortices according to many studies) near the middle of the cylinder are observed, which are induced by the strong downwash from the free end of the cylinder. The leg of the horseshoe vortices can be seen near the ground. The numerical and experimental flows show the same “dipole” behavior, i.e., only longitudinal vortices caused by downwash are observed. In summary, the flow field is in good qualitative agreement between the numerical and experimental results. Aerodynamic drag and lift (FD and FL ) are estimated based on the time-dependent pressure series recorded at the probe points on the cylinder walls. The friction-stress contributions are neglected, which will be considered in future simulations. The force coefficients are defined by

Eqs. (8)–(10). Dominant peaks corresponding to the vortex shedding frequency fvs are observed in the power spectra of lift coefficients. Accordingly, the Strouhal number is defined in Eq. (10). It should be noted that the mean inflow velocity (UH ) at the cylinder height is selected as a reference velocity for normalizing the forces and St, which is common way in wind engineering. For AR ¼ 3 and AR ¼ 4, the reference velocities are slightly different. The mean and r.m.s forces (including CD , CL , C'D , and C'L ) and St are collected in Table 1, together with the reference velocities. Table 1 also includes data from previous experiments (Sakamoto and Oiwake, 1984; Sakamoto, 1985; McClean and Sumner, 2014). The influence of aspect ratio on the bulk parameters is discussed first. The reference velocity UH¼3 is consistently used to

7

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Journal of Wind Engineering & Industrial Aerodynamics 188 (2019) 1–18

Fig. 11. Comparison of r.m.s pressure distributions with the experiment conducted by Katsumura (Maruyama et al., 2013) at α ¼ 0
. Note that AR ¼ 4 in the numerical and experimental cases.

Fig. 12. Comparison of the mean flow field with PIV measurements when AR ¼ 3: (a) mean streamlines above the top wall with a distance of 0.016D; (b) streamlines in the y-z plane in the far wake indicating longitudinal vortices and horseshoe vortices. Table 1 Summary of aerodynamic forces and St, and a comparison with previous experiments when α ¼ 0
. Ureference is the reference velocity used to normalize the forces and St.

present

McClean and Sumner (2014) Sakamoto and Oiwake (1984)

AR

δ=H

Re

CD

CL

C'D

C'L

St

Ureference

3 4 4 3 5 3 4

6.7 5.0 5.0 0.5 0.3 0.7 0.7

5  104 5  104 5  104 7  104 7  104 6  104 6  104

1.23 1.42 1.28 1.29 1.43 1.20 1.27

– – – – – – –

0.142 0.147 0.132 – – 0.0779 0.0842

0.353 0.358 0.322 – – 0.286 0.296

0.094 0.090 0.085 0.102 0.105 0.096 0.097

UH¼3 UH¼3 UH¼4 U∞ U∞ U∞ U∞

normalize the forces and St; see the present results at AR ¼ 3 and AR ¼ 4 when Ureference ¼ UH¼3 in Table 1. CD , C'D , and C'L increase slightly when AR increases from 3 to 4. This trend is reasonable because the three-dimensionality of the cylinder becomes weaker as AR increases. Sakamoto (1985) and McClean and Sumner (2014) studied the effects of aspect ratio on aerodynamic forces in a relatively thin boundary layer and found the same tendency. On the other hand, St changes only slightly, and this phenomenon is consistent with the results in Sakamoto (1985) and McClean and Sumner (2014). CD ¼

FD 0:5ρU 2H DH

(8)

CL ¼

FL 0:5ρU 2H DH

(9)

St ¼

fvs D UH

(10)

The influence of boundary layer conditions is discussed next. The reference velocity at cylinder heights is considered to be more suitable. That is, Ureference ¼ UH¼3 for AR ¼ 3, and Ureference ¼ UH¼4 for AR ¼ 4. The present boundary layer is significantly thicker relative to the height of the cylinder. However, the experiments in Table 1 have boundary layers that are thinner than the cylinder heights. According to Table 1, CD seems to be less sensitive to changes in boundary layer conditions than C'D and C'L . The r.m.s forces tend to be greater in the present thick boundary layer than the experiments. However, St is slightly smaller in the present thick boundary layer.

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3.2. Wall-mounted square cylinder at α ¼ 15


to the behavior of the separated shear layer on the flow-reattachment side. The dynamic behaviors of the shear layer are affected by the inflow conditions (e.g., turbulence characteristics in the boundary layer). The differences in dynamic inflow characteristics between this study and the experiment are another possible cause for the discrepancy in σ p at α ¼ 15
. However, it is believed that quantitative differences in r.m.s values do not significantly affect the discussion regarding time-averaged wall streamlines and their relationship to time-averaged wall pressures. The mean far-wake pattern is qualitatively compared with the available PIV measurements by Unnikrishnan et al. (2017) in Fig. 15. The term “qualitatively” is used because the inflow conditions and aspect ratio between this simulation and the experiment are different. δ/D in the present case is much greater than that from Unnikrishnan et al. (2017). AR ¼ 4 here, while AR ¼ 5 in the experiment. According to Fig. 15, the primary flow patterns look similar, including the longitudinal vortices induced by downwash and the horseshoe vortex legs. However, the

Comparisons of the time-averaged and r.m.s pressure coefficients with the data in the TPU experimental database for α ¼ 15
are shown in Figs. 13 and 14. The time-averaged pressure distributions are consistent. However, this numerical simulation tends to underestimate the r.m.s pressures on the frontal and flow-reattachment walls (i.e., face 01 and face 34). In comparison, the data for α ¼ 0
are consistent. The reasons for the discrepancy in the r.m.s pressures are not fully known. Numerically speaking, the body walls at α ¼ 0
are parallel or perpendicular to the grid lines, whereas the walls at α ¼ 15
and the grid lines cross each other (but not perpendicular). In the latter case, the IBM approach may perform worse when the magnitude of the velocity near the wall is high. The performance of the immersed boundary method in the case of inclined or curved geometry will be examined in future research. Moreover, the aerodynamic pressures at this glancing angle are very sensitive

Fig. 13. Comparison of time-averaged pressure distributions with experiments and the TPU data at α ¼ 15
. Note that AR ¼ 4 in the numerical and experimental cases.

Fig. 14. Comparison of r.m.s pressure distributions with experiments and the TPU data at α ¼ 15
. Note that AR ¼ 4 in the numerical and experimental cases. 9

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Fig. 15. Comparison of mean far-wake flow in the y-z plane with the PIV measurement by Unnikrishnan et al. (2017), which indicates longitudinal and horseshoe vortices.

difference from the case at α ¼ 0
arises from the asymmetry in the longitudinal vortices on both sides of the cylinder. Specifically, the size and height of the longitudinal vortex on the full-separation side (i.e., right-hand side) are greater than on the flow-reattachment side (i.e., left-hand side). Table 2 shows the aerodynamic forces and St at α ¼ 15
compared to previous experiments. As investigated by Sakamoto (1985), McClean and Sumner (2014), and Sohankar et al. (2018), the critical angle (approximately 15
) is associated with the maximum St, the minimum mean drag, and maximum mean lift magnitude values. Variations in the forces and St in this study are consistent with those observed in the aforementioned experiments when the incidence angle changes from 0
to 15
. This means that the force and St tendencies caused by the attack angle are unaffected by the thickness of the boundary layer. In particular, the increase in St as α increases is due to reattachment of the shear layer and the subsequent narrower and shorter wake. From Table 2, the primary difference between the results in this study and the experiments is that the present fluctuating forces are greater than those in the experiments. This phenomenon may arise because the cylinder is subject to the fluctuations of the turbulent boundary layer in this study.

patterns. Three typical instants are selected corresponding to the trough and peak in the time series of CL, as well as the intermediate instant between them (marked with the circles in Fig. 16(a)). The instantaneous cross-sectional flows around the middle of the cylinder are shown in Fig. 16(b)-(d). They correspond to the instants marked in Fig. 16(a). The wake is dominated by anti-symmetric Karman vortex shedding. Accordingly, the oscillation of CL is a periodic sinusoidal waveform with considerable amplitude. Nevertheless, the anti-symmetry differs in time and height. The vortices are schematically indicated in (b)-(d); the antisymmetry of the two vortices behind the cylinder in (d) is weaker than in (b). The flow pattern in (c) shows the transition state from (b) to (d) as the orientation of the near wake and the size of two vortices change. Table 1 shows that C'L ¼ 0.353 when AR ¼ 3 and remains nearly unchanged when AR ¼ 4. Moreover, a clear peak is observed in the lift power spectra (not shown here). The above evidence shows that the critical AR is less than 3 in the present conditions. Critical AR values in previous studies vary widely. Sakamoto and Oiwake (1984) observed a sharp change in the slope of the relationship between C'L and AR near AR ¼ 2.5 and suggested that vortex shedding changed from arch type to Karman type. Above AR ¼ 2.5, C'L was stable near 0.30. McClean and Sumner (2014) reported that the critical AR ¼ 3 because they found that the trends in CD , CL , and St with changes in the incidence angle are significantly different from the trend when AR ¼ 5 to AR ¼ 11. Examples of instantaneous limiting streamlines are presented in Fig. 17(a)-(c). The time-averaged field is shown in Fig. 17(d). The wallnormal distance of the streamlines is 0.005D. Topological differences and position discrepancies at the critical points may arise as the distance to the wall changes (Depardon et al., 2005). In Depardon et al. (2005), the change from 0.0025D and 0.017D to 0.008D (the minimum value tested) did not induce a topological difference upstream from the cube, and there is no significant position discrepancy at the critical points. No visible difference in the limiting streamlines was observed when the wall spacing was 0.008D and 0.005D. The first nearest cell does not need to be used thanks to the numerical treatment used to stabilize the pressure near the wall. Therefore, a wall distance of 0.005D was chosen as this is considered small enough to approximate the skin friction lines. Fig. 17(a)-(c) show the same instant as those in Fig. 16(b)-(d),

4. Wall pressures and surface flow patterns at α ¼ 0
A cylinder of finite height provides strong three-dimensional flow separation and vortical structures due to interference from the ground and the free end. Three-dimensionality refers to a change along the vertical direction. The wall streamlines are imprints of the outer flow on the cylinder surfaces and exhibit distinct changes along the vertical direction. The three-dimensional effect of limiting streamlines will be described in what follows. Later, we attempt to distinguish the relationship between the limiting streamlines and wall pressures in the mean sense. 4.1. Limiting streamlines on the cylinder Despite focusing on the time-averaged field, the study first illustrates the instantaneous surface flows and their corresponding vortex shedding

Table 2 Summary of aerodynamic forces and St, and a comparison with previous experiments when α ¼ 15
. Ureference is the reference velocity used to normalize the forces and St.

present McClean and Sumner (2014) Sakamoto (1985)

AR

δ=H

Re

CD

CL

C'D

C'L

St

Ureference

4 3 5 3 4

5.0 0.5 0.3 0.7 0.7

5  104 7  104 7  104 6  104 6  104

1.18 1.15 1.21 1.03 1.04

0.462 0.435 0.621 0.420 0.465

0.136 – – 0.0558 0.0580

0.129 – – 0.109 0.098

0.117 0.115 0.120 0.109 0.115

UH¼4 U∞ U∞ U∞ U∞

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Fig. 16. (a) Time series of CL at α ¼ 0
and AR ¼ 3; (b) vortex shedding at the instant when CL has a trough near the middle of the cylinder (i.e., z/D ¼ 1.45); (c) vortex shedding at the instant when CL ¼ 0; (d) vortex shedding at the instant when a CL peak arises.

Fig. 17. Surface flow patterns on the side wall: (a) at the instant when CL has a trough; (b) at the instant when CL ~ 0; (c) at the instant when a CL peak arises; (d) time-averaged pattern. In (a)–(c), vectors are colored with the positive (red) or negative (blue) streamwise velocity, the instants are the same as those in Fig. 16, and the wall is the upper side wall in Fig. 16(b)-(d). (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

respectively. The primary flow direction points from left to right. The cylinder wall corresponds to the upper side wall in Fig. 16(b)-(d). The instantaneous flows in Fig. 17(a)-(c) are colored with the streamwise velocity component: red for positive and blue for negative. This allows the mean flow to be related to the instantaneous flow, especially the important features. The focus “F” observed in the mean flow is also seen in the three instantaneous fields, although the instants have a maximum phase difference of π (assuming that one vortex shedding has a total phase of 2π). The type of flow below the saddle point “S1” is similar in the mean and instantaneous fields. In the middle zone, the flow pattern is too complicated to conclude the simplified type shown in Fig. 17(a). However, in Fig. 17(b)-(c), the flapping shear layer is found closer to this side wall, pushing the fluid between the shear layer and the side wall downstream and upstream. The magnitude of the induced velocity is much larger than that shown in Fig. 17(a). Above the height of “S2”, the upward flow is clearly on the upstream portion of the side wall when the shear layer is far from the wall (see Fig. 17(a)), while the downward flow is dominant in the downstream portion when the shear layer is close to

the wall (see Fig. 17(c)). In general, the phase of the shear layer close to this side wall has a larger velocity magnitude and a greater contribution to the mean field. As a result, the mean surface flow tends to be more similar to Fig. 17(b)-(c). The similarity between the instantaneous and mean surface flows motivates further study to emphasize the mean field before attempting to analyze complex transient flows. Fig. 18 shows the limiting streamlines on each cylinder wall. The separation/attachment lines and the primary critical points (including focus, nodal points, and saddle point) are also shown in Fig. 18. To the best of the authors’ knowledge, the limiting streamlines on the surfacemounted square cylinder fully immersed in the thick turbulent boundary layer at Re ¼ o(104) are reported here for the first time. Overall, significant three-dimensionality in the vertical direction is observed on all cylinder walls. The stagnation point on the front wall (i.e., the nodal point “N1”) is at z/H ¼ 0.75. Below and above this height, the vertical velocity component reverses its direction. A bifurcation line (i.e., the attachment line “a.l-1”) passes through “N1” and extends over the majority of the cylinder height. It forms at the center of the front wall and 11

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Fig. 18. Mean limiting streamlines just adjacent to the walls of the cylinder at α ¼ 0
at a distance 0.005D from the walls. (a) and (b) show views from different perspectives. Critical points: S: saddle points; N: nodal points; F: foci. The bifurcation lines (dashed) are: s.l: separation lines; a.l: attachment lines.

critical points are of particular importance. The two saddle points (“S1” and “S2”) divide the separation/attachment lines, which will be emphasized when discussing their relationship to pressure classification. The focus (“F”) near the bottom of the cylinder is an imprint of the tornado-like vortex created next to the side wall (or “inverted conical vortices” according to Okuda and Taniike. (1993)). An attachment line (“a.l-4”) on the back wall forms in the middle of the cylinder surface, and the flow diverges from this line. Above this line, the flow is directed

splits the streamlines into two separate parts along the positive and negative y-directions. The streamlines converge to the separation line “s.l-1” near the bottom of the front wall. The separation line “s.l-1” implies flow separation occurs and the secondary horseshoe vortex arises. The flow patterns on the side walls look more complicated. Only the primary flow-topology features are discussed here. Separation lines (“s.l2” and “s.l-3”) and the attachment lines (“a.l-2” and “a.l-3”) are observed near the leading and trailing edges, respectively. In addition, three

Fig. 19. (a)–(c) Pressure distribution categories along the vertical direction. (d) Mean wall streamlines on the side face. The red circles are the saddle points “S1” and “S2”. The dashed lines are the separation/ attachment lines. (e) Mean streamline patterns on the cross sections in different regions. The flow reattachment point corresponds to the attachment line “a.l2(L)” when z/H ¼ 0.01. The reattachment point arises from the trailing-edge vortex and corresponds to “a.l2(U)” when z/H ¼ 0.13. The reattachment point corresponds to “a.l-2(U)” when z/H ¼ 0.60. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

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Fig. 20. Comparison between the 2D-like region for cylinders with finite and infinite height at α ¼ 0
: (a) mean pressure distributions, where the line of the infinite cylinder is shifted upward by 0.6; (b) mean flow patterns. Note that the mean flow on the infinite cylinder is numerically obtained using a body-fitted unstructuredgrid large-eddy simulation at Re ¼ 2.2  104 (see the case of “CWR” in Cao and Tamura, 2016).

0.85 ≲ z/H ≲ 1.0. The pressure distributions of the three categories are shown in Fig. 19(a)-(c). Different characteristics or tendencies are observed in each region and will be discussed in detail separately. The spatial relationship between the mean pressure categories and the streamline patterns are shown in Fig. 19(d). Interestingly, the different pressure regions are divided based on the saddle points (“S1” and “S2”) in the mean streamlines. The separation line is cut into “s.l-2” and “s.l-3” by saddle point “S1”. The attachment line is cut into “a.l-2” and “a.l-3” by saddle point “S2”. Several rectangles are included in Fig. 19(d) to indicate the pressure recovery upstream from the trailing edge of the side wall. In the junction-influence region, pressure recovery is found upstream from the trailing edge of the side wall. Pressure recovery occurs via two different mechanisms. The first sub-region is below z/H ≲ 0.1, which marked by the lower red rectangle in Fig. 19(d). In this sub-region, the pressures progressively increase throughout the spatial extent between the center and trailing edge of the side wall, which is due to reattachment of the flow upstream from the trailing edge. An example of flow reattachment when z/H ¼ 0.017 is shown in Fig. 19(e). The reattachment point belongs to the attachment line “a.l-2(L)” shown in Fig. 19(d). However, a local maximum is present in the second sub-region (0.1 ≲ z/ H ≲ 0.4) (indicated by an ellipse in Fig. 19(a)), which is associated with the small-scale trailing-edge vortices. The streamlines in the cross section at z/H ¼ 0.133 in Fig. 19(e) illustrate the trailing-edge vortex, which imprints the attachment line “a.l-2(U)”. The attachment line “a.l-2(L)” in the first sub-region appears curved, in correspondence with the flow reattachment. However, “a.l-2(U)” in the second sub-region is straight, in correspondence with the formation of trailing-edge vortex. In the 2D-like region, the pressure distributions remain nearly unchanged, even at such small aspect ratio. Moreover, the flow pattern beside the side faces is very similar to that of the two-dimensional infinite square cylinder (see Fig. 19(e), where z/H ¼ 0.6). A detailed comparison

upward, which is associated with the roll-up of flow separation from the free end (called “near-wake vortex”). The above description is believed to be sufficient to consider the relationship between the vertical variation in the mean pressures and the wall streamlines. 4.2. Pressure and streamline patterns on the side faces The side face is first investigated regarding the limiting streamlines and wall pressures for AR ¼ 3. After careful examination of the threedimensional effect, the pressure distributions from the ground to the top of the cylinder can be divided into three categories: “junction influence region”, “2D-like region”, and “free-end influence region”. The vertical range for each category is 0 ≲ z/H ≲ 0.4, 0.4 ≲ z/H ≲ 0.85, and

Fig. 21. Relationship between the mean base pressures and the formation length of the near wake when α ¼ 0
.

Fig. 22. Mean flow streamlines in the symmetry plane and the limiting streamlines on the back wall, and their relationship with the base-pressure regions. 13

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between the finite-height cylinder and the infinite cylinder will be performed later. In particular, a large recirculation vortex forms between the mean shear layer and the side wall, while two small vortices form downstream from the frontal edge and upstream from the trailing edge. Pressure recovery in the free-end influence region is again observed upstream from the trailing edges. The average lateral size of the recirculation zone near the side walls is compressed. The sizes of trailing-edge vortices are implied in Fig. 19(d) by the location of the attachment lines (“a.l-3”) upstream from the trailing edge. These are marked with green and red rectangles in the 2D-like region and the free-end influence region, respectively. The trailing-edge vortices become smaller and closer

to the trailing edge in the free-end influence region than in the 2D-like region. The 2D-like region is recognized, even in the short cylinder when AR ¼ 3 or AR ¼ 4. The similarity between the 2D-like region and the infinite square cylinder deserves further discussion. First, the mean pressure distributions in the 2D-like region and the infinite square cylinder are gathered in Fig. 20(a). Data for the infinite square cylinder comes from a previous experiment (Nishimura and Taniike, 2000). Note that the mean pressure coefficients for the infinite square cylinder are shifted upward by 0.6. It is clear that the shapes of the 2D-like region and the infinite cylinder are very similar. However, Cp in the 2D-like region is

Fig. 23. (a) CL over time when α ¼ 15
and AR ¼ 4; (b) vortex shedding at the instant when CL has a trough in the cross-section in the middle of the cylinder (z/ D ¼ 2.0); (c) vortex shedding at the instant between the CL trough and peak; (d) vortex shedding at the instant where a CL peak arises. Fig. 24. Surface flow patterns on the flow-reattachment wall: (a) at the instant when CL has a trough; (b) at the instant between the CL trough and peak; (c) at the instant when a CL peak arises; (d) time-averaged pattern. In (a)–(c), the streamlines are colored as positive (red) or negative (blue) streamwise velocities. The instants are the same as those in Fig. 23(b)-(d) and the wall is the upper side wall in Fig. 23(b)-(d). (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

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Fig. 24, together with the time-averaged pattern. The color in Fig. 24 (a)(c) indicates the positive (red) or negative (blue) streamwise velocity. Interestingly, the three instantaneous patterns are very similar when α ¼ 15
, even though they have a maximum phase difference of π. As a result, the flow features in the instantaneous fields are well-preserved in the time-averaged surface flow. The attachment and separation lines are indicated with dashed lines, which could be used to identify the size of the flow reattachment zone. Fig. 24(a) and (c) show that the size of separation bubble is smaller at the CL-peak instant than at the CL-trough instant, especially in the conjunction and free-end regions. The mean flow reattachment line ultimately lies between the two instantaneous reattachment lines. The mean limiting streamlines on the cylinder walls at α ¼ 15
are shown in Fig. 25 for the first time. Flow direction and bifurcation lines are drawn schematically. The critical points are indicated and numbered. First, the flow topology on the front wall is similar to that at α ¼ 0
. The difference is that the attachment line (“a.l-1”) moves toward the No. 0(4) cylinder edge. The flow topology looks very complicated on side face 12 and back face 23. A focus “F1” forms near the bottom of face 12 and may be associated with a tornado-like vortex. A clear saddle point “S1” is located near the middle of face 12, which clearly distinguishes upward and downward flow. The attachment line “a.l-2” on face 23 is found to be close to the corner of face 12. However, this attachment line is apparent only in the middle region of the cylinder. Strongly three-dimensional flow reattachment occurs on face 34. Two foci (i.e., “F2” and “F3”) are found near the free end and the bottom inside the separation bubble. The

approximately 0.8 higher than for the infinite cylinder. Furthermore, Fig. 20(b) compares the mean flow topologies in the cross sections between the 2D-like region of the finite-height cylinder and the infinite cylinder (Cao and Tamura, 2016). The two cases produce similar flows. Nevertheless, two notable quantitative differences can be found. The first is that vertical flow (perpendicular to the cross section in Fig. 20(b)) in the finite-height cylinder is common along one direction, i.e., towards the free end. Vertical flow can be seen in the surface flow pattern in Fig. 19(b). However, there is no spanwise flow in the mean flow field for the infinite cylinder. The other difference is that the recirculation zone of the finite-height cylinder is much longer and wider than for the infinite cylinder. This is due to the increased influx of momentum induced by downwash and upwash at both ends of the finite-height cylinder. As expected, the larger recirculation zone for the finite-height cylinder results in smaller pressure suction on the side and rear walls. The observation of a longer and wider recirculation zone behind the finite-height cylinder is consistent with the experimental measurements by Sumner et al. (2017) and Sohankar et al. (2018), although the AR and the boundary layer thickness differ from each other. Furthermore, the size of the recirculation zone can be quantified by the formation length Lf (defined as the distance between the cylinder centroid and the location where the sign of the mean streamwise velocity changes from negative to positive) and the local wake width W (i.e., the lateral distance between two turbulent kinetic energy peaks at x/D ¼ 1.0). Lf/D ¼ 1.9 and W/D ¼ 1.62 in the present condition (AR ¼ 3, δ/D ¼ 20.1). Both values are smaller than those presented by Sumner et al. (2017) (Lf/D ¼ 2.5 when AR ¼ 3 and δ/D ¼ 1.7) and Sohankar et al. (2018) (W/D ¼ 2.32 when AR ¼ 7 and δ/D ¼ 0.5) when the Reynolds numbers are of the same order of magnitude. This indicates that a thick boundary layer with shear effects and turbulence effects across the entire cylinder height tends to shrink the recirculation zone. 4.3. Pressures on back face associated with wake The relationship between the mean base pressure coefficients (Cpb ) and the mean formation lengths (Lf =D) in the near-wake region is shown in Fig. 21. Cpb is multiplied by 4 to match the scale of the formation length. The trends in Cpb and Lf =D generally coincide as the vertical height increases. However, unlike the clear categories on the side-wall pressures, the base pressures seem to be divided into only two categories, i.e., the lower region and the near-wake vortex influence region. In the lower region, the base pressures remain nearly constant. The “handle” shape in the distribution of formation lengths when z/H ≲ 0.05 does not significantly influence the base pressures. The border of base pressure categories is applied to the wall streamlines on the back wall and the mean flow streamlines in the symmetry plane, as shown in Fig. 22. The saddle point (“S”) in the nearwake is indicated in the symmetry plane. Some spatial relationships can be found: (1) the border is higher than the saddle point in the near-wake, and (2) the border is near the upper end of the attachment line (“a.l”) on the back-wall streamlines. 5. Wall pressures and surface flow patterns at α ¼ 15
Just as in the case where α ¼ 0
, we attempt to identify the wall pressure regions along the vertical direction at α ¼ 15
and relate these to the limiting streamlines on the cylinder. 5.1. Limiting streamlines at α ¼ 15
The instantaneous flows in the cross section at the middle of the cylinder are shown in Fig. 23(b)-(d). Three typical instants with a CL trough and peak are selected and marked in Fig. 23(a). The wake in Fig. 23(b)-(d) appears to exhibit anti-symmetric Karman vortex shedding. The instantaneous limiting streamlines at the instants are shown in

Fig. 25. Mean limiting streamlines just adjacent to the cylinder walls at α ¼ 15
with distance 0.005D from the walls. Critical points: S: saddle points; N: nodal points; F: foci. The bifurcation lines (dashed) are: s.l: separation lines; a.l: attachment lines. 15

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curvature along the vertical direction, especially near the junction and the free end. The regions with different curvatures roughly correspond to various pressure categories, i.e., the junction influence (low) region at z/ H  [0, 0.20], 2D-like (middle) region at z/H  [0.20, 0.75], and free-end influence (upper) region at z/H  [0.75, 1]. It is worth noting that the saddle point (“S2” at z/H  0.7) is located near the border between the middle and upper regions. The wall pressures between the 2D-like region of the finite-height cylinder and the infinite cylinder at α ¼ 15
are compared in Fig. 27(a). The pressure distribution of the infinite cylinder (the experiment by Nishimura and Taniike, 2000) is shifted upward by 0.6. Fig. 27(a) shows that pressure suction on the reattachment wall of the finite-height cylinder is much weaker than the infinite cylinder. As shown in Fig. 26(a)-(c), the reattachment points are at the end of the rapid pressure

two foci should be the imprints of the standing and inverted conical vortex observed by Okuda and Taniike (1993). 5.2. Pressures and streamline patterns on the flow-reattachment face Flow reattachment occurs in the trailing portion of side face 34 (see the example in Fig. 26(e)). This is accompanied by pressure recovery. Three categories could be identified, as shown in Fig. 26(a)-(c). The pressure distributions in the middle region are close to each other. However, pressure recoveries often have local maxima that are located downstream of the reattachment line in the low and upper regions (note that the flow reattachment points are marked in red in Fig. 26(a)-(c)). The surface streamlines and the positions of the reattachment points are shown in Fig. 26(d). One can see that the reattachment line has different

Fig. 26. Pressure categories on the reattachment wall and relationship with the wall streamlines: (a)–(c) pressure categories from the lower to upper regions; (d) location of the reattachment line; (e) mean flow streamlines on the side wall; (f) quasi-streamlines at z/H ¼ 0.538.

Fig. 27. Comparison between the 2D-like region of the finite and infinite cylinders at α ¼ 15
: (a) mean pressure distributions, upward shift of 0.6 in the line of the infinite cylinder; (b) mean flow patterns. Note that the mean flow for the infinite cylinder was determined from PIV measurements by van Oudheusden et al. (2008). 16

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recovery for the cylinder with finite height. According to this estimation method, the reattachment point in the experiment from Nishimura and Taniike (2000) should be closer to the trailing edge (i.e., the corner of No. 3) than the 2D-like region of the cylinder with finite height. The smaller separation bubble for the finite-height cylinder may be associated with regions near the ground and the free end, where reattachment occurs earlier. Furthermore, the mean flow pattern in the cross section is compared with the PIV measurement by van Oudheusden et al. (2008) in Fig. 27(b). The flow topology is generally the same in both cases. However, the recirculation zone behind the finite-height cylinder is more elongated and wider than the infinite cylinder. Finally, the influence of the attack angle is discussed by comparing the wake widths at α ¼ 0
and α ¼ 15
obtained in this study. The wake width W/D ¼ 1.57 at α ¼ 15
is smaller than W/D ¼ 1.62 at α ¼ 0
. This tendency was widely observed in previous studies on cylinders with finite height (e.g., Unnikrishnan et al., 2017; Sohankar et al., 2018). Moreover, the present wake width at α ¼ 15
is smaller than that reported by Sohankar et al. (2018) (W/D ¼ 2.01 when AR ¼ 7 and δ/D ¼ 0.5).

dominant near-wake vortex. 6. Conclusions BCM-IBM was used to simulate high-Re turbulent flow around a surface-mounted square cylinder with AR ¼ 3 or 4 in thick turbulent boundary layers. Two typical attack angles are considered, namely α ¼ 0
and α ¼ 15
. Numerical methods are first examined in qualitative and quantitative approaches. The mean limiting streamline patterns are subsequently described, and their relationship with the wall pressures are discussed. We offer the following concrete conclusions: (1) The numerical and experimental mean pressures are quantitatively consistent when the inflow conditions are consistent, regardless of the attack angle. The r.m.s pressures at α ¼ 0
are accurately predicted, while the r.m.s pressures at α ¼ 15
on the flow-attachment wall tend to be underestimated. Furthermore, the mean flow fields around the cylinder wall and the far wake in different flow conditions are qualitatively validated against available PIV measurements. (2) The primary features of the wall streamlines are described as an imprint of the three-dimensional flow separation and vortical structures on the cylinder walls, including critical points and separation/attachment lines. The three-dimensional effects of the mean pressures and wall streamlines along the vertical direction are clarified.

5.3. Pressures on back faces associated with wake The near-wake vortex plays a dominant role in determining the pressure distributions in the upper region for faces 12 and 23, similar to the base pressure at α ¼ 0
. However, it is very difficult to define the base pressure when α ¼ 15
. This study uses the two-point averaged pressure to roughly examine the distribution along the vertical direction in Fig. 28(c). The two-point average is considered to be an approximation of the local spatial average. These two points are located 0.05D and 0.15D from the edge of the cylinder. Three typical areas are selected: one that surrounds the vertical attachment line “a.l-2” on face 23 (marked with the black rectangle in Fig. 28(b)); the other that is near the trailing and leading edges of face 12 (marked with the blue and red rectangles in Fig. 28(a)). A common feature is that the pressures tend to decrease as z increases when approaching the free end (e.g., z/H > 0.6), which is related to the shorter formation length. In the lower regions, the variation along the vertical direction becomes much smaller. For the “a.l-2” region on face 23 and the region around the leading edge of face 12 (i.e., the black and red lines in Fig. 28(c)), a small ridge arises at the middle of the cylinder, which roughly corresponds to the height of the saddle point “S1” and the attachment node “N2”. A more detailed analysis of the relationship between the wall streamlines and the pressures seems to be much more complicated than the case when α ¼ 0
. This may arise because the flow structures behind the two faces are not directly connected (i.e., there are many isolated flow structures), except for the

 At α ¼ 0
, the mean pressure distributions on the side walls are classified into three categories: the junction influence region, 2D-like region, and free-end influence region. The borders are found to be consistent with the saddle points in the side-wall streamlines. Strong two-dimensional effects arise locally in the 2D-like region. The mean base pressures on the back wall and the formation lengths of the recirculation zone behind the cylinder are closely related to each other and are divided into two regions: the lower region and the nearwake vortex influence region. The border is above the wake saddle point in the symmetry plane and is located near the upper end of the attachment line in the back-wall streamlines.  At α ¼ 15
, the mean pressures on the flow-reattachment wall are divided into three regions along the vertical direction: the junction influence region, 2D-like region, and free-end influence region. The pressure distributions in the 2D-like region are close to each other, while higher pressure recovery in the other regions. The pressure categories are strongly associated with the curvature of the reattachment line, which has a larger curvature near the junction and the free end. The wall streamlines and mean pressures on the two leeward walls are complicated. Nevertheless, it is found that the mean

Fig. 28. (a)–(b) Limiting streamlines on faces 12 and 23; (c) two-point averaged pressures along the vertical direction, where the rectangles in (a)–(b) indicate the locations of two points. 17

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Journal of Wind Engineering & Industrial Aerodynamics 188 (2019) 1–18

pressure decreases when approaching the free end. Moreover, slight pressure recovery is observed near the salient attachment node and the saddle point on the two leeward walls.  The similarities between the 2D-like region of the finite-height square cylinder and the infinite square cylinder at two typical angles are observed, including the shape or tendency of the pressure distribution on the side wall and the mean flow topologies in cross sections. However, the quantitative pressure in the 2D-like region for the finite cylinder is much higher than for the infinite cylinder. The size of the recirculation region in the 2D-like region is much larger than the size of the infinite cylinder due to the increased momentum influx caused by the downwash and upwash in the case of a finite cylinder.

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Acknowledgments The study was funded by the Ministry of Education, Culture, Sports, Science, and Technology (MEXT), Japan project: “Advancement of meteorological and global environmental predictions utilizing observational ‘Big Data’” of the social and scientific priority issues (Theme 4) to be tackled by using the post-K computer of the FLAGSHIP2020 Project (hp160229, hp170246, hp180194). The authors are very grateful to Dr. Keiji Onishi, Dr. Rahul Bale and Prof. Makoto Tsubokura for the technical support and fruitful discussions about the numerical methods. References Bourgeois, J.A., Sattari, P., Martinuzzi, R.J., 2011. Alternating half-loop shedding in the turbulent wake of a finite surface-mounted square cylinder with a thin boundary layer. Phys. Fluids 23 (9), 095101. Cao, Y., Tamura, T., 2016. Large-eddy simulations of flow past a square cylinder using structured and unstructured grids. Comput. Fluids 137, 36–54. Castro, I.P., Dianat, M., 1983. Surface flow patterns on rectangular bodies in thick boundary layers. J. Wind Eng. Ind. Aerod. 11 (1–3), 107–119. Delery, J., 2013. Three-dimensional Separated Flow Topology: Critical Points, Separation Lines and Vortical Structures. John Wiley & Sons. Depardon, S., Lasserre, J.J., Boueilh, J.C., Brizzi, L.E., Boree, J., 2005. Skin friction pattern analysis using near-wall PIV. Exp. Fluid 39 (5), 805–818. Dianat, M., Castro, I.P., 1984. Fluctuating surface shear stresses on bluff bodies. J. Wind Eng. Ind. Aerod. 17 (1), 133–146. Kawai, H., Okuda, Y., Ohashi, M., 2012. Near wake structure behind a 3D square prism with the aspect ratio of 2.7 in a shallow boundary layer flow. J. Wind Eng. Ind. Aerod. 104, 196–202. Komatsu, K., Soga, T., Egawa, R., Takizawa, H., Kobayashi, H., Takahashi, S., Sasaki, D., Nakahashi, K., 2011. Parallel processing of the building-cube method on a GPU platform. Comput. Fluids 45 (1), 122–128. Lund, T.S., Wu, X., Squires, K.D., 1998. Generation of turbulent inflow data for spatiallydeveloping boundary layer simulations. J. Comput. Phys. 140 (2), 233–258. Martinuzzi, R., Tropea, C., 1993. The flow around surface-mounted, prismatic obstacles placed in a fully developed channel flow (data bank contribution). J. Fluid Eng. 115 (1), 85–92. Maruyama, Y., Tamura, T., Okuda, Y., Ohashi, M., 2013. LES of fluctuating wind pressure on a 3D square cylinder for PIV-based inflow turbulence. J. Wind Eng. Ind. Aerod. 122, 130–137.

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