2.1. Numerical method

This study uses the code ‘CUBE’ developed by the Riken Centre for Computational Science (Jansson et al. Reference Jansson, Bale, Onishi and Tsubokura2019; Onishi & Tsubokura Reference Onishi and Tsubokura2021). It combines the building cube method (BCM) with the topology-independent immersed boundary method (IBM). The topology-independent IBM was proposed and implemented by Onishi et al. (Reference Onishi, Obayashi, Nakahashi and Tsubokura2013, Reference Onishi, Ando, Nakasato and Tsubokura2018) under the framework of BCM (Nakahashi Reference Nakahashi2002, Reference Nakahashi2005). This method is simple in all flow simulation stages (i.e. mesh generation, solution algorithm and post-processing). Owing to space limitations, the numerical details are not repeated in this study (interested readers are referred to the aforementioned references). The incompressible continuity and Navier–Stokes equations are solved based on the Cartesian grid without any explicit subgrid-scale (SGS) modelling for turbulence. The convective and diffusion terms are spatially discretised using the second-order central difference scheme, and 5 % of the first-order upwind scheme is blended to estimate the convective flux on the cell face. The fractional-step method is applied for time marching. The semi-implicit Crank–Nicolson method is used to treat the convective and diffusion terms. The semi-implicit velocity and Poisson equations are solved using the red/black successive over-relaxation method.

The numerical process and set-up are the same as those of Cao et al. (Reference Cao, Tamura and Kawai2019). A model of the surface-mounted square cylinder immersed in a thick boundary layer is schematically shown in figure 1(a) when $Re = 5\times 10^4$. The aspect ratio of the cylinder is $AR=3$. The thickness of the turbulent boundary layer is $\delta /D=20.1$. The origin of the axis is placed at the centre of the bottom of the cylinder. The computational domain size is $32D\times 8D\times 32D$ in the streamwise (x), lateral (y) and vertical (z) directions. The domain size is examined in Appendix A. The grid system in the symmetry plane is shown in figure 1(b). The x–y plane at the middle height of the cylinder is shown in figure 1(c). The minimum cell size is determined from the thickness of the boundary layer on the cylinder ($\delta _B$). According to White (Reference White2006), $\delta _B \approx 5.5\times (0.5D)/\sqrt {Re_{0.5D}}= 0.0174D$. In this study, the cell size in the wall-normal direction is $\varDelta = 0.00195D$, which means that the boundary layer spans approximately nine cells. In other directions near the cylinder walls, the cell size is the same as $\varDelta$ because the elemental cells are in the shape of a cube, which means that approximately $512 \times 512$ cells are distributed in an area of $1D\times 1D$ on the surface of the cylinder. This resolution is much finer than that of most numerical simulations or experimental PIV measurements previously performed, and has the advantage of capturing finer flow structures. In addition, $\varDelta$ is the smallest cell size throughout the computational domain and is considered the reference grid resolution used to describe other regions, as indicated in figures 1(b) and 1(c). The region from the ground surface to $z/D=1$ has a cell size of $8\varDelta$ before it approaches the cylinder. A total of 460 million cells were used in this study. The sufficiency of the current cell sizes is verified in Appendix A. The results showed that the cell sizes in the regions of the shear layer and the near and moderate wakes were nearly of the same order of magnitude as the Kolmogorov length scale estimated by Tennekes & Lumley (Reference Tennekes and Lumley1972). Although an examination of strict mesh independence is preferable, the use of the present mesh is necessary to ensure an affordable computational cost. Jiang & Cheng (Reference Jiang and Cheng2020) observed the quantitative variation in the separation and reattachment points around an infinite-length square cylinder at $Re = 10\text {--}400$ when refining mesh resolutions by a factor $F_{mr}$. The shift in critical points was less than $0.003D$ and less than $0.04D$ when $F_{mr}$ increased from 4 to 6 and from 1 to 6, respectively. Considering the purpose of the topological description, two topology patterns are said to be topologically equivalent if they can be distorted into another by a stretching process but without tearing (Tobak & Peake Reference Tobak and Peake1982; Perry & Chong Reference Perry and Chong1987; Délery Reference Délery2001). Accordingly, in this study, the slight quantitative shift in the critical-point locations as observed by Jiang & Cheng (Reference Jiang and Cheng2020) will not significantly influence the topological description of the flow.

Figure 1. (a) Model of a surface-mounted square cylinder in a turbulent boundary layer when $Re=5\times 10^4$. (b) Grid system in the symmetry plane. (c) Grid system in the x–y plane in the middle of the cylinder. The domain sizes are normalised by D. The cell sizes are also tabbed in different regions relative to the minimum cell size $\varDelta$ immediately near the immersed boundary. Panels (b) and (c) are reproduced from Cao et al. (Reference Cao, Tamura and Kawai2019).

The time step ${\rm \Delta} t^{*}=( {\rm \Delta} t U_{\infty } ) /D$, (where $U_{\infty }$ is the free-stream velocity) is $2\times 10^{-4}$, which results in a maximum Courant number of approximately 0.3. Statistical analysis begins after the flow becomes statistically stationary. The duration of the statistical average is approximately $200t^{*}$, which corresponds to approximately 20 periods of Kármán vortex shedding. The temporal convergence of statistical pressures, forces and velocity is examined in Appendix B and by Cao et al. (Reference Cao, Tamura and Kawai2019). The computations were performed on a K-computer, one of the highest calibre supercomputers known. One computational node has a SPARC64^TM VIIIfx 2-GHz CPU, 128-GF performance and 16-GB memory. Each node contains eight cores. The network is based on the Tofu Interconnect (6D Mesh/Torus). In this case, 2881 computational nodes were used. The wall-clock time was approximately 7 days for 1 300 000 time steps.

The effects of small-amount numerical dissipation are examined when no explicit model is used for turbulence modelling. The energy spectra of streamwise velocity are compared in figure 2 between the present study and the DNS study by Trias, Gorobets & Oliva (Reference Trias, Gorobets and Oliva2015). Note that Trias et al. (Reference Trias, Gorobets and Oliva2015) simulated the uniform flow past an infinite-length square cylinder at a $Re$ similar to that in the present case. The productivity of small-scale turbulent motion is verified by comparing the high-frequency region. In figure 2, the black vertical dashed-dotted line represents the cut-off frequency of local cells in the current study and the blue dashed line indicates the slope of $-5/3$, which indicates the inertial subrange. Clearly, the present spectrum decays earlier than in the DNS study. Furthermore, the earlier decay commences from the inertial subrange, particularly around the cut-off frequency. The present spectrum appears similar to the filtered one from the DNS results, where the filter width is approximately the cut-off wavenumber. In other words, a small numerical dissipation exists to a certain degree in time and space functions such as the SGS model, although the numerical dissipation differs fundamentally from that of the SGS model. However, the high-frequency range is reproduced sufficiently up to several hundred vortex-shedding frequencies ($f_{vs}$). The absence of the peak of vortex-shedding frequency ($St =f_{vs}D/U_H=0.094$) in the streamwise velocity spectra in the centreline of the wake is similar to the phenomenon in which the peak is much less pronounced in the streamwise velocity than that in the lateral velocity in the wake centreline of bluff bodies (Ong & Wallace Reference Ong and Wallace1996; Parnaudeau et al. Reference Parnaudeau, Carlier, Heitz and Lamballais2008; Cao & Tamura Reference Cao and Tamura2015). If the probe point is far from the cylinder in the lateral direction, the vortex shedding frequency can be observed clearly (Mcclean & Sumner Reference Mcclean and Sumner2014; Rastan et al. Reference Rastan, Shahbazi, Sohankar, Alam and Zhou2021).

Figure 2. One-dimensional energy spectra of the streamwise velocity component in the wake. The measurement station in the present case is located at $(x/D, y/D, x/D) = (1.5, 0, 1.5)$, and those of Trias et al. (Reference Trias, Gorobets and Oliva2015) are located along the centreline of the wake when $x/D = 1.0$ and 2.5. The vertical dash–dotted line represents the grid cut-off frequency and the blue dashed line represents a slope of $-5/3$.

2.2. Numerical validation

The inflow conditions were first examined by comparing the numerical results with the experimental results by Katsumura (as cited by Maruyama et al. Reference Maruyama, Tamura, Okuda and Ohashi2013). In this study, the Lund method (Lund, Wu & Squires Reference Lund, Wu and Squires1998) was used to generate a fully developed turbulent boundary layer (TBL) on a smooth ground plane. The Lund method was implemented in another in-house code based on the finite difference method. The non-dimensional Navier–Stokes equations were solved, where the friction velocity and boundary-layer thickness were the characteristic velocity and length, respectively. The sampled TBL data were scaled to fit the desired TBL thickness, which imitates the atmospheric boundary layer approaching buildings. They were then input into the main domain with a finite-height square cylinder through the inlet boundary condition. Additional details can be found in Nozawa & Tamura (Reference Nozawa and Tamura2002) and Nozu & Tamura (Reference Nozu and Tamura2012). Figure 3(a) shows the mean velocity and turbulence intensity profiles at a location close to the inlet (i.e. $x/D=-7.4$) and figure 3(b) shows the power spectrum density compared with the Kármán spectrum. The turbulence intensity is defined as $I_u(z)=\sigma _u(z)/\bar {U}(z)$ , where $\sigma _u(z)$ is the root mean square (r.m.s.) of the streamwise velocity and $\bar {U}(z)$ is the mean velocity at the height of $z$. The experimental and numerical results were generally consistent. Moreover, this consistency was maintained before approaching the cylinder.

Figure 3. (a) Time-averaged velocity and turbulence intensity profiles near the inlet as compared with the experiment by Katsumura (Maruyama et al. Reference Maruyama, Tamura, Okuda and Ohashi2013). (b) Power spectrum density at $(x/D, y/D, z/D) =(-7.4, 0, 4)$, where the cell size at this position is approximately $0.03D$. The figures are reproduced from Cao et al. (Reference Cao, Tamura and Kawai2019).

To validate the method presented in this work, figure 4 compares the distributions of time-averaged and r.m.s. pressure coefficients (denoted $\bar {C}_p$ and $\sigma _p$, respectively) with the experimental results by Katsumura (as cited by Maruyama et al. Reference Maruyama, Tamura, Okuda and Ohashi2013) under similar inflow conditions. The subplots show distributions at different cylinder heights. The measurement heights are indicated in figure 4(a), where the black lines on the surfaces of the cylinder denote the locations of measurement in the present study and the red lines denote the experimental locations. The letters ‘(b)’ and ‘(c)’ with numbers, which appear next to the cylinder in figure 4(a), correspond to the subplots in figure 4(b) and 4(c), respectively. The current results agreed well with Katsumura's measurements. A comparison of total forces, Strouhal number and mean flow fields with the available experiments was conducted by Cao et al. (Reference Cao, Tamura and Kawai2019). In general, the flow field showed good qualitative agreement between the numerical and experimental PIV results (Sumner et al. Reference Sumner, Rostamy, Bergstrom and Bugg2017; Unnikrishnan et al. Reference Unnikrishnan, Ogunremi and Sumner2017). Concerning the symmetry along the plane of $y=0$ in figure 4, the general symmetry of mean values was obtained even though the symmetry of the r.m.s. values was broken at some points. Obtaining the so-called ‘strictly’ symmetric distributions of physical quantities is particularly difficult for a surface-mounted cylinder immersed in a thick turbulent boundary layer at a high $Re$. The asymmetry can be readily found in the previous experimental studies at a high $Re$, particularly for the r.m.s. values, as with the infinite-length cylinder of Lee (Reference Lee1975) and Noda & Nakayama (Reference Noda and Nakayama2003) and the finite-length cylinder of Wang et al. (Reference Wang, Zhao, He and Zhou2017). For the wake velocity, the r.m.s. values of streamwise and lateral components along the lateral direction were examined (not shown here for brevity), which showed very good symmetry in the wide region of the near wake.

Figure 4. (a) Measurement heights in the present study (black lines) and the experiment (red lines). The letters ‘(b)’ and ‘(c)’ with numbers, which appear next to the cylinder, correspond to the subplots in panels (b) and (c), respectively. (b)–(c) Comparison of time-averaged and r.m.s. pressure distributions with the experiment, respectively, which are reproduced from Cao et al. (Reference Cao, Tamura and Kawai2019).

3.1. Classification of critical points

The term ‘skin-friction line’ is used in this study to represent the surface flow pattern, which is defined as the trajectory of the surface shear stress (or skin friction, which is the non-dimensional form of the shear stress). The classification of critical points in skin-friction lines is briefly reviewed following Délery (Reference Délery2001, Reference Délery2013). Here, ($x_1$, $x_2$) denotes the orthogonal coordinates in a two-dimensional (2-D) space. In this study, the 2-D space is the surface of the cylinder or ground surface. For simplicity, the origin of ($x_1$, $x_2$) is located at the critical point under consideration and ($\tau _{w1}$, $\tau _{w2}$) denotes the vector of the surface shear stress in the space of ($x_1$, $x_2$). In the neighbourhood of the critical points, the first-order Taylor series expansion determines the behaviour of the vector field. The matrix of the velocity derivatives (i.e. Jacobian matrix) is shown in (3.1):

(3.1)\begin{equation} {\boldsymbol{\mathsf{F}}} = \begin{bmatrix} \dfrac{\partial \tau_{w1}}{\partial x_1} & \dfrac{\partial \tau_{w1}}{\partial x_2} \\ \dfrac{\partial \tau_{w2}}{\partial x_1} & \dfrac{\partial \tau_{w2}}{\partial x_2} \end{bmatrix}.\end{equation}

Let $\lambda _1$ and $\lambda _2$ be the eigenvalues of ${\boldsymbol{\mathsf{F}}}$, ${\boldsymbol{\mathsf{I}}}$ the identity matrix and ${\textbf { {e}}}$ the eigenvector,

(3.2)\begin{equation} ( {\boldsymbol{\mathsf{F}}} - \lambda {\boldsymbol{\mathsf{I}}} ) {\textbf{{e}}}=0.\end{equation}

The eigenvalues can be determined by solving the characteristic equation $det({\boldsymbol{\mathsf{F}}}-\lambda {\boldsymbol{\mathsf{I}}})=0$, which is expressed as

(3.3)\begin{equation} \lambda^2+p\lambda+q=0,\end{equation}

where the discriminant is $p^2-4q$ and

(3.4)\begin{gather} p={-} \left( \frac{\partial \tau_{w1}}{\partial x_1} +\frac{\partial \tau_{w2}}{\partial x_2} \right) ={-}tr({\boldsymbol{\mathsf{F}}}) ={-}(\lambda_1+\lambda_2), \end{gather}

(3.5)\begin{gather}q= \left( \frac{\partial \tau_{w1}}{\partial x_1}\frac{\partial \tau_{w2}}{\partial x_2} -\frac{\partial \tau_{w1}}{\partial x_2}\frac{\partial \tau_{w2}}{\partial x_1} \right) =det({\boldsymbol{\mathsf{F}}})=\lambda_1\lambda_2. \end{gather}

The characteristics of the skin-friction lines around the critical point are determined by the eigenvalues $\lambda _1$ and $\lambda _2$. The properties of $\lambda _1$ and $\lambda _2$ are represented in the chart of p–q. Figure 5 shows the classification of the critical points in a 2-D space, which is available in other publications (Perry & Chong Reference Perry and Chong1987; Délery Reference Délery2013). First, the node type is defined in the region below the parabola $q=p^{2}/4$ and above the axis $q=0$. At the node point, all skin-friction lines have a common tangent that corresponds to one of the eigenvectors, excluding one line corresponding to the other eigenvector. A node is referred to as an attachment node when $\lambda _1$ and $\lambda _2$ are negative or as a separation node when $\lambda _1$ and $\lambda _2$ are positive. The special case is an isotropic node when $q=p^{2}/4$ (i.e. $\lambda _1=\lambda _2$). Second, the saddle point type is defined in the region below the axis $q=0$. All skin-friction lines avoid the critical point appearing in a hyperbolic shape, excluding the two lines passing through the saddle point. The skin-friction lines passing through the saddle are known as separation lines or separatrices. The occurrence of the saddle point is a major sign of flow detachment from the wall. According to Délery (Reference Délery2013), the flow is detached if the surface flow pattern contains at least one saddle point. Third, the focus type is defined in the region above the parabola $q=p^{2}/4$. The focus type is stable or unstable based on whether the skin-friction lines spiral into or out of the critical point, respectively. A special case is the centre, which is located on the axis $p=0$. On the boundaries of the $p$–$q$ chart (i.e. the parabola and axes), the critical points degenerate with structural instability; that is, an infinitesimal change in some parameters changes the type or direction of the critical point. In this study, the nomenclature of the critical points and separation/attachment lines generally follows that of Délery (Reference Délery2013). In other words, $F$, $N$, $S$, $S^{\prime }$, $N^{\prime }$, ($A$) and ($S$) represent foci, nodes, saddle points, half-saddle points, half-node points, attachment lines and separation lines, respectively.

Figure 5. Critical-point classification in the chart of p–q, reproduced following Délery (Reference Délery2001).

3.2. Basic rules of critical points

The surface flow patterns should obey the basic rules (see details in Délery Reference Délery2013). For example, the skin-friction lines must originate at one or several attachment nodes and terminate in either a focus or separation node. This excludes the separation lines of the saddle points. As a confirmation step, the formula determined by counting the number of critical points should be applied to a closed 3-D body, which is known as the ‘Poincaré–Bendixson theorem’ or ‘hairy sphere theorem’ and was introduced by Davey (Reference Davey1961) to address fluid-flow problems:

(3.6)\begin{equation} \sum N - \sum S =2,\end{equation}

where $\sum N$ and $\sum S$ are the numbers of nodal points (nodes and foci) and saddles, respectively.

Note that (3.6) applies to a simply connected surface with a complexity equal to zero. The complexity of a surface is defined as zero if any closed curve traced on the surface can be reduced to a point by a continuous deformation without leaving the surface (Délery Reference Délery2013). The complexity of the body is denoted by $\mathcal {P}$. If a body has a hole through it (such as a torus), the body has a complexity of one (i.e. $\mathcal {P}=1$). If a body has two holes, then $\mathcal {P}=2$. When the complexity $\mathcal {P}$ is considered, (3.6) is revised to (3.7) to be applicable to a body with or without holes passing through it:

(3.7)\begin{equation} \sum N - \sum S =2-2\mathcal{P}.\end{equation}

For a 3-D body B placed on a plane P, several topological rules were recognised by Hunt et al. (Reference Hunt, Abell, Peterka and Woo1978). Surfaces B and P are treated as the upper surfaces of an imaginary 3-D body. There must be two nodes, one upstream and one downstream, at infinity on the imaginary lower surface of the imaginary 3-D body. For practical wind tunnels or numerical investigations of 3-D obstacles (as in the present study), the surface flow patterns on both the body and plane should satisfy the topological rule shown in (3.8). In this study, the surface-mounted square cylinder has no holes; thus, the total number of nodes should be the same as that of saddles on the combined surface of the square cylinder and the ground surface, that is, $( \sum N - \sum S )_ {( P+B )} =0$.

(3.8)\begin{equation} \left( \sum N - \sum S \right)_ {( P+B )} ={-}2\mathcal{P}.\end{equation}

Equations (3.6)–(3.8) require the entire skin-friction field obtained from experiments or numerical simulations. Caution is required when applying (3.6)–(3.8) because most practical applications obtain only a part of the skin-friction field and rarely fulfil the aforementioned requirements. A useful practical rule was proposed by Hunt et al. (Reference Hunt, Abell, Peterka and Woo1978) based on the 2-D plane sections of the flow, which are easily obtained in experiments or simulations. The half-nodes ($N^{\prime }$) and half-saddles ($S^{\prime }$) on the surfaces are introduced. The topological rule that obeys the kinematic principle for a 2-D section of the flow is expressed as

(3.9)\begin{equation} \left( \sum N + \frac{1}{2} \sum {N^{\prime}} \right) - \left( \sum S + \frac{1}{2} \sum {S^{\prime}} \right) = 1-n,\end{equation}

where this 2-D slice of the flow is n-tuply connected. For a single-connected region, $n=1$; for a doubly connected region, $n=2$.

3.3. Approximation of skin-friction lines

In terms of surface flow patterns, the skin-friction lines are physical notions that should be adopted as much as possible. However, in practice, the limiting streamlines are sometimes used to approximate the skin-friction lines. In this study, a one-sided two-point finite difference is used to estimate the velocity derivative and skin friction. The feasibility and sufficiency are examined in Appendix C by comparing schemes with different accuracies. Hereafter, ‘skin-friction lines’ is the term used to describe the surface flow patterns. In addition, this study focuses on the time-averaged field and attempts to construct a mean separated flow organisation that satisfies the topological rules. One major reason for this is the similarity between some instantaneous vortex-shedding phases and the time-averaged field. This was illustrated in our previous study (see § 4.1 of Cao et al. Reference Cao, Tamura and Kawai2019). When the shear layer is close to the side wall of the cylinder, the surface flow pattern on the side wall is similar to the time-averaged pattern. This means that the salient critical points in the time-averaged field can be found in their counterparts in the instantaneous fields. This motivates our study to emphasise the mean field before attempting to analyse complex transient flows. In addition, a large range of motion scales coexist in the transient flows at high $Re$ values. The application of critical-point theory to complete instantaneous flow topology is very difficult (if not impossible).

5.1. Surface flow pattern

The skin-friction lines on the walls of the cylinder are shown in figure 8. The arrow vector represents the direction of the in-plane velocity. The borders between the junction influence and 2-D-like regions and between the 2-D-like and free-end influence regions consist of two saddle points on the side wall, as indicated by $S$ in figure 8(a). Furthermore, figure 8(c) zooms in to the local regions on the side wall (labelled as Zones D, E and F in figure 8a), where the arrows are plotted at every grid cell.

Figure 8. Skin-friction lines on the walls of the cylinder when $Re=5\times 10^4$. Panels (a) and (b) show two perspectives. Critical points are $S$, saddle points; $N$, nodal points; and $F$, foci. The dashed lines denote the ($S$) separation and ($A$) attachment lines. (c) Close-up view of the local regions on the side wall, including Zones D, E and F, where blue indicates positive $\bar U$ and red indicates reverse $\bar U$.

The aforementioned patterns of skin-friction lines on the walls of the cylinder are interpreted in figure 9 using the critical-point concept. At the centre of the frontal wall, as shown in figure 9(a), a pronounced attachment line ($A_1$) emanates from an attachment point ($z/D \approx 2.25$, out of the range of this figure). The attachment line ($A_1$) splits most of the skin-friction lines of the frontal wall into two separate portions along the positive and negative $y$ directions. The attachment line ($A_1$) eventually connects with the saddle point $S_1$. The separation line ($S_1$) passing through the saddle point $S_1$ is the trace of the detachment of the downward flow induced by the stagnation effect of the ground surface. This detachment actually rolls to the secondary horseshoe vortex in front of the cylinder.

Figure 9. Surface flow pattern on the walls of the cylinder in the junction influence region: (a) frontal wall; (b) side wall, with flow from left to right; (c) back wall; (d) close-up view of the local region on the side wall.

As a comparison, the surface flow pattern appears to be more complicated on the side wall, as shown in figure 9(b) and the close-up view in figure 9(d), where the flow is from left to right. First, the notable saddle point $S_2$ is located around the middle height of the cylinder (at $z/D\approx 1.2$) and is the border of the junction region with the 2-D-like region according to Cao et al. (Reference Cao, Tamura and Kawai2019). The separation lines ($S_3$) and ($S_4$) pass through the saddle point $S_2$. In particular, flow separation occurs from the separation line ($S_4$) as the skin-friction lines converge onto ($S_4$). The flow separation is confirmed again in terms of flow topology in the cross-section. The induced small-scale vortex is called the secondary separation vortex ($SSV$). The separation line ($S_3$) starts from node $N_2$ and continues roughly parallel to the attachment line ($A_2$) before turning towards the direction of the saddle point $S_2$. The positive streamwise velocity indicates the vortex upstream of the trailing edge between the attachment line ($A_2$) and trailing edge. Close to the leading edge, an attachment line ($A_3$) is observed, which emanates from node $N_1$. The separation line ($S_4$) winds towards focus $F_1$. Here, $F_1$ is a salient focus in the junction influence region because the skin-friction lines on the side wall of the cylinder that spiral into $F_1$ occupy a large portion of the junction influence region. This focus is the trace of the tornado-like vortex on the side of the cylinder (or ‘inverted conical vortices’ according to Okuda & Taniike Reference Okuda and Taniike1993). The skin-friction lines in the region enclosed by the separation or attachment lines ($S_3$), ($S_4$) and ($A_2$) start from node $N_2$, bend in their direction and terminate at focus $F_1$. Near the ground surface, the saddle–focus–saddle–node connection $S_3$–$F_1$–$S_4$–$N_2$ is shown in figure 9(d). In this region, two sources of skin-friction lines exist. One source is the obvious attachment node $N_2$ and the other is the half-node point $N_1^{\prime }$ on the side wall. Moreover, this combination of $S_3$–$F_1$–$S_4$–$N_2$ ensures the side-wall surface flow is consistent with the surface flow pattern on the ground surface, as shown in figure 7(d). The half-saddle and half-node points identified on the side wall and ground surface are consistent in locations, that is, $N_1^{\prime }$, $N_2^{\prime }$ and $S_1^{\prime }$ on the side wall corresponding to $S_4^{\prime }$, $S_6^{\prime }$ and $N_1^{\prime }$ on the ground surface, as shown in figure 9(d).

The surface flow pattern on the back wall in the junction region is shown in figure 9(c). The attachment line ($A_4$) arising from the attachment node $N_3$ divides the attachment flows into two families moving towards two sides of the y coordinate. Below $N_3$, the sufficient height of the attachment region provides the condition for the formation of 3-D reattachment. The typical organisation of critical points is observed, that is, the saddle-node combination $N_3$–$S_5$–$N_4$–$S_6$ emphasised by Délery (Reference Délery2013), which belongs to the axis of $q=0$ in the p–q chart. From the separation line ($S_5$) of the saddle point $S_6$ in the region located close to the ground surface, flow separation occurs and rolls up to generate a small-scale vortex immediately behind the cylinder. This small vortex corresponds to the focus $F_3$ in the symmetry plane when the vertical height is near the ground surface, as shown in figure 17.

5.2. Three-dimensional separated flow topology

For 3-D separation, the concept of a separation/attachment surface is adopted and denoted by ($\varSigma$) following the work of Délery (Reference Délery2013). The traces of separation/attachment surfaces on the body or ground surface are the separation/attachment lines. Similar to the surface flow patterns, we first visualise the numerical results. A 3-D separated flow topology is then constructed, which has topological consistency with the surface flow patterns and pseudo-streamlines in a cross-section.

The three-dimensionality in the junction region is so strong that the vortical structures and their relationships cannot be easily identified. Flow is visualised in a relatively neat manner in figure 10. Only half of the flow ($y \geq 0$) is shown, considering the symmetry in the mean field. Figure 10 shows several basic components: the pseudo-streamlines in the horizontal cross-section perpendicular to the cylinder axis (x–y plane); pseudo-streamlines on a vertical slice perpendicular to the x coordinate; surface flow patterns on the side wall; and 3-D streamlines twining around the vortex cores. The horizontal cross-sections are located at $z/D=0.05$, 0.5, 1.0 and 1.5, and the vertical slice is located at $x=0$. The vortex cores from the 3-D streamline swirl are identified using eigenmode analysis (Sujudi & Haimes Reference Sujudi and Haimes1995). Three primary vortices are emphasised and named: the side vortex, which originates from the focus on the side wall $F_1$ ($SV$); the trailing-edge vortex immediately upstream of the trailing edge of the cylinder ($TEV$); and the near-wake vortex behind the cylinder that emerges from the focus behind the cylinder ($WV$). The $SSV$ is located just downstream of the leading edge and is relatively weak. The saddle point $S_2$ represents the border between the junction and 2-D-like regions.

Figure 10. Flow visualisation of 3-D separation in the junction region. Panels (a)–(d) are from different perspectives. Here, $S_2$, saddle point on the side wall; $F_1$, focus on the side wall; $SV$, side vortex; $WV$, near-wake vortex; $TEV$, trailing-edge vortex; $SSV$, secondary separation vortex.

The pseudo-streamlines in additional cross-sections are shown in the left-hand column of figure 11 to present the gradual evolution of the flow topology in the vertical direction. The flow topologies are constructed in the right-hand column of figure 11 using the critical-point theory to provide more rigorous insights. The number of critical points is shown. All flow topologies were checked and found to obey the topological rules (3.9) proposed by Hunt et al. (Reference Hunt, Abell, Peterka and Woo1978), where $n=2$.

Figure 11. Flow topologies projected in the cross-sections at different heights above the secondary horseshoe vortex: (a) $z/D=0.16$; (b) $z/D=0.20$; (c) $z/D=0.28$; (d) $z/D=0.36$. The left-hand column shows the pseudo-streamlines from the numerical results; the right-hand column indicates the flow topology constructed using critical-point concepts, where the bold lines denote the separation lines. The number of critical points is also shown.

Figures 11(a) and 11(b) show the flow topologies in the cross-sections at $z/D=0.16$ and 0.20. The insulation of the free stream from the side wall is observed in the ground-surface flow patterns, and is characterised by the existence of $S_2$ in figure 7(d). However, it is eliminated when the horizontal sectional plane is sufficiently removed vertically from the ground surface. As figure 11(a) shows, when $z/D=0.16$, the free stream reattaches directly to the trailing portion of the side wall, and is characterised by flow reattachment on the half-saddle points $S_7^{\prime }$ and $S_8^{\prime }$ on the side walls. Moreover, the heights of $z/D=0.16$ and 0.20 are above the secondary horseshoe vortex. The saddle point $S_1$ in figures 11(a) and 11(b) occurs in front of the cylinder, which is thought to be caused by the rapid increase in the thickness of the boundary layer attached to the frontal wall of the cylinder when approaching the separation line which maintains the secondary horseshoe vortex. The thick boundary layer flow is divided into two components: the inner component, which detaches from the leading edge of the cylinder and rolls up into the focus of the side vortex; and the outer component, which reattaches to the side wall upstream of the trailing edge and finally detaches again from the trailing edge. These two components are separated by separation lines ending at the half-saddle points $S_7^{\prime }$ and $S_8^{\prime }$ on the side walls. Figures 11(a) and 11(b) are topologically equivalent. However, the boundary layer thickness in figure 11(b) is obviously thinner, and the side and near-wake vortices are much wider. Because of the latter, the pseudo-streamlines near the flow reattachment region of the wall have greater curvature. When the height increases to $z/D=0.28$ in figure 11(c), the reattachment region becomes shorter, the side and near-wake vortices are further widened, and the pseudo-streamlines near the reattachment region are curved more considerably and pushed outwards from the cylinder. Moreover, the saddle point in front of the cylinder is barely visible.

When $z/D=0.36$ in figure 11(d), flow reattachment is no longer observed. Instead, a saddle point $S_1$ in figure 11(d) occurs between the side and near-wake vortices. The formation of the saddle point $S_1$ indicates the completeness of the changeover from the vertical flow region with to without reattachment. The separated flow from the leading edge of the cylinder is the primary source of the near-wake vortex. However, the separation line starting from the leading edge of the cylinder (i.e. starting from the half-saddle $S_2^{\prime }$ in the subpanel of figure 11d) blocks the passage of the free stream into the side vortex foci $F_1$ and $F_2$. Instead, the main source of $F_1$ and $F_2$ is the reverse flow from the central wake. The reverse flow from the wake is easily detached from the trailing edge and forms trailing-edge vortices (see $F_8$ in the subpanel of figure 11d). The attachment points (attachment lines in 3-D space) of the trailing-edge vortices are indicated using half-saddle points $S_8^{\prime }$ and $S_9^{\prime }$ in figure 11(d). The reverse flow subsequently detaches from the half-saddle points $S_6^{\prime }$ and $S_7^{\prime }$, such that focus $F_5$ is generated in the subpanel of figure 11(d). Note that $F_5$ is the trace of the SSV.

The flow topology shown in figure 11(d) is applicable to the 2-D-like region of the surface-mounted and infinite-length square cylinders. Huang et al. (Reference Huang, Lin and Yen2010) proposed a topological flow pattern based on visualisations of both their surface oils and limited-resolution PIV (van Oudheusden et al. Reference van Oudheusden, Scarano, van Hinsberg and Roosenboom2008). However, the study failed to consider the trailing-edge vortex. The newly proposed flow topology resembles the general flow pattern obtained by van Oudheusden et al. (Reference van Oudheusden, Scarano, van Hinsberg and Roosenboom2008) and Huang et al. (Reference Huang, Lin and Yen2010), and allows for small-scale vortices used in the previous high-resolution numerical simulations by Trias et al. (Reference Trias, Gorobets and Oliva2015), Cao & Tamura (Reference Cao and Tamura2016) and Cao, Tamura & Kawai (Reference Cao, Tamura and Kawai2020).

The flow topologies in the cross-sections of figure 11 are consistent with the surface flow patterns on the walls of the cylinder, as shown in figure 9. The corresponding relationship between the critical points in the flow topologies in the cross-sections and the separation/attachment lines on the walls of the cylinder is summarised as follows, where $\Leftrightarrow$ represents the corresponding relationship:

1. (1) $N_1^{\prime }$ in figures 11(a) and 11(b) and $S_1^{\prime }$ in figures 11(c) and 11(d) $\Leftrightarrow$ attachment line ($A_1$) on the frontal wall in figure 9(a);

2. (2) $S_3^{\prime }$ and $S_4^{\prime }$ in figures 11(a) and 11(b) and $S_4^{\prime }$ and $S_5^{\prime }$ in figures 11(c) and 11(d) $\Leftrightarrow$ attachment line ($A_3$) on the side wall in figure 9(b);

3. (3) $S_5^{\prime }$ and $S_6^{\prime }$ in figures 11(a) and 11(b) and $S_6^{\prime }$ and $S_7^{\prime }$ in figures 11(c) and 11(d) $\Leftrightarrow$ separation line ($S_4$) on the side wall in figure 9(b);

4. (4) $S_7^{\prime }$ and $S_8^{\prime }$ in figures 11(a) and 11(b) and $S_8^{\prime }$ and $S_9^{\prime }$ in figures 11(c) and 11(d) $\Leftrightarrow$ attachment line ($A_2$) on the side wall in figure 9(b);

5. (5) $S_{11}^{\prime }$ in figures 11(a) and 11(b) and $S_{12}^{\prime }$ in figures 11(c) and 11(d) $\Leftrightarrow$ attachment line ($A_4$) on the back wall in figure 9(c).

Based on the aforementioned observations, the 3-D separation surfaces that constitute the side and near-wake vortices are shown in figure 12, where panel (a) is covered by the outer separation surface and panel (b) focuses on the inner flow with the removal of the outer separation surface. In addition, the secondary horseshoe vortex proceeds out of the side vortex and then infinitely downstream. Because of the rotation effects of the side vortex and the curved but upward direction of the side vortex core, the separation node ($N_4$ in the ground-surface flow pattern in figure 7d) is formed by the attraction effect from the side vortex. Consistency is confirmed among the separation surfaces, the surface flow patterns on the ground plane and walls of the cylinder and the projected flow field in the cross-sections.

Figure 12. Three-dimensional separation surfaces in the junction region, including the near-wake, side, trailing-edge and secondary horseshoe vortices: (a) covered by the outer separation surface; (b) with the removal of the outer separation surfaces.

6.1. Surface flow pattern

The skin-friction lines on the walls of the cylinder in the free-end influence region are shown in figures 8(a) and 8(b) and are interpreted in figure 13. The border between the 2-D-like and free-end regions is indicated by saddle point $S_7$ in figure 13(c).

Figure 13. Surface flow pattern on the walls of the cylinder in the free-end influence region: (a) upper wall; (b) frontal wall; (c) side wall, with flow from left to right; (d) back wall.

Node point $N_5$ in figure 13(b) is the stagnation point on the frontal wall, which is located at $z/D=2.25$. All skin-friction lines spread out from $N_5$. Two attachment lines ($A_1$) and ($A_2$), which originate from $N_5$, correspond to two orthogonal eigenvectors of the node point and divide the streamlines. The flow topology on the side wall is shown in figure 13(c). The existence of separation and attachment lines ($A_3$), ($S_4$) and ($A_4$) in the junction influence region implies two small-scale vortices next to the side wall of the cylinder (i.e. the $SSV\ F_5$ and the trailing-edge vortex $F_8$ in figure 11d). Above the saddle point $S_7$, a trailing-edge vortex is obtained from the flow separation and reattachment between the attachment line connecting $S_7$ and $N_6$ and the trailing edge of the cylinder. However, the fluid inside the trailing-edge vortex flows downwards. The size of the trailing-edge vortex in the free-end region becomes considerably smaller than that in the 2-D-like region. In the upstream portion of the side wall, most of the skin-friction lines emanate from node $N_6$ and change their direction from downwards to upwards. The skin-friction lines on the back wall in figure 13(d) are dominated by the near-wake vortex, and consequently flow upwards. The skin-friction lines are assumed to be symmetric around the attachment line ($A_5$) which emerges from node point $N_3$ in figure 9(c).

The surface flow patterns on the frontal, side and back walls are consistent with the top wall of the cylinder, as shown in figure 13(a). Towards the leading edge, the skin-friction lines curve outwards from the centreline of the prism. For simplicity, the flow topology is assumed to be symmetric around the separation line ($S_5$) connecting saddle points $S_8$ and $S_9$. However, the symmetry holds only theoretically because of the structural instability of the saddle-to-saddle connection $S_8$-to-$S_9$. The attachment line ($A_6$) implies the attachment of the reverse flow from the back wall which is driven by the near-wake vortex. The reverse flow enters the recirculation zone and separates again from the separation line ($S_6$). The process is the same as in the generation of vortex foci $F_5$ and $F_8$ in figure 11(d). Most of the skin-friction lines have the same tendency as the PIV measurements of a surface-mounted cube by Depardon et al. (Reference Depardon, Lasserre, Boueilh, Brizzi and Borée2005) and of a surface-mounted square cylinder in a thin turbulent boundary layer by Sumner et al. (Reference Sumner, Rostamy, Bergstrom and Bugg2017). The differences are observed only in regions close to the cylinder edges, where these regions are sensitive to the measurement technique and distance from the wall. In this study, the flow topology in figure 13(a) is constructed after a special close-up view of the local regions is examined near the edges of the cylinder when considering the topological consistency with the surface flows on the other walls of the cylinder. The same combination $N_7$–$S_8$–$S_9$ in the centreline of the top wall is consistent with Depardon et al. (Reference Depardon, Lasserre, Boueilh, Brizzi and Borée2005) when $h=0.008D$, but it was not observed by Sumner et al. (Reference Sumner, Rostamy, Bergstrom and Bugg2017) when $h=0.016D$.

6.2. Three-dimensional separated flow topology

Figure 14 presents a flow visualisation of the free-end influence region. The horizontal cross-sections are located at $z/D=2.1$, 2.4 and 2.7, and the vertical slices are positioned at $y/D=0$, 0.2 and 0.4. Saddle point $S_7$ represents the border between the 2-D-like and free-end influence regions. The cores of vortex structures wrapped by 3-D streamlines are emphasised and named in figure 14. Two vortices are produced by the main flow separation, namely, the side ($SV$) and near-wake ($WV$) vortices. Several small-scale vortices are produced by the secondary flow separation inside the recirculation region: the trailing-edge vortex next to the side wall ($TEV$), the vortex induced by the secondary separation next to the side wall ($SSV$), the trailing-edge vortex above the top wall ($TEV$-$t$), the side-edge vortex above the top wall ($SEV$-$t$) and the vortex induced by the secondary separation above the top wall ($SSV$-$t$). The side-edge vortex ($SEV$-$t$), which became weaker with a decreasing aspect ratio of the prism, was also reported by Sumner et al. (Reference Sumner, Rostamy, Bergstrom and Bugg2017). Based on the flow visualisation, the 3-D separated flow topology is shown in figure 15.

Figure 14. Flow visualisation of 3-D separation in the free-end region. Panels (a)–(c) are from different perspectives. Here, $S_7$, saddle point on the side wall; $SV$, side vortex; $WV$, near-wake vortex; $TEV$, trailing-edge vortex; $SSV$, secondary separation vortex. The vortex name followed by ‘-$t$’ denotes the vortex above the top wall (i.e. $TEV$-$t$, side-edge vortex above the top wall; $SSV$-$t$, $SSV$ above the top wall; $SEV$-$t$, side-edge vortex above the top wall.)

Figure 15. Three-dimensional separated flow topology in the free-end influence region. The abbreviations of vortices are referred to the caption of figure 14.

The near-wake vortex ($WV$) behind the cylinder is first discussed. The near-wake vortex is composed of two vortex segments: one from the roll-up of the separated flow from the leading edge on the side wall and the other from the leading edge on the top wall. Interestingly, these two segments are connected. However, the vortex segment from the top wall is closer to the cylinder than that from the side wall. With the shape similarity, the near-wake vortex is also called an ‘arch-type vortex’ (Sakamoto & Arie Reference Sakamoto and Arie1983; Wang & Zhou Reference Wang and Zhou2009; Bourgeois et al. Reference Bourgeois, Sattari and Martinuzzi2011; Rastan et al. Reference Rastan, Sohankar and Alam2017), with two standing legs corresponding to the roll-up from the sides of the cylinder and the head corresponding to that from the free end. The side vortex ($SV$) is maintained until it connects with the vortex inside the recirculation zone above the top wall. The $SV$ originates from the focus $F_1$ on the lower upstream corner of the side wall. The sectional size of the side vortex is compressed when it is located near the side edge of the top wall. The compression propels the fluid near the side edge of the top wall towards the two sides, such that the skin-friction lines on the top wall curve towards the centreline and those on the side wall curve downwards.

Of the small-scale vortices, the trailing-edge vortex next to the side wall ($TEV$) warrants further discussion because its location is another signal of the classification between the 2-D-like and free-end influence regions, in addition to the aforementioned signal of saddle point $S_7$. From figure 14(b), the trailing-edge vortex shows an obvious downstream movement above saddle point $S_7$, and is induced by the push of the fluid near the side edge of the top wall.

The topological rule proposed by Hunt et al. (Reference Hunt, Abell, Peterka and Woo1978) is examined for symmetry-plane flow in this study. For completeness, the numbering of critical points is derived again for the symmetry-plane flow following Hunt et al. (Reference Hunt, Abell, Peterka and Woo1978). Figure 16(a) shows the symmetry-plane flow above the ground surface. Only one of the node, saddle, half-node and half-saddle points are illustrated for simplicity of explanation, which does not influence the following derivation. The ground surface is denoted as OP, and the upper line XY indicates the upper boundary of the symmetry-plane flow. First, the space above the ground surface OP is mapped into the region OPXY, as shown in the upper part of figure 16(b), by flattening the square cylinder. The nature of each singular point remains unchanged. Next, consider the image space ${O}_i {P}_i {X}_i {Y}_i$ formed by a mirror and containing images of the nodes and saddles. The two spaces are then connected along OP and ${O}_i {P}_i$ and wrap the surface into a cylinder until XY is connected with ${X}_i {Y}_i$. The cylinder is shown in figure 16(c). The streamlines above the ground surface are regarded as the skin-friction lines covering the cylinder. Any skin-friction line must follow the principle in which it must terminate in either a focus or detachment node. Therefore, two imaginary nodes are created: upstream attachment and downstream detachment at infinity. The Poincaré—Bendixson theorem (i.e. where the number of nodes exceeds that of saddle points by two) is written in (6.1) for the skin-friction lines on the closed cylinder in figure 16(c) with a complexity of zero. Here, $\sum N$, $\sum N^{\prime }$, $\sum S$ and $\sum S^{\prime }$ are the numbers of critical points in the symmetry flow above the surface, as shown in figure 16(a). Additionally, $\sum N_i$ and $\sum S_i$ denote the numbers of nodes and saddles in the image space ${O}_i {P}_i {X}_i {Y}_i$. The upstream and downstream nodes at infinity are considered by adding the two nodes to the first term on the left-hand side of (6.1). The relationships of $\sum N=\sum N_i$ and $\sum S= \sum S_i$ hold. Thus, (6.1) is rewritten as (6.2), which is used to check the number of critical points in the symmetry plane:

(6.1)$$\begin{gather} \left( 2+ \sum N + \sum N_i + \sum N^{\prime} \right) - \left( \sum S + \sum S_i + \sum S^{\prime} \right) = 2, \end{gather}$$

(6.2)$$\begin{gather}\left( \sum N + \frac{1}{2} \sum {N^{\prime}} \right) - \left( \sum S + \frac{1}{2} \sum {S^{\prime}} \right) = 0. \end{gather}$$

Figure 16. Schematic of mapping the flow above the surface to a closed body: (a) actual flow above the surface, where only one of the node, saddle, half-node and half-saddle points are illustrated; (b) flow region OPXY mapped from the actual flow by flattening OP, and the image system ${O}_i {P}_i {X}_i {Y}_i$ formed by a mirror; (c) closed body with a complexity of zero created by wrapping the region OPXY and ${O}_i {P}_i {X}_i {Y}_i$, where the imaginary upstream and downstream nodes at infinity are added to satisfy the topological principle.

Figure 17 shows the flow topology in the symmetry plane of the high-$Re$ flow around surface-mounted square cylinder. The number of critical points is also shown in the figure and satisfies the topological rule, as shown in (6.2), that is, $( \sum N + \frac {1}{2}\sum {N^{\prime }} )- ( \sum S + \frac {1}{2}\sum {S^{\prime }} ) = 0$, where $\sum N=8$, $\sum N^{\prime }=2$, $\sum S=2$ and $\sum S^{\prime }=14$. The flow topology is consistent with the surface flow patterns on the ground plane, the walls of the cylinder and the 3-D separated flows. Half-saddle point $S_2^{\prime }$ coincides with node $N_1$ on the ground-surface flow pattern in figure 7. The half-saddle points $S_{13}^{\prime }$ and $S_{14}^{\prime }$ on the ground surface, as viewed in the symmetry plane, coincide with the flow separation from $S_4$ and attachment on $N_6$ in figure 7(c). The half-node $N_2^{\prime }$ behind the cylinder corresponds to the saddle point $S_5$ in the surface flow pattern of the ground plane in figure 7(a). In particular, for correspondence with vortices, the foci $F_6$ and $F_8$ correspond to the segment of the side vortex above the top wall and near-wake vortex ($WV$), respectively. The foci $F_5$ and $F_7$ correspond to the small-scale vortices $SSV$-$t$ and $SEV$-$t$, respectively.

Figure 17. Flow topology in the symmetry plane when $Re=5\times 10^4$. The number of critical points is also shown.