2.1.1 Large-eddy simulations

Since the velocities in this study are low, with the maximum Mach number less than 0.2, the air may be assumed to be incompressible. The filtered equations for continuity and momentum for an incompressible fluid with constant properties are

(2.1) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\bar{U}_{i}}{\unicode[STIX]{x2202}x_{i}}=0,\end{eqnarray}$$

and

(2.2) $$\begin{eqnarray}\unicode[STIX]{x1D70C}\frac{\unicode[STIX]{x2202}\bar{U}_{i}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D70C}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{j}}(\bar{U}_{i}\bar{U}_{j})=-\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}x_{i}}+\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}_{ij}}{\unicode[STIX]{x2202}x_{j}}+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{j}}(\unicode[STIX]{x1D70E}_{ij}),\end{eqnarray}$$

where $\unicode[STIX]{x1D70C}$ is the density and $\bar{U}$ is the filtered velocity. The stress tensor due to molecular viscosity, $\unicode[STIX]{x1D70E}_{ij}$ , is defined by

(2.3) $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{ij}=\left[\unicode[STIX]{x1D707}\left(\frac{\unicode[STIX]{x2202}U_{i}}{\unicode[STIX]{x2202}x_{j}}+\frac{\unicode[STIX]{x2202}U_{j}}{\unicode[STIX]{x2202}x_{i}}\right)\right]-\frac{2}{3}\unicode[STIX]{x1D707}\frac{\unicode[STIX]{x2202}\bar{U}_{l}}{\unicode[STIX]{x2202}\bar{x}_{l}}\unicode[STIX]{x1D6FF}_{ij}\end{eqnarray}$$

and $\unicode[STIX]{x1D70F}_{ij}$ is the subgrid-scale stress defined by

(2.4) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{ij}=\unicode[STIX]{x1D70C}\overline{U_{i}U_{j}}-\unicode[STIX]{x1D70C}\bar{U}_{i}\bar{U}_{j}.\end{eqnarray}$$

The subgrid-scale model used in the simulations was proposed by Smagorinsky (Reference Smagorinsky1964). Here, the model employs the Boussinesq hypothesis to compute subgrid-scale turbulent stresses,

(2.5) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{ij}-{\textstyle \frac{1}{3}}\unicode[STIX]{x1D70F}_{kk}\unicode[STIX]{x1D6FF}_{ij}=-2\unicode[STIX]{x1D707}_{t}\bar{S}_{ij}.\end{eqnarray}$$

The rate of strain tensor, $\bar{S}_{ij}$ , is

(2.6) $$\begin{eqnarray}\bar{S}_{ij}=\frac{1}{2}\left(\frac{\unicode[STIX]{x2202}U_{i}}{\unicode[STIX]{x2202}x_{j}}+\frac{\unicode[STIX]{x2202}U_{j}}{\unicode[STIX]{x2202}x_{i}}\right)\end{eqnarray}$$

and the subgrid-scale turbulent viscosity, $\unicode[STIX]{x1D707}_{t}$ , is modelled by

(2.7) $$\begin{eqnarray}\unicode[STIX]{x1D707}_{t}=\unicode[STIX]{x1D70C}L_{s}^{2}\unicode[STIX]{x1D6E5}^{2}|\bar{S}|,\end{eqnarray}$$

where $L_{s}$ is the mixing length for subgrid scales and $|\bar{S}|=\sqrt{2\bar{S}_{ij}\bar{S}_{ij}}$ . $L_{s}$ is computed using

(2.8) $$\begin{eqnarray}L_{s}=\text{min}(\unicode[STIX]{x1D705}y_{wall},C_{s}\unicode[STIX]{x1D6E5}),\end{eqnarray}$$

where $\unicode[STIX]{x1D705}$ is the von Kármán constant, $y_{wall}$ is the distance to the closest wall, $C_{s}$ is the Smagorinsky constant and the subgrid filter width, $\unicode[STIX]{x1D6E5}$ , is defined as

(2.9) $$\begin{eqnarray}\unicode[STIX]{x1D6E5}=V^{1/3},\end{eqnarray}$$

where $V$ is the volume of a cell in the computational domain. In this application, $C_{s}$ is evaluated using the dynamic procedure based on the Germano identity (Germano et al. Reference Germano, Piomelli, Moin and Cabot1991), which uses a solution proposed by Lilly (Reference Lilly1992). Details of the dynamic procedure and its implementation have been presented by Kim (Reference Kim2004). The interaction between the oncoming boundary layer formed on the wall and the boundary layer formed on the surface of the cylinder is important in the formation of the junction flow structures. As such, the flow in the boundary-layer region was resolved in the LES and no wall model was implemented.

These equations were solved with the aid of ANSYS FLUENT 17.2 finite-volume solver. The temporal discretisation scheme is advanced using a second-order implicit method and the convection term is discretised using a second-order central-differencing scheme. The SIMPLEC method (Van Doormaal & Raithby Reference Van Doormaal and Raithby1984) was used to handle the iterative process of pressure–velocity coupling. The convergence criteria at each time step were set at scaled residual of $10^{-4}$ for the continuity equation and each directional component of the velocity, which was typically attained in less than 30 iterations per time step.

The time step size, $\unicode[STIX]{x0394}t$ , was $5\times 10^{-6}$  s, which corresponds to a Nyquist frequency of 100 kHz. This $\unicode[STIX]{x0394}t$ was selected to keep the Courant–Friedrichs–Levy number along the streamwise direction, $\text{CFL}=U_{\infty }\unicode[STIX]{x0394}t/\unicode[STIX]{x0394}x\approx 1$ around the cylinder, with $\unicode[STIX]{x0394}x$ being the grid spacing on the surface of the cylinder along the streamwise direction. The simulation for each aspect ratio was run for a total of 16 000 time steps or 0.1 s in physical time. All time-averaged results shown in § 3 were obtained by averaging the data over the last 12 000 time steps or 0.075 s, which corresponds to 343 flow-through times across the width of the cylinders. Computations were performed on the Raijin cluster of the Australian National Computational Infrastructure (NCI) on a single computation node, which consists of 28 Intel Xeon E5-2690v4 2.6 GHz processor cores with 128 GB physical memory.

2.1.2 Boundary conditions

The oncoming free stream is simulated using a velocity inlet of $U_{\infty }=32~\text{m}~\text{s}^{-1}$ as the boundary condition at approximately $140W$ or 1 m upstream of the FWMC. To allow the wake to develop and avoid backflow, a pressure outlet condition of zero gauge pressure was applied $80W$ or 0.57 m downstream of the cylinder, which corresponds to more than four times the span of the cylinder with the highest aspect ratio ( $L/W=18.6$ ). A no-slip condition is applied to the mounting wall and the faces of the cylinder. Periodic boundary conditions are applied to the cross-stream boundaries of the domain, which are $30W$ or 0.21 m apart. The boundary conditions are summarised in figure 2.

Figure 2.

Schematic diagram of the computational domain. The type and value of the boundary conditions are specified.

Forced transition was necessary in this study to create a turbulent boundary layer with parameters similar to those in the experiments. To generate the turbulent boundary layer, a uniform jet injected through a 10 mm or $1.4W$ wide slot along the wall was used, as shown in figure 2. This is akin to the boundary-layer tripping method employed by Wolf & Lele (Reference Wolf and Lele2012), except no suction is employed in this study which adds a small amount of momentum to the flow. However, due to its placement, ${\approx}100W$ upstream of the FWMC, and the large cross-section of the domain the local momentum injection is deemed to have enough time and space to dissipate into the global flow.

An inlet velocity of $1~\text{m}~\text{s}^{-1}$ or $U/U_{\infty }=0.03$ was applied to the slot area. The desired boundary-layer thickness, 9.1 mm, was then found to occur 1 m downstream of the main inlet of the computational domain, hence the FWMC was placed at this location. Jet velocities other than $1~\text{m}~\text{s}^{-1}$ were also tested: $0.5~\text{m}~\text{s}^{-1}$ , $1.2~\text{m}~\text{s}^{-1}$ and $1.5~\text{m}~\text{s}^{-1}$ . While higher velocities accelerate the growth, and therefore allow a smaller computational domain, using a value of $1~\text{m}~\text{s}^{-1}$ was found to yield the profile closest to the law of the wall. Detailed parameters of the boundary-layer profile are presented in § 2.4.

2.1.3 Computational grid

Computational grids were created for four cylinders with aspect ratios of: $L/W=1.4$ , 4.3, 10 and 18.6. A H-grid blocking scheme was employed to mesh the computational domain, with an O-grid in the immediate vicinity of the cylinder acting as an inflation layer. The total number of cells in the computational domain varies from $12.7$  million for the lowest aspect ratio of $L/W=1.4$ to $27.1$  million for the highest aspect ratio of $L/W=18.6$ . In a similar fashion, the number of elements across the span of the cylinder, $N_{z}$ , ranges from $N_{z}=40$ for $L/W=1.4$ to $N_{z}=250$ for $L/W=18.6$ . The number of elements on the cylinder along the streamwise, $N_{x}$ , and cross-stream, $N_{y}$ , directions are consistently 70 for all aspect ratios. Table 3 provides a summary of these details. Based on these grid parameters, the average $z^{+}$ value (spanwise direction) on the cylinder varies from 31 to 66 and the $x^{+}$ value (streamwise direction) is constantly 22, which puts these simulations in the medium-to-high resolution LES regime according to the criteria by Wagner, Hüttl & Sagaut (Reference Wagner, Hüttl and Sagaut2007). The size of these spanwise elements is reduced near the wall junction and the free-end tip to properly capture the more complex flow in these regions. There are 40 elements across each side of the square cross-section, this is consistent for all aspect ratios. The $y^{+}$ value on the surface of the wall and all sides of the cylinder is less than 2.

From the size of the computational domain shown in figure 2, the gap between the cylinder’s top surface and the top boundary is $2L$ for the cylinder the highest aspect ratio ( $L/W=18.6$ ). Behera & Saha (Reference Behera and Saha2019) performed numerical simulations of the flow past a finite wall-mounted square cylinder, where they examined three different gap heights of $1.3L$ , $1.7L$ and $2.1L$ , and found that the flow field predicted in the three domains is almost identical. To confirm the domain independence in the current study, a simulation was run using a computational domain with a gap height of $L$ . Figure 3 compares the time-averaged pressure coefficient on the back face (or ‘base’, facing downstream) obtained using the domains with $2L$ and $L$ . Unless stated, the pressure data herein are presented as a pressure coefficient, $C_{p}(z,t)$ , defined as

(2.10) $$\begin{eqnarray}C_{p}(z,t)=\frac{p(t)-\overline{p_{\infty }}}{0.5\unicode[STIX]{x1D70C}{U_{\infty }}^{2}},\end{eqnarray}$$

where $p(z,t)$ is the pressure time recorded at a specific spanwise location, $\overline{p_{\infty }}$ is the time-averaged free-stream static pressure. Although the gap height has been reduced from $2L$ to $L$ , the base pressure coefficient maintains a consistent profile across its span. This suggest that the minimum gap of $2L$ employed in the simulations in this paper is sufficient to ensure domain independence. The maximum blockage of the cylinder with the highest aspect ratio of $L/W=18.6$ is 0.6 %, which is much lower than the recommended maximum of 3 % by Tominaga et al. (Reference Tominaga, Mochida, Yoshie, Kataoka, Nozu, Yoshikawa and Shirasawa2008).

Figure 3.

Time-averaged base pressure coefficient obtained from the computational domains where the gap between the cylinder top surface and the top boundary is $L$ and $2L$ .

Three grid resolutions were used to demonstrate grid spacing convergence, which is presented below for the aspect ratio $L/W=4.3$ . The grid convergence index (GCI), following the method proposed by Roache (Reference Roache1997), was computed for each grid refinement to estimate the spatial discretisation errors. The refinement was focused on the immediate vicinity of the cylinder and the wake region, but with the aforementioned $y^{+}$ values maintained in all three grids. As such, the grid refinement was not isotropic and the refinement ratio was estimated as $\unicode[STIX]{x1D6FC}=(N_{j}/N_{j+1})^{1/3}$ , where $N$ is the number of cells in the computational domain and subscript $j=1,2,3$ indicates the coarse, medium and fine mesh, respectively.

The first parameter used to assess the grid convergence was the time-averaged drag force coefficient, $\overline{C_{Fx}}$ . Force coefficient in this paper is defined as

(2.11) $$\begin{eqnarray}C_{F_{i}}=\frac{F_{i}}{0.5\unicode[STIX]{x1D70C}U_{\infty }^{2}LW},\end{eqnarray}$$

where $F_{i}$ is the force acting on the cylinder in direction $i=x,y,z$ and the planform area is evaluated by $LW$ . The second parameter considered here was the time-averaged velocity in the wake region. The time-averaged velocity is spatially averaged along the line formed by the intersection of the $x/W=4$ and $y/W=0.6$ planes. Spatially averaged velocity was preferred over single-point velocity, because of the unsteady and three-dimensional nature of the wake flow.

The results of the grid convergence study for $L/W=4.3$ are summarised in table 1; $\text{GCI}_{2,1}$ and $\text{GCI}_{3,2}$ are the grid convergence indices for the coarse-to-medium and medium-to-fine refinements, respectively. The discretisation error was quantified by $\unicode[STIX]{x1D700}$ , which is the percentage error between the value predicted by the LES with the medium resolution grid and that predicted by the Richardson extrapolation method when the grid spacing approaches zero. The $\text{GCI}_{3,2}$ of both $\overline{C_{Fx}}$ and $\overline{U}$ are lower than their respective $\text{GCI}_{2,1}$ , which indicates that the flow variables are converging to a value as the grid spacing is reduced. The spatial discretisation error for the medium grid $\overline{C_{Fx}}$ and $\overline{U}$ are 5.17 % and 3.34 %, respectively. The medium mesh was deemed to have a good balance between the discretisation error and the computational cost, and therefore was used to obtain the LES results. The mesh for the cylinders with $L/W=1.4$ , 10 and 18.6 also exhibit similar convergence behaviours.

Table 1.

The results of the grid convergence study for $L/W=4.3$ , showing the grid convergence indexes and the spatial discretisation error obtained using Richardson’s extrapolation method.

The validation of the LES using experimental data is presented concurrently with the discussion of the results. Validations are available in § 3.1 for the time-averaged velocity field, §§ 3.3.1 and 3.3.2 for the cylinder surface pressure (both time evolution and time averaged) and § 3.3.3 for the frequency content of the wake velocity.

2.2.1 Hot-wire measurement

Flow-induced wake velocity measurements were taken in the University of Adelaide’s Anechoic Wind Tunnel (AWT). The air flows through an outlet with a height of 75 mm and a width of 275 mm. The flow is exhausted into an anechoic chamber with internal dimensions of $1.4~\text{m}\times 1.4~\text{m}\times 1.6~\text{m}$ . The walls of this chamber are lined with pyramid shaped foam wedges which provide a near-anechoic environment above 250 Hz. The experiments conducted in the AWT were performed at a free-stream velocity of $U_{\infty }=32~\text{m}~\text{s}^{-1}$ corresponding to a Reynolds number of $Re_{W}=U_{\infty }W/\unicode[STIX]{x1D708}=1.4\times 10^{4}$ .

A test rig was designed to automatically change the aspect ratio of the square cylinder in small increments. A stepper motor is used to rotate a 300 mm threaded rod with an 8 mm pitch via a helical shaft coupling. The cylinder is driven linearly through a square hole in a flat plate made from 10 mm thick aluminium. The aluminium plate measures 150 mm in the streamwise ( $x$ ) direction and 300 mm in the cross-stream ( $y$ ) direction while the hole is located in the centre of the plate at a position of 43 mm downstream of jet exit plane. For the hot-wire wake surveys, the results for four different aspect ratio FWMCs are included in this paper: $L/W=1.4$ , 4.3, 10, 18.6.

Hot-wire anemometry was used to perform a wake survey of the streamwise velocity downstream of the FWMCs. A Dantec Dynamics miniature hot-wire (model 55P16) controlled by an IFA300 Constant Temperature Anemometer system was used for the wake survey. The hot-wire is made from platinum plated tungsten, with a diameter of $5~\unicode[STIX]{x03BC}\text{m}$ and a wire length of 1.25 mm. The measurement plane behind each aspect ratio cylinder was a grid of points that measured 50 mm in the streamwise ( $x$ ) direction and the length of the cylinder in the spanwise ( $z$ ) direction. The measurement plane began at approximately 0.2 mm from the trailing edge of the cylinder and 2 mm above surface of the plate. Additionally, the measurement plane was located at $y/W=0.6$ to avoid entering the reverse flow region. The grid increments in the $x$ and $z$ directions were $\unicode[STIX]{x0394}x_{hw}=\unicode[STIX]{x0394}z_{hw}=2$  mm. A hot-wire velocity record was taken at each node in the grid. A National Instruments PXI Express Chassis (Model PXIe-1082) with PXIe-4499 simultaneous sample and hold ADC cards was used to collect data from the hot-wire. These cards have built-in antialiasing low-pass filters with cutoff of DC to 0.4 of the sampling frequency. Velocity measurements were sampled at $2^{15}$  Hz for 15 s. The complete description of the method employed for the hot-wire measurements is available in Porteous (Reference Porteous2016) and Porteous et al. (Reference Porteous, Moreau and Doolan2017).

2.2.2 Surface pressure measurements

Surface pressure measurements were performed in a wind tunnel facility at the University of New South Wales (UNSW), known as the ‘18 inch Wind Tunnel’ (18WT). The 18WT is an open-circuit, suction-type, closed-test-section wind tunnel with a working section that measures $460~\text{mm}\times 460~\text{mm}$ and a maximum velocity of $29~\text{m}~\text{s}^{-1}$ . Air first passes through a series of smoothing screens before entering a contraction with a contraction ratio of 5.5 and subsequently passing through the test section. The air is drawn through the wind tunnel by an 8-blade, 5.5 kW axial fan located downstream of the test section. Afterwards, the air is ejected through a vertical diffuser. The longitudinal turbulence intensity at $15~\text{m}~\text{s}^{-1}$ (the speed for which the surface pressure and flow visualisation experiments were conducted) is approximately 0.5 %, which is comparable to the turbulence intensity measured in the AWT.

The FWMC models used for surface pressure measurements were three-dimensionally (3-D) printed out of ABS plastic using an ‘Ultimaker $^{2}$ ’ 3-D printer. Two models were printed with width of 12 mm and aspect ratio of $L/W=4$ and 12.5. Each model was printed with a series of internal passages that ran from the base of the cylinder to a number of pressure taps located on one of the cylinder side faces. The pressure taps each had a diameter of 1 mm and the internal passages had an inner diameter of 1.5 mm. The $L/W=4$ cylinder has eight pressure taps while the $L/W=12.5$ FWMC has seven pressure taps. The pressure taps were evenly spaced in the spanwise direction of the cylinders and located directly in the centre of the side face, as illustrated in figure 1. The spanwise location of the pressure taps for each cylinder considered in this paper is provided in table 2. The spanwise position axis has been normalised by the total span of each cylinder, such that $z/L=0$ is the wall junction and $z/L=1$ is the free-end tip. The FWMCs were printed such that they could be pressed through a $12~\text{mm}\times 12~\text{mm}$ hole located in the floor of the working section. The square hole was such that the cylinder and the hole made an interference fit. The hole was located 150 mm from the end of the contraction. The aspect ratio of each FWMC could be lowered by reducing the length of the cylinder exposed to the flow. Once an aspect ratio had been selected, the cylinder was clamped outside of the wind tunnel to prevent any movement. It should be noted that because each of the two printed FWMC models had a fixed number of pressure taps, the number of pressure taps exposed to the flow changed with variation in aspect ratio.

Table 2.

The spanwise location of the pressure taps on the cylinder side faces.

The Reynolds number for experiments in the 18WT was slightly lower than AWT at $Re_{W}=1.1\times 10^{4}$ owing to a slightly lower velocity ( $U_{\infty }=15~\text{m}~\text{s}^{-1}$ ) and slightly higher cylinder width ( $W=12$  mm). According to Bourgeois et al. (Reference Bourgeois, Sattari and Martinuzzi2011), the fixed separation point on a two-dimensional (2-D) square cylinder makes the wake dynamics of the ensuing vortex street largely independent of Reynolds number. Bourgeois et al. (Reference Bourgeois, Sattari and Martinuzzi2011) also shows that the Strouhal number of vortex shedding for square FWMCs with $L/W=4$ remains constant between $Re_{W}=0.5\times 10^{4}$ to $2\times 10^{4}$ . It is thus reasonable to assume that the vortex shedding process at $Re_{W}=1.1\times 10^{4}$ and $Re_{W}=1.4\times 10^{4}$ will be similar for the test cases studied here. Hence, the surface pressure measurements performed in the 18WT at a lower Reynolds number should be indicative of the processes occurring in the AWT at higher Reynolds numbers.

In the 18WT surface pressure measurements, the aspect ratios presented in this paper are: $L/W=1.5$ , 4, 9.9 and 12.5. Note that these values are slightly different than the aspect ratios for AWT and LES, due to the different test rig set-up. Nonetheless, these aspect ratios were selected to be as close as possible to the those selected for the LES.

Static and fluctuating pressures were measured using a 16 channel Scanivalve digital sensor array (DSA), model number 3217/16 Px (Serial No. 10136). Each channel has a temperature compensated piezoresistive pressure sensor and measures differential pressure relative to a common reference pressure port. The module also contains a 16-bit ADC and pressures are recorded at a maximum rate of 500 samples/second/channel. The differential input pressure range of the DSA is $\pm 2.5~\text{kPa}$ .

In this study, each pressure port in the FWMC was connected to a channel of the DSA via PVC tubing with an inner diameter of 1.53 mm and length of 600 mm. The reference pressure port was connected to a static wall pressure tap on the roof of the 18WT test section. According to the DSA calibration sheet, static readings of each channel of the DSA is accurate to within $\pm 0.09\,\%$ of the output. The dynamic response of the pressure tubing system was estimated using the approach of Holman (Reference Holman2010). The natural frequency of the system is calculated to be approximately $\unicode[STIX]{x1D714}_{n}=1300$  Hz while the damping ratio is approximately $h_{damp}=0.063$ . Bendat & Piersol (Reference Bendat and Piersol2010) recommend that for systems with low damping, the frequencies of interest must be less than 20 % of the natural frequency. For the current system, this corresponds to a usable frequency range of less than 260 Hz. For the surface pressure experiments conducted in this study, the frequencies of interest where vortex shedding occurs, lie between 120 to 240 Hz depending on the aspect ratio. Thus, the system is suitable for dynamic pressure measurements in this case. The experimental error of the amplitude of the pressure measured by the Scanivalve is approximately $\pm$ 3.5 % full scale across all frequencies.