1 Introduction

Predicting global climate change requires accurate estimations of the air-sea momentum transfer, heat, and CO2 flux. The air-sea momentum exchange in global climate models is generally expressed using the drag coefficient for a wind speed at 10 m above the sea surface (Suzuki et al., 2022). Some researchers (Charnock, 1955; Large & Pound, 1981; Smith, 1980; Takagaki et al., 2020, 2022; Yelland & Taylor, 1996) have estimated the drag coefficient from the friction velocity and wind speed; however, adequate models for the drag coefficient have not yet been developed. To estimate the exact drag coefficient, both the wind speed and friction velocity over the sea surface need to be measured accurately.

Ships, buoys, and meteorological towers are typically used to measure wind speed over the ocean. Measuring wind speed on ships can be difficult. With regard to the exposure of wind measuring instruments, the disturbance of the airflow produced by the superstructure of ships must be considered because the obstruction of the airflow by the hull of the ship is not negligible (Suzuki et al., 2019). In addition, the pitching and rolling motions of a ship cause a measurement error owing to the difference between the wind direction and the direction of the ship’s oscillation. In the measurement of wind speed on a floating buoy, the buoy motion affects the accuracy of the wind speed measurement. To measure the exact wind speed, appropriate correction of the observed data considering the buoy motion effects is required. Compared with ships and buoys, meteorological towers are unaffected by pitching and rolling and are capable of long-term observations. However, the disturbance of the air generated by the tower structure affects the wind speed around the measurement position. To obtain high-precision measurements of wind speed, anemometers should be mounted at a suitable position undisturbed by tower structure.

Meteorological towers typically consist of open-lattice or cylindrical structure. In open-lattice towers, anemometers are commonly mounted on top of a lattice or at multiple heights on an instrumented boom. If anemometers are not adequately away from the tower, the wind speed recorded by the anemometers might include the flow distortion caused by the tower structure itself. Moses and Daubek (1961) measured the wind speed using anemometers mounted on a lattice-type tower, and a substantial reduction in the wind speed was observed when the anemometers were located downwind of the tower. For certain wind directions, the wind speeds measured by the anemometer were overestimated, and wind direction measurements were also affected. The International Electrotechnical Committee (IEC, 2005) provides an appropriate arrangement of anemometers on the metrological masts. These recommendations based on free-field wind speed analysis by two-dimensional Navier–Stokes computation and actuator disc theory specify that wind speed decreases upwind of the tower, increases around the tower, and decreases downwind in the wake of the tower. Fabre et al. (2014) implemented a numerical simulation for a triangular-lattice meteorological mast and showed that the IEC recommendation overestimated the flow distortion zone generated by the lattice mast. In addition, their simulation results suggest much shorter boom lengths to place the anemometers.

Meteorological tower with a cylindrical structure is also common. Orlando et al. (2011) investigated the effect of tower shadowing on anemometer measurements mounted on solid cylindrical towers through wind tunnel experiments and found a significant wind speed deficit (up to 35% velocity reduction) when the anemometer was located in the wake of the tower. Kondo and Naito (1972) investigated the pattern of the scalar velocity from field observations of a circular cylinder and the 1/12-scale model of a real marine tower (Hiratsuka observation tower). The tower is a cylindrical structure with a diameter of 2 m and height of 21.6 m, and the control room at the top part of the tower is a nearly circular cylinder with a diameter of 7.6 m. Reductions in the wind speed were observed at the windward side of the tower, and the high-velocity region on the transverse side was dragged to leeward. Recently, Suzuki et al. (2017) implemented a numerical simulation for the flow around the Hiratsuka observation tower and showed that the locations unaffected by the tower were at 8 m above the top of the tower.

In both the open-lattice and cylindrical towers, previous research has gradually provided clarity on the recommended positions for estimating the wind speed; however, the effect of the tower structure on the Reynolds shear stress remains unclear. The uncertainty in the Reynolds shear stress affects the accuracy of the drag coefficient.

In the present study, a large-eddy simulation (LES) is implemented for the flow around a cylindrical observation tower, and the effect of the tower structure on the wind speed, Reynolds shear stress, and drag coefficient is analyzed. In addition, the appropriate position enabling accurate estimation of the drag coefficient is investigated from the LES results.

2 Computational setup

Figure 1 shows a schematic diagram of the computational domain in the main region with dimensions of 1000 m × 500 m × 1500 m for the x × y × z grid. The streamwise, spanwise, and vertical directions are the x-, y-, and z-axes, respectively, and the origin of the coordinate axis is at the bottom surface at the center of the tower. The center of the Hiratsuka observation tower is located 500 m downwind of the inlet boundary of the computational domain. The height of the tower is 21.6 m (= H), the control room at the upper part of the tower is a circular cylinder with a diameter of 7.6 m (= R), and the passageway around the control room is 1.78 m wide. A fully developed turbulent boundary layer flow developed in the driver region was imposed at the inlet of the main computational domain. The inflow turbulent boundary layer was generated using a recycling-rescaling method proposed by Lund et al. (1998). The computational domain in the driver region of 4000 m × 500 m × 1500 m for the x × y × z grid. The recycling station is 3000 m downwind of the domain inlet. The boundary layer thickness is 500 m and the free velocity is set as 5 m/s. The governing equations and numerical scheme are the same as those used for the main region. Figure 2 shows the vertical distributions of the mean streamwise velocity component \(<\overline{U }>\) and the standard deviations of the fluctuations in the streamwise velocity component urms at the main domain inlet. These statistics were normalized by the freestream velocity Uref (= 5.0 m/s). The power index of the power law for the mean velocity profile is 1/7, which is approximately equal to the value over the sea (1/10). The values of \({u}_{rms}/{U}_{ref}\) near the surface exceed 0.1, indicating fully developed turbulent flow.

Fig. 1
figure 1

Computational domain

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Fig. 2
figure 2

Vertical distributions of the mean streamwise velocity component \(<\overline{U }>\), and the standard deviations of the fluctuations in the streamwise velocity component, urms at the main domain inlet

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Four numerical simulations were implemented in the main region. The computational mesh comprises a hex grid, and a uniform grid spacing of the hex grid with dx = dy = dz = 0.5 m was used in the areas of \(-10\le x \le 10\text{ m}, -10\le y \le 10\text{ m}, z \le 32.5\text{ m}\). The grid was geometrically stretched away from the area toward the upper, lower, and both side boundaries of the computational domain. In Run 0, the tower was not represented. Then, three runs with the tower were implemented to examine the influence of the computational grid size on the mean wind velocity and turbulent intensities. In Run 1, the same grid system was used as the one used in Run 0, with the exceptions that the mesh was snapped to the surface of the tower, and three additional layers of hexahedral cells aligned to the tower surface were introduced, as shown in Fig. 3a. In Run 2, the computational mesh sizes were approximately half of those in Run 1 only around the tower (Fig. 3b). In Run 3, the mesh refinement was performed by halving the cell sizes in each direction (dx = dy = dz = 0.25 m) for \(-10\le x \le 10\text{ m}, -10\le y \le 10\text{ m}, z \le 32.5\text{ m}\), and three additional layers of hexahedral cells aligned to the tower surface were introduced (Fig. 3c).

Fig. 3
figure 3

Computational grids for a Run 1, b Run 2, and c Run 3

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The filtered continuity, momentum, and mass conservation equations are as follows,

$$\frac{\partial \overline{{U}_{i}}}{\partial {x}_{i}}=0,$$
(1)
$$\frac{\partial \overline{{U}_{i}}}{\partial t}+\frac{\partial \overline{{U}_{j}} \, \overline{{U}_{j}}}{\partial {x}_{j}}=-\frac{1}{\rho }\frac{\partial \overline{P}}{\partial {x}_{i}}+\frac{\partial }{\partial {x}_{j}}\left\{\left(\nu +{\nu }_{t}\right)\left(\frac{\partial \overline{{U}_{i}}}{\partial {x}_{j}}+\frac{\partial \overline{{U}_{j}}}{\partial {x}_{i}}\right)\right\},$$
(2)

where an overbar denotes a filtered value, Ui is a velocity component, P is the pressure, \(\rho\) is the density, and \(\nu\) is the kinetic viscosity of air (= 1.5 × 10−5 m2/s). The subgrid eddy viscosity \({\nu}_{t}\) is modeled using the standard Smagorinsky model with a Smagorinsky constant of 0.1 (Deardorff, 1970). The governing equations were solved directly using the PIMPLE solver in an open-source code package (OpenFOAM 2.4) which uses a finite-volume method. This solver was also used in the LES studies conducted by Michioka et al., (2016, 2017, 2018, 2019, 2023) and Michioka (2018, 2022).

No-slip boundary conditions were applied to the bottom and tower surfaces, and slip boundary conditions were imposed on the velocity components at the upper boundary. Under slip boundary conditions, the velocity normal to the free-slip wall is zero. To eliminate the effects of the side wall, periodic boundary conditions were imposed on the velocity components in the spanwise direction. The standard log-law-based wall functions are used to model the wall stress at the bottom and tower surfaces.

A second-order central scheme was used in the space. A first-order Euler implicit temporal discretization was used for the time derivative term, and a time step of 2.0 × 10−3 s was used. The pressure-implicit with the splitting of operators (PISO) algorithm was used for the resolution of the governing equations (Issa, 1986).

3 Results

3.1 Velocity statistics

Figure 4 shows the vertical distributions of the mean streamwise velocity component \(<\overline{U }>\), the standard deviations of the fluctuations in the streamwise velocity component, urms, and the Reynolds shear stress, < uw > at x/R =  − 4/3, − 1, 0, 1, and 4/3 in Runs 1, 2, and 3. The mean streamwise velocities in all runs are quite similar for x/R < 0, but after x/R = 1, the mean streamwise velocities in Run 1 are smaller than those in Runs 2 and 3. This is due to the fact that the size of the recirculation flow generated near the circular cylinder is overestimated in Run 1. The overestimation results in a difference in both urms and < uw > for x/R > 1. In addition, the difference in the mean streamwise velocity between Runs 1 and 2 indicates that the grid resolution in the immediate vicinity of the circular cylinder affects the recirculation flow. Because the values of urms and < uw > do not include the subgrid-scale contributions, the magnitude of these values in Run 2 is slightly smaller than those in Run 3 for x/R \(\ge 0\). The grid resolution affects the flow behavior near the structure, but it does not significantly affect the flow statistics for \(z-H\ge 1.5\text{ m}\). Therefore, in the present study, the effect of the tower on the velocity statistics was investigated without the flow statistics for \(z-H<1.5\text{ m}\) immediately above the tower (\(-1\le x/R\le 1\)), and the flow behavior is discussed using the results from Run 3.

Fig. 4
figure 4

Vertical distributions of a the mean streamwise velocity component \(<\overline{U }>\), b the standard deviations of the fluctuations in the streamwise velocity component, urms, and c the Reynolds shear stress, < uw > at x/R =  − 4/3, − 1, 0, 1, and 4/3 in Runs 1, 2, and 3

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Figure 5 shows the contour of the mean streamwise velocity and the Reynolds shear stress for the x–z cross-section. Kawamura et al.(1984) conducted flow visualization experiments around a finite circular cylinder on a flat plate and showed that a prominent mean recirculation flow was generated on the free-end surface. Gao et al. (2018) implemented Reynolds-averaged Navier–Stokes (RANS) simulation for the flow around a finite circular cylinder with two free ends and showed that the flow separates the end surface at the leading edge and attracts a wall-attached backflow from the rearward portion. Irrespective of the number of free ends, a recirculation flow is generated at the free end of a finite cylinder. As shown in Fig. 5, flow separation occurs at the leading edge of the circular cylinder, and a recirculation flow is generated above the cylinder surface. Above the circular cylinder, except for the immediate vicinity of the cylinder surface, the mean streamwise velocities become larger than that upwind of the tower. The Reynolds shear stress is negative upwind of the tower, but it changes to positive above the upwind part of the circular cylinder. A positive value indicates inverse momentum transfer. In the latter half of the surface, the sign of the Reynolds shear stress becomes negative and the magnitude increases compared to that upwind of the tower because of the steep velocity gradients as shown in Fig. 4a, indicating that the Reynolds shear stress in this part is produced by the vortices generated on the circular cylinder and the separated flow downstream of the cylinder.

Fig. 5
figure 5

Mean streamwise velocity and the Reynolds shear stress for x–z cross-section

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3.2 Velocity statistics at the anemometer locations

Two 3-axis sonic anemometers (with two horizontal components and one vertical component; Wind Master II, Gill) were installed in the Hiratsuka Tower. The first anemometer (A) was mounted on a handrail close to the west-southwest side edge of the circular cylinder at approximately 3.71 m from the center at z – H = 1.75 m, and the second anemometer (B) was mounted on a pole positioned on the southeast side at approximately 3.13 m from the center at z – H = 3.75 m as shown in Fig. 6 (Suzuki et al., 2017). The accuracy of the sonic anemometers was less than 1.5% for the wind velocity.

Fig. 6
figure 6

Location of the two sonic anemometers on the tower

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Figure 7 shows the streamwise distributions of the scalar velocity (\({U}_{S}=\sqrt{<\overline{U}{>}^{2}+<\overline{V}{>}^{2}+<\overline{W}{>}^{2}}\)) and the Reynolds shear stresses at z – H = 1.75 m and 3.75 m, whose heights correspond to the installation height of the two anemometers. At both heights, the scalar velocity slightly decreases just upwind of the tower and increases above the tower owing to flow separation at the edge of the circular cylinder. The scalar velocity has a peak above the tower and then returns to the value of the upwind scalar velocity. From x =  − 8 m, the magnitude of the Reynolds shear stress gradually decreases and reaches a minimum above the circular cylinder. The Reynolds shear stress at z – H = 3.75 m gradually returns to the upwind value, whereas the magnitude of the Reynolds shear stress at z – H = 1.75 m dramatically increases on the latter half of the tower because of the steep vertical velocity gradient as shown in Fig. 4. Thus, the scalar velocity and Reynolds shear stress at both heights were affected by the tower.

Fig. 7
figure 7

Streamwise distributions of the scalar velocity and the Reynolds shear stress

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To investigate the effect of the tower on the scalar velocity and Reynolds shear stress for each of the 16 wind directions, 16 LESs are required, in which the orientation of the tower is changed with respect to the wind direction. This results in a high computational cost. Because the top part of the Hiratsuka tower is a circular cylinder, both the anemometers are repositioned relative to the wind direction instead of rotating the tower in the present simulation, as shown in Fig. 6b. Figure 8 shows the scalar velocity and Reynolds shear stress at positions A and B for each of the 16 wind directions. Here, \({U}_{S0}\) and \(<uw{>}_{0}\) represent the scalar velocity and Reynolds shear stress, respectively, at the same heights of positions A and B in Run 0. To quantify the comparison, the normalized mean bias (NB) and the normalized root-mean-squared error (NRMSE) were estimated:

$$NB=\frac{\sum_{i=1}^{n}({{E}_{i}-P}_{i})}{\sum_{i=1}^{n}({P}_{i})},$$
(3)
$$NRMSE=\frac{\sqrt{\sum_{i=1}^{n}{\left({E}_{i}-{P}_{i}\right)}^{2}}}{\sum_{i=1}^{n}{P}_{i}},$$
(4)

where n (= 16) is the number of the wind directions, Ei is the variable for the case with the tower, and Pi is the variable for the case without the tower. In addition, to estimate the effect of the wind direction on the statistics, the normalized root-mean-squared error toward the averaged values (NRMSEWD) was calculated.

Fig. 8
figure 8

Scalar velocity and the Reynolds shear stress for each of 16 wind directions at the positions A and B

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$$NRMS{E}_{WD}=\frac{\sqrt{\sum_{i=1}^{n}{\left({E}_{i}-\sum_{i=1}^{n}{E}_{i}\right)}^{2}}}{\sum_{i=1}^{n}{E}_{i}}.$$
(5)

Table 1 shows the NB, NRMSE, and NRMSEWD for the scalar velocity, Reynolds shear stress, and drag coefficient. The NB for the scalar velocity is 0.148 at position A and 0.075 at position B, and the scalar velocity is overestimated for all wind directions at both positions A and B because of the flow separation at the edge of the circular cylinder. The maximum value of \({U}_{S}/{U}_{S0}\) is 1.18 for SSE (\(\theta \approx {90}^{\text{o}}\)) at position A and 1.08 for WSW (\(\theta \approx {-112.5}^{\text{o}}\)) at position B. Here, \(\theta\) is the angle of the location of the anemometer with respect to the wind direction. The minimum values of \({U}_{S}/{U}_{S0}\) are 1.08 for WSW (\(\theta \approx {0}^{\text{o}}\)) at position A and 1.04 for SE (\(\theta \approx {0}^{\text{o}}\)) at position B. Figure 9 shows the contour of the scalar velocity and Reynolds shear stress at the heights of both positions A and B. The patterns of the scalar velocity are quite similar to the field observations by Kondo and Naito (1972) for the 1/12-scale model of the tower on the bare soil surface. They showed that the wind speed decreases on the windward side of the tower, and high-velocity regions are observed on the transverse side. Thus, the wind velocity at the transverse side of the tower was mainly affected by the flow separation.

Table 1 Normalized mean bias (NB), normalized root-mean squared error (NMSE), and the normalized root-mean squared error toward the averaged values (NMSEWD) for the scalar velocity (Us), the Reynolds shear stress (\(\it <\text{uw}>\)) and the drag coefficient (\(\it {\text{C}}_{\text{D}}\))
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Fig. 9
figure 9

The contour of the scalar velocity and the Reynolds shear stress at both positions A and B

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At position A, the NB and NRMSEWD for the Reynolds shear stress are − 0.278 and 0.118, respectively, and the Reynolds shear stress at position A is sensitive to the wind direction. The maximum and minimum values of \(<{u}{w}> /<{u}{w}{>}_{0}\) are 1.577 for ENE (\(\theta \approx {180}^{\text{o}}\)) and 0.006 for WSW (\(\theta \approx {0}^{\text{o}}\)), respectively. At the upwind of the tower, the sign of the Reynolds shear stress changes from negative to positive owing to flow separation, as shown in Fig. 5. Since the position A for WSW is around the point where the sign changes, the Reynolds shear stress becomes quite small. At position B, the maximum and minimum values of \(<{u}{w}> /<{u}{w}{>}_{0}\) are 0.700 for NE (\(\theta \approx {90}^{\text{o}}\)) and 0.621 for SSE (\(\theta \approx {25}^{\text{o}}\)), respectively. Because the NRMSEWD for the Reynolds shear stress at position B is much smaller than that at position A, the Reynolds shear stress at position B is less sensitive to the wind direction. However, the NB for the Reynolds shear stress is − 0.336, which shows that the Reynolds shear stress was underestimated.

The wind stress is generally expressed using the drag coefficient CD for a wind speed at a height of 10 m (\({z}_{10}\)) above the sea surface, U10 as follows.

$$\tau =\rho {C}_{D}{U}_{10}^{2},$$
(6)

where \(\rho\) is the air density. Alternatively, the wind stress is expressed using the friction velocity \({u}_{*}\) as,

$$\tau =\rho {u}_{*}^{2},$$
(7)

The values of u* are estimated using the following expressions,

$$u_\ast=\sqrt{-<{u}{w}>}\,\mathrm{at}\,z=z_p,$$
(8)

where \({z}_{p}\) is the height at positions A or B. The magnitude of \(<{v}{w}>\) in the present simulation becomes larger near the tower, but the values of \(<{v}{w}>\) are too small to change the magnitude and direction of the vertical shear stress in an atmospheric boundary layer. To minimize the influence of the tower on \({u}_{*}\), \({u}_{*}\) is based solely on \(<{u}{w}>\). The wind velocity profile over rough boundary is expressed as

$$U(z)=\frac{{u}_{*}}{\kappa }\text{ln}\left(\frac{z}{{z}_{0}}\right),$$
(9)

where \(\kappa\)(= 0.4) is the Von Karman constant and z0 is the roughness length (Takagaki et al., 2022). The value of z0 is estimated using \({z}_{p}\) and \(U\left({z}_{p}\right).\) From Eqs. (6), (7), and (9), the drag coefficient is expressed as

$${C}_{D}=\frac{{u}_{*}^{2}}{{U}_{10}^{2}}=\frac{{\kappa }^{2}}{{\left(\text{ln}\left(\frac{{z}_{10}}{{z}_{0}}\right)\right)}^{2}} .$$
(10)

Figure 10 shows the drag coefficient for each of the 16 wind directions at positions A and B. Here, \({C}_{D0}\) represents the drag coefficient at the same heights of positions A and B in Run 0. The NB for the drag coefficient is − 0.500 at position A and − 0.527 at position B, and the drag coefficient is underestimated for most of the wind directions at both positions A and B. At position A, the minimum and maximum values of \({C}_{D}/{C}_{D0}\) are 0.004 for WSW (\(\theta \approx {0}^{\text{o}}\)) and 1.193 for ENE (\(\theta \approx {180}^{\text{o}}\)), respectively. Because both positions correspond to the positions where the magnitude of Reynolds shear stress has the minimum and maximum values, the drag coefficient is strongly linked to the Reynolds shear stress. At position B, the NRMSEWD for the drag coefficient is quite small, and thus the drag coefficient is less sensitive to the wind direction, but the drag coefficient estimated at position B becomes almost half the value irrespective of the wind direction. When comparing the two locations, position B is the better one for measuring the CD because it is less sensitive to the wind direction.

Fig. 10
figure 10

Drag coefficient for each of 16 wind directions at the positions A and B

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3.3 Appropriate position for estimating the drag coefficient using one anemometer

As described in Section 3.2, the scalar velocity and Reynolds shear stress at the present positions are affected by the tower, and it is difficult to estimate the drag coefficient accurately. In this section, the appropriate position for measuring the scalar velocity and Reynolds shear stress is determined using an anemometer with the velocity statistics obtained by LES. Figure 11 shows the installation positions considered in the present study. The installation positions are the most upwind position for the north wind at 1/3R, 2/3R, R, 4/3R, 5/3R, and 2R, and the height is varied from z − H = 1.5 m to z − H = 15 m at equal intervals of 0.5 m.

Fig. 11
figure 11

Installation position considering in the present study

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Figure 12 shows the vertical distribution of NB, NRMSE, and NRMSEWD for the scalar velocity, Reynolds shear stress, and drag coefficient. The NB for the scalar velocity is positive irrespective of the installation position because the flow separation above the circular cylinder increases the wind velocity. As the height increases, the NB and NRMSE become smaller and approach the constant value of 0.02. It should be noted that the NB and NRMSE do not become zero at z – H > 10 m because the wind velocity far away from the tower is not the same as the wind velocity without the tower (Run 0). The difference is attributed to computational error, implying that the scalar velocity obtained through LES has a margin of error of approximately 2%. When the height of the installation position is located at over 10 m above the circular cylinder (z – H > 10 m), the scalar velocity is not strongly affected by the circular cylinder irrespective of the installation position. The velocity is approximately the same as that at 8 m through RANS simulation (Suzuki et al., 2017). As the horizontal positions are away from the circular cylinder, the NB and NRMSE decrease, whereas the NRMSEWD tends to increase. This indicates that the scalar velocity outside the tower (4/3R, 5/3R, and 2R) is less affected by the circular cylinder, but it is sensitive to the wind direction. When the installation position is located upwind of the tower, the scalar velocity is not strongly affected by the circular cylinder. However, it is affected by the circular cylinder when the installation position is located downwind of the tower.

Fig. 12
figure 12

Vertical distributions of NB, NRMSE, and NRMSEWD for the scalar velocity, the Reynolds shear stress, and the drag coefficient

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The NB for the Reynolds shear stress is negative near the circular cylinder, and the values approach zero as the height increases. At z – H = 1.5 m, the magnitude of the NB for the Reynolds shear stress is small. As shown in Fig. 5b, the Reynolds shear stress in the downwind region of the circular cylinder is generated by the circular cylinder, and the values are incidentally close to the values without the tower. For z – H > 10 m as well, the Reynolds shear stress is not strongly affected by the circular cylinder. As the installation position is away from the circular cylinder, the NB and NRMSE for the Reynolds shear stress decrease. In addition, the NB and NRMSE for the drag coefficient are similar to those for the Reynolds shear stress. The magnitude of NB at any location is smaller than 0.15 for z – H > 7 m, and the magnitudes are smaller than 0.1 for z – H > 10 m. If the sonic anemometer is installed at r = 2R, the height z – H = 1.5 m also seems to be the best position, but the Reynolds shear stress at the downwind part of the circular cylinder is strongly affected by the circular cylinder, as shown in Fig. 5b. It is therefore difficult to justify the appropriate position at the lower height when r = 2R.

As discussed above, when one anemometer is used to measure the drag coefficient, a higher location of z – H > 10 m is desirable irrespective of the sonic location.

3.4 Appropriate positions for estimating the drag coefficient using two anemometers

In this section, the appropriate positions for measuring the drag coefficient using two anemometers are investigated. Figure 13 shows the installation positions using the two anemometers for two cases. In both cases, anemometer 1 was mounted at the most upwind position for the north wind at 5/3R and 2R. Because the flow above the circular cylinder is strongly affected by the circular cylinder, the locations at 1/3R, 2/3R, R, and 4/3R were not investigated. The height is varied from z–H = 1.5 m to z–H = 15 m at equal intervals of 0.5 m. In Case 1, the anemometer 2 was mounted at the most downwind position for the north wind, as shown in Fig. 13a. To exclude the position in the wake region of the tower, the flow data at anemometer 1 were used for wind directions from west to east, and those at anemometer 2 were used for the other wind directions. In Case 2, to avoid the deceleration of the wind speed by the tower, anemometer 2 was mounted on the east side of the tower, as shown in Fig. 13b. To avoid the wind direction blowing from the front and rear sides with respect to the tower, the flow data at anemometer 1 were used for the wind directions from west-southwest to northwest and from northeast to east-southeast, and those at anemometer 2 were used for the other wind directions.

Fig. 13
figure 13

Installation positions using two sonic anemometers

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Figure 14 shows the vertical distribution of NB, NRMSE, and NRMSEWD for the scalar velocity, Reynolds shear stress, and drag coefficient. When the anemometers are mounted for z – H > 5 m, the values of these statistics do not have an apparent difference in all cases and are almost equivalent to those using one anemometer. For z – H < 5 m, these statistics are affected by the installation positions. Since all the three statistics for the scalar velocity in Case 2 become larger toward the top surface, the circular cylinder strongly affects the scalar velocity. In Case 1, the magnitude of NB for the scalar velocity is less than 0.015 irrespective of the height, and its magnitude is smaller than that in Case 2, but the NRMSE and NRMSEWD for the two cases are comparable. This implies that the scalar velocity in Case 1 depends on the wind direction. In contrast to the scalar velocity, the Reynolds shear stress in Case 2 is less affected by the circular cylinder. The NB, NRMSE, and NRMSEWD for both the Reynolds shear stress and the drag coefficient in Case 2 are smaller than those in Case 1 for z – H < 5 m. To obtain these values, the position in Case 2 is appreciated. Comparing the two horizontal locations (r = 5/3R, and 2R), the effect of the tower on the drag coefficient becomes smaller as the horizontal location is further away from the circular cylinder.

Fig. 14
figure 14

Vertical distributions of NB, NRMSE, and NRMSEWD for the scalar velocity, the Reynolds shear stress, and the drag coefficient

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To clarify which wind direction reduces the accuracy of these statistics, the scalar velocity, Reynolds shear stress, and drag coefficient are obtained for each of the 16 wind directions at r = 2R and z – H = 4 m, as shown in Fig. 15. When the wind direction is north or south, the selected anemometer in Case 1 is located upwind of the tower (\(\theta \approx {0}^{\text{o}}\)), and the scalar velocity is underestimated. Under easterly or westerly winds, the selected anemometer in Case 1 is located at the lateral side of the tower (\(\theta \approx {90}^{\text{o}}\)), and the scalar velocity is slightly overestimated. Thus, the scalar velocity in Case 1 depends on the wind direction. In Case 2, the normalized scalar velocities are over 1.0 for all directions and they are less affected by the variation in the wind direction. As shown in Fig. 15b, the Reynolds shear stress in Case 1 decreases when the wind directions are north and south. Under these wind directions, the selected anemometer is located upwind of the tower (\(\theta \approx {0}^{\text{o}}\)). The Reynolds shear stress decreases in the upwind area and is less affected by the tower on the lateral side of the tower, as shown in Fig. 9b. This means that the upwind area is unsuitable for the measurement of Reynolds shear stresses. The NRMSE and NRMSEWD for the Reynolds shear stress are related to those of the drag coefficient. The decrease in the magnitude of the Reynolds stress, as observed in north and south winds, directly reduces the drag coefficients. In Case 2, the magnitudes of NB, NRMSE, and NRMSEWD for the drag coefficient are smaller than 0.15, and the magnitudes in Case 2 are smaller than those in Case 1. Therefore, the location of the two anemometers in Case 2 is suitable for measuring the drag coefficient.

Fig. 15
figure 15

The scalar velocity, the Reynolds shear stress, and the drag coefficient for each of 16 wind directions at r/R = 2

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The accuracy of the drag coefficient did not show dramatic improvement when using two anemometers compared to that when using one anemometer. It should be noted that when the anemometer is located downwind of the tower, the Reynolds shear stress is generated by the circular cylinder, as shown in Fig. 5b, and the values are not generated by an atmospheric boundary layer over the sea. It is therefore difficult to accurately measure the Reynolds shear stresses using one anemometer when the anemometer is located near the tower. In the present study, the location of the anemometers was investigated for \(r/R\le 2\), and the locations of the anemometers at \(r/R=2\) in Case 2 were the best positions. If the anemometers can be installed for \(r/R>2\), the accuracy of the drag coefficient can be improved.

4 Conclusions

An LES was implemented for the flow around a cylindrical observation tower, and the effects of the tower structure on the wind speed and Reynolds shear stress were investigated. In addition, the appropriate position for estimating the drag coefficient was investigated from the LES results.

The grid resolution near the circular cylinder at the top of the tower affected the streamwise mean velocities and the standard deviation of the fluctuations in the streamwise velocity component. However, the effect was insignificant above 1.5 m from the top surface of the circular cylinder. The effect of the tower on the velocity statistics at 1.5 m above the circular cylinder was investigated. The mean streamwise velocities accelerate above the tower, except near the circular cylinder, because flow separation occurs at the leading edge of the circular cylinder. The Reynolds shear stresses change from negative to positive within the recirculation zone generated near the top of the circular cylinder and return to a negative value on the latter half of the surface because of the steep velocity gradients near the circular cylinder. To accurately estimate the drag coefficient, an accurate measurement of the Reynolds stress is required. The streamwise velocities and drag coefficient below 10 m above the circular cylinder were affected by the circular cylinder.

The appropriate position using one anemometer was investigated for the installation positions from the center of the circular cylinder to twice its radius. When the installation height was over 10 m above the circular cylinder, the scalar velocity was not strongly affected by the circular cylinder, irrespective of the installation position. The scalar velocity outside the tower was less affected by the circular cylinder, but it was sensitive to the wind direction because the scalar velocity downwind of the tower was strongly affected by the circular cylinder. The Reynolds shear stresses in the downwind region of the circular cylinder were generated by the circular cylinder. This indicates that the downwind location near the circular cylinder is not appropriate for measuring the Reynolds shear stress. When one anemometer is used to measure the drag coefficient, a location at a height of over 10 m above the circular cylinder is desirable.

Appropriate positions to measure the drag coefficient using two anemometers were investigated for two cases. The magnitudes of NB, NRMSE, and NRMSEWD for the drag coefficient were smaller than 0.1, for the case in which one anemometer was located in a northerly direction and the other was in the easterly direction. Although the effects of the tower on the drag coefficient are not entirely removed, the use of two anemometers is a promising approach for measuring the drag coefficient in a tower.

This study is the first step in identifying better locations for measuring the drag coefficient in a meteorological tower using the Hiratsuka Tower as a case study, and further work is required to determine the optimal location of a sonic anemometer in different observation towers.