4.1.1. Surface pressure and friction

The behaviour of the forces exerted on the bodies, which are contributed by the pressure and viscous shear stress, is an important issue associated with the interaction between the body and turbulent wake. An illustration of the force behaviour of the rod–airfoil configuration is given by the distribution of the mean pressure on the surface of the cylinder and airfoil, shown in figure 10(a,b), respectively. As shown in figure 10(a), the airfoil behind the cylinder leads to the pressure increase in the base range of the cylinder. When the distance between the airfoil and the cylinder continues to increase, the influence of the airfoil is weakened correspondingly on the surface pressure distributions of the cylinder, and gradually converges to the distribution on the single cylinder. Moreover, in figure 10(a), the present mean pressure distributions on the surface of the cylinder are compared with the experimental results of Apelt & West (Reference Apelt and West1975), and those of Szepessy & Bearman (Reference Szepessy and Bearman1992) to complement the verification of the present simulation. The present distributions from the simulation of the single cylinder demonstrate that the angular position (70°) for the mean pressure minimum is very close to that of the experimental measurement of Szepessy & Bearman (Reference Szepessy and Bearman1992). Although the present mean pressure distributions in the base region provide an unsatisfactory comparison with the experimental data, a significant dispersion of the experimental results is observed.

Figure 10.

The calculated mean pressure distributions compared with the experimental data and LES results: (a) on the surface of the cylinder, (b) on the surface of the airfoil. Here, the experimental data from Apelt & West (Reference Apelt and West1975) and Szepessy & Bearman (Reference Szepessy and Bearman1992) are measured for the circular cylinder with $\mathit{Re}_{d}=4\times 10^{4}$ ; the experimental data from ECL (Jacob et al. Reference Jacob, Boudet, Casalino and Michard2005) and the LES results of Giret et al. (Reference Giret, Sengissen, Moreau, Sanjosé and Jouhaud2015) are obtained for the rod–airfoil configuration with $L/d=10$ .

As shown in figure 10, the pressure distributions with $L/d=2$ are obviously different from the others. This behaviour is related to the flow phenomena of the separated shear layers from the upstream cylinder impinging onto the leading edge of the airfoil. The main separated vortices impact the leading edge of the airfoil directly, and von Kármán vortex shedding is suppressed, which will be discussed in detail in § 4.2. The increasing $L$ , followed by the weakening interaction between the turbulent wake and airfoil, is also consistent with the pressure distributions on the airfoil surface tending to be identical. In figure 10(b), the present mean pressure distributions on the airfoil surface are also compared with the experimental data from ECL (Jacob et al. Reference Jacob, Boudet, Casalino and Michard2005) and the LES results of Giret et al. (Reference Giret, Sengissen, Moreau, Sanjosé and Jouhaud2015). It can be seen that the calculated solutions for the rod–airfoil configuration with $L/d=10$ are very close to the LES results of Giret et al. (Reference Giret, Sengissen, Moreau, Sanjosé and Jouhaud2015), however, a large gap between the numerical results displayed and the experimental data can be seen. According to the discussion of Giret et al. (Reference Giret, Sengissen, Moreau, Sanjosé and Jouhaud2015), the experimental dip is most likely caused by a curvature discontinuity in the mockup leading-edge region.

Continuing the investigation of the skin friction, figure 11(a) shows the present skin friction distributions on the cylinder, together with the experimental data of Achenbach (Reference Achenbach1968) and the LES results of Giret et al. (Reference Giret, Sengissen, Moreau, Sanjosé and Jouhaud2015). It shows that the skin friction increases continuously from the stagnation point until the maximum value is obtained at the location near ${\it\theta}=45^{\circ }$ , this trend then reverses, and it finally achieves a zero or negligible negative value near ${\it\theta}=80^{\circ }$ where the flow is separated. The positive skin friction occurs obviously in the attached flow region and drops quickly at the mean separation location. Moreover, the absolute value of the skin friction observed in the attached flow region is usually much higher than that in the separated region. Although the calculated skin friction maximum of the single cylinder is underpredicted, with a relative error of 11 %. It can be seen that the present simulations for the single cylinder and the rod–airfoil configuration with $L/d\geqslant 4$ provide similar shapes and maximum positions, as well as mean separation positions, in comparison with the experimental data of Achenbach (Reference Achenbach1968) and the LES results of Giret et al. (Reference Giret, Sengissen, Moreau, Sanjosé and Jouhaud2015). For the case of $L/d=2$ , however, the peak skin friction is lower in the present simulations, furthermore, the mean separation in this case is earlier than in the other cases.

Figure 11.

Mean skin friction distributions on the surface of bodies: (a) cylinder and (b) airfoil. Here, the calculated distributions on the surface of the cylinder are compared with the experimental data of Achenbach (Reference Achenbach1968) for the circular cylinder with $\mathit{Re}_{d}=10^{5}$ and the LES results of Giret et al. (Reference Giret, Sengissen, Moreau, Sanjosé and Jouhaud2015) for the rod–airfoil configuration with $L/d=10$ ; moreover, the mean skin friction in this figure is non-dimensional according to the formulation given by Achenbach (Reference Achenbach1968).

To clearly demonstrate the effect of the distance $L$ on the skin friction of the airfoil, figure 11(b) shows the mean friction distributions on the airfoil surface. It is not surprising to see that the distributions are highly distinguishable at the leading edge with $L/d=2$ . A similar phenomenon is also identified in the surface pressure distributions shown in figure 10(b). The distinguishable distributions, i.e. the negative mean friction values observed in figure 11(b) are caused by the large separated region in the rod–airfoil configuration with $L/d=2$ , which is in figure 14. The main vortex separated from the cylinder with negative $z$ -vorticity entrains the boundary layer and causes early separation at the leading edge with $L/d=2$ . With increment of the distance $L$ , the skin friction tends to be indistinguishable. When $L/d\geqslant 4$ , the skin friction reaches its maximum value at the leading edge, and then drops quickly, furthermore, the maximum skin friction increases as the distance $L$ is enlarged. This behaviour may correspond to the phenomenon of the influence of the turbulent wake on the airfoil fading gradually as the distance $L$ increases.

Table 1.

The mean and fluctuating integral force for the single cylinder and rod–airfoil system. (a) The calculated mean and fluctuating integral force on the single cylinder compared with experimental data and other numerical results. Here, $\langle C_{D}\rangle _{t}$ represents the time-averaged drag coefficient, $\langle C_{L}\rangle _{t}$ denotes the time-averaged lift coefficient and $C_{Lrms}$ is the r.m.s. of the lift coefficient, $\mathit{St}$ of the present study is calculated from the figure 13. (b) The force behaviour of the rod–airfoil configuration. Here, $\langle C_{D\text{-}rod}\rangle _{t}$ represents the time-averaged drag coefficient on the cylinder, as well as $\langle C_{D\text{-}tol}\rangle _{t}$ , $\langle C_{L\text{-}tol}\rangle _{t}$ and $C_{Lrms\text{-}tol}$ are the time-averaged drag coefficient, the time-averaged lift coefficient and the r.m.s. of the lift coefficient for the whole configuration, respectively.

4.1.2. Mean and fluctuating integral force

To assess quantitatively the integral force on the rod–airfoil system, table 1(a) lists the calculated mean and fluctuating integral force on the single cylinder compared with the experimental data and other numerical results, while table 1(b) lists the force behaviour of the rod–airfoil system. There is, in general, a considerable scattering in the results of the force on the cylinder surface, the present solutions however are in good agreement with the experimental data of Szepessy & Bearman (Reference Szepessy and Bearman1992). As the Reynolds number considered here is $O(10^{5})$ , the pressure force plays a dominant role in the total forces on the cylinder, additionally, the pressure increases in the base region as the distance $L$ is shortened (see figure 10 a), therefore we can see in table 1(b) that the mean drag of the cylinder falls as the airfoil approaches the cylinder. The drag varys on the whole rod–airfoil system, as shown in table 1(b). The regulation of the drag varying of the is same as that of the cylinder in the configuration when the distance $L$ increases.

Under the circumstances, it is obvious that the drag of cylinder plays a dominant role in the total drag of the rod–airfoil configuration.

As shown in table 1(b), the mean lift of the rod–airfoil system almost vanishes, however, the instability feature of this configuration can be investigated by the r.m.s. value of the lift fluctuation $C_{Lrms\text{-}tol}$ . When $L/d=2$ , the value of $C_{Lrms\text{-}tol}$ is the lowest of all the five rod–airfoil configurations, and even less than that of a single cylinder, see table 1(a). This demonstrates that the lift fluctuation can be suppressed if the distance $L$ is very short. According to the analysis above, the total forces are dominated by the surface pressure and it is natural to investigate the lift fluctuation by examining the pressure oscillation on the body surface. Figure 12 shows the distributions of r.m.s. value of the pressure fluctuation on the cylinder and airfoil. The pressure fluctuation on the rod–airfoil configuration with $L/d=2$ is suppressed, which is consistent with the least lift fluctuation shown in table 1(b). Moreover, it is interesting to find that a hump is shown in the distributions of the pressure fluctuations on the airfoil between $x/c=0.4$ and 0.8 (see figure 12 b). A similar shape can be observed from the typical vortex-blade interactions in counter-rotating open rotors (e.g. Roger & Carazo Reference Roger and Carazo2010; Carazo, Roger & Omais Reference Carazo, Roger and Omais2011).

Figure 12.

Root-mean-square value of pressure fluctuation on the surface of bodies: (a) cylinder and (b) airfoil.

The PSD of the time-dependent lift coefficient of the rod–airfoil and cylinder are shown in figure 13. As exhibited in this figure, the smallest peak value corresponding to the least lift fluctuation given in table 1(b) is obtained by the configuration with $L/d=2$ . Usually, the force fluctuation on a cylinder is associated with vortex shedding in the wake (e.g. Oertel & Affiliation Reference Oertel and Affiliation1990; Owen & Bearman Reference Owen and Bearman2001), and the power spectral density of the cylinder in figure 13 can be used to identify the frequency of vortex shedding (Xu et al. Reference Xu, Chen and Lu2010). The primary frequency corresponding to the highest peak for the single cylinder is approximately 0.2. For the configurations with $L/d\geqslant 4$ , it is interesting to find that the peak value in figure 13 periodically increases and decreases with the distance $L$ increasing. This behaviour is most likely caused by the fluid resonant oscillations, which will be discussed in § 4.3.2. Furthermore, the $\mathit{St}$ corresponding to the peak value increases with increments in the distance $L$ . This feature can be reasonably related to the higher-base-pressure distribution on the cylinder with the shorter distance $L$ in figure 10(a), and the vortex shedding suppression in the interaction region of $L/d=2$ , which will be discussed in § 4.2.

Figure 13.

Profiles of power spectral density of time-dependent lift coefficient on the rod–airfoil and cylinder. Here, the time-dependent lift coefficients are collected every 10 time steps in the last 20 000 time steps for all the cases.

Based on the above analysis, the force on the surface of the rod–airfoil configuration is affected by the distance $L$ . In particular, the force is obviously different when $L/d=2$ . The differences involving surface pressure and friction lead to the reduction of mean and fluctuating forces for the flow past the rod–airfoil configuration. The distance $L$ may also have an effect on the flow structures and we thus pay more attention to the relevant flow characteristics in the next section.

4.2.1. Mean flow structures in the interference region

We investigate the mean flow structures by first examining the vorticity consisting of the streamwise ${\it\omega}_{x}$ , vertical ${\it\omega}_{y}$ and spanwise ${\it\omega}_{z}$ vorticity. The time-averaged ${\it\omega}_{y}$ becomes zero because of the periodic boundary condition applied in the spanwise direction. The increasing vertical gradients of the streamwise $u$ and spanwise $w$ velocity, together with the gradients of vertical velocity $v$ in the horizontal $(x,z)$ -plane, give rise to a streamwise ${\it\omega}_{x}$ and spanwise ${\it\omega}_{z}$ vorticity. It is important to note that these two quantities will appear in the instantaneous turbulent flow field, but in the present cases, the time-averaged quantity ${\it\omega}_{x}$ is negligible compared with that of ${\it\omega}_{z}$ .

Time-averaged spanwise vorticity ${\it\omega}_{z\text{-}mean}$ together with the corresponding streamlines projected onto the $(x,y)$ -plane in the interference region for $L/d=2$ and $L/d=4$ are shown in figure 14. Clearly, two counter-rotating vortices evolve along the axis of the cylinder. The mean streamlines are symmetric about the horizontal center-line, whereas the ${\it\omega}_{z\text{-}mean}$ contours exhibit an antisymmetry. Visualizations of the flow field in the interference region have revealed that at short enough distances, the Kármán-street type of shedding is replaced by a non-shedding mode (see e.g. Munekata et al. Reference Munekata, Kawahara, Udo, Yoshikawa and Ohba2006), with two counter-rotating vortices with opposite circulation connecting to each side of the leading edge of the airfoil. In the non-shedding mode, the negative mean friction values are observed at the leading edge (see figure 11 b) due to the main vortex separated from the cylinder with a negative $z$ -vorticity connecting to the airfoil. Kármán-street shedding is also observed in the other three cases. Two patterns which will be further discussed in § 4.2.2 are then outlined: Kármán-street shedding for the long distances and Kármán-street suppressing for the short distances.

Figure 14.

Time-averaged spanwise vorticity and streamlines in the interaction region taken from the flow field with $U_{\propto }=72~\text{m}~\text{s}^{-1}$ , $Re_{d}=4.8\times 10^{4}$ , $L/d=2$ (a) and $L/d=4$ (b).

Due to the fact that counter-rotating vortices are connected to the airfoil, the distributions of ${\it\omega}_{z\text{-}mean}$ for $L/d=2$ are evidently different from those of other cases at the leading edge of the airfoil. This also explains why the distributions of pressure and friction, shown in figures 10(b) and 11(b) respectively, are obviously distinguishable on the surface of the airfoil for $L/d=2$ . When the cylinder is set away from the airfoil, the counter-rotating vortices with opposite circulation emanate from each side of the cylinder and do not touch the airfoil. The interaction is relatively weak between the turbulent wake and airfoil, reasonably related to the fact that the distributions of mean friction tend to be identical on the surface of the cylinder (see figure 11 a) and the leading edge of the airfoil (see figure 11 b).

Figure 15(a) shows the contours of time-averaged streamwise velocity $u_{m}$ in the interference region, where the counter-rotating vortices connecting with the leading edge of airfoil lead to negligible or negative values of $u_{m}$ for the case of $L/d=2$ ; the flow behind the counter-rotating vortices is accelerated until it impinges on the leading edge of airfoil for the cases with $L/d\geqslant 4$ (the other three cases are not shown in figure 15 a). The phenomena exhibited in this figure can be explained by the fact that in this interaction region, the flow is the result of two tendencies. The influence of the airfoil on the steady flow tends to accelerate it, whilst the counter-rotating vortices are connected with the leading edge of airfoil due to the approaching cylinder, which tends to slow the flow down.

Figure 15.

Contours of time-averaged velocity taken from the flow field with $U_{\propto }=72~\text{m}~\text{s}^{-1}$ , $Re_{d}=4.8\times 10^{4}$ , $L/d=2$ (a,c) and $L/d=4$ (b,d). (a,b) Streamwise component $u_{m}$ ; (c,d) vertical component $v_{m}$ , here solid lines denote positive values and dashed lines negative ones.

Figure 16.

The r.m.s. value of streamwise velocity taken from the flow field with $U_{\propto }=72~\text{m}~\text{s}^{-1}$ , $Re_{d}=4.8\times 10^{4}$ , $L/d=2$ (a) and $L/d=4$ (b).

In figure 15(b), we plot the contours of time-averaged vertical velocity $v_{m}$ for $L/d=2$ and $L/d=4$ . It can be seen that the positive $v_{m}$ presents below the leading edge of the airfoil with $L/d=2$ , and the negative one is observed above. However, the contrary distribution is shown in this region with $L/d=4$ . When the incoming flow approaches the upper part of the leading edge of the airfoil with $L/d=2$ , it experiences the downdraft depicted by the contours of $v_{m}$ (see also the streamlines in figure 14), which produces negative horizontal gradients in $v_{m}$ in this region, thereby giving rise to the negative ${\it\omega}_{z\text{-}mean}$ . The positive one however is generated symmetrically about the center-line, as illustrated by the contours of ${\it\omega}_{z\text{-}mean}$ in figure 14.

Finally, the contours of r.m.s. value of streamwise velocity $u_{rms}$ are plotted in figure 16 for $L/d=2$ and $L/d=4$ . It can be seen that the $u_{rms}$ values of $L/d=2$ behind the cylinder and near the central line $y=0$ are much lower than those of $L/d=4$ . This phenomenon is consistent with the fact that the non-shedding mode is observed in the rod–airfoil configuration with $L/d=2$ , and the negligible or negative $u_{m}$ generated in the interaction region (see figure 15 a) is caused by the main separated vortices connecting to the airfoil (see figure 14) in this case. It is expected that the turbulent fluctuations in the interaction region of $L/d=4$ are much more intensive than those of $L/d=2$ , and the underlying reason will be further discussed in § 4.2.3 for the generation of lower-level turbulent fluctuations in the rod–airfoil configuration with $L/d=2$ .

Figure 17.

Vortical structures by iso-surface of the $Q$ -criterion ( $Q=1000$ ). The instantaneous vertical velocity $v$ at the symmetry $x,z$ -plane  $(y=0)$ is included to highlight the alternating pattern of the main vortices. (a) Taken from the flow field with $U_{\propto }=72~\text{m}~\text{s}^{-1}$ , $Re_{d}=4.8\times 10^{4}$ , $L/d=2$ ; (b) $L/d=4$ ; (c) $L/d=6$ ; (d) $L/d=8$ ; (e) $L/d=10$ .

4.2.2. Instantaneous flow structures and interference flow patterns

Figure 17 shows instantaneous snapshots of the flow field depicted by iso-surface of the $Q$ criterion (Jeong & Hussain Reference Jeong and Hussain1995)

where ${\it\Omega}_{ij}=(u_{i,j}-u_{j,i})/2$ and $\unicode[STIX]{x1D61A}_{ij}=(u_{i,j}+u_{j,i})/2$ are the antisymmetric and the symmetric components of the curl of the velocity, respectively. A positive value of $Q$ represents the regions in which the rotation exceeds the strain. It should be recalled that the criterion (4.1) is only applied to the resolved scales obtained by the HILES and that the vortical structures could be different if the whole flow field is considered. It can be seen that the vortices in the rod wake impinge onto the airfoil, and then partly split at the leading edge. The flow topology exhibits two different patterns in the interaction region depicted by figure 17. The first one observed in this region with $L/d=2$ includes only the primary vortical structures, roughly oriented along the axis of the cylinder. These primary structures originate from the rolling-up of the detached shear layers at each side of the cylinder. The other one presented by the rod–airfoil configuration with $L/d=4,6,8$ and 10 comprises both the primary vortical structures and also secondary vortical structures, exhibiting a wide range of scales and orientations. These secondary structures come from the Kármán-street vortex shedding. For rod–airfoil configurations, Munekata et al. (Reference Munekata, Kawahara, Udo, Yoshikawa and Ohba2006) also observed two flow patterns. In the first, the Kármán-street is not generated for a short enough distance $L$ , whereas in the other, the Kármán vortex is steadily generated for a long distance $L$ . Based on this observation, the authors suggested that changing the distance $L$ led to the flow pattern altering. In figure 17, the instantaneous vertical velocity $v$ , highlighting the alternating pattern of the main vortices, indicates that two flow patterns exist in the present study, supporting the observation made by Munekata et al. (Reference Munekata, Kawahara, Udo, Yoshikawa and Ohba2006) who observed the pattern dependence of the interference flow on the distance $L$ .

Figure 18.

Juxtapositional view of the unsteady spanwise vorticity ${\it\omega}_{z}$ of vertical $(x,y)$ -planes taken from the flow field of five rod–airfoil configurations at the central slice $z=0.015~\text{m}$ of the computational domain.

Based on the discussion above, a possible categorization for the flow patterns associated with the interference region is given as follow: (i) for $L/d\leqq 2$ , there is the Kármán-street suppressing mode. The eddies propagate downstream without being shed in this mode. In the present study, the Kármán-street is not generated by the free shear layers separated from the upstream cylinder in the interference region with $L/d=2$ ; (ii) for $L/d\geqslant 4$ , Kármán-street occurs. This is classified as the Kármán-street shedding mode and occurs in the present study at $L/d=4,6,8$ and 10. As the Reynolds number considered here is large enough, the eddies arise from the rolling-up of the shear layers at an almost fixed position and the vortex shedding then occurs when one eddy becomes strong enough to cut the opposite eddy from the supply of the circulation from the shear layer (Zdravkovich Reference Zdravkovich1997), furthermore, the distance $L$ is too long to suppress the vortex shedding in the interaction region; (iii) for $2<L/d<4$ , the altering mode may exist. This flow pattern is not observed in the present study, however, it has been reported by Munekata et al. (Reference Munekata, Kawahara, Udo, Yoshikawa and Ohba2006) that the two flow patterns, i.e. Kármán-street shedding mode and Kármán-street suppressing mode alternate at aperiodic time intervals when $L/d=3.1$ .

A different illustration of the instantaneous flow structures in the interaction region is given by the snapshots of the spanwise vorticity ${\it\omega}_{z}$ in the $(x,y)$ -plane, shown in figure 18. This component of the vorticity complements the $Q$ -iso-surfaces by enabling the visualization of the shear layers that detach from both sides of the cylinder and their further roll-up to form primary vortices, as well as the turbulent wake impinging onto the airfoil and then partly splitting at the leading edge. In the latter process, the turbulent wake containing Kármán-street shedding is clearly seen in all the vertical planes in figure 18, with the exception for the case of $L/d=2$ , suggesting that with reducing distance $L$ , the shear layer is less susceptible to rolling up.

In figure 18, the four vorticity planes with $L/d\geqslant 4$ exhibit in the interaction region a clear Kármán-street shedding. Due to entrainment of free stream fluid, this turbulent wake expands as it evolves downstream until it hits the leading edge of the airfoil. For $L/d=2$ , however, the Kármán-street shedding in the interaction region seems to be suppressed due to the short distance $L$ . These observations suggest that vortex shedding in the present cases is influenced by the distance $L$ between the two bodies. It is obvious that the wake from the cylinder upstream is blocked by the airfoil downstream. The blocking effect is minor for long distances between the two bodies, but becomes more evident at short distances, which may lead to the base pressure rising abnormally on the cylinder upstream (see figure 10 a). It is natural to assume that the blocking effect takes main responsibility for the Kármán-street suppression in the rod–airfoil configuration.

4.2.3. Turbulent fluctuations and their suppression

The turbulent fluctuations around a body are closely associated with the fluctuating forces exerted on it (Wu, Lu & Zhuang Reference Wu, Lu and Zhuang2007). To describe the turbulent fluctuations around the rod–airfoil configurations, the iso-contours of spanwise averaged mean value of turbulent-kinetic-energy (TKE), i.e. $\text{TKE}_{m}$ , are shown in figure 19. The $\text{TKE}_{m}$ is relatively smaller in the region around the airfoil with $L/d=2$ . This feature leads to correspondingly lower fluctuating surface pressure (see figure 12 b). Moreover, the suppression of the turbulent fluctuations is also observed in the interaction region.

Figure 19.

The iso-contours of spanwise averaged mean value of TKE. (a) taken from the flow field with $L/d=2$ ; (b) $L/d=4$ ; (c) $L/d=6$ ; (d) $L/d=8$ ; (e) $L/d=10$ .

Additionally, the profiles of $\text{TKE}_{m}$ in the interaction region along the line of symmetry of the cylinder are shown in figure 20. From the distribution behind the cylinder with $L/d=2$ , the $\text{TKE}_{m}$ is significantly lower in the range of approximately $x/c<0.1$ , and increases gradually because of the development of primary vortical structures, reaching a peak value in the range of approximately $x/c=0.15$ . It then falls as the wake propagates downstream, finally undergoing a severe jump and gaining a maximal value since the vortices in turbulent wake impinge onto the airfoil and partly split at the leading edge. It is expected in figure 20 that the reduction and jump are also present in the distributions of $\text{TKE}_{m}$ from the flow fields with $L/d=4,6,8,10$ . However, it may be noted that in these distributions, the maximal value of $\text{TKE}_{m}$ is not observed after the significant jump, but is exhibited at the first peak. This difference indicates that the suppression is associated with the primary vortical structures.

Figure 20.

Profiles of $\text{TKE}_{m}$ in the interaction region along the symmetry line of the cylinder.

To understand the characteristics relating to the suppression of turbulent fluctuations of the rod–airfoil configuration with $L/d=2$ , we analyse the underlying reason for the generation of higher-level turbulent fluctuations in the other four cases. As shown in figure 17, for the configuration with $L/d=4,6,8,10$ , the vortex shedding occurs when one eddy becomes strong enough to cut the opposite eddy off from the supply of the circulation from the shear layer, and the formed vortex is released into the wake. The flow structures appear to be stronger in the vortex formation region which corresponds approximately to the location of the primary vortex. This observation is consistent with numerical simulation of a turbulent wake behind a curved circular cylinder (Gallardo et al. Reference Gallardo, Andersson and Pettersen2014), and correspondingly, the maximal value of turbulent fluctuations is present at this location (see figure 20). Further downstream, but before the airfoil, the size of the structures tends to increase as their concentration decreases, this behaviour is related to the varying of $\text{TKE}_{m}$ in figure 20 which goes down until the blocking effect is met, also consistent with the report of Mansy, Yang & Williams (Reference Mansy, Yang and Williams1994) that the amplification of the larger scales and attenuation of the smaller scales occur as the wake propagates downstream. The shedding wake containing massive turbulent fluctuations requires impingement onto the airfoil to create the second peak of turbulent fluctuations. The larger before the impact the $\text{TKE}_{m}$ is, the higher the second peak is, as shown in figure 20. It is clear that the primary vortex shedding and the shedding wake impingement onto the airfoil play a dominant role in the production of turbulent fluctuations. However, for the case of $L/d=2$ , primary vortex shedding is suppressed, correspondingly, the intensity of impingement is weakened, resulting in a significant suppression of the turbulent fluctuations.

4.3.1. Acoustic analysis

From the preceding analysis of the flow structures and turbulent fluctuations, it is reasonably well understood that the distance $L$ plays an important role in the flow features relevant to the flow pattern altering and the turbulent fluctuations being suppressed. The flow features are closely associated with the noise radiation. We further analyse the influence of distance $L$ on the noise radiation.

For rod–airfoil configurations with $L/d=6$ and $L/d=8$ , figure 21 shows the comparison between the overall SPLs (OASPLs) calculated directly from the HILES solutions and provided by the FW-H analogy to complement the verification with respect to the acoustics outcome. The OASPLs on the line $y=0.4~\text{m}$ are chosen to ensure the mesh resolution is fine enough for the HILES with the HDCS-E8T7 capturing the acoustic noise. Although small differences are observed between the two results, the same varying trend of the OASPLs is predicted by the two methods. In the following, we will apply the FW-H analogy to discuss the acoustic behaviour in the far-field.

Figure 21.

Comparison between the OASPLs calculated directly from the HILES solutions and provided by the FW-H analogy on the line $y=0.4~\text{m}$ for the configuration with $L/d=6$ and $L/d=8$ .

Figure 22.

Directivities for the OASPL of the rod–airfoil configurations at $1.85~\text{m}$ . Here, the location for the observation angle 0 is $x=1.85~\text{m},y=0$ .

Figure 23.

The SPL spectra at the location $(x=0.68~\text{m},y=1.74~\text{m})$ for all the five rod–airfoil configurations.

In order to show the dominant noise radiation directions of different configurations, figure 22 plots the directivities of OASPL at $1.85~\text{m}$ for all five rod–airfoil configurations. It is clear that the dominant noise radiation directions are between the observation angles of 60° and 90°. For a further investigation of the noise radiation in the dominant radiation direction, figure 23 shows the SPL spectra at the location $(x=0.68~\text{m},y=1.74~\text{m})$ , i.e. the observation angle 70° for all the five rod–airfoil configurations. Although a relatively narrow range around the peak frequency of the noise spectra is presented in this figure, it is clearly observed that the noise spectra is separated into two different types. The first one is a broadband noise without a manifestly distinguishable peak, which is radiated by the rod–airfoil configuration with $L/d=2$ . The second type is present when $L/d\geqslant 4$ . With a dominant peak, the broadband noise and the tonal noise are exhibited simultaneously in this type. Moreover, the levels of the peaks in the second type are much higher than that in the first, which is not surprising since the noise radiation in the far-field is closely associated with the flow features in the near-field. As reported by Boudet et al. (Reference Boudet, Grosjean and Jacob2005), the interaction between the wake from the cylinder upstream and the airfoil downstream is mainly responsibile for the noise radiation of the rod–airfoil configuration. In particular, the spectrum of a single cylinder may be approximately 10 dB lower than that of the rod–airfoil system (Jacob et al. Reference Jacob, Boudet, Casalino and Michard2005), which is also supported by the observation of Munekata et al. (Reference Munekata, Kawahara, Udo, Yoshikawa and Ohba2006). Based on the discussion of flow features above, vortex shedding is suppressed for a very short distance $L/d=2$ , leading to significant suppression of the pressure fluctuations and the turbulent fluctuations, as well as the weakening of the impingement intensity. Thus, the lower levels of noise are observed for the rod–airfoil configuration with $L/d=2$ .

To exhibit the influence of the distance $L$ on the peak frequency of the radiative noise, a peak Strouhal number, $\mathit{St}_{p}$ , is analysed as performed by Munekata et al. (Reference Munekata, Kawahara, Udo, Yoshikawa and Ohba2006) and is defined as $\mathit{St}_{p}=f_{p}d/U_{\infty }$ , where $f_{p}$ represent the peak frequency given in figure 23. Quantitatively, compared with the primary frequency of the vortex shedding behind a single cylinder, represented by the $\mathit{St}$ which is calculated from the PSD of the time-dependent lift coefficient (see figure 13), the $\mathit{St}_{p}$ s of all the rod–airfoil configurations are linked to form a line are and plotted in figure 24 with a exception of the case of $L/d=2$ , in which there is no manifestly distinguishable peak frequency. The $\mathit{St}_{p}$ increases with the distance $L$ increasing, finally approaching the primary frequency of the single cylinder, supporting the observation made by Munekata et al. (Reference Munekata, Kawahara, Udo, Yoshikawa and Ohba2006) who observed the peak frequency dependence of the noise radiation on $L/d$ . This feature may be reasonably related to the higher-base-pressure distribution on the cylinder with shorter distance $L$ in figure 10(a), and vortex shedding suppressing in the interaction region in figure 17.

Figure 24.

Effect of $L/d$ on $\mathit{St}$ based on peak frequency.

4.3.2. Fluid resonant oscillation

The fluid resonant oscillation within the flow interaction between the turbulent wake from a cylinder and the bodies have been observed in previous studies (e.g. Mochizuki et al. Reference Mochizuki, Kiya, Suzuki and Arai1994). This feature is usually associated with the impingement of the turbulent wake onto the surface of the body and a feedback loop system (e.g. Mochizuki et al. Reference Mochizuki, Kiya, Suzuki and Arai1994; Fitzpatrick Reference Fitzpatrick2003; Munekata et al. Reference Munekata, Kawahara, Udo, Yoshikawa and Ohba2006). The fluid resonant oscillation means that the vortex generation in the rod–airfoil configuration is promoted by the acoustic field induced from the wake interference between the shedding vortex behind the cylinder upstream and the airfoil downstream (Munekata et al. Reference Munekata, Kawahara, Udo, Yoshikawa and Ohba2006). In the quantitative observation made by Munekata et al. (Reference Munekata, Kawahara, Udo, Yoshikawa and Ohba2006), it was found that the peak SPL depended on $L/d$ , and increases and decreases periodically, relating to the fluid resonant oscillation. Mochizuki et al. (Reference Mochizuki, Kiya, Suzuki and Arai1994) also observed the fluid resonant oscillation in the configuration of two circular cylinders in tandem, and these authors proposed a model to describe the feedback loop system as illustrated in figure 25. According to this feedback model (Mochizuki et al. Reference Mochizuki, Kiya, Suzuki and Arai1994), the feedback frequency $f_{f}$ , defined by the inverse of time taken to feed back for one loop, and is described by the following equation:

where $L_{f}$ represents the characteristic length of feedback loop, $U_{c}$ is the convective velocity of shedding vortex and $c$ is the speed of sound. If the constraint $f_{p}=nf_{f}$ is satisfied, the effect of fluid resonant oscillation becomes stronger. Here, $n$ is a positive integer.

Figure 25.

Schematic diagram of feedback system for fluid resonant oscillation.

Figure 26.

The peak SPL and OASPL for the second type of noise spectra.

As shown in figure 23, there are two types of noise spectra due to the varying of the distance $L$ . The first one is caused by the flow features related to vortex shedding suppression in the interaction region. Regarding the second one, although a similar dominant peak is present in this type of noise spectra, the value of the dominant peak varies periodically. To show this varying regularity clearly, figure 26 plots the peak SPL and OASPL for the second type of noise spectra exhibited in figure 23. The periodic increasing and decreasing of the peak SPL and OASPL are clearly exhibited in this figure and are similar to the phenomenon of fluid resonant oscillation observed by Munekata et al. (Reference Munekata, Kawahara, Udo, Yoshikawa and Ohba2006). It should be noted that the regulation of the peak SPL from all the five rod–airfoil configurations in the dominant noise radiation directions, i.e. between an observation angle of 60° and 90° is similar to that at the observation angle of 70°, however, in the other directions, this regulation is not observed.

Based on the above analysis, we suggest that fluid resonant oscillation is observed in the present study. To further describe the fluid resonant oscillation, the feedback model (4.2) is adopted to analyse this phenomenon. The characteristic length $L_{f}$ in (4.2) is reasonably well given by the horizontal distance between the leading edge and the mean separation location on the cylinder surface, corresponding to the resolved skin friction stress vanishing (Munekata et al. Reference Munekata, Kawahara, Udo, Yoshikawa and Ohba2006), and can be determined from the figure 11(a). The values of $L_{f}$ are listed in table 2 for $L/d=4,6,8$ and 10, respectively.

Table 2.

The corresponding parameters of the feedback model for fluid resonant oscillation.

We here deal with the convective velocity of the shedding vortex, which is used in the feedback model of the fluid resonant oscillation. Two-point cross-correlation of the unsteady pressure can be used to quantitatively determine the propagation speed of the pressure disturbances along a given path (Xu et al. Reference Xu, Chen and Lu2010). A covariance coefficient $C_{ij}$ for two pressure signals $p_{i}(t)$ and $p_{j}(t)$ with time delay ${\it\tau}$ can be defined as

where $\langle ~\rangle _{t}$ denotes time average. Within the interaction region, the cross-correlation analysis is conducted for probes A and B shown in figure 25 and the results are given in figure 27. Positive time delays are obtained, indicating that the pressure disturbances within the interaction region propagate downstream toward the airfoil leading edge. The convective velocity of the shedding vortex $U_{c}$ can then be calculated by dividing the horizontal distances between the neighbouring probes by the time delays between the peaks of the corresponding cross-correlations. Due to the differences of the grid point distributions, the horizontal distances between the probes A and B, i.e. $L_{AB}$ may be different. For $L/d=4$ , $L/d=6$ , $L/d=8$ and $L/d=10$ , the values of $L_{AB}$ are listed in table 2, where the corresponding speed $U_{c}$ and the calculated feedback frequency $f_{f}$ are also presented.

Figure 27.

Cross-correlation of the downstream propagating pressure waves for (a) $L/d=4$ ; (b) $L/d=6$ ; (c) $L/d=8$ and (d) $L/d=10$ .

Figure 28 shows the relation between $f_{p}$ and $f_{f}$ for the four different interaction distances. When $L/d=6$ and 10, the exhibited data are periodically close to the solid lines, which means that the effect of fluid resonant oscillation becomes stronger. This behaviour corresponds to the periodic change of the peak SPL and OASPL shown in figure 26. We recognize the limitations of the simplified analysis of the fluid resonant oscillation based on the feedback model; nevertheless, the results obtained from the model are of help in understanding the physical mechanism of the fluid resonant oscillation involved in this flow.

Figure 28.

The relation between $f_{p}$ and $f_{f}$ for the four different interaction distances. Here, the solid lines mean the equation of resonant condition.