3.1 Flow regimes

The flow regimes are mapped out in figure 5 at $e/D=0.25$ , 0.5, 1 and 2 in $(KC,\unicode[STIX]{x1D6FD})$ -space for $3\leqslant KC\leqslant 12$ and $\unicode[STIX]{x1D6FD}\leqslant 300/KC$ , while a representative flow pattern for each identified regime is given in figure 6. The $KC$ and $\unicode[STIX]{x1D6FD}$ resolutions used to generate figure 5 range from 0.1 to 1 and from 1 to 10 respectively. Fine resolution is employed to identify the regime boundaries.

Figure 5.

Classifications of oscillatory flow regimes around a circular cylinder above a plane boundary at different gap ratios based on 2-D DNS results. The boundary lines are obtained based on the flow identification at discrete data points, which are marked with different symbols and colours along with regime names. The synchronous and asynchronous regimes with regard to the oscillatory flow are distinguished by the bold dashed line (– – – blue). Only the red solid lines (—— red) represent an abrupt change between regimes, while all others represent a gradual transition between adjacent regimes. The upper limit of $Re=300$ for the current research is represented by dot-dashed lines (– ⋅ – ⋅ –). The open triangle marker (▵) on the horizontal axis indicates the corresponding Stokes number where the thickness of undisturbed Stokes boundary layer calculated by (1.5) is identical to the gap size, $e$ .

Figure 6.

Flow patterns represented by streaklines in different flow regimes for oscillatory flow around a circular cylinder above a no-slip plane boundary: (a) regime $\text{A}^{\ast }$ at $(e/D,KC,\unicode[STIX]{x1D6FD})=(2,4,20)$ ; (b) A at (2, 5, 30); (c) $\text{HA}^{\ast }$ at (0.25, 4, 10); (d) HA at (0.25, 10, 10) (see movie 1 available at https://doi.org/10.1017/jfm.2018.164); (e) transitional regime A/GVS at (0.5, 4, 20); (f) GVS at (0.5, 6, 20) (see movie 2); (g) GVS-A at (0.5, 6, 30) (see movie 3); (h) D at (0.5, 4, 50); (i) D-A at (1, 6, 25); (j) E at (2, 6, 30) (see movie 4); (k) $\text{F}^{\prime }$ at (0.25, 11, 20) (see movie 5); (l) F at (2, 10, 20) (see movie 6); ( $m_{i}$ ) I at (0.5, 11, 20) and ( $m_{ii}$ ) I at (2, 11, 20).

A number of distinctive flow regimes are observed, as shown in figures 5 and 6. The naming of flow regimes follows the conventions previously reported in the literature as much as possible. Among these, regimes $\text{A}^{\ast }$ , A, D, E, F are similar to those found in oscillatory flow around an isolated circular cylinder (Tatsuno & Bearman Reference Tatsuno and Bearman1990), and regimes GVS and D-A are similar to those identified for two side-by-side cylinders (Zhao & Cheng Reference Zhao and Cheng2014). Although these flows are not strictly identical to those found previously, the naming convention is retained to avoid creating too many new regime names. The regimes $H\text{A}^{\ast }$ , HA, GVS-A and I are unique for oscillatory flow around a circular cylinder near a plane boundary. Here the prefix ‘H’ stands for ‘half’ because of its spatial similarity to a half of regime $\text{A}^{\ast }$ or A structure for an isolated cylinder. The superscript ‘ $^{\ast }$ ’ indicates no vortex shedding, which is inherited from the definition of regime $\text{A}^{\ast }$ in previous studies and ‘I’ represents ‘irregularity’, where no clear cycle-to-cycle periodicity is observed. A transitional region between regimes $\text{HA}^{\ast }/\text{HA}$ and GVS is marked with A/GVS in the regime map at $e/D=0.25$ and 0.5, because the flow possesses the features of both regimes. In general, an abrupt change is only observed along boundary lines between regime GVS to GVS-A and D to D-A (or E) within the parameter space.

The classification of flow regimes is primarily based on visual observations of instantaneous vorticity contours and the streaklines of the 2-D DNS results, with the aid of information derived from hydrodynamic forces. Each of the flow regimes is characterized by unique vortex shedding features, force characteristics, synchronization and symmetry states. The main criteria used for the classification are summarized in table 1.

Table 1.

The features and the criteria of identification for each flow regime.

A prominent feature is that the flow regimes are highly dependent on $e/D$ . The flow regime map for $e/D=2$ tends to resemble the map of an isolated cylinder shown in figure 1, suggesting the weak influence of the plane boundary at $e/D>2$ . As $e/D$ is reduced, the regime maps change significantly, especially at the lower end of the $\unicode[STIX]{x1D6FD}$ range. The newly identified flow regimes ( $\text{HA}^{\ast }$ , HA, GVS-A and I) are either associated with small $e/D$ or/and low $\unicode[STIX]{x1D6FD}$ values. The typical flow features in each regime will be explained below based on vorticity contours and hydrodynamic forces. In the present study, the inline force coefficient $C_{x}$ and lift coefficient $C_{L}$ are used to plot Lissajous phase diagrams. The force coefficients are defined as $C_{x}=F_{x}/(0.5\unicode[STIX]{x1D70C}DU_{m}^{2})$ and $C_{L}=F_{y}/(0.5\unicode[STIX]{x1D70C}DU_{m}^{2})$ , where $\unicode[STIX]{x1D70C}$ is the fluid density. $F_{x}$ and $F_{y}$ are the inline and transverse forces respectively and are obtained by integrating pressure and viscous forces over the surface of the cylinder. The flow states of symmetry can be identified based on Lissajous phase diagram of $C_{x}-C_{L}$ : (i) a closed loop suggests cycle-to-cycle periodicity; (ii) a horizontal line implies $K_{x}$ symmetry; (iii) a symmetric Lissajous phase diagram about $C_{x}=0$ means $H_{1}$ symmetry; and (iv) a symmetric Lissajous phase diagram about ( $0,0$ ) indicates $H_{2}$ symmetry.

Figure 7.

From top to bottom are ( $i$ ) the instantaneous contours of $\unicode[STIX]{x1D714}_{z}$ between $\unicode[STIX]{x1D714}_{z}=1$ (light colours) and $+1$ (dark ones) with cutoff level $\pm 0.2$ (this applies to other vorticity contours unless otherwise specified), and the phase $\unicode[STIX]{x1D70F}_{0}$ of sampling is indicated by $\bullet$ in the sinusoidal velocity signal; ( $ii$ ) the transient trace of $C_{L}$ (solid black lines) along with scaled free-stream velocity $u^{\ast }=u_{\infty }/5$ (dashed red lines); and ( $iii$ ) Lissajous phase diagram of $C_{x}{-}C_{L}$ . From left to right are (a) regime $\text{A}^{\ast }$ at $(e/D,KC,\unicode[STIX]{x1D6FD})=(2,4,20)$ ; (b) regime A at (2, 5, 30); (c) regime H $\text{A}^{\ast }$ at (0.25, 4, 10); and (d) regime HA at (0.25, 10, 10). Due to the strict $H_{1}$ symmetry, only two snapshots at $\unicode[STIX]{x1D70F}_{0}=0$ and 1/4 are shown in ( $i$ ) and those at $\unicode[STIX]{x1D70F}_{0}=1/2$ and 3/4 can be obtained by (1.3).

3.1.1 Regimes $A^{\ast }$ , A, $HA^{\ast }$ and HA

As shown in figure 7(a,b), the flows in regime $\text{A}^{\ast }$ /A resemble their counterparts around an isolated cylinder (figure 1). Due to the proximity of the plane boundary, there is a slight asymmetry in the flow about the oscillating axis, which gives rise to a non-zero lift in regime A.

The newly reported regimes $\text{HA}^{\ast }$ /HA (figure 7 c,d) only exist for $\unicode[STIX]{x1D6FF}/e>1$ and are similar to the flow observed by Sumer et al. (Reference Sumer, Jensen and Fredsøe1991) at $e/D=0$ and $KC=4$ . Regime $\text{HA}^{\ast }$ is characterized by vortices attached to the upper half of the cylinder surface, without discernible vortex shedding. The streaklines (figure 6 c) from the upper surface are convected in a similar way to the upper half of regime $\text{A}^{\ast }$ (figure 6 a), but the streaklines on the bottom half of the cylinder surface are quite different and form a narrow horizontal strip moving along with the Stokes boundary layer. The particles released from the upper and lower cylinder surfaces do not mix, suggesting a rather weak interaction between the shear layers formed on the cylinder surface. Figure 7(c) for regime $\text{HA}^{\ast }$ flow illustrates a regular and smooth temporal variation of the lift with a positive non-zero mean. The amplitude of $C_{x}$ is larger than its counterpart in regime A/ $\text{A}^{\ast }$ .

Figure 8.

Same as figure 7 but (a) transitional regime A/GVS at $(e/D,KC,\unicode[STIX]{x1D6FD})=(0.5,4,20)$ ; (b) regime GVS at (0.5, 6, 20); and (c) regime GVS-A at (0.5, 6, 30).

The flow features observed in regime HA (figure 7 d) are characterized by the shedding of one vortex from the top side in each half of an oscillation period, while the formation of the vortex on the gap side is suppressed due to the Stokes boundary layer. The streaklines (figure 6 d) show that fluid particles are transported nearly horizontally but with an upwards inclination. The variation of $C_{L}$ (figure 7 d) is strong and the value is almost always positive.

3.1.2 Regimes GVS, GVS-A, D and D-A

Regimes GVS and GVS-A only emerge at $e/D\leqslant 1$ , where the plane boundary has a significant influence on the flow around the cylinder. These flow patterns, along with the transitional regime A/GVS, are illustrated in figure 8. By comparing figure 8(a, b), it is obvious that the gap vortex shedding strengthens with increasing $KC$ . The streakline patterns, shown in figure 6(e,f), also show that the particles are convected horizontally with a slight upwards inclination due to the gap flow.

Regimes GVS and A may coexist due to the influence of wall proximity, forming a combined regime GVS-A. The flow is in regime GVS at the left-hand side in figure 8(c), but remains in regime A at the other side. The flow structure is asymmetric about $x=0$ , which was also observed by Sumer et al. (Reference Sumer, Jensen and Fredsøe1991) and An et al. (Reference An, Cheng and Zhao2010) for $4<KC<7$ and $e/D<1$ .

Figure 8(b,c) compares the shedding and movement of a pair of vortices, A and B (shed from the gap in the previous half-cycle) and a vortex C (attached to the upper cylinder surface) for GVS and GVS-A in a typical oscillation period. After the flow reversal, vortex A is split into two vortices $\text{A}^{\prime }$ and $\text{A}^{\prime \prime }$ . The vortex C is also split into two parts during the acceleration phase with the major part $\text{C}^{\prime }$ flowing through the gap and the minor part $\text{C}^{\prime \prime }$ washing over the upper surface. The major difference between GVS and GVS-A at $\unicode[STIX]{x1D70F}=t/T=100.25$ is that $\text{C}^{\prime }$ is stronger than $\text{C}^{\prime \prime }$ in GVS, opposite to GVS-A. The vortices $\text{A}^{\prime }$ and B remain at the left-hand side while the vortices $\text{A}^{\prime \prime }$ and $\text{C}^{\prime }$ are convected to the right-hand side. For GVS flow, vortices $\text{C}^{\prime }$ and $\text{A}^{\prime \prime }$ in the second half of the cycle are spatio-temporal mirror images of the vortices A and B in the previous half-cycle. Successive vortex pairs are convected away from the gap, resulting in a vortex street at either side of the cylinder, as shown in figure 6(f). On the other hand, the vortices $\text{C}^{\prime }$ and $\text{A}^{\prime \prime }$ roll up at $\unicode[STIX]{x1D70F}=100.5$ for GVS-A, merging with vortex $\text{C}^{\prime \prime }$ , and are washed back towards the cylinder during the flow reversal. The flow in regime GVS-A is still strictly cycle-to-cycle periodic, as can be seen from the closed loop Lissajous phase diagram shown in figure 8(c), but breaks $H_{1}$ symmetry. Two cycles of lift fluctuation are observed in each flow period with two peaks of different magnitudes.

Regime D features V-shaped oblique vortex shedding (figure 9 a) while regime D-A flow exhibits the characteristics of both regimes D and A of an isolated cylinder (figure 9 b). By comparing regimes GVS and D with GVS-A and D-A, two major differences are found: (i) the shed vortices are convected horizontally along the boundary layer in regime GVS/GVS-A while in regime D/D-A they are ejected away with an upwards inclination angle; (ii) the mean lift force in regime GVS/GVS-A is approximately zero and that of regime D/D-A is almost always negative.

Figure 9.

Same as figure 7 but (a) regime D at $(e/D,KC,\unicode[STIX]{x1D6FD})=(0.5,4,50)$ ; and (b) regime A-D at (1, 6, 25).

A fast Fourier transform (FFT) of the time series of the lift coefficient in these four regimes is shown in figure 10. The frequency is normalized by the flow oscillation frequency as $f/f_{\infty }=fT$ . The amplitude at $fT=0$ represents the amplitude of the mean lift force, which is approximately zero for GVS/GVS-A and non-zero for D/D-A. Their primary non-zero frequencies all occur at $fT=2$ frequency as expected, and the extra peaks observed at $fT=1$ for GVS-A and D-A flows suggest $H_{1}$ symmetry breaking.

Figure 10.

Comparison of the spectrum of the lift coefficient in regimes GVS, GVS-A, D, D-A and E at the selected parameters as shown in figures 8, 9 and 11 with a close-up view around $fT=2$ . The amplitudes $(C_{L}^{A})^{\ast }$ have been normalized by the peak value $(C_{L}^{A})_{max}=1.41$ from regime GVS-A at $fT=2$ .

3.1.4 Regimes $F^{\prime }$ and F

In regime $\text{F}^{\prime }$ , a vortex pair is shed horizontally from the upper surface of the cylinder every half-cycle. An example is shown in figure 12(a). Vortex A shown at $\unicode[STIX]{x1D70F}=100$ was developed in the previous flow cycle and forms a vortex pair with the newly generated vortex B at $\unicode[STIX]{x1D70F}=100.25$ . This pair of vortices is shed from the cylinder and a new vortex C begins to develop at the end of the first half-cycle of the flow. Since the vortex shedding and development process in the second half of the flow cycle approximately mirror those observed in the first half of the flow cycle, regime $\text{F}^{\prime }$ generally possess a $H_{1}$ symmetry. Similar flow patterns have also been found in the experimental study of Sumer et al. (Reference Sumer, Jensen and Fredsøe1991) at $e/D=0$ and $KC=10$ . As shown in figure 13(a), the value of $C_{L}$ is almost always positive and is characterized by four peaks every cycle of flow oscillation. The Lissajous phase diagram shows the periodicity and rough $H_{1}$ symmetry of the flow.

Regime F flow around a circular cylinder close to a plane boundary is also characterized by diagonal vortex shedding (figure 12 b), similar to its counterpart for an isolated cylinder. However the shed vortices cannot be convected to the far field due to the plane boundary confinement. The cycle-to-cycle variations of the flow are evident in regime F (figure 13 b). A similar observation has also been made by Scandura et al. (Reference Scandura, Armenio and Foti2009) at $(e/D,KC,\unicode[STIX]{x1D6FD})=(1.5,10,20)$ and (0.625, 10, 20). The dominant frequency component of regime $\text{F}^{\prime }$ is $fT=4$ (figure 14), in contrast to that of regime F at $fT=3$ due to the $H_{1}$ symmetry breaking in regime F.

Figure 11.

(a,c) Instantaneous vorticity contours at different instants and (b,d) the transient trace of $C_{L}$ along with the Lissajous phase diagram; ( $a,b$ ) regime E at $(e/D,KC,\unicode[STIX]{x1D6FD})=(2,6,30)$ and regime E at (2, 6, 40).

Figure 12.

Instantaneous vorticity contours in different flow regimes. The red dot on a scale bar above each contour represents the phase angle of the free oscillatory flow.

Figure 13.

Transient trace of $C_{L}$ (left) and Lissajous phase diagram of $C_{x}{-}C_{L}$ (right) for the same cases as shown in figure 12. (a) Regime $\text{F}^{\prime }$ at $(e/D,KC,\unicode[STIX]{x1D6FD})=(0.25,11,20)$ ; (b) regime F at (2, 10, 20); (c) regime I at = (0.5, 11, 20); and (d) regime I at (2, 11, 20).

Figure 14.

Comparison of the spectrum of the lift coefficient in regimes E, $\text{F}^{\prime }$ , I and F at the selected parameters as shown in figures 12 and 13 with close-up view around $fT=2$ and 3. The amplitudes $(C_{L}^{A})^{\ast }$ have been normalized by the peak value $(C_{L}^{A})_{max}=1.16$ from regime F at $fT=3$ .

The temporal vortex shedding motions of regime $\text{F}^{\prime }$ and F in half a cycle are similar to their counterparts in steady flow past a near-wall cylinder (Rao et al. Reference Rao, Thompson, Leweke and Hourigan2013, Reference Rao, Thompson, Leweke and Hourigan2015). This is possibly because $KC/2$ ( $=U_{m}T/(2D)\sim 5$ ) in the oscillatory flow is close to the inverse of the Strouhal number of the steady flow ( $St=fD/U=D/(UT)\sim 0.2$ ) since a vortex pair is shed every half-cycle for the oscillatory flow and every cycle for the steady flow.

3.1.5 Regime I

A regime I flow features irregular and unpredictable cycle-to-cycle variations. Figure 12(c,d) shows two typical examples of flow patterns in regime I at $e/D=0.5$ and 2 respectively. The irregular variation of the corresponding transient traces of $C_{L}$ indicates the irregularity of both flows (figure 13 c,d). For the first case, it is observed that the pattern of vortex motions is similar to that in regime $\text{F}^{\prime }$ in the cycle prior to $\unicode[STIX]{x1D70F}=120$ . However, at $\unicode[STIX]{x1D70F}=130$ , the flow pattern resembles that in regime HA. Further, at $\unicode[STIX]{x1D70F}=140$ and 150, the flow field is characterized by a vortex pair shed from the gap and a single vortex shed from the upper surface. The dominant frequency component of $C_{L}$ for regime I is $fT=4$ , while multiple peaks occur around $fT=3$ as shown in figure 14. This suggests that the flow irregularly switches among several states.

For the second case (figures 12 d and 13 d), the irregularity of the flow is much stronger, and the spectrum of $C_{L}$ (figure 14) becomes broadbanded. In the first 30 cycles, the flow is a typical regime F and diagonal vortex shedding is aligned in the south-west and north-east directions. However, the flow becomes unstable and large cycle-to-cycle variations of $C_{L}$ are also observed. Although the flow displays the features of regime F sometimes, it also shows irregular transitions, such as those observed at $\unicode[STIX]{x1D70F}=42$ and 68 (figures 12 d and 13 d).

Figure 15.

Classifications of the flow regimes at (a) $KC=3$ on an $e/D$ - $\unicode[STIX]{x1D6FD}$ map and (b) $\unicode[STIX]{x1D6FD}=20$ on an $e/D$ - $KC$ map.

Figure 16.

The streamwise velocity components $u$ at five streamwise locations for different gap ratios at $(KC,\unicode[STIX]{x1D6FD})=(5,20)$ at time $\unicode[STIX]{x1D70F}=100.25$ . The thickness of Stokes boundary layer by (1.5) is indicated at $x/D=10$ .

3.2 Influence of wall proximity

The influence of the plane boundary on flow behaviour is considered in this section. The regime maps shown in figure 5 are re-plotted in figure 15 in the ( $e/D,\unicode[STIX]{x1D6FD}$ )-space at $KC=3$ and in the $(e/D,KC)$ -space at $\unicode[STIX]{x1D6FD}=20$ , with more data from the two small gap ratios $e/D=0.03125$ and 0.125.

The boundaries of regimes $\text{HA}^{\ast }/\text{HA}$ are dependent on both $e/D$ and $\unicode[STIX]{x1D6FD}$ and specifically, depend largely on $\unicode[STIX]{x1D6FF}/e$ , where $\unicode[STIX]{x1D6FF}$ is inversely proportional to the square-root of $\unicode[STIX]{x1D6FD}$ as shown in (1.5) and figure 3. These flows are mostly observed at low $\unicode[STIX]{x1D6FD}$ values and small gap ratios. This is expected because the occurrence of H $\text{A}^{\ast }$ and HA flows is primarily due to the interaction of the shear layer developed on the gap side of the cylinder surface and the Stokes boundary layer developed on the plane boundary. The shear layers developed on the gap side of the cylinder surface and on the plane boundary are of opposite sign and their interactions become stronger as $e/D$ is reduced. This is the primary reason regime H $\text{A}^{\ast }$ and HA flows are observed for $\unicode[STIX]{x1D6FF}/e\geqslant 1$ for the gap ratios investigated. This can be further demonstrated by the velocity profiles shown in figure 16, where the streamwise velocity on the gap side around the cylinder is much smaller than that on the top side on the counterpart for larger gap ratios. Similar to regime HA* and HA, regime $\text{F}^{\prime }$ only occurs at large $\unicode[STIX]{x1D6FF}/e$ where the vortex shedding through the gap is largely suppressed.

Figure 17.

Comparison of the amplitude of the lift coefficient $C_{L}^{A}$ at frequency $fT=2$ by FFT, the mean lift coefficient $\overline{C}_{L}$ , and the difference between $C_{L}^{A}$ and $|\overline{C}_{L}|$ .

Figure 18.

Variation of (a) the drag and (b) the inertia coefficients with $KC$ at a fixed stokes number $\unicode[STIX]{x1D6FD}=20$ . The drag coefficient $C_{D}$ and inertia coefficient $C_{M}$ are determined by the Morison equation $F_{x}(t)=1/2\unicode[STIX]{x1D70C}DC_{D}|u(t)|u_{\infty }+1/4\unicode[STIX]{x1D70C}D^{2}C_{M}du_{\infty }/dt$ , where $\unicode[STIX]{x1D70C}$ is the density of fluid. The least squares method is adopted to determine the coefficients from the inline force over 100 cycles.

As the cylinder is placed further away from the Stokes boundary layer with $\unicode[STIX]{x1D6FF}/e<1$ , the influence of the latter becomes weaker and the gap flow is enhanced due to the plane boundary blockage effect (Anagnostopoulos & Minear Reference Anagnostopoulos and Minear2004). This leads to a stronger shear layer on the gap side of the cylinder surface, as shown by the velocity profile for $e/D=0.5$ in figure 16. This explains why the regimes GVS/GVS-A appear at $\unicode[STIX]{x1D6FF}/e<1$ and $e/D\leqslant 1$ . When the gap ratio is further increased, the flow behaviour asymptotically resembles that of an isolated cylinder.

The flow irregularity increases with $KC$ and $Re$ and is dependent on $e/D$ . At $e/D=0.25$ , almost no transverse vortex develops within the parameter space and the overall flow pattern is regular. The proximity of the plane boundary stabilizes the flow, especially at $KC\leqslant 6$ , because the gap effect is so strong at small gap ratios that it dictates the convection direction of vortices. For intermediate gap ratios, the nonlinear interaction between separated vortices and the Stokes boundary layer gives rise to more irregularity in flow pattern at $KC>6$ , compared to the cases that are regarded as regime F for an isolated cylinder. At $e/D=2$ , the confinement effect of the plane boundary becomes rather weak and irregularity is more attributed to the vortex shedding pattern around the cylinder itself.

The plane boundary also has a significant influence on the lift coefficient, as shown in figure 17. In general, the lift force is positive (away from the wall) and is largest in regime H $\text{A}^{\ast }$ , which occurs at small gap ratios. The mean lift force is approximately zero in regime GVS with relatively strong variation at intermediate gap ratios. The instantaneous lift force vanishes for regime A/ $\text{A}^{\ast }$ flows. A more careful examination of figure 17 reveals that the amplitude of variation of the lift coefficient ( $C_{L}^{A}$ ) is more sensitive to $KC$ , while the mean lift coefficient ( $|\overline{C}_{L}|$ ) is more sensitive to $\unicode[STIX]{x1D6FD}$ . Based on the above observation, an index $C_{L}^{A}-|\overline{C}_{L}|$ is designed to indicate the strength of gap vortex shedding. At a fixed combination of $e/D$ and $KC$ , the vortex shedding becomes stronger with increasing $\unicode[STIX]{x1D6FD}$ . On the other hand, at a fixed $\unicode[STIX]{x1D6FD}$ , the critical $e/D$ for the strongest gap vortex shedding increases with $KC$ . In addition, the drag and inertia coefficients for $\unicode[STIX]{x1D6FD}=20$ are shown in figure 18. Both coefficients increase with decreasing gap ratio, while drag coefficient decreases with increasing $KC$ , with an exception at $e/D=0.03125$ .

Figure 19.

Comparison of the regime map obtained by different initial conditions at $e/D=0.5$ . (a) The regime map based on simulations initialized from zero homogeneous condition (same as the figure given in figure 5 b). (b) The regime map based on simulations initialized by a fully developed flow field from the adjacent lower $\unicode[STIX]{x1D6FD}$ number. That is, for instance at $KC=6$ , the simulation at $\unicode[STIX]{x1D6FD}=30$ was started from an instantaneous flow field at $\unicode[STIX]{x1D6FD}=29$ . (c) The regime map based on simulations initialized by the fully developed flow from the adjacent smaller $KC$ number. The starting point and direction for successive initialization is marked by the dark grey outline and grey arrow, respectively, in (b) and (c).

4.1 The influence of initial conditions

Hysteresis, where the flow develops into different flow regimes with different initial conditions, was found for oscillatory flow around a circular cylinder near a plane boundary. To quantify the hysteresis phenomenon, the regime map presented in figure 5 at $e/D=0.5$ (obtained with zero initial conditions) is reproduced in figure 19(a), by successively initializing the flow with a fully developed flow field obtained either at the adjacent lower $\unicode[STIX]{x1D6FD}$ but constant $KC$ , as shown in figure 19(b), or alternatively, at the adjacent lower $KC$ but a constant $\unicode[STIX]{x1D6FD}$ in figure 19(c). The interval of $\unicode[STIX]{x1D6FD}$ or $KC$ is not a fixed value and is determined by trial and error, with fine resolutions approaching the regime boundaries. The minimum intervals for $KC$ and $\unicode[STIX]{x1D6FD}$ used in the simulations are 0.1 and 1, respectively. To facilitate discussions, the zero and non-zero initial conditions (IC) are referred to as zero IC and non-zero IC, respectively.

The results shown in figure 19 confirm the existence of hysteresis where the locations of regime boundaries between GVS and GVS-A, D and D-A and GVS-A and D-A are not only dependent on initial conditions but also on the route of parameter change. In the following discussion, the hysteresis phenomenon is illustrated by increasing $\unicode[STIX]{x1D6FD}$ at a fixed $KC$ only, although other options in changing the parameters are also available.

Figure 20.

The marginal curve of the $H_{1}$ symmetry at $KC=6$ by 2-D Floquet analysis and 2-D DNS with zero initial condition (IC) and non-zero IC. The red dashed lines denote the gap $\unicode[STIX]{x0394}\unicode[STIX]{x1D6FD}_{cr}$ between the lower critical Stokes $(\unicode[STIX]{x1D6FD}_{cr})_{min}$ and upper critical Stokes number $(\unicode[STIX]{x1D6FD}_{cr})_{max}$ .

Figure 21.

(a) The variation of Floquet multiplier $|\unicode[STIX]{x1D707}|$ for marginal stability of the $H_{1}$ symmetry at $KC=6$ for an isolated cylinder. The critical Stokes number $\unicode[STIX]{x1D6FD}_{cr_{\infty }}=28.3$ is obtained by extrapolating the points with $|\unicode[STIX]{x1D707}|>1$ . (b) Comparison on the marginal curves of the $H_{1}$ symmetry for $0.03125\leqslant e/D\leqslant 2$ at $KC=6$ from the Floquet analysis and from 2-D DNS with non-zero initial condition.

It is observed that the regime boundaries obtained using non-zero IC shift to the right (higher $\unicode[STIX]{x1D6FD}$ ), relative to those obtained using zero IC. For instance, at $KC=6$ , the transition from regime GVS to GVS-A occurs at $\unicode[STIX]{x1D6FD}=24$ with zero IC (figure 19 a), but does not occur until $\unicode[STIX]{x1D6FD}=39.5$ with non-zero IC (figure 19 b). This is attributed to the larger disturbances introduced by zero IC than those by the non-zero IC. For the sets of initial conditions, the base flow in regime GVS with $KC=6$ is unconditionally stable with respect to $H_{1}$ symmetry for $\unicode[STIX]{x1D6FD}<24$ and absolutely unstable for $\unicode[STIX]{x1D6FD}>39.5$ . The results demonstrate the subcritical nature of bifurcations of oscillatory flow around a circular cylinder near a plane boundary.

4.2 The influence of gap ratios and boundary conditions

The influence of $e/D$ on the hysteresis is also investigated through 2-D DNS in figure 20. Over a range of $0.03125\leqslant e/D\leqslant 2.0$ at a fixed $KC$ of 6, two marginal stability curves are obtained by initiating the simulations with non-zero IC and zero IC, respectively. The two marginal stability curves are clearly different over $0.25<e/D<1.0$ . The one obtained using non-zero IC lies above that obtained using zero IC, and they are referred to as the upper and lower marginal stability curves correspondingly. The gap between these marginal curves is referred to as the hysteresis range between $(\unicode[STIX]{x1D6FD}_{cr})_{min}$ and $(\unicode[STIX]{x1D6FD}_{cr})_{max}$ , denoted by $\unicode[STIX]{x0394}(\unicode[STIX]{x1D6FD}_{cr})=(\unicode[STIX]{x1D6FD}_{cr})_{max}-(\unicode[STIX]{x1D6FD}_{cr})_{min}$ .

Figure 21 further explores the physical mechanisms responsible for the two distinct variation trends through Floquet analysis. The time-stepper approach reported by Barkley & Henderson (Reference Barkley and Henderson1996) is employed in the Floquet analysis. The convergence tolerance for the Floquet multiplier is set as $10^{-6}$ . An algorithm for stabilization used to obtain the base flow with enforced $H_{1}$ symmetry is detailed in the appendix. The critical Stokes number $\unicode[STIX]{x1D6FD}_{cr}$ is estimated by extrapolation of the Floquet multipliers $|\unicode[STIX]{x1D707}|$ that are larger than 1.0 to $|\unicode[STIX]{x1D707}|$ = 1 as illustrated in figure 21(a), in the same way as that reported in Elston et al. (Reference Elston, Blackburn and Sheridan2006). The marginal stability curve from the Floquet analysis at $KC=6$ over a range of $e/D$ is shown in figure 21(b), together with that from DNS with non-zero IC. It is seen that the marginal stability curves obtained from the Floquet analysis and DNS agree well. This is attributed to the fact that the infinitesimal disturbances introduced in the Floquet analysis are comparable to the small finite disturbances (less than 1% in terms of overall energy) introduced in DNS with successive initializations. The predicted $\unicode[STIX]{x1D6FD}_{cr_{\infty }}$ for an isolated cylinder at $KC=6$ is 28.3, and is very close to the value of $\unicode[STIX]{x1D6FD}_{cr_{\infty }}\approx 28$ reported by DNS with non-zero IC.

Figure 22.

Comparison on the marginal curves of the $H_{1}$ symmetry at $KC=6$ by DNS with different conditions. NSPB, no-slip plane boundary condition; SPB, slip plane boundary condition; IC, initial condition.

The existence of the no-slip plane boundary possibly affects the stability of the flow through the following two mechanisms. (i) The blockage effect: which introduces a geometric asymmetry about the $x$ -axis ( $y=0$ ) and the gap flow; and (ii) the interaction between the Stokes boundary layer developed above the no-slip plane boundary and the shear layers developed over the cylinder surface (along with the resultant shed vortices). Although those two flow mechanisms are prime candidates responsible for the distinct variation trends of $\unicode[STIX]{x0394}\unicode[STIX]{x1D6FD}_{cr}$ and $\unicode[STIX]{x1D6FD}_{cr}$ observed in this study, their individual roles are not clear. To quantify their individual contributions to the observed trends, additional DNS are carried out with the same parameters except that a slip plane boundary condition (i.e. $\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}y=0$ , $v=0$ ), denoted as SPB, is imposed on the plane boundary. Under SPB, the influence of the Stokes boundary layer is removed while the blockage effect is retained. The marginal stability curves obtained for oscillatory flow above an SPB through DNS with zero IC and non-zero IC are shown in figure 22, together with their counterparts from the no-slip plane boundary (NSPB). It is seen that the overall variation trends of the marginal stability curves with $e/D$ obtained using an SPB are similar, but the hysteresis range $\unicode[STIX]{x0394}\unicode[STIX]{x1D6FD}_{cr}$ for flow above an SPB is much smaller than its counterpart above a NSPB and remains almost a constant over the entire range of $e/D$ . This suggests that the existence of the Stokes boundary layer over the plane boundary is mainly responsible for the large values of $\unicode[STIX]{x0394}\unicode[STIX]{x1D6FD}_{cr}$ over $0.25<e/D<1$ , possibly through the interaction between the Stokes boundary layer and the shear layers developed over the cylinder. This is indirectly supported by the fact that $\unicode[STIX]{x0394}\unicode[STIX]{x1D6FD}_{cr}$ values obtained using the two sets of boundary conditions are almost identical for $e/D>1.0$ , where the interaction is expected to be weak. The interaction between the Stokes boundary layer and the flow developed around the cylinder is illustrated qualitatively in figure 23 for $0.125\leqslant e/D\leqslant 1.0$ , in comparison with the flow around the cylinder over an SPB in the same range of $e/D$ .

The variation of $\unicode[STIX]{x1D6FD}_{cr}$ with $e/D$ for the case with a NSPB (shown in figure 20) is also explained. The observed decrease of $\unicode[STIX]{x1D6FD}_{cr}$ with $e/D$ over $0.875\leqslant e/D<2.0$ is due to peak flow velocity increase through the gap, induced by the blockage effect of the plane boundary. The flow instability over the range of $e/D$ is expected to be dominated by the flow around the gap side. Therefore, as the cylinder moves closer to the plane boundary, the blockage effect keeps increasing, leading to a further reduction of $\unicode[STIX]{x1D6FD}_{cr}$ from $\unicode[STIX]{x1D6FD}_{cr_{\infty }}$ . The increase of peak flow velocity through the gap with reduction in $e/D$ is quantified in figure 16 ( $x/D=0$ ) till $e/D=0.5$ , although it is sampled at a fixed point. It is also seen the peak velocity over the top of the cylinder barely changes. The observed trend in peak velocity is expected to be typical for the ( $KC,\unicode[STIX]{x1D6FD}$ )-space. When $e/D$ is reduced below a critical value (about 0.875), $\unicode[STIX]{x1D6FD}_{cr}$ starts to increase with reduction in $e/D$ , peaking at $e/D=0.25$ for zero IC and 0.375 for non-zero IC.

Figure 23.

Comparison of instantaneous contours of $\unicode[STIX]{x1D714}_{z}$ between slip plane boundary condition (SPB, upper) and no-slip boundary condition (NSPB, lower).

4.3 The favourable pressure gradient

The observed variation of $\unicode[STIX]{x1D6FD}_{cr}$ with $e/D$ over $0.375\leqslant e/D<0.873$ is attributed to the stabilization introduced by the favourable pressure gradient over the gap. The favourable pressure gradient over the gap increases as $e/D$ is reduced, as shown in figure 24 where the variation of pressure gradient over two points on the plane boundary with $e/D$ is quantified at $(KC,\unicode[STIX]{x1D6FD})=(6,30)$ . It is well known from stability theory that a favourable pressure gradient often improves the stability of the flow. Therefore, the effect of the favourable pressure gradient on the flow around the gap side of the cylinder is to improve its stability, counter-balancing the effect of the peak flow velocity increase mentioned above. The lower peak of $\unicode[STIX]{x1D6FD}_{cr}$ near $e/D=0.875$ signals the onset of the effect of favourable pressure gradient. As $e/D$ is further reduced, the increase in peak velocity through the gap starts to diminish while the favourable pressure gradient starts to increase rapidly, causing a change of flow instability from occurring on the gap side to the top side of the cylinder, at a gap ratio where $\unicode[STIX]{x1D6FD}_{cr}$ peaks at $(e/D)_{\mathit{peak}}=0.25$ and 0.375 for zero IC and non-zero IC, respectively. The reduction of $\unicode[STIX]{x1D6FD}_{cr}$ with $e/D$ for $e/D<(e/D)_{peak}$ is because the flow over the dominant side (the top side) for flow instability is enhanced as $e/D$ is reduced. The peak velocity above the cylinder increases with reduction of $e/D$ for $e/D<(e/D)_{peak}$ as shown in figure 16 ( $x/D=0$ ).

4.4 Eigenmodes by linear stability analysis

Figure 25 illustrates the influence of wall proximity on the eigenmodes. It is seen that the leading eigenmodes are scattered around the gap side of the cylinder for $e/D>0.25$ and over the top side for $e/D<0.25$ . The transition of the initiation location of flow instability thus takes place at $e/D=0.25$ . This is consistent with the trends of marginal stability curves obtained by DNS shown in figure 22.

Figure 24.

Variation of pressure drop between the detecting points A and B as a function of gap ratios at $KC=6$ and $\unicode[STIX]{x1D6FD}=30$ .

4.5 Discussion on the marginal stability curves

The flow mechanisms responsible for the differences between the marginal stability curves obtained using a NSPB and an SPB are explained based on the above results. Three major differences between the marginal stability curves obtained using a NSPB and an SPB are observed in figure 22: (i) the marginal stability curves obtained using an SPB shift slightly to the left of those obtained using a NSPB; (ii) the difference between the maximum $(\unicode[STIX]{x1D6FD}_{cr})_{max}$ and the minimum $(\unicode[STIX]{x1D6FD}_{cr})_{min}$ for using an SPB is larger than its counterpart obtained using a NSPB; and (iii) the observed difference in $\unicode[STIX]{x1D6FD}_{cr}$ values at $e/D=0.03125$ . These differences can all be attributed to the existence of the Stokes layer for a NSPB and the absence of the Stokes layer for an SPB. The absence of the Stokes layer for an SPB causes an increase of flow rate through the gap side in the case with a NSPB at the same gap ratio, resulting in a smaller gap ratio that is required to pass the same gap flow. This causes the marginal stability curves obtained using an SPB to shift to the left of those obtained using a NSPB. The increased flow rate through the gap for an SPB also leads to a smaller $(\unicode[STIX]{x1D6FD}_{cr})_{min}$ and a larger $(\unicode[STIX]{x1D6FD}_{cr})_{max}$ than their counterparts for a NSPB. The difference in $\unicode[STIX]{x1D6FD}_{cr}$ values at $e/D=0.03125$ is because the gap still has considerable influence on $\unicode[STIX]{x1D6FD}_{cr}$ even at such a small gap size for an SPB while such an influence is diminished for a NSPB due to the influence of the Stokes boundary layer.

5.1 Three-dimensional Floquet stability analysis

Three-dimensional Floquet stability analysis is conducted at $(e/D,KC)=(0.5,6)$ , following the same approach of Henderson & Barkley (Reference Henderson and Barkley1996) and Elston et al. (Reference Elston, Blackburn and Sheridan2006). The purpose of 3-D Floquet stability analysis is mainly to provide guidance to 3-D DNS in determining the critical Stokes number for onset of the 3-D instability, and the spanwise length of the computational domain through determining the spanwise wave length of the 3-D instability. The Floquet analysis results are shown in figure 26(a,b), where the base flows are obtained using zero IC and non-zero IC (by increasing $\unicode[STIX]{x1D6FD}$ ), respectively. The fully developed base flow initialized with zero IC is in GVS-A while that initialized with non-zero IC evolves into a GVS flow. The critical Stokes number for the 3-D instability for the regime GVS-A base flow is $\unicode[STIX]{x1D6FD}_{cr,\text{3-D}}=24.8$ and for the GVS base flow is $\unicode[STIX]{x1D6FD}_{cr,\text{3-D}}=33.9$ . This indicates that the flow would remain two-dimensional up to $\unicode[STIX]{x1D6FD}=24.8$ with zero IC and $\unicode[STIX]{x1D6FD}=33.9$ with non-zero IC. For $24<\unicode[STIX]{x1D6FD}_{cr,\text{3-D}}=33.9<39.5$ (where $\unicode[STIX]{x1D6FD}=24$ and 39.5 denote the lower and upper $\unicode[STIX]{x1D6FD}_{cr}$ for 2-D $H_{1}$ symmetry breaking), there must exist a range of $\unicode[STIX]{x1D6FD}$ where 3-D instability is not involved in the hysteresis phenomenon. However, whether the upper $\unicode[STIX]{x1D6FD}_{cr}$ for 2-D $H_{1}$ symmetry breaking is affected by 3-D instability remains to be checked by fully 3-D simulations.

5.2 Three-dimensional DNS

The purpose of 3-D DNS is to confirm the credibility of the hysteresis results and regime maps that are identified by 2-D simulations, especially for the flow in regime GVS. A convergence test on the choice of spanwise length $L_{z}$ and resolution $2N$ in the 3-D DNS is carried out at $e/D=0.5$ with $\unicode[STIX]{x1D6FD}=30$ and $KC=6$ and the results are shown in figure 27. Figure 27(a) shows that for both $L_{z}/D=4$ and 8 the primary wavenumber is $2k/\unicode[STIX]{x03C0}$ = 3, despite the lowest wavenumber extending to $2k/\unicode[STIX]{x03C0}=0.5$ for $L_{z}/D=8$ . Therefore, a conservative selection of $L_{z}/D=8$ is made to allow for a wider spectrum being included. The spanwise resolution is further checked at maximum Reynolds number ( $Re=300$ , or $\unicode[STIX]{x1D6FD}=50$ ). Different from that at $\unicode[STIX]{x1D6FD}=30$ , the amplitude of the lift force monotonically decreases at higher wavenumber, which is different from the case of $\unicode[STIX]{x1D6FD}=30$ , but similar to the case reported by Scandura et al. (Reference Scandura, Armenio and Foti2009). Since the results obtained with $2N=96$ and 144 are very similar, $2N=96$ is chosen to allow for a timely completion of the 3-D simulations up to $Re=300$ .

Figure 27.

Comparison of characteristic wavenumber obtained by FFT of the sectional lift force at (a) $\unicode[STIX]{x1D6FD}=30$ for spanwise length $L_{z}/D=4$ and 8 with fixed $2N=96$ , and (b) $\unicode[STIX]{x1D6FD}=50$ for spanwise resolution $2N=96$ and 144 with fixed $L_{z}/D=8$ . Results are averaged for 10 oscillation periods from $\unicode[STIX]{x1D70F}=50$ to 60 when the three-dimensional instabilities have been well developed. The original wavenumber is normalized by $2\unicode[STIX]{x03C0}/L_{z}$ for comparison of different spanwise length. The amplitudes $(\overline{C}_{L}^{A})^{\ast }$ have been normalized by the peak value.

Figure 28 shows results of 3-D DNS between $\unicode[STIX]{x1D6FD}=30$ and 50 with $(e/D,KC)=(0.5,6)$ , visualized by using the $Q$ -criterion (Hussain Reference Hussain1986). Based on the 2-D regime maps shown in figure 19(a), the fully developed flows initiated with zero IC are in regime GVS-A. Although the corresponding 3-D results shown in figure 28(a–c) clearly indicate that the flows are 3-D at $\unicode[STIX]{x1D6FD}=30$ , 40 and 50, their structures resemble 2-D regime GVS-A structures, and are also similar to the 3-D structures shown in figure 20 of Scandura et al. (Reference Scandura, Armenio and Foti2009) at $(e/D,KC,\unicode[STIX]{x1D6FD})=(0.625,10,20)$ . If the spanwise-averaged flow is used to classify the flow, the flows at $\unicode[STIX]{x1D6FD}=30$ , 40 and 50 will be categorized in regime GVS-A.

Figure 29 compares the spanwise-averaged flow and corresponding trace of force coefficients of 3-D results with those obtained using 2-D simulation at $\unicode[STIX]{x1D6FD}=50$ , where the three-dimensionality is strongest among the three cases examined. There is no significant difference between them, suggesting that similar flow regime maps to those obtained using 2-D simulations (figure 19 a) would have been obtained if 3-D DNS were employed. In addition, the hysteresis is also observed in the 3-D DNS, as shown in figure 28(d) where the flow remains in regime GVS and two-dimensional at $\unicode[STIX]{x1D6FD}=30$ when it is initiated with non-zero IC. Figure 28(e) shows that the flow is weakly three-dimensional at $\unicode[STIX]{x1D6FD}=38$ but maintains the features of regime GVS. The obvious regime GVS-A is observed at $\unicode[STIX]{x1D6FD}=40$ in figure 28(f) with non-zero IC, indicating the upper limit for $H_{1}$ symmetry breaking is consistent with $\unicode[STIX]{x1D6FD}_{cr}=39.5$ predicted by the 2-D model. This gives us confidence that although the regime maps and hysteresis range obtained using 2-D simulation may not be the exactly same as those obtained using 3-D simulations, 2-D results do reflect the real flow mechanisms well.

Figure 28.

The isosurface of $Q=0.25$ of the 3-D DNS results at $(e/D,KC)=(0.5,6)$ : GVS-A flow at (a) $\unicode[STIX]{x1D6FD}=30$ , (b) 40 and (c) 50 by zero IC; and GVS flow at (c) $\unicode[STIX]{x1D6FD}=30$ , (d) 38 and (f) 40 by successive iteration.

Figure 29.

Comparison of (a) lift force coefficient and (b) inline force coefficient between 3-D (- - - - - red) and 2-D (——) results, and corresponding instantaneous vorticity contours (c) and (d) at $(e/D,KC,\unicode[STIX]{x1D6FD})=(0.5,6,50)$ . The 3-D results are averaged along the spanwise direction.