Research on the effects of in-line oscillatory flow on the vortex-induced motions of a deep draft semi-submersible in currents

A Deep Draft Semi-submersible (DDS) under certain flow conditions could be subjected to Vortex-Induced Motions (VIM), which significantly influences the loads on and life fatigue of the moorings and the risers. To investigate the VIM of a DDS with four rectangular section columns in waves coupled with a uniform current, a numerical study using the computational fluid dynamics (CFD) method was conducted. The issues of the VIM of multi-column floaters can be conveniently converted to the issues of oscillating cylinders in fluid cross flows. This paper looks into the CFD numerical simulation of infinite cylinders having rectangular sections in a two-dimensional sinusoidal timedependent flow field coupled with a uniform current. The resulted hydrodynamic forces and motion responses in different oscillatory flows plus currents both aligned in the same direction for the incidence of 135° of the DDS relative to the flow are compared with the ones in current only cases. The results show that the VIM response of this geometric arrangement of a DDS with four rectangular columns in a current combined with oscillatory flows is more evident than that in the current only case. The oscillatory flows and waves have the significant influence on the VIM response, forces and trajectory, in-plane motions of the DDS.


Introduction
Studies have demonstrated that vortex shedding may happen around a bluff body under certain fluid flow conditions and which will cause a pulsating surface pressure on the body and potentially leading to the Vortex-Induced Vibration (VIV) or Vortex-Induced Motion (VIM). In the real sea states, floating structures such as Spar, TLP and semisubmersible, are subjected to ocean waves and currents. The potential for the VIM of multi-column floaters, typically, a Deep Draft Semi-submersibles (DDS) in the flows and waves is very obvious and is thus worthy of investigation. Izhar et al. (2014) studied the VIV of a cylinder and showed that the 2D as well as 3D analyses are performed well using air as the fluid. He et al. (2010) simulated the cylinder in the current with a two-dimensional model and found that the interactions between fluid and structure produce some weakly energetic vortices which induce the modulations of amplitude and frequency.
Oscillatory flow passing a multi-column floater is a very common flow phenomenon and it has been proved to be a challenging area for research, since it provides the basis of a simplified tool for the investigation of the flow around a multi-column floater that is immersed in a wave environment (Iliadis and Anagnostopoulos, 1998). Sinusoidal oscillating flow around cylinders has long been of special interest to fluid dynamicists and offshore engineers (Sarpkaya, 1986). In the same way, oscillating flow coupled with a uniform flow around a multi-column arrangement is frequently used to simulate waves coupled with an ocean current acting on a floating platform with several vertical columns.
The investigations into oscillatory flow started very early. Justesen (1991) carried out several two-dimensional (2D) numerical simulations in the range of 0<KC<26 (the Keulegan-Carpenter number) and found that good agreements between the numerical and experimental results of the force coefficients were achieved. The vortex-shedding regimes observed in the experiments were captured by the 2D mathematical model. Düstch et al. (1998) used 2D numerical simulations also to perform laboratory experiments for study on oscillating flow around a circular cylinder and found that the 2D results of the velocity field and force coefficients agreed well with the experimental data. Nehari et al. (2004) simulated oscillating flow around a circular cylinder using both 2D and 3D numerical models. Their investigation covered two vortex shedding regimes: namely the asymmetric transverse-street regime and the single pair regime. It was reported that the cross-sectional vortex streets appeared to be unstable, and the constant switching of a vortex pattern to its mirror-image mode occurs. The VIM of multi-column floating structures under a uniform flow or under wave actions has been extensively studied during the earlier years. Ishihara et al. (2001) used numerical simulation in the time domain to study floating production platform motions in directional waves and they found that the time domain simulation is an important tool when non-linear effects are presented and that the influences of wave spreading and the second order slow motions are very important and are to be included in the platform motion behavior analyses. Bultema et al. (2007) compared the VIM responses of multi-column platforms in currents approaching in turn from three different directions relative to the platform through model tests in a towing tank. It was observed that the largest motions were observed at a 45° towing direction (135° seas also) and that at a reduced higher velocity (U r ) a combination of galloping and VIM occurred. Rijken and Leverette (2008) performed an experimental study on the VIM response of semi-submersibles with square section columns and a vortex lock-in phenomenon was found to develop when the overall motion was nearly harmonic. Hong et al. (2008) carried out a model test of a DDS in a simulated wind, wave, and current environment. The results show that a DDS may experience significant VIM in the direction that is normal to that of the current, and its amplitude was strongly dependent on not only the current speed but also the wave excited particle velocity. Gonçalves et al. (2011) conducted model tests and concluded that the VIM response appears to be at the maximum for 45° current approach incidence and at a reduced velocity of U r =7.0-8.0. Gonçalves et al. (2012a) conducted other laboratory experiments and came to the conclusion that when compared with the situation with current only, the motion amplitudes in the transverse direction decreased appreciably when the combined regular waves and currents were applied around the platform. Even so, the research given above on the VIM of the multi-column floaters is still relatively rare, especially the study through numerical simulation. In this paper, a numerical simulation is conducted by using a simplified model of a representative multi-column floating platform in a typical uniform current combined with different superimposed oscillatory flows, in order to obtain a better understanding of the mechanism of VIM as ocean currents combined with waves acting on multi-column floaters.

Methodology
Increasing computer capacity and the development of progressively more efficient numerical methods allow the solution of the Reynolds-averaged Navier-Stokes equations for practical fluid and aerodynamic applications. Currently, most of the numerical studies on the VIM focuse on the flow around fixed structures or on the flow-induced motions of floating single-column structures. However, reports on the numerical simulation by using CFD techniques on the VIM of a multi-column semi-submersible are relatively scarce. In this paper, in order to improve on the understanding of the VIM characteristics of a typical multi-column floater, the numerical research is conducted by solving the unsteady Reynolds-averaged Navies-Stokes (RANS) equations based on the application of the Finite Volume Method (FVM) as the RANS method ensures the efficiency and accuracy of the calculations.

Reynolds-averaged Navies-Stokes equations
In the Cartesian coordinate system, the transient governing equations for incompressible flow can be written as: u, v, and w ρ υ where u is the velocity vector; represent each of the three velocity components in x, y, and z directions respectively; p denotes the pressure; and are the density and kinematic viscosity respectively of the fluid.
The time-averaging method has been widely used to solve the above equations. The instantaneous fluid velocity can be expressed as , i.e. the sum of the time-averaged velocity component and the fluctuating velocity component . The time-averaged velocity component can be defined as .
The advantage of this approach is of the relatively low computational cost associated with the determination of the turbulent viscosity. The SST (Shear Stress Transport) turbulence model described in the following section is based on this hypothesis.
Two-equation eddy-viscosity turbulence models have been widely used. The so-called model and the various developed forms that are based on it are the most popular ones. The model, developed by Wilcox (1988), is one of the more successful models that have been developed from model and it is designed to overcome some of the well-known shortcomings of the originated model (Menter, 1992). The model of Wilcox was chosen for this study for two reasons. First, it gave superior results for the adverse pressure gradient flows computed by Wilcox ω (1988) when compared with other two-equation models. Second, because of its mathematical simplicity, it does not need damping functions in the low Reynolds number region close to the mathematic boundary wall, and the equation has an exact boundary condition at the wall.

Computational mesh domain
For the development of the solution of the problem, various 2D computational meshes were used, with the exact fine mesh configuration depending on the particular flow case under investigation and the turbulence model for the simulation. All meshes consist of triangular and quadrilateral elements. The appropriate upstream, downstream and side distances and boundaries of the analysis domain were selected in order to provide a good compromise between accuracy of the solution and computational cost. In the part of the solution domain near the column surfaces, more refined grids were used. This was done because more refined grids should be used for better accuracy of the field variables within the boundary layer near the cylinders or columns. The element size in the vicinity of the cylinders or columns was made very small and increases in sight gradually with the distance from the cylinders or columns. The grids close to the structures are thus made small enough to ensure that the grids satisfy . In the wake flow region, the field has little influence, and thus sparser grids were used to improve the calculation efficiency. A mesh convergence study was undertaken in order to decide the final overall mesh plan which can ensure the sufficient accuracy and at the same time minimize the computational cost. Similarly timestep in the calculations was also selected in order to provide a good compromise between the accuracy of the solution and the computational cost.

Scope
As the potential for the VIM of the DDS will be induced by the vortex shedding from the four rectangular columns, clearly the shape and proportions of the columns themselves will play the most important role in the VIM (Gonçalves et al., 2012b). A simplified model consisting of four rectangular section columns of a given proportion and spacing of the DDS was used in the numerical simulation omitting the pontoons and all other typical appendages. The assumed simplification made it possible to use a 2D horizontal plane model in the numerical simulation for the sake of saving computer resources. The above simplifications avoid some complex 3D problems.
The DDS employed in this study consists of four rectangular columns and four rectangular pontoons interconnecting the columns. Table 1 shows the main dimensions that are relevant to this study.
From an earlier study (Bai et al., 2013), the responses of the DDS in a uniform current from different headings were observed that the largest VIM amplitude in the transverse direction appears at the 135° incidence for a reduced velocity U r =7 compared with the response amplitudes resulted from other current directions and reduced velocities. Hence a 135° incidence heading for the reduced velocity of U r =7 was chosen for this study as the uniform current to be coupled with an oscillatory flow at different relative current numbers (U/U m ) and Keulegan-Carpenter (KC) numbers. The cases with different KC numbers (KC=U m T/D) and relative current numbers U/U m of the incoming flows which were studied are shown in Table 2. U m is the maximum velocity of the water molecules in the wave alone, i.e. the source of the oscillatory flow in the horizontal 2D plane.
, where a is the wave amplitude, T is the wave period, D is the hydraulic diameter, and U is the speed of the uniform current. The time-dependent free stream velocity of the oscillatory component flow is defined in terms of the maximum flow velocity as , where is the wave frequency.

VIM responses
The experimental results show that the motions in developed VIM for a DDS of this type in the transverse and in-line directions, relative to the incoming flow are predominant (Bai et al., 2013). Hence, the CFD models were mathematically allowed to move only in these two directions and the four columns were constrained to behave together as a single rigid body in the simulation of VIM. The motion responses were predicted by solving the motion equations coupled with the fluid forces in the two directions as follows (Sarpkaya, 2004): where m represents the mass of the platform; c is the structural damping coefficient of the system; K Y and K X are the stiffnesses of the mooring system in the transverse and inline directions respectively, and they are adjusted to realist- ic values so that the natural rigid body periods in the two directions are the same as in the real platform upon which this study is based on. F Y and F X are the total fluid forces in the transverse and in-line directions and obtained by solving the RANS equations. The excitation force terms are assumed to be constant within a sufficiently small time step. Thus, the motion equations presented above can be integrated in time using a fourth order Runge-Kutta algorithm. The position of the rigid body with the flow field surrounding it is updated within each time step and the next time step that follows.
Here c was set as zero since the structural damping coefficient of the system is very small. And K X and K Y were set as 20.06 t/m. The natural periods of the DDS are 187 s and 202.5 s in the surge and sway directions, respectively. The corresponding natural frequencies are 0.034 rad/s and 0.031 rad/s.

Meshes for the numerical simulations
The whole fluid domain and a locally enlarged view ad-y + ≈ 1 jacent to a column are shown in the example that is shown in Fig. 1. In order to ensure that the grid satisfies in the whole calculation process, the meshes in the boundary layer region (the very fine quadrilateral grids in the enlarged view (b)) remain in a relatively static state close to the columns. Also the larger triangular grids are the merely flow affected dynamic meshes. The quantity of the grids would also change with the physical motions of the columns, noting again that the four columns are rigidly connected to each other and thus must share the same motions, in the calculation process in order to maintain the grid definition at an appropriate size and thus ensure the accuracy of the calculations.

Re = UD/ν
Different meshes around a single column of the DDS at Re=20000, and the numerical results are listed in Table 3. Here , where U is the velocity of the current, and ν is the kinematic viscosity coefficient. It can be seen that mesh model No. 2 has achieved good convergences and thus been selected for further simulations.

Verification of numerical model
Many researches including numerical simulations and actual model tests on the vortex shedding from the circular cylinders held in a fixed position in current are carried out, and many meaningful results and conclusions are available. The flow field around rectangular or square columns, with large corner radii or simple chamfers and immersed in current, is similar to that with a circular cylinder. Therefore in this paper, the numerical simulations of a single circular cylinder fixed in the current at Re=20000 and of a four circular cylinders group fixed at Re=30000 are used to investigate the selected method for the simulation of the DDS in the combined current and oscillating flow.

Single cylinder in current
Computations were carried out for a circular cylinder in current at Re=20000. The meshes and the time step were both tested and proved to be satisfactory. The meshes used for these calculations are shown in Fig. 2.
The results of the numerical simulation with the SST turbulence model were compared with the model test results of Yokuda and Ramaprian (1990) and the numerical results (Lu et al., 1997) at Re=20000, as shown in Table 4. is the time-averaged drag coefficient given bȳ and St obtained from the current numerical simulation seem to agree well with the published results, proving that the current proposed simulation method is reliable.  Fig. 3 displays the meshing of the whole domain and of a locally enlarged region near a cylinder. Computations of the current around the four rigidly fixed circular cylinders were made at Re=30000, and the results were compared with those from model tests carried out by Sayers (1990), as shown in Table 5.

Four cylinders fixed in current
According to Table 5, the force coefficients and the Strouhal numbers from the numerical simulation agree well with the experimental results of Sayers (1990), further indicating that the numerical method produced credible results.

Numerical results and analysis
The results of the VIM calculations will be presented in terms of the maximum response amplitude and the nominal response amplitude of the four column body combination. The induced sway and surge motions, i.e. transverse and inline motions can be described as: Maximum response amplitude: Nominal response amplitude: where A y and A x are the amplitudes of the sway motion and the surge motion, respectively; and are the standard deviations of and , respectively. Statistical and spectral analyses were carried out for a simulated time series in order to facilitate the simulation result analyses, so that the vortex-shedding frequency could be obtained.

Body motions in the transverse direction
After a series of computations with different KC and U/U m value pairs, for this DDS with asymmetric rectangular column sections, the sway motion response amplitudes of the example of DDS in oscillatory flow coupled with a current were found to be larger than that in the current only, as shown in Fig. 4. It can be seen that the trends of the max-  imum and the nominal response amplitudes of the sway motion are similar. For KC=0.6, the sway motion response amplitudes decrease at first and then increase with the increasing U/U m . The results obtained by Chakrabarti et al. (2007) showed that at a wave period of about 8.84 s, the surge-exciting force RAO of a typical semi-submersible platform is at the minimum value. In this study, the calculated oscillatory flow period is 8.122 s at KC=0.6 with U/U m =0.8 and the corresponding sway motion response amplitude is at the minimum point. With the increasing values KC=0.9 and KC=1.2, the calculated sway motion response amplitudes increase with the increase of the relative current numbers. The KC numbers in this range appear to have little influence on the VIM in the transverse direction. For KC=1.5 and KC=1.8, with the relative current number increasing, the sway motion response amplitudes decrease at U/U m <0.6, then increase again at 0.6<U/U m <0.8 and finally decrease at U/U m >0.8. Justesen (1991) showed that the lift force on the circular cylinders is at the minimum value for KC values around 1.6, which agrees well with the experimental data of Obasaju et al. (1988), and it indicates that a KC number around 1.6 makes a difference to the vortexshedding behaviors. According to Sarpkaya (1986), the force coefficient is at the minimum point when KC=1.5 for different Re values. The combined action of KC and the relative current number makes the motion response amplitudes in the transverse direction to be changeable. Fig. 5 presents the comparison between the vortex-shedding frequencies for a range of KC values of the DDS for three flow cases, i.e. a current coupled with oscillatory flows with the relative current number, a current only and an oscillating flow only. According to these results, oscillatory flows at different frequencies are almost insignificant in their affection of the vortex-shedding frequency. The same conclusions were presented by Gonçalves et al. (2011). At KC=0.6 and 0.9, the frequencies of the vortex-shedding of  the DDS in oscillatory flow only have an almost effect equal to that of the vortex-shedding frequency in a current on its own. At KC>0.9, the vortex-shedding frequency decreases slightly with the oscillatory flow period and the relative current number increasing. It is also seen that in the range of 0.6<U/U m <1.0 and for KC values of 0.9, 1.2, 1.5 and 1.8, at least one vortex-shedding frequency with oscillatory flow to be equal to the frequency with only the current. For all KC values, at U/U m ≥1.0, the vortex-shedding frequency of the DDS immersed in the combined oscillatory flows and current is a little smaller than that in the current alone. Fig. 6 shows the time history of the non-dimensional sway motion Y/D and the spectrum of the sway motion in the current coupled with oscillatory flow at KC=0.9. The time series of the VIM in the transverse direction fluctuates around Y/D=0. According to Fig. 6, the relative current number makes little difference to the time history of the sway motion at KC=0.9. The results of statistical and spectral analyses show that the energy component in the low frequency is observed in the sway motion spectrum, owing to the slowly varying oscillation of the four-column unit. These results are similar to those of another semi-submers- ible platform which has four square section columns in a square arrangement (Ishihara et al., 2001). However at U/U m =0.8, the vortex-shedding frequency is seen to be relatively larger. Fig. 7 shows the surge motion response amplitudes for different KC and relative current number combinations. According to the curves, the surge motion response amplitudes in the oscillatory flow plus current case is larger than that in the current only case except at KC=0.6. The relative current number is seen to have little influence on the surge motion. However, with the increasing KC, the surge motion becomes more prominent. The shape of the surge motion response amplitude curves is much similar to that of the sway motion shown in Fig. 4. For KC=0.6 and KC=1.5, the influence of U/U m on the maximum response amplitude of the surge motion is much greater than that with other KC values. At KC=0.9, 1.2 and 1.8, the relative current number makes little difference to the VIM in the in-line direction, which is similar to that for the sway motion. For KC=1.8, the main trend inflection point of the surge motion is at U/U m =1.0, not at U/U m =0.8 as seen in the sway motion results. The similar inflection of the nominal response amplitude of the motion in the in-line direction for the same KC number is much less pronounced than that in the maximum response amplitude plot. Both the maximum and the nominal response amplitudes of the sway motions are about twice those of the surge motions. Similarly the influence of the oscillatory flow on the sway motion is greater than that on the surge motion. Again it must be appreciated that this is for a specific current and wave heading relative to the DDS. Fig. 8 shows the time history of the surge motion at KC=0.9. According to Fig. 8, the relative current number, U/U m , had little influence on the amplitude of the surge motion but obvious influence on the period of the time series of the surge motion. There are two typical frequencies observed in the surge motions, one is the frequency of the oscillatory flow and the other is the natural frequency of surge. This indicates that the in-line motion is dominated by the oscillatory flow (i.e. wave) instead of the vortex shedding. Therefore, the frequencies of the in-line motion are different from that of the transverse motion which is dominated by the vortex shedding. Compared with the time history in the in-line  WU Fan et al. China Ocean Eng., 2017, Vol. 31, No. 3, P. 272-283 279 direction, the VIM of the DDS in the transverse direction shown in Fig. 6 is more close to being a sinusoidal curve. It is also noted that the time-averaged value of the surge motion is much larger than that in the sway motion. However, the fluctuating component of the surge motion is much smaller than that in the sway motion. The curves of the time history are quite similar to the experimental results of Gonçalves et al. (2012c) for the incidence of 45° (the reciprocal heading of 135° is made in these calculations) for the reduced velocity, corresponding to a region before the eddy shedding synchronization region for the transverse motion. The oscillatory flow clearly makes the time history of the VIM in the XY plane to be more violent.

Trajectories in the XY plane
The VIM trajectories at KC=0.9 of the DDS with four rigidly connected rectangular asymmetrical columns are presented in Fig. 9, representing the typical VIM response. Compared with the current only case in this study, similar results are presented and that the sway amplitude is larger than the surge amplitude. However, considerable differences are also presented. The range in both the in-line and transverse directions with current flow is wider than that in the situation without oscillatory flow. The trajectories are not perpendicular to the in-line direction anymore because of the effects of oscillatory flows. And as U/U m increases, it tends to be perpendicular. Besides, different from the current only case that Bai et al. (2013) conducted model tests, and the trajectories are raised again, but no longer circles as in the current. Fig. 10 presents the results of non-dimensional characteristic values in terms of the lift and drag coefficients of the forces on the four cylinders group for different KC and relative current numbers. As shown in Fig. 10, both the overall four column body mean lift and mean drag force coefficients are smaller when the DDS is under the combined oscillatory flow and current condition than those under the current only condition. Conversely, the fluctuation lift and drag coefficients become larger. The shape of the trend inflection line of the time-averaged forces in XY plane for KC=0.6 is similar to the maximum response amplitudes of the VIM in the same directions. The relative current number makes significant difference to the time-averaged lift force in terms of the coefficients at KC=0.9 and KC=1.2 which is different from the VIM in the XY plane. For KC=0.9 and 1.2, U/U m =1.0 is an inflection point in the curves. As U/U m <1.0, the time-averaged lift coefficient decreases with the increasing relative current number. For KC=1.5, the shape of the trend inflection line of the timeaveraged force coefficients is similar to the VIM in the XY plane. However, the minimum point is at U/U m =0.8 not at 0.6. Compared with the time-averaged drag coefficient, the time-averaged lift coefficient is more sensitive to the relative current number. For 0.9<KC<1.5, U/U m value makes little difference to the time-averaged drag coefficient. In this KC range, when U/U m <1.0, the time-averaged lift coefficient decreases with the increase of KC. But when U/U m > 1.0, as KC increases, the time-averaged lift force also increases. Also for the time-averaged forces in both the in-line and the transverse directions are smaller with oscillatory flow only compared with the situation of the DDS in the current only. For U/U m <1.0, the fluctuation components of both drag and lift forces decrease with the increasing U/U m at the same KC and also increases with the KC number at the same relative current number. For U/U m ≥1, both KC and U/U m have little influence on the fluctuation of both the drag and lift forces. It can also be seen that the fluctuation components are larger with the DDS in oscillatory flow compared with those in the situation of current. Additionally, both the time-averaged and the fluctuation components of the drag force are much larger than those of the lift force.

Lift and drag forces on the DDS
The time histories of the drag and lift coefficients for KC=0.9 are shown in Fig. 11. According to Fig. 11, the shape of drag coefficient (C D ), time history is more similar to the sinusoidal curve than that of the lift coefficient, (C L ), time history. Also it is should be noted that the amplitude of C D is about ten times that of C L . Also the relative current number makes an appreciable difference to the amplitudes and periods of both C D and C L time histories. Fig. 12 presents the flow field around the DDS in the current for U r =7 and oscillatory flow plus current at U/U m =0.6, 1.0 and 1.4 for KC=0.9 (t 1 and t 2 mean two different time during the flow attacking the DDS). In the current, the vortex shedding of the columns is in-phase with one another and occurs around the far corners away from the column centerline. In the oscillatory flow at U/U m =0.6, the resultant velocity varies from a negative value to a posit-ive one, i.e., the direction of the current velocity may change at some particular moments. As a result, the vortex is likely to shed from each corner of columns as shown in Fig. 12b. The flow field for U/U m =0.6 also shows a slightly weaker vortex strength at some points due to the possible smaller magnitude of the resultant velocity. In addition, for the cases of U/U m =1.0 and 1.4, the vortex shedding is similar to that of U/U m =0.6, owing to the resultant velocity being a small positive one and even equal to zero when the  directions of the uniform current and the oscillatory flow are opposite to each other at some moment. However, when the directions of the oscillatory flow and the current are the same, a large positive current velocity appears; consequently, the strong vortex shedding occurs around the far corners away from the column centerline. Therefore, the vortex shedding of the columns in the oscillatory flow shows random characteristics.

Conclusions
(1) The VIM response of the DDS with four rectangular asymmetrical columns in oscillatory flow coupled with a current is more significant than that in a current only. For the particular asymmetrical section and arrangement of the columns, the addition of oscillatory flows did not decrease the VIM of the DDS in current.
(2) Oscillatory flows have little influence on the vortexshedding frequency or the frequency of the motion in the sway direction, which is similar to that of a DDS with symmetrical square columns in symmetrical square arrangement. The in-line motion is dominated by the oscillatory flow (i.e. wave) and its frequencies are different from those of the transverse motion which is dominated by the vortex shedding.
(3) When compared with the forces on the DDS in current only, the time-averaged forces decrease in the oscillatory flow combined with a current condition. However, the associated fluctuation components increase at the same time. Additionally, the fluctuations in the drag and lift forces decrease as the KC number increases at the same rel- evant current number, and also at the same KC number as the relevant current numbers increase, the fluctuations decrease.
(4) The sway amplitude is larger than the surge amplitude. Also the motion trajectories are not perpendicular to the current flow direction because of the effect of oscillatory flow. As the relevant current number increases, the motion trajectories tend to be more perpendicular to the in-line surge direction.
(5) The flow field around the DDS in oscillatory flow coupled with current is more complex than that in the current. The vortices become more changeable owing to the influence of oscillatory flow.