## Abstract

In the present study, we apply proportional (P), proportional–integral (PI) and proportional–differential (PD) feedback controls to flow over a circular cylinder at *Re*=60 and 100 for suppression of vortex shedding in the wake. The transverse velocity at a centreline location in the wake is measured and used for the feedback control. The actuation (blowing/suction) is provided to the flow at the upper and lower slots on the cylinder surface near the separation point based on the P, PI or PD control. The sensing location is varied from 1*d* to 4*d* from the centre of the cylinder. Given each sensing location, the optimal proportional gain in the sense of minimizing the sensing velocity fluctuations is obtained for the P control. The addition of I and D controls to the P control certainly increases the control performance and broadens the effective sensing location. The P, PI and PD controls successfully reduce the velocity fluctuations at sensing locations and attenuate vortex shedding in the wake, resulting in reductions in the mean drag and lift fluctuations. Finally, P controls with phase shift are constructed from successful PI controls. These phase-shifted P controls also reduce the strength of vortex shedding, but their results are not as good as those from the corresponding PI controls.

## 1. Introduction to proportional–integral–differential control and its application to flow over a bluff body

Flow control is one of the main issues in fluid mechanics, and there have been many different approaches of flow control for the purpose of drag reduction, lift enhancement, mixing enhancement, etc. (for reviews, see [1–5]). In general, control methods are classified as passive, active open-loop and active closed-loop (feedback and feed-forward) controls. The merits and demerits (or limitations) of these control methods are well summarized in [1–5]. Among these control methods, in this paper, we introduce an active feedback control method called linear proportional (P)–integral (I)–differential (D) control and its application to flow over a bluff body.

The PID control is a control law based on only the output being available for feedback, and is a very common and practical control method used in the control society. In the PID control (figure 1), the controller is composed of a simple gain (P control), an integrator (I control), a differentiator (D control) or some weighted combination of these possibilities [6,7]:
1.1
where *ψ* is the control input, *e*(*t*) is the error, and *α*,*β* and *γ* are the proportional, integral and differential gains, respectively. The error *e* may be a sensing velocity *v*_{s} at a location in a flow field that we want to drive to zero through the feedback control.

The proportional part in the PID control (first term in equation (1.1)) adjusts the output signal in direct proportion to the controller input that is the error signal. When the proportional gain *α* increases, the system under consideration yields a fast response, small steady-state errors and a highly oscillatory response. The integral part (second term in equation (1.1)) corrects for any error that may occur between the desired value and the process output over time. Thus, the steady-state error becomes zero owing to the integral part. However, it is slow in response and may induce system instability. The differential part (last term in equation (1.1)) uses the rate of change of the error signal and introduces an element of prediction into the control action and can force the error to zero without oscillations having excessive amplitudes. For the details of the PID control, see Rowland [7] and Johnson *et al.* [6].

Although the PID control is well established in the control society, most previous studies in fluid mechanics have focused on the P control with/without phase shift. In the P control without phase shift, the control input (actuation) *ψ* is linearly proportional to the sensing variable *v*_{s} as
1.2
The P control with phase shift is another expression of the PID control when the system under consideration is linear. For example, if the sensing velocity is expressed as (*b*=2*π*/*T* and *T* is the period), the actuation velocity from the PID control (equation (1.1)) is expressed as
1.3
where *A*^{2}=*α*^{2}+(*bγ*−*β*/*b*)^{2} and . The phase shift *c* is a function of the gains and frequency. For most fluid-mechanics problems, the system is nonlinear and thus the P control with phase shift is not the same as the PID control in general.

Table 1 summarizes the previous studies on the application of the PID control to flow over a bluff body, where the Reynolds number, sensing variable, actuation type and feedback law (P control, P control with phase shift, PI and PID) are described. The purpose of control in these studies was the attenuation or annihilation of vortex shedding behind a bluff body. As shown in this table, most studies involved the P control with or without phase shift except that of Zhang *et al.* [8]. Those studies indicated that the results of P control are very sensitive to the sensing location and the amount of phase shift. With proper choices of these variables, the P controls were quite effective in annihilating the vortex shedding or reducing its strength. Also, it was shown that increasing the proportional gain gives more reductions of the velocity fluctuations in the wake and the strength of vortex shedding, but a large gain results in system instability [9–12]. Furthermore, it was noted that the control based on a single-point sensing does not completely stabilize the wake at high Reynolds numbers [10,13]. Zhang *et al.* [8] is the only study where the PI and PID controls were applied to the flow over a bluff body. They measured both the transverse cylinder displacement *Y* and the streamwise fluctuating velocity *u* at (*x*/*d*,*y*/*d*)=(1.6,−2.5). The actuation was the transverse cylinder displacement. They showed that both the streamwise velocity fluctuations at (*x*/*d*,*y*/*d*)=(2,1.5) and the cylinder vibration amplitude are reduced by PID-*Y* , PID-*u* and PID-*Y* *u* controls. However, the effect of sensing position was not considered in their study.

The proportional feedback control is simple and easy to apply. However, its result is quite sensitive to the sensing location and feedback gain *α*. It is known from the control theory that these disadvantages of the P control should be overcome by adopting the I and D controls. However, as reviewed above, there are a very limited number of studies dealing with the PID control in the literature for fluid-mechanics problems. On the other hand, there have been quite a few studies dealing with the P control with phase shift, but it has not been carefully studied as to how much the results from this control are different from those of PI, PD and PID controls for fluid-mechanics problems. Therefore, in the present study, we apply the P, PI and PD controls to the flow over a circular cylinder at low Reynolds numbers for the reductions of the mean drag and lift fluctuations, and investigate their effectiveness. Then, we compare some of the successful results from the PI controls with those from the corresponding P control with phase shift, and discuss whether the PID control can be represented as the P control with phase shift for the present fluid flow. In this paper, the control method and numerical details are given in §2. The control results such as the sensitivities on the feedback gain and sensing location and the drag and lift variations are shown and discussed in §§3–5. The comparison to the results from the P control with phase shift is given in §6, followed by conclusions in §7.

## 2. Control method and numerical details

Let us consider the unsteady two-dimensional flow over a circular cylinder at low Reynolds number. The PID control is given as (figure 2)
2.1
where *ψ* is the radial actuation velocity (blowing/suction) from the upper and lower slots on the cylinder surface, *v*_{s} is the measured velocity at the sensing position *x*_{s} (centreline of the cylinder wake) and is the maximum value of the sensing velocity for *τ*≤*t*. The P control (the proportional part in equation (2.1)) is the same as that used by Park *et al.* [13]. As shown in figure 2, the actuation phases from the upper and lower slots are 180° out of phase and thus zero net mass flow rate is satisfied during the control. In the present study, we set the slot locations as *θ*_{1}=100° and *θ*_{2}=120° just before the separation points for *Re*=60 and 100. Various sensing locations are tested such as *x*_{s}/*d*=1–4 by increments of 0.5. The target of the PID control is to reduce the error, that is the sensing velocity at the centreline location of the cylinder wake. Therefore, a successful control should reduce the root mean square (r.m.s.) transverse velocity fluctuations at the sensing location, which in turn possibly attenuates vortex shedding and reduces the mean drag and lift fluctuations.

We solve the incompressible Navier–Stokes and continuity equations in Cartesian coordinates using an immersed boundary method proposed by Kim *et al.* [14]:
2.2
and
2.3
where *t* is the time, *x*_{i} are the Cartesian coordinates, *u*_{i} are the corresponding velocity components, *p* is the pressure, *f*_{i} and *q* are the momentum forcing and mass source/sink, respectively, and is the Reynolds number. Here, all the variables are non-dimensionalized by the free-stream velocity and the cylinder diameter *d*. In this simulation, we use a fractional step method to decouple the velocity and pressure, and the second-order central difference for all the spatial derivative terms on the staggered grid system. That is, *u*_{i} and *f*_{i} are defined at the cell surfaces, and *p* and *q* are defined at the cell centre. The roles of *f*_{i} and *q* are to satisfy no slip on the immersed boundary (cylinder surface) and mass conservation for the cell containing the immersed boundary, respectively. For the details of determining *f*_{i} and *q*, see Kim *et al.* [14]. The accuracy of this method applied to flow over a circular cylinder has been confirmed by our previous studies [14,15].

The computational domain size is −50.5*d*≤*x*≤19.5*d* and −49.5*d*≤*y*≤49.5*d*. Dirichlet boundary conditions () are applied at the inflow and far-field boundaries and a convective boundary condition (∂*u*_{i}/∂*t*+*c*∂*u*_{i}/∂*x*=0) is used for the outflow boundary, where *c* is the line-averaged streamwise velocity at the exit. We consider two different Reynolds numbers, *Re*=60 and 100, and the numbers of grid points used are 391(*x*)×295(*y*). The computational time step is . The numerical accuracy is confirmed by increasing the number of grid points in each direction.

## 3. P control

The parameters to consider for the P control are the proportional gain *α* and sensing location *x*_{s}. Given the sensing location (*x*_{s}/*d*=1–4), we vary the proportional gain *α* to find out the best proportional gain for maximum reduction of the sensing velocity fluctuations *v*_{sr.m.s.}.

Let us first consider the case of *Re*=60. Figure 3*a* shows the variations of *v*_{sr.m.s.} with the proportional gain for different sensing locations *x*_{s}. The values of *v*_{sr.m.s.} at *α*=0 correspond to those without control. The range of *α* at which *v*_{sr.m.s.} is decreased by the control varies depending on the sensing location. That is, negative *α* reduces *v*_{sr.m.s.} for *x*_{s}≤1.5*d*, positive *α* for 2*d*≤*x*_{s}≤3.5*d* and negative *α* again for *x*_{s}=4*d*. Especially, *v*_{sr.m.s.} completely disappears, indicating complete attenuation of vortex shedding, at *α*≤−0.2 for *x*_{s}=*d*, at −0.2≤*α*≤−0.1 for *x*_{s}=1.5*d* and at 0.1≤*α*≤0.3 for *x*_{s}=3*d*. Large magnitudes of proportional gain (for example, *α*≤−0.25 for *x*_{s}=1.5*d* and *α*≥0.35 for *x*_{s}=3*d*) increase *v*_{sr.m.s.} for most sensing locations. Also, it is clear that the performance of the P control becomes less effective when the sensing location is farther away from the cylinder centre, as observed by previous studies [10,13]. Figure 3*b* shows the time histories of the sensing velocity *v*_{s} (directly representing the actuation velocity in the case of P control) for three different P controls. As expected from figure 3*a*, the sensing velocity becomes zero after some transient period for the case of (*x*_{s}=1.5*d*,*α*=−0.15). The sensing velocities are attenuated and amplified for the cases of (*x*_{s}=2.5*d*,*α*=0.1) and (*x*_{s}=4*d*,*α*=0.1), respectively. The great benefit from the feedback control over an open-loop control is that the control input is negligible for successful control and then the control efficiency becomes very large.

Figure 4*a* shows the variations of r.m.s. lift fluctuations with the proportional gain *α* for several sensing locations. The variations of *C*_{Lr.m.s.} with *α* are similar to those of *v*_{sr.m.s.} but are not the same. For example, for *x*_{s}=2*d* and 0<*α*≤0.3, *v*_{sr.m.s.} decreases but *C*_{Lr.m.s.} increases, and for *x*_{s}=3.5*d* and *α*=0.1 and 0.2, *v*_{sr.m.s.} increases but *C*_{Lr.m.s.} decreases. These different behaviours of *v*_{sr.m.s.} and *C*_{Lr.m.s.} for some control cases are attributed to the fact that the velocity information at a single location does not perfectly represent the vortex-shedding process. Nevertheless, as shown here, the present P control based on the single-sensor measurement in the wake successfully reduces the r.m.s. lift fluctuations for most cases. Figure 4*b* shows the coefficients of the mean drag and r.m.s. lift fluctuations for the best case of the P control (in terms of minimum *v*_{sr.m.s.}) given each sensing location (see figure 3*a* for optimal *α*). The mean drag is significantly reduced when the sensing locations are *x*_{s}/*d*=1,1.5,2.5 and 3. On the other hand, it is reduced only slightly when *x*_{s}/*d*=2,3.5 and 4. Similar reductions of r.m.s. lift fluctuations are also observed for *x*_{s}/*d*=1,1.5 and 2.5–3.5.

Figure 5 shows the contours of instantaneous spanwise vorticity for different sensing locations and proportional gains. The P control with *x*_{s}=1.5*d* and *α*=−0.15 results in no vortex shedding in the wake (figure 5*b*). The P controls of the cases of (*x*_{s}=2.5*d*,*α*=0.1) and (*x*_{s}=4*d*,*α*=0.1) produce an attenuation and enhancement of vortex shedding, respectively (figure 5*c*,*d*).

As indicated by the study of Park *et al.* [13], the P control that successfully annihilates the vortex shedding at *Re*=60 stabilizes the primary vortex-shedding mode at *Re*=80 but destabilizes a secondary mode. To study this further, we consider a higher Reynolds number of 100. Figure 6*a* shows the variations of *v*_{sr.m.s.} with the proportional gain for different sensing locations *x*_{s}. The values of *v*_{sr.m.s.} at *α*=0 correspond to those without control. As for the case of *Re*=60, the range of *α* at which *v*_{sr.m.s.} is decreased by the control varies depending on the sensing location. That is, negative *α* reduces *v*_{sr.m.s.} for *x*_{s}=*d*, positive *α* for 1.5*d*≤*x*_{s}≤2.5*d* and negative *α* again for *x*_{s}=3.5*d* and 4*d*. The sensing location of 3*d* does not produce any reduction of *v*_{s}. Also, the performance of the P control becomes less effective when the sensing location is farther away from the cylinder centre, and the sensing velocity cannot be reduced to zero by any P control at this Reynolds number, indicating that complete attenuation of vortex shedding is not possible with the present P control. Figure 6*b* shows the coefficients of the mean drag and r.m.s. lift fluctuations for the best case of the P control given each sensing location (see figure 6*a* for optimal *α*). The mean drag is again significantly reduced by 7–12% for *x*_{s}/*d*=1,2,2.5 and 4. On the other hand, it is reduced only slightly when *x*_{s}/*d*=1.5 and 3.5. Note that the overall shapes of and *C*_{Lr.m.s.} are shifted upstream as compared with the case of *Re*=60 because of the reduced vortex-formation region with increasing Reynolds number.

## 4. PI control

In this section, we consider adding an I control to the P control, which is the PI control. The parameters to consider for the PI control are the proportional gain *α*, integral gain *β* and sensing location *x*_{s}. In this study, we do not look for the best configuration of *α*,*β* and *x*_{s}. Instead, we look for the possibility of having a better result by introducing an I control for the case where the P control does not provide large suppression of *v*_{s}. Therefore, we consider the sensing locations of *x*_{s}/*d*=2,2.5,3.5 and 4 for *Re*=60 (figure 3).

Figure 7*a* shows the variations of *v*_{sr.m.s.} with the integral gain *β* for different sensing locations *x*_{s}, where the proportional gains are the ones that provide the maximum reductions of *v*_{sr.m.s.} from the P control (figure 3*a*). The values of *v*_{sr.m.s.} at *β*=0 correspond to those of the best cases of the P controls. As shown in figure 7*a*, the PI control enhances the performance of reducing *v*_{sr.m.s.} for the sensing locations of *x*_{s}/*d*=2,3.5 and 4. Especially, for *x*_{s}/*d*=2 and 3.5, *v*_{sr.m.s.} becomes nearly zero. On the other hand, the PI control is not effective for *x*_{s}=2.5*d*. The addition of the I control increases *v*_{sr.m.s.} for this sensing location (as we show later in this paper, an addition of the D control further reduces *v*_{sr.m.s.} at this sensing location). Figure 7*b* shows the coefficients of the mean drag and r.m.s. lift fluctuations for the best case of the P or PI control given each sensing location (see figures 3*a* and 7*a* for optimal *α* and *β*). Except for *x*_{s}/*d*=2.5, the PI control significantly attenuates the velocity fluctuations at the sensing location, thus reducing the mean drag and lift fluctuations.

Figure 8 shows the temporal variations of the sensing and actuation velocities for the P and PI controls, where *x*_{s}=2*d*,*α*=0.1 and *β*=0.1. The sensing velocity maintains a certain level of fluctuations for the P control, whereas it becomes nearly zero but fluctuates around a very small positive value for and eventually approaches zero for the PI control. In the PI control, the I component produces negative actuation velocity for the present initial flow field (this depends on the initial flow field for the control) and *ψ* reaches nearly zero after a long time. Therefore, the flow field receives the suction and blowing from upper and lower slots (but decreasing in magnitude in time) for a long time and finally reaches the state having nearly no vortex shedding. Figure 9 shows this temporal variation of the flow field for the PI control.

Let us consider a higher Reynolds number of *Re*=100. At this Reynolds number, the P control was not so effective as compared with that at *Re*=60. Hence, we consider the sensing locations of *x*_{s}/*d*=1.5–4 with the PI control. Figure 10*a* shows the variations of *v*_{sr.m.s.} with the integral gain *β* for different sensing locations *x*_{s}, where the proportional gains are the ones that provide the maximum reductions of *v*_{sr.m.s.} from the P control (figure 6*a*; note that *α*=0 for *x*_{s}=3*d*). The values of *v*_{sr.m.s.} at *β*=0 correspond to those of the best cases of the P controls. As shown in figure 10*a*, the PI control enhances the performance of reducing *v*_{sr.m.s.} for *x*_{s}/*d*=1.5,2,3 and 3.5. Especially, for *x*_{s}/*d*=1.5 and 2, *v*_{sr.m.s.} becomes very small. On the other hand, the PI control is not effective for *x*_{s}/*d*=2.5 and 4. Figure 10*b* shows the coefficients of the mean drag and r.m.s. lift fluctuations for the best case of the P or PI control given each sensing location (see figures 6*a* and 10*a* for optimal *α* and *β*). The PI control significantly attenuates the velocity fluctuations at *x*_{s}/*d*=1.5 and 2, thus reducing the mean drag and lift fluctuations. However, for *x*_{s}/*d*=3 and 3.5, the r.m.s. lift fluctuations increase.

Now let us explain, using the concept of the P control with phase shift, why the addition of the I control to the P control enhances the control performance for the cases of *x*_{s}/*d*=2 and 3.5 (*Re*=60), but not for the case of *x*_{s}/*d*=2.5. For the case of *x*_{s}=2*d*,*α*=0.1 and *β*=0.1, *A*=0.1545 and *c*=−1.02 from equation (1.3), where *b*=2*π*/*T* and at *Re*=60 [16]. According to the temporal variation of the transverse velocity (figure 11), the sensing velocity with the phase shift of *c*=−1.02 at *x*_{s}=2*d* approximately corresponds to the velocity at *x*_{s}=2.5*d*, and thus this PI control becomes similar to the P control with *α*=0.1545 based on the sensing at *x*_{s}=2.5*d* and produces a better control result. On the other hand, for *x*_{s}=2*d*,*α*=0.1 and *β*=−0.1, *A*=0.1545 and *c*=1.02 from equation (1.3). Then, this PI control corresponds to the P control with *α*=0.1545 based on the sensing velocity at *x*_{s}≈1.5*d*. This control produces an increase of *v*_{sr.m.s.} (figure 3*a*) and thus the PI control with *β*=−0.1 does not perform well. Similarly, the PI control with *α*=0.05 and *β*=−0.05 for *x*_{s}=3.5*d* produces a much better result than the P control, because the phase shift from the I control is *c*=1.02 and then the corresponding control is the P control with *α*=*A*=0.077 based on the sensing at *x*_{s}≈3*d* (figure 3*a*). On the other hand, for *x*_{s}=2.5*d*, the PI control with *α*=0.15 and *β*=−0.1 gives *A*=0.191 and *c*=0.784, whose corresponding velocity is at *x*_{s}≈2*d*. The performance of the P control with *α*=0.191 at *x*_{s}=2*d* is not better than that at *x*_{s}=2.5*d* and thus the PI control performs worse than the P control alone. However, not all the results shown in figure 7*a* are able to be explained in terms of the phase shift, because some control cases involve relatively large control amplitudes and introduce strong nonlinearity to the system.

## 5. PD control

As discussed above, the PI control was not so effective for the sensing location of *x*_{s}=2.5*d* (*Re*=60). Therefore, in this section, we add a D control to the P control. Figure 12*a* shows the variation of *v*_{sr.m.s.} with the differential gain *γ* for *x*_{s}=2.5*d* and *α*=0.15. The value of *v*_{sr.m.s.} at *γ*=0 corresponds to that of the P control. As shown, the PD control enhances the performance of reducing *v*_{sr.m.s.} for negative *γ*, where *v*_{sr.m.s.} becomes nearly zero. Figure 12*b* shows the temporal variations of the sensing and actuation velocities for the P and PD controls for *x*_{s}=2.5*d*,*α*=0.15 and *γ*=−0.1. Unlike the case of the PI control, the PD control shortens the transient period and makes the sensing velocity quickly go to zero.

The phase analysis for the present PD control with *α*=0.15 and *γ*=−0.1 based on the sensing velocity at *x*_{s}=2.5*d* gives *A*=0.172 and *c*=−0.607 from equation (1.3), with which this PD control corresponds to the P control with *α*=0.172 based on the sensing at *x*_{s}/*d*=2.5–3 and thus produces a better control result (figure 3*a*). On the other hand, with *γ*>0, the PD control becomes the P control based on the sensing at *x*_{s}/*d*=2–2.5 and produces poorer performance than the P control without phase shift.

## 6. PI versus P control with phase shift

As mentioned earlier, the PI or PD control is represented as a P control with phase shift when the system under consideration is linear (see equations (1.1) and (1.3)). In previous sections, we have briefly explained why the introduction of an I or a D control to the P control provides better and worse results than that of the P control alone in terms of the phase shift. In this section, we compute a few cases of the P control with phase shift constructed from successful PI controls and compare their results with those from the corresponding PI controls.

First, we consider the case of *x*_{s}=2*d*, *α*=0.1, *β*=0.1 and *Re*=60. In this case, the P control alone provided some reduction of the sensing velocity fluctuations (figure 3), whereas the PI control resulted in nearly zero sensing velocity fluctuations (figure 7). For these feedback gains and the sensing velocity signal from uncontrolled flow, the corresponding P control with phase shift is *ψ*(*t*)=*Av*_{s}(*t*+*c*), where *A*=0.1545 and *c*=−1.02. Figure 13 shows the temporal variations of the sensing velocity and lift coefficient for the PI control and the P control with phase shift. As shown, the result of the P control with phase shift reduces the sensing velocity and lift fluctuations, but not as much as the PI control does. Another case we show here is *x*_{s}=1.5*d*,*α*=0.2 and *β*=0.4 for the higher Reynolds number of *Re*=100. For the P control with phase shift, *A*=0.4351 and *c*=−1.06. This P control with phase shift again reduces the sensing velocity fluctuations (though not as much as the PI control does) but rather increases the lift fluctuations (figure 14).

We considered a few other cases for comparison and observed differences in the control results between the PI control and the P control with phase shift. However, the results from most of the P controls with phase shift were quite successful, indicating that the flow under consideration is weakly nonlinear at this low Reynolds number range. It is not straightforward to judge the effect of the Reynolds number, because the optimal values of control parameters change for different Reynolds numbers. Nevertheless, we expect that the P control with phase shift should produce more and more different results at higher Reynolds numbers as compared with the results from the corresponding PID control.

## 7. Conclusions

In the present study, we applied the P, PI and PD controls to flow over a circular cylinder at the Reynolds numbers of 60 and 100. The transverse velocity at a centreline location in the wake was measured for sensing, and the actuation velocity was determined from the P, PI or PD control. The actuation was given from the upper and lower slots on the cylinder surface and provided zero net mass flow rate to the flow. In this study, we showed that the P control itself is quite effective for the present flow because this flow is subject to global instability as discussed in Park *et al.* [13]. Although the control was sensitive to the proportional gain, it completely suppressed the vortex shedding for some sensing regions at *Re*=60. For a higher Reynolds number of 100, the effective sensing region for successful control was shifted upstream and became narrower than that for *Re*=60. The addition of an I or a D control to the P control, for the cases for which the P control did not completely suppress the vortex shedding or was not so successful in reducing the sensing velocity fluctuations, successfully reduced the velocity and lift fluctuations and the mean drag.

In the present study, we showed that the PI and PD controls can be formulated in terms of the P control with phase shift when the system under consideration is linear. The P controls with phase shift constructed from successful PI controls were tested and compared with those PI controls. Owing to the low Reynolds number range considered in the present study, the P control with phase shift also produced successful attenuations of the vortex shedding and lift fluctuations, although the results were not as good as those of the PI control. However, most flows at high Reynolds number contain multiple dominant frequencies and their nonlinear interactions. The simple analysis conducted here may not be applicable to those flows and thus it should be interesting to apply the present PID control to nonlinear flows such as turbulent channel flow.

Choi *et al.* [17] introduced the opposition control in turbulent channel flow for skin friction reduction. The sensing variable *v*_{s} was the wall-normal velocity near the wall and the actuation *ψ* was the blowing and suction at the wall: *ψ*=−*v*(*y*^{+}≈10), where *y*^{+}=*yu*_{τ}/*ν*, *y* is the wall-normal distance, *u*_{τ} is the wall shear velocity and *ν* is the kinematic viscosity. The skin friction on the wall was significantly reduced by 25 per cent. This control is a P control whose control results are sensitive to the sensing location. Even for the most successful case in their study, the sensing velocity fluctuations did not go to zero; in other words, there existed a steady-state error owing to the characteristics of the P control. Therefore, an addition of I, D or ID control to this P control may be useful in further reducing the skin friction on the wall. Recently, Kim & Choi [18] applied a PI control to turbulent channel flow and showed that the steady-state error is significantly reduced. Further studies are needed to examine the applicability and effectiveness of the PID control to nonlinear fluid flows.

Although we obtained successful results from the P, PI and PD controls for flow over a bluff body, it is still not very clear why and how the sensing velocity fluctuations and vortex-shedding strength were reduced by these controls. For example, at *Re*=60 and *x*_{s}=1.5*d*, the P control with *α*=−0.2 provided a complete attenuation of vortex shedding, but that with *α*=−0.3 increased the sensing velocity fluctuations. Although this behaviour with the feedback gain agrees with the general trend of the P control, the detailed reason is yet to be found. The answer may be searched for from the understanding of the flow system, its modelling and its response to the controls (e.g. [4,19–22]), which should also significantly reduce the efforts on finding optimal values of feedback gains and sensing positions. This line of research is currently under way.

## Acknowledgements

This work was supported by the NRL (grant no. R0A-2006-000-10180-0) and WCU (grant no. R31-2008-000-10083-0) programmes of KRF, MEST, Korea.

## Footnotes

One contribution of 15 to a Theme Issue ‘Flow-control approaches to drag reduction in aerodynamics: progress and prospects’.

- This journal is © 2011 The Royal Society