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Feedback control of the laminar-turbulent transition onset in a boundary layer by suction
Experimental Thermal and Fluid Science 16 (1998) 22±31
Feedback control of the laminar-turbulent transition onset in a boundary layer by suction Philippe Mouyon *, Gregoire Casalis, Alain Seraudie, Sylvain Prudhomme ONERA/CERT, B.P. 4025, F31055 Toulouse Cedex, France Received 14 October 1996; received in revised form 28 February 1997; accepted 3 March 1997
Abstract The boundary layer on a ¯at plate is laminar in a region close to the leading edge and becomes turbulent further downstream. The streamwise position of the onset of the transition depends on dierent parameters among them the magnitude of the incident velocity. We assume that the ¯at plate is equipped with a porous zone which allows a suction of the boundary layer. Again, the onset of transition depends strongly on the suction velocity. Acting on this velocity allows an active control of the position of the transition. Only the transition location is assumed to be measured. The paper develops a controller to keep the transition point ®xed in a slightly unsteady ¯ow by controlling the suction velocity on the panel. The aerodynamical modelling is presented and approximations are physically justi®ed. Low cost numerical simulation aspects are discussed. A low order simpli®ed model of the ¯ow is identi®ed from numerical simulations. It is used for the control law design. The system is then computationally tested. Feasibility of the active control is proved, but the closed loop behavior is not conform to the one expected. We conjecture that the simpli®ed model does not retain suciently rich information to allow good previsions of the closed loop properties. Correlatively the closed loop seems to be very sensitive to dierent varying quantities such as delay, unmodelized actuator and sensor dynamics, and control sampling period. Ó 1998 Elsevier Science Inc. All rights reserved. Keywords: Boundary layer; Laminar-turbulent transition; Feedback control
1. Introduction An in®nitely thin ¯at plate is assumed to be placed without incidence in a time varying one-dimensional ¯ow. The incident velocity U1 is assumed to be suciently small in order to neglect compressibility. If U1 is constant, then the laminar boundary layer becomes linearly unstable: for selected frequencies the Tollmien±Schlichting (TS) waves grow downstream. Some of these waves reach an important amplitude which is sucient to generate strong nonlinear resonances. The instability waves become fully nonlinear and break down ®nally into turbulent spots. It is the classical picture of the onset of the laminar-turbulent transition of the ¯at plate boundary layer. On the other side it is well known that the skin friction is higher for turbulent boundary layer than for laminar boundary layer. Therefore for drag reduction problems, it is clear that the objective is to maintain *
Corresponding author. Tel.: 33 0562252785; fax: 33 0562252564; e-mail: [email protected]. 0894-1777/98/$19.00 Ó 1998 Elsevier Science Inc. All rights reserved. PII: S 0 8 9 4 - 1 7 7 7 ( 9 7 ) 1 0 0 0 4 - 8
the boundary layer laminar as long as possible. However, when U1 varies in time, the position XT of the laminar-turbulent transition can move on a large distance. In some cases it can be useful to be able to control XT , for example to ®x it at a prescribed value XTd in spite of the variations of U1 . The ¯at plate is assumed to be equipped with a suction panel that allows a suction of the boundary layer and with sensors that deliver the transition location. The problem addressed in this paper is the design of an output feedback law to control XT using the suction velocity Vp . The study is fully numeric and prepares an experimental work devoted to the same task. Our control approach involves several dierent steps: statement of a detailed aerodynamical modelling of the experiment, development of a low time consuming numerical simulation tool, identi®cation of a low dimensional linear model, control law design and tuning, performances and robustness analysis. Let us recall that experiments with this kind of control law have already been presented in [1]. The sensors used were microphones that measure the passing of turbulence spots.
P. Mouyon et al. / Experimental Thermal and Fluid Science 16 (1998) 22±31
The main interest of a numerical approach is the possibility of easily analyzing the impact of dierent parameters (geometrical and aerodynamical characteristics, sensors and actuators performances, control law) on the transition location. Also it could lead to a better understanding of the dynamic extend of the laminar boundary layer. In particular we show that although the simulation may be used for the feedback control design, reducing the model order implies taking care of high frequency unretained modes, even for the control of such a global event. 2. Aerodynamical aspects The general con®guration is reported on Fig. 1, where the boundary layer height has been sketched out in dotted lines. The x coordinate denotes the streamwise position and y the distance from the wall. The ¯ow is assumed to be two-dimensional and the ¯uid incompressible. Let us recall that the incident velocity U1 can vary in time. For simpli®cation, we assume that the inviscid velocity is everywhere identical to U1 , consequently, the inviscid velocity is a function of the time t only. Its variations are assumed to be low in terms of time scale. More precisely, the typical frequency will be of the order of 1 Hz, which is small compared to the scale of the unstable TS waves frequencies. This assumption allows us to numerically integrate the unsteady boundary layer equations and, at each time step, to make a classical stability analysis for which the previous boundary layer is assumed to be steady (at the corresponding time value). 2.1. Inviscid region The Euler equations of the inviscid region must be solved in order to give the external conditions for the boundary layer calculation. Taking into account the above mentioned assumptions, the Euler equations reduce to Eq. (1), where q and P are the ¯ux density and the pressure. U1 is assumed to be known, so that the pressure can be calculated, @U1 1 @P 0: @t q @x
Fig. 1. General con®guration.
1
23
2.2. Boundary layer With respect to the x and y coordinate, the velocity components are denoted by U and V . m is the kinematic viscosity. The boundary layer equations are written for a two-dimensional incompressible ¯ow. Then the pressure is eliminated by Eq. (1). We obtain @U @t
U
@U @V @x @y @U V @U @x @y
2
ÿ m @@yU2
0;
@U1 @t
:
2
The ®rst boundary conditions associated to Eq. (2) are the no slip conditions at the wall, except on the porous zone where suction can be imposed: Vp results from the control law 8t; x U x; 0; t 0;
3 V x; 0; t Vp t: Then there is the classical link condition: the ¯ow is inviscid outside the boundary layer and link must be written with the inviscid ¯ow values (4). The system (2) is parabolic in x, so that a speci®ed velocity pro®le must be imposed at a prescribed ®rst station in x. Noting x0 this ®rst station, the x-initial condition is given by Eq. (5). Finally, concerning the time dependence, Eq. (6) gives an initial value of the complete ®eld for t 0: 8t; x
lim U x; y; t U1 t;
y!1
4
~ x0 ; y; t U ~0 y; t; U 5 ~ ~ 8x; y U x; y; 0 Uinit x; y: 6 The system (2) coupled with boundary conditions (3)± (5) and the initial condition (6) is numerically solved, so that at this step U and V are assumed to be known at each time integration step, when U1 and Vp are speci®ed. 8t; y
2.3. Linear stability analysis The previous boundary layer calculation is fully laminar and does not take into account instability waves which are always present. The linear stability analysis [2] is applied to calculate these ¯uctuations. A small perturbation is added to the main boundary layer ¯ow previously calculated. It is the unknown of the stability problem. In order to simplify its evaluation, two assumptions are made: the quasi parallel ¯ow and the quasi steady ¯ow approximations. The ®rst approximation is justi®ed by noting that the main boundary layer ¯ow has a weak x dependence compared to the rapid y dependence, at least suciently far away from the leading edge. This approximation is completely well-founded for the ¯at plate under consideration in this work. The second approximation is not so classical. The typical frequencies of the time evolution of the inviscid ¯ow and thereby of the main boundary layer ¯ow is of the order of 1 Hz. In steady conditions and for U1 in the order of 50 m/s, the typical frequencies of the TS waves are of the order of 1000 Hz. For this reason, stability analysis can
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P. Mouyon et al. / Experimental Thermal and Fluid Science 16 (1998) 22±31
be carried out as if the main boundary layer ¯ow was strictly time independent. With these two approximations for linear stability analysis, it is possible to seek for the perturbation in the mathematical form of a normal mode (7). In the spatial theory, which is considered in this study, a is a complex number and x is a real one. Separating real ar and imaginary part ai of a, the form (7) can be written as Eq. (8): ^ exp i ax ÿ xt ~ u y ufluct ~ ^ exp ÿai x exp i ar x ÿ xt: ~ u y ufluct ~
7
8 The ®rst factor, the amplitude function u^, represents the evolution of the ¯uctuation across the boundary layer. The last term is periodic and shows the wave nature of the ¯uctuation: ar is a wave number, x=2p gives the frequency of the perturbation. These waves are the TS waves. Finally the second term is real and represents the growth rate of the perturbation. It indicates if the perturbation is increasing downstream (the main ¯ow is unstable) or decreasing (the main ¯ow is stable) depending on the sign of ai . It is possible to de®ne an amplitude A of the perturbation by 1 dA : 9 A dx Introducing the form (7) of the perturbation in the linearized Navier±Stokes equations leads to the famous Orr±Sommerfeld equation (the main boundary layer plays the role of data in this problem). It is an eigenvalue problem. A nonzero solution exists only if a and x satisfy a certain relation (10) which implicitly depends on x. a is completely determined at each x position for a speci®ed frequency. Therefore the question of the stability of the main ¯ow is given by solving Eq. (10), ÿ ai
a F x; x:
10
2.4. Laminar-turbulent transition The linear stability analysis is not sucient to predict the onset of the laminar-turbulent transition. For steady ¯ows this prediction can be achieved with the semi-empirical criterion: the en -method [3]. As indicated by Eq. (10), ai depends on x and x. Let x0 x be the ®rst abscissa for which the wave of circular frequency x starts to be ampli®ed. Then the n factor is de®ned by Eq. (11). Moreover since ai represents a growth rate (Eq. (9)), n is also given by Eq. (12) where A0 is the amplitude of the wave at x x0 x: Zx n x sup ÿ ai X ; x dX ; 11 x
x0 x
A x : 12 A0 A0 is generally assumed to be independent of x. Then the idea of the so-called en -method is to assume that the transition occurs for the abscissa XT given by Eq. (13) where nT characterizes the quality of the ¯ow
n x ln
(noise level, turbulence level. . .) of the wind tunnel where the experiment is carried out. The more quiet the installation is, the higher nT is, n XT nT : 13 We justify now the use of the en -method for the unsteady ¯ow considered here. Let us ®rst recall that due to the relatively low frequency variations of the main ¯ow, the ¯uctuations are well approximated by the normal mode decomposition (7), and so a depends slowly on the time. Note also that for subsonic two-dimensional main ¯ows, the velocity of the TS waves convection is in the order of U1 =3. Therefore, if U1 is around 45 m/s, the characteristic time length for the TS waves to be convected from the leading edge up to the onset of the transition (which occurs typically for x close to 0:5 m) is of the order of 0:033 s. Since this time scale is small compared to the main ¯ow variations time scale (1 s), then the main ¯ow may be considered as constant during the convection of the TS waves. Consequently, n may be calculated at each time step as if the mean ¯ow was strictly constant (but with the constant reevaluated at each time step). The validity of this second approximation is not so clear as the previous one about the normal mode decomposition. If the frequency of the U1 variations exceeds a few hertz, the main ¯ow can not be assumed to be constant during the propagation of the TS waves and the validity of the proposed modelling and the numerical procedure break down. 2.5. Numerical procedure The unsteady boundary layer equations are solved with a classical ®nite volume method [4,5]. The main dif®culty related to this numerical integration is the evaluation of the nonlinear convection terms. It is usually done with an extrapolation which gives satisfactory results for steady ¯ows. But for our unsteady problem, this procedure implies a very small time integration step: CFL number less than 0:1. We introduce an iterative procedure to evaluate the nonlinear convection terms that allows to increase it by twenty. In regard to the determination of the growth rates ai , solving the Orr±Sommerfeld equation leads to gigantic CPU time. Fortunately a special simpli®ed method (the parabola method) has been developed [6]. The growth rates are estimated with analytical functions, the coecients of which have been determined from exact stability computations. The dierent numerical parameters are ®xed as follows. The initial ®eld (Eq. (6)) is given by the Blasius similarity pro®le. The same pro®le is used at x0 0:05 m for the ®rst x-station, see boundary condition (5). Suction can be applied between x 0:2 m and x 0:25 m. Viscosity is equal to m 0:1515 10ÿ4 m2 /s. Finally transition is assumed to occur when the n factor reaches the value nT 4, (Eq. (13)). We give some results in order to indicate the general behavior of the numerical code with a prescribed suction
P. Mouyon et al. / Experimental Thermal and Fluid Science 16 (1998) 22±31
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velocity Vp t (open loop simulations). In Fig. 2 sharp suction velocity modi®cations are imposed at t 0:1 s and at t 0:25 s while U1 is ®xed to 42 m/s. The stabilizing eect of suction is evident. Note that 0.05 s is necessary for XT to be ®xed: it is the transient phase duration. When suction is stopped, XT comes back to the initial position but two times more rapidly. Fig. 3 shows the evolution of XT , when U1 t is a periodic function, the suction velocity being ®xed to zero. It can be observed that XT t is not exactly periodic. The small irregularities correspond to the times when dU1 =dt, and hence the pressure (see Eq. (1)), are maximum. 3. Control The feedback control design has two main objectives. First we want to bring XT to a desired location XTd , then we want to maintain XT at this value in spite of U1 ¯uctuations. The ®rst objective is a transition problem from an equilibrium point to another, while the second is a disturbance rejection problem. In order to design the control law, we ®rst need to de®ne, from an automatician point of view, what is the system to control, then to specify our performance requirements. Next a set of open loop numerical simulations is carried out which leads us to identify a simpli®ed model of the ¯ow (more precisely of the phenomena under interest, that is to say of the evolution of XT versus Vp and U1 ). This model enables us to specify a control law structure. The tuning of this law is exposed. With the help of another set of numerical simulations (closed loop simulations) a performance analysis is performed as well as a robustness analysis to delay and sampling rate.
Fig. 2. XT open loop response to Vp step variations. U1 42 m/s.
Fig. 3. XT open loop response to U1 periodic variations. Vp 0.
3.1. Speci®cations We consider that XT , the variable to be controlled, is measured and that the control variable is the vertical velocity Vp on the plate along the suction area (Fig. 1). The block diagram is shown in Fig. 4, note that XT and Vp are not available in practice. For instance the control variable will be in fact the desired suction rate sent to the servovalve mechanism. It will result in a vertical velocity through an unmodelized dynamic. In regard to the measure, an array of hot-®lms will be used. Thus XT will have to be estimated from the friction measurements, which will also introduce delay and inaccuracy. So the problem under investigation is idealized, and robustness to unmodelized phenomena will have to be checked. After the system de®nition, the speci®cation of the various elements frequency pass bands is another important stage. The numerical simulation of the ¯ow that we used in this study takes into account the variations of
Fig. 4. Closed loop block diagram.
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P. Mouyon et al. / Experimental Thermal and Fluid Science 16 (1998) 22±31
U1 up to a few hertz. The pass-band of the sucking up mechanism that will be used to produce Vp will also be equal to a few hertz. Consequently the study is restricted to the frequency band 0; fmax with fmax 5 Hz. The control law is designed in the continuous time domain. Unless otherwise informed, the control sampling rate used, is equal to the time integration step of the ¯ow. Nevertheless robustness to a slower control sampling rate is analyzed in the end. Several simulations have been done with constant U1 and Vp . As it is shown in Figs. 5 and 6, the in¯uence of U1 and Vp on the steady state value of XT is almost linear along the range under interest: 40; 50 m/s for U1 and 0; ÿ5 10ÿ3 m/s for Vp . This gives a static model of the ¯ow that can be written as 14 XT a bU1 cVp : Coecients a, b and c are estimated by a least mean square approach over the above mentioned range. We found the following values: a 0:884, b ÿ0:009 and c ÿ18:839. The accuracy of the static model is about 3%, and its validity decreases when the ratio Vp =U1 decreases. It is worth noting that the speci®ed range of variations for U1 induces a maximum of 9 cm static change for XT while Vp produces variations of at most 9.4 cm. Thus the static control ability is well dimensioned. 3.2. Dynamic model Previous simulations (Fig. 2) show that the 95% response time of the transfer function XT =Vp is about 0.02 s, which corresponds to a natural frequency of about 25 Hz. But the static model seems to be valid at most up to a few hertz. A more accurate model must be found for a control design. For this purpose, a step response identi®cation procedure is carried out around a nominal equilibrium point characterized by a mean ¯ow velocity of 45 m/s, and Vp about 5 10ÿ3 m/s. The following model structure is assumed
Fig. 5. Equilibrium study. U1 40, 45 and 50 m/s.
Fig. 6. Equilibrium study. Vp 0, ÿ1; . . . ; ÿ510ÿ3 m/s.
g Vp : 15 1 ss The coecients g and s result from an optimization scheme that minimizes the output error between the complete and the simpli®ed simulations. Optimal values are: g ÿ17:5 and s 0:006. The cut-o frequency of the identi®ed transfer is equal to 25 Hz. Comparison between the ¯ow simulation and the identi®ed model (15) is done in Fig. 7 when U1 45 m/s and in Fig. 8 when U1 50 m/s. Positive and negative steps with magnitude 10ÿ3 m/s are applied to Vp , for dierent initial conditions on XT . In regard to the dynamic response to U1 variations, we note that the validity of such a model is restricted to rather slow and small excursions around the nominal point. Fig. 9 shows a typical open loop time response to ®ltered step disturbance. Starting from 45 m/s, U1 is increased up to 48 m/s following an integrated squared XT a bU1
Fig. 7. XT =Vp step responses. Comparison of aerodynamical and ®rst order models. U1 45 m/s.
P. Mouyon et al. / Experimental Thermal and Fluid Science 16 (1998) 22±31
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3.3. Control law design With the ®rst order dynamic model, a proportional integral controller is necessary to assign all the closed loop eigenvalues. The control law structure is given by Eq. (16), where XTr is the reference value of XT . Then Eq. (17) gives the closed loop transfer function ki Vp ÿ kp XT ÿ XTr ; s g kp s ki XTr XT 2 ss 1 gkp s gki s 2 a bU1 : ss 1 gkp s gki
Fig. 8. XT =Vp step responses. Comparison of aerodynamical and ®rst order models. U1 50 m/s.
sine variation: U1 t U1 dU1 s ÿ 0:5 sin 2ps with s t ÿ t0 fmax if t0 < t < 1=fmax . This expression is chosen because its second order derivative exists everywhere, and the simulation is rather sensitive to discontinuous variations of dU1 =dt. Besides this kind of disturbance may modelize a blast. Even for variations in the order of a few meters per second with a limited bandwidth of a few Hertz, this simulation clearly shows a highly nonlinear and nonminimum phase behavior, with a very large deviation (almost 20 cm). In fact it seems that the most important parameter is not U1 but dU1 =dt. Nevertheless in a ®rst approach the in¯uence of the perturbation U1 on the measured variable XT needs not to be precisely modelized. And so we have not tried to improve the XT =U1 transfer function modelling.
16
17
Local stability of the closed loop system is ensured if the denominator has the following structure s2 2nxc s x2c , with xc 2pfc . So we take: ki sx2c =g and kp 2nxc s ÿ 1=g. In order to completely reject disturbances up to fmax 5 Hz, let us choose fc 25 Hz. We obtain ki ÿ8:46 and kp ÿ0:056. This choice is interesting since it does not modify the system dynamic in comparison to the open loop. Furthermore since fc > fmax , very good performances are expected over all the frequency range 0; fmax . The most sensitive parameter is the ratio ki =kp that corresponds to the zero introduced by the controller. Here it is chosen rather close to the null pole, which explains its prominent part. It can be chosen slower in order to take into account an unmodelized dynamic. For example with a 5 Hz pass band actuator, ki =kp is chosen in the order of 30. Then kp is tuned to get the desired closed loop pass band (kp ÿ0:06 gives fc 25 Hz). Nevertheless this does not signi®cantly modify the following results. From Eq. (17) note that the pass-band towards XTr is the stop-band towards U1 . To better reject the dynamic disturbance eect (increase the attenuation in the stopband), we are interested in increasing fc , or equivalently the controller gain. In order to avoid an excessive suction when a large equilibrium transition is asked for, we introduce a dynamic reference model. In other words XTr is not equal to XTd but is a time dependent trajectory XTr t which asymptotically tends towards XTd . The dynamic reference model used in the simulations is a second order system with unit damping factor and a cuto frequency equal to 80 rad/s (its 95% response time is 0.05 s). Finally, in order to avoid training error and also to improve the stability in case of control saturation, the integral term of the control law is frozen when the control is close to the saturation. Typically if Vp is greater than 90% of its nominal value, then the integration is stopped. 4. Results
Fig. 9. Response to a ®ltered step disturbance. Vp 0.
With the initial tuning of the control law, the dierence between the actual and the reference values of XT is generally less than one centimeter. This is true at
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P. Mouyon et al. / Experimental Thermal and Fluid Science 16 (1998) 22±31
low frequency, as well in regulation with small magnitude disturbances as in transition from one equilibrium to another. 4.1. Equilibrium change The control law was found to be highly successful in reference following with constant U1 . Fig. 10 depicts the system response to a requisite transition from the nonsucked equilibrium (Vp 0) to the desired value XTd 0:573 m (which corresponds to Vp ÿ0:005 m/ s). The overall variation of XT is greater than 10 cm. The transition location follows the reference with a small training error. We check that the closed loop pass-band is large enough to follow the reference trajectory. The same reference variation was simulated with dierent incident velocities (U1 40; 45 and 50 m/s). Maximal dierences between XT and its reference are 0.7, 1.5 and 2.2 cm respectively (Fig. 11). This error increases with U1 , and a rather slow mode appears.
Fig. 11. Impact of U1 on the reference following.
4.2. Step disturbance rejection The closed loop system behavior is now studied in the presence of a ®ltered step disturbance (cf. Section 3.2). The objective is to maintain the transition location at the desired value XTd . Controller and system responses to a change in mean ¯ow velocity from 45 to 46 m/s are shown in Fig. 12. The step duration corresponds to a 4 Hz frequency. The deviation from the desired value XTd 57:3 cm is less than 0.5 cm. This deviation increases with the U1 step magnitude. It reaches 1.1 cm when the step magnitude is 3 m/s. Fig. 13 depicts the case where the step duration is two times shorter (the relevant frequency is 8 Hz). The deviation of XT is more important and reaches 3.4 cm. But it is still three times smaller than for the uncontrolled ¯ow.
Fig. 12. Response to a slow and small step disturbance.
Fig. 13. Response to a large and fast step disturbance.
4.3. Periodic disturbance rejection
Fig. 10. Reference following.
A sinusoidal variation of U1 was then simulated with frequency 1 Hz and amplitude 1 m/s. The system response shows only a few residual high frequency oscillations (Fig. 14). When the frequency is increased up to 5 Hz, residual oscillations on XT reach 2.5 cm. But with
P. Mouyon et al. / Experimental Thermal and Fluid Science 16 (1998) 22±31
Fig. 14. Slow sinusoidal disturbance rejection.
the same disturbance the open loop oscillation magnitude exceeds 6 cm. A comparison of the time histories is illustrated on Fig. 15. The left side of the ®gure is concerned with the open loop behavior, while the right side deals with the closed loop behavior. So we have shown that for rather low frequency oscillations of U1 , the controller reduces the variations of XT within a reasonable level. Previous analysis is now performed at dierent frequencies. The reduction rate, in comparison with the noncontrolled ¯ow depends on the U1 oscillation frequency, but is of the order of 80% below 5 Hz (Fig. 16). However, it is noticeable that the disturbance rejection is not obtained all over the frequency band 0; fc as expected from the control design. In fact, simulations show that higher frequency components are found in the ¯ow when the control is used. It seems that the complexity of the controlled ¯ow is greater than would originally be suggested from consideration of the uncontrolled ¯ow. Attempts to control the ¯ow at low frequency tend
Fig. 15. Open and closed loop response with sinusoidal disturbance.
29
Fig. 16. Closed loop attenuation of XT variations.
to excite high frequency components. It is a kind of modal spillover phenomena that has already been observed in active control of turbulent ¯ow [7]. Nevertheless note that the validity of the simulation is not so clear when the frequency goes above a few hertz. 4.4. Robustness analysis The experiment will involve several systems that have not yet been modelized. A more realistic block diagram of the experiment is shown in Fig. 17, where q is the suction rate, qc its desired value computed by the controller and X^T the transition location estimated from hot-®lms measurements. So robustness of the closed loop must be investigated. In regard to delay, Fig. 18 shows an important deterioration of the response in the presence of a rather small delay (10 ms). The U1 oscillation frequency is 4 Hz. There are three curves for XT . The dotted lines represent the open loop and the nominal closed loop (no delay). The solid line depicts the closed loop behavior with delay. Once again, performances degradation is due to the appearance of high frequencies components. A sampling of the control is then introduced (Fig. 19). U1 is always a 4 Hz sine curve. Even at a rather high sampling rate (66 Hz), the XT response is very much aected. This study was carried out at 4 Hz only.
Fig. 17. Detailed block diagram.
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P. Mouyon et al. / Experimental Thermal and Fluid Science 16 (1998) 22±31
Fig. 19. Robustness to sampling rate.
Fig. 18. Robustness to delay.
At lower frequencies the degradation is not so important. Nevertheless the closed loop seems not to be robust enough and must be improved. 5. Practical signi®cance This research shows that the development of a low time consuming numerical simulation for the laminarturbulent transition study in unsteady ¯ow is now possible. This kind of simulation may be easily used for feedback control design. However, a lot of attention must be paid to the ampli®cation of high frequency components when the control is used in order to reduce low frequency ¯uctuations. 6. Conclusions In this paper we present an aerodynamical modelling of the boundary layer that develops along a ¯at plate and more especially of the laminar-turbulent transition. A numerical procedure is proposed which allows closed loop simulations of suction experiments. Computation time is reasonable (in the order of 10 mn for 1 s) on a standard computer station.
A simpli®ed dynamic model is identi®ed between the transition location XT and the suction velocity Vp . Then a control law is designed on this basis. Simulations show that under generic experimental conditions (measurement of the current transition location, and control of the vertical velocity along the suction area), the control law leads to a 80% reduction of the XT ¯uctuations when U1 presents harmonic variations up to 5 Hz. Nevertheless high frequency components are found in the ¯ow when the control is used. They lead to limited performances, and could explain the lack of robustness. This discussion raises the question of whether it is possible to identify a low order model of the ¯ow which retains enough information for a robust control design or whether another approach must be found. Nomenclature fc kp ki n nT P
closed loop desired cut-o frequency, Hz proportional controller gain, dimensionless integral controller gain, dimensionless maximal growth rate over frequencies, dimensionless growth rate value at the transition, dimensionless pressure, Pa
P. Mouyon et al. / Experimental Thermal and Fluid Science 16 (1998) 22±31
t ~ ufluct U U1 V Vp x XT XTd XTr t y ai m q
time, s ¯ow ¯uctuation, m/s x-velocity component of the mean ¯ow, m/s incident velocity, m/s y-velocity component of the mean ¯ow, m/s suction velocity, m/s downstream coordinate, m laminar turbulent transition location, cm desired transition location, cm reference transition trajectory, cm distance from the wall, m growth rate of the ¯uctuation, % kinematic viscosity, m2 /s ¯ux density, kg/m3
References [1] J.-L. Rioual, P.A. Nelson, M.J. Fisher, Experiments on the automatic control of boundary-layer transition, Journal of Aircraft 31 (6) (1995) 1417±1418.
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[2] L.M. Mack, Boundary layer stability theory, Jet Propulsion Lab., California Inst. of Techn., Pasadena, California, 1969. [3] D. Arnal, Boundary layer transition: predictions based on linear theory, Special course Agard on Progress in Transition Modeling, 1993. [4] R. Michel, J. Cousteix, R. Houdeville, Unsteady turbulent shear ¯ows, in: IUTAM Symposium Toulouse, Springer, Berlin, 1981. [5] J. Cousteix, A. Desopper, R. Houdeville, Recherche sur les couches limites turbulentes instationnaires, La Recherche Aerospatiale, 3, 1977. [6] D. Arnal, Transition prediction in transonic ¯ow, in: Zierep et Oertel, (Ed.), IUTAM Symposium Transonicum III, Gottingen, Springer, Berlin, 1988. [7] K.Y. Tang, W.R. Graham, J. Peraire, Active ¯ow control using a reduced order model and optimum control, 27th AIAA Fluid Dynamics Conference, New Orleans, LA, 1996.
Feedback control of the laminar-turbulent transition onset in a boundary layer by suction Philippe Mouyon *, Gregoire Casalis, Alain Seraudie, Sylvain Prudhomme ONERA/CERT, B.P. 4025, F31055 Toulouse Cedex, France Received 14 October 1996; received in revised form 28 February 1997; accepted 3 March 1997
Abstract The boundary layer on a ¯at plate is laminar in a region close to the leading edge and becomes turbulent further downstream. The streamwise position of the onset of the transition depends on dierent parameters among them the magnitude of the incident velocity. We assume that the ¯at plate is equipped with a porous zone which allows a suction of the boundary layer. Again, the onset of transition depends strongly on the suction velocity. Acting on this velocity allows an active control of the position of the transition. Only the transition location is assumed to be measured. The paper develops a controller to keep the transition point ®xed in a slightly unsteady ¯ow by controlling the suction velocity on the panel. The aerodynamical modelling is presented and approximations are physically justi®ed. Low cost numerical simulation aspects are discussed. A low order simpli®ed model of the ¯ow is identi®ed from numerical simulations. It is used for the control law design. The system is then computationally tested. Feasibility of the active control is proved, but the closed loop behavior is not conform to the one expected. We conjecture that the simpli®ed model does not retain suciently rich information to allow good previsions of the closed loop properties. Correlatively the closed loop seems to be very sensitive to dierent varying quantities such as delay, unmodelized actuator and sensor dynamics, and control sampling period. Ó 1998 Elsevier Science Inc. All rights reserved. Keywords: Boundary layer; Laminar-turbulent transition; Feedback control
1. Introduction An in®nitely thin ¯at plate is assumed to be placed without incidence in a time varying one-dimensional ¯ow. The incident velocity U1 is assumed to be suciently small in order to neglect compressibility. If U1 is constant, then the laminar boundary layer becomes linearly unstable: for selected frequencies the Tollmien±Schlichting (TS) waves grow downstream. Some of these waves reach an important amplitude which is sucient to generate strong nonlinear resonances. The instability waves become fully nonlinear and break down ®nally into turbulent spots. It is the classical picture of the onset of the laminar-turbulent transition of the ¯at plate boundary layer. On the other side it is well known that the skin friction is higher for turbulent boundary layer than for laminar boundary layer. Therefore for drag reduction problems, it is clear that the objective is to maintain *
Corresponding author. Tel.: 33 0562252785; fax: 33 0562252564; e-mail: [email protected]. 0894-1777/98/$19.00 Ó 1998 Elsevier Science Inc. All rights reserved. PII: S 0 8 9 4 - 1 7 7 7 ( 9 7 ) 1 0 0 0 4 - 8
the boundary layer laminar as long as possible. However, when U1 varies in time, the position XT of the laminar-turbulent transition can move on a large distance. In some cases it can be useful to be able to control XT , for example to ®x it at a prescribed value XTd in spite of the variations of U1 . The ¯at plate is assumed to be equipped with a suction panel that allows a suction of the boundary layer and with sensors that deliver the transition location. The problem addressed in this paper is the design of an output feedback law to control XT using the suction velocity Vp . The study is fully numeric and prepares an experimental work devoted to the same task. Our control approach involves several dierent steps: statement of a detailed aerodynamical modelling of the experiment, development of a low time consuming numerical simulation tool, identi®cation of a low dimensional linear model, control law design and tuning, performances and robustness analysis. Let us recall that experiments with this kind of control law have already been presented in [1]. The sensors used were microphones that measure the passing of turbulence spots.
P. Mouyon et al. / Experimental Thermal and Fluid Science 16 (1998) 22±31
The main interest of a numerical approach is the possibility of easily analyzing the impact of dierent parameters (geometrical and aerodynamical characteristics, sensors and actuators performances, control law) on the transition location. Also it could lead to a better understanding of the dynamic extend of the laminar boundary layer. In particular we show that although the simulation may be used for the feedback control design, reducing the model order implies taking care of high frequency unretained modes, even for the control of such a global event. 2. Aerodynamical aspects The general con®guration is reported on Fig. 1, where the boundary layer height has been sketched out in dotted lines. The x coordinate denotes the streamwise position and y the distance from the wall. The ¯ow is assumed to be two-dimensional and the ¯uid incompressible. Let us recall that the incident velocity U1 can vary in time. For simpli®cation, we assume that the inviscid velocity is everywhere identical to U1 , consequently, the inviscid velocity is a function of the time t only. Its variations are assumed to be low in terms of time scale. More precisely, the typical frequency will be of the order of 1 Hz, which is small compared to the scale of the unstable TS waves frequencies. This assumption allows us to numerically integrate the unsteady boundary layer equations and, at each time step, to make a classical stability analysis for which the previous boundary layer is assumed to be steady (at the corresponding time value). 2.1. Inviscid region The Euler equations of the inviscid region must be solved in order to give the external conditions for the boundary layer calculation. Taking into account the above mentioned assumptions, the Euler equations reduce to Eq. (1), where q and P are the ¯ux density and the pressure. U1 is assumed to be known, so that the pressure can be calculated, @U1 1 @P 0: @t q @x
Fig. 1. General con®guration.
1
23
2.2. Boundary layer With respect to the x and y coordinate, the velocity components are denoted by U and V . m is the kinematic viscosity. The boundary layer equations are written for a two-dimensional incompressible ¯ow. Then the pressure is eliminated by Eq. (1). We obtain @U @t
U
@U @V @x @y @U V @U @x @y
2
ÿ m @@yU2
0;
@U1 @t
:
2
The ®rst boundary conditions associated to Eq. (2) are the no slip conditions at the wall, except on the porous zone where suction can be imposed: Vp results from the control law 8t; x U x; 0; t 0;
3 V x; 0; t Vp t: Then there is the classical link condition: the ¯ow is inviscid outside the boundary layer and link must be written with the inviscid ¯ow values (4). The system (2) is parabolic in x, so that a speci®ed velocity pro®le must be imposed at a prescribed ®rst station in x. Noting x0 this ®rst station, the x-initial condition is given by Eq. (5). Finally, concerning the time dependence, Eq. (6) gives an initial value of the complete ®eld for t 0: 8t; x
lim U x; y; t U1 t;
y!1
4
~ x0 ; y; t U ~0 y; t; U 5 ~ ~ 8x; y U x; y; 0 Uinit x; y: 6 The system (2) coupled with boundary conditions (3)± (5) and the initial condition (6) is numerically solved, so that at this step U and V are assumed to be known at each time integration step, when U1 and Vp are speci®ed. 8t; y
2.3. Linear stability analysis The previous boundary layer calculation is fully laminar and does not take into account instability waves which are always present. The linear stability analysis [2] is applied to calculate these ¯uctuations. A small perturbation is added to the main boundary layer ¯ow previously calculated. It is the unknown of the stability problem. In order to simplify its evaluation, two assumptions are made: the quasi parallel ¯ow and the quasi steady ¯ow approximations. The ®rst approximation is justi®ed by noting that the main boundary layer ¯ow has a weak x dependence compared to the rapid y dependence, at least suciently far away from the leading edge. This approximation is completely well-founded for the ¯at plate under consideration in this work. The second approximation is not so classical. The typical frequencies of the time evolution of the inviscid ¯ow and thereby of the main boundary layer ¯ow is of the order of 1 Hz. In steady conditions and for U1 in the order of 50 m/s, the typical frequencies of the TS waves are of the order of 1000 Hz. For this reason, stability analysis can
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P. Mouyon et al. / Experimental Thermal and Fluid Science 16 (1998) 22±31
be carried out as if the main boundary layer ¯ow was strictly time independent. With these two approximations for linear stability analysis, it is possible to seek for the perturbation in the mathematical form of a normal mode (7). In the spatial theory, which is considered in this study, a is a complex number and x is a real one. Separating real ar and imaginary part ai of a, the form (7) can be written as Eq. (8): ^ exp i ax ÿ xt ~ u y ufluct ~ ^ exp ÿai x exp i ar x ÿ xt: ~ u y ufluct ~
7
8 The ®rst factor, the amplitude function u^, represents the evolution of the ¯uctuation across the boundary layer. The last term is periodic and shows the wave nature of the ¯uctuation: ar is a wave number, x=2p gives the frequency of the perturbation. These waves are the TS waves. Finally the second term is real and represents the growth rate of the perturbation. It indicates if the perturbation is increasing downstream (the main ¯ow is unstable) or decreasing (the main ¯ow is stable) depending on the sign of ai . It is possible to de®ne an amplitude A of the perturbation by 1 dA : 9 A dx Introducing the form (7) of the perturbation in the linearized Navier±Stokes equations leads to the famous Orr±Sommerfeld equation (the main boundary layer plays the role of data in this problem). It is an eigenvalue problem. A nonzero solution exists only if a and x satisfy a certain relation (10) which implicitly depends on x. a is completely determined at each x position for a speci®ed frequency. Therefore the question of the stability of the main ¯ow is given by solving Eq. (10), ÿ ai
a F x; x:
10
2.4. Laminar-turbulent transition The linear stability analysis is not sucient to predict the onset of the laminar-turbulent transition. For steady ¯ows this prediction can be achieved with the semi-empirical criterion: the en -method [3]. As indicated by Eq. (10), ai depends on x and x. Let x0 x be the ®rst abscissa for which the wave of circular frequency x starts to be ampli®ed. Then the n factor is de®ned by Eq. (11). Moreover since ai represents a growth rate (Eq. (9)), n is also given by Eq. (12) where A0 is the amplitude of the wave at x x0 x: Zx n x sup ÿ ai X ; x dX ; 11 x
x0 x
A x : 12 A0 A0 is generally assumed to be independent of x. Then the idea of the so-called en -method is to assume that the transition occurs for the abscissa XT given by Eq. (13) where nT characterizes the quality of the ¯ow
n x ln
(noise level, turbulence level. . .) of the wind tunnel where the experiment is carried out. The more quiet the installation is, the higher nT is, n XT nT : 13 We justify now the use of the en -method for the unsteady ¯ow considered here. Let us ®rst recall that due to the relatively low frequency variations of the main ¯ow, the ¯uctuations are well approximated by the normal mode decomposition (7), and so a depends slowly on the time. Note also that for subsonic two-dimensional main ¯ows, the velocity of the TS waves convection is in the order of U1 =3. Therefore, if U1 is around 45 m/s, the characteristic time length for the TS waves to be convected from the leading edge up to the onset of the transition (which occurs typically for x close to 0:5 m) is of the order of 0:033 s. Since this time scale is small compared to the main ¯ow variations time scale (1 s), then the main ¯ow may be considered as constant during the convection of the TS waves. Consequently, n may be calculated at each time step as if the mean ¯ow was strictly constant (but with the constant reevaluated at each time step). The validity of this second approximation is not so clear as the previous one about the normal mode decomposition. If the frequency of the U1 variations exceeds a few hertz, the main ¯ow can not be assumed to be constant during the propagation of the TS waves and the validity of the proposed modelling and the numerical procedure break down. 2.5. Numerical procedure The unsteady boundary layer equations are solved with a classical ®nite volume method [4,5]. The main dif®culty related to this numerical integration is the evaluation of the nonlinear convection terms. It is usually done with an extrapolation which gives satisfactory results for steady ¯ows. But for our unsteady problem, this procedure implies a very small time integration step: CFL number less than 0:1. We introduce an iterative procedure to evaluate the nonlinear convection terms that allows to increase it by twenty. In regard to the determination of the growth rates ai , solving the Orr±Sommerfeld equation leads to gigantic CPU time. Fortunately a special simpli®ed method (the parabola method) has been developed [6]. The growth rates are estimated with analytical functions, the coecients of which have been determined from exact stability computations. The dierent numerical parameters are ®xed as follows. The initial ®eld (Eq. (6)) is given by the Blasius similarity pro®le. The same pro®le is used at x0 0:05 m for the ®rst x-station, see boundary condition (5). Suction can be applied between x 0:2 m and x 0:25 m. Viscosity is equal to m 0:1515 10ÿ4 m2 /s. Finally transition is assumed to occur when the n factor reaches the value nT 4, (Eq. (13)). We give some results in order to indicate the general behavior of the numerical code with a prescribed suction
P. Mouyon et al. / Experimental Thermal and Fluid Science 16 (1998) 22±31
25
velocity Vp t (open loop simulations). In Fig. 2 sharp suction velocity modi®cations are imposed at t 0:1 s and at t 0:25 s while U1 is ®xed to 42 m/s. The stabilizing eect of suction is evident. Note that 0.05 s is necessary for XT to be ®xed: it is the transient phase duration. When suction is stopped, XT comes back to the initial position but two times more rapidly. Fig. 3 shows the evolution of XT , when U1 t is a periodic function, the suction velocity being ®xed to zero. It can be observed that XT t is not exactly periodic. The small irregularities correspond to the times when dU1 =dt, and hence the pressure (see Eq. (1)), are maximum. 3. Control The feedback control design has two main objectives. First we want to bring XT to a desired location XTd , then we want to maintain XT at this value in spite of U1 ¯uctuations. The ®rst objective is a transition problem from an equilibrium point to another, while the second is a disturbance rejection problem. In order to design the control law, we ®rst need to de®ne, from an automatician point of view, what is the system to control, then to specify our performance requirements. Next a set of open loop numerical simulations is carried out which leads us to identify a simpli®ed model of the ¯ow (more precisely of the phenomena under interest, that is to say of the evolution of XT versus Vp and U1 ). This model enables us to specify a control law structure. The tuning of this law is exposed. With the help of another set of numerical simulations (closed loop simulations) a performance analysis is performed as well as a robustness analysis to delay and sampling rate.
Fig. 2. XT open loop response to Vp step variations. U1 42 m/s.
Fig. 3. XT open loop response to U1 periodic variations. Vp 0.
3.1. Speci®cations We consider that XT , the variable to be controlled, is measured and that the control variable is the vertical velocity Vp on the plate along the suction area (Fig. 1). The block diagram is shown in Fig. 4, note that XT and Vp are not available in practice. For instance the control variable will be in fact the desired suction rate sent to the servovalve mechanism. It will result in a vertical velocity through an unmodelized dynamic. In regard to the measure, an array of hot-®lms will be used. Thus XT will have to be estimated from the friction measurements, which will also introduce delay and inaccuracy. So the problem under investigation is idealized, and robustness to unmodelized phenomena will have to be checked. After the system de®nition, the speci®cation of the various elements frequency pass bands is another important stage. The numerical simulation of the ¯ow that we used in this study takes into account the variations of
Fig. 4. Closed loop block diagram.
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P. Mouyon et al. / Experimental Thermal and Fluid Science 16 (1998) 22±31
U1 up to a few hertz. The pass-band of the sucking up mechanism that will be used to produce Vp will also be equal to a few hertz. Consequently the study is restricted to the frequency band 0; fmax with fmax 5 Hz. The control law is designed in the continuous time domain. Unless otherwise informed, the control sampling rate used, is equal to the time integration step of the ¯ow. Nevertheless robustness to a slower control sampling rate is analyzed in the end. Several simulations have been done with constant U1 and Vp . As it is shown in Figs. 5 and 6, the in¯uence of U1 and Vp on the steady state value of XT is almost linear along the range under interest: 40; 50 m/s for U1 and 0; ÿ5 10ÿ3 m/s for Vp . This gives a static model of the ¯ow that can be written as 14 XT a bU1 cVp : Coecients a, b and c are estimated by a least mean square approach over the above mentioned range. We found the following values: a 0:884, b ÿ0:009 and c ÿ18:839. The accuracy of the static model is about 3%, and its validity decreases when the ratio Vp =U1 decreases. It is worth noting that the speci®ed range of variations for U1 induces a maximum of 9 cm static change for XT while Vp produces variations of at most 9.4 cm. Thus the static control ability is well dimensioned. 3.2. Dynamic model Previous simulations (Fig. 2) show that the 95% response time of the transfer function XT =Vp is about 0.02 s, which corresponds to a natural frequency of about 25 Hz. But the static model seems to be valid at most up to a few hertz. A more accurate model must be found for a control design. For this purpose, a step response identi®cation procedure is carried out around a nominal equilibrium point characterized by a mean ¯ow velocity of 45 m/s, and Vp about 5 10ÿ3 m/s. The following model structure is assumed
Fig. 5. Equilibrium study. U1 40, 45 and 50 m/s.
Fig. 6. Equilibrium study. Vp 0, ÿ1; . . . ; ÿ510ÿ3 m/s.
g Vp : 15 1 ss The coecients g and s result from an optimization scheme that minimizes the output error between the complete and the simpli®ed simulations. Optimal values are: g ÿ17:5 and s 0:006. The cut-o frequency of the identi®ed transfer is equal to 25 Hz. Comparison between the ¯ow simulation and the identi®ed model (15) is done in Fig. 7 when U1 45 m/s and in Fig. 8 when U1 50 m/s. Positive and negative steps with magnitude 10ÿ3 m/s are applied to Vp , for dierent initial conditions on XT . In regard to the dynamic response to U1 variations, we note that the validity of such a model is restricted to rather slow and small excursions around the nominal point. Fig. 9 shows a typical open loop time response to ®ltered step disturbance. Starting from 45 m/s, U1 is increased up to 48 m/s following an integrated squared XT a bU1
Fig. 7. XT =Vp step responses. Comparison of aerodynamical and ®rst order models. U1 45 m/s.
P. Mouyon et al. / Experimental Thermal and Fluid Science 16 (1998) 22±31
27
3.3. Control law design With the ®rst order dynamic model, a proportional integral controller is necessary to assign all the closed loop eigenvalues. The control law structure is given by Eq. (16), where XTr is the reference value of XT . Then Eq. (17) gives the closed loop transfer function ki Vp ÿ kp XT ÿ XTr ; s g kp s ki XTr XT 2 ss 1 gkp s gki s 2 a bU1 : ss 1 gkp s gki
Fig. 8. XT =Vp step responses. Comparison of aerodynamical and ®rst order models. U1 50 m/s.
sine variation: U1 t U1 dU1 s ÿ 0:5 sin 2ps with s t ÿ t0 fmax if t0 < t < 1=fmax . This expression is chosen because its second order derivative exists everywhere, and the simulation is rather sensitive to discontinuous variations of dU1 =dt. Besides this kind of disturbance may modelize a blast. Even for variations in the order of a few meters per second with a limited bandwidth of a few Hertz, this simulation clearly shows a highly nonlinear and nonminimum phase behavior, with a very large deviation (almost 20 cm). In fact it seems that the most important parameter is not U1 but dU1 =dt. Nevertheless in a ®rst approach the in¯uence of the perturbation U1 on the measured variable XT needs not to be precisely modelized. And so we have not tried to improve the XT =U1 transfer function modelling.
16
17
Local stability of the closed loop system is ensured if the denominator has the following structure s2 2nxc s x2c , with xc 2pfc . So we take: ki sx2c =g and kp 2nxc s ÿ 1=g. In order to completely reject disturbances up to fmax 5 Hz, let us choose fc 25 Hz. We obtain ki ÿ8:46 and kp ÿ0:056. This choice is interesting since it does not modify the system dynamic in comparison to the open loop. Furthermore since fc > fmax , very good performances are expected over all the frequency range 0; fmax . The most sensitive parameter is the ratio ki =kp that corresponds to the zero introduced by the controller. Here it is chosen rather close to the null pole, which explains its prominent part. It can be chosen slower in order to take into account an unmodelized dynamic. For example with a 5 Hz pass band actuator, ki =kp is chosen in the order of 30. Then kp is tuned to get the desired closed loop pass band (kp ÿ0:06 gives fc 25 Hz). Nevertheless this does not signi®cantly modify the following results. From Eq. (17) note that the pass-band towards XTr is the stop-band towards U1 . To better reject the dynamic disturbance eect (increase the attenuation in the stopband), we are interested in increasing fc , or equivalently the controller gain. In order to avoid an excessive suction when a large equilibrium transition is asked for, we introduce a dynamic reference model. In other words XTr is not equal to XTd but is a time dependent trajectory XTr t which asymptotically tends towards XTd . The dynamic reference model used in the simulations is a second order system with unit damping factor and a cuto frequency equal to 80 rad/s (its 95% response time is 0.05 s). Finally, in order to avoid training error and also to improve the stability in case of control saturation, the integral term of the control law is frozen when the control is close to the saturation. Typically if Vp is greater than 90% of its nominal value, then the integration is stopped. 4. Results
Fig. 9. Response to a ®ltered step disturbance. Vp 0.
With the initial tuning of the control law, the dierence between the actual and the reference values of XT is generally less than one centimeter. This is true at
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P. Mouyon et al. / Experimental Thermal and Fluid Science 16 (1998) 22±31
low frequency, as well in regulation with small magnitude disturbances as in transition from one equilibrium to another. 4.1. Equilibrium change The control law was found to be highly successful in reference following with constant U1 . Fig. 10 depicts the system response to a requisite transition from the nonsucked equilibrium (Vp 0) to the desired value XTd 0:573 m (which corresponds to Vp ÿ0:005 m/ s). The overall variation of XT is greater than 10 cm. The transition location follows the reference with a small training error. We check that the closed loop pass-band is large enough to follow the reference trajectory. The same reference variation was simulated with dierent incident velocities (U1 40; 45 and 50 m/s). Maximal dierences between XT and its reference are 0.7, 1.5 and 2.2 cm respectively (Fig. 11). This error increases with U1 , and a rather slow mode appears.
Fig. 11. Impact of U1 on the reference following.
4.2. Step disturbance rejection The closed loop system behavior is now studied in the presence of a ®ltered step disturbance (cf. Section 3.2). The objective is to maintain the transition location at the desired value XTd . Controller and system responses to a change in mean ¯ow velocity from 45 to 46 m/s are shown in Fig. 12. The step duration corresponds to a 4 Hz frequency. The deviation from the desired value XTd 57:3 cm is less than 0.5 cm. This deviation increases with the U1 step magnitude. It reaches 1.1 cm when the step magnitude is 3 m/s. Fig. 13 depicts the case where the step duration is two times shorter (the relevant frequency is 8 Hz). The deviation of XT is more important and reaches 3.4 cm. But it is still three times smaller than for the uncontrolled ¯ow.
Fig. 12. Response to a slow and small step disturbance.
Fig. 13. Response to a large and fast step disturbance.
4.3. Periodic disturbance rejection
Fig. 10. Reference following.
A sinusoidal variation of U1 was then simulated with frequency 1 Hz and amplitude 1 m/s. The system response shows only a few residual high frequency oscillations (Fig. 14). When the frequency is increased up to 5 Hz, residual oscillations on XT reach 2.5 cm. But with
P. Mouyon et al. / Experimental Thermal and Fluid Science 16 (1998) 22±31
Fig. 14. Slow sinusoidal disturbance rejection.
the same disturbance the open loop oscillation magnitude exceeds 6 cm. A comparison of the time histories is illustrated on Fig. 15. The left side of the ®gure is concerned with the open loop behavior, while the right side deals with the closed loop behavior. So we have shown that for rather low frequency oscillations of U1 , the controller reduces the variations of XT within a reasonable level. Previous analysis is now performed at dierent frequencies. The reduction rate, in comparison with the noncontrolled ¯ow depends on the U1 oscillation frequency, but is of the order of 80% below 5 Hz (Fig. 16). However, it is noticeable that the disturbance rejection is not obtained all over the frequency band 0; fc as expected from the control design. In fact, simulations show that higher frequency components are found in the ¯ow when the control is used. It seems that the complexity of the controlled ¯ow is greater than would originally be suggested from consideration of the uncontrolled ¯ow. Attempts to control the ¯ow at low frequency tend
Fig. 15. Open and closed loop response with sinusoidal disturbance.
29
Fig. 16. Closed loop attenuation of XT variations.
to excite high frequency components. It is a kind of modal spillover phenomena that has already been observed in active control of turbulent ¯ow [7]. Nevertheless note that the validity of the simulation is not so clear when the frequency goes above a few hertz. 4.4. Robustness analysis The experiment will involve several systems that have not yet been modelized. A more realistic block diagram of the experiment is shown in Fig. 17, where q is the suction rate, qc its desired value computed by the controller and X^T the transition location estimated from hot-®lms measurements. So robustness of the closed loop must be investigated. In regard to delay, Fig. 18 shows an important deterioration of the response in the presence of a rather small delay (10 ms). The U1 oscillation frequency is 4 Hz. There are three curves for XT . The dotted lines represent the open loop and the nominal closed loop (no delay). The solid line depicts the closed loop behavior with delay. Once again, performances degradation is due to the appearance of high frequencies components. A sampling of the control is then introduced (Fig. 19). U1 is always a 4 Hz sine curve. Even at a rather high sampling rate (66 Hz), the XT response is very much aected. This study was carried out at 4 Hz only.
Fig. 17. Detailed block diagram.
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P. Mouyon et al. / Experimental Thermal and Fluid Science 16 (1998) 22±31
Fig. 19. Robustness to sampling rate.
Fig. 18. Robustness to delay.
At lower frequencies the degradation is not so important. Nevertheless the closed loop seems not to be robust enough and must be improved. 5. Practical signi®cance This research shows that the development of a low time consuming numerical simulation for the laminarturbulent transition study in unsteady ¯ow is now possible. This kind of simulation may be easily used for feedback control design. However, a lot of attention must be paid to the ampli®cation of high frequency components when the control is used in order to reduce low frequency ¯uctuations. 6. Conclusions In this paper we present an aerodynamical modelling of the boundary layer that develops along a ¯at plate and more especially of the laminar-turbulent transition. A numerical procedure is proposed which allows closed loop simulations of suction experiments. Computation time is reasonable (in the order of 10 mn for 1 s) on a standard computer station.
A simpli®ed dynamic model is identi®ed between the transition location XT and the suction velocity Vp . Then a control law is designed on this basis. Simulations show that under generic experimental conditions (measurement of the current transition location, and control of the vertical velocity along the suction area), the control law leads to a 80% reduction of the XT ¯uctuations when U1 presents harmonic variations up to 5 Hz. Nevertheless high frequency components are found in the ¯ow when the control is used. They lead to limited performances, and could explain the lack of robustness. This discussion raises the question of whether it is possible to identify a low order model of the ¯ow which retains enough information for a robust control design or whether another approach must be found. Nomenclature fc kp ki n nT P
closed loop desired cut-o frequency, Hz proportional controller gain, dimensionless integral controller gain, dimensionless maximal growth rate over frequencies, dimensionless growth rate value at the transition, dimensionless pressure, Pa
P. Mouyon et al. / Experimental Thermal and Fluid Science 16 (1998) 22±31
t ~ ufluct U U1 V Vp x XT XTd XTr t y ai m q
time, s ¯ow ¯uctuation, m/s x-velocity component of the mean ¯ow, m/s incident velocity, m/s y-velocity component of the mean ¯ow, m/s suction velocity, m/s downstream coordinate, m laminar turbulent transition location, cm desired transition location, cm reference transition trajectory, cm distance from the wall, m growth rate of the ¯uctuation, % kinematic viscosity, m2 /s ¯ux density, kg/m3
References [1] J.-L. Rioual, P.A. Nelson, M.J. Fisher, Experiments on the automatic control of boundary-layer transition, Journal of Aircraft 31 (6) (1995) 1417±1418.
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[2] L.M. Mack, Boundary layer stability theory, Jet Propulsion Lab., California Inst. of Techn., Pasadena, California, 1969. [3] D. Arnal, Boundary layer transition: predictions based on linear theory, Special course Agard on Progress in Transition Modeling, 1993. [4] R. Michel, J. Cousteix, R. Houdeville, Unsteady turbulent shear ¯ows, in: IUTAM Symposium Toulouse, Springer, Berlin, 1981. [5] J. Cousteix, A. Desopper, R. Houdeville, Recherche sur les couches limites turbulentes instationnaires, La Recherche Aerospatiale, 3, 1977. [6] D. Arnal, Transition prediction in transonic ¯ow, in: Zierep et Oertel, (Ed.), IUTAM Symposium Transonicum III, Gottingen, Springer, Berlin, 1988. [7] K.Y. Tang, W.R. Graham, J. Peraire, Active ¯ow control using a reduced order model and optimum control, 27th AIAA Fluid Dynamics Conference, New Orleans, LA, 1996.