CONTROL OF LAMINAR SEPARATION BUBBLES

Transcription

1 CONTROL OF LAMINAR SEPARATION BUBBLES Ulrich Rist, Kai Augustin Institut für Aerodynamik & Gasdynamik Universität Stuttgart Pfaffenwaldring 21 D-755 Stuttgart, Germany ABSTRACT In the present paper active separation control of a transitional laminar separation bubble is investigated by means of direct numerical simulations. The specific influence of different forcing amplitudes and disturbance modes on the size and shape of the laminar separation bubble are illustrated. These consist of steady and unsteady two- and three-dimensional Tollmien Schlichting-(TS)-like boundary layer disturbances which are amplified by instability mechanisms within the laminar boundary layer and in the separation bubble itself leading to an upstream shift of laminar-turbulent transition such that the streamwise etent of the laminar separation bubble is controlled. It is shown that an appropriate selection of the forcing frequency together with a suitable location and length of the disturbance strip for eciting sinusoidal disturbance waves can be found with the help of linear stability theory. The present results demonstrate that unsteady forcing is much more efficient than steady forcing of the flow in order to provoke laminar-turbulent transition to shorten a laminar separation bubble. For an active control method based on these results a criterion to detect the etension of the bubble is investigated as well and several control schemes are suggested and compared to each other. The paper ends with an outlook on their realization possibilities for future applications. 1. INTRODUCTION A transitional laminar separation bubble (LSB) is characterized by laminar separation (S) because of an adverse pressure gradient, laminar-turbulent transition (T), and turbulent reattachment (R). Despite the occurrence of negative skin friction within the bubble LSBs may cause an undesired drag rise because of a considerable influence on the global pressure distribution of the airfoil. Unfortunately, their spatial etend is difficult to predict by theory or in wind-tunnel eperiments, because of their Reynolds number dependence and their sensitivity to background disturbances which are difficult to take into account. Small environmental changes may have a sudden influence and operating an airfoil at off-design conditions can make it prone to LSBs. However, such considerations will become completely obsolete once a suitable separation bubble control becomes available for operational use. The key of LSB control is to control laminar-turbulent transition since an earlier transition will move the re-attachment upstream. Gad-el-Hak [4] has termed this the easy task of flow control (compared to turbulent separation control). Mainly because it suffices to provoke laminar-turbulent transition by some appropriate means without an urgent need for some highly sophisticated controller. In fact, a simple switch that turns LSB control on or off when appropriate could be sufficient, once the 1

2 z,w y,v integration domain δ u S R disturbance strip separation bubble buffer domain flat plate Figure 1: Integration domain for the DNS of a separation bubble with 2-D and 3-D disturbance control. S = separation; R = re-attachment.,u different flow situations are understood well enough. As a result, the main purpose of the present contribution is to illustrate the underlying mechanisms in a rather detailed manner. We hope that this will help the practitioner to get a better understanding of the basic mechanisms which he might need for an optimization of his ideas. Thus, the present contribution will present a summary of the work described in [1, 2, 3, 22] and it will be based on the view of people who have worked in basic boundary-layer transition research for quite a while. It is not intended to provide an overview on all previous or current (mainly eperimental) research on laminar separation control, nor will it treat the much larger area of turbulent boundary-layer separation, at all. The reader who is interested in the latter subject could start with the reviews by Greenblatt & Wygnanski (2) [5] or the overview by Gad-el-Hak (2) [4], for instance. In the following sections the numerical method used (i.e. direct numerical simulation, DNS) will be outlined, followed by investigations of different forcing strategies in section three. Section four is devoted to the problem of detecting an LSB for an automatic control and it will present preliminary results of applying a feedback control to a LSB. The paper ends with conclusions and an outlook in section five. 2. NUMERICAL MODEL To investigate laminar separation bubbles spatial direct numerical simulations (DNS) of a flat plate boundary layer with a 2-D base flow and an adverse pressure gradient applied at the free-stream boundary are performed. The DNS code has been developed for the investigation of instability and laminar-turbulent transition in boundary layers [1, 13, 14, 15, 23]. Figure 1 sketches the rectangular integration domain. The complete Navier Stokes equations for incompressible flow are written in a vorticity-velocity formulation ω t + y (vω uω y )+ z (wω uω z ) = ω, (1) ω y t + (uω y vω )+ z (wω y vω z ) = ω y, (2) ω z t + (uω z wω )+ y (vω z wω y ) = ω z. (3) 2

3 Once the vorticity-transport equations have been solved, the remaining velocity components can be computed from three Poisson equations 2 u u z 2 = ω y z 2 v, (4) y v = ω z ω z, (5) 2 w w z 2 = ω y 2 v. (6) y z Non-dimensionalization is performed with reference to the free stream velocity Û, a characteristic length ˆL and the kinematic viscosity ˆν. Thus, the non-dimensional variables and a modified Laplace operator Re = Û ˆL/ˆν, β 1 =2π ˆf ˆν Re/Û 2, (7) =ˆ/ˆL, y = Re ŷ/ˆl, z =ẑ/ˆl, (8) u =û/û, v = Re ˆv/Û, w =ŵ/û (9) = Re y Re z 2 (1) can be derived. The numerical method used is 4 th -order accurate in time and space due to finite differences in streamwise and wall-normal direction and a Runge Kutta scheme in time. For the spanwise direction a spectral ansatz implying periodic boundary conditions in z is used f (, y, z, t) = K k= K F k (, y, t) e ikγz, γ = 2π λ z. (11) With eqn. (11) the Poisson equations for the streamwise and spanwise velocity reduce to ordinary differential equations. The remaining Poisson equation for the wall-normal velocity is solved by a line relaation method using a multi-grid algorithm. All Poisson equations can be solved separately for each spanwise spectral mode k allowing effective parallelisation. At the inflow boundary a Blasius boundary layer solution with Re = 1722 is prescribed. To avoid nonphysical reflections at the outflow boundary the disturbance vorticity is artificially damped to zero in a buffer domain [11]. The potential flow at the free-stream upper boundaryis decelerated by 1% of Û and the displacement effects of the LSB on the potential flow can be captured by a viscous-inviscid boundary-layer interaction model [16, 28] at every time step of the calculation. On the surface of the plate the no-slip condition is applied ecluding the disturbance strip upstream of the LSB, where arbitrary 2-D and 3-D forcing can be imposed by means of suction and blowing at the wall K v(,,z,t)= A vk v a () e i(kγz β kt) k= K, (12) according to which each Fourier mode in eqn. (11) can be controlled independently by its particular frequency β k and amplitude A vk. The function v a () is point-symmetric with respect 1 Note: throughout this paper we use β for the normalized circular frequency because ω is used for the vorticity 3

4 a) 8 y b) 8 y S R dist. strip 12 S 13 R 15 Figure 2: Streamlines, boundary layer velocity profiles and vorticity contours of the timeaveraged flow field. to the center of the disturbance strip such that the net mass flu of the disturbance generator becomes zero. A forcing of the spanwise mode k = introduces two-dimensional fluctuations and k is responsible for the generation of oblique waves with frequency β k that run at angles ϕ = atan(kγ/α k )relativetothe ais. Here, α is the streamwise wave number. Steady 3-D modes can be generated as well. For that purpose, it suffices to set β k = and to replace v a () by a symmetric function v s () which is more efficient for zero frequency [23]. 3. INVESTIGATIONS OF SEPARATION CONTROL The present quantitative investigations are performed for a pressure-induced LSB in a flat-plate boundary layer where a Blasius boundary layer solution with Re = 1722 is given at inflow and a streamwise velocity deceleration by 1% is prescribed at the free-stream boundary (cf. Fig. 2 and U e in Fig. 3). The other parameters of the flow are Re = 1 (for =1), =.8376, y =.2296, and t = This case is derived from the midchord-bubble considered in greater detail in [2]. The streamlines and vorticity contours of the mean flow in Fig. 2 are for a forcing with A v =1 6, β = 5. Although the present illustration is from a pure 2-D simulation only [K = in eqn. (11)] it captures the qualitative features of the LSB surprisingly well. A long and shallow bubble appears with a much more rapid re-attachment than separation due to the enhanced transversal miing after transition, followed by a considerably increased wall shear downstream of (R). Also note the matching of ψ =andω w = at separation (S) and reattachment (R). The free-stream velocity and the disturbance development are compared with a number of boundary layer parameters in Fig. 3. Due to separation the wall pressure 1 p w develops the characteristic plateau inside the separation bubble which is followed by a rapid pressure rise at re-attachment, because (R) becomes the stagnation point of the re-attaching flow. The velocity perturbations u ma introduced via suction and blowing ehibit a dramatic amplification by five to si orders of magnitude until their non-linear saturation which corresponds to transition. According to the wall shear ω z the re-attachment vorte at the downstream end of the bubble is rather strong in the present case. Note however, that such features of the time-averaged flow have nothing to do with the real and instantaneous flow which ehibits a periodical vorte shedding starting near (T) [21]. Compared to the leading-edge bubble in [2, 21] the changes in boundary layer displacement thickness and shape factors are much more etreme in the 4

5 U e, 1 p w 1.9 U e p w 2 1 δ u ma ω w δ /Θ δ /Θ y 2 2 y 1 1 dist. strip ψ = ψ = A 13 W A 13 W Figure 3: Comparison of boundary layer parameters for the bubble in Fig. 2. δ, Θ, δ : displacement, momentum, and energy thickness, respectively. present case. It will be an appealing task to control the present bubble. 3.1 Effect of Unsteady 2-D Forcing In the following we show results from simulations of the above midchord bubble where the viscous-inviscid interaction model of Maucher et al. has been used [16]. The main effect of this is to modify the free-stream velocity distribution at the upper boundary of the integration domain, see Fig. 4 where the resulting streamwise velocity distribution at the upper boundary as a result from the interaction model is compared to the initial potential flow prescribed at the startup of the computation. The characteristic pressure plateau formed by the LSB now also develops at the free-stream boundary in contrast to Fig. 3. A comparison of the disturbance amplification from the DNS with linear stability theory u e 1 potential flow DNS with interaction model Figure 4: Streamwise free-stream boundary condition with potential distribution at the start-up of the DNS and the distribution developing from the interaction model. 5

6 1 A DNS 1-1 LST S T disturbance strip Figure 5: Comparison of streamwise 2-D disturbance amplification from DNS with LST. (LST) in Fig. 5 confirms the good agreement between LST and DNS already found in [21], for instance. The amplification of u ma for the fundamental disturbance mode evidently coincides with LST until close to transition, which shows that the LST is a proper tool to investigate the disturbance development in flows with LSBs further. In Fig. 6 the corresponding linear stability diagram α i (, β) reveals a narrow band of amplified frequencies β only upstream of the LSB, comparable to a Blasius flow, whereas the amplified frequency band broadens further downstream towards the separation line. At the same time the maimal amplification rate α i increases about si-fold. Thus, the LST allows to estimate the necessary initial disturbance amplitude A for influencing the LSB. By integrating the stability diagram from Fig. 6 in one obtains the amplification curves A/A (, β). These can be normalised to any A, i.e. amplitude at some = const. In Fig. 7 such a standardisation has been performed with respect to 1 =13.66 with the idea to illustrate the frequency dependence of the initial amplitude A of an actuator placed at =1.86 that is needed to attain unit amplitude at 1. It is obvious that a suitable frequency should be chosen from within the amplified frequency band to obtain an efficient control. Otherwise, the initial disturbance amplitudes at might become ecessively large. For β>11.5 the disturbance is damped in such a manner that the necessary amplitude A would be larger than the desired amplitude A 1 at 1, which is obviously etremely inefficient. However, if a proper frequency like β 5 is selected for the disturbance, a 1-fold amplification can be easily achieved. For β = 5 the amplification of the u disturbance maima for varying forcing amplitude A vk ; k = are compared in Fig. 8. Here, the forcing amplitude is denoted A v for brevity. In the simulation with A v = an etremely large LSB has developed that remained unsteady despite steady boundary conditions all over the domain. In the upstream part of the bubble the fluctuations which correspond to the instability frequency are too small to be seen because of a low-frequency flapping of the separating boundary layer which spoils the periodicity of the present data for their Fourier analysis. The actual travelling wave-like disturbances are thus hidden by an according aliasing phenomenon. But with increasing forcing amplitude the 6

7 β damping S α i amplification disturbance strip Figure 6: Amplification rate α i for 2-D disturbances (stability diagram from linear stability theory, LST). a) b) log 2 1 (A) β = = log 1 (A) = = β Figure 7: Amplification factors A/A for 2-D disturbances according to LST. a) overview and b) cross-sections at and 1. 7

8 log 1 u /u disturbance strip A v = A v =1-7 A v =1-6 LST A v =1-6 A v =1-5 A v = Figure 8: Disturbance maima for different levels A v of ecitation with frequency β =5. bubble gets more and more controlled and this effect vanishes. Thus, the curves with lower forcing appear larger in the upstream part than their counterparts obtained for increasing A v. A comparison of the amplification curve for A v =1 6 yields the same ecellent agreement between LST and DNS, as before. However, it should be noted that LST in these comparisons is always based on the base flow from the according simulation, such that any non-linear effect (and part of the non-parallel effects) of the disturbances on the mean flow are included. Strictly speaking, our comparisons are only post diction, because the base flow is not predictable based on linear models. Construction of appropriate methods for LSB prediction which might be based on Reynolds averaged Navier Stokes (RANS) equations and appropriate transition or turbulence models are a point of ongoing research. The amplification curves in Fig. 8 clearly show an upstream shift of transition (amplitude saturation) with increasing A v. Since the re-attachment (R) will move upstream accordingly, it can be epected that this upstream shift will reduce the size of the LSB also. This strong influence of rather small disturbances (A v 1) on the shape and the size of the time averaged separation bubble is displayed in Fig. 9. On the left hand side by streamlines including the separation line (emphasized) versus the streamwise coordinate and on the right hand side by the mean skin friction distribution ω zw for the five cases A v =toa v =1 4,already considered in the preceding figure. The initially very large separation bubble with its large recirculation vorte at its rearward end is gradually reduced in size when forcing is increased. At the end of the present series with a disturbance level of A v =1 4 only a very small bubble remains. Moreover, the strong displacement of the boundary layer by the large separation bubble, which would affect the pressure distribution of an airfoil and cause additional drag, almost vanishes with an increased 8

9 disturbance level. Both the separation and re-attachment point are shifted leading to a size reduction of the bubble. The re-attachment point moves upstream under the growing influence of the disturbance ecitation and the thereby earlier transition. In addition to that the separation point moves further downstream towards the re-attachment because of upstream effects of the transition location on the wall-pressure (cf. [2]). In other words, the bubble becomes shallower with increased forcing and eerts less displacement on the potential flow. The influence on the size of the bubble is also present in the local mean velocity profiles of the streamwise velocity u at = const positions. In Fig. 1 the velocity profiles at the point of separation a) and at the point of re-attachment b) of the case A v =1 4 (with the smallest LSB) are shown for comparison with the other considered cases. The reference case without disturbance ecitation shows the greatest differences. The boundary layer has separated upstream and is still separated in b) where a fully developed separation profile with reverse flow (u <) close to the wall is visible. All other cases demonstrate the same properties at these positions but with higher disturbance levels the size and the strength of the region of reverse flow becomes smaller and smaller as the overall size of the LSB reduces. 3.2 Effect of Unsteady 3-D Forcing The previous subsection has shown that a control based on the most amplified disturbance frequency can be very efficient in the 2-D case. However, in practice a purely two-dimensional flow doesn t eist and a perfectly 2-D forcing is much more difficult to realize in practice than a 3-D forcing. Therefore, we investigate now the reaction of the previous LSB to unsteady 3-D forcing by a pair of (weakly) oblique waves. The reasoning behind this is that weakly oblique modes are nearly as amplified as the purely two-dimensional ones, such that not much efficiency is lost. On the other hand such modes may initiate the so-called oblique transition scenario which is much more efficient in producing turbulence than two-dimensional forcing (see [21]). Therefore, a pair of oblique (ϕ 1 o ) unsteady modes (1, ±1) (bold solid line with squares in Fig. 11) is introduced into the same base flow as before. In the notation (h, k) the inde h denotes harmonic modes with multiples of the fundamental frequency β (eqn. (7) ) while k means spectral modes of the spanwise wave number γ (eqn. (11) ). The initial disturbance amplitude has been set to A v1 =1 5, the frequency is β 1 =5 and γ =5.45. An unsteady 2-D background disturbance (1, ) (solid line) is also present with A v =1 6, β = β 1. For the present case this one doesn t participate in the disturbance development but for the case with steady 3-D forcing in the net subsection it will be needed. For a sound comparison of the two cases later on, it is already included here. For verification purposes of the simulation results the development of the 2-D mode (1, ) is compared to linear stability theory. Due to its very low amplitude even inside the LSB the mode shows very good agreement with the theory up to saturation. As in the previous purely 2-D investigations the size of the LSB is reduced due to a downstream and an upstream shift of (S) and (R), respectively. Additional simulations show that the bubble vanishes totally with a forcing level of A v1 = Effect of Steady 3-D Forcing In the second case a steady 3-D disturbance mode (, 1) is ecited at the disturbance strip with a wall-normal amplitude A v1 =1 3, β 1 =andγ = 15. Thus, the spanwise wavelength of the steady 3-D disturbance corresponds to λ z =.419. Figure 12 compares amplification curves of the disturbance velocity u versus for different 2-D and 3-D spectral modes in logarithmic scale. Compared to the unsteady motions the disturbance amplitude of mode (, 1) is weakly damped at first and then weakly amplified far into the bubble. Only at 14.2 it grows close to the point of non-linear saturation 9

10 y/ Re A v = ω zw LSB y/ Re A v = ω zw LSB y/ Re A v = ω zw LSB y/ Re A v = ω zw LSB y/ Re A v = ω zw LSB Figure 9: Streamlines and separation line of the time-averaged flow with separation bubble and corresponding skin friction distribution ω zw. 1

11 a) y/ Re.1 u(a v =.) u(a v = ) u(a v = ) u(a v = ) u(a v = ) b) u.5 1 u Figure 1: Wall normal distribution of the streamwise time-averaged velocity u at a) separation of the A v =1 4 case and b) re-attachment of the A v =1 4 case. b) log(u ) (1,) (2,) -1 (,1) -2 (1,1) (,2) -3 LST (1,) disturbance strip S R Figure 11: Amplification of the disturbance velocity u of dominant 2-D and 3-D disturbance modes for unsteady 3-D forcing via mode (1, 1). 11

12 a) -1-2 log(u ) disturbance strip S R (1,) (2,) (,1) (1,1) (,2) Figure 12: Amplification of the disturbance velocity u of dominant 2-D and 3-D disturbance modes for steady 3-D forcing via mode (, 1). which marks the transition. A higher spanwise harmonic mode (, 2) (dash-dot-dotted line with deltas) is generated as a (spanwise) higher harmonic of mode (, 1). At the disturbance strip an additional 2-D mode (1, ) (solid line) of fundamental frequency has been ecited to mimic background disturbances with an initial amplitude A v =1 6, β = 5, three orders of magnitude below the amplitude of the 3-D mode (, 1). This mode becomes strongly amplified by base-flow instability and finally eceeds the amplitude of the 3-D mode (, 1) at =13.8. An oblique fundamental mode (1, 1) is generated by continuous non-linear interaction of the (1, ) and (, 1) modes and finally reaches the amplitude of the 3-D steady mode. Without an unsteady background, mode (, 1) would not have developed unsteady harmonics and no transition would have happened. Because of its quasi-neutral amplification, the 3-D steady forcing had to be introduced on a comparatively large level (compared to the much more efficient unsteady disturbances). However, its initial amplitude is less than 1% which is still very weak compared to the free-stream velocity. Due to its insignificant amplitude changes with the steady 3-D disturbance may be introduced at any convenient place upstream of the LSB (i.e., where the wall shear is large for enhanced receptivity with respect to roughness) Comparison of Unsteady and Steady 3-D Forcing The above results show a significantly different behaviour of the LSB with respect to steady and unsteady disturbances. In contrast to the steady case the wall-forced unsteady disturbance mode (1, 1) is strongly amplified by boundary layer instability and continues to be the most dominant mode for all. Although equally amplified, the 2-D mode (1, ) stays below the oblique one due to its lower initial amplitude. Because of the strong amplification of mode (1, 1) non-linear stages of the disturbance development ( 1% U ) are reached somewhat further upstream than in the steady case at =14.. The point of laminar-turbulent transition and thus the reattachment is shifted upstream likewise. Figure 13 illustrates the differences between unsteady 2 placing the roughness at (S) is not a good idea because of the small du/dy there which minimizes the receptivity of the boundary layer 12

13 a) separation stream surface Ψ y/ Re z -.2 λ z,stead b) y/ Re z Figure 13: Separation stream surface Ψ for steady 3-D forcing (a) and unsteady 3-D forcing (b). and steady 3-D forcing on the basis of a perspective view at the time averaged separation stream surfaces Ψ of the two cases. The steady disturbances have carved streamwise grooves into the bubble in Fig. 13 a), as if a spanwise row of circular roughness elements with period λ z(,1) were present at separation. This is the effect of the large-amplitude (, 1) forcing in Fig. 12. In the unsteady case in Fig. 13 b) the LSB is much shorter, of lower height and only slightly modulated in spanwise direction. The different bubble sizes as they appear in the time and spanwise averaged separation streamlines Ψ are compared to each other in Fig. 14. As a reference the case with only 2-D background forcing at A v =1 6 is included and labelled undisturbed case. The stronger influence of unsteady forcing now becomes fully evident. Steady forcing (dashed line) is hardly discernible from the reference case, despite its comparatively large amplitude (1 3 instead of 1 5 in the unsteady 3-D case), whereas the unsteady case (bold dash-dotted line) reduces the height and the length of the bubble to about 75% of its original size. This clearly shows the superiority of unsteady control to steady control. 4. TOWARDS AN AUTOMATIC BUBBLE CONTROL The above results directly led to the idea whether a mechanism can be constructed that detects and controls a laminar separation bubble in an automatic manner. Such a concept is sketched in 13

14 .3 undisturbed 3D steady disturbance 3D unsteady disturbance y/ Re Figure 14: Comparison of bubble shapes for three different cases. Actuator and sensor system U A C S Figure 15: Concept for an automatic LSB control. Fig. 15. It requires an appropriate disturbance generator ( ), an actuator (A), measurements of the size of the LSB via some sensor array (S), and a controller (C). The controller will read the sensor signals and determine the size of the LSB by time averaging over a certain phase, e.g., ten disturbance cycles. The controller is necessary to avoid ecessive disturbance amplitudes, to react on non-linearities in the control loop, and to enable the mechanism to respond to changing flow conditions where a LSB might not be present any longer. 4.1 Separation Bubble Detection Different shear stress sensors for the present purpose are currently under development, but not yet fully operational for a real environment. It seems that a reliable detection of re-attachment is easier than the detection of the separation point. The ideal way to determine the length of the LSB would be based on the time averaged skin friction ω z,wall. Other methods [6, 7] rely on the detection of wall-pressure or wall-shear fluctuations. Developing appropriate surface sensors is an area of active research in the field of Micro Electro-Mechanical Systems (MEMS) [12, 18, 19]. A good alternative to hot-films, at least for the laboratory, are so-called surface hot-wires which are mounted flush with the surface over small grooves [27] and which generate highly accurate data. Figure 16 compares the time averaged ω z at the wall for the two 3-D controlled cases considered above. The attached flow upstream of separation which belongs to positive wall shear can be observed, as well as spanwise periodic structures in and outside the LSB. The grey-scale 14

15 coding of the contours has a large contrast at ω z,wall = in order to emphasize the separation and re-attachment lines. This makes the separation line clearly stand out, together with its spanwise modulation by λ z in the first case, which could already be seen in Fig. 13 a). Unlike the separation line the re-attachment is hidden in an area of large gradients of the skin friction caused by remnants from the highly unsteady processes connected to large-amplitude vorte shedding in the re-attachment region of the separation bubble. To resolve this problem, Augustin et al. [2] suggested to use histograms of the spanwise skin friction at discrete streamwise positions and derive an easy-to-implement bubble detection criterion. In those histograms ω z,wall is decomposed into intervals. The number of occurrences within each interval are counted and finally normalized by the total number of analysed points N. Time averaging is done by performing this analysis for all time steps over a fied period of time. In the present study four cycles of the fundamental disturbance frequency have been used. The upper row of plots in Fig. 17 contains histograms for eleven intervals in the range.1 ω z,wall.1 atthree positions. At the first streamwise position =13.25 [also marked in Fig. 16 b) as a dashed line] the histogram indicates a sharp peak at weakly negative values. This obviously belongs to a position inside the separation bubble. Going further downstream this peak is smeared out more and more, and it is hardly possible to decide whether a point inside or outside the bubble is encountered. An etra simplification can provide a clearer view by restricting the analysis to only two intervals of ω z,wall, one below and one above zero, as illustrated in the lower row of Fig. 17. If the number N of points with ω z,wall < eceeds the number of points with ω z,wall >, then the considered streamwise position lies within the bubble. These properties can be used to define a binary separation bubble criterion { 1 N ωz,wall < N ωz,wall > C ωz,wall () =, (13) N ωz,wall < >N ωz,wall > which becomes 1 for points located inside the bubble and for all other streamwise points. Applied to the cases shown above in Fig. 14, the streamwise etent of all bubbles can be confidently determined by the above criterion (compare Fig. 18 with Fig. 14). However, a shortcoming of this easy-to-use method might be that only the streamwise epansion of the LSB can be detected, not its height which could be also interesting for fine-tuning the control. Sometimes a long but shallow separation is desired because of the low skin friction inside the bubble leading to a reduced total drag. Nevertheless, the present bubble criterion seems well suited for a controller which then changes the amplitude of the disturbance input upstream in order to reduce the LSB. 4.2 Skin-Friction Signal Feedback The above results indicated the merits of using unsteady 2-D or 3-D disturbances rather than steady disturbances. In order to provide the necessary unsteady disturbance amplitude directly from the flow, the system in Fig. 15 can be slightly modified by taking the unsteady signal of one sensor as a disturbance source instead of the disturbance generator ( ). In this case, instantaneous amplitude signals of the skin-friction downstream of the LSB will provide unsteady input to the actuator at the disturbance strip via a controller that simply modifies the signal s amplitude (Fig. 19) A (t l )=C v ω z,wall ( n,t l 1 ). (14) In eqn. (14) instantaneous skin-friction data from a previous time step t l 1 at a certain streamwise position n is used as forcing amplitude A for the disturbance strip of the current time step t l. Due to the above mentioned fact that 2-D or weakly 3-D disturbances are most 15

16 Figure 16: Time averaged skin friction ω z,wall for a) the steady and b) unsteady 3-D forcing. 16

17 4 3 % 2 1 = ω z,wall 1 < > 15 % 1 5 = ω z,wall 1 < > % 4 2 = ω z,wall 1 < > % 5 % 5 % ω z,wall ω z,wall ω z,wall Figure 17: Histograms of the spanwise distribution of the skin friction ω z,wall at three streamwise positions for the case with unsteady 3-D control. 1 undisturbed 3D steady disturbance 3D unsteady disturbance C ωz,wall Figure 18: Etent of the separation bubbles of the three different cases as detected by the binary bubble criterion C ωz,wall (). Disturbance strip Separation bubble C v Signal feedback Skin friction sensor Figure 19: Sketch of the skin-friction signal feedback mechanism. 17

18 Figure 2: Time trace of ω z,wall at the sensor position. effective in influencing the LSB, only 2-D DNS have been performed so far to investigate the properties of the suggested feedback mechanism. To reduce the large sensor amplitudes from the non-linear region to the desired small-amplitude level at the disturbance strip, the amplitude gain is set to C v << 1. It will later be automatically adapted by a controller according to a detected length difference of the separation bubble with respect to the desired length. As a broad spectrum of disturbance frequencies becomes desirable, data from the quasi turbulent flow downstream of the re-attachment are well suited, especially since they will contain the most unstable frequency as well, because this one gets most amplified in the upstream region of LST. To start the simulations, the flow has been disturbed by a background perturbation of the fundamental mode (1, ) of frequency β =5andA v =1 6 for a start-up period of 12 cycles (T ). Although the initial fundamental disturbance period becomes non-representative in the further self-controlled flow, time will still be indicated by multiples of the original fundamental period T after the feedback mechanism has been switched on at t = 12T. For the present investigation of the feedback mechanism, the viscous-invsicid boundary-layer interaction model at the free-stream boundary has been switched off. Therefore, the location of the LSB in this case is shifted downstream again, and the bubble is of lower height compared to the unforced case in Fig. 14. The skin-friction sensor has been placed at n =14.48 which is about 1.15 λ of the fundamental 2-D disturbance downstream of the re-attachment line. After the start-up phase, the signal feedback has been turned on for 8 (pseudo-)periods T. The amplitude gain has been set to C v = As the level of the disturbance signal at n =14.48 is of order 1, the chosen factor C v compares to the disturbance level where the bubble finally disappears. Fig. 2 indicates the time-trace of the skin-friction signal which has been used for feedback. The according frequency spectrum is shown in Fig. 21 together with two other stations further upstream. Data from the time interval 16T t 2T have been used. This choice avoids the capturing of any transient effects from the start-up of the feedback. A rather broad spectrum of frequencies is present in the flow at each station, but the largest amplitudes occur for the frequency β 4 which is right in the middle of the instability region of the upstream flow, see Fig. 22. The result of laminar-turbulent transition in Fig. 21 is to amplify a broad spectrum of higher harmonics. Once these are fed back at 11 they are damped according to LST 18

19 Figure 21: Frequency spectra of the skin-friction ω z,wall at three different streamwise positions. because they fall outside the instability region there. Thus, the boundary layer approaching the LSB acts like a low-pass filter to the feedback data. As the disturbances amplify downstream the higher harmonic content is successively regenerated. The effectiveness of the feedback mechanism can be shown by the spanwise vorticity ω z in Fig. 23. Here, contours of the instantaneous vorticity are plotted versus streamwise and wallnormal coordinates at two different time steps of the calculation. At t = 12T,whereonly the small background disturbance is ecited at the disturbance strip, a large separation bubble is present. The shape of the separation bubble is outlined by the time averaged separation streamline Ψ. In contrast to the time-averaged picture (e.g., Fig. 9 or [21]) the instantaneous flow-field is subject to 2-D vorte shedding. The transition location is marked by the onset of vorte shedding. Applying the skin-friction signal feedback with the given factor C v,the transition location is shifted far upstream in Fig. 23 b) which is taken 4T later compared to Fig. 23 a) and the time-averaged separation streamline Ψ disappears in the controlled case. Figure 24 provides an -t-diagram with contours of the skin friction ω z,wall for 3T after switching the disturbance generation to skin-friction signal feedback. The separation bubble emerges as a white area at the lower end of the figure, whereas the region of higher skin-friction upstream and downstream of the bubble appears dark grey or black. Five T after the feedback has been switched on, the region of strong positive skin friction is shifted almost instantly to = The onset of vorte shedding, traced by oblique lines of alternating light and dark shades, is being shifted upstream to = 11, as well. A region of evenly distributed negative skin friction, the LSB, is no longer present. After the new flow pattern has been established, an almost constant vorte shedding frequency sets in. Occasional phase jumps like the one in 13T t 135T are due to the feedback which is not synchronised to the phase speed of the disturbances. However, this is not necessary for the present purpose. The data at hand show that the LSB reacts very quickly on changes of the feedback amplitude. 5. CONCLUSIONS AND OUTLOOK Operated in a low Reynolds number regime the drag characteristics of laminar airfoils at offdesign conditions can be considerably deteriorated by laminar separation bubbles (LSB). The 19

20 Figure 22: Amplification rate α i according to linear stability theory of the time averaged flow. y/ Re y/ Re Figure 23: Comparison of the spanwise vorticity before (t = 12T ) and after onset of feedback (t = 16T ). 2

21 Figure 24: t diagram of the skin-friction ω z,wall after onset of feedback control at t/t = 12. LSBs form due to the inability of the laminar boundary layer to overcome the stronger adverse pressure gradient caused for eample by a higher angle of attack or a deflected trailing edge flap. Such an effect is shown in Fig. 25, where the lift coefficient c L is plotted versus the drag coefficient c D for a laminar airfoil at two characteristic operating conditions: Low speed characterised by the chord Reynolds number Re =.7 1 6, and high speed at Re = The airfoil can be adapted to its operation conditions via a deflectable trailing-edge flap which is moved upwards for high speed and downwards at low speed when high lift is required. The high-lift configuration is subject to a LSB on the lower side in the angle of the flap. However, this LSB can be removed by a turbulence trip mounted upstream of the bubble. The effect of this is to reduce the low-re drag accordingly (dashed line with white diamonds). On the other hand, when the aircraft flies fast, the turbulator (dashed line with white squares) will cause an etra drag penalty due to the presence of a turbulent boundary layer instead of a laminar one. Thus, if the separation bubble control were adaptive, like those considered above, one could turn it on in the low-speed regime to remove the LSB, and turn it off for high-speed to gain some etra speed. For a robust practical implementation of the actuation concept presented in section 3 it might become necessary to replace actuation via suction and blowing by a controlled deflection of a membrane. For this we provide here an estimation of the required deflection amplitude and frequency (with respect to the parameters used in section 3). Under the assumption of a harmonic oscillation, the surface deflection amplitude h w is related to the forcing amplitude A vk in eqn. (12) via h w = A vk /β, (15) i.e., it depends only on the circular frequency. For β = 5 one obtains ĥw = A vk 1 4 / Re [µm], 21

22 C L Re= Re= with turbulator Re= Re= with turbulator C D Figure 25: Lift vs. drag coefficients for the laminar-flow airfoil W99K137 (W. Würz, IAG). Table 1: Estimation of required dimensional frequencies f for different free-stream velocities Ũ and β =5. m Û s ˆf Hz which means that a harmonic surface deflection of 1 µm will already produce a disturbance corresponding to A vk = in the above investigations. This seems largely sufficient since the LSB disappeared at A vk 1 3. However, these estimations are only valid under the assumption that the streamwise length of the actuator is tuned to the streamwise wave length of the TS-wave that is going to be ecited. Otherwise the efficiency of the actuator would be reduced. An estimation of the appropriate dimensional disturbance frequencies ˆf = βû /(2π ˆL) for maimum amplification is collected in Table 1. In contrast to the amplitude the dimensional frequency depends on the free-stream velocity Û as well. The estimation shows that a frequency range up to 1 khz is required for a successful realisation of the above concept for free-stream speeds until Û = 216 km/h. Based on the present results, a Piezo-ceramics actuator has been built and tested [17]. It is interesting to note, that a similar device has already been constructed and evaluated in the 196s (based on that time s technology) [24], or more recently by Sinha (21) [26]. More recently, wind-tunnel eperiments have been performed on the use of piezoelectric actuators for airfoil separation control, e.g. by Seifert et al. [25]. Suchconcepts arein considerable contrast to more traditional vorte-generator- [9] or jet-based separation control techniques [8]. Our actuator showed the capability of a maimum frequency of more than 1 khz and a 22

23 maimum amplitude of the surface deformation of several µm. This demonstrates that it is possible to build a suitable actuator for the concepts presented in the previous sections. More work is now necessary to improve the sensor technology in order to see whether the LSB can be detected accurately enough for a reliable control. This is the area of ongoing research, as already said above. References [1] K. Augustin, U. Rist, S. Wagner (21): Active control of a laminar separation bubble, in: P. Thiede (Ed.), Aerodynamic Drag Reduction Technologies, Proc. CEAS/DragNet Conference, Potsdam, , NNFM Vol. 76, Springer-Verlag, Berlin, Heidelberg, [2] K. Augustin, U. Rist, S. Wagner (22): Active control of separation bubbles eploiting laminar base-flow instabilities, Proc. 22 ASME-European FED Summer Annual Meeting, No. 3147, Montreal, Quebec, Canada, July [3] K. Augustin, U. Rist, S. Wagner (23): Investigation of 2D and 3D boundary-layer disturbances for active control of laminar separation bubbles, AIAA [4] M. Gad-el-Hak (2): Flow Control: Passive, Active, and Reactive Flow Management, Cambridge University Press, London, United Kingdom. [5] D. Greenblatt, I.J. Wygnanski (2): The control of flow separation by periodic ecitation, Progr. Aerospace Science 36, [6] E. Greff (1991): In-flight measurement of static pressures and boundary-layer state with integrated sensors, J. Aircraft 28, [7] B.J. Holmes, C.J. Obara (1992): Flight research on natural laminar flow applications, R.W. Barnwell, M.Y. Hussaini (Eds.), Natural laminar flow and laminar, flow control, Springer, Berlin, New York, [8] Horstmann, K.H.; Quast, A.; Boermans, L.M.M. (1984): Pneumatic turbulators a device for drag reduction at Reynolds numbers below 5 1 6, AGARD-CP-365, [9] M. Kerho, S. Hutcherson, R.F. Blackwelder, R. H. Liebeck (1993): Vorte generators used to control laminar separation bubbles, J. Aircraft 3, [1] M. Kloker (1993): Direkte numerische Simulation des laminar-turbulenten Strömungsumschlages in einer stark verzögerten Grenzschicht, Dissertation, Universität Stuttgart. [11] M. Kloker, U. Konzelmann, H. Fasel (1993): Outflow boundary conditions for spatial Navier Stokes simulations of transitional boundary layers, AIAA J. 31 (4), [12] L. Löfdahl, M. Gad-el-Hak (1999): MEMS applications in turbulence and flow control Progr. Aerospace Science 35, [13] U. Maucher (21): Numerische Untersuchungen zur Transition in der laminaren Ablöseblase einer Tragflügelgrenzschicht, Dissertation Universität Stuttgart. [14] U. Maucher, U. Rist, M. Kloker, S. Wagner (2): DNS of laminar-turbulent transition in separation bubbles, in: E. Krause, W. Jäger (Eds.), High Performance Computing in Science and Engineering 99, Springer-Verlag, Berlin, Heidelberg, pp

24 [15] U. Maucher, U. Rist, S. Wagner (1999): Transitional structures in a laminar separation bubble, in: W. Nitsche, H.J. Heinemann, R. Hilbig (Eds.), New Results in Numerical and Eperimental Fluid Mechanics II, NNFM Vol. 72, Vieweg, [16] U. Maucher, U. Rist, S. Wagner (2): A refined interaction method for DNS of transition in separation bubbles, AIAA J. 38, [17] J. Müller (1999): Konzeptstudie für ein aktives Oberflächenfeld zur Verhinderung laminarer Strömungsablösung. Studienarbeit, Bereich Flugzeugentwurf, Fakultät Luft- und Raumfahrttechnik, Universität Stuttgart. [18] A. Padmanabhan, M. Sheplak, K.S. Breuer, M.A. Schmidt (1997): Micromachined sensors for static and dynamic shear-stress measurements in aerodynamic flows, Proc. IEEE Transducers 97 Conference, Chicago, IL, June [19] T. von Papen, H.D. Ngo, E. Obermeier, M. Schober, S. Pirskawetz, H.-H. Fernholz (21): A MEMS surface fence sensor for wall shear stress measurement in turbulent flow areas, in: E. Obermeier (ed.), Transducers 1 / Eurosensors XV, Springer-Verlag, Berlin, Heidelberg. [2] U. Rist (1999): Zur Instabilität und Transition in laminaren Ablöseblasen, Habilitation Universität Stuttgart, Shaker, Aachen, Maastricht, [21] U. Rist (23): Instability and transition mechanisms in laminar separation bubbles, VKI/RTO-LS Low Reynolds Number Aerodynamics on Aircraft Including Applications in Emerging UAV Technology, Rhode-Saint-Genèse, Belgium, November 23. [22] U. Rist, K. Augustin, S. Wagner (21): Numerical simulation of laminar separation bubble control, in: S. Wagner, U. Rist, H.J. Heinemann, R. Hilbig (Eds.), New Results in Numerical and Eperimental Fluid Mechanics III, Proc. 12. DGLR-Fachsymposium AG STAB, Stuttgart, , NNFM Vol. 77, Springer-Verlag, Berlin, Heidelberg, [23] U. Rist, H. Fasel (1995): Direct numerical simulation of controlled transition in a flat-plate boundary layer, J. Fluid Mech. 298, [24] W. Schilz (1965): Untersuchungen über den Einfluss biegeförmiger Wandschwingungen auf die Entwicklung der Strömungsgrenzschicht, Acustica 15, 6 1. [25] A. Seifert, S. Eliahu, D. Greenblatt, I. Wygnanski (1998): Use of piezoelectric actuators for airfoil separation control, AIAA J. 36, [26] S.K. Sinha (21): Flow separation control with microfleural wall vibrations, J. Aircraft 38, [27] D. Sturzebecher, S. Anders, W. Nitsche (21): The surface hot wire as a means of measuring mean and fluctuating wall shear stress, Ep. Fluids 31, [28] A.E.P. Veldman (1981): New, quasi-simultaneous method to calculate interacting boundary layers, AIAA J. 19,