DNS of Ative Control of Disturbanes in a Blasius Boundary Layer Christoph Gmelin, Ulrih Rist, and Siegfried Wagner Universitat Stuttgart, Institut fur Aerodynamik und Gasdynamik Pfaenwaldring 21, D-755 Stuttgart, e-mail: gmelin@iag.uni-stuttgart.de Abstrat. Three dierent ontrol onepts to attenuate disturbanes in dierent stages of the laminar-turbulent transition are presented. Ative ontrol via FIR lter is a linear onept based on the priniples of wave superposition suitable for linear and weakly nonlinear disturbanes. The onept of v-ontrol originates from turbulene ontrol where it was basially intended to suppress quasi-steady longitudinal disturbanes. In our ase the resonant behavior of nonlinear waves was strongly altered. As a third onept the vortiity-ontrol is a novel approah, sensing the spanwise vortiity (shear stress) at the wall and presribing v at the wall in phase. This strategy yields a diret attenuation of both linear and nonlinear waves as well as a hange in their resonant behavior. 1 Introdution Many onepts with the objetive to atively delay the laminar-turbulent transition are urrently under investigation. In ontrast to rather mathematial approahes based on optimal ontrol theory, we follow a more pratial path here, whih is based on the use of a FIR-lter as in the eperiments of Baumann & Nitshe [1], the \v-ontrol" as in the work on turbulene ontrol of Hammond [6] and Choi et al [2] or the \vortiity-ontrol", a novel approah in transition ontrol (Fig. 1). Two questions are addressed: To what etent an a linear tehnique be used in the non-linear regime of transition and to what etent an the feedbak of instantaneous signals from the ow eld to the wall be used for transition ontrol, i.e. transition delay. Moreover, the underlying ontrol mehanisms are of major interest. 2 Numerial Method All simulations were performed in a retangular integration domain with the spatial DNS-ode developed by Konzelmann, Rist and Kloker [7]. The ow is split into a steady 2D-part (Blasius base ow) and an unsteady 3Dpart. The -(streamwise) and y-(wall-normal) diretions are disretized with nite dierenes of fourth-order auray and in the spanwise diretion z a spetral Fourier representation is applied. Time integration is performed by the lassial fourth order Runge-Kutta sheme. The utilized variables are
2 C. Gmelin, U. Rist and S. Wagner a) b) ) wall normal veloity v measured in the ontrol layer (y = onst.) integration domain U1 spanwise vortiity ω z v v wall wall FIR ontrol strip ontrol strip sensor ontrol strip initial disturbane strip Fig. 1. Dierent ontrol strategies: a): FIR lter, b): v-ontrol, ): vortiity ontrol normalized with U ~ 1 = 3 m s, ~ = 1:5 1;5 m s 2 dimensional variables): and ~ L = :5m (~ denotes = ~ ~ L y = ~ ~ L p Re z = ~z ~L t = ~t ~ U 1~L u = ~U ~u v = ~v p Re w = ~w 1 ~U ~U1 Re = U ~ 1 L ~ 1 ~ =1 5 where u, v and w are the omponents of the unsteady veloity disturbanes. This leads to the dimensionless Frequeny = 2 f ~ L ~U1 ~, where f ~ is the Frequeny in [Hz] and the dimensionless spanwise vortiity! z = @u @y ; 1 @v Re @. 3 Control Conepts and their Appliation The (linear) FIR-lter is an on-line onept whih has to be trained for eah ow ondition or has to be adapted ontinuously. Superposition of anti-phase disturbanes is used to eliminate the initial perturbations. In our ase, the lter is trained one using data of simulations with suessful anellation to obtain the lter oeients for the subsequent runs avoiding time-onsuming alulations. Appliation of the FIR-lter to linear and weakly nonlinear disturbanes show a redution in amplitude of about 1.5 to 2.5 orders of magnitude at a ed position downstream [3]. After obtaining eellent results in the linear ase, ative ontrol via FIR-lters was applied to fundamental resonane whih would lead to a K- breakdown (Fig. 2). Due to the onset of fundamental resonane the phases of the interating modes (1,) and (1,1) (the rst inde gives multiples of the frequeny, the seond multiples of the basi spanwise wave number ) are synhronized, i.e. their phase speed beomes equal (Fig. 3 (a)). With appliation of ontrol to the 2D mode the phase oupling is broken up and the waves evolve independently (Fig. 2 and Fig. 3 (b)). As a result, the ontrol based on the wave superposition priniple is most eetive when applied at an early stage of transition.
DNS of Ative Control of Disturbanes in a Blasius Boundary Layer 3 1-1 1-2 Amp(U ma ) 1-3 1-4 1-5 1-6 (1;) (;1) Fig. 2. Seondary instability: ontrol of the fundamental 2D mode via FIR lter at =2:96, dotted lines: unontrolled ase. (frequeny = 1, spanwise wave number = 2) a) b).5.45.4.35 (1;).45.35.3.3.5.4 Fig. 3. Seondary instability: phase veloities (a) unontrolled ase, (b) ontrolled ase, dashed lines: phase veloities aording to the linear stability theory Another approah, suitable for nonlinear disturbanes is the \v-ontrol" originally applied to turbulent ows. In this ase the instantaneous wallnormal veloity at a suient distane from the wall (y ) is presribed 3 with opposite sign as a boundary-ondition on the wall at the net time step of the simulation. Several simulations have been performed applying v-ontrol to the fundamental 2D modes (Fig. 4(a)) in the K-breakdown senario already shown in Fig. 2 (dotted lines). Despite an inreased amplitude of the 2D mode (1,) they showed a delayed onset of resonane between the fundamental 2D mode and the resonant 3D modes with the same frequeny. Looking at the phase speed of the interating modes (Fig. 4(b)) a strong aeleration of the ontrolled 2D mode is observed. The resonant mode (1,1) is not able to attain the same phase speed as the fundamental one. Therefore, phase oupling between the waves is impossible and fundamental resonane is delayed. A novel approah reduing the amplitude of both linear and nonlinear disturbanes and simultaneously hanging the resonant behavior of the interating modes is the vortiity-ontrol. The instantaneous spanwise vortiity! z at the wall is presribed in phase as v-omponent atthewall multiplied by an amplitude fator of 7.5 (with respet to the dimensionless quantities used here). This onept requires only informations about the shear stress at the wall whih are easily avaliable in pratie. Applying the vortiity-ontrol to the fundamental resonane senario mentioned above, a strong damping eet on the ontrolled mode an be observed (Fig. 5(a)). Moreover, an eet similar to that obseved applying the v-ontrol an be found. A strong deelleration of the ontrolled mode (in Fig. 5(b) the fundamental 2D mode (1 )) is
4 C. Gmelin, U. Rist and S. Wagner Amp(U ma ) 1-1 1-2 1-3 1-4 (1;) 1-5 (;1) 1-6 (a) u ma Amplitudes.5.45.4.35 (1;).3 (b) phase veloities Fig. 4. Seondary instability: ontrol of the fundamental 2D mode via v-ontrol at = 2:55 :::5:4, dotted lines: unontrolled ase. initiated by appliation of the vortiity ontrol. This suppresses a phase oupling between the interating modes and redues the seondary growth of the 3D disturbanes. An eplanation for the damping of the unsteady 2D modes whih is not some kind of wave anellation is obtained by the dimensionless energy balane equation (for two-dimensional, linear disturbanes): d d U u2 2 + v2 2Re dy = @U ;u v @y + 1 @V Re @ {z } ; 1 d Re d R v! z dy ; dy ;! z 2 dy 1 d ; d (u 2 ; 1 Re v2 ) @U @ dy + v wall p wall p u dy (1) where apitals stand for steady quantities and overbars denote the temporal average over one period in time. A detailed disussion of the energy balane equation an be found in [4] and [5]. The term on the left hand side of (1) represents the spatial inrease of the utuation energy at a ed - position i.e. the ampliation or attenuation of the disturbanes. The major ontribution to the right hand side is supplied by the rst two terms. The energy prodution term (rst term on the right hand side) indiates whether energy is transfered from base ow to disturbane ( R Rdy >) or vie versa ( R Rdy < ) and it dominates in ombination with the dissipation term R! 2 dy the energy balane. The sign of R is determined by the sign of the Reynolds stress ;u v.thisimplies for wave like disturbanes that the phase-dierene between u and v is the most important property of these variables for the energy balane. Figures 6(b) and (e) show besides the phases of u and v the phase dierene between both. In the unontrolled ase = j u ; v j > 2 for y=p Re < :2 (Fig. 6(e)) is observed. This leads to
DNS of Ative Control of Disturbanes in a Blasius Boundary Layer 5 a Reynolds stress ;u v > (Fig. 6(f)) and hene to an ampliation of the disturbane. On the other hand, appliation of vortiity ontrol hanges the phases of u and v in a way that the Reynolds stress ;u v beomes negative (Fig. 6()) and therefore leads to a derease in amplitude. Although the energy balane equation mentioned above has been derived for disturbanes with linear amplitude we an show, that it is valid for nonlinear 2D waves, too, as long as saturation of the amplitude is not reahed. 1-1.5 Amp(U ma ) 1-2 1-3 1-4 1-5 1-6 (1;) (;1).45.4.35 (1;).3 (a) u ma Amplitudes (b) phase veloities Fig. 5. Seondary instability: ontrol of the fundamental 2D mode via vortiityontrol at = 2:55 :::5:4, dotted lines: unontrolled ase. 4 Summary Three dierent unsteady ontrol approahes to ontrol disturbanes were applied to both linear and nonlinear disturbanes. Best results in the nonlinear ase were obtained by using vortiity-ontrol, a onept whih follows a very simple yet eetive ontrol law. This approah is very robust in the sense that it is allmost independent of the amplitude of the ontrolled wave. In the ases we investigated a superposition of some kind of \anti-phase disturbane" ould not be observed. Rather the hanged phases of u and v are the reason for the weakening of the ontrolled modes. Aknowledgements The support of this researh by the Deutshe Forshungsgemeinshaft DFG under ontrat number Be 1192/7-1 is gratefully aknowledged.
6 C. Gmelin, U. Rist and S. Wagner a) b) ).6.4.2 Amp(u) Amp(v).6.4.2 Θ(u) Θ(v) Θ(u)-Θ(v).6.4.2 uv/u 2 ma R/u 2 ma ω z 2 /u 2 ma.1.2.3 ; ; -1-5 5 1 Amplitude 2 Θ 2 d) e) f).6.6.6.4.2.4.2.4 y/vre.2.1.2.3 Amplitude ; ; 2 Θ 2-1 -5 5 1 Fig. 6. Energy relevant quantities vs. y. (a)...() ontrolled ase, (d)...(f) unontrolled ase. (a), (d) veloity proles in wallnormal and streamwise diretion (b), (e) orresponding phases (), (f) energy properties Referenes 1. M. Baumann and W. Nitshe (1996) Investigations of Ative Control of Tollmien-Shlihting Waves on a Wing. In Henkes and van Ingen(Eds.), Transitional Boundary Layers in Aeronautis, 46:89{98. KNAW, Amsterdam 2. H. Choi, P. Moin, and J. Kim (1994) Ative Turbulene Control for Drag Redution in Wall-bounded Flows. J. Fluid Meh., 262:75{11 3. C. Gmelin, U. Rist, and S. Wagner (1999) Investigations of Ative Control of Wave Pakets and Comparable Disturbanes in a Blasius Boundary Layer by DNS. In Notes on Numerial Fluid Mehanis. Vieweg-Verlag, Braunshweig 4. Franis R. Hama and Stephan de la Veau (198) Energy balane equations in the spatial stability analysis of boundary layers with and without parallel-ow approimation. Prineton University, Prineton, New Jersey, unpublished, 198. 5. Franis R. Hama, David R.Williams, and Herrmann Fasel (1979) Flow eld and energy balane aording to the spatial linear stability theory of the blasius boundary layer. In Laminar-Turbulent Transition. Springer-Verlag, 198. IUTAM Symposium Stuttgart/Germany 6. E. P. Hammond, T. R. Bewley, and P. Moin (1998) Observed mehanisms for turbulene attenuation and enhanement in opposition-ontrolled wall-bounded ows. Phys. Fluids, vol.1 7. U. Rist and H. Fasel (1995) Diret numerial simulation of ontrolled transition in a at-plate boundary layer. J. Fluid Meh., 298:211{248