IIIfl ifl +Ijili I IQ MqO O

Transcription

1 N A D-A TION PAGE IIIfl ifl +Ijili I IQ MqO O j fiij A Form-Appr-ved I O0JW N "jp n.+ -.n Ul ' - - t - got ~ *IIr jf V10i UiIf Of I tofltsona Silvid ommel reqarumt ul de numme M mptef *ran El o f 1,, ol iiii 1111Iiili iitl I I WaSMIWPqO "OaiQUMOI Seftiu iorgnorat* for InfonwaIo~ 0, ain aqemflf OMog a ASOR ano 1udqe Paowvom ia 1111; 9tn., Redutioi 1'otet (Ole" I). Weibtwol. O 20S0L 1.".... T 3..EPORT TYPE AND DATES OVERED SDe 1991 Final 1 Aug, 8) - 31 Jill Tit AND SUBITLrE (U) S. FUNDING NUMBERS PE F Feedbak Stabilization of Hydrodynami Instabilities PR SA - BS. AUTHOR(S) 0 - Avn'QP RQ-nl!Q Professor M. Gaster 7. PERFORMING ORGANIZATION NAME(S) AND AOORESS(ES). PERFORMING ORGANIZATION REPORT NUMBER University of ambridge Engineering Dept AFOSR-TR-!'4 2 O( Trumpington Street ambridge B2 1PZ 9. SPONSORING/MONITORING AGENY NAME(S) AND ADOARES I 10. SPONSORING/MONITORING AG EN Y AFOSR/NA F,"' REPORUMBER Boiling AFB D MAYJ ,/ 11. SUPPLEMENTARY NOTES 12. DISTRIBUTION / AVAILAIUITY STATEMENT 12b. DISTRIBUTION O01 Approved for Publi Release; Distribution unlimited 13. ABSTRAT (Ma,,mum 200 words) The stabilization of boundary layers via ontrol ompared to passive boundary layer ontrol tehniques. To date, all experimental disturbanes in the laminar boundary layer. To further improve the level of disturbane attenuation in the boundary layer requires an inherently three dimensional approah. The goal of this study was to ontrol the boundary layer response to random three dimensional disturbanes introdued near the leading edge of a flat plate. This goal was not fully ahieved in the shortened time frame of this investigation. The basi flow struture to be ontrolled is the three-dimensional wave paket, i.e. the boundary later response to loalized pulse exitation. Sine the ontrol is to take plae in the linear region of the transition zone, any oneivable flow disturbane an be synthesized and hene anelled by appropriate wave paket superposition. 14. SUBJET TERMS IS. NUMBER OF PAGES boundary layer, turbulent flow, transition PRIE ODE 17. SEURITY LASSIFIATION IL5 SEURITY LASSIFIATION 19. SEURITY LASSIFIATION 20. LIMITATION OF ABSTRAT OF REPORT OF THIS PAGE OF ABSTRAT Unlassified -Unlassified I Unlassified NSN 754"1.2O0-S500 Standard Form 293 (Rev. 2-89)

2 AFOSR Feedbak Stabilization of Hydrodynami Instabilities. Roland A.E. Heinrih and Mike Gaster F.R.S. Final Report August 1, 1989 to July 31, 1991 Summary The stabilization of boundary layers via ative ontrol promises drag redution with relatively small power requirements ompared to passive boundary layer ontrol tehniques. To date, all experimental ative ontrol investigations have onentrated on two-dimensional disturbanes in the laminar boundary layer. To further improve the level of disturbane attenuation in the boundary layer requires an inherently three-dimensional approah. The goal of this study was to ontrol the boundary layer response to random three-dimensional disturbanes introdued near the leading edge of a flat plate. This goal was not fully ahieved in the shortened time frame of this investigation. The basi flow struture to be ontrolled is the threedimensional wave paket, i.e. the boundary layer response to loalized pulse exitation. Sine the ontrol is to take plae in the linear region of the transition zone, any oneivable flow disturbane an be synthesized and hene anelled by appropriate wave paket superposition. The projet was divided into two main parts, (i) numerial modelling of the ontrol proess, and (ii) experimental investigation of the most promising ontrol arrangement identified in (i). The numerial modelling was an integral part of our approah and allowed us to a priori assess the best possible ontrol performane ahievable with a given design. The numerial model allows rapid testing of different spanwise and streamwise distributed detetor and atuator arrays. Using this model, a promising ontrol onfiguration was identified, implemented in the ambridge University low turbulene wind tunnel faility and tested in part (ii) of this work. The experimental investigation ahieved signifiant ontrol of single mode disturbanes. In fat, ontrol to about twie the natural bakground level was possible for frequenies within the unstable T-S band (attenuation up to 32dB). The disturbane detetion was based on a differene mirophone arrangement in order to rejet aousti disturbanes in the faility. This mirophone arrangement proved to be ideally suited for the urrent purpose. The atuators, embedded miniature loudspeakers, proved apable of ontrolling moderate sized instability waves. Wave paket ontrol was attempted but failed due to inadequate real time behaviour of the eletroni filtering iruit. This was traed to exeedingly large rise and settling times of the eletroni iruit. i mmmmmmm mm

3 -1- AFOSR Feedbak Stabilization of Hydrodynami Instabilities Roland A.E. Heinrih and Mike Gaster F.R.S. Final Report August 1, 1989 to July 31, Introdution The possibility of "ative" ontrol of instability waves in boundary layers has attrated the renewed attention of sientists sine the early 80s. Ative ontrol denotes a onept in whih the manifestations of flow instability, i.e. the Tollmien-Shlihting (T-S) waves are influened by diret means using wave superposition priniples. It is inherently different fromr "passive" methods of ontrol whih modify mean flow features to ahieve stability with regard to small disturbanes. The potential benefit of ative rather than passive ontrol of external fluid flows (i.e. boundary layers) lies in the redued power requirement of the former. Even though passive means of flow ontrol have been suessful in flight experiments in delaying transition by, say, boundary layer sution or wall ooling (air), the undeniable benefit in drag redution for pratial appliations is partly negated by the power requirement for the speifi

4 -2- tehnique employed (Antonatos, 1966; Whites, Sudderth, and Wheldon, 1966; Pfenninger and Reed, 1966; Nenni and Gluyas, 1966; Wagner and Fisher, 1983). The power requirement for an ative ontrol onfiguration is proportional to the energy ontent of the flow disturbanes and thus is small ompared to a passive ontrol implementation. Nossenhuk (1982) inreased the transition length in a flat plate boundary layer in water by 25% using an ative ontroller with 1 Watt power input. The same delay using passive ontrol (wall heating) required 1900 Watts! Though not as widely known as the passive flow ontrol methods mentioned above, ative flow ontrol has been explored sine the mid 60s (Wehrmann, 1965; Shilz, 1965/66) and re-emerged in the early 80s (Milling, 1981; Liepmann, Brown and Nosenhuk, 1982; Thomas, 1983; Strykowski and Sreenivasan, 1985; Maestrello, 1985; Ladd and Hendriks, 1988; Ladd, 1990). Improved data-aquisition tehniques and advanes in miro-eletronis are at least partially responsible for the renewed fous on these tehniques. The early experiments by Wehrmann (1965) and Shilz (1965/66) demonstrated the experimental viability of ative ontrol by anelling two-dimensional instability waves via a downstream outof-phase ontroller input (through flexible wall or sound exitation). Milling (1981) used the vibrating ribbon tehnique to exite T-S waves in his water hannel experiment. A seond wire

5 -3- downstream introdued an "anti-wave", i.e. a wave with phase and amplitude adjusted to anel the downstream propagating T-S wave. Liepmann, Brown and Nosenhuk (1982) used a novel tehnique to exite T-S waves in their water tunnel experiment. Here, flush mounted surfae heaters were ativated to exite the T-S waves in a zero pressure gradient flat plate boundary layer. Downstream anellation was attempted via a seond set of heater strips driven with the appropriate anti-wave. A redution in disturbane amplitude and a delay of transition was ahieved. Liepmann and Nosenhuk (1982) extended the tehnique to the ontrol of natural transition, i.e. the T-S waves were not artifiially introdued as in the above mentioned experimental studies but were generated naturally by the bakground free-stream disturbane. A hot-film probe was used as a sensor downstream of the heater strip whih served as the ontrol atuator. The sensor output was analyzed and a signal was synthesized to drive the atuator. This feedbak ontrol system showed T-S wave attenuation or reinforement depending on the relative phase of sensor and atuator signal. Thomas (1983) studied the influene of vibrating ribbon ontrol (two-dimensional) on the evolution of the three-dimensional strutures in the transition region. He showed that even though transition ould be delayed with two-dimensional ative ontrol, the weak three-dimensional bakground disturbane field interated with the remainder of the two-dimensional primary field to restart the transition proess. A modified vibrating ribbon tehnique was used in a wind tunnel experiment by Strykowski and Sreenivasan (1985) to exite and then anel the T-S wave. In this experiment,

6 -4- the ribbons where supported in slots in the surfae of the plate and hene reated disturbanes more like loalized sution and blowing. The experiment was suessful in ontrolling the TS-wave. They also tried to use the heater strip tehnique of Liepmann et al. to generate and anel T-S waves in air. However, they were unable to observe any flow perturbations using this tehnique. This suggests that the heater strip tehnique might not be suitable for ative ontrol atuation in zero pressure gradient airflow appliations. In boundary layers with streamwise pressure gradients, however, this might be different. In fat, Maestrello (1985) has used heating strips to generate instability waves in boundary layers with favorable (I) pressure gradients. He also used an external sound field to generate an out-of-phase ontrol signal whih led to T-S wave redution. The reent experimental investigations by Ladd and Hendriks (1988) ari Ladd (1990) are onerned with the ative ontrol of artifiially exited and natural ouring instability waves on an axissymetri body. The experiments were arried out in a water tunnel using hot-film detetor and exiter arrays. A feedbak loop using an adaptive ontrol algorithm was used to suessfully ontrol two-dimensional disturbanes. The authors disuss the strong three-dimensional (azimuthal) disturbane omponents and envision improved results if a three-dimensional ontrol sheme were implemented. Numerial simulations of ative boundary layer ontrol have been performed by various investigators. Most of them integrate the Navier-Stokes equations numerially in one form or another. A

7 -5- review [see Kral (1988)1 of these is outside the sope of this work, it suffies to mention that until reently the majority dealt only with two-dimensional disturbanes and essentially modelled vibrating ribbon experiments [but see three-dimensional numerial simulations of ative ontrol in boundary layers by Laurien and Kleiser (1985), Laurien (1985) and Zang and Hussaini (1985a, 1985b)]. All studies onfirm the feasibility of ative ontrol transition delay or enhanement. 2. Sope of Investigation The above survey highligths the ahievements and limitations of urrent ative ontrol boundary layer researh. One serious limitation of all the reported experimental investigations is their restrition to two-dimensional ative ontrol. This limitation is a heritage of Shubauer and Skramstad's (1947) experimental verifiation of the nearly two-dimensional nature of T-S waves in the early stage of the transition proess. By now, however, it is well established that the transition proess is inherently threedimensional and the onentration on two-dimensional waves seems unjustified. In fat the above ited studies by Thomas (1983) and Ladd (1990) disuss the need of ontrolling the three-dimensional strutures in order to advane the ative ontrol onept. The lassial vibrating ribbon annot generate a three-dimensional ontrol input and hene a different onfiguration is required. The tehnique of flush mounted surfae heaters used by Liepmann et al. _m mmm, mmmm m mmmmmmm m m m mmm

8 -6- is in priniple suited to three-dimensional ontrol atuation, however, the relative large size of the surfae heaters and the rather limited power input possible might prelude their suessful use in air flow appliations. The urrent investigation is based on analytial and experimental work on three-dimensional wave pakets in the laminar boundary layer by Gaster (1968, 1975, 1981, 1985). Starting oneptually from a loal three-dimensional disturbane in the boundary layer produes an approah more suited to the ontrol problem at hand. Sine it is possible (in linear theory) to superpose point soures to synthesize arbitrary foring fields, the pulse response an be viewed as a generi flow disturbane. A loalized pulse generates a wave paket whih propagates downstream in the boundary layer and may form into a turbulent spot. Isolated turbulent spots grow and merge to form a fully developed turbulent boundary layer. Gaster (1975) has shown that the wave paket development in its early stage an be modelled aurately by linear mode superposition. Hene, extending the ideas of T-S wave anellation via an anti-wave, the anellation of the evolving wave paket by superposition with an "anti-paket" should be possible. Viewing the linear wave paket as fundamental building blok of the transition proess we propose the use of wave paket superposition to delay boundary layer transition due to artifiially exited and - ultimately - randomly exited disturbanes. In order to establish the feasibility of three-dimensional random

9 -7- disturbane ontrol, the following issues were investigated in the ourse of this projet: (a) sensor/atuator seletion for ontrol problem, (b) sensor/atuator arrangement based on numerial model of ontrol system inluding boundary layer response, () disturbane detetion in a "noisy" environment. Finally, a wind tunnel experiment was arried out in order to evaluate the seleted ontrol arrangement. This report is divided into three setions. In Setion 3 below the omputer model of the linear three-dimensional ontrol proess it desribed. One a promising ontrol onfiguration was identified through the modelling proess, the required hardware was built and an experimental investigation was launhed. In Setion 4 this experimental investigation is desribed in detail. Final, in Setion 5 onlusions are drawn from the experiene gained during this investigation and a ourse of future researh is suggested. 3. omputer Model of Ative ontrol Proess in order to minimize the amount of experimental "trial and error" neessary to find a viable arrangement of loal surfae detetors and ontrol atuators, full use was made of omputer simulation to test ideas and optimize (up to a ertain point) the setup before the ative ontrol experiment was performed. In Subsetion 3.1, the speifi flow ontrol onfiguration will be presented. The ontrol task will be disussed. Subsetion 3.2 deals with the model of the

10 -8- flow field response based on the three-dimensional Orr-Sommerfeld stability equation. In Subsetion 3.3 we digress and ompare our numerial flow field response with the experimental flow field response obtained in a preliminary wind tunnel investigation. In Subsetion 3.4 we use the model of our flow field response to takle the boundary layer ontrol problem by modelling various spanwise detetor and atuator arrangements. This modelling leads to the definition of the most "suitable" ontrol arrangement for the ontrol experiment desribed in Setion 4. (Listings of the omputer programmes developed in the ourse of this projet an be found in Appendix E.).. 1 ontrol Task Definition At the outset it is neessary to speify the system to be modelled and ontrolled. As stated above, the unstable T-S disturbanes propagating in a flat plate Blasius boundary layer are to be ontrolled. Figure 1 shows a shemati of the flat plate boundary layer and introdues the ontrol problem. In typial flow situations, disturbanes whih are naturally present in the flow environment will enter the boundary layer and generate T-S instability waves. The proesses by whih these external disturbanes enter the boundary layer - the "reeptivity" problem - are still not fully understood and urgently require further researh. Our urrent understanding suggests, however,

11 -9- that - in the absene of any wall roughness or wall humps - the external disturbanes enter the boundary layer predominantly in the leading edge region. In order to model this proess properly, a distributed disturbane generation in the region downstream of the leading edge would be neessary. For the urrent investigation, however, the disturbane generation proess is assumed to take plae at a ertain fixed distane downstream of the leading edge. It is important to emphazise that the artifiial disturbane generation downstream of the leading edge is an aid in the modelling and experimental phase of the urrent investigation. Ultimately the flow disturbanes should be generated by "natural" means, i.e. turbulene grids upstream of the plate. One the disturbanes have entered the boundary layer, they will be damped or amplified as they propagate downstream (see Subsetion 3.2) and will reah the detetor loation in the flow. Here a spanwise array of detetors will measure the disturbanes present in the flow. In essene, the ontinuous time-dependent and spanwise varying flow field will be "disretized" with a finite number of spanwise detetors. (Sine an analogue ontrol iruit will be used no time disretization takes plae.) After detetion of the signal it will be analyzed and filtered appropriately. A ontrol signal will be synthesized and injeted into the flow at disrete spanwise atuator loations. The ontrolled flow field will be filtered by the boundary layer and the ontrol performane will be monitored downstream of the

12 - 10- ontrol loation in the "target" zone. It is oneivable that a separate row of detetors in this target zone provides the neessary information for true losed loop feedbak. In the present study, however, signals obtaine-d at the target loation are only used to monitor the ontrol performane and are not fed bak into the ontrol iruit. The equivalent of the just desribed physial system is represented in blok diagram form in Fig. 2a. Sript H denotes a system transfer funtion and round and square brakets identify ontinuous and disrete representations respetively. H1(k,) and H2(k,w) denote the boundary layer response to the external foring upstream and downstream of the ontrol loation. H12(k,w) is the boundary layer response in between the physial loation of ontrol detetion and atuation. The bottom branh of the loop in the blok 1,agi--- int±'-ies deteilor or:trolier and atuator transfer funtions. Detetor and atuator transfer funtions are eah split into t',o distint parts, the first (1) taking into ao-nt the disrete spanwise nature of the detetion and atuation, the seond part (2) dealing with the finite frequeny response of the physial omponents. Inluded in the figure are some possible ontrol strategies, namely gain sheduling or adaptive ontrol. None of those however will be onsidered in this report. A simplified blok diagram is presented in Fig. 2b. Here, the loop's omplexity is redued by negleting the disturbane modifiation due to the flow field inbetween detetor and atuator.

13 This simplifiation is justified if detetor and atuator are physially lose together as will be the ase in the onfiguration hosen for our study. Furthermore, in this simplified ase, the finite frequeny response of the detetors and ontrol atuators is negleted, i.e. these omponents are presumend to have an infinite bandwidth flat frequeny response harateristi. This assumption is justified for the urrent investigation sine embedded mirophones and loudspeakers were hoosen for detetion and atuation respetively and these respond essentially flat in the relevant T-S frequeny band [150 Hz to 400 Hz]. 3.2 Boundary Layer Impulse response The modelling of the ontrol iruit desribed in the previous subsetion requires knowledge of the boundary layer response, i.e. the system transfer funtions HI, H12 and H2 (Fig. 2). Sine the early stage of the boundary layer transition proess is desribed by linear theory, the impulse response of the boundary layer provides the required information. Linear systems theory allows to onstrut the flow response to any arbitrary input by simple onvolution of this input with the impulse response funtion. Sine an impulsive input is modelled as a Dira spike, the initial ondition for the numerial model of the boundary layer impulse response onsists of a flat spetrum of unit amplitude. The downstream development of eah individua 1 Fourier omponent an be

14 -12- determined from lassial linear stability theory. The pulse response at a ertain downstream loation is obtained by superposing the ontribution of all Fourier omponents at this point. The linear theory desribing the three-dimensional disturbane development in a flat plate boundary layer is based on the Orr-Sommerfeld equation invoking Squire's transformation to redue the three-dimensional problem to an equivalent twodimensional one (Mak, 1969). Solving the resulting eigenvalue problem provides the omplex wavenumber "a" as a funtion of loal Reynoldsnumber and foring frequeny (J. The impulse response at a ertain downstream loation x as a funtion of spanwise position is thus desribed by u'(time,z-span) =2Erexp [i Ja x) dx: + b. z - W 1 t)]i Here u' denotes the disturbane veloity omponent parallel to the wall in mean flow diretion. Approximately eigenvalues "a" need to be alulated in order to represent the temporal and spanwise development properly. The integral in the exponent is evaluated using a Romberg integration proedure. In order to rapidly alulate the large number of required eigenvalues, an effiient algorithm (Gaster, 1978) was used whih expresses the eigenvalue dispersion relationship in a omplex double series, s~~~~. (o -Z.,.a# 2 n _2 UX ; = 1 P 3

15 -13- The spatial eigenvalue evaluation required Newton-Raphson iteration. Using the Gaster double series, the eigenvalues were obtained in about 15 minutes on a SUN-Spar station. It should be noted that only eigenvalues falling within a ertain area surrounding (and of ourse inside) the neutral stability urve were onsidered sine the other highly damped modes do not ontribute signifiantly to the impulse response far away from the soure. Using the desribed proedure, the required boundary layer impulse response funtions Hi and H2 were alulated. Figure 3 shows ontour plots of the spanwise distribution of the disturbane u- veloity. Also shown is the onvolution Hi*H2, orresponding to the flow response at the target loation (see Fig. 1) due to pulse foring at the exiter loation. The modelled boundary layer impulse response produes the expeted wave paket shape (Gaster and Grant, 1975; Gaster; 1975). In the following subsetion, this predited wave paket shape will be ompared more losely with the one obtained from experiment. 3.3 omparison of Model and Experiment The above desribed boundary layer response alulation forms the building blok for the modelling of the ontrol arrangement. Hene, before starting extensiv simulations based on this model it is sensitive to ompare the predited impulse response of the

16 -14 - boundary layer flow with suitable experiments. To this end a test plate was built and experiments were performed in the ambridge University low turbulene researh wind tunnel. Details of the hardware are desribed in Appendix and the experiment itself is presented in Appendix D. The experiment was set up to model the parameters of our numerial model, i.e. disturbane generation at Re, = 870, and measuring position at Reep 1235, orresponding roughly to the fititious "ontrol" loation of ReF Measurements of the flutuating u-veloity omponent where obtained with a onstant temperature hot-wire positioned just outside the boundary layer where the T-S eigenfuntions exhibit an outer (flat) maxima. In the experiment, the flow was exited by a short duration (0.5ms) omputer generated pulse via an embedded loudspeaker. The results are presented as isometri views and ontour plots for the u-veloity flutuation. Figure 4 shows the experimental wave paket results from ensemble averaged reords. The equivalent flow field alulation based on our numerial model is shown in Fig. 5. Sine the model based on the Orr-Sommerfeld eigenvalue problem does not provide absolute amplitude levels, the amplitude level in the simulation was adjusted to the experimental maximum. omparing experimental and numerial result shows remarkable agreement. In partiular the loation of the wave paket within the time reord is very well reprodued by the model. This an be seen even more learly in Fig. 6 whih shows a omparison of experimental result and numerial result on the paket entre line. onsidering that the model is

17 -15- based on a loally parallel boundary layer approximation, the agreement is exellent. 3.4 ontrol Simulation Having established that the numerial model of the flow response and the atual flow response obtained by wind tunnel experiment are nearly idential, the simulation of the ontrol arrangement introdued in Subsetion 3.1 an be performed General Performane onsiderations oneptually, the ontrol iruit (Fig. 1) onsists of the following omponents (in flow diretion): (i) event detetors, (ii) ontrol atuators, and (iii) residue disturbane detetors whih are loated in the "target" zone downstream of the atuators. The event detetors upstream of the atuators are required sine the disturbane wave pakets (events) appear at random times. We are searhing for viable ontrol geometries, i.e., we use the numerial model to rapidly explore the advantages and disadvantages of different streamwise and spanwise detetor/atuator arrangements. The numerial "ontrol" proess proeeds as follows: the flow response at the detetor loation (Re ) due to any time dependent input disturbane f(t) is obtained by onvoluting f(t) with the apropriate system transfer funtion (Hi). As anonial

18 -16- ase, we hoose the pulse as input disturbane, i.e. f(t) - J(t). At the ontrol loation we "detet" the disturbane and an "ontrol" by arbitrarily modifying the numerial signal at different spanwise loations. This "ontrolled" disturbane is then onvoluted with the sytem transfer funtion H2 in order to provide the disturbane flow in the target zone at Re.= 1780 (Fig. 1). In this first attempt to establish a viable ontrol geometry the individual detetor and atuator dynamis is not taken into aount. Hene, "ideal" ontrol is simulated, i.e. the omponents of the ontrol iruit are assumed to have an undistorted, infinitely wide frequeny response with zero response delay (see Fig. 2b). Three different open-loop ontrol onfigurations are explored: - ase 1, the ideal ontroller is loated at the entre-line, - ase 2, in addition to the ontroller at the entre-line, two ontrollers are plaed symmetrially 12mm to eah side of the entre-line, and - ase 3, a total of four ontrollers are used whih are plaed symmetrially at 6mm and 21mm above and below the entre-line. In order to assess the ontrol performane, the pseudo-energy (u' ) in the target zone is integrated over the whole disturbane flow field. The ost funtion in the target zone is minimized by "optimal" hoie of ontroller gain and phase. Without any ontrol, Fig. 7a shows the modelled flow response at the target loation. The pseudo disturbane energy in the field is This serves as the referene ase to assess the ontrol performane. Figure 7b shows the best possible ontrol for ase 1, a single ontroller on the entre-line. Even though the fringes of the paket an not be

19 -17- ontrolled, the disturbane energy is redued by a fator of 3 to 5.0. Figures 8a, b show the best ontrol possible for the three ontrollers and four ontrollers onfiguration respetively. With four ontrollers, the residual disturbane energy is redued to 0.6 whih is just 4% of the unontrolled level. The sensitivity of our open loop ontrol to variation in amplifier gain and ontroller phase is shown in Fig. 9. As expeted, proper phase ontrol is ruial for a suessful ative ontrol implementation. It is interesting to note that the minimum in the amplifier gain urve (Fig. 9a) is rather shallow indiating a moderate ost penalty for a ontroller gain mismath. For omparison, the dotted line in Fig. 9a, b indiates the ontrol sensitivity for the ase of a purely two-dimensional single mode wave ontrolled with an ideal twodimensional ontrol arrangement. Obviously, a ontrol amplifier gain of unity allows omplete anellation for this hypothetial ase. A properly mathed amplifier gain is more ritial for this ase than for the three-dimensional wave paket ases presented above, while the phase sensitivity is about equal. From the simulations desribed in this subsetion, it was deided that a spanwise spaing of about 1 0mm would provide an aeptable balane of ahievable ontrol performane and hardware omplexity. U _

20 Performane of hosen onfiguration The "ideal" ontrol iruit used above did not take into aount the physial spaing of detetors and atuators. It was based on the simplified blok diagram in Fig. 2b. The streamwise spaing of detetor and atuator array an be inluded into our numerial model. Furthermore, the detetion was modelled as an ideal point detetion. Embedded mirophones were hoosen to provide disturbane pressure detetion. Experiene from previous own experiments and the work by Kendall (1990) shows, that aousti disturbanes present in the wind tunnel environment produe wall pressure flutuations an order of magnitude larger than those assoiat.ad with the instability waves. These aousti pressure flutuations have to be eliminated from our signal in order to allow detetion of the relevant disturbanes, the instability waves. This was done using two mirophones spaed a "small" (on an aousti sale) distane apart in streamwise diretion and swithed suh that the signals are substrated from eah other. The streamwise distane was in effet hosen to be roughly (0.5*lambda), "lambda" being the wavelength of the most amplified instability wave omponent at the detetor loation. Hene this double mirophone arrangement will effetively rejet the long aousti wavelength but will enhane the detetability of disturbanes with wavelengths in the relevant instability wave range. The behaviour to be expeted for this arrangement is shown in a Bode plot in Fig. 10. The frequeny for whih the differene mirophone arrangement is tuned is denoted by F. and a 6dB

21 -19- amplifiation is ahieved for this partiular frequeny. For lower frequenies orresponding to longer wavelengths, the amplitude response is lower, asymptotially approahing a typial first order attenuation of 20dB per frequeny deade. It is oneptually helpfull to view the double mirophone arrangement from a finite differene standpoint. Essentially the differene mirophone approximates the derivative of the passing signal at the midpoint between the mirophone loation. Integrating the differene mirophone signal eletronially should reover the exat signal at the midpoint (without the onstant ontribution from the aousti ontamination) providing the "ideal" response indiated in the graph. The atual ombined response of differene mirophones and integrator is onstruted by simple addition of the appropriate urves in the Bode plot. It should be noted that in priniple the signal deteted with the differene mirophone arrangement is distorted by folding (aliasing) due to higher disturbane frequenies with shorter wavelength than an be resolved with the seleted streamwise separation of the mirophones. However, sine the unstable frequeny band is rather narrow and frequenies whih exeed F o by a fator of 2 or higher are strongly damped, this spatial aliasing is insignifiant. In order to minimize interferene between atuator loudspeaker and detetor mirophone, it was deided to plae the atuator loudspeakers staggered in the spanwise diretion with respet to the mirophone array (Fig. 10). It is then sensible to synthesize the ontrol atuator input by a weigthed average of the nearest

22 differene mirophone detetors Inluding the differene mirophone with integrator and the spanwise staggering of the detetor/atuator arrangement in our numerial model and taking furthermore the boundary layer flow development in the ontroller region (impulse response funtion H12 in Fig. 2a) into aount, the performane of the full ontrol system was analyzed. Figure 11 a shows the unontrolled model flow field at the target loation due to exitation at the exiter loation near the leading edge. The input at the exiter loation was hosen as a random funtion in time and spanwise distribution in order to present a more realisti flow piture. (Of ourse, testing the ontrol performane with a single pulsed point soure exiter reveals exatly the same information sine the system is modelled linearly.) As before, Fig. 11a shows a ontour plot of the disturbane u-veloity distribution as a funtion of time and spanwise position. For the sake of larity, the enter time trae and the spanwise distribution at a partiular point in time are extrated and plotted separately below and to the right of the ontour plot respetively. onsidering the temporal struture first, the signal looks plausible in omparison to typial osillosope traes of natural (i.e. random) transition experiments. onentrating on the spanwise distribution it is apparent that the struture is predominantly two-dimensional in nature, i.e. the wave fronts are aligned almost perpendiular to the mean flow diretion. This again is expeted from linear stability theory sine two-dimensional disturbanes are in

23 priniple the most amplified ones. However, the struture shows suffiient spanwise modulation to uin any ontrol attempt based solely on a disturbane detetion at one spanwise loation. In other words, this plot onfirms the neessity of the distributed ontrol approah hosen in the urrent investigation if anellation of the disturbane struture over a signifiant surfae area is to be suessful. The same flowfield at the target loation with the ontrol ative is shown if Fig. 1lb. Obviously no total anellation is ahievable with the hoosen detetor/atuator arrangement. However, the maximum amplitude in the whole flow field was redued to 18% of its unontrolled value and the overall pseudo disturbane energy was redued to about 4% of the unontrolled result. The ahieved ontrol has to be viewed as the best possible ontrol in an ideal setting and it is to be expeted that the implementation in a wind tunnel experiment will degrade the performane. Nevertheless, the predited amplitude redution to about one-fifth of its unontrolled value justifies the implementation of the ontrol system in a wind tunnel experiemnt.

24 Ative ontrol Experiment An insert for the test plate was designed and built whih inorporated the detetor and atuator arrangement arrived at by numerial modelling as desribed above. The mehanial and eletroni hardware used is desribed in more detail in Appendix. The experimental investigation was divided into three different stages desribed below. In Subsetion 4.1, the suitability of the embedded miniature loudspeakers for the ontrol of instability waves was tested. Subsetion 4.2 deals with the important issue of disturbane detetion and noise rejetion. Implementation of the full ontrol arrangement inluding both mirophone detetors and loudspeaker atuators is desribed in Subsetion External ontrol As a first step to a suessful ontrol of boundary layer instability waves, it is important to asertain that the ontrol atuators, i.e. the miniature loudspeakers embedded in the plate an indeed produe the disturbane level required for anellation at the atuator loation. In order to do so, an "external" ontrol experiment was devised, bypasssing disturbane detetion entirely. A ontinuous single mode disturbane of known fixed frequeny was introdued at the exiter loation. At the ontrol loation, the detetors were unoupled from the iruit. The ontrol signal driving the ontrol loudspeakers was generated externally with a

25 frequeny generator set to the same frequeny as the exiter frequeny. Finally the phase and amplitude of this external "ontroller" was manually adjusted until disturbane anellation was ahieved as monitored by a hot-wire on the entre-line at the target loation. This proedure was repeated for different frequenies and exiter amplitudes in order to establish the maximum disturbane amplitude whih ould be anelled using the hosen atuator arrangement. It transpired that the hosen atuators ould ontrol disturbanes of moderate amplitude. This moderate amplitude disturbanes were generated by driving the exiter speaker near the leading edge with roughly 40% of the maximal voltage allowable for linear disturbane generation. In other words, driving the exiter speaker 2.5 times as hard produes a disturbane 2.5 times as big (hene the system response linearly) but the ontrol atuators were not able to anel it at the ontroller loation downstream. Figure 12a shows a hot-wire trae of an unontrolled disturbane generated by simultaneously feeding three disrete frequenies into the exiter speaker. Prior to this superposition, eah of these three frequenies had been ontrolled individually by external ontrol as desribed above. If the system is linear, swithing on the externally tuned ontrollers of all three frequenies simultaneously should eliminate the signal. Figure 12b shows the hot-wire trae of the ontrolled result and omparison with the unfored referene ase (not shown) reveals that the residual disturbane in the ontrolled ase is twie the size of

26 - 24- the natural bakground rnoise. Hene from this experiment we ould onlude: (a) the ontrol atuators are suitable for ontrol of moderatly sized instability waves, and (b) tlb iystem behaves linearly as expeted, i.e. linear superposition holds. As a next step, the spanwise distribution of the ontrol "wedge" generated by a finite number of spanwise atuators was examined in. an external ontrol setting. In Fig. 13, the spanwise distribution of the hot-wire rms integrated over the relevant T-S " is shown. The partiular ase represents superpose foring and ontrol of two distint frequenies. The external ontreller was driving four ontrol atuators simultaneaously (indiated by blak triangles on the abissa), only half of the flow field is shown in the graph. In the unontrolled ase we see the expeted roll-off in the spanwise diretion. In the ontrolled onfiguration, the very loal nature of the ontrol near the entre-line is apparent. It is interesting to note that there is no performane penalty at the fringes of the ontrolled region due to possible in-phase amplifiation of the flow disturbane. 4.2 Disturbane Detetion The quality of the sensor signal is of ruial importane for good ontrol performane. This requires proper sensing of disturbane

27 amplitude and phase as well as adequate noise rejetion. The later point is partiularly important due to the dominant aousti disturbane field present in the wind tunnel environment. As desribed above, a double mirophone arrangement was hosen as sensor, the small (aousti sale) st.eamwise spaing of the mirophones allowing rejetion of aousti disturbanes. In order to asertain the "oherene" of the relevant flow disturbane, i.e. T-S instability wave, and detetor signal, a hot-wire was plaed at the entre of the double mirophone in streamwise position but displaed sideways in spa&i by 8mm. This spanwise displaement was hosen in order to avoid any influene of the loal flow field generated by the hot-wire support struture on the mirophone signal. A pulse disturbane was generated at the exiter loation near the leading edge and three hannels were reorded simultaneously providing (i) the hot-wire signal, (ii) the upstream mirophone signal of the double mirophone arrangement, and (iii) the downstream mirophone signal of the same arrangement. Figures 14a and 14b show a single realization of the hot-wire trae and the ensemble average of 64 realizations of the hot-wire signal respetively. The hot-wire trae shows a wave paket riding on a strong bakground disturbane field of omparable magnitude. The generated wave paket is learly disernible after ensemble averaging the signal. However, a single hot-wire at the detetor loation would not be suitable for disturbane detetion due to insuffiient signal to noise ratio. Figures 15a and 15b show a single realization of a single mirophone and of the double

28 mirophone arrangement respetively. This data were taken simultaneously with the single shot hot-wire trae shown in Fig. 14a. No wave paket an be deteted in the single mirophone signal, rendering a single mirophone totally unsuitable for our purpose of instability wave detetion. However, using the mirophone in onjuntion with the seond mirophone plaed downstream suh that the differene signal is reorded, a spatial filter of exellent quality is obtained. In the single realization double mirophone trae shown in Fig. 15b, the instability wave paket learly rises above the bakground noise level. The signal to noise ratio is similar to the one obtained by ensemble averaging the hot-wire with 64(0) realizations. 4.3 Open Loop ontrol Performane Given the ontrol atuator performane disussed in Subsetion 4.1 and the signal quality of a single shot differene mirophone arrangement disribed in the previous subsetion, the ative ontrol of instability waves in the boundary layer should be ahievable with the present hardware onfiguration. n this subsetion, we desribe the ontrol performane ahieved using the desribed arrangement. We onentrate on the ontrol performane at the entre-line. Due to time onstraints, no spanwise traverses were reorded. The whole system thus looks as follows (see Fig. 1). Exitation takes plae with a single "point soure" exiter on the plate entre-line. At the ontrol loation detetion via one set of

29 embedded double mirophones on the entre-line takes plae. This signal is then filtered (see Appendix for hardware details) and fed to four ontrol atuators simultaneously whih are symmetrially distributed aross the entre-line and are 10mm apart. The ontrol performane is assessed in the target zone downstream of the ontrol loation with a hot-wire plaed above the plate entre-line at the outer edge of the boundary layer. In the first set of experiments single mode ontrol was attempted. using the full ontrol arrangement. Figure 16a and b show the hotwire trae at the target loation with ontrol off and on respetively. The almost omplete anellation of the disturbane signal is apparent. In fat, taking the Fourier transform of the ontrolled signal and omparing the integrated rms-value over the T-S frequeny band of the ontrolled and the unontrolled ase reveals a redution of about a fator of 10 in disturbane amplitude, i.e., a fator of 100 in disturbane energy. omparing the ontrolled ase with a referene ase without any disturbane exitation (not shown) reveals that the ontrol ahieves attenuation of the disturbane amplitude to about twie the natural bakground level. It is thus apparent that our ontrol arrangement is as good as the external ontrol arrangement desribed in Subsetion 4.1 in eliminating the instability wave. It should be pointed out, that in order to ahieve this good ontrol performane, a phase shifter within the ontrol iruit (see Appendix ) had to be adjusted quite arefully. The ontrol iruit was tested for different exiter frequenies and amplitudes and

30 ontrol to twie the bakground level was onsistently possible if the internal phase shifter was adjusted to a slightly different phase lag for eah frequeny. However, even if the phase shifter was not preisely tuned, i.e., it was set to a onstant average value, the ontrol of single mode frequenies over the whole unstable T-S band was still signifiant, i.e., a fator of 4 in u- disturbane amplitude for eah frequeny. In arrying out these experiments, it beame apparent that the proximity of disturbane detetor and atuator leads to undesirable rosstalk, i.e., the two ontrol atuators nearest the detetor loation influene the detetor reading. The extent of interferene is shown in Fig. 17. Detetor output as a funtion of ontrol atuator driving voltage is shown. In the absene of any ross-talk, the detetor output should be independent of the atuator driving level. In Fig. 17a the interation is apparent. Using, however, two ontrol atuators further apart from the detetor mirophones, Fig. 17b shows that the interferene problem is eliminated. Sine the ontrol performane using the arrangement of Fig. 17a did not deteriorate profoundly, no attempt was made to eliminate this interferene problem by further filtering. The seond problem whih was unexpeted was the varying phase lag requirement in the ontrol iruit for different frequeny omponents. From the beginning it was expeted that a phase shift would be required to take aount of the loal reeptivity behaviour at the atuator loation. In other words, sine it is not known what the (omplex) "transmission" oeffiient is between a

31 pressure disturbane generated at the wall and the downstream disturbane u-veloity omponent if would be fortuitous if it were identially zero. It was unexpeted that this transmission oeffiient is a funtion of frequeny. areful measurements of this transmission oeffiient were performed suh that a proper phase shifting all-pass filter an be designed at a later stage. As the last experiment in this investigation full wave paket ontrol was attempted. The differene between ontrolling single mode disturbanes and a full wave paket is the appearane of the latter at a ertain a priori unknown point in time. Hene, the time onstant of the eletroni ontrol iruit plays a signifiant role. The influene of this time onstant was underestimated in our ontrol arrangement and led to the failure in ontrolling the wave paket. It was however not only the larger than expeted rise time of our eletronis whih aused this failure. In addition, the settling time was exeeding the passing time of the wave paket thus ausing a ringing of the ontrol atuator whih introdued additional disturbanes into the boundary layer flow. The severity of this problem an be seen from Fig. 18. The top trae shows the detetor signal and the wave paket an be identified in the trae. The spikes seen in this trae are due to a ground onnetion whih was left floating by mistake. This was orreted in later attempts and should be ignored for the urrent argument. The bottom trae shows the ontrol atuator signal driving the embedded ontrol speakers. The delayed start of the ontrol signal and the long settling time after the signal has passed are apparent. This long

32 settling time was traed to the narrow band filtering assoiated with the employed phase shifting devie and ould not be orreted in the time available for this experiment. The time delay due to the Bessel filters was found to be in the order of 1mse whih is too large for the system to be able to ontrol frequenies in the neighbourhood of 300Hz (period 3mses) effetively. It is important to note, that the failure of the attempted wave paket ontrol is not based on any unexpeted flow behaviour but purely due to the inadequate eletroni iruit. Fortunately, having onluded this experiment and having speified the iruit requirements it is feasible to onstrut an improved eletroni ontrol iruit taking into aount the lessons learned. 5. onlusions and Outlook The stabilization of boundary layers via ative ontrol promises drag redution with relatively small power requirements ompared to passive boundary layer ontrol tehniques. To date, all experimental ative ontrol investigations have onentrated on two-dimensional disturbanes in the laminar boundary layer. However, to further improve the level of disturbane attenuation in the boundary layer requires an inherently three-dimensional approah. The work presented in this report addresses this issue. The goal of this study was to ontrol the boundary layer response to random three-dimensional disturbanes introdued near the leading edge of a flat plate. This goal was not fully ahieved in

33 the shortened time frame of this investigation. (The original proposal set out a three year programme, however, two years were funded.) The basi flow struture to be ontrolled is the three-dimensional wave paket, i.e. the boundary layer response to loalized pulse exitation. Sine the ontrol is to take plae in the linear region of the transition zone, any oneivable flow disturbane an be synthesized and hene anelled by appropriate wave paket superposition. The projet was divided into two main parts, (i) numerial modelling of the ontrol proess, and (ii) experimental investigation of the most promising ontrol arrangement identified in (i). The numerial modelling was an integral part of our approah and allowed us to a priori assess the best possible ontrol performane ahievable with a given design. The numerial model allows rapid testing of different spanwise and streamwise distributed detetor and atuator arrays. Using this model, a promising ontrol onfiguration was identified, implemented in the ambridge University low turbulene wind tunnel faility and tested in part (ii) of this work. The experimental investigation ahieved signifiant ontrol of single mode disturbanes. In fat, ontrol to about twie the A natural bakground level was possible for frequenies within the unstable T-S band. The disturbane detetion was based on a

34 differene mirophone arrangement in order to rejet aousti disturbanes in the faility. This mirophone arrangement proved to be ideally suited for the urrent purpose. The atuators, embedded miniature loudspeakers, proved apable of ontroling moderate sized instability waves. Slight ross talk between atuator and detetor arrangement was observed but was not so severe as to adversely affet the ontrol performane. Wave paket ontrol was attempted but failed due to inadequate real time behaviour of the eletroni filtering iruit whih ould not be retified in the experimental time remaining. This failure was due to exeedingly large rise and settling times of the eletroni iruit. The former aused the ontrol to "kik in" after about half the wave paket had passed the atuator loation, while the latter aused exessive "ringing" of the atuator after the wave paket had passed thus in itself reating a disturbane in the flow field. It should be emphasized that the failure to ontrol the wave paket is not due to any unforseen flow behaviour but solely due to inadequate eletroni iruitry. Redesigning the eletroni filters is therefore the neessary first step in ahieving wave paket ontrol loally at the plate entre-line. Inreasing the physial distane between ontrol detetors and atuators is advisable in view of (a) eliminating ross talk and (b) allowing for more proessing time in the ontrol loop. After ontrol at the entre-line is ahieved, it is suggested to arefully explore the

35 spanwise distribution and ompare this with the numerial model developed in the ourse of this projet. Afterwards, several spanwise detetor/atuator bloks an be ombined to provide omplete spanwise ontrol of distributed random disturbane fields.

36 Appendix A: Figures List of Figures Fig. 1 Shemati of boundary layer ontrol arrangement. Fig. 2 Blok diagram representation of boundary layer ontrol system. (a) full system (b) simplified basi system Fig. 3 Simulation of flow response due to point soure exitation at Re,= 870. Dashed lines orrespond to negative values. STI - System transfer funtion from Re,= 870 to ST2 - System transfer funtion from Rear= 1235 to STI *ST2 - Disturbane flow response at Rej due to point soure exitation at Re6,= 870 (STI onvoluted with STII). Fig. 4 Streamwise veloity r utuations (u') measured at outer edge of bor-ip ry layer (Rea= 1235). Single Pulse exitation at Rea = 870. Fig. 5 Numerial mrodelling of streamwise veloity flutuation (Re,= 1235). Single Pulse exitation at Res Fig. 6 omparison of disturbane u-veloity flutuation along wave paket entre-line. Fig. 7 Simulated flow response (u') in target zone (Re 6.= 1780), dashed lines orrespond to negative values. (a) no ontrol (dash-boxed insert of Fig. 3 resaled) (b) ase (1): single ontroller on entre-line, ontour levels as in (a) Fig. 8 Simulated flow response (u') in target zone (Ress 1780), dashed lines orrespond to negative values. ontour levels as in Fig. 10a. Relative size of wave paket at ontrol loation Rej,= 1235 and spanwise ontroller positioning is indiated in the left part of plot area. (a) ase (2): three ontrollers onfiguration (b) ase (3): four ontrollers onfiguration. Fig. 9 Performane sensitivity to ontroller gain and phase mismath for different ontrol arrangements. (a) gain mismath (b) phase mismath

37 Fig. 10 Bode plot of amplitude and phase behaviour of differene mirophone arrangement inluding ideal integrator. (a) amplifiation vs. normalized frequeny (b) phase shift vs. normalized frequeny Fig. 11 Simulated flow response (u') in target zone (Rej,= 1780), due to random disturbane exitation in spanwise position and time. Dashed lines orrespond to negative values. Time trae along entre-line and spanwise distribution at intermediate time extrated to bottom and right of plot respetively. (a) unontrolled referene ase (b) ontrolled ase Fig. 12 entre-line hot-wire trae at target loation. Exitation with three disrete frequenies. (a) unontrolled ase (b) "external" ontrol on Fig. 13 Integrated hot-wire rms in T-S frequeny band at target loation versus spanwise probe position. "External" ontrol of two superposed frequenies. Atuator loation at ontrol loation indiated by full triangles on abissa. Fig. 14 Hot-wire trae at detetor loation (entred in streamwise position between double mirophone but displaed sideways by 8mm to avoid interferene). Pulse exitation near leading edge. (a) single realization hot-wire reord (b) ensemble average of 64 realizations Fig. 15 Mirophone trae at detetor loation on entre-line. Pulse exitation near leading edge. Simultaneously reorded with hot-wire trae in Fig. 14a. (a) Single realization of single mirophone (b) Single realization of differene mirophone Fig. 16 entre-line hot-wire trae at target loation. Single mode exitation near leading edge. (a) unontrolled signal (b) ontrolled signal using ontrol loop.

38 Fig. 17 Atuator/detetor ross-talk. Detetor output as a funtion of externally driven atuator foring level. (a) two atuators symmetrial to detetor mirophones, spanwise displaed by 5mm. (b) two atuators symmetrial to detetor mirophones, spanwise displaed by 15mm. Fig. 18 Settling time of ontrol iruit. (a) Detetor trae showing passing of wave paket. [Spurious spikes in signal are due to floating ground onnetion in this partiular ase and should be ignored. ] (b) Synthesized ontrol atuator driving signal showing long settling time. Fig. 1 Aluminium test plate for ative ontrol experiments. Fig. 2 ambridge University Low Turbulene Researh Tunnel. Fig. 3 Eletroni hardware for ative ontrol experiment. Fig. DI Boundary layer response in paket entre-line at Re = Data analogoue filtered between 20Hz and 2000Hz. (a) single realization (b) ensemble average of 189 realizations Fig. D2 Fig. D3 Power spetral density versus frequeny plot of ensemble averaged data shown in Fig. D1b. Measured u-flutuations at outer boundary layer edge at Re,= Ensemble averaged (189) and digitally filtered between 80Hz and 500Hz. (a) perspetive view (b) ontour plot, negative ontours dashed Fig. D4 Measured u-flutuations in the boundary layer at Re = 1235 for twin pulse interation experiment. Two pulses of same strength buth opposite sign exite flow at Ree= 880. Ensemble averaged data digitally filtered between 80Hz and 500Hz. (a) perspetive view (b) ontour plot, negative ontours dashed Fig. D5 Measured u-flutuations in the boundary layer at Re r = 1235 for triple pulse interation experiment. Three pulses of same strength (3m apart spanwise) exite flow at Reem 880. Middle pulse of opposite sign. Ensemble averaged data digitally filtered between 80Hz and 500Hz. (a) perspetive view (b) ontour plot, negative ontours dashed

39 go 0.0 4J 0 1 E U4% u 00U 0 m4-000 II 0 * 0 $ VI 0 0 u,.4r- U4.. U ~GJ0 0,4 r A 1.4J u.) 4w4 P40 0"4 Eu 0 V %4 1

40 LZ - anshduig datv oto (a) 1 M(a) ullsyseiz (b) bai impifie sste (,x

41 STI * il.'.. T I I,.T, S ITS " SI TI I --,,&,, Fig. 3 Simulation of flow response due to point soure exitation at Rest= 870. Dashed lines orrespond to negative values. ST1 - System transfer funtion from R%, to ST2 -System transfer funtion from Reja to ST1 *ST2 - Disturbane flow response at Rejo due to point soure exitation at Re,,- 870 (STI onvoluted with STII).

42 EXPERIMENTAL RESULTS -SINGLE SOURE ( 04 Fig. 4 Streasuwise veloity flutuations (u') measured at outer edge of boundary layer (Rep- 1235). Single Pulse exitation at Reiss 870.

43 SIMULATION RESULTS -SINGLE SOURE U f 0a Fig. 5 Numerial modelling of streamwise veloity flutuation (Ret. 1235). Single Pulse exitation at Rer

44 -42- U, 1 A. 0,1er m n time Fig. 6 omparison of disturbane u-veloity flutuation along wave paket entre-line.

45 -43- % D; stwr ofn PiteI (ID~s. Eukerly 15.6 (oao t~m mm Fig. 7 Simulated flow response (ul) in target zone (Rep-= 1780), dashed lines orrespondl to negative values. (a) no ontrol (dash-boxed insert of Fig. 3 resaled) (b) ase (1) : single ontroller on entre-line, ontour levels as in (a)

46 Dist. Erer, 1.1 ( -,,, - Dist. Eiey 0.6 I Z --" Fig. 8 Simulated flow response (u') in target zone (Ref- 1780), dashed lines orrespond to negative values. ontour levels as in Fig. 10a. Relative size of wave paket at ontrol loation Red and spanwise ontroller positioning is indiated in the left part of plot area. (a) ase (2): three ontrollers onfiguration (b) ase (3): four ontrollers onfiguration.

47 - - -no on.m'. ase 1!ase S i 2 0 ' g. T 10. Amplifier G n ase/ ase 3 e 2 ase i Single ontroller on entreline. ase 2 : Three ontrollers total, entreline and at 12 ma above and below entreline. ase 3 Four ontrollers total, 6am -I 21mm above and below entreline. - -:Two-di.mmmional single mode ontrol. I 'I S I * i Los ovk,'ol tvani Lead x O.23uIS.. Fig. 9 Performane sensitivity to ontroller gain and phase mismath for different ontrol arrangements. (a) gain mismath (b) phase mismath

48 Atuvator 'Detet O' o 0 I ombined response 2 response\ E -5- ff. - MIS.toglo (f/o (6) Integrator 0.1 ***%% Iogo(f I/F 0 ) Fig. 10 Bode piot of amplitude and phase behaviour of differene mirophone arrangement inluding ideal integrator. (a) amplifiation vs. normalized frequeny (b) phase shift vs. normalized frequeny

49 t " Fig. 11 Simulated flow response (ul) in target zone (Rej#- 1780), due to random disturbane exitation in spanwise position and time. Dashed lines orrespond to negative values. Time trae along entre-line and spanwise distribution at intermediate time extrated to bottom and right of plot respetively. (a) unontrolled referene ase (b) ontrolled ase

50 EXTERNAL ONTROL MUMRE FREQUENIES WPL Fig. 12 entre-line hot-wire trae at target loation. Exitation with three disrete frequenies. (a) unontrolled ase (b) "external" ontrol an

51 SPANWISE DISTRIBUTION * Foring at Re,. 870, ontrol at Re.. = 1260, target zone at Re,, External ontrol frequenies 250/286 Hz X., measured W0mm downstream S 30- Of ontrol loation unontrolled 20- "noise" f loor spanwise position [m] Fig. 13 Integrated hot-wire rms in T-S frequeny band at target loation versus spanwise probe position. "External" ontrol of two superposed frequenies. Atuator loation at ontrol loation indiated by full triangles on abissa.

52 -So- (ai) Fig. 14 Hot-wire trae at detetor loation (entred in streamwise position between double mirophone but displaed sideways by 8.m to avoid interferene). Pulse exitation near leading edge. (a) single realization hot-wire reord (b) ensemble average of 64 realizations

53 -5'- Fig. 15 Mirophone trae at detetor loation on entre-line. Pulse exitation near leading edge. Simultaneously reorded with hot-wire trae in Fig. 14a. (a) Single realization of single mirophone (b) Single realization of differene mirophone

54 (i) GsL Fig. 16 entre-line hot-wire trae at target loation. Single mode exitation near leading edge. (a) unontrolled signal (b) ontrolled signal using ontrol loop.

55 -53- I t- I 0 ". 0 To- U. 0H a= - o.o~ 4re- tj 0 2: o S Or. i j 4 VOLTS10- A r- 00r o.o.-4.. I = "= S.= -o o- t_ S (a) tw *rni iuato,-vlts utr ymtia t eetrmohns Fig 17 Atuato /dto ros-tal. Deeo output a Fig. 17 Atuator/detetor ross-talk. Detetor output as a funtion of externally driven atuator foring level. (a) two atuators symmetrial to detetor mirophones, spanwise displaed by 15mm. (b) two atuators symmetrial to detetor mirophones, spanwise displaed by 15ram.

56 VLS a.- 0. a V O L T S. " N A " " BUF. 2 SE"ONDS (ba A$AA 8A AaAAA -l 0 VOLT VV V V V V V V V V V -V VVVVV V 0.-o'o,ID " SEONDS Fig. 18 Settling time of ontrol iruit. (a) Detetor trae showing passing of wave paket. [Spurious spikes in signal are due to floating ground onnetion in this partiular ase and should be ignored.] (b) Synthesized ontrol atuator driving signal showing long settling time.

57 EXPERIMENTAL SETUP 3/5 "... R.t;..a. At,,..r Plate "Ep.llf (3:1) 16 -goo50 / 16 Pe.... Tpp; jj (00-3) i _ 0 SFod Wok t (I 0.3). -. das..j.,"n4 UQ 2 Ir,..t -(200) Dimensionse ;fn-i.i ", 17 Vo * One and twin soure foring. Measuring station at Rv 1235., ov,,o tov,,t ot ', * 170. Fig. 1 Aluminium test plate for ative ontrol experiments.

58 EXPERIMENTAL, FAILITY -.U.E.D. low turbulene researh tunnel (former NFL tunnel). " Exhangeable working setions 3ft x 3ft x 6ft. * 3-D omputer ontrolled traverse " Miro-omputer based d.a. system " Turbulene Intensity ( 0.01% (20m/s, 4 Hz to 2000 Hz). Honeyomb and hangeable sreen~ 3- t x3-ft) 9:1 ontration Fig. 2 ambridge University Low Turbulene Researh Tunnel.

59 =TTANDY Preision Shieled Pro-Amrp Preision Shielded Pre-An Gain Gain Low-Pass Bessel Filter Low - Pass Bessel Filter Krohn -Hite 3343 Krohn -Hite 3343 L- I IntegratorI - - -j Phase Shifter Feedbak VPO 602 Power Ampliir Arriron D 150A Mini - Speaker Knwls EH 3044 Fig. 3 Eletroni hardware for ative ontrol experiment.

60 (a) ses o... (b) ses Fig. D1 Boundary layer response in paket entre-line at Re Data analogoue filtered between 20Hz and 2000Hz (a) single realization (b) ensemble average of 189 realizations

61 8 6 L Frequeny (Hz) Fig. D2 Power spetral density versus frequeny plot of ensemble averaged data shown in Fig.b 1b.

62 -60- (a) 25 mse 19.6 m 0 0 (b) m -I 0% 0 25 mse Fig. D3 m~easured u-flutuations at outer boundary layer edge at er Ensemble averaged (189) and digitally filtered ween 8Hz and 500Hz. (a) perspetive view (b) ontour plot, negative ontours dashed

63 -61- (a) 25 mse 15.6 m m (b) I / -, It I t I mse,i y7.,-; Fig. D4 measured u-flutuations in the boundary layer at RejA 1235 for twin pulse interation experiment. Two pulses of same strength buth opposite sign exite flow at Rer= 880. Ensemble averaged data digitally filtered between 80Hz and 500Hz. (a) perspetive view (b) ontour plot, negative ontours dashed

64 -62- (a) 25 mse m m (b) r il 0,,.*_,.,~_,.-~-. t mse Fig. D5 Measured u-flutuations in the boundary layer at Rer for triple pulse interation experiment. Three pulses of same strength (3m apart spanwise) exite flow at Re, Middle pulse of opposite sign. Ensemble averaged data digitally filtered between 80Hz and 500Hz. (a) perspetive view (b) ontour plot, negative ontours dashed

65 Appendix B: Projet Plan Phase One - Thvitnw1s - Review of Literature. - Familiarization (R.H.) with wind tunnel setup, omputer ontrolled data-aquisition system. - Test plate design. - Three-dimensional traverse. Phase Too - Trarfer fhnxdos - Seletion of method for detetion and ontrol. - Experimental determination of detetor and ontroller transfer funtion. Phase Tire - Mii'al moddliqg - Implement and experiment with omputer model of various ontrol arrangements using Gaster's (1978) rapid series eigenvalue expansion method. - Based on above modelling identify the most promising ontrol ar dngements. Phase For - AUi'e awd erpe*nwt - Experimental study of the most promising ontrol arrangements identified in phase three using a small number of detetors and ontrollers. - Prepare final report.

66 Appendix : Hardware Development The experiments to be performed require a flat plate whih is suited for the testing of several different exiter, detetor and ontroller arrangements. A plate (Fig. l) whih allows for hangeable inserts has been designed, build and installed in the ambridge University Engineering Department (.U.E.D.) low turbulene wind tunnel (Fig. 2). The plate allows disturbane exitation at five spanwise loations (3m apart) via embedded loudspeakers 20.5m downstream of the leading edge. An important feature of the plate is the provision for exhangeable inserts whih allow the testing of various different ontrol arrangements. Inserts an be exhanged while the plate is positioned in the tunnel. Great are was taken that the edge of the insert was flush with the working side of the plate in order to avoid the reation of flow disturbanes. The gap between insert and plate was filled and polished in order to ahieve the required surfae ontinuity. ontrol atuators and disturbane detetors had to be seleted aording to frequeny response harateristi, ease of installation, physial robustness and availability. The 'buried loudspeaker" tehnique had been suessfully used in previous work (Gaster and Grant, 1975) for disturbane exitation in a similar setting. It was deided to use the same tehnique for ontrol atuation. In order to allow lose spanwise spaing of the ontrol atuators a physially small speaker was required. Various earphone speakers were tested. The mini-speakers EH3030 of Knowles Eletronis o. (73 Vitoria Road, Burgess Hill, West Sussex, RH15 9LP, England) were found to be the most suitable in view of their frequeny response harateristis, their relative high power output and their small physial dimensions (7mm x 3mm x 3mm).

67 Disturbane detetion was originally envisaged to be done with either hot-film sensors flush mounted on the plate surfae or embedded mirophones a la Kendall (1990). The hot-film sensors, however, were eliminated beause of mehanial diffiulties. It was very time onsuming to attah flying leads to the sensor without produing protrusions on the plate working side whih ause unaeptable flow interferenes. In addition, the eletroni requirements, i.e. a bridge iruit for eah sensor and possibly drift ompensation, is rather omplex. Hene, embedded mirophone detetors were used. The mirophone detetor hosen was a 9.5mm diameter, 10mm high model TANDY No The detetor ommuniated with the plate working side via a 1mm diameter hole. Voltage output reading was readily available. The output signal level was only about 0.2mV and areful sreening was required. Eletroni Hardware Figure 3 shows diagramatially the eletroni ontrol iruuit employed in the final experiments. Exluding the integrator whih was only used during part of the experiment, and the preision shielded op-amps, the omponents employed were standard of the shelf omponents. The time onstants of the Bessel Filters were to large to respond to a passing wave paket in time and were partially responsible for the failure to ahieve wave paket ontrol. The Phase Shifters aused narrow bandwidth filtering around the priniple frequeny leading to ontrol atuator ringing after the detetor signal had eased to exist.

68 Appendix D: Preliminary Wave Paket Experiment Using the newly designed test plate, some preliminary wave paket experiments were arried out. The purpose of these experiments was threefold: (a) to show that wave pakets ould be generated with the embedded loudspeaker tehnique, (b) to demonstrate that the bakground disturbane level in the.u.e.d. low turbulene faility was indeed low enough to allow easy detetion of a single wave paket in the boundary layer and () to investigate if loudspeaker exitation was "reversible," i.e., does a sign hange in the driving voltage pulse reate a wave paket of same amplitude but 180 out of phase? Point (b) is espeially important for the implemer -ation f the ontrol system sine some filtering will have to be employed to isolate the relevant wave paket features from "noise" pres6 nt in the boundary layer. In addition to these three main objetives, the experiments were required to test the new plate setup for (near) zero streamwise pressure gradient and to hek the three-dimensional traverse gear and data aquisition software. Two different sets of experiments were performed. The tunnel speed for all experiments was 20m/se and measurements were taken using a onstant temperature boundary layer hot-wire. A 'hear zero" streamwise pressure gradient was ahieved by adjusting a flap extending downstream from the main plate. In order to avoid flow separation at the leading edge a seond small trailing edge flap was adjusted to move the stagnation point slightly to the working

69 side of the plate. Following work by Gaster and Grant (1975), the hot-wire was positioned just outside the boundary layer at 1.2 boundary layer thiknesses from the plate surfae, where the Tollmien-Shlihting eigenfuntions show an outer (flat) maximum. In the first set of experiments, the downstream and spanwise development of a single wave paket was investigated. The exiter loation at 20.5m from the leading edge orresponds to Re = 880 (Reynolds number based on displaement thikness) and the flow responses were measured at Ree= 1235 (40.5m) and 1380 (50.5m), respetively. Typially 45 spanwise "data-slies" roughly six displaement thiknesses apart were reorded. The exiting pulse was omputer generated and ould be varied in amplitude and width. For the experiments desribed in this report a onstant pulse width of 0.Smse was used. To allow ensemble averaging of the data, the pulse generation was synhronised with the data aquisition system. The sampling rate was 4kHz and the signal was analogue filtered between 20Hz and 2kHz. Eah reord ontains 512 data points and olletion time per reord was thus 0.13se. An ensemble average was formed using 189 individual reords. Figure Dla shows a typial single realization of the boundary layer response at Re,,t= Figure D1b shows the same data after ensemble averaging. A power spetral density versus frequeny plot of the ensemble averaged data is shown in Fig. D2. The quite satisfatory signal-to-noise ratio is apparent. The Tollmien- Shlihting frequeny band shows power spetral densities of about

70 dB above the bakground noise level. A perspetive view and a ontour plot of the exited wave paket at the two different Reynolds numbers are shown in Figs. 4 and D3. In order to emphasize the Tollmien-Shlihting wave response, the ensemble averaged reords presented in the last two figures were hanned and digitally filtered between 80Hz and 500Hz. At the higher Reynolds number, the stronger three-dimensional modulation of the signal and the inreased paket "length" are apparent. The first set of experiments established that (a) the embedded loudspeaker tehnique is suited to the generation of wave pakets in laminar boundary layers and (b) the.u.e.d. low turbulene wind tunnel provides the desired flow quality for this type of experiments. The seond series of experiments involved the interation of wave pakets generated simultaneously at different spanwise loations. Only the signal at Re.= 1235 was reorded. First, two pulses 3m apart of same amplitude but opposite sign were introdued at the exiter loation to disturb the flow. The experiment was then repeated using three spanwise drivers 3m apart. Again all pulses generated had the same amplitude but the middle pulse was of opposite sign. The data aquisition and data redution performed was as desribed above. Here, however, 55 and 63 spanwise "slies" were olleted in order to map out the interation region for the "twin" pulse and "triple" pulse experiments respetively.

71 Figures D4 and D5 show perspetive views and ontour plots of the flow response to twin pulse and triple pulse exitation. quality of the data obtained is not as high as in the first The set of experiments reported above. This was espeially so in the triple pulse experiment whih showed an underlying modulation of about 500Hz. This was traed to vibrations of the hot-wire when plaed in a shearflow, a phenomenon traed to lak of wire tension. The ontour plots of both experiments show the symmetry of the flow response for "positive" and "negative" pulse exitation. It should be pointed out that the lines represent the same absolute values for positive and negative ontours drawn. The ontour plots for both experiments show narrow regions of wave anellation between adjaent wave pakets of opposite sign. The seond series of experiments ahieved objetive () formulated above, i.e., sign swithing of the driving pulse leads to sign swithing of the resulting wave paket.

72 Appendix E: Programme Listings

73 Program mainon **********************************************t************** ontrol of instability waves marh 31, 91 modified to allow phase shifting in order to asses sensitivity to off-timing hanged to read in unformatted data files to speed up run (marh 29, 15:00) july 16, 91 mainontr2.f with double loop (110,105) to map out ampgain, phaseshift spae (Hh). july 18, 91 mainontr3.f implements one detetor driving several (three) atuators spanwise where the two sideatuators are driven with amplitude weight of 0.5 of the middle atuator amplitude (Hh) july 25, 91 mainontr4.f implements random foring in spae and time. Note that FUNTAION DRAN2 had to be orreted. (Hh) option of reading unformatted Fourier oeffiients of these from data diretory DAT/Fsy... Hene onvol2d is modified in order to take advantage of this. (Hh) Furthermore pogram was modulized into more subroutines and ommon blok is used to redued ompiled file storage requirements. july 26, 91 mainontr5.f speeds up onvolution with known system transfer funtions by using sept 5, 91 modified just for target field superposition of whole fields (in sontr.f) (mainontr6.f) Subroutines required: sin - prepare physial spae data field at loation of mirophone detetors written in array Heightl onvol2d - 2D fast Fourier transform of 512x128 double preision fields. If both fields are in physial spae (inl = in2 = 0), Heightl then physial spae result is in Heightl. If arrays are in proper fourier form (ini and/ in2 = 1) then array templ and/or temp2 are use and onvolution is speeded up signifiantly. onvoluted result is always returned in Height (physial spae). FOURN - 1D fast Fourier transform sontr - ontrol of disturbane field and ontinuation of ontrolled field to target loation. Unontrolled physial spae input field in Heightl, ontrolled ouput field also in Heightl (Height2 is dummy field -destroyed). sanaly - Analyze ontrol performane at target loation in ertain unspoiled part (wraparound) of field Height2. sout - output of resulting field to harddis diretor./dat/<filename> Data files from diretory./dat: FsystrI - file with Fourier spae representation of system transfer funtion I (unformatted double preision).

74 - 72- FsystrlI - file with Fourier spae representation of system transfer funtion II (double preisi unformatted) Roland Heinrih ambridge,marh 27, 1991 ************************************************************w impliit real (a-b, d-h, o-y), omplex(, z) integer(i-n) harater*16 fname harater*l A parameter (inmx=-l 0, iampmx=22 ) parameter (inmx=l, iampmx=2) parameter (ndim = 2, itmax=-512,kzmax=128, ive=itmax*kzmax*2) double preision psen, ampgain,pi, fatl, fat2, fat3, fsale,help, + ampdel dimension psve (1: iampmx, -inmx: inmx) double preision templ, temp2,height1, Height2 ommon/fld/templ (ive), temp2 (ive), + Heightl (itmax, kzmax), Height2 (itmax, kzmax) pi = 4.dO*datan(1.dO) print*,'pi = ',pi idum = -1 fname(1:4) ='DAT/' ***** set phys. spae disturbane field at Detetor loation I all sinf ***** ontrol by setting seletive spanwise values to zero enterline is at 1,use 1 < nontrol < 64 1 print*,' What ontrol amplifier gain? ' read(*,*)am.pgain print*,' Timeshift nshift*.25mses, nshift?i read(*, *) nhift print*,' print*, 'al pgain ',ampgain nshift = 's print*, 'nr-nift ',nshift all sont,. (ampgain,nshift) goto 555 ***** Pull whole field forward by npull suh that nonorrupted field (by wraparound) starts at ntime =1 to ntime = 420. Note also that in spanwise diretion nonorrupted field lies between iz=35 and iz=93. Important for power alulation. 333 npul = 360 do ik = i,kzmax ii =0 do it npul,itmax ii =ii + 1 Height2(iiik) = Heightl(it,ik) end do end do do ik = l,kzmax ii = itmax-npul+l do it = l,npul-1 ii = ii+l Height2(iiik) = Heightl(it,ik) end do end do **&*** anrilysp rnntrol porforrnanr' t!--ite i-it~n

75 -73- in un-spoiled area of field Height2 itl = 1 ith = 420 izl = 35 izh = itl - 1 ith izl - 1 izh = 128 do ik = l,kzmax do it = l,itmax Height2 (it, ik) = Heightl (it, ik) end do end do all sanaly(itl,ith,izl, izh) * write out result all sout (Height2) pause 'new ontrol onfiguration?' goto print*,'error in reading ',fname,' start over?' pause goto print*, 'End reahed in reading,,fname,' start over?' pause gotol 999 end Subroutine sinf prepares Heightl(itmax,kzmax) to orrespond to the unontrolled disturbane field at the loation of the detetor (and perharps ontroller array). Output : Heightl (itmax, kzmax) Subroutines required: onvol2d Fourn Roland Heinrih ambridge, July 31, 91 ************************************************************** harater*16 fname harater*1 A parameter (itmax=512, kzmax=128, ive=itmax*kzmax*2) double preision temp1,temp2,heightl,height2 ommon/fld/templ(ive),temp2(ive), + Heightl (itmax, kzmax), Height2 ( itmax, kzmax) fname(l:4) = 'DAT/' 1 print*,' Do you have double preision phys. spae ' print*,' input file orresponding to disturbane I print*,' flow field at ontrol loation (1) or do I print*,' you have Fourier spae input data (2) at I print*,' exiter loation (all produt of ranfield' print*,' input 1 or 2 :?' read(*,*) infld if(infld.eq.l) then print*,' Phys. dist. field at ontrol loation (doubl.pre.)? ' read(*,*) fname(5:16) print*, fname open (1, f ile=f name, form--"un formatted") read( 1, ERR=-900, END=910) Heightl lose(l) goto print*,'error in reading ',fname,' try again I pause

76 goto print*, 'End of file enountered in ', fname, ' try again' pause goto 1 else if (infld.eq.2) then open (1, file="dat/fsystriun", form="unformatted") read(l, ERR=900, IED=910) temp2 lose(l) * Read in fourier transform of input field (512x128) whih was generated with ranfield. f print*,' Fourier spae field of disturbane input? ' read(*,*) fname(5:16) open (1, file=fname, form="unformatted") read(i) templ lose (l) * onvolute random field with systri inl=l in2=l all onvol2d(inl,in2) else pause 'Wrong infield integer, try again' goto 1 endif 10 print*,,phys. dist. field at detetor loation' print*, '(Heightl) set.' return end subroutine onvol2d (inl, in2) ************************************************************* Use 2-d Fast Fourier transform (FOURN) to onvolute arrays of itnax t-points * kzmax z-point of data. July 26, 1991 modified to allow onvolution and baktransform if fields are passed down as appropriate fourier oeffiients in vetor templ and/or tenp2 as indiated by ontrollparameter inl and In2. inl = 0 : data in datinl need transform inl = 1 : fourier tranformed data in templ are ready for multipliation. in2 = 0 and in2 = 1 as above for inl. (Hh) templ,datinl : holding onvoluted result upon return temp2,datin2 : unaltered by subroutine Roland Heinrih ambridge, marh 26,91 impliit real (a-h,o-y) double preision pi,helpr,helpi harater*12 fname parameter(ndim = 2, itmax =512,kzmax=128, ive=itmax*kzmax*2) dimension nve(ndim) double preision templ,temp2,datinl,datin2 ommon/fld/templ (ive), temp2 (ive), + datinl (itmax, kzmax), datin2 (itmax, kzmax) nve(l) = itmax nve(2) - kzmax pi4 = datan(-i.do)

77 -75- pi = 4.dO* pi4 '** prepare o a p 1 e x data vetors for subroutine FOURN if neessary as indiated by dummy argument inl,in2 If(inl.eq.0) then ii = 1 do 10 ia = 1,kzmax do 15 it = 1,itmax tenpl (jj) = datinl (it, ik) 15 ontinue io ontinue templ(jj+1)= O.dO jj = jj + 2 * hek max dimension if((jj-1).ne.ive) stop,onvol2d jj n.e. ive *** transform into fourier all FOURN(tep1,nve,ndim,+1) end if If(in2.eq.0) then ji = 1 do 11 ik = 1,kzax do 17 it = 1,itmax temp2(jj) temp2(jj+l) = O.dO jj = jj ontinue 11 ontinue = datin2(it,ik) * hek max dimension if((jj-1).ne.ive) stop,onvol2d jj n.e. ive *** transform into fourier all FOURN(temp2,nve,ndim,+I) end if *** onvolute data in fourier spae, templ(jj) holds onvolution this is start point if inl and in2.ne. 0 indiating that templ,temp2 hold allready fourier oeffiients in proper form do 16 jj = l,ive,2 helpr = templ(jj)*temp2(jj) - templ(jj+l)*temp2(jj+l) helpi = templ(jj)*temp2(jj+l) + temp2(jj)*templ(jj+l) templ(jj) = helpr templ(jj+1) = helpi print*, 'ornvoluted,j,j+l,', jj, templ (jj),templ (jj+l) 16 ontinue *** baktransform into physial spae all FOURN(templ,nve,ndim, -1) fat = l.do/dfloat(nve(1)*nve(2)) ***** rearrange data vetor in (time,spae) array (see arrangement onvention 'Num. Reipes' pg 451) Note that the imaginary part of onvoluted sum in physial spae should be idential zero. hek! ji = 1 dsumim = O.dO do 20 ik = l,kzmax do 25 it = 1,itmax datinl(it,ik) = fat*templ(jj) dsumim = dsumim + dabs(templ(jj+1)) ii = jj ontinue 20 ontinue

78 -T - print*, Isum of imag. part should be zero ',dsumim return end Subroutine FOURN (data, nn, ndim, isign) Replaes DATA by its NDI-dimensional disrete Fourier transform if ISIGN is input as 1. NN is an integer array of length NDIM,ontaining the lengths of eah dimension (number of omplex values), whih MUST all be powers of 2. DATA is a real array of length twie the produt of these lengths, in whih the data are stored as in a multidimen sional omplex FORTRAN array. I ISIGN is input as -1, DATA is replaed by its inverse transform times the produt of the lengths of all dimension. from Press, Flannery, Teukolsky, and Vetterling, "Numerial Reipes", ambridge University Press,! 986, pg onverted to overall double preision (R.Heinrih) Roland Heinrih ambridge, Febr. 8, 1991 Impliit real (a,b,d-h,o-y), omplex(,z), + integer(i-n) Double Preision wr,wi,wpr,wpi,wtemp,theta,tempr,tempi, + data Dimension nn (ndim), data(*) *** ompute total number of omplex values. ntot = 1 do 11 Idim = l,ndim ntot = ntot*nn(idim) 11 ontinue print*,'fourn ntot,ndim ',ntot,ndim do 500 i = l,ntot*2 print*, 'fourn-i,data ',i,data(i) 500 ontinue *** Main loop over the dimensions. nprev = 1 do 18 idim = l,ndim n = nn(idim) nrem= ntot/(n*nprev) ipl = 2*nprev ip2 = ipl*n ip3 = ip2*nrem *** This is the bit reversal setion of the routine i2rev = 1 do 14 i2=1,ip2,ipl if (i2. 1t. i2rev) then do 13 il=i2,i2+ipl-2,2 do 12 i3=il,ip3,ip2 i3rev - i2rev+i3-i2 tenpr = data(i3) tempi = data(i3+1) data(13) = data(i3rev) data(i3+1) = data(i3rev+l) data(i3rev) = tenpr data(i3rev+l) = tempi 12 ontinue

79 13 ontinue endif -77- ibit = ip2 / 2 1 if ((ibit.ge.ipl).and. (i2rev.gt.ibit)) then i2rev = i2rev-ibit ibit = ibit/2 goto I endif i2rev = i2rev + ibit 14 ontinue *** Here begins the Danielson-Lanzos setion of the routine. ifpl = ipl 2 if(ifpl. It.ip2) then ifp2 = 2 * ifpl *** Initialize for trig reurrene theta = isign * d0/(ifp2/ipl) wpr = -2.dO*dsin(O.5d0*theta)**2 wpi = dsin(theta) wr =.do Wi = O.dO do 17 i3 = l,ifpl,ipl do 16 il = i3,i3+ipl-2,2 do 15 i2 = il,ip3,ifp2 ki = i2 k2 = kl + ifpl tempr = wr*data (k2)-wi*data (k2+l) tempi = wr*data (k2+1) +wi*data (k2) data(k2) = data(kl)-tempr data(k2+1) = data(kl+l) - tempi data(kl) = data(kl) + tempr data(kl+l) = data(kl+l) + tempi 15 ontinue 16 ontinue *** trigonometri reurrene. wtemp = wr wr = wr*wpr - wi*wpi + wr Wi = wi*wpr + wteup*wpi + wi 17 ontinue ifpl = ifp2 goto 2 endif nprev = n*nprev 18 ontinue return end Subroutine sontr(ampgain, ni) **************************************************************** ontrols field aording to user interation with regard to ontrol inputs and ontrol amplitude and possible phase shift fieldi = unontrolled input field holds ontrolled field upon return field2 = dummy field whih is destroyed in routine Roland Heinrih ambridge, july 29, 1991 **************************************************************** harater*16 fname double preision ampgain,help parameter(ndim = 2,itmax 512,k7max- 12, v it'* :2)....

80 -78- double preision tempi, tem2, Fieldl, Field2 ommon/fld/templ (ive), temp2 (ive), + Fieldl (itmax, kzmax), Field2 (itmax, kzmax) **** Preset Field2 with O.do do ik = 1,kzmax do it - i,itnax Field2(it,ik) = 0.do end do end do * ontrol by setting seletive spanwise values to zero enterline is at 64,use 1 < nontrol < 128 ontrol atuator input is 2*3mn to either side of detetor line with weight 0.5 print*, 'ontrol phase - Set seleted mirophone position' print*,' Atuate at position +/- 6mm.' 90 print*,'enter miontrol > 1000 if you' print*,'have set all ontrol points. 3 <= nontrol <=126.' print*, 'Enter mi-ontrol =? ' read*,nontrol if (nontrol.eq.0) then ***** no ontrol do ik=- l,kzmax do it = 1,itmax Field2(itik) = Fieldl(it,ik) end do end do end if if (nontrol.gt.1000) goto 95 do it = l,itmax help = Fieldl(it,nontrol) Field2(it,nontrol-2) = Field2(it,nontrol-2) - 0.5d0*help Field2 (it,nontrol+2) = Field2 (it, nontrol+2) d0*help end do goto 90 onvolute ontrolled Field2 with SystrII (Fieldi) 95 ***** open (3, file="dat/ FsystrIIun", form="un formatted") read(3) templ lose(3) inl = 1 in2 = 0 all onvol2d(inl, in2) amplifier gain and off-timing of ontrol signal in ni*0.25 mse if(ni.gt.0) then do it - 1,itmax-ni do ia = l,kzmax Field2 (it, ik) = Fieldl (it+ni, ik) end do end do do it - 1,ni do ik = i,kzmax Field2 (itmax-ni+it, ik) = Fieldl (it, end do end do elseif (ni.lt.0) then ik)

81 *** do it - 1, -ni do ik - 1,kzmax Field2 (it, 1k) =Fieldi (itmax+ni+it, 1k) end do end do do it =-ni+1,itmax do Ak = 1,kzmax Field2(it,ik) = Fieldl(ni+it,ik) end do end do else do it = l,itnax do ik = 1,kzmax Field2(it,ik) =Fieldl(it,ik) end do end do end if now read in unontrolled field whih is to be superposed with ontroll output 200 fname=' DAT/pta rgrani open (3, f ile=fname, form="unformatted") read (3, ERR-=900, END=910) Fieldi lose(3) 196 do ik = l,kzmax do it = 3,itnmax Fieldi (it,ik) = ampgain*field2 (it, ik) 1 + Fieldl(it,ik) end do end do return 900 print*,error reading',fname,' try again' pause goto print*,'end of file ',fnaxe,' try again V' pause goto 200 end Subroutine sanaly(itl, ith, izi, izh) ******************************** Analyze data in in subset iti-ith, izl-izh of field.2 itmax x kzrnax (pseudopower per point ) field2 = ontrolled input field Roland Heinrih ambridge, july 30, 1991 double preision psen,rpsen parameter(ndim = 2, itnmax-512,kzmax=-128, ive--itlax*kzmax*2) double preision tempi, tenlp2, Fieldi, Field2 ommon/ fld/templ (ive), teop2 (ive), + Fieldl (itmax, kzmax), Field2 (itnax, kzmax) * find pseudo disturbane energy in field psen = 0.dO iount = 0 do Ak = izl,izh do it = itl,ith iount =iount+l

82 -goend do end do pzen = dsqrt(psen) ***** alulate residual power per point rpsen = psen/dfloat(iount) Print*,' Power in ontrolled subfield ',psen Print*,' Power per pint in ontr.field', rpsen print*,' psve(iamp,ni) ',psve(iamp,ni) test l 110 go to 101 ontinue l 105 l ontinue print*, ' Enter filename for Power vetor psve(iamp,in) l fname(1:4)='dat/' l read(*,107) fname(5:16) 107 format(a12) i open (2, f ile=fname, form=' formatted') l write(2,*) ((iamp,in,psve(iamp,in), l lose(2) in=-innlx, inmx), iamp=l, iapmx) return end Subroutine sout (Heightl) ************************************************************ Write out data fields for possible post-proessing via graphis routines. Hene hange field data to single preision for ountourplotting ontsim*.f Roland Heinrih ambridge,marh impliit real (a-b,d-h,o-y),omplex(,z),integer(i-n) harater*16 fname harater*1 A parameter (ndim = 2, itmax=512, kzmax=128, ive=itmax*kzmax*2) double preision psen, psve, Heightl dimension Heightl (itmax,kzmax) dimension sheight (itmax,kzmax) ***** onvert Height2 to single preision for later use in ontour plot routine do ik = l,kzmax do it = 1,itmax sheight(it,ik) = sngl(heightl(it,ik)) end do end do 100 print*,' Do you want to store field single preision (1), ' print*,' double preision (2) or both (3)? Enter 1,2 or 3: ' read(*,*) istore if (istore.eq.1) then print*,' filename to write ontrolled result (sngl.pre.)' print*,' at target loation = (in subdir DAT)' print*,' (unformatted store) fname(l:4) = 'DAT/' read*, fname(5: 16) open (4, file=fname, ERR=-900, form="un formatted") write (4, ERR=900) write (4, ERR=900) sheight psen lose(4) else if(istore.eq.2) then print*,' filename to write ontrolled resuilt (double pr' t~rint*, ',+t t,1r',,+, Th*,iti-- ".,..,, f..t

83 print*,' (unformatted store) fname(l:4) = 'DAT/' read*, fname(5: 16) open (4, file=fname, ERR=-900, form="un formatted") write(4,err=-900) Heightl write(4,err=-900) psen lose (4) else if (istore.eq.3) then print*,' filename to write ontrolled result (sngi.pre.)' print*,' at target loation = (in subdir DAT)' print*,' (unformatted store) fname(1:4) = 'DAT/' read*, fname(5: 16) open (4, file=fname, ERR=-900, form=" unformatted") write(4,err=-900) sheight write (4,ERR=900) psen lose(4) print*,' filename to write ontrolled result (double pre.)' print*,' at target loation = (in subdir DAT)' print*,' (unformatted store) fname(l:4) = 'DAT/' read*, fname(5: 16) open (4, file=fname, ERR=900, form="unformatted') write (4, ERR=-900) Heighti write(4,err=900) psen lose(4) else pause ' error in storage integer istore ' goto 100 end if return 900 print*,'error in write operation file ',fname print*,' Try again? I pause goto 100 end

84 program ranfld ************************************************************ Programm to set the input data field (itmax*kzmax) used as input for ontrol modelling (mainontr5.f). Output is written out either as physial spae field at detetor loation (this requires onvolution) with systri or as field Fourier omponent at exiter loation. In either ase is the resulting file written as unformatted double preision. August 2, 1991 random field is set in Fourier spae with random phase and unit amplitude of fourier oeffiients. (Hh) irand = starting integer for random number gen. Subroutines required : FOURN sranfld dran2 onvol2d Roland Heinrih ambridge,july ************************************************************* impliit real (a-b,d-h,o-y),omplex(, z), integer(i-n) harater*16 fname harater*1 A parameter (ndim=2, itmax=512, kzmax=128, ive=itmax*kzmax*2) double preision templ, temp2,datinl,datin2 ommon/fld/templ (ive), temp2 (ive), + datinl(itmax,kzmax),datin2(itmax,kzmax) dimension nve(ndim) nve(l) = itmax nve(2) = kzmax irand = -1 fname(i:4)='dat/' all sranfld (irand, templ) 1 print*,' Enter I to store Fourier oeffiients of field' print*,' at exiter loation or 2 to alulate phys. field' print*,' at dft-o-tor loation downstream ' read(*,*) ifield if (ifield.eq.l) then print*,'*un*formatted file for Fourier oef.' print*,' of random field (double pre. )in DAT?' read*, fname(5:16) print*, fname open (1, file=fname, form="unformatted") write(l) templ lose(l) else if (ifield.eq.2) then open (1, file="dat/fsystriun", forta="unformatted") read(l) temp2 lose(l) inl = 1 in2 = 1 all onvol2d(inl, in2) print*,'*un*formatted file for phys. field at print*,' detetor loation downstream (doubl. pre.) in fat?' rn d *, fn-oi(5:16)

85 print*, fname open (1, file=fname, form-"unformatted") write(l) datini lose(l) else pause ' wrong integer in ifield, try again ' goto 1 end if 999 end Subroutine sranfld ( idum, temp) Preset Fourier vetor temp with random phase and unity amplitude oeffiients. Roland Heinrih ambridge, July 24, 1991 Double omplex help, im Double Preision temprand,pi parameter(ndim = 2, itmax =512,kzmax=128, ive=itmax*kzmax*2) Dimension temp(ive) pi = 4.dO*datan(l.dO) im =(O.dO,l.dO) do ii=l,ive,2 rand = dran2 (idum) help = zexp(im*rand*2.do*pi) temp(ii) = dble(help) temp(ii+l) = dimag(help) end do return end Double Preision Funtion dran2(idum) Random Funtion generator, see Numerial Reipes, Press et al. page 192. (Made double preision by Roland Heinrih) Returns a uniform random deviate between 0.0 and 1.0. Set "idum" to any negative value to initialize or reinitialize the sequene. Roland Heinrih ambridge, Feb 5, 1991 Dimension ir(97) double preision rm Parameter(m=714025, ia = 1366, i=150889,rm=1.do/m) Data iff /0/ if (idum.lt.0.or.iff.eq.0) then iff =1 idum = mod(i-idum,m) do 11 j=1,97 idum = mod(ia*idum+i,m) ir(j) = idum 11 ontinue idum = mod(ia*idum+i,m) iy = idum endif j = 1 + (97*iy)/m if(j.gt.97.or.j.it.1) pause iy = ir(j) dran2 = iy*rm idum = mod(ia*idum+i,m) ir(j)= idum

86 return end

87 PROGRAM dstwpd Stability program for parallel three-dimensional Blasius boundary layer (spatial stability). This program is based on Mike Gasters rapid series expansion eigenvalue alulation using Shanks transformation to aelerate the onvergene proess (qqsum). This program alls the following subroutines qnewt multi-dimensional newton raphson root finder ludmp l.u. deomposition for qnewt lubksb used with ludmp to solve linear system of eqts. userfun provides funtion and derivatives for qnewt qfsub rapid series eigenvalue expansion The error-integer "ierr" should be zero for normal performane. It is given the value 1000 if the iteration iter in this main program part fails to onverge within tolerane tol. For eah all to the newton subroutine qnewt whih leads to non-onvergene within the subroutine in ntrial attempts, ierr is inremented by 1. The first three subroutines are adapted from "Numerial Reipes" William H. Press, Brian P. Flannery, Saul A. Teukolsky, William T. Vetterling, ambridge University Press, Otober 9, 1990 modified to double preision and use of qfsub (Roland Heinrih) Otober 17, 1990 modified for "B" version,i.e. outer loop is eliminated and dfuserb is used (hh). Otober 18, 1990 this program dstwpb is modified dstab3db to allow mapping of Re, spanwise wavenumber br spae. The physial experimental parameters are set in DATA statement. Make toleranes of onvergene test value-dependent (atol, omegatol) November 6, 1990 extrapolate guess for alpha from previous alulated values (Hh). November 10, 1990 preset all eigenvalues "a" to (1.0,1.0) and alulate only up to frequny ( in frequeny loop iomega) where tolerane riterium is missed for the first time. Also "taper off" the last (in omega) value s.t. a smooth frequeny window is used -- > avoids ringing. At same time set ierr to -99 as flag. Read runparameter from file HeadwpD. (Hh) ROLAND HEINRIH AMBRIDGE, MAY 29,1990 parameter (np=2) impliit real*8 (a-h,o-y) double preision rel(32,32), ig(32,32),eta,delta,dnu double preision arenr, areni, arrad, asenr, aseni, asrad double preision pomst, pomdel,pbdel double omplex im,a,astrt,astrtl,hlp,hlpl,an,aml 1,astrtlmlastrtml double omplex oalp(9,20,55) dimension rre(9,20,55),rbr(9,20,55),romr(9,20,55) dimension ierarr(9,20,55) dimension x(np) ommon /inp2/ Rehat,Reihat, omsetr,omseti,br ommon /head04/ 1 arrad,asenr,aseni,asrad,m, im, istep ommon /oef04/ rel,img eta,delta,epso,epsl,arenr, areni, j ****** The following data qtatement ontainr, *rrwt'... Ir

88 performed aug14, 90, UO = 20 n/s, flu e-5 zmr/s. Variables starting sith P are Physial varialbes whih have to be nondimensionalized for run using this program. **~** DATA UO,dnu /20.dO,l.57d-5/ DATA Restart, Redel, Pbstart, astrt/1260.-do, 65.dO, 1 0.dO,(.15d0,.005d0)/ read in run-parameters open (5, file= IHeadwpD',form= I formatted') read(5,950) Pomst read(5,950) Pomdel read(5,950) Pbdel read(5,960) iomax read(5,960) ibmax read(5,960) itermax lose (5) 950 format(1ox,dlo.4) 960 format(10x,15) print*,'pomst = ',Pomst,' Pomdel =,Pomdel,' Pbdel =',Pbdel, 1 ' omax =', iornax,' ibmax = ',ibmax,' itermax =,itermax im =(0.dO,1.dO) dum = dnu/uo/uo PRe =Restart astrtl = astrt Pnr = Pomst 1 tolfa =1.d-4 iremax = 9 ierr = 0 **** Preset all eigenvalues with (l.,1.0) do i =1,iRemax do ii = libmax do iii = 1, iomax alp(i,ii,iii) = (l.do,1.do) ierarr(i,ii,iii) = -99 end do end do end do Header in eigmap, '~write open (1O,file = 'HeadO4',orn ='formatted') read (l0,*) m,eta,im read (10,*) delta, epso, epsl, istep read (10,*) arenr,areni,arrad read (10,*) asenr,aseni,asrad lose (10) open (1O,file = 'foef',form = 'formatted') read (10,15) ((rel(i,j),i=1,im),j=1,in), 1 ((xug(i~j),i=1,im),j=1,imd) lose (10) 15 format(2x,e24.16e3) open(13,file = 'eigmap2',form = 'formatted') write(13, 300) Restart, Redel, Pomst, Pomdel, Pbstart, Pbdel 300 format (2X, 'restart = ',f6.0,2x,' delre - ',F6.0,/,2X 2,1 Pomstar't =I 1,f8.3,2X,' Pdelomr = ',f6.3,/,2x,' Pbstart = 1,F6.3, 3 2X,' Pbdel = lf., ~ O* UTR loop 490 in Re 'iren do 490 ire~n 1,iRe~max

89 Rehat = Pre Reihat -0.dO astrt = astrtl dt-lpl = astrtl V 7- Program parameter Re dependent bst = Pre * Pb * dum*uo bdel = Pre * Pbdel * dun*uo omrdel = Pre * Pouxdel * dum omstrt = Pre * Poflr *dum keep loop starting value br =bst ************loop 550 in spanwise wavenuniber fib' **** do 550 ib = l,ibmax a = astrt ami = astrt hip = astrt atol = zabs (astrt) *tol fa keep loop starting value =M~ omstrt *******inner frequeny loop 'iomega' ********* do 551 iomeg = l,iomax omsetr = omr omseti = 0.dO omtol = omsetr*tolfa ****start Newton-Raphson* with 3-D paramters***** ierr = 0 an = 2.dO*a - ami x (1) = dble(an) x(2) = dimag(an) all dqnewt(itermax,x,2,atol,omtolierr) ami a = a = dmplx(x(l),x(2)) write (13,900) PMe, omr, test4 write(13,910) a,ierr alp(iren,ib,iomeg) = a, rbr(iren,ib,iomeg) = br romr(iren,ib,iomeg) = omr Rke(iren,ib,iomeg) = PRe ierarr(iren, ib, iomeg)= ierr if (iomeg.eq.l) then astrtml = hip hip = a astrt = 2.dO*hlp - astrtml if (ib.eq.1) then astrtlml = hipi hipi = a astrtl = 2.dO*a - astrtlml end if end if ***if ierr = 1, end omega loop and jump to next "b" value sine series is starting to have onvergene problems. remezmber the higher omega values lead to stronger deaying eigenvalues alpha (provided I am above stability "banana" as has to be the ase for this exit proe~dure to sb-e sensible).

90 if (ierr.eq.1) then iomfin = iomega goto 552 end if **** Update omega our = omr + omrdel 551 ontinue iomfin = iomax *** Window off eigenvalues smoothly with os(45,90,135). 552 alp(iren,ib,iomfin-2) = d0*alp(iren,ib,iomfin-2) alp(iren,ib,iomfin-1) =.5d0*alp(iren,ib,iomfin-1) alp(iren,ib,iomfin) = d0*alp(iren,ib,iomfin) ***** update spanwise wavenumber br =br + bdel write(13,*) '************** br =,br,******* 550 ontinue ***** update physial Reynoldsnumber PRe = PRe + Redel write(13,*) '#II#I#I#### PRe = ',PRe, 'f####' 490 ontinue open (1, file="eigwp", form="unformatted") write(l) alp write(l) rotr write(l) rbr write(l) RRe write(l) ierarr lose(l) 900 format (3x,F6.0,3X, F6.3,3X, E12.3) 910 format(3x,2f9.6,3x,i6) lose(13) END SUMRUTINE dqnewt (NTRIAL, X, N, TOLX, TOLF, ierr) Given an initial guess X for a root in N dimension, take NTRIAL Newton- Rahpson steps to improve the root. Stop if the root onverges in either sum variable inrements TOX or suimmed funtion values TOLF. INPUT : A(ij) = del(fi)/del(xj) B(i) =- fi Ref: Numerial Reipes, W.H. Press et al. pg. 273 version for eigenvalue alulation gaster series ierr is error indiator. Should be zero for normal run. Eah time the tolerane riterium is not met for ntrial attempts, ierr is inremented by one. September 24,1990 modified for double preision (Hh) Roland Heinrih ambridge, May 29, 1990 PARAMETER (NP = 2)

91 impliit real*8 (a-h,o-z) DIMENSION X(NP), ALPHA(NP,NP), BETA(NP), INDX(NP),xinit(NP) onmon /inp2/ Rehat, Reihat, osetr, omseti,br ***** keep initial guess for error message do 2 i = l,np xinit(i) = x(i) 2 ontinue DO 13 K = 1,NTRIAL ALL duserfu (X,ALPHA, BETA) ERRF = 0.d0 DO 11 I=1,N ERRF = ERRF+ABS(BETA(I)) 11 ONTINUE IF (ERRF.LE.TOLF) RETURN ALL dwdmp(alpha, N, NP, INDX, D) ALL dlujbksb(alpha,n,np, INDX, BETA) ERRX = 0.dO DO 12 I = 1,N ERRX = ERRX + ABS(BETA(I)) X(I) = X(I) + BETA(I) 12 ONTINUE IF (ERRX.IE.TOLX) RETURN 13 ONTINUE ****** write message if tolerane riteria not met ** print*, ' Tolerane riteria in qnewt not' print*, ' met within NTRIAL attempts for ase' print*, ' Rehat = ',rehat,' omsetr = ',omsetr,' br = ',br print*, ' ntrial= ', ntrial print*, ' errf = ',errf,' tolf = ',tolf print*, ' errx = ',errx,' tolx =,,tolx print*,' The initial guess for a was ',xinit(l),xinit(2) print*, ' The failed a value = ',x(l),x(2) print*,'a value reset to initial guess before return' do 800 i = l,np x(i) = xinit(i) 800 ontinue ierr = ierr + 1 RETURN END SUBROUTINE duserfu (X, ALPHA, BETA) User supplied funtion and derivative for multiple dimensional newton-rahpson aording to NUMERIAL REIPES by W.H. Press p this program is used for eigenvalue alulation Gaster series September 22,1990 modified to double preision (Hh) Otober 9, 1990 modified to use dqfsum (Hh) $$$$$ Otober 16, 1990 modified to take variation of Re(2-d) into $$$$$ aount when alulation domega/dx. Here $$$$$ X is 3d wavenumber "a" a4 ALPHA $$$$$ ontains derivatives dmega/da (Hh) Roland Heinrih ambridge, May 29, 1990 PARAMEMER (NP-2) impliit real*8 (a-h,o-z) double preision dxr,dxi,drer,drei,

92 1 delxr omplex*16 a,al,or,orn,alp,alpl,rerei DIMENSION X(NP), ALPHA(NP,NP), BETA(KP) ommn~on / inp2/ Rehat, Reihat, orsetr, omseti,br delxr =.0005d0 DO 20 J = 1, NP DO 10 I = 1, NP ALPHA(I,J) = ONTINUE BETA(J) = ONTINUE ****** alulate orresponding 2-d parameters aording to parallel flow Squire's transform a = drplx(x(l),x(2)) aip = dsqrt(a*a + br*br) re = Rehat*a/alp print*,' 2d-parameters re,bet,alp print*, 're = ',re print*, Ibet= ',bet print*, 'alp= ',alp ** drer = dble(re) drei = dimag(re) dxr = dble(alp) dxi = dimag(alp) note 10 has to be 14,18,22,... for subroutine dqfsum whih uses full shanks IQ = 22 all dqf sum (dxr, dxi, drer, drei, IQ, 1, dbetr, dbeti) ~ alulate 3-d omega orn a/alp*dmplx(dbetr,dbeti) BETA(l) = (dble (om) -ornsetr) BETA(2) = (dimag(om) -omseti al =diplx (x(1) +delxr, x(2) ) alpl = dsqrt(al*a1 + br*br) Rel =Rehat*al/alpl drerl = dble(rel) dfleil = dimag(rel) dxri = dble(alpl) dxii = dimag(alpl) all dqfsurn(dxrl,dxil, drerl, dreil, IQ,l1,dbetrl,dbetil) **~alulate 3-d omega omi =a1/alp1*dnplx(dbetrl,dbeti1) fprr =(dble(oml - orn)) /delxr fpri =(dble(oml1 - orn)) /delxr afprr = dabs(fprr) afpri = dabs(fpri) if((afprr.lt.1.d-8).or. (afpri.lt.l.d-8)) then print*,'pause in duserfun.f, slope smalli print*,' fprr = ',fprr,' fpr i= I jfpri

93 pause else if ((afprr.gt.l.d5).or.(afpri.gt.l.d5)) then print*,'pause in duserfun-f, slope large print*,' fprr = ',fprr,' fpri= ',fpri. pause end if ALPHA(1,l) = fprr alpha(l,2) = -fpri alpha(2,1) = -alpha(1,2) alpha(2,2) = alpha(1,l) RE1RM END ************************************** Program QFSum ************************************** Fast version of series program Feb 1990 : uses Quik summation but Full matrix shanks SUBROUTINE DQFSUM (-ATPHAR, ALPHAI, REYR,REYI, IQ, INUM, OMEGaR, OMEGaI) impliit real*8 (a-h,o-z) Double Preision Real(32,32), Imag(32,32), r(32), i(32) Double Preision Omegar, Omegai, Realsum, Imagsum Double Preision Eta, Delta, Br, Bi Double Preision ARenr, AReni, ARrad Double Preision ASenr, ASeni, ASrad Double Preision Alphar, Alphai, Reyr, Reyi Double Preision Ur, Ui, Vr, Vi, Zr, Zi, Temp Double Preision lr, li, tr, ti, lmag, r, i onmmon /head04/ eta, delta, epso, epsl, arenr, areni, 1 arrad,asenr,aseni,asrad,m,im, istep oimmon /oef04/ real, imag Integer R Tol=1.d-1O IR=IQ+ 1 Do 100 Inr-l,Inum Ur=-Alphar*Alphar-Alpha i*alphai-asenr Ui=2.ODO*Alphar*Alphai-ASeni Vr=-Alphar*Reyr-Alphai*Reyi-ARenr Vi=Alphar*Reyi+Alphai*Reyr-AReni Ur=Ur/ASrad Ui=Ui/ASrad Vr=-Vr/ARrad Vi=Vi/ARrad Temp =Ur*Ur+Ui*Ui Br=Ur/ Temp Bi=-Ui/Tentp Temp=Vr*Vr+Vi*Vi Zr= (Ur*Vr+Ui*Vi) /Temp Zi= (-Ur*Vi+Ui*Vr) /Temp Diagonal summation of series r=0. ODO i=0. ODO Do 20 R--l,IR

94 Tenmp=Br*Vr-Bi*Vi Bi=Br*vi+Bi*vr Br=-renip Realsum=O. ODO Imnagsumn=0.ODO Do 10 J=R,1,-1 I=R-J+l Realsum=-Realsum+Real (I,J) Imagsumf=Imagstim+Imag (I,J) Teznp=Realsum*Zr-Imagsun*Z i Imagsum=-Real sumi* Z i+ixnagsun*z r Realsum-TemIp 10 ontinue r--r+br*realsum-bi* Imagsum i=i+br*imagsum+bi*realsum Write (*,*) R, r, i r(r)=r i (R)=i 20 ontinue Shanks Transformation -- Full Matrix L= 0 M =IQ 40 Do 50 J=2,M Jpl =J+1 lr=-r (Jmn) +r (Jpl) -r (J) -r (J) l i=i (Jmn) +i (Jpl) -i (J) -i (J) lxaag=lr*lr+1 i*1 i If (imag.lt. Tol) Then r (Jmn) =r (J) i (Jnin) =i (J) Else tr--r (Jmn) *r(jpl) -i (Jmn) *i (Jpl) -r (J) *r(j) +i (J)*i (J) ti=r (Jmn) *i (Jpl) +i (Jmn) *r (Jpl) -2. ODO*r (J) *i (J) r (Jmnn) = (tr*lr+ti*l i) /lmag i (Jmn) =(-tr*li+ti*lr) /lmag End If 50 ontinue M M-2 if (M.Ne.0) goto 40 if (L.eq.1) goto 80 Mh IQ/2-1 11M= M+1 Do 70 J=2,IM r(j) = r(2*j-1) 70 i(j) = i(2*j-1) Goto O)Megar = r(1)*alphar -i(l)*alphai Oinegai = r(l)*alphai + i(1)*alphar 100 ontinue return End SUJBROUTINE du)bksb (A, N,NP, INDX, B)

95 -13- Solves the set of N linear equations A.X = B. Here A is input, nat as the but rather as its "LU" deomposition, determined by the routine "LLDMP. is input as the permutation vetor returned by IDDMP. B is input as the r side vetor B and returns with the solution vetor X. A, N, NP and INDX ar modified by this routine and an be left in plae for suessive alls wit different right-hand sides B. This routine takes into aount the possibli that B will begin with many zero elements, so it is effiient for use in matrix inversion Ref.: Numerial Reipes, W.H. Press et al. pg 37 September 22, 1990 modified for double preision (Hh) Roland Heinrih ambridge, May 29, 199 impliit real*8 (a-h,o-z) DIMENSION A(NP,NP), INDX(N), B(N) II = 0 DO 12 I=l,N LL = INDX(I) SUM = B(LL) B(LL) = B(I) IF (II.NE.0) THEN DO 11 J=II,I-i SUM = SUM - A(I,J)*B(J) 11 ONTINUE ELSE IF (SUM.NE.0.) THEN II = I ENDIF B(I) = SUM 12 ONTINUE DO 14 I=N,1,-l SUM = B(I) IF(I.LT.N) THEN DO 13 J=I+1,N 13 SUM = SUM-A(I,J)*B(J) ONTINUE ENDIF B(I) = SUM/A (I, I) 14 ONTINUE reurn END SUBROUTINE dludmp(a,n,np, INDX, D) Given an N xn matrix A, with physial dimension NP, this routine replaes "LU" deomposition of a rowwise permutation of itself. A and N are input., arranged as in equation (2.3.14) of "Numerial Reepies " W. H. Press e pg. 35.; INDX is an output vetor whih reords the row permutation effet the partial pivoting; D is output as +/- 1 depending on whether the number interhanges was even or odd, respetively. This routine is used in ombin with IUBKSB to solve linear equations or invert a matrix. September 22, 1990 modified for double peision (Hh) Roland Heinrih ambridge, May 29, 1990 PARAMETER (NMAX=100, TINY=l.Od-20) impliit real*8 (a-h,o-z) DIMENSION A(NP,NP),INDX(N),VV(NMAX) D = 1. DO 12 I = I,N AAMAX=0. do DO 11 J = 1,N IF (ABS(A(I,J)).GT.AAMAX) AAMAX=ABS(A(I,J)) 11 ONTINUE

96 IF (AAI4AX. EQ. 0. ) then PAUSE 'SINGuLhR MATRIX. end i f VV(I) -1.I(AAMAX) 12 ONTINUE DO 19 J=1,N DO 14 I-,- SUM=-A (I, J) DO 13 K=1,I-i "U 2 SUM - A (I, K)*A (K, 13 J) ONTINUE A(I,J) = SUM 14 ONTINUE AAKAX = 0. DO 16 1=J,N sum = A(I,J) DO 15 K=,J-1 SUM = SUM-A (I, K) 15 *A(K,J) ONTINUE A(I,J) =SUM DUM = WV(I)*ABS(SuM) IF (DUM.GT.AAMAX) THEN IMAX =I AAMAX = 1111 ENDIF 16 ONTINUE IF (J.NE.IMX)THEN DO 17 XK1,N IXJI=A (14AX, K) A(IMA,x) = A(J,K) A (J, K) = DuM 17 ONTINUE D= -D VV(IMAX) = VV(j) ENDIF INDX (J)=IMA IF(A(J,J).EQ.o.) THEN A (J, J) = TINY PRINT *, 'A(J,J) SINGUIAR (SET TINY) FOR j,j,,j ENDIF IF(J.NE.N) THEN UM=.1. /A(J,J) DO 18 I=J+1,N 18 ONTINUE 19 ENDIF ONTINUE RETRN END 1000, 10.ODO, , 1.E-20, 1.E5, ODo, 0.ODO, 255.0W kt 0.063d0, 0.0WO, DW Pomst = 500.do Pomdel = 28.do h w jr.pbdel - 20.do imax - 55 ibmax - 10 itermax= 300

97 omplex Fourier oeffiients, ' if for wave paket eigenvalue alulation i,j, real part, imag. part E E-0l E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E O E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E F-003

98 E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E OOE E a O0E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-005

99 E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E b E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-007

100 E E E-0O E E E E E E E E E E E E E E O E O E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-I E F E E E E E E E E E E E E E E E E E E E E E E E Z E E E E E E E E E E E E E E E E E E E E E E E E-010

101 E E E E E E E E E E E E E E E OE E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E OE E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-011

102 OE E E E E E E E E E E E E E OE E E E E E E E OOE E OE E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E OE E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E OE E E E E E E E E E E E E E E E E E-004

103 -l E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E OOE E E E )943400E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E OE E E E E E E E E E E E E OOE E E E E E E E E E E E E E F O7

104 E E E E E E E E E E E E OOE E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E oE E E E E E E E E E E OE E OE E E E E E E E E E E E E E E E E E E E E E E E Z E E E E E E OE E E E E E E E E E E E E E E E E E E E E-009

105 E E E E E E E E E E E E E E E E E E E-O E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-010

106 -I E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E i e e E E E OE E E E E E E E E E E E E E E E E E E E E E E E E E E E-013

107 E OE E E ' E E E E-U E E E E E E E E E E E E E E E E E E E E E OE E E E E E E-0O.., E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-" E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-013

108 E E E E E E E E E E E E E E E OE E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E OOE E E E E E E E E E E E-008

109 E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-010