IN THE past decade, aerodynamic shape optimization has been the

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1 AIAA JOURNAL Vol. 45, No. 7, July 27 Optmum Shape Desgn for Unsteady Flows wth me-accurate Contnuous and Dscrete Adjont Methods Sva K. Nadarajah McGll Unversty, Montreal, Quebec H3A 2S6, Canada and Antony Jameson Stanford Unversty, Stanford, Calforna 9435 DOI:.254/ hs paper presents an adjont method for the optmal control of unsteady flows. he goal s to develop the contnuous and dscrete unsteady adjont equatons and ther correspondng boundary condtons for the tmeaccurate method. Frst, ths paper presents the complete formulaton of the tme-dependent optmal desgn problem. Second, we present the tme-accurate unsteady contnuous and dscrete adjont equatons. hrd, we present results that demonstrate the applcaton of the theory to a two-dmensonal oscllatng arfol. he results are compared wth a multpont approach to llustrate the added beneft of performng full unsteady optmzaton. Nomenclature A = flux Jacoban matrx B = boundary b = boundary velocty component c = chord length D = artfcal dsspaton flux E = nternal energy F = Euler numercal flux vector f = Euler flux vector G = gradent I = cost functon, j = cell ndces M = Mach number N = outward-facng normal p = pressure R = resdual S = shape functon S = face areas of the computatonal cell s = arc length = tme perod t = tme t f = fnal tme u = velocty (physcal doman) V = cell volume w = state vector x = coordnates (physcal doman) = angle of attack = adjustable constant for artfcal dsspaton scheme = step sze = coordnates (computatonal doman) = densty = Lagrange multpler Presented as Paper 5436 at the 9th AIAA/ISSMO Symposum on Multdscplnary Analyss and Optmzaton Conference, Atlanta, GA, 4 6 September 22; receved 29 March 26; revson receved 29 November 26; accepted for publcaton 8 January 27. Copyrght 27 by the Amercan Insttute of Aeronautcs and Astronautcs, Inc. All rghts reserved. Copes of ths paper may be made for personal or nternal use, on condton that the coper pay the $. per-copy fee to the Copyrght Clearance Center, Inc., 222 Rosewood Drve, Danvers, MA 923; nclude the code -452/ 7 $. n correspondence wth the CCC. Assstant Professor, Computatonal Flud Dynamcs Lab, Department of Mechancal Engneerng, 688 Sherbrooke Street West, Room 7; sva.nadarajah@mcgll.ca. homas V. Jones Professor of Engneerng, Department of Aeronautcs Astronautcs, Durand Buldng, 496 Lomta Mall. Fellow AIAA. 478! r = reduced frequency I. Introducton IN HE past decade, aerodynamc shape optmzaton has been the focus of attenton due largely to advanced algorthms that have allowed researchers to calculate gradents cheaply and effcently. he majorty of work n aerodynamc shape optmzaton has been focused on the desgn of aerospace vehcles n a steady flow envronment. Investgators have appled these advanced desgn algorthms, partcularly the adjont method, to numerous problems, rangng from the desgn of two-dmensonal arfols to full arcraft confguratons to decrease drag, ncrease range, and reduce sonc boom [ ]. hese problems were tackled usng many dfferent numercal schemes on both structured and unstructured grds [2 5]. Unlke fxed-wng arcraft, helcopter rotors and turbomachnery blades operate n unsteady flows and are constantly subjected to unsteady loads. herefore, optmal control technques for unsteady flows are needed to mprove the performance of helcopter rotors and turbomachnery and to allevate the unsteady effects that contrbute to flutter, buffetng, poor gust and acoustc response, and dynamc stall. Recently, the desgn of blade profles usng unsteady technques was attempted. Dverse methods were employed n the desgn of rotorcraft and turbomachnery blades. he followng are a selected number of papers on ths topc. Ghayour and Baysal [6] solved the unsteady transonc small-dsturbance equaton and ts contnuous adjont equaton to perform an nverse desgn at Mach.6. Aerodynamc shape optmzaton of rotor arfols n an unsteady vscous flow was attempted by Yee et al. [7] usng a responsesurface methodology. Here the authors used an upwnd-basedfactorzed mplct numercal scheme to solve the Reynoldsaveraged Naver Stokes equatons wth a Baldwn Lomax turbulence model. A response-surface methodology was then employed to optmze the rotor blade. he objectve functon was a sum of the L=D at three dfferent azmuth angles and was later redefned to nclude unsteady aerodynamc effects. Florea and Hall [8] modeled a cascade of turbomachnery blades usng the steady and tme-lnearzed Euler equatons. Gradents for aeroelastc and aeroacoustc objectve functons were then computed usng the dscrete adjont approach. Both the flow and adjont equatons were solved usng a fnte volume Lax Wendroff scheme. he gradents were then used to mprove the aeroelastc stablty and acoustc response of the arfol.

2 NADARAJAH AND JAMESON 479 radtonally, a multpont desgn approach s one possble technque for the optmzaton of blade profles n an unsteady flow envronment. hs approach only requres a small extenson of a steady flow desgn code to redesgn a blade or arfol profle for multple flow condtons. Because the steady flow equatons are used to desgn the blades, uncertantes stll preval surroundng the performance of these blades n an unsteady flow envronment. In ths paper, we develop a framework n whch to perform two major tasks: frst, to perform senstvty analyss n a nonlnear unsteady flow envronment; second, to further modfy the shape of the object to acheve the objectve of the desgn usng a full unsteady optmzaton method based on control theory. Optmal control of tme-dependent trajectores s generally complcated by the need to solve the adjont equaton n reverse tme from a fnal boundary condton usng nformaton from the trajectory soluton, whch n turn depends on the control derved from the adjont soluton. In ths work, we extend the adjont method to unsteady perodc flows usng a tme-accurate approach. he tme-accurate unsteady adjont equatons are based on Jameson s [9] cell-centered multgrddrven fully mplct scheme wth upwnd-based, blended, frst- and thrd-order artfcal-dsspaton fluxes. he goal of ths research s to develop both the tme-accurate contnuous and dscrete adjont equatons and use them n the redesgn of the RAE 2822 arfol undergong a ptchng oscllaton to acheve lower tme-averaged drag whle keepng the tme-averaged lft constant. hs technque s compared wth the multpont and steady adjont approaches to gauge the effectveness of the method. II. Governng Equatons he Euler equatons for a rgdly translatng control volume, defned by wth an outward-facng normal N, can be wrtten n ntegral form as I d w dx dx 2 f N ds () he state vector w and a component of the nvscd flux vector, f, can be wrtten as u >< >= >< b >= u w u ; f u u b p >: 2 u >; >: 2 u b 2 p >; E Eu b pu In these equatons, x and x 2 are the Cartesan coordnates; u and b are the Cartesan velocty components of the flud and boundary, respectvely; and E s the total energy. he results presented n ths paper are based on transonc flow calculatons n whch the deal gas equaton s applcable. Consequently, the pressure can be expressed as p E u 2 u A. Numercal Dscretzaton of Governng Equatons A fnte-volume methodology s used to dscretze the ntegral form of the conservaton laws. When usng a dscretzaton on a body-conformng structured mesh, t s useful to consder a transformaton to the computatonal coordnates and 2,defned by the K j ; J j he smulatons contaned n ths research are restrcted to rgd mesh translaton. Consequently, the volumetrc ntegral from Eq. () can be approxmated as the product of the cell volume and the temporal dervatve of the soluton at the cell center. he Euler equatons can then be wrtten n computatonal space (2) where the convectve flux s now defned wth respect to the computatonal cell faces by F S j f j, and the quantty S j JKj represents the projecton of the cell face along the x j axs. When Eq. (2) s formulated for each computatonal cell, a system of frstorder ordnary dfferental equatons s obtaned. Equaton (2) can then be wrtten for each computatonal cell n semdscrete form as Rw (3) @F 2 f 2 ;j f 2 ;j f 2 f 2 (4) he notaton ndcates that the quantty s calculated at the flux 2 faces. o stablze the scheme, a blended mx of frst- and thrd-order fluxes frst ntroduced by Jameson et al. [2] s added to the convectve flux at each cell face and can be defned as d 2 ;j 2 2 ;jw w 4 2 ;jw 2;j 3w 3w w (5) he frst term n Eq. (5) s the frst-order dffuson term, where 2 s 2 ;j proportonal to the normalzed second dervatve of pressure. hs term s domnant n the vcnty of shocks and serves to damp oscllatons and overshoots assocated wth ths dscontnuty n the soluton feld. he 4 coeffcent scales the magntude of the thrdorder dsspatve flux. he coeffcent s scaled so that t becomes the 2 ;j domnant term away from the shock, elmnatng the odd even decouplng assocated wth central-dfference schemes. he tme-dervatve term can be approxmated by a kth-order mplct backward-dfference formula (BDF) such as d dt t X k q q q (6) where w n w n. A second-order expanson of Eq. (6) wll result n the followng equaton for the semdscrete form (3) of the governng equatons: 3 V 2t wn 2 t wn 2t wn R w n (7) Equaton (7) represents a set of hghly nonlnear, coupled, ordnary dfferental equatons and can be solved at each tme step usng the explct multstage modfed Runge Kutta scheme. We defne a new modfed resdual R w as 3 R w V 2t wn 2 t wn 2t wn R w n where the modfed resdual R w s the sum of the steady-state resdual and a source term that orgnates from the second-order dscretzaton of the tme dervatve. he modfed resdual s then marched to pseudosteady state n a fcttous tme t, as follows: dw R w dt he procedure advances the soluton forward n tme from the t nt to t nt. he fnal tme-accurate soluton wll be composed of a sequence of pseudotme steady-state solutons. (8)

3 48 NADARAJAH AND JAMESON III. General Formulaton of the me-dependent Optmal Desgn Problem he aerodynamc propertes that defne the cost functon are functons of the flowfeld varables, w, and the physcal locaton of the boundary, whch may be represented by the functon S. We then ntroduce the cost functon: I Lw; S dt Mwt f (9) he cost functon s a sum of a tme-averaged functon Lw; S and a functon M that s a functon of the soluton wt at the fnal tme. A change n S then results n a @S wt f () n the cost functon. Usng control theory, the governng equatons of the flowfeld are now ntroduced as a constrant n such a way that the fnal expresson for the gradent does not requre reevaluaton of the flowfeld. o acheve ths, w must be elmnated from Eq. (). From Eq. (3), a varaton of the semdscrete form of the governng equatons can be wrtten @w Next, ntroduce a Lagrange multpler to the tme-dependent flow equaton and ntegrate t over tme to w w Subtract Eq. () from the varaton of the cost functon to arrve at the followng @S wt w w S Next, collect the w and S terms and ntegrate w by parts, to @w V t f @S S Choose to satsfy the adjont wth the termnal boundary condton V hen the varaton of the cost functon reduces to where G I @R Optmal control of tme-dependent trajectores s generally complcated by the need to solve the adjont equaton n reverse tme from a fnal boundary condton usng data from the trajectory soluton, whch n turn depends on the control derved from the adjont soluton. he senstvtes are determned by the soluton of the adjont equaton n reverse tme from the termnal boundary condton and the tme-dependent soluton of the flow equaton. hese senstvtes are then used to get a drecton of mprovement and steps are taken untl convergence s acheved. IV. me-accurate Contnuous Adjont Equaton o formulate the tme-accurate contnuous adjont equaton, we frst ntroduce the cost functon. In ths work, the cost functon s chosen to be the tme-averaged drag coeffcent and s defned as I C d dt p 2 M 2 c C a cos C n sn dt @ sn d dt where C a and C n are the axal and normal force coeffcents, respectvely. he desgn problem s now treated as a control problem n whch the control functon s the arfol shape, whch s chosen to mnmze I subject to the constrants defned by the flow equatons. A varaton n the shape causes a varaton p n the pressure and, consequently, a varaton n the cost I p sn p 2 M 2 B sn d dt Because p depends on w through the equaton of state, then the varaton p s determned from the varaton w. From Eq. (2), the varaton of the governng equaton can be wrtten where F C w S j f j. he Euler Jacoban matrx n the computatonal doman C s defned as C S j A j, where =@w s the Euler Jacoban matrx. Multplyng by a costate vector, also known as a Lagrange multpler, and ntegratng over the space and tme produces w S j f j dd dt If we confne the smulatons to rgd mesh translaton, then we can separate the equaton nto two terms and swtch the order of the doman and tme ntegrals for the frst term, to yeld w dt C w S j f j dd dt If s dfferentable, then the two terms n the precedng equaton can be ntegrated by parts, to gve D V w t f w dt n C w S j f j db C w S j f j dd dt he next procedure s to rearrange the terms n the equaton such that

4 NADARAJAH AND JAMESON 48 ntegrands that are multpled by the varaton of the state vector, w, are grouped together and terms that are multpled by the varaton of the metrc terms are separated nto a dfferent ntegral. hs procedure s crucal to solate the ntegral that wll produce the tmeaccurate contnuous adjont equaton: D V t f wt f V w w dd dt n F db B n S j f j db B D j f j dd w Because the left-hand expresson equals zero, t may be subtracted from the varaton of the cost functon (3) to I p p 2 M 2 c B p cos sn D V t f wt f V w dd D w dd dt n F db n B S j f j db S j f j dd dt (4) Because s an arbtrary dfferentable functon, t may be chosen n such a way that I no longer depends explctly on the varaton of the state vector, w. he gradent of the cost functon can then be evaluated drectly from the metrc varatons wthout havng to recompute the varaton w resultng from the perturbaton of each desgn varable. he varaton w can then be elmnated by solvng for the Lagrange multpler by settng the transpose of the ntegrand of the second ntegral n the thrd lne of Eq. (4) to zero to produce a dfferental adjont system governng @ C n D he convectve flux of the tme-accurate contnuous adjont equaton (5) s dscretzed usng a second-order central spatal dscretzaton. he temporal dscretzaton s based on a second-order backward-dfference formula. Artfcal dsspaton based on the Jameson et al. [2] scheme, smlar to that employed for the flow solver, s added to stablze the scheme. Refer to Nadarajah [2] for a more detaled overvew of the numercal dscretzaton. A pseudotme dervatve s added and the tme-accurate adjont equaton s marched to a perodc steady-state soluton. At the outer boundary, ncomng characterstcs for correspond to outgong characterstcs for w. Consequently, we can choose boundary condtons for such that n C w If the coordnate transformaton s such that S s neglgble n the far feld, then the only remanng boundary term s F 2 d B W hus, by lettng satsfy the boundary condton, jn j p 2 M 2 sn on W (6) where n j are the components of the surface normal. Because the ntal condton for the Lagrange multplers are set to zero, then V w. If the problem s perodc n nature and the cost functon used for ths problem s not dependent upon t f, then V t f wt f. Equaton (4) fnally reduces to the I p p 2 M 2 c B n S j f j db B S j f dt sn ddt he precedng equaton s then used to solve for the gradent, whch can then provde a drecton of mprovement to reduce the objectve functon. V. me-accurate Dscrete Adjont Equaton As n the case of steady flow, the tme-accurate dscrete adjont equaton s obtaned by applyng control theory drectly to the set of tme-accurate dscrete feld equatons. he resultng equaton depends on the type of scheme used to solve the flow equatons. o formulate the dscrete adjont equaton, we frst take a varaton of Eq. (8) wth respect to w and S (only terms that are multpled to w are shown): 3 R n wv 2t wn 2 t wn 2t wn R n w (7) Next, multply the precedng equaton by the transpose of the Lagrange multpler and sum over the doman and tme, to yeld X t f X t n2 R n2 R n R n n3 R n3 (8) Substtute Eq. (7) nto the n, n 2, and n 3 terms of the modfed resdual n the precedng equaton, to yeld X t f X t 2t wn R R n 2t wn R n2 2t wn R n3 n 3 V n2 V n3 V 2t wn 3 3 2t wn2 2t wn3 2 t wn 2 t wn 2 t wn2 Keepng only the n terms, the precedng equaton reduces to X t f X t 2t R V n3 w n 3 2t n 2 t n2 n R n (9) Next, we ntroduce the dscrete cost functon for the drag mnmzaton problem as X t f I c C d t t p 2 M 2 c X t f X t f t X UE t LE C a cos C n sn t p ;W x 2 cos x sn t where LE s the lower tralng edge, UE s the upper tralng edge,

5 482 NADARAJAH AND JAMESON and p ;W s the wall pressure. In ths research, the wall pressure s defned as p ;W 2 p ;2 p ; where p ;2 and p ; are the values of the pressure n the cell above and below the wall. A varaton n the cost functon wll result n a varaton p n the pressure and varatons x 2 and x n the metrcs. he varaton of the cost functon for drag mnmzaton can be wrtten as I c p 2 M 2 c Xt f X UE 2 x 2 cos w ;2 w ; t LE XUE 2 p ;2 p ; p cos x 2 LE sn x t (2) he tme-dependent dscrete Euler equatons can now be ntroduced nto I as a constrant, to produce I I c X t f t X R w (2) Substtute Eqs. (9) and (2) nto the precedng expresson, whch can then be rearranged nto two man categores: frst, terms that are multpled by the varaton of the state vector, w; second, terms that are multpled by the varaton of the shape functon, S. he tmeaccurate dscrete adjont equaton can now be defned n 3 2t n 2 t n2 2t n3 n w R n (22) he frst n =@, s added to march the soluton n a fcttous tme to a perodc steady-state soluton. he second term represents a second-order forward-dfference formula for the temporal dscretzaton of the tme-accurate adjont equaton. he term demonstrates that to acqure the soluton of the adjont equaton at the tme step n requres the soluton at n 2 and n 3. herefore, the adjont equaton s ntegrated backward n tme from a termnal boundary condton. he last term requres the varaton of the resdual as a functon of the state vector. Here, R n denotes the dscrete representaton of the resdual whch s the sum of the dscrete convectve and dsspatve fluxes n each cell. A full dfferentaton of the equaton would nvolve dfferentatng every term that s a functon of w. A collecton of all terms that are multpled to w wll produce the dscrete adjont convectve and dsspatve fluxes. Refer to Nadarajah [2] for a complete dervaton of the dscrete adjont fluxes. o acqure the dscrete adjont boundary condton, Eq. (22) can be expanded at cell ; 2 to create the n ;2 3 ;2 2t 2 h A n B n ; ;2 n ;2 2 t n ;2 n ;2 n ;3 n ;2 n2 ;2 2t n3 ;2 A n n n 2 ;2 ;2 ;2 (23) where A and B are the Euler Jacobans n the and drectons, and s the source term for drag mnmzaton, x n 2 2 ;2 x n 3 ;2 x 2! cos t All of the terms n Eq. (23), except for the source term, scale as the square of x. herefore, as the mesh wdth s reduced, the terms n the source term must approach zero as the soluton reaches a steady state. One then recovers the contnuous adjont boundary condton, as stated n Eq. (6). Equaton (2) then smplfes to the followng form: I 2 p ;2 p ; p x 2! cos x! sn t n f R n he precedng equaton represents the total gradent obtaned usng the tme-accurate dscrete adjont approach to reduce the total drag of a ptchng arfol, and f R n represents the varaton of the resdual as a functon of the shape functon. VI. Desgn Process he objectve of ths work s to change the shape of the arfol to mnmze ts tme-averaged coeffcent of drag. Gven the dervaton provded n prevous sectons, the adjont boundary condton can easly be modfed to admt other fgures of mert. he shape of the arfol s constraned such that the maxmum thckness-to-chord rato remans constant between the ntal and fnal desgns. In addton, the mean angle of attack o s allowed to vary to ensure that the tmeaveraged coeffcent of lft remans constant between desgns. he arfol undergoes a forced ptchng oscllaton about the quarterchord. he angle of ncdence s gven by t o m sn!t where o deg, and the maxmum angle of attack s m : deg. One perod of oscllaton s defned from t to t 2. o compute the entre unsteady flow soluton for each perod, t s dvded nto N dscrete ponts or tme steps. he ndvdual steps wthn each teraton of the desgn process for the full unsteady desgn and multpont approaches are outlned next. A. Full Unsteady Desgn he followng steps detal the procedures for the full unsteady contnuous and dscrete adjont-based desgn optmzaton problem:. Unsteady Flow Calculaton A multgrd scheme s used to drve the unsteady resdual at each tme step to a neglgble value. he duraton of the physcal tme hstory (typcally quantfed n the number of oscllatory perods) depends on the physcs of the flowfeld and the accuracy requrements of the calculaton. Generally, t requres fve perods before a lmt cycle s acheved. Here, a perod refers to one full oscllaton. Durng the last perod, the flow soluton at each tme step s saved n memory. Ffteen multgrd cycles were used for each tme step. If 24 tme steps are used for each cycle and fve cycles are used to acheve the lmt cycle, then a total of 8 multgrd cycles are requred to obtan the unsteady soluton. o mantan the tmeaveraged lft coeffcent, the mean angle of attack o s perturbed. However, o s only modfed every three perods, because t requres at least three perods for the global coeffcents such as tme-averaged lft and drag to converge to an accuracy level of E-3. A total of 5 perods are needed nstead of fve to acheve the desred tmeaveraged lft coeffcent. hs multples the total cost by three tmes. 2. Unsteady Adjont Calculaton he unsteady adjont equaton, ether the dscrete or contnuous verson, requres ntegraton n reverse tme. he same numercal

6 NADARAJAH AND JAMESON 483 scheme employed to solve the unsteady flow s used here, as well wth mnor adjustments n the code to allow ntegraton n reverse tme. Only three perods are needed before the lmt cycle s acheved. Ffteen multgrd cycles are used at each tme step, whch translates to a total of 8 cycles to acheve a lmt cycle for the adjont equaton. 3. Gradent Evaluaton An ntegral over the last perod of the adjont soluton s used to form the gradent. hs gradent s then smoothed usng an mplct smoothng technque. hs ensures that each new shape n the optmzaton sequence remans smooth and acts as a precondtoner, whch allows the use of much larger steps. he smoothng leads to a large reducton n the number of desgn teratons needed for convergence. Refer to Nadarajah and Jameson [22] for a more comprehensve overvew of the gradent smoothng technque. An assessment of alternatve search methods for a model problem s gven by Jameson and Vassberg [23]. 4. Arfol-Shape Modfcaton he arfol shape s then modfed n the drecton of mprovement usng a steepest-descent method. Let F represent the desgn varable and G the gradent. An mprovement can then be made wth a shape change F G where s determned by tral and error to ensure a fast rate of convergence whle guaranteeng that the objectve functon convergence to an error level of at least E-6. he step-sze value s constant for all cases presented n ths work. 5. Grd Modfcaton he nternal grd s modfed based on perturbatons on the surface of the arfol. he method modfes the grd ponts along each grd ndex lne projectng from the surface. he arc length between the surface pont and the far-feld pont along the grd lne s frst computed, then the grd pont at each locaton along the grd lne s attenuated proportonal to the rato of ts arc length dstance from the surface pont and the total arc length between the surface and the far feld. 6. Repeat Desgn Process he entre desgn process s repeated untl the objectve functon converges. he problems n ths work typcally requred between 9 and 25 desgn cycles. Each desgn cycle requred 8 multgrd cycles to compute the flow soluton and 8 cycles for the adjont soluton. and adjont solvers, and the total computatonal cost s 72 multgrd cycles. able llustrates a cost comparson between the unsteady and multpont desgn approaches. Here, the mddle two columns contan the total number of multgrd cycles used to compute the Euler and adjont equatons. he numbers n the last column sgnfy the rato of cost of the unsteady desgn method wth respect to the multpont approach. Usng the full unsteady desgn approach requres four tmes the computatonal cost as dong the multpont approach. However, f the tme-averaged lft s constraned, an addtonal four perods are requred for the multpont approach to converge the tme-averaged lft coeffcent and an addtonal ten are needed for the unsteady case. hs brngs the total number of multgrd cycles requred by the flow solver per desgn cycle to 8 for the multpont case and 54 for the unsteady. he cost of the adjont solver remans the same, because the mean angle of attack s only modfed durng the flow-solver stage. he unsteady approach s then three tmes the computatonal cost of the multpont approach. he dfference n cost between one steady Runge Kutta teraton and one unsteady Runge Kutta teraton was not factored nto the computng cost, because the dfference s mnmal, requrng only the addton of the tme dervatves of the flow varables for the mplct tme-steppng algorthm. VII. Results he followng subsectons present results of the tme-averaged drag mnmzaton problem for a two-dmensonal arfol undergong a perodc ptchng moton. he frst subsecton contans a codevaldaton study. he second subsecton s dedcated to a temporalresoluton study. he thrd subsecton demonstrates the redesgn of the RAE 2822 arfol to reduce the tme-averaged drag coeffcent whle mantanng the tme-averaged lft. he last subsecton presents a comparson between the multpont and unsteady desgn approaches. A. Code Valdaton An Euler soluton s computed on a grd and the lft coeffcent versus angle of attack s compared wth the expermental NACA 64A C6 [24] data. A spatal-resoluton study [2] was conducted for grds of varous types and szes to determne the mnmum number of grd ponts needed to establsh an accurate hysteress loop. It was deemed that a grd, as shown n Fg., was adequate for nvscd solutons. B. Multpont Desgn In the multpont desgn approach, the unsteady flow and adjont solvers are replaced wth a steady flow and adjont solver at each tme step. he gradent s an average of the gradents from each tme step, defned as g N X N! t g where g s the gradent and! t s the weght at each desgn pont. In ths work, the weghts are chosen to be unty. If the tme-averaged lft s not constraned, then only one perod s requred for both the flow able Comparson of computatonal cost between the desgn approaches Method Euler (multgrd cycles) Adjont (multgrd cycles) Cost Multpont Full unsteady Fg. NACA 64A mesh.

7 484 NADARAJAH AND JAMESON..8 me Accurate, 23 me Steps Expermental.75.7 Steady ω =. r.6.65 = =.2 Lft Coeffcent.2.2 Lft Coeffcent, c l = Angle of Attack Fg. 2 Comparson of lft hysteress wth NACA 64A C6 [24] expermental data at M :78 and! r : Angle of Attack Fg. 4 Lft hysteress at varous reduced frequences for the RAE 2822 at M :78. he computatons are performed at a freestream Mach number of.78, a mean angle of attack of o deg, a maxmum angle of attack of m : deg, and at a reduced frequency of! r :22. Fve perods of computaton are requred n order for the perodc flow to be establshed and to allow the tme-averaged lft and drag coeffcents to converge to an error level of E-3. Fgure 2 llustrates the hysteress loop and the results reproduce the expermental results wth suffcent accuracy. In Fg. 2, the lft coeffcent versus angle of attack loop moves n a counterclockwse drecton. he nonlnear behavor due to the movement of the shock causes an ampltude reducton of the lft coeffcent and a phase lag. he prmary goal of ths work s to nvestgate the benefts of aerodynamc shape optmzaton of arfols undergong unsteady moton va an adjont-based unsteady optmzaton approach. he tradtonal approach s to perform a multpont desgn usng solutons from varous phases of the unsteady moton. he followng subsectons wll establsh that under specfc condtons, there s an advantage to an unsteady optmzaton approach. Frst, a temporalresoluton study wll be examned. Second, an optmzaton of a ptchng arfol to reduce drag wll be demonstrated. Fnally, a thorough nvestgaton of the advantage of an unsteady optmzaton approach over a multpont wll be presented. he chosen test case s that of a ptchng RAE 2822 arfol n transonc flow at a Mach number of.78, a mean angle of attack of o deg, and a reduced frequency of! r :2 on a grd. Fgure 3 llustrates the convergence hstory for the unsteady flow solver and the unsteady contnuous and dscrete adjont Unsteady Flow Solver Unsteady Contnuous Adjont Unsteady Dscrete Adjont equatons. he contnuous and dscrete unsteady adjont equatons have the same convergence rate, and the resduals reach machne zero wthn multgrd W-cycles. For unsteady transonc flows, as the arfol oscllates at a small angle of attack, the shock wave moves back and forth about a mean locaton and s closely snusodal and lags the arfol moton. hs lag s evdent n the lft hysteress loop, n whch the maxmum lft does not occur at the maxmum angle of attack. he nonlnear behavor of nvscd unsteady transonc flows s prmarly due to the movement of the shock. Accordng to McCroskey [25], however, the shock wave moton s greatest at moderate reduced frequences. He further clarfes that for small-ampltude oscllatons for reduced frequences approxmately below.5, large shock moton s seen. hs moton gradually reduces as the reduced frequency ncreases. It s also expected that at very low reduced frequences, the flow characterstcs are very smlar to that of steady-state computatons. Lft hysteress loops were computed for several dfferent reduced frequences at the same Mach number and are compared wth the steady lft curve n Fg. 4. As expected, at a reduced frequency of., the lft-curve slope s smlar to the steady soluton and the slope reduces as the reduced frequency ncreases. B. emporal Resoluton o quantfy the requred number of tme steps per perod, a temporal convergence study was performed. he code was run wth 5,, 23, 47, 95, and 9 tme steps per perod. At each tme step, the modfed resdual R w from Eq. (8) was drven to machne zero Log(Error) 8 cl error Multgrd Cycles Fg. 3 Convergence hstory of the unsteady flow solver and unsteady contnuous and dscrete adjont equatons; mesh; RAE 2822 arfol at M : Number of me Steps Per Perod Fg. 5 C lerror as a functon of the number of tme steps per perod usng a logarthmc vertcal scale.

8 NADARAJAH AND JAMESON 485 able 2 me-averaged coeffcent of lft as a functon of temporal resoluton. me steps per perod C l o guarantee temporal convergence, both the tme-averaged lft and drag coeffcents were montored. Generally, 6 tme perods were needed to elmnate errors due to ntal transents and to ensure a machne-zero convergence of the tme-averaged lft and drag. Fgure 5 llustrates the C lerror as a functon of the temporal resoluton. he case wth 9 tme steps per perod was used as the control soluton, and each pont on the fgure represents the dfference between the tme-averaged lft coeffcent of the control soluton and the current soluton. Fgure 5 llustrates that the error decay rate s proportonal to t 2. able 2 shows that 23 tme steps per perod s suffcent to obtan a lft coeffcent accurate to four decmal places. o perform optmum shape desgn for unsteady flows, t s mportant to access the requred temporal resoluton to obtan accurate senstvty of the objectve functon to the desgn varable. In ths work, the objectve functon s the tme-averaged drag coeffcent, and the desgn varables are the ponts on the surface of the arfol. Fgure 6 llustrates the L 2 -norm of the gradent error as a functon of temporal resoluton for both the contnuous and dscrete adjont approaches, usng the gradent computed wth 9 tme steps as the control soluton. he gradents of the two adjont approaches converge at the same rate. o ensure accurate gradents, the resduals at each tme step for both the flow and adjont solvers were drven to machne zero, and 6 tme perods were used for both solvers to ensure elmnaton of transent errors. he decay rate of the L 2 -norm of the gradent error s proportonal to t 2. he equvalent convergence decay rates between the gradent (Fg. 6) and tmeaveraged lft (Fg. 5) confrm the proper mplementaton of both the contnuous and dscrete adjont equatons. Fgure 7 shows the gradent at each pont on the surface of the arfol. he fgure llustrates that the gradent s almost dentcal to engneerng accuracy for a vast range of ponts, except n the regon of the shock wave between grd ponts 8 and. A closer nspecton of the data reveals that these ponts converge as the temporal resoluton s ncreased. hs s further llustrated n able 3, whch shows the convergence of the L 2 -norm of the gradent error. he gradent error for the case wth 23 tme steps per perod s approxmately e-4, provdng an equvalent level of accuracy to that observed for the tme-averaged lft. hus, 23 tme steps per perod s deemed suffcent for ths partcular test case. Fgure 8 shows a comparson of the dscrete and contnuous adjont gradents for the case wth 23 tme steps per perod. A closer error 2 3 Dscrete Contnuous Gradent.5. 5 SPP SPP 23 SPP 47 SPP 95 SPP Grd Pont Fg. 7 Comparson of dscrete unsteady adjont gradents for varous temporal resolutons. examnaton of the dscrete and contnuous adjont solutons reveals that the excellent gradent comparson s prmarly due to the comparable adjont solutons between the two approaches. A comparson of the second adjont varable along the wall s demonstrated n Fg. 9. In Fg., both the second and thrd adjont varables are shown. Values are plotted for ponts along a grd lne extendng from the arfol surface to the far-feld boundary. he two methods dffer n the manner that the adjont convectve and dsspatve fluxes are dscretzed, as well as n the enforcement of the boundary condton. In the case of the contnuous adjont approach, the boundary condton s computed for the cells below the wall along j ; however, n the case of the dscrete adjont method, the boundary condton appears as source terms added to the adjont convectve and dsspatve fluxes along the cells above the wall (j 2). herefore, the dscrete adjont approach does not requre values of adjont varables along j ; however, they are set to be equal to the values n j 2. he effect of ths dfference n the enforcement of the boundary condton can be observed n a close-up vew of the adjont values close to the boundary, as llustrated n Fg.. Despte the dfference n the boundary condtons and the slght dfference n the adjont soluton, the two approaches produce vrtually ndstngushable gradents, as demonstrated n Fg. 8. C. me-averaged Drag Mnmzaton of RAE 2822 Arfol wth Fxed me-averaged Lft Coeffcent Fgure llustrates the ntal and fnal geometry for the RAE 2822 arfol. he sold lne represents the ntal arfol geometry and the dashed lne llustrates the redesgned arfol. A dstnctve feature of the new arfol s n the drastc reducton of the upper-surface curvature. A reduced curvature leads to a weaker shock and thus a lower wave drag, but t also leads to a reducton n arfol camber, resultng n a loss n lft. o mantan the tme-averaged lft coeffcent C l, the mean angle of attack o s perturbed to a new value. he mpact of ths decson resulted n a need to compute more perods to allow the C l and C d to converge. In ths work, o was perturbed every three perods. hs allowed the C l to converge to a new value before the angle of attack 4 able 3 L 2 -norm of the gradent error as a functon of temporal resoluton Number of Samples Per Perod Fg. 6 Gradent error L 2 -norm as a functon of the number of tme steps per perod usng a logarthmc vertcal scale. me steps per perod L 2 -norm e e e e-5

9 486 NADARAJAH AND JAMESON Gradent.5.2. Dscrete Contnuous.4 Intal Fnal Grd Pont Fg. 8 Comparson of dscrete and contnuous unsteady adjont gradents for 23 tme steps per perod Fg. Intal and fnal geometres for the RAE 2822 arfol at M :78,! r :2, and o deg. E 3.5 Dscrete Contnuous.4.3 me Averaged Drag I Second Adjont Varable..5.2 me Averaged Drag E 4 E 5 I Grd Pont Fg. 9 Comparson of the second adjont varable at the wall for the dscrete and contnuous unsteady adjont methods for 23 tme steps per perod. Grd Number n the j Drecton Second Adjont Varable Dscrete Contnuous...2 hrd Adjont Varable Fg. Comparson of the second and thrd adjont varables for the dscrete and contnuous unsteady adjont methods for 23 tme steps per perod E 6 Fg. 2 Convergence of the tme-averaged drag coeffcent for the RAE 2822 arfol at M :78,! r :2, and o deg. was perturbed any further. A total of 5 perods were used for each desgn cycle. Fgure 2 llustrates the convergence rate of C d, whch s reduced by 43% from 32 drag counts to 75 drag counts wthn 2 desgn cycles. In Fg. 2, the change n the objectve functon or I, where I s the objectve functon, s also shown. Durng the frst 6 desgn cycles, I converges lnearly, as expected. Lnear convergence s characterstc of a steepest-descent-type method. As the fnal arfol profle s realzed, the convergence I ncreases rapdly. he code s automatcally stopped as soon as a change of E- 6 s detected. hs level of change corresponds to a change to the sxth decmal place of the drag coeffcent and ths s suffcent for engneerng accuracy. Fgure 3 llustrates the ntal and fnal pressure contours at a 8- deg phase. he sonc lne represented by a dashed lne s overplotted on each fgure. It s clearly vsble that the strong shock on the upper surface of the ntal geometry was consderably reduced. Fgures 4a 4d llustrate the upper and lower surface nstantaneous pressure coeffcents for the ntal and fnal desgn. In Fg. 4a, a comparson of the ntal nstantaneous pressure dstrbuton versus the fnal at a -deg phase shows an almost complete reducton of the wave drag. he strong shock on the sucton sde of the arfol s weakened at all other phases of the oscllaton. D. Multpont Versus Unsteady Optmzaton A multpont desgn approach was often the method of choce for optmzaton of arfols n an unsteady flow envronment, due to ts

10 NADARAJAH AND JAMESON 487 Fg. 3 Pressure contour plot for the RAE 2822 arfol at grd of 92 32, M :78,! r :2, and fxed C l :5. lower computatonal and memory cost. In ths subsecton of the paper, we make the argument that even f a multpont desgn approach s cheaper, t cannot replace a full unsteady optmzaton technque. hrough the use of fnal arfol profles, convergence hstores, and gradent comparsons, the followng results show that there are benefts to unsteady optmzaton. o compare the unsteady optmzaton approach to that of the multpont technque, unsteady desgn cases are computed at reduced.8.8 Pressure Coeffcent Pressure Coeffcent a) Phase = deg b) Phase = 9 deg Pressure Coeffcent Pressure Coeffcent c) Phase = 8 deg d) Phase = 27 deg Fg. 4 Intal and fnal pressure coeffcents at varous phases for the RAE 2822 arfol at M :78,! r :2, and o deg ( s ntal pressure and s fnal pressure).

11 488 NADARAJAH AND JAMESON frequences rangng from. to.45 at a Mach number of M :78. he tme-averaged lft coeffcent for all reduced frequences s fxed at C l :5. Lft hysteress loops for the varous reduced frequences are compared wth the steady lft curve n Fg able 4 Intal and fnal tme-averaged drag for varous desgn approaches Case Intal C d Fnal C d Reducton Multpont, mp % Multpont, mp % Multpont, mp % Full unsteady,! r : % Full unsteady,! r : % Full unsteady,! r : % Full unsteady,! r : % =. =.5 =.22 =.45 mp Fg. 5 Comparson of fnal arfol geometres between varous reduced frequences and the multpont approach at M :78, deg, and fxed C l :5. Y Coordnate mp 5 mp mp 23 =. =.5 =.22 = Fg. 6 Close-up vew of the fnal arfol geometry at M :78, deg, and fxed C l :5. me Averaged Drag Coeffcent, C d mp 5 mp mp 23 ω c =. ω c =.5 ω c =.22 ω c = Desgn Cycles Fg. 7 Convergence hstory of the tme-averaged drag coeffcent for varous desgn approaches for the RAE 2822 arfol at M :78, o deg, and fxed C l :5. Fgure 5 llustrates a comparson of fnal arfol geometres between arfols desgned usng the full unsteady optmzaton approach at varous reduced frequences and the multpont technque. he arfols are desgned at a Mach number of M :78, a mean angle of attack of o deg, and an angle of attack devaton of : deg. We show n Fg. 5 that the arfol desgned usng the multpont approach, usng fve tme steps per perod, desgnated as mp 5 s almost dentcal to the one desgned usng a full unsteady optmzaton approach at a reduced frequency of! r :. Fgure 6 demonstrates a close-up vew of the fnal upper surface between the and 75% chord locaton for varous desgn approaches. he fnal arfol based on the multpont approach usng varous number of tme steps per perod vrtually produces dentcal profles. As expected and seen n the prevous fgure, the unsteady desgn at a reduced frequency of. s very smlar to that of the multpont approaches. As the reduced frequency ncreases, the fnal arfol profle departs from that produced by the smple multpont approach. However, the dfferences are very small, except n areas on the upper surface, for whch a greater reducton n the curvature s seen for hgher reduced frequences. Fgure 7 further supports ths fact, wth almost dentcal convergence hstores of the objectve functon (tme-averaged drag) between the varous multpont approaches and the unsteady desgn at! r :. able 4 contans a comparson of the ntal and fnal tme-averaged drag counts for the varous desgn approaches performed at varous reduced frequences. he table further llustrates that for the four cases (mp 5, mp, mp 23, and! r :), the ntal tme-averaged drag s approxmately 47 drag counts and the fnal drag s counts. As the reduced frequency ncreases, the ntal drag reduces by a small amount due to nonlnear effects prmarly due to the movement of the shock; however, the fnal drag count reduces substantally. herefore, the three unsteady cases showcased n able 4 and Fgs. 5 and 7 represent the lower, mddle, and upper lmts of the reduced-frequency range. Fgure 8 demonstrates the change n I, where I s the objectve functon and I converges lnearly untl the fnal arfol profle s realzed and the convergence accelerates. All desgn cases are automatcally halted once I reaches E-6. As llustrated n both Fgs. 6 and 7, the convergence of the unsteady case at! r : s very smlar to the multpont cases. As the reduced frequency ncreases, the number of desgn cycles requred to obtan the fnal desgn ncreases. We conjecture that ths s due to the level of unsteadness of the problem. In Fgs. 9 and 2, we demonstrate the comparson of the gradents between the adjont-based multpont and unsteady approaches. he gradents are plotted n a clockwse drecton from the lower tralng edge to the upper tralng edge. Fgure 9 llustrates that the gradents between the multpont approach usng 23 tme steps per perod are very smlar to the unsteady desgn approach for varous reduced frequences for a majorty of the grd ponts, except for grd ponts between 8 and, as shown n Fg. 2. hs range of grd ponts concdes wth the shock footprnt and produces the domnant gradents n ths desgn problem. he fgure llustrates once agan that the gradents produced by the unsteady test case at a reduced frequency of! r : are smlar to that produced by the multpont approach. At hgher reduced frequences, the gradents dffer greatly n ths range of grd ponts and ths s largely due to the ncrease n the nonlnear behavor of the unsteady transonc flow. In summary, Fgs. 5 2 demonstrate that at very low reduced frequences, the desgn convergence hstores, fnal arfol profles,

12 NADARAJAH AND JAMESON =. ω =.5 r =.22 4 mp 5 4 =.45 I mp mp 23 Error 6 5 ω c =. ω =.5 c 8 ω c =.22 ω c = Desgn Cycles Fg. 8 Convergence of I for varous desgn approaches for the RAE 2822 arfol at M :78, o deg, and fxed C l : Number of me Perods Fg. 2 Convergence of tme-averaged lft coeffcent for varous reduced frequences. Gradent mp 23 =. =.5 =.22 = Grd Pont Fg. 9 Comparson between unsteady dscrete adjont gradents for varous reduced frequences and the multpont approach for the RAE 2822 arfol at M :78, o deg, and fxed C l :5. Gradent mp 5 mp mp 23 =. =.5 =.22 = Grd Pont Fg. 2 Close-up vew of the comparson between unsteady dscrete adjont gradents for varous reduced frequences and the multpont approach for the RAE 2822 arfol at M :78, o deg, and fxed C l :5. and gradents are very smlar to that produced by a multpont desgn approach. At moderate reduced frequences between.5 and.45, however, these trends begn to devate. he greater the nonlnear behavor of the unsteady transonc flow, the larger the dfference from the multpont approach. Fgure 2 further supports ths observaton by llustratng the addtonal number of tme perods requred for the hgher reduced frequency cases to obtan a converged tme-averaged drag coeffcent to machne accuracy. he addtonal perods are requred to dmnsh the transent solutons due to the ncrease n the nonlnear behavor, to arrve at a perodc steady-state soluton. he fgure demonstrates that only four perods are requred to obtan a converged drag coeffcent at a reduced frequency of.; however, 6 perods were requred for the.2 case and for the.45 case. Ultmately, to compare the behavor of the fnal arfol desgned by the two approaches, the arfol desgned usng the multpont approach was computed usng the tme-accurate flow solver for the three moderate reduced frequences. able 5 lsts the fnal tmeaveraged drag coeffcents computed at a Mach number of.78 and at a fxed lft coeffcent of.5. At! r :5, the dfference between the unsteady and multpont approaches s.2%. At a reduced frequency of.2 and.45, the dfference grows to 6.4 and 8.3%, resultng n lower drag coeffcents for the arfols desgned usng the unsteady approach. Fgure 22 llustrates a comparson of the pressure dstrbuton at a reduced frequency of.45 between the arfols desgned usng the multpont and unsteady approaches. It s clearly seen that the 8.3% mprovement n the drag coeffcent for the unsteady technque s due to the weaker shock on the upper surface of the arfol. he multpont approach certanly provdes an nexpensve alternatve to the unsteady desgn technque and has produced an optmzed arfol wth a lower tme-averaged drag coeffcent over a range of reduced frequences. However, f an arfol, turbne, or rotor blade s desgned for a specfc range of reduced frequences, the adjont-based unsteady optmzaton technque may provde an added beneft. VII. Conclusons hs paper presents a complete formulaton of the contnuous and dscrete unsteady nvscd adjont approaches to automatc able 5 Comparson of C d between the multpont and unsteady desgn approaches Case Unsteady Multpont Improvement! r : %! r : %! r : %

13 49 NADARAJAH AND JAMESON.8.8 Pressure Coeffcent Pressure Coeffcent a) Phase = deg b) Phase = 9 deg Pressure Coeffcent Pressure Coeffcent c) Phase = 8 deg d) Phase = 27 deg Fg. 22 Comparson of pressure dstrbutons between arfols desgned va multpont and unsteady approaches at varous phases for the RAE 2822 arfol at M :78,! r :45, and o deg ( s unsteady and s multpont). aerodynamc desgn. A 46% reducton n the tme-averaged drag coeffcent was acheved for the RAE 2822 arfol at a reduced frequency of.45 whle mantanng the tme-averaged lft coeffcent. A comparson between fnal arfol profles, convergence hstores of the objectve functon, and gradents demonstrate that at low reduced frequences, there s no added beneft of performng aerodynamc shape optmzaton for unsteady flows va an adjontbased unsteady optmzaton technque; however, at moderate reduced frequences, an unsteady optmzaton technque produced fnal arfol profles wth tme-averaged drag coeffcents between 6.4 to 8.3% mprovement over the multpont approach. he framework was establshed to extend ths method to vscous domnated flows n whch secondary flow effects are present. Acknowledgments hs research benefted greatly from the generous support of the U.S. Ar Force Offce of Scentfc Research (AFOSR) under grant number AF F and the Department of Energy under contract number LLNL B3449 as part of the Accelerated Strategc Computng Intatve (ASCI) program. References [] Jameson, A., Computatonal Aerodynamcs for Arcraft Desgn, Scence, Vol. 245, No. 496, July 989, pp [2] Reuther, J., Clff, S., Hcks, R., and van Dam, C. P., Practcal Desgn Optmzaton of Wng/Body Confguratons Usng the Euler Equatons, AIAA Paper , 992. [3] Gallman, J., Reuther, J., Pfeffer, N., Forrest, W., and Bernstorf, D., Busness Jet Wng Desgn Usng Aerodynamc Shape Optmzaton, 34th Aerospace Scences Meetng and Exhbt, Reno, NV, AIAA Paper , 996. [4] Reuther, J., Alonso, J. J., Martns, J. R. R. A., and Smth, S. C., A Coupled Aero-Structural Optmzaton Method for Complete Arcraft Confguratons, 37th Aerospace Scences Meetng and Exhbt, Reno, NV, AIAA Paper 99-87, 999. [5] Kasdt, L., and Jameson, A., Case Studes n Aero-Structural Wng Planform and Secton Optmzaton, 22nd AIAA Appled Aerodynamcs Conference, Provdence, RI, AIAA Paper , 24. [6] Nadarajah, S., Jameson, A., and Alonso, J. J., An Adjont Method for the Calculaton of Remote Senstvtes n Supersonc Flow, 4th Aerospace Scences Meetng and Exhbt, Reno, NV, AIAA Paper 22-26, 22. [7] Mavrpls, D., Multgrd Soluton of the Dscrete Adjont for Optmzaton Problems on Unstructured Meshes, AIAA Journal, Vol. 44, Jan. 26, pp [8] Soto, O., Lohner, R., and Yang, C., An Adjont-Based Desgn Methodology for CFD Problems, Internatonal Journal of Numercal Methods for Heat and Flud Flow, Vol. 4, No. 6, 24, pp [9] Gles, M. B., Duta, M., Muller, J., and Perce, N., Algorthm Developments for Dscrete Adjont Methods, AIAA Journal, Vol. 5, No. 5, 23, pp [] Nemec, M., ngg, D., and Pullam,. H., Multpont and Mult- Objectve Aerodynamc Shape Optmzaton, AIAA Journal, Vol. 42, No. 6, 24, pp