Actuator Disc Model Using a Modified Rhie-Chow/SIMPLE Pressure Correction Algorithm Rethore, Pierre-Elouan; Sørensen, Niels

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1 Aalborg Univrsitt Actuator Disc Modl Using a Modifid Rhi-Chow/SIMLE rssur Corrction Algorithm Rthor, irr-elouan; Sørnsn, Nils ublishd in: EWEC 2008 Confrnc rocdings ublication dat: 2008 Documnt Vrsion ublishr's DF, also known as Vrsion of rcord Link to publication from Aalborg Univrsity Citation for publishd vrsion (AA): Rthor, -E., & Sørnsn, N. (2008). Actuator Disc Modl Using a Modifid Rhi-Chow/SIMLE rssur Corrction Algorithm: Comparison with Analytical Solutions. In EWEC 2008 Confrnc rocdings Th Europan Wind Enrgy Association. Gnral rights Copyright and moral rights for th publications mad accssibl in th public portal ar rtaind by th authors and/or othr copyright ownrs and it is a condition of accssing publications that usrs rcognis and abid by th lgal rquirmnts associatd with ths rights.? Usrs may download and print on copy of any publication from th public portal for th purpos of privat study or rsarch.? You may not furthr distribut th matrial or us it for any profit-making activity or commrcial gain? You may frly distribut th URL idntifying th publication in th public portal? Tak down policy If you bliv that this documnt brachs copyright plas contact us at vbn@aub.aau.dk providing dtails, and w will rmov accss to th work immdiatly and invstigat your claim. Downloadd from vbn.aau.dk on: juli 26, 208

2 Actuator disc modl using a modifid Rhi-Chow/SIMLE prssur corrction algorithm. Comparison with analytical solutions irr-elouan Réthoré,2,3, Nils N. Sørnsn,2 Wind Enrgy Dpartmnt, Risø National Laboratory for Sustainabl Enrgy, DTU - Tchnical Univrsity of Dnmark, DK-4000 Roskild, Dnmark 2 Dpartmnt of Civil Enginring, Aalborg Univrsity, Sohngaardsholmsvj 57, DK-9000 Aalborg, Dnmark 3 pirr-louan.rthor@riso.dk Abstract: An actuator disc modl for th flow solvr EllipSys (2D&3D) is proposd. It is basd on a corrction of th Rhi-Chow algorithm for using discrt body forcs in collocatd variabl finit volum CFD cod. It is compard with thr cass whr an analytical solution is known. Ky words: CFD, acuator disc, Rhi-Chow, SIMLE, numrical wiggls, wind turbin, wak Introduction Th flow going through a wind turbin can b modld using Computational Fluid Dynamics (CFD) by solving th Navir Stoks quations. Th influnc of th turbin in th quation can b implmntd as a body forc acting against th flow. Th Navir Stoks quations ar ssntially composd of vlocity trms, of prssur gradint trms and of body forcs. Whn discrtizd ovr a msh using a finit volum mthod, if th prssur and vlocity trms ar rprsntd at th sam plac, a dcoupling might occur that can lad to numrical oscillations of th prssur (or prssur wiggls). On way of daling with this issu is to kp th vlocitis at th cll facs and th prssur trms at th cll cntrs, so that th prssur gradint trms, drivd from thm, ar locatd at th sam plac as th vlocity trms (.g. th cll cntrs). This mthod is calld th staggrd grid mthod. Th othr standard way is to kp th prssur and vlocity trms at th cll cntrs (known as th collocatd variabl mthod), and uss a spcial tratmnt of th prssur, to avoid th prssur/vlocity dcoupling. This mthod, which was first introducd by Rhi-Chow [], was nvr intndd to tak car of th prssur/vlocity dcoupling introducd by inputting a suddn prssur jumps, or discrt body forcs. EllipSys, th in-hous curvilinar CFD cod dsignd at Risø National Laboratory for Sustainabl Enrgy (Risø-DTU) [2] and th Fluid Mchanics dpartmnt of th Tchnical Univrsity of Dnmark (MEK-DTU) [3] is basd on a collocatd variabl arrangmnt using th Rhi-Chow prssur corrction algorithm. Discrt body forcs ar, in th prsnt contxt, usd in ordr to modl th influnc of wind turbins on th flow. In ordr to ovrcom th prssur wiggls introducd by discrt body forcs, on approach is to smooth th body forcs out by using a Gaussian distribution instad of a Dirac dlta distribution [4]. This mthod rquirs that th prssur jump is mad ovr svral clls which can bcom computationally xpnsiv on larg problm lik simulating a wind turbin farm. In ordr to sav computational tim, a modification of th Rhi-Chow algorithm is proposd to trat th spcial cas of discrt prssur jumps. In th prsnt papr, an algorithm to discrtiz an actuator disc ovr a msh is brifly introducd. Scondly, th problm of th prssur wiggls is prsntd for a D xampl of a rgular Cartsian msh with a spcial cas of uniform vlocity ovr th domain. Th proposd algorithm is thn dscribd in th contxt of th curvilinar CFD cod EllipSys3D. For th sak of clarity, th sam notations usd in

3 y WW W w E EE x Figur : D msh th original thsis dscribing EllipSys ar usd (s Sørnsn [2]). Finally thr diffrnt tst cass, for which an analytical solution is known, ar prsntd and compard with Ellipsys rsults. Dscription of th algorithm. Forc allocation Th basic ida of th forc allocation algorithm usd is to, first, sarch for all th clls which ar crossd by th disc. Thn, in ordr to dtrmin th quivalnt body forc that will b allocatd to th cll, to calculat th intrsctional surfac btwn th disc and ach cll, and to intgrat th forc distribution of th actuator disc ovr it. Finally, to apply th prssur jump corrction, dscribd in th following sction, to rdistribut th forcs ovr th nighboring clls, and to driv th corrsponding cll facs prssur jumps. Th Navir Stoks quations ar thn solvd using th body forcs in th momntum quation, and using th prssur jumps in th Rhi-Chow algorithm..2 rssur jump corrction In ordr to undrstand th ncssity of th th prssur jump corrction, a simpl D cas, with uniform vlocity flow is usd. Basd on this xampl, th principl of th Rhi-Chow algorithm is brifly introducd. Thn, prssur wiggls ar shown to b prsnt whn discrt body forcs ar applid. Finally, th basic ida bhind th prssur jump corrction is prsntd, and its application to th CFD cod EllipSys is dscribd. Origin of prssur wiggls Th D Navir Stoks quations can b writtn as ρu t whr F is a volumic forc [N/m 3 ]. + ρuu = + In ordr to hav an quation for th prssur, th Continuity quation is usd ( µ U ) + F, () ρu = 0. (2) In th finit volum formulation, drivativs can b discrtizd by intgrating thm ovr a control volums. Using a CDS schm ovr th D msh prsntd in Figur, th following ruls can b applid. Ò Ψ dxdy ΨΨ dxdy ( ) Ψ dxdy = = (Ψ Ψ w ) y = (Ψ E Ψ W ) y 2, (3) = (Ψ Ψ Ψ wψ w ) y = (Ψ E (Ψ E + Ψ ) Ψ W(Ψ W + Ψ ) + Ψ (Ψ E Ψ W [( ) ( ) Ψ Ψ w )) y 4, (4) ] y = (Ψ W + Ψ E 2Ψ ) y x. (5)

4 whr th indicats that th trm is known from a prvious tim stp. Applying ths ruls on th Continuity quation (2) givs which thrfor lads to a scond rlationship, y (ρu ρu w ) = 0, (6) U W = U E. (7) Assuming a stady flow and thrfor dropping th unstady trm of Equation () givs ρ [(UU ) (UU ) w ] y 4 = ( W E ) y 2 + µ (U W + U E 2U ) y x + F x y. (8) Equation (8) can b rwrittn into a gnral formulation by linarizing and assuming that on of th U s is known in th convctiv trm (UU trm) A U = A nb U nb + ( W E ) y 2 + F x y, (9) with nb (W, E), and A W = µ y x + ρ y 4 (U W + U ), A E = µ y x ρ y 4 (U E + U ), Ò A = 2µ y x + ρ y 4 (U E U W). (0) Not that Continuity stats that ρ yu W = ρ yu E, and so thrfor A = A W + AE. Th cll cntr vlocity obtaind from th Navir Stoks quations (9) can b intrpolatd, using th midpoint rul, at th cll fac in ordr to apply th Continuity U = 2 (U + U E ). () A simpl cas can b applid, whr th vlocity is assumd to b uniform in th domain (i.. U WW = U W = U = U E = U EE ). Combind with th assumption of a rgular Cartsian msh, all th vlocity trms and cofficints A ar vntually cancling ach othr. Insrting Equation (9) into Equation (7) thn givs ( WW ) y 2 + F W x y = ( EE ) y 2 + F E x y WW 2 + EE = (F E F W ) x. (2) If thr ar no body forcs in th domain, th rlationship btwn th prssur at ach cll bcoms = 2 ( WW + EE ). (3) As th prssur of a cll is not dpndnt of its dirct nighboring clls prssur, this rlationship can b satisfid by a prssur wiggl solution (.g ) Th Rhi-Chow algorithm Th Rhi-Chow algorithm is adrssing this issu by sparating th prssur trms from th rst of th momntum trms, whn th fac vlocitis ar drivd. Instad of intrpolating th prssur gradint at th cll facs using th prssur gradints at th clls cntr, thy ar dirctly drivd from th prssur at th closst clls cntr. U = 2 (Ũ + ŨE ) + y A ( E ), (4)

5 rssur 2F y WW W w E EE x Figur 2: rssur jump with wiggls whr Ũ = A ( A nb U nb + F x y ), Ò A = 2 (A + A E ). (5) Insrting Equation (4) into th Continuity Equation (2) thn brings a diffrnt rlation (ŨE ŨW) + y ( E ) y ( W ) = 0. (6) 2 A A w Applying th sam simpl cas of uniform vlocity in th ntir domain, with a rgular Cartsian msh, all th vlocity trms cancl ach othr, which brings a rlationship btwn th prssur and th body forcs 2 (F E F W ) x y + y (2 W E ) = 0. (7) If thr ar no body forcs in th domain, thn th prssur in th cll is rlatd with its dirct nighboring clls prssur. In this cas, th oscillations prssur fild is not a solution. = 2 ( W + E ). (8) Applying a discrt forc or a prssur jump Howvr, if thr is a discrt forc F applid in th cll, alrady from Equation (9), thr is a problm. If th vlocity is th sam ovr th domain, thn all th vlocity trms ar cancling ach othr which givs a rlationship btwn th body forc and th prssur. E W = 2F x. (9) Similarly, applying Equation (??) on th cll W and E shows that thr is a prssur wiggl solution WW = 2F W x = 0 Ò EE = 2F E x = 0. (20) So vn using th Rhi-Chow corrction, applying a suddn prssur jump into this schm causs th apparanc of numrical prssur wiggls (s Figur 2). Basic ida of th modification In ordr to corrctly handl discrt forcs in th Navir Stoks and th Continuity quations, th forcs ar dfind at th fac of th clls in th sam way th prssur gradint trms ar. Th body forc in th cll is splittd into two prssur jumps: on on th wst fac j w and on on th ast fac j rspctivly. Equation (9) can thn b rwrittn as A U = A nb U nb + ( W E ) y 2 + ( j w + j ) y 2. (2)

6 rssur 2F y WW W w E EE x Figur 3: rssur jump without wiggls Th prssur jumps ar thn tratd in th sam way as th prssur gradints during th drivation of th fac vlocity whr U = 2 ( U + U E ) + y A ( E ) + j A y, (22) U = A nb U nb, Ò A = 2 (A + A E ). (23) Th Continuity quation (6) is thn giving ( ) y UE U W + ( E ) y ( W ) + y j 2 A A w A y w j = 0. (24) A w Applying th sam simpl cas (uniform vlocity and rgular Cartsian msh) brings a rlationship btwn th prssur and th body forcs. Th Continuity, in Equation (24), thn givs W + E 2 = j j w. (25) Furthrmor, by cancling th vlocity trms, Equation (2) givs Finally, combining Equation (25) and (26) givs E W = j w + j. (26) W = j w Ò E = j, (27) which is th corrct rsult without prssur wiggls (s Figur 3). Implmntation in EllipSys Th SIMLE algorithm [5] of EllipSys is using th prdictd vlocity, obtaind from th Navir Stoks quations, in ordr to find th prssur, through th Continuity quation. This prssur is usd to corrct th prdictd vlocity so that it complis with th Continuity quation (it is thn not complying with th Navir Stoks). Th itration gos on until th vlocity convrgs to a solution that satisfis both th Navir Stoks quations and th Continuity quation. Th Continuity quation can b xprssd using th divrgnc oprator ρu = 0. (28) Using th notation of Sørnsn [2]-Eq.28 for a curvilinar grid, this quation can b rwrittn as J (ρuα ξx + ρv α ξy + ρwα ξz ) ξ + J (ρuα ηx + ρv α ηy + ρwα ηz ) η + J (ρuα ζx + ρv α ζy + ρwα ζz ) ζ = 0, (29)

7 whr J is th Jacobian of th curvilinar to Cartsian transformation matrix, and th α s ar diffrntial aras of th cll facs projctd in th Cartsian coordinats. Equation (29) can b writtn in a mor compact way to [2]-Eq.7 whr C = ρ U (α ξx ) + ρ V (α ξy ) + ρ W (α ξz ). J [(C C w ) + (C n C s ) + (C t C b )] = 0, (30) Th prdictd vlocity, drivd from th Navir Stoks quations, is composd of implicit trms (A nb U nb ) and an xplicit trms (S U mom ) [2]-Eq.65. U = S U mom A nb U nb A,U, (3) whr th xplicit trms (S U mom ) contain th cross diffusion trms, th prssur trms and th body forcs. In ordr to apply th Continuity, it is ncssary to find th vlocity at th cll facs. Th usual collocatd approach is to intrpolat th vlocity U at th cll facs. This lads to th prssur wiggls, as xplaind prviously. Th ida of th Rhi-Chow algorithm is to sparat th prssur gradint trms from th rst, and to dirctly stimat it at th cll fac. ) (( ) ( ) ( ) αξx αηx αζx ( = J ξ + η + ζ ). (32) Th normal gradints ar dirctly computd using a scond ordr accurat cntral diffrnc schm [2]-Eq.42. ( ) αξx = ( E )(α ξx ). (33) ξ Th cross-trm gradints ar computd as th intrpolation btwn two cntral diffrncs [2]- Eq.43. ( ) αηx = η 4 [( N S ) + ( NE SE )](α ηx ) (34) ( ) αζx = ζ 4 [( T B ) + ( TE BE )](α ζx ). (35) Thrfor, instad of intrpolating dirctly (3), th prssur gradint is stimatd at th cll facs [2]-Eq.69 U = ( SU mom ) A nb U nb A,U ( ) ( + (α ξx ) ( E ) A + 4 (α ηx) [( N S ) + ( NE SE )] + ) 4 (α ζx) [( T B ) + ( TE BE )], (36) whr th first trm in th Right Hand Sid (RHS) is th linar intrpolation at th cll fac of all th momntum trms xcpt th prssur gradint trms. In th modification of th Rhi-Chow algorithm, th body forcs ar also xtractd from th momntum trms. Thy ar thn transformd into prssur jumps locatd at ach cll facs, in a similar mannr as proposd by Mncingr and Zun [6].

8 U = ( SŨ mom ) A nb U nb A,U + 4 (α ηx) [( N S ) + ( NE SE )] ( ) ( + (α ξx ) ( E ) A + 4 (α ζx) [( T B ) + ( TE BE )] + j, x ), (37) whr, j x is th prssur jump at th ast cll fac in th x dirction, and S Ũ mom is now th momntum sourc without th prssur trms and th body forcs. In ordr to b consistnt with th th original body forc applid in th cll, th prssur jump nds to satisfy th following proprty. F dv = n j ds. (38) V This rlationship can b projctd on th Cartsian coordinat systm and discrtizd ovr th currnt cll. For th x-dirction, this corrsponds to A F, x V = nb n nb, x S nb j nb, x, (39) whr nb ar th nighboring facs, S is th fac surfac ara and V th cll volum, n nb, x is th normal vctor of th fac nb in th x dirction. On solution to this rlationship is to wight ach facs accordingly to its normal vctor and fac surfac ara. Th following rlationship is complying with Equation (39) j nb, x = F, x V n nb, x S nb nb (n nb, x S nb ) 2. (40) Th prssur jump contributions from th two clls adjacnt facs ar addd up. Th final prssur jump can thn b usd dirctly in Equation (37) j, x = F, x V n, x S nb, (n nb, x S nb ) 2 + F E, x V E n, x S nb, E (n nb, x S nb ) 2. (4) Finally, th forcs usd in th Navir Stoks quations ar rcomputd at th cll cntr using th fac prssur jumps and dividd by two, so that ach nighboring clls carry out th prssur jump qually. F, x V = n nb, x S nb j nb, x (42) 2 nb Thrfor, th nw F, x is not xactly th sam as th original F, x. In practic th forc has bn smard ovr th narst nighboring clls, so that th jump of prssur, corrsponding to th body forc, is occurring at th cll facs. Th nw fac vlocity can b usd in th fac mass flux cofficints from Equation (30), which is thn usd to comput th prssur and to corrct th vlocity, in ordr to satisfy Continuity. 2 Analytical validation In ordr to study th validity of th actuator disc modl, and th impact of th prssur jump corrction, thr cass, whr th analytical formula ar known, ar compard with th modl, with and without th prssur jump corrction. Th modl is both implmntd in 2D and in 3D. As th rsults of th 2D cod and th 3D cod ar giving similar rsults within th convrgnc prcision rquird, only th 3D rsults ar prsntd hr.

9 Symmtry BC Symmtry BC Inlt BC F Outlt BC Inlt BC F Outlt BC Symmtry BC Symmtry BC 2D cas 3D cas Figur 4: 2D infinit lin.02 u U [-] D with corrction D without corrction Analytical solution Normalizd prssur [-] x D [-] Figur 5: Infinit lin/plan cas 2. 2D Infinit lin and 3D Infinit plan Th first cas studid is a channl flow similar to th xampl prsntd in th prvious sctions. Th boundary conditions ar takn to b symmtric on th sid, so that no xpansion is possibl. An homognous forc, opposd to th flow dirction, is applid along a lin (in 2D) or a plan (in 3D) (s Figur 4. This stup insurs that th flow dirction rmains D and constant bcaus of Continuity. Only th prssur is xpctd to vary along th domain, incrasing in a discrt mannr from on sid to th othr of th lin/plan, as it was dscribd in th prvious sctions. Th rsults from th EllipSys (s Figur 5) ar in agrmnt with th thory prsntd in th prvious sctions. Using th uncorrctd algorithm, th prssur prsnts som wiggls, visibly dampd aftr 5-6 clls both bfor and aftr th jump. Th vlocity is also prsnting wiggls on th sam clls whr th prssur is fluctuating. Using th corrctd algorithm, th prssur follows a clan jump carrid ovr thr clls, in good agrmnt with th analytical solution. Thr ar no visibl wiggls on th prssur, nor on th vlocity rsults D actuator strip and 3D actuator infinit ribbon Th scond cas is a 2D actuator strip undr a rctangular inflow profil. In ordr to modl it in 3D, th top and bottom facs of th domain ar takn as symmtric boundary condition, whil th north

10 Farfild BC Symmtry BC Inlt BC F Outlt BC Inlt BC F Farfild BC Outlt BC Farfild BC 2D cas Farfild BC Symmtry BC 3D cas Figur 6: 2D actuator strip and south facs ar takn as farfild boundary conditions. Th actuator strip is thn rprsntd as an infinitly long ribbon of homognous forc going through th domain from top to bottom (s Figur 6). Th analytical solution for lightly loadd actuator strip, drivd by Madsn [7], is usd as followd. p(x, y, p, R) = p ( ( ) ( )) R y R + y tan + tan (43) 2π x x v x (x, y, p, R) = u p(x, y, θ, p, R) ρu p ρu }{{} ÓÒÐÝ Ò Ø Û Th assumptions mad to driv this quation ar only valid for a vry lightly loadd actuator disc (C T ). For a vry lightly loadd actuator strip (C T = 0.0), th numrical rsult, using th corrction, is in good agrmnt to th analytical solution (s Figur 7). Similarly to th prvious cas, th numrical rsult without th corrction is prsnting quit important vlocity and prssur wiggls in th axial dirction, both bfor and aftr th position of th actuator strip. Howvr, thr ar no visibl wiggls in th radial dirction D actuator disc with axis symmtry Finally th cas of an actuator disc in 3D is studid. In ordr to modl th flow appropriatly, th boundary condition on th sid facs (south, north, top, bottom) ar all takn as farfild (s Figur 8). Th analytical solution for an axis symmtric lightly loadd actuator disc in cylindrical coordinats, drivd by Koning [8], can b numrically intgratd using th following quations. p(x, r, θ, p, R) = p 4π R 2π v x (x, r, θ, p, R) = u 0 0 (44) r x [r 2 + r 2 + x 2 2r r cos(θ θ)] 3/2 dr dθ (45) p(x, r, θ, p, R) ρu p ρu }{{} ÓÒÐÝ Ò Ø Û Th assumptions mad to driv this quation ar only valid for a vry lightly loadd actuator disc (C T ). For a vry lightly loadd actuator disc (C T = 0.0), th numrical rsult, using th corrction, is also in good agrmnt to th analytical solution. Th bhavior of th numrical rsult, without th corrction, looks vry similar to th prvious actuator strip cas. Vlocity and prssur wiggls ar clarly visibl both bfor and aftr th position of th body forcs, in th axial dirction, but not in th radial dirction. It is intrsting to not that th wiggls do not sm to affct th ovrall solution. Thy only giv an rror at th local position of th actuator disc. It is nonthlss rathr intrsting to obtain a corrct vlocity and prssur at th actuator disc position, as this information can b usd, for xampl, to dtrmin th nrgy xtraction of th wind turbin modld by th actuator disc. For a mor havily loadd actuator disc (C T = 0.89), a similar convrgnc is achivabl, but it is irrlvant to compar it with th analytical solution. (46)

11 u [-] U x-dirction, y = 0.0D y-dirction, x =.0D 3D with corrction 3D without corrction 2D analytical solution p (ct ρu 2 ) [-] x D [-] y D [-] Figur 7: Infinitly long actuator ribbon (3D) compard with a 2D analytical solution for an actuator strip 3 Conclusion An actuator disc modl using a corrction of th Rhi-Chow algorithm for th usag of discrt body forcs is prsntd. Th corrctd algorithm show a clar improvmnt of th solution around th location whr th body forcs ar applid. This altrnativ way of trating body forcs as prssur jump can rduc significantly th numbr of clls ndd to modl a prssur jump. In th prsnt rsarch contxt, this can potntially lad to chapr modling of wind turbin nar wak rgion, and thrfor opning th possibility to modl largr wind farms. Th rsults from th actuator disc modl show a good agrmnt with analytical solutions, for som lightly loadd cass. 4 Acknowldgmnt Th currnt hd projct is supportd by th Danish ublic Srvic Obligation (SO) rsarch projct Flaskhals. Farfild BC Inlt BC F Outlt BC Farfild BC Figur 8: 3D actuator disc

12 u [-] U x-dirction, y = 0.0D y-dirction, x =.0D 3D without corrction 3D with corrction Analytical axisymmtric solution p (ct ρu 2 ) [-] x D [-] y D [-] Figur 9: Actuator disc (3D) compard with an analytical axisymmtric solution Rfrncs [] C.M. Rhi and W.L. Chow. Numrical study of th turbulnt fow past an airfoil with trailing dg sparation. AIAA Journal, 2: , 983. [2] N.N. Sørnsn. Gnral urpos Flow Solvr Applid to Flow ovr Hills. hd thsis, Tchnical Univrsity of Dnmark, 994. [3] J.A. Michlsn. Basis3d - a platform for dvlopmnt of multiblock pd solvrs. Tchnical rport afm 92-05, Tchnical Univrsity of Dnmark, Lyngby, 992. [4] R. Mikklsn. Actuator Disc Mthods Applid to Wind Turbins. hd thsis, Tchnical Univrsity of Dnmark, Mk dpt, [5] S.V. atankar and D.B. Spalding. A calculation procdur for hat, mass and momntum transfr in thr-dimnsinoal parabolic flows. Int. J. Hat Mass Transfr, 5: , 972. [6] J. Mncingr and I. Zun. On th finit volum discrtization of discontinuous body forc fild on collocatd grid: Application to vof mthod. Journal of Computational hysics, 22: , [7] H.A. Madsn. Application of actuator surfac thory on wind turbins. IEA R&D WECS, Joint action on Arodynamics of wind turbins, Lyngby, Dnmark, nov 988. [8] C. Koning. Arodynamic thory: A gnral rviw of progrss - Vol.IV. Division M, Influnc of th propllr on othr parts of th airplan structur, p.366. tr Smith, 976.

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