DYNSTALL: Subroutine Package with a Dynamic stall model

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1 FFAP-V-110 DYNSTALL: Subroutie Package with a Dyamic stall model Aders Björck Steady data Experimet Simulatio FLYGTEKNISKA FÖRSÖKSANSTALTEN THE AERONAUTIAL RESEARH INSTITUTE OF SWEDEN

2 FLYGTEKNISKA FÖRSÖKSANSTALTEN THE AERONAUTIAL RESEARH INSTITUTE OF SWEDEN Box 11021, SE Bromma, Swede Phoe , Fax

3 Summary A subroutie package, called DYNSTALL, or the calculatio o 2D usteady airoil aerodyamics is described. The subrouties are writte i FORTRAN. DYNSTALL is a basically a implemetatio o the Beddoes- Leishma dyamic stall model. This model is a semi-empirical model or dyamic stall. It icludes, however, also models or attached low usteady aerodyamics. It is complete i the sese that it treats attached low as well as separated low. Semi-empirical meas that the model relies o empirically determied costats. Semi because the costats are costats i equatios with some physical iterpretatio. It requires the iput o 2D airoil aerodyamic data via tables as uctio o agle o attack. The method is iteded or use i a aeroelastic code with the aerodyamics solved by blade/elemet method. DYNSTALL was writte to work or ay 2D agles o attack relative to the airoil, e.g. low rom the rear o a airoil. 3

4 4

5 otet 1 INTRODUTION THE BEDDOES-LEISHMAN MODEL ATTAHED FLOW SEPARATED FLOW THE MODEL IN STEPS SEMI-EMPIRIAL ONSTANTS THE DYNSTALL UNSTEADY AIRFOIL AERODYNAMIS MODEL A HANGE TO WORK IN THE WIND REFERENE SYSTEM ATTAHED FLOW The agle o attack Attached low equatios (LPOTMETH=3) Shed wake eects with accout or a varyig lit curve slope (LPOTMETH=4) SEPARATED FLOW A shit i the agle o attack Lit as uctio o the separatio poit positio The Øye separatio poit model (lmeth=2) A versio o the Kircho low (lmeth=4) Vortex lit (lvormeth=2) A alterative versio o vortex lit (lvormeth=1) UNSTEADY DRAG UNSTEADY PITHING MOMENT WAYS TO MAKE THE MODEL WORK FOR ALL ANGLES OF ATTAK ROBUSTNESS OF THE DYNSTALL-METHODS Methods that work or all agles o attack INPUT TO THE FORTRAN SUBROUTINE PAKAGE SUBROUTINES ONVENTIONS FOR VARIABLE NAMES ABOUT HISTORY OF VARIABLE NAMES OMMUNIATION WITH THE SUBROUTINES OMMON AREAS IN DYNL_.IN Storage o static airoil data Rage o agle o attack i iput airoil tables Lit curve slope ad zero lit agle o attack hoices betwee submodels oeiciets ad time costats Variables used or state variables ad other variables eeded or iteral commuicatio

6 4.5.5 Variables i commo areas that must be set at each time step outside o the DYNSTALL-package GENERATION OF FNSTIN, NINVIN AND NSEPIN TABLES SUB-METHODS AND VALUES FOR SEMI-EMPIRI ONSTANTS AN OPTIMIZATION STUDY TO FIND OPTIMUM VALUES LDDYN EFFETS OF VARYING VELOITY ON THE SEPARATION POINT REOMMENDED VALUES OF SUB-MODELS AND SEMI-EMPIRIAL PARAMETERS Eect o aerodyamic dampig, atigue ad extreme loads Dampig o lap-wise vibratios i stall Dampig o edge-wise vibratios i stall Extreme loads

7 Symbols ad otatios c hord cv Vortex eed (equatio (2.2.3) ad ( )) N Lit curve slope.[1/rad] Same as L L Lit curve slope.[1/rad] Same as N L D M T N Lit coeiciet Drag coeiciet Momet coeiciet Tagetial orce coeiciet. Normal orce coeiciet. Note that capital subscripts or lit, drag, pitchig momet, tagetial ad ormal orce coeiciets are used eve though it reers to two-dimesioal quatities. apital letters or the subscripts is ot to mix with other idices, e.g. or time step umber, ' separatio poit locatio IB IR q Idex or blade umber IB Idex or radial elemet IR Pitch rate [rad/s] u p Pluge velocity (equatio )) Vrel w s Velocity relative to two-dimesioal airoil elemet. Normal compoet o velocity relative to the airoil (equatio ( )) Dimesioless time, Dimesioless with hal-chord ad V rel t E Time Agle o attack Eective agle o attack (equatio (2.1.6) ad equatio ( )) i Shed wake iduced agle o attack (equatio (2.1.5)) 7

8 g Geometric agle o attack (equatio ( )) 75 ¾ chord agle o attack (equatio ( )) 2 Γ irculatio [ m / s ] Tp T Time costat or leadig edge pressure lag (equatio (2.3.1)) Time costat or separatio poit positio lag (equatio (2.2.2)) Tv Time costat or vortex lit decay (equatio (2.2.4)) 2D 3D Two-dimesioal Three- dimesioal Subscripts qs I c pot v stat dy quasi steady values time step Free stream coditios or coditios at iiity Impulsive loads irculatory loads Potetial loads (sum o impulsive ad circulatory loads) Vortex part o lit, drag et.c. Related to separatio poit locatio Static (or quasi-static) coditios Usteady coditios 8

9 1 Itroductio This report describes the usteady proile aerodyamics model used i the blade elemet/mometum (BEM) code AERFORE [1]. Usteady aerodyamics is o great importace or wid turbies. Both or low i the attached low agle o attack regio ad or low i the separated low regio. Most attetio is ote give to the usteady separated low aects, commoly amed dyamic stall. The stallig behavior o a airoil will be quite dieret rom its steady stallig behavior i the agle attack is chaged ast eough. The maximum lit coeiciet ca e.g. sigiicatly exceed its static value. For wid turbies, dyamic stall also largely aects the aerodyamic dampig i stall. Geerally the dampig would be largely uder-predicted i steady stall data were used i aeroelastic calculatios. Also, or yawed turbies, dyamic stallig plays a great role with icreased ad dieret phase L, max o the aerodyamic orces as a uctio o rotor azimuth agle relative to usig steady data. May papers ad reports have bee writte dealig with dyamic stall or wid turbies. A recet paper is [2] i which urther reereces ca oud. Dyamic stall has also bee subject o two E research projects [3] ad [4]. Usteady aerodyamics is, however, also importat or attached low. Usteady aerodyamics eects are importat e.g. or stability calculatios at turbie over-speedig, or blade-tower iteractios ad or ast pitchig actio. The model i the DYNSTALL-package is a implemetatio o the Beddoes-Leishma dyamic stall model described i e.g. [5], [6] ad [7]. This model is a 2D semi-empirical model or dyamic stall. It icludes, however, also models or attached low usteady aerodyamics. It is thus complete i the sese that it treats attached low as well as separated low 9

10 1.1 Structure o the report The Beddoes-Leishma model i its origial versio i preseted i chapter 2. hages to this model that are itroduced i the FFA DYNSTALL versio are described i chapter 3. I chapter 4 details i the Fortra implemetatio are give. Fially some recommedatios o the use are give i chapter 5. 10

11 2 The Beddoes-Leishma model The Beddoes-Leishma model is a semi-empirical model. Semiempirical meas that the model relies o empirically determied costats. Semi because the costats are costats i equatios with some physical iterpretatio. The model is based o tables o steady aerodyamic data ( L ( ), D () ad M ( ) ). The usteady behavior is described by dieret processes as e.g. the shed wake eect o the ilow ad a dyamic delay i the separatio process. The dieret dyamic processes are modeled as dieretial equatios that eed user-set values or costats to get best agreemet with experimets or theoretical models. The Beddoes-Leishma model ca be described as a idicial respose model or attached low exteded with models or separated low eects ad vortex lit. The orces are computed as ormal orce, tagetial orce ad pitchig momet. Idicial respose meas that the model works with a series o small disturbaces. The attached low part reproduces the liear theory o [9] or orce respose or a pitchig airoil, a plugig airoil ad the wid gust case. See e.g. [8], sectio 2.6 or a geeral descriptio o the attached low solutio. The usteady 2D airoil aerodyamic respose scales with time made dimesioless with speed relative to the airoil ad the chord o the airoil. The time i the equatios 1 is thereore replaced with dimesioless time: 1 Oe exceptio is the impulsive loads which scale with the chord ad the speed o soud 11

12 Vrel t s = (2.1) c 2 The model uses steady L ( ) ad D ( ) data as iput or a speciic airoil or which the dyamic orces are to be calculated. I the Beddoes-Leishma model, the orces i the body ixed system, N ad T, are used, where: = cos( ) + si( ) (2.2) N L D T = cos( ) si( ) (2.3) D L The Beddoes-Leishma model also treats the usteady eects o the pitchig momet. Dyamic stall has a dramatic eect o the pitchig momet ad o the dampig or pitch-oscillatios. This is very importat or helicopter rotors but is less importat or wid turbie blades. The treatmet o the pitchig momet is thereore let out i the descriptio below. The descriptio below o the Beddoes-Leishma model ollows as ar as how to solve equatios, the methods described i [5]. Alterative solutios usig a state-space model are described i [7]. 2.1 Attached low For attached low, two dieret aspects are modeled. Whe the boud vorticity o the airoil is time varyig, vorticity is shed i the wake. This shed vorticity iduces a low over the airoil, so that the airlow sesed by the airoil is ot the ree stream velocity. I or example the lit (ad the boud circulatio o the airoil) has bee icreasig or some time, the the shed wake will cause a dowwash over the airoil, resultig i less lit tha would be aticipated i steady low. 12

13 The other eect is the impulsive load eect. This eect is also called the apparet mass eect. This eect causes a lit orce or a ast pitch or ast dowward pluge motio. The shed wake eects ilueces the circulatory lit. The impulsive loads are added to get the total attached low lit (lit is here the same as ormal orce sice the agle o attack is assumed small or the attached low part) The low coditios are simulated by the superpositio o idicial resposes. irculatory part For the circulatory part this is writte as: [ ( M ) φ ( s, ] = ) (2.1.1) N, c N c M The idicial respose or the shed wake eects is approximated by φ c = A exp( b βs) A exp( b β ) (2.1.2) s where β = 2 1 M Suggestios or costats A 1, A 2, b 1 ad b 2 are give i [5] as A 1 =0.3 A 2 =0.7 b 1 =0.14 ad b 2 =0.53 The shed wake eect is computed by usig a lagged eective agle o attack: E = i where i ca be see as the shed wake iduced agle o attack. The umerical method to compute step: X i give i [5] is or the :th time = X 1 exp( b1 β s) + A1 exp( b1 β s / 2) (2.1.3) 13

14 Y = Y 1 exp( b2β s) + A2 exp( b2β s / 2) (2.1.4) = X + Y (2.1.5) i, The circulatory lit (or really the ormal orce) is take as ( ) (2.1.6) N, c, = N E 0 where E = i Impulsive load The impulsive lit is calculated rom pisto theory: 4 N, I = φ I ( s, M ) (2.1.7) M The traser uctio is approximated by I t K T = e I φ ( t) (2.1.8) where c T I = where c is the chord ad a is the speed o soud. a K is a uctio o the mach umber as give i [5]. Equatio (2.1.7) ad (2.1.8) are solved to give the impulsive lit at time step as N 4 K c, I, = D (2.1.9) V t rel where the deiciecy uctio is give by 14

15 D t t K T I 1 2KT I = D 1 e + e (2.1.10) t Total attached low lit The total attached low lit (potetial low lit) is the obtaied by addig the circulatory ad impulsive lit. N, pot N, c + N, I = (2.1.11) 2.2 Separated low The simulated separated low eects are: 1) a dyamic delay i the movemet o the boudary layer separatio positio ad 2) the dyamic leadig edge stall with a vortex travellig dowstream alog the chord. The irst part is based o the physical cocept that N is a uctio o the separatio poit positio ad that the movemet o the separatio poit has a dyamic delay. The secod part results i vortex lit that is added to the ormal orce. The vortex lit has a dramatic eect o the pitchig momet. Much attetio has thereore bee give to estimate the coditios at which the vortex lit is shed over the airoil ad the speed at which it travels. The coditio at which the dyamic stall vortex is shed, is i the model cotrolled by the coditio that the ormal orce coeiciet exceeds a certai value. The coditio at whe leadig edge stall occurs, is really cotrolled by a critical leadig edge pressure coeiciet. To lik this critical leadig edge pressure coeiciet to, the model works with a lagged. (See N e.g. [5] or details ad [10] or commets o the dierece betwee pluge ad pitchig motio). N 15

16 Delay i the movemet o the separatio poit. To model the eect o the delay i the movemet o the separatio poit positio requires a N ( ) relatioship. Oe such relatioship is the Kircho low model as described i [5]. The relatioship betwee ad N i the Kircho low model is: 2 ( 1+ ( )) ( ) ( = N ) N (2.2.1) Where 0 is the agle o attack or zero lit ad N is the attached low lit curve slope. is the chord-wise positio o separatio. is made dimesioless with the chord legth ad = 1 represets ully attached low ad = 0 represets ully separated low. The static ( ) ca i priciple be obtaied as a table by solvig equatio (2.2.1) with the steady N ( ) curve. However, i the Beddoes-Leishma model as described i [5], a parametric ( ) - curve that is a best it to the steady ( ) curve is suggested. N I order to obtai the dyamic value o the positio o separatio poit, the model is that the dyamic lags behid its steady value accordig to a irst order lag equatio: d ds T stat = (2.2.2) s is here the dimesioless time rom equatio (2.1). T is a semiempirical time costat (time costat i the s-space). By usig the dimesioless time, s, the model or the lag i the separatio poit works or ay requecy o oscillatio or ay time history with T as a costat. The dyamic lit, N,, is obtaied rom the Kircho low equatio with the dyamic value o the separatio poit positio. 16

17 Vortex lit The model urther icludes a model or vortex lit, with the amout o vortex lit determied by a time-lag uctio with a time costat, T. It is assumed that the vortex lit cotributio ca be viewed as a v excess circulatio, which is ot shed ito the wake util some critical coditio is reached. The vortex lit,, is determied by the ollowig equatios: N v The eed o vortex lit is proportioal to the dierece i usteady circulatory lit ad the o-liear lit give rom the Kircho low equatio with the dyamic value o the separatio poit positio. c v N, c N, = (2.2.3) At the same time, the total accumulated vortex lit is allowed to decay expoetially with time, but may also be updated with ew vortex lit eed. d N, v = (2.2.4) T dcv N, v ds ds v The last equatio is solved at the :th time step as: s s Tv 2T v N, v, N, v, 1e + ( cv, cv, 1 ) e = (2.2.5) oditios or whe the additio o ew vortex lit eed should stop ad coditios or how T v should vary with ad the positio o the travellig vortex, are give i [5]. 2.3 The model i steps The model works as a ope loop model so that the whole model is programmed i steps i which the output rom earlier sub-models are iput to the ext sub-model. 17

18 Step 1 alculate attached low eects (impulsive loads ad shed wake eects) => eective agle o attack, ad impulsive orce E N, imp Step 2 ompute a shit i agle o attack due to the lag i leadig edge pressure respose. d ' N, pot = (2.3.1) T ' N, pot N, pot ds p Where T p is a empirical time costat. ' N, pot is the used to get substitute value o the eective agle o attack: = ' N, pot N 0 (2.3.2) Step 3 Use stat = stat ( ) ad compute the usteady separatio poit positio, dy, rom the lag equatio or the separatio poit. Use dy i the Kircho ( ) relatioship to obtai a dyamic value o the ormal orce coeiciet,,. N Step 4 ompute vortex lit rom equatios (2.2.3) ad (2.2.4) Step 5 Add compoets to get total ormal orce: N = N, I + N, + N, v 18

19 2.4 Semi-empirical costats The mai semi-empirical costats are the three time costats or the leadig edge pressure lag, T p, the lag i separatio poit movemet, T, ad or the vortex lit decay, T v. Values or these costats are suggested i [5]. I the Beddoes-Leishma model, semi-empirical costats are also eeded to cotrol the coditios or the dyamic stall vortex sheddig ad to cotrol its travel speed. For a explaatio o these costats, N, 1 ad T vl, see [5]. I [5], several coditios are also suggested or which some o the time costats should be halved or doubled depedig o e.g. the value o the separatio poit positio, or the positio o the travelig vortex. 19

20 20

21 3 The DYNSTALL usteady airoil aerodyamics model The model i the DYNSTALL-package is a implemetatio o the Beddoes-Leishma model, but with a ew chages made relative to the descriptio above ad as give i [5], [6] ad [7]. The chages to the model are made to make it more robust or use i wid turbie aeroelastic simulatios with turbulet wid. The model ow works or agles o attack i the whole rage [-180,180 ]. Some chages were also itroduced durig the Stallvib project [3] where the usteady aerodyamics or lead-lag blade oscillatios where studied. I the ollowig text, whe there is metio o the curret versio o DYNSTALL it is reerred to the subrouties i the iles: dycl_c., v_esti_c. ad vo_sub_c. 3.1 A chage to work i the wid reerece system The Beddoes-Leishma model as described i [5] works i the body ixed rame. Forces are described i the directio ormal to the airoil,, ad i the chord-wise directio,. The attached low N part origiates rom liearised theory at small agles o attack where si( ) ta( ), so that o distictio ca really be made betwee N ad L i that theory. For the o-liear part (separated low) the Beddoes-Leishma model also works with N ad T. There might be advatages by settig up the separated low equatios i the body ixed system but diiculties arise whe the dyamic behavior o T at high agles o attack should be modeled with the model rom [5]. T The curret versio o DYNSTALL thereore uses the wid reerece system. Most equatios i i the Beddoes-Leishma N 21

22 model are treated as beig equatios i L. Oe exceptio is that the vortex lit still is treated as beig ormal to the airoil chord. Oe advatage o workig i the wid reerece system is that the delay i the separatio poit positio automatically aects the tagetial orce. This happes sice T has a rather large compoet o L at stall agles o attack. Most o the eect o the dyamic delay o the separatio poit o T is thereore automatically a result o the delay i the lit. The eect o the dyamic delay o the separatio poit o D is also modeled i DYNSTALL. Still the mai eect o comes rom the lit part at stall agles o attack. T The vortex lit is, however, assumed to act oly i the ormal orce directio. A icrease o T will icrease the dyamic stall loop width o the N ( ) -curve as well as the T ( ) -curve, whereas a icrease o T V will icrease oly the width o the N ( ) - curve ad leave the T ( ) - curve uaected. The act that T V oly aects N makes it possible to tue the time costats, T ad T V, to obtai a good it to both the ( ) as well as the ( ) -curve. 3.2 Attached low T The attached low equatios are basically the same as i [5] ad give i chapter 2 above. N I DYNSTALL, two methods are however available or how to calculate the shed wake eect. I the alterative method explaied i sectio the shed wake eect is made a uctio o the actual circulatio history ad ot the circulatio history that would occur or attached low as i the origial model The agle o attack The agle o attack that is iput to the equatios icludes the pitch rate eects. 22

23 A geometrical agle o attack, g, is deied as the agle betwee the chord-lie ad the relative velocity to the airoil. The relative velocity to the airoil icludes here the traslatio motio o the airoil. No dierece is hece made betwee the airoil movig i still air or the airoil beig at a stad-still at a icidece i a airlow. u y V x q u x V y V rel Figure 1 I the picture above, the airoil is at a geometrical icidece ad also has a heavig velocity, u y, ad a lead lag velocity, u x. g is the calculated rom the agle o the relative velocity to the chord lie as: Vy u y ta( g ) = ( ) V u x x I order to automatically iclude the pitch rate eect i the lit, the agle o attack at the ¾ chord positio ca be used. The DYNSTALL package thereore also works with the agle o attack at the ¾ chord positio. This agle o attack is called 75. With the velocity relative to the airoil, V rel, ad the pitch rate q, the c q 75 = g + ( ) 2 V rel 23

24 75 is the used or i equatios or the circulatory lit i sectio Attached low equatios (LPOTMETH=3) The attached low equatios are available i two versios. The choice o versio is cotrolled by a iput parameter LPOTMETH. (LPOTMETH=1 used to be the attached low equatios i sectio 2.1. but is o loger implemeted sice it is equivalet to LPOTMETH=3 or costat V rel ) irculatory part The equatios are reormulated to work with the ormal velocity at the ¾ poit chord positio. This is the same as reormulatig the equatios i circulatio rather tha i lit coeiciet. With this ormulatio, the circulatory eect will be correct or lead-lag motios ad or a varyig wid speed. Equatio (2.1.1) the becomes c Γ = L φc w ( ) 2 w is the ormal compoet o the relative velocity to the airoil at the ¾ poit chord positio. I this case the ormal directio is take as ormal to the zero-lit lie ad the small agle approximatio is used. ( 75 0 V rel ) w = ( ) The equatios to solve or the eective agle o attack are the at the :th time step X = X 1 exp( b1 β s) + A1 w exp( b1 β s / 2) ( ) Y = Y 1 exp( b2 β s) + A2 w exp( b2 β s / 2) ( ) 24

25 ( X Y V + ( ) E = 75 ) / rel 0 This method is obtaied by choosig LPOTMETH=3. 2 β = 1 M is i the curret versio set to uity sice Mach umber eects or wid turbies or the shed wake eect are assumed small. I order to avoid queer results whe the liearised theory is used at high agles o attack, w i equatios ( ) ad ( ) are actored with a actor ade. Where ade = cos ( 75 ) ( ) 2 Impulsive load The impulsive load should be calculated rom the airoil pluge velocity at the ¾ poit chord positio. This pluge velocity could come rom airoil plugig but could also be due to a pitch rate iduced velocity at the ¾ poit chord positio. With, u p as the velocity o the ¾ poit chord positio directed i the egative y-directio, the equatios or impulsive loads at the :th time step are writte: L, I, 4 K c V U u p t, = D rel, mea ( ) with the deiciecy uctio give by D t t K T u p, u p, 1 I 2KT I = D 1 e + e t ( ) ad U mea is the mea o V rel durig the curret ad the previous time step. 25

26 K is i the curret versio o DYNSTALL set to which is the value take rom [5] or M=0.15) I the curret versio the impulsive ormal orce rom equatio ( ) is take as the impulsive lit orce eve though this is strictly correct oly or = 0. I order to avoid queer results whe the liearised theory is used at high agles o attack, u i equatios ( ) ad ( ) are actored with the actor ade rom equatio ( ). p Pitchig momet The attached low eects o the pitchig momet are icluded by addig the impulsive load compoet to the pitchig momet L, I M, I = ( ) Shed wake eects with accout or a varyig lit curve slope (LPOTMETH=4) The above equatios (used with LPOTMETH=3) are valid or attached low. The shed wake eect is the calculated as i the low were attached with a lit curve slope, which is the lit curve slope aroud the zero lit agle o attack. L At agles o attack where the airlow starts to separate, the steady lit curve slope is o loger the same as or attached agles o attack. The lit may eve decrease or a icrease i agle o attack above stall. It is the questioable i the shed wake eect (circulatory lit) could be calculated with a costat. L O way to overcome this would be to calculate the shed wake eects rom the circulatio history ( or lit history or costat V rel ) istead. 26

27 Oe problem is, however, that the lit is ot kow util the dyamic eects o the separatio poit positios is determied. This is solved by irst calculatig a estimate o L,. This estimate, L,, est, is calculated as i step 2 ad 3 o the Beddoes-Leishma model but as explaied i the sectios o separated low below. The shed wake eect is the made a uctio o the dierece i,, istead o the dierece i 75 : L est ( ) L,, est, = L,, est, L,, est, 1 Further, L, est, must be based o the agle o attack without the shed wake eects i the model should give correct results or attached low with the equatios correspodig to ( )-( ). is thereore set to 75 whe the estimate o the dyamic lit, E L, est,, is calculated. The method used to accout or the act that a reduced lit curve slope due to separated low give reduced shed wake eects is: exchage i equatios (2.1.1)-(2.1.4) with L,, est L. To, at the same time accout or a varyig circulatio due to a varyig velocity, w is used istead o with w as i equatio ( ) w to be put i equatios ( )-( ) is the calculated, by dieretiatio, as w = L,, est, mea L V rel + L,, est L U mea ( ) L, est, mea, with, beig the mea o L, est at the curret ad the previous time step. As or LPOTMETH=3 equatio ( ) w is multiplied with the actor ade rom 27

28 The so calculated eective agle o attack will be exactly the same as calculated with LPOTMETH=3 i the low is attached. Attached low meas here that =1 ad ( ) = ( ). L, E L, stat E For separated low it will approximately accout or the act that the shed wake history is a uctio o the lit history rather tha o the agle o attack history. The lit that is used to estimate the chage o lit excludes the vortex lit sice the vortex lit is cosidered ot to aect the shed wake history. The extra computig or LPOTMETH=4 relative to LPOTMETH=3, is that step 2 ad 3 has to be ru a extra time every time step to obtai L,, est. 3.3 Separated low A shit i the agle o attack The separatio poit positio is statically a uctio o the agle o attack. What really ilueces the separatio positio is the pressure gradiets i the boudary layer. To use the shited agle o attack,, as calculated rom equatio (2.3.2) is a way to accout or the usteady pressure gradiet. I DYNSTALL, the shit i the agle o attack to compesate or the lag i the leadig edge pressure is used just as i the origial model. What is ew to the FFA versio, is to accout or chages i the usteady pressure gradiet due to a varyig ree stream velocity. This was itroduced durig the STALLVIB project where lead-lag airoil oscillatios were studied [3], [11]. By studyig the usteady Beroulli equatio it ca be see that the pressure gradiet alog the airoil will be dieret i the relative velocity is costat, icreasig or decreasig (see [3] or [11]). I Vrel 28

29 is icreasig, the the eective usteady pressure gradiet is oud to be more avorable which i priciple should result i less separatio. The questio is how this eect o the pressure gradiet ad the correspodig eect o the separatio poit should be icorporated i the dyamic stall model. Oe way to do this is to see the resultig pressure gradiets as gradiets occurrig at a dieret agle o attack. (A lower agle o attack or a acceleratig ree stream.) The poit o separatio,, could the be determied usig a corrected agle o attack as step 2b i the dyamic stall model. The assumptio is that, at some agle o attack, = +, the pressure gradiet or the case with varyig V rel, is similar to the pressure gradiet at a agle o attack, or the case with costat V rel. The equatios to derive a estimate o the shit are give i [11] ad [3]. The result is that the shit should be proportioal to the dimesioless velocity chage rate c Vrel γ = ( ) 2 2 V t rel The amout o shit is urther cotrolled by a empirical parameter,. u A estimate o = 0. 5 is give i [11]. u Step 2, as i the descriptio o the Beddoes-Leisha model i sectio 2.3, is i the FFA model: Equatio (2.3.1) is solved at the :th time step as dp = dp 1 e s Tp + s 2 T p ( ) e N, pot, N, pot, 1 ( ) ad 29

30 , ade dp ( ) ' N pot, = N, pot, The actor ade rom equatio ( ) is icluded sice the cocept ayway breaks dow at high agles o attack. The shited agle o attack is take rom equatio (2.3.2) I the FFA-model the agle o attack is also shited due to a varyig velocity. is thereore take as ' N, pot = 0 γ N u ( ) Lit as uctio o the separatio poit positio The steady l ( ) curve is i the dyamic stall model exchaged with a static ( ) relatioship as explaied i sectio 2.2. The dyamic value (usteady value) o the lit coeiciet is the obtaied by puttig i a dyamic value o i the ( ) relatioship. Two dieret types o ( ) ca be used i DYNSTALL. The choice betwee models is cotrolled by a variable lmeth. Oe model is the relatioship itroduced by Øye [12]. The other is a versio o the Kircho low model. I either case, the dyamic value o is calculated by solvig equatio (2.2.2) The Øye separatio poit model (lmeth=2) The ormula likig L, ad origiatig rom Øye is used. L L, iv + ( 1 ) L, sep = ( ) 30

31 This relatioship requires steady curves o L iv ( ) ad L sep ( ) i order to obtai either a steady L, stat ( ) curve rom a steady ( ) or a dyamic L rom a dyamic.,, Equatio ( ) ca be used to id a static ( ) relatioship. stat L, stat ( ) L, sep ( ) ( ) = ( ) ( ) ( ) L, iv L, sep ( ), ( ) ad ( ) are iput to the program as L, iv L, sep stat tabulated values as uctio o. should perhaps ot be see strictly as the separatio poit positio, but rather as a iterpolatio actor betwee ( ) ad ( ). L, iv L, sep L, iv could be take as L ( 0 ) ad L, sep as the curve or ully separated low. The choice or the latter curve is more arbitrary. Øye [12] suggests a curve a curve startig with a slope o hal value o the useparated curve ad gradually ittig to the steady L ( ) curve at approximately 30 agle o attack. ' To solve the dieretial equatio (2.2.2), a value o is obtaied rom the stat ( ) table with the agle o attack as iput. (2.2.2) is the solved at the :th time step as d = d 1 e s T + s ' ' 2 T ( ) e 1 ( ), d ( dy = ', is obtaied rom equatio ( ) as L ) = ( ) + (1 ) ( ) ( ) L, ( g dy L, iv E dy L, sep E 31

32 A versio o the Kircho low (lmeth=4) The Kircho low equatio writte i equatio (2.2.1) is used, but with some modiicatios. First, is used rather tha. L N With =1 the umodiied Kircho low equatio with L will give rather high values o L at large agles o attack. The DYNSTALL model should work over the whole rage o agles o attack, ad modiicatios are itroduced to limit queer results at high agles o attack. The Kircho low equatio is thereore modiied to limit L or =1. Aother ix is itroduced to overcome the act that i equatio (2.2.1) (with N replaced by L, stat ) is used to solve or the static ( ), the this will oly work as log as stat 1 L, stat < L ( 0 ). 4 With ( ) L, iv = L 0 ad L, sep = L, iv 4, the Kircho equatio ca be writte to use the expressios L, iv ( ) ad ( ) as i the Øye model. L, sep 1 = L, sep + L, iv ( 2 ) ( ) 4 L + To limit I c L or =1, equatio ( ) is modiied or > c. >, the the liear ( ) is exchaged or a L, iv = L 0 sie curve that beds dow almost to zero at 90 : 1 = d ( ) L, iv si( d1 ( 0 )) + d1 2 d 1 is a costat, somewhat arbitrarily, chose to be 1.8. The costat d 2 assures cotiuity o the two L, iv expressios at = c with 32

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