FFAP-V-110 DYNSTALL: Subroutie Package with a Dyamic stall model Aders Björck 2 1.8 1.6 Steady data Experimet Simulatio 1.4 1.2 1 0.8 0.6 0.4 0 5 10 15 20 25 FLYGTEKNISKA FÖRSÖKSANSTALTEN THE AERONAUTIAL RESEARH INSTITUTE OF SWEDEN
FLYGTEKNISKA FÖRSÖKSANSTALTEN THE AERONAUTIAL RESEARH INSTITUTE OF SWEDEN Box 11021, SE-161 11 Bromma, Swede Phoe +46 8 634 10 00, Fax +46 8 25 34 81
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
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otet 1 INTRODUTION... 9 2 THE BEDDOES-LEISHMAN MODEL...11 2.1 ATTAHED FLOW... 12 2.2 SEPARATED FLOW... 15 2.3 THE MODEL IN STEPS... 17 2.4 SEMI-EMPIRIAL ONSTANTS... 19 3 THE DYNSTALL UNSTEADY AIRFOIL AERODYNAMIS MODEL... 21 3.1 A HANGE TO WORK IN THE WIND REFERENE SYSTEM... 21 3.2 ATTAHED FLOW... 22 3.2.1 The agle o attack... 22 3.2.2 Attached low equatios (LPOTMETH=3)... 24 3.2.3 Shed wake eects with accout or a varyig lit curve slope (LPOTMETH=4)... 26 3.3 SEPARATED FLOW... 28 3.3.1 A shit i the agle o attack... 28 3.3.2 Lit as uctio o the separatio poit positio... 30 3.3.2.1 The Øye separatio poit model (lmeth=2)... 30 3.3.2.2 A versio o the Kircho low (lmeth=4)... 32 3.3.3 Vortex lit (lvormeth=2)... 34 3.3.4 A alterative versio o vortex lit (lvormeth=1)... 34 3.4 UNSTEADY DRAG... 35 3.5 UNSTEADY PITHING MOMENT... 37 3.6 WAYS TO MAKE THE MODEL WORK FOR ALL ANGLES OF ATTAK... 37 3.7 ROBUSTNESS OF THE DYNSTALL-METHODS... 38 3.7.1 Methods that work or all agles o attack... 39 4 INPUT TO THE FORTRAN SUBROUTINE PAKAGE... 41 4.1 SUBROUTINES... 41 4.2 ONVENTIONS FOR VARIABLE NAMES... 42 4.3 ABOUT HISTORY OF VARIABLE NAMES... 42 4.4 OMMUNIATION WITH THE SUBROUTINES... 42 4.5 OMMON AREAS IN DYNL_.IN... 43 4.5.1 Storage o static airoil data.... 43 4.5.1.1 Rage o agle o attack i iput airoil tables.... 45 4.5.1.2 Lit curve slope ad zero lit agle o attack... 45 4.5.2 hoices betwee submodels... 46 4.5.3 oeiciets ad time costats... 47 4.5.4 Variables used or state variables ad other variables eeded or iteral commuicatio... 47 5
4.5.5 Variables i commo areas that must be set at each time step outside o the DYNSTALL-package 48 4.6 GENERATION OF FNSTIN, NINVIN AND NSEPIN TABLES...48 5 SUB-METHODS AND VALUES FOR SEMI-EMPIRI ONSTANTS...49 5.1 AN OPTIMIZATION STUDY TO FIND OPTIMUM VALUES...49 5.2 LDDYN...50 5.3 EFFETS OF VARYING VELOITY ON THE SEPARATION POINT...50 5.4 REOMMENDED VALUES OF SUB-MODELS AND SEMI-EMPIRIAL PARAMETERS...50 5.4.1 Eect o aerodyamic dampig, atigue ad extreme loads...51 5.4.1.1 Dampig o lap-wise vibratios i stall...51 5.4.1.2 Dampig o edge-wise vibratios i stall...51 5.4.1.3 Extreme loads...52 6
Symbols ad otatios c hord cv Vortex eed (equatio (2.2.3) ad (3.3.3.1)) 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 3.2.2.7)) Vrel w s Velocity relative to two-dimesioal airoil elemet. Normal compoet o velocity relative to the airoil (equatio (3.2.2.2)) 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 (3.2.2.5)) i Shed wake iduced agle o attack (equatio (2.1.5)) 7
g Geometric agle o attack (equatio (3.2.1.1)) 75 ¾ chord agle o attack (equatio (3.2.1.2)) 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
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
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
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
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
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) 1 1 1 2 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
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
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
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 0.25 0 (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
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
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
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
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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
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 3.2.2 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. 3.2.1 The agle o attack The agle o attack that is iput to the equatios icludes the pitch rate eects. 22
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 ) = (3.2.1.1) 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 + (3.2.1.2) 2 V rel 23
75 is the used or i equatios or the circulatory lit i sectio 2.1. 3.2.2 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 (3.2.2.1) 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 = (3.2.2.2) 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) (3.2.2.3) Y = Y 1 exp( b2 β s) + A2 w exp( b2 β s / 2) (3.2.2.4) 24
( X Y V + (3.2.2.5) 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 (3.2.2.3) ad (3.2.2.4) are actored with a actor ade. Where ade = cos ( 75 ) (3.2.2.6) 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 (3.2.2.7) with the deiciecy uctio give by D t t K T u p, u p, 1 I 2KT I = D 1 e + e t (3.2.2.8) ad U mea is the mea o V rel durig the curret ad the previous time step. 25
K is i the curret versio o DYNSTALL set to 0.846 which is the value take rom [5] or M=0.15) I the curret versio the impulsive ormal orce rom equatio (3.2.2.7) 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 (3.2.2.7) ad (3.2.2.8) are actored with the actor ade rom equatio (3.2.2.6). 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 = (3.2.2.9) 4 3.2.3 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
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 (3.2.3.1) 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 (3.2.2.3)-(3.2.2.5). 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 (3.2.1.2) w to be put i equatios (3.2.1.3)-(3.2.2.5) is the calculated, by dieretiatio, as w = L,, est, mea L V rel + L,, est L U mea (3.2.3.2) L, est, mea, with, beig the mea o L, est at the curret ad the previous time step. As or LPOTMETH=3 equatio (3.2.2.6) w is multiplied with the actor ade rom 27
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 3.3.1 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
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 = (3.3.1.1) 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 (3.3.1.2) ad 29
, ade dp (3.3.1.3) ' N pot, = N, pot, The actor ade rom equatio (3.2.2.6) 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 (3.3.1.4) 3.3.2 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) 3.3.2.1 The Øye separatio poit model (lmeth=2) The ormula likig L, ad origiatig rom Øye is used. L L, iv + ( 1 ) L, sep = (3.3.2.1.1) 30
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 (3.3.2.1.1) ca be used to id a static ( ) relatioship. stat L, stat ( ) L, sep ( ) ( ) = (3.3.2.1.2) ( ) ( ) 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 (3.3.2.1.3), d (3.3.2.1.4 dy = ', is obtaied rom equatio (3.3.2.1.1) as L ) = ( ) + (1 ) ( ) (3.3.2.1.5) L, ( g dy L, iv E dy L, sep E 31
3.3.2.2 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 ) (3.3.2.2.1) 4 L + To limit I c L or =1, equatio (3.3.2.2.1) 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 (3.3.2.2.2) 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
1 d = si( ) 2 La ( c 0 ) c 0 (3.3.2.2.3) d1 The origial Kircho curve is thus retaied uchaged or < c. c =30 is used i the AERFORE package [1]. I order to get the equatio to work or agles o attack where 1 L ( 0 ) > L, stat, L, sep is the set to L, stat ad stat is the 4 set to zero. To costruct tables o ( ), ( ) ad ( ) or the L, iv modiied Kircho low model: I < c, L ( ), iv = L 0 L, sep stat Else L, iv is take rom equatio (3.3.2.2.2). Equatio (3.3.2.2.1) is used to solve or stat i 1 L, stat > L ( 0 ). 4 The = 4 => L, sep L, iv 2 4 L, stat 1 stat = (3.3.2.2.4) L, iv 1 I L, stat < L ( 0 ), the 4 stat = 0 ad L, sep = L, stat. I DYNSTALL,, is the obtaied rom equatio (3.3.2.2.1) as L 33
1 L, ( g ) = L, sep ( E ) + L, iv ( E ) ( dy + 2 dy ) (3.3.2.2.5) 4 with dy rom equatios (3.3.2.1.3) ad (3.3.2.1.4) 3.3.3 Vortex lit (lvormeth=2) The vortex lit is calculated with equatios equivalet to equatios (2.2.3)-(2.2.5) but with equatio (2.2.3) replaced with c v L, c L, = (3.3.3.1) The eed i vortex lit is urther oly allowed as log as the agle o attack is icreasig. So that the secod term i equatio (2.2.5) is zero i the agle o attack is decreasig. Furthermore, egative eed o vortex lit is ot allowed. So that i ( c c 1) 0, the secod term i equatio (2.2.5) is agai set to v, v, < zero. The eed i vortex lit is urthermore disregarded i the agle o attack is too high. This is cotrolled by a iput variable. shit, ds I AERFORE [1] shit, ds is set to 50 (0.87 rad). 3.3.4 A alterative versio o vortex lit (lvormeth=1) The subroutie vo_sub_c. a also icludes a versio o vortex lit calculatios where the eed i vortex lit stops ater a certai time delay ater a critical coditio has bee obtaied. N This versio o vortex lit calculatios also icludes criteria or whe vortex eed ca start agai ater havig bee termiated. Oly primary vortex sheddig is cosidered. The method is cosidered less robust or simulatios with stochastic wid or wid turbies. No urther explaatio o the method is thereore give i this report, ad the reader is reerred to the code ad to [5]. 34
3.4 Usteady drag Usteady drag eects are implemeted i DYNSTALL. The usteady eects o the drag are: D, id. The shed wake iduced drag D, sep. The chage i drag due to the separatio poit positio beig dieret rom its static positio. D, vor. Vortex drag. A compoet o the vortex lit must be added to the drag sice vortex lit is assumed to act ormal to the chord. The total drag is take as: D D, stat + D, id + D, sep + D, vor = (3.4.1) Iduced drag, is calculated as the compoet o lit iclied the shed wake D id iduced agle o attack ) = si( ) ( ) (3.4.2) D, id ( g 75 E L, g Separatio drag I the separatio poit is orward o its static positio, the it is assumed that the drag will be less tha i the separatio poit was at its static positio ad vice versa. This is modeled, more or less as suggested by Motgomerie i [13]: ) = A ( ( ) ( )) (3.4.3) D, sep ( g cd L, stat E L, g A cd is a empiric costat. Vortex drag The vortex drag is calculated as 35
) = ( ) si( ) (3.4.4) D, vor ( g N, v g g With the vortex lit take as ) = ( ) cos( ), (3.4.5) L, vor ( g N, v g g the tagetial orce compoet o vortex lit will the be zero. Versios o how to add dyamic drag eects A iput parameter cotrols how the usteady drag eects should be calculated. I LDDYN=0, the the drag is take as the static drag D ( g ) = D, stat ( g ) I the usteady drag is modeled the it is ot quite clear i the static drag rom the tables o D, stat ( ) should be take or the geometric agle o attack or or the eective agle o attack, E. There are thereore two versios: LDDYN=1: The static drag is evaluated at the geometric agle o attack: = (3.4.6) D ( g ) D, stat ( g ) + D, id + D, sep + D, vor LDDYN=2: The static drag is evaluated at the eective agle o attack: = (3.4.7) D ( g ) D, stat ( E ) + D, id + D, sep + D, vor 36
3.5 Usteady pitchig momet Usteady eects o the pitchig momet or separated low eects are ot icluded i DYNSTALL. The pitchig momet that is retured is thus the static pitchig momet corrected or attached low usteady eects. = (3.5.1) M ( g ) M, stat ( g ) + M, I with, rom equatio (3.2.2.9) M I 3.6 Ways to make the model work or all agles o attack For simulatio o wid turbie loads i desig calculatios, the wid ca i priciple come rom ay directio. The airoils could thereore be blow at rom the rear. Oe problem i itsel is to geerate sesible airoil data or the whole rage o agles o attack. The other problem is to make the dyamic stall model to work or low (that is 2D low) at ay agle. First all equatios i sectios 3.1-3.4 are writte so that the dyamic stall works or egative agles o attack i the rage [0,-90 ]. This comes automatically ad oly eeds some special treatmet i the vortex lit calculatios. With a decrease i agle o attack i sectio 3.3.3 is really meat a decrease o absolute values o agle o attack ad egative eed o vortex lit reers to the case whe the agle o attack is positive. Moreover, the equatios should work or the agle o attack rages [-180,-90 ] ad [90,180 ]. The agle o attack rage [-90,90 ] is called = 1 i the code. case 37
The agle o attack rage [-180,-90 ] is called = 2 i the code. The agle o attack rage [90,180 ] is called = 3 i the code. case case The agle o attack rages or = 2 ad = 3 could also be case put together i a cotiuous agle o attack rage with the trasorm case! 2 = +180 i case = 2! 2 = 180 i case = 3 (3.6.1) (3.6.2) I this ew agle o attack, both = 180 ad = + 180, is represeted by 2 = 0. A secod lit curve slope ad a secod zero lit agle o attack, L, 2 ad 0, 2 must thereore be used i is i the rage [-180,-90 ] or [90,180 ]. For every agle o attack there, thus, also exist a dyamic stall agle o attack that goes i the equatios or the dyamic stall model. Sice the dyamic stall model icludes some table lookig, the, origial agle o attack also is passed to the model. The dyamic stall model is thereore ed with g, g, ds ad 75, ds. g, ds ad 75, ds are the 2 agles o attack rom equatios (3.5.1) ad (3.5.2) i = 2 or = 2. I case = 1, the g ds = g, ad 75, ds = 75. case case L ad 0 are also chose or the respective case. 3.7 Robustess o the DYNSTALLmethods Dieretial equatios are solved such that rather log time steps could be used. To get the method work well, the time step should o course be shorter tha the time-costats i the equatios. The equatios are, however, writte such that i the time costat is small 38
i relatio to the time step, the the dyamic eects just becomes abset. See e.g. equatios (3.3.1.2) ad (3.3.1.3) or the equatios to calculate the shed wake eects, equatios (3.2.2.3) ad (3.2.2.5). I order to simulate the attached low eects well the time step should be approximately shorter tha correspodig to s = 0. 5 ( s = 0. 5 is the step legth or the airoil to move a ¼ chord). 3.7.1 Versios that work or all agles o attack I the dyamic stall model should be used or agles o attack above the rage [-90,90 ], the Øye model (lmeth=2) does ot work. The modiied Kircho model (lmeth=4) works or the whole rage o agles o attack. 39
40
4 Iput to the FORTRAN subroutie package The use o the subrouties or dyamic stall is partly described i the descriptio o the AERFORE blade-elemet/mometum code program [1]. Much o the same iormatio ad some urther iormatio is give below. 4.1 Subrouties The mai subroutie i the DYNSTALL-package is the dycl_c subroutie. This subroutie the i tur calls subroutie v_esti_c i ile v_esti_c. ad the appropriate subroutie or vortex lit i the ile vo_sub_c.. The subrouties work with a umber o state-variables. E.g. d p ad at the curret time step ad at the pervious time step (see N, pot equatio (3.3.1.2). Methods to solve dieretial equatios are such that data oly rom the previous ad curret time step are eeded) I order to update values at the curret time step to values at the ext time step subroutie cu_2_old_aer_c i the ile cu_2_old_aer_c. is called. The DYNSTALL-package is iteded or use i a bladeelemet/mometum code ad it is called or oe blade elemet at the time. Matrix idices or blade ad radial elemet umber are thereore eeded (variables IB ad IR). At the irst time step, values or state-variables at the previous time step must be set. A example o how this ca be made is give i the subroutie clcd_irstime i the AERFORE subroutie package. 41
4.2 ovetios or variable ames. For the solutio o dieretial equatios, variable values at the previous time step is eeded. Variable ames edig with _1 reers to variable values at the previous time step. 4.3 About history o variable ames I the code, variable ames still bear marks rom the irst versios o DYNSTALL which worked with ad istead o. A variable NF should perhaps better be called LF ow, ad NSEPIN should perhaps better be called LSEPIN. The history o the code is hece, ot or best readability, still visible. 4.4 ommuicatio with the subrouties The basic iput to the mai DYNSTALL subroutie, dycl_c, is N L Airoil data Agle o attack, chord ad time-step The basic output is Lit, drag ad pitchig momet coeiciets Some o the variables are commuicated as ormal parameters to dycl_c, but the mai part are commuicated via commo areas. The ormal parameters i the subroutie call are explaied i the code. The dycl_c subroutie is called rom subroutie clcdcalc_c i the AERFORE subroutie package ad this gives some iormatio o the use. 42
4.5 ommo areas i Dycl_c.ic The ile dycl_c.ic icludes commo-areas with variables that are eeded by the dyamic stall subrouties e.g. storage o airoil aerodyamic coeiciets. 4.5.1 Storage o static airoil data. The aerodyamic static airoil data should be stored i data tables L (), D ( ) ad M ( ). The subroutie package is iteded or use with a blade-elemet/mometum code where track o a speciic airoil table or each blade elemet is required. Airoil data are thereore give or oe or or more airoils. These airoil data tables are stored i the commo area /proidata996/ Each airoil is associated with oe table. Data or at least oe airoil is eeded. Static airoil data,, L, stat, D stat, ad M, stat are stored i variables alasti, clsti, cdsti ad cmsti. For calculatio with the dyamic stall model (lccl=2), the static data or is however ot used rom the, variable, but rom L L stat the variables stat, L, iv ad L, sep as explaied i sectio 3.3. I the Øye ( ) relatioship is used (LFMETH=2), the the iput airoil data tables should iclude tabulated values o stat, L, iv ad L, sep as iput. The tabulated values o L, stat are the igored. I the modiied Kircho low ( ) relatioship is used (LFMETH=2), the same also holds. The tables o stat, L, iv ad, could, however, the automatically be geerated by a call to L sep the subroutie kirchmake_c. The subroutie kirchmake_c ills the tables sti, civi ad csepi ( stat, L, iv ad L, sep ). 43
Airoil data are stored i matrices alasti(ia,ip), clsti(ia,ip) etc.. The secod idex reers to a table umber. i.e. which airoil it is. The irst idex reers to the row umber (agle o attack idex) i the airoil data table. Apart rom matrices alasti, clsti, cdsti, cmsti, sti, civi ad csepi, data or the zero lit agle o attack lit curve slope is also eeded. The zero lit agle o attack ad the lit curve slope are stored i the variables ala0 ad c_ala. (c_ala should be i uits 1/radias ad ala0 i radias). The zero lit agle o attack ad the lit curve slope aroud ±180 agle o attack is also eeded as iput i the agle o attack exceeds ±90. These are stored i variables c_ala2 ad ala02. Furthermore the /proidata996/ area cotais c1pos ad c1eg which is the critical coditios eeded or the vortex N lit calculatios i the method LVORMETH=1 is used (see sectio 3.3.4). A example o oe airoil data table is show below 2 Ala l d m -10.0-0.8600 0.0150-0.0375-4.0-0.1283 0.0067-0.0785-2.0 0.1252 0.0068-0.0834 2.0 0.6273 0.0073-0.0925 4.0 0.8748 0.0079-0.0964 6.0 1.1190 0.0086-0.0997 8.0 1.3525 0.0102-0.1014 10.0 1.5349 0.0154-0.0973 12.0 1.5899 0.0271-0.0843 2 Note that the agle o attack as iput to alasti(.,.) should be i radias ad ot degrees 44
14.0 1.5957 0.0511-0.0837 16.0 1.5963 0.0797-0.0868 This table cotais 11 rows. I it would reer to airoil umber 1, the the 11 L -values should be stored i clsti(i,1) with i rom 1-11. 3 Additioal to alasti, clsti, cdsti, cmsti, sti, civi ad csepi the variable i(ip) (i the same commo area as alasti et.c.) is also eeded. i tells how may rows that are used i each table. I the example above, i(1)=11 should be set. 4.5.1.1 Rage o agle o attack i iput airoil tables. The rage o agles o attack i each airoil data table must cover the agles o attack that will be ecoutered durig the calculatios. I a agle o attack is calculated outside the iput rage, the the program will halt (i subroutie iterp999) whe L, D ad M values are sought. To cover the ull 360 degrees rage o agles o attack, should be give i the rage [-180,180] degrees. 4.5.1.2 Lit curve slope ad zero lit agle o attack The zero lit agle o attack ad the lit curve slope are stored i the variables ala0, c_ala, c_ala2 ad ala02. The zero lit agle o attack ad the lit curve slope are however iput to the subroutie dycl_c as ormal parameters together with the variable (acase) as explaied i sectio 3.6. case The variables ala0_used ad clala_used should be set to the appropriate values o ala0 c_ala, c_ala2 ad 3 The table must o course iclude the ull rage o agles o attack that will be ecoutered durig the calculatios. I a agle o attack is calculated to e.g. 20 degrees ad the table with ala[-10,16] is used, the the program will stop. 45
ala02. Noe o the variables ala0, c_ala, c_ala2 ad ala02 is, however, actually used i the subroutie dycl_c or subrouties directly called by dycl_c. ala0_used ad clala_used should be take as the zero lit agle o attack ad the lit curve slope or the civi curve, ( ) -curve. L, iv 4.5.2 hoices betwee submodels Variables to store choices or which sub-model to use or the dyamic stall calculatios are stored i the commo-area /typeparam/ These variables eed to be set outside o the dycl_c subroutie. Lccl Lpotmeth Lmeth This parameter should be set to 2 i the dyamic stall model is used. Lccl=1 was a old, ow removed method. lpotmeth should be set to 3 or 4 to obtai methods as explaied i sectio 3.2. lmeth should be set to 2 or 4 to obtai methods as explaied i sectio 3.3.2. Lvormeth lvormeth should be set to 2 or 1 to obtai methods as explaied i sectio 3.3.3 or 3.3.4 Lcddy lcddy should be set to 0, 1 or 2. I lcdy=0, the D will be retured as static D. The choices betwee lcdy=1 ad lcdy=2 are explaied i sectio 3.4 Ldut I ldut=0, the the eect o a varyig velocity o the separatio poit positio is eglected. The same eect is also obtaied by settig u to zero. 46
4.5.3 oeiciets ad time costats The area /potcoe/ cotais the coeiciets or the iviscid circulatory lit respose. /tparam/ ad /vorparam/ cotai iput variables or the separatio lag part ad the vortex lit part o the dyamic stall model. /tparam/ tp Time costat T p. See sectio 3.3.1 t_i Time costat T. See sectio 3.3.2 par_i ad par_d are ot used ay more. crit is ot used ay more uac ostat u, see sectio 3.3.1 acd ostat A cd, see sectio 3.4 /vorparam/ tvo Time costat T v. See sectio 3.3.3 tvl Time costat used with lvormeth=1. See sectio 3.3.4 ad the code tvs Time costat used with lvormeth=1. See sectio 3.3.4 ad the code 4.5.4 Variables used or state variables ad other variables eeded or iteral commuicatio The areas /curval/ ad /oldval/ cotai variables used or the calculatio o the dyamic stall (state variables). Values at the curret time step is stored i the variables i /curval/ ad the values at the previous time step is stored i /oldval/. Noe o the variables i /curval/ or /oldval/ eed to be set outside o the dycl_c subroutie package except or that start-coditios or the /oldval/- variables must be set as explaied i sectio 4.1. 47
4.5.5 Variables i commo areas that must be set at each time step outside o the DYNSTALL-package The two variables vrel ad up i the commo area /curval/ must be set at each time step. vrel(ib,ir) is V rel or blade elemet IB ad radial positio IR (see sectio 3.2.1). up(ib,ir) is the plugig velocity u p (see sectio 3.2.2). 4.6 Geeratio o FNSTIN, NINVIN ad NSEPIN tables The tables o stat ( ), L, iv ( ) ad L, sep ( ) are eeded or the calculatio o a dyamic L. I the Øye ( ) relatioship is used (LFMETH=2), the the airoil data or stat, L, iv ad L, sep must be icluded i iput iles. Values o ala0_used ad clala_used should the be take as the zero lit agle o attack ad the lit curve slope or the civi curve. Note that the Øye method, i the curret implemetatio, oly works or the ±90 rage o agels o attack. I the modiied Kircho low ( ) relatioship (lmeth=4) is used, tables o stat, L, iv ad L, sep ca be automatically geerated by a call to the subroutie kirchmake_c. The subroutie kirchmake_c ills the tables sti, civi ad csepi ( stat, L, iv ad L, sep ). I the AERFORE package, kirchmake_c is called at the irst time step to set up stat, L, iv ad L, sep -tables i lmeth=4. The subroutie kirchmake_c also ills the variables o ala0 c_ala, c_ala2 ad ala02 rom which appropriate values o ala0_used ad clala_used ca be selected 48
5 Sub-methods ad values or semi-empiric costats Reported use o the FFA dyamic stall model is the use i the Stallvib project [3], [11]. The model is also used i a sesitivity aalysis [14]. 5.1 A optimizatio study to id optimum values Optimal parameters or the dyamic stall model was sought by optimizatio i [15], [16]. I this work experimetal data rom several sources were used. Values or the semi-empirical costats i the dyamic stall model that would miimize the dierece betwee simulatio ad experimetal results were sought. I [15] ad [16] the Kircho low model was used. The result was that there is a substatial spa i the result o optimum values or the parameters or dieret airoils ad dieret cases. T v ad T Oe coclusio is that with T 2 ad T 5, a reasoable good agreemet was obtaied or a large umber o cases. v A cd Optimum values semi-empirical parameter A cd was oud i the rage rom 0 to 0.43. Most values were below 0.1. A value o A 0.05 to A 0.1 seams reasoable to use. cd cd T p 49
The value or T p =0.8 was used i most cases i the optimizatio study [16]. A optimum value was also sought, but the spa o optimum values was substatial so that it is hard to recommed a value based o this study. The value T p =0.8 was take rom [10] or plugig motio. 5.2 lcddy I calculatios or the stallvib project it was oud that lcddy=2 gave better agreemet with experimetal data tha usig lcddy=1 5.3 Eects o varyig velocity o the separatio poit The model to accout or chages i the usteady pressure gradiet due to a varyig ree stream velocity was itroduced durig the STALLVIB project. Very limited ivestigatios [11] oud a value o the semi-empiric costat u =0.5 to be reasoable. u should, however, likely be a uctio o agle o attack or a uctio o the separatio poit positio. At high agels o attack, a value o u =0.5 might be too high. 5.4 Recommeded values o submodels ad semi-empirical parameters The dyamic stall model captures several eatures o usteady airoil aerodyamics ad dyamic stall. The model is ote able to quite well predict the L (t) ad D (t) behavior. The model is, however, ot a solutio o the low or the dyamic stall process. Dieret airoils might require dieret values o semi-empirical costats to get a good it to the real dyamic stall behavior. Based o experiece with the semi-empirical dyamic stall model ad based o the experiece rom the optimizatio study [16] it 50
seems a bit uwise to recommed values or the semi-empirical costats or a geeral airoil. Nevertheless, such values will be asked or so why ot give a best guess. For use i aeroelastic calculatios with commo wid turbie blade airoils the author s curret deault settig is: T p =0, T =5, T v =2, Acd 0.08, u =0.25 The choice o sub-models should the be: lccl=2, lpotmeth=4, lmeth=4, lvormeth=2, lcddy=2, ldut=1 5.4.1 Eect o aerodyamic dampig, atigue ad extreme loads Sometimes it is good to be a bit o the coservative side i load calculatios. How the do the values o the semi-empirical costats aect loads? Uortuately that is hard to say i geeral. Some rules o thumb could, however, be give. 5.4.1.1 Dampig o lap-wise vibratios i stall Icrease i T p => icrease Icrease i T => icrease Icrease i T v => icrease A cd hardly aects the lap-wise dampig u does ot aect the lap-wise dampig lpotmeth=4 istead o lpotmeth=3 does ot make a big chage 5.4.1.2 Dampig o edge-wise vibratios i stall A icrease i dampig here meas more positive or less egative dampig. Icrease i T p => icrease Icrease i T => icrease T v hardly aects the edge-wise dampig 51
Icrease i A cd => icrease Icrease o u => decrease lpotmeth=4 istead o lpotmeth=3 icreases the dampig. 5.4.1.3 Extreme loads I the extreme loads icrease with icreasig maximum values o, the ollowig holds. L Icrease i T p => icrease Icrease i T => icrease Icrease i T v => icrease Acd u hard to say hard to say lpotmeth=4 istead o lpotmeth=3: hard to say 52
Reereces [1] Aders Björck. AERFORE: Subroutie Package or usteady Blade-Elemet/Mometum alculatios, FFA TN 2000-07, The Aeroautical Research Istitute o Swede, Jue 2000. [2] F. Rasmusse, J. Thirstrup Peterse ad H. Aagaard Madse, Dyamic stall ad aerodyamic dampig, J. Solar Eergy Eg. (1999) 121, 150-155 [3] Jörge Thirstrup Peterse, Helge Aagard Madse, Aders Björck, Peter Evoldse, Stig Öye, Has Gaader och Day Wikelaar. Predictio o Dyamic Loads ad Iduced Vibratios i stall, Risoe-R-1045(EN), Risoe Natioal Laboratory, Demark, 1998 [4] A. Björck, Dyamic Stall ad Three-dimesioal Eects, Fial report or the E DGXII Joule II Project, JOU2-T93-0345, FFA TN 1995-31, Jauary 1996, The Aeroautical Research Istitute o Swede [5] J.G. Leishma ad T.S. Beddoes, "A Geeralised Model or Airoil Usteady Aerodyamic Behaviour ad Dyamic Stall Usig the Idicial Method", Preseted at the 42d Aual Forum o the America Helicopter Society, Washigto D.. Jue 1986 [6] J.G. Leishma ad T.S. Beddoes,, T.S. A Semi-Empirical Model or Dyamic Stall. (A revised versio o a paper preseted at the 42d Aual Forum o the America Helicopter Society), Joural o the America Helicopter Society, July 1989 [7] J.G. Leishma. ad G.L. rouse. State-Space Model or Usteady Airoil Behaviour ad Dyamic Stall. Preseted 53
as paper 89-1319 at the AIAA/ASME/ASE/AHS/AS 30th Structures, Structural Dyamics ad Material oerece, Mobile Alabama, 1989 [8] Bered Va der Wall ad Gordo Leishma, "The iluece o Varaiable Flow Velocity o Usteady Airoil behaviour", 18th Europea Rotorcrat Forum, Avigo, Frace 1992 [9] Y.. Fug, A itroductio to the theory o aeroelasticity, Joh Wiley ad sos, 1955. [10] J. T. Tyler ad J.G. Leishma, Aalysis o Pitch ad Pluge Eects o Usteady Airoil Behaviour, Joural o the America Helicopter Society, July 1992 [11] A. Björck, "The FFA Dyamic Stall Model. The Beddoes- Leishma Dyamic Stall Model Modiied or Lead-Lag Oscillatios". IEA 10th Symposium o Aerodyamics o Wid Turbies i Edigburgh, 16-17 December 1996. Editor Maribo Pederse, DTU Damark [12] Stig Øye, Dyamic stall simulated as time lag o separatio, Preseted at the IEA Fourth Symposium o the Aerodyamics o Wid Turbies, Rome November 1990, Edited by McAulty, ETSU-N-118 [13] Björ Motgomerie, Dyamic Stall Model alled "Simple, Rep. No. EN---95-060, Netherlads Eergy Research Foudatio EN, Jue 1995. [14] A. Björck et al omputatios o Aerodyamic Dampig or Blade Vibratios i Stall Paper Preseted at the Europea Wid Eergy oerece, EWE 97 Dubli, Irelad, October 1997 54
[15] A. Björck, M. Mert ad H. Aagard Madse, Optimal parameters or the FFA-Beddoes dyamic stall model Proceedigs o the Europea Wid Eergy oerece, 1-5 March 1999, Nice, Frace [16] Murat Mert, Optimizatio o Semi-Empirical Parameters i the FFA-Beddoes Dyamic Stall Model, FFA TN 1999-37, The Aeroautical Research Istitute o Swede, Jue 1999 55
Issuig orgaizatio Flygtekiska Försöksastalte Box 11921 S - 161 11 Bromma Swede Sposorig agecy Swedish Natioal Eergy Admiistratio Documet o. FFAP-V-110 Date Jue 2000 Reg. No. 948/98-23 Project o. VU0313 Security Uclassiied No. o pages 56 Order/otract STEM P11556-1 Title DYNSTALL: Subroutie Package with a Dyamic stall model. Author Aders Björk hecked by Approved by Björ Motgomerie Sve-Erik Thor Abstract 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 semiempirical 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. Keywords/Nyckelord Aerodyamics, Wid Eergy, Usteady Aerodyamics, Dyamic Stall, Wid Turbie Distributio 56