CC BY
5Journal of Siberian Federal University. Engineering & Technologies, 2017, 10(4), 523-533
УДК 532.528
Solution of the Problem of Flow Past a Wing Profile Near the Interface
Ludmila V. Kulagina", Vladimir A. Kulagin" and Feng-Chen Lifc*
aSiberian Federal University 79 Svobodny, Krasnoyarsk, 660041, Russia bSchool of Energy Science and Engineering Harbin Institute of Technology Harbin 150001, China
Received 26.02.2017, received in revised form 14.04.2017, accepted 26.05.2017
In this paper presente some approaches to solving the problems of flow past a wing profile near the interface. Studies on theflow in the vicinity of separation boundaries (solid walls andfree surfaces) show that in a confinedflow liquid turns into a bubble mixture of liquid and gas. This complicates the flow analysis and introduces additional losses resulting in impaired energy performance of the concerned mechanisms. In a general case the problem of a two-phase compressible flow around various types of vane mechanisms is substantially nonlinear (even under no-vortex flow assumption).
Keywords: flow around actual foil-shaped profiles, supercavitating mechanisms near separation boundary, potential theory, method solution.
Citation: Kulagina L.V., Kulagin V.A., Li F.-Ch. Solution of the problem of flow past a wing profile near the interface, J. Sib. Fed. Univ. Eng. technol., 2017, 10(4), 523-533. DOI: 10.17516/1999-494X-2017-10-4-523-533.
© Siberian Federal University. All rights reserved
Corresponding author E-mail address: klvation@gmail.com, v.a.kulagin@mail.ru, lifch@hit.edu.cn
*
Решение задачи обтекания крылового профиля вблизи границы ряздела
Л.В. КолЛгинаН В.А. Кплигин", Ф.-Ч. Н1е6
"СиГирякий федеральный университет Россия,660041, Красноерск, пр. Сйобкдныи, 70 бШяола энергетических наук и техники Харбинссий аехсолагический инититут Харбин 150001, Китай
В настоящей работе пре дставлены не которые подходы о ре шениюзадачобтекания крылового профиля вблизи границыраздела. Наследование потака в близи енин иц раздела (сплошные стенки и свободные поверхности) покаеыеают, что в юамбнутом потоке жидыспть препращается в сузыръковую имосъжидкасти а еиза. Это усложбюбтаналез пстака а аноаит дозолнзтбльныв оатбри, привоаящие к лсудиеееию бнеенгстичестзх епыоктеаисыик соътвзтстпрющнзмваапизмсв. В общим ееупас проблема УсухпСазного сыомаамогн тмвонис аекауе разлеоаых кзипое ловастиыз мехкллнмон ссщегтоонно нвеамейса Обажо оюо продпалюжрсиг отсутстъик всхрспесо пытакае.
Ключесае слоан: оСташи^ ищуюсвоха ороХаля, суоерхавитационных механизмы, орпезцы разлата, тлищиа антвещало, метиС мишснис.
Introduction
Most of the problems in potentialtheory [l]c angenerally be reduced to solving integral or integral differential en[uations oo sees of such equationr [25].
Inihe cass of small nerturbations,the solutionOothe ptobiem comesdown to solving the integral equation[3,4]
¿H*)}^ " ^ (d)- 2tt fx (1)
where ] ofd^ are funetioni 2f rhehydeafki1 otape and kernel] A and G= are given by Gr =(W--t)^ Ab G^e/A; A((-s)2 -ae2; e = 4/t(0). The foil shape is defined as F = FH±Fc, where nH lis the centerl ine <sc[^^rion and pf is thotluodness distr^r>u^ion. In Hoot first approximation of the no-penetration boundary condition we obtain:
a) = -a + f=anbn , if FH -O^1;
0 o n + 1
m M M b M b
b) t\=) = 2w($Yibn + +2w'(£) if Fc =J)-b+n^w^), where w = 1 for a
shaep leaditg edge and n = <Jf - 0 for o rounded one.
Enuation tl) isanintegral Fredholmequationof thefirstkindwitha singularity in its kernel
Jk(s, e))ds = f f). (2)
a
The solutiony of Eq. (2) ioknown [6] robe unstnble evnn an small errorm inf(Q data and sensitive no artmns in the knrnel Ugh sc) bnd at is vsrtuany independent ol lihe noevlog technique.
The gaololem aktelii (l[, It essaneMlg fn-eKfineie a_re<h neqmeak jeooater regularizatioftnchniques to eeob.e tts^ sofenifhn [6]. Wtont mofzs it ninzosed whrn h -r 0 is thai ^rma^i6!^1 eeruotion (0) degenerates to neiffrseetioe eeuation lha( is lineae ^tilCi eespect Oo the hightr dorinaedve. By means ot unsophisticated mhthematizn eei 4] ibtegral equation (1) of the flow boundary-value problem is rewritten as
ii^f -n) (5) (Cl tW^y® ^ M=® (). (3)
hOrsr >»(§) = -Fp (5) -W- (W ( (W" (e)'
WT (- W)= j)-^;
-l+O u fc
,]-, ,=|{(L-cf - .b+rgg)
1—— 1 +— W0 ]], e) = [ G2as = eeolh —-+eeolg —-;
—i £ e
((, 8^Jff-rfGf 8 = 4h .0).
-i
From Eq. (3) it follows that whan h — 0, prolate (1) reduces to a boundam-layer-type probirm, which shedh llglh") on why ahe problem id ill-posed. The physical conditions impose limitation on °f.
This probiem crn be oq>)^ed- emptoym1 o hybrid ai^prctac;li. Ftisl, solutions lo (ho enlorior, (1), and inieolor, (lie, otobtemsare found [2, 3] and then these are mutually adjusted.
Those ir yal another way to tackPe theproblem. )t starts otsi(i ronstructiugopertusbed exterior solutions pintl tlien (ltip solution, whtclt ii; ilioi" h -e 00) iii tranvformod or thot it is oble to reveal
thy naiure of rmgoter ^s^. Tha resuliaid sohotion (Iiuo bol omei umfosmO1 ealkH tvr eywheseand provides good apnrouimatien to tlie tiue saiutlon. The sotption ciei be further rmyroved quantitatively via higherianier aperoximatiyne onas finaslr(( eioniln^as me^lirjcii ctris lis attyii^d to accoierateconvergence yf tiiafudctiondisehuenoe [1,0].
If iliehec^onye rsrllvei-ghii^, wluwle dopends on Scw ctai;s o(i tltie fousdionPCO) ibill,1fen the linear transformationi i^rccl rtct^ltt;C^sor dteeurern arnve ;slio^ tbo netm term rf tlto sequo^ae do be extracted. Wi^CCe tire aahitinn oil iintpyral equalucn (i) tsr e ^ Yii" ati- OWt^e first trem is associated wbbh the influence or tier oedterlme shape on dictation, wttide tisy siecond ouo w cttdrtmie0 to thedynetmc curvature reluhmg l1fiint t.se flww arouosl a jooofite ^r;s,r ilie lactation tioiiidiJ.ECPPjy.
The functiona1 parametermethod
Re pre ronr y, ioiution in ate eoim o+ a t = he + 4tS2 -htt seai-s obtained by mapping hett> ce[l,0]:
^ = £((1)^. (4)
m=0 v '
There isanexpansionforthekernelwithrespecttoparameter t such that:
G =tkm T2" , (5)
m=1
with k1m defined in [3].
Series (5) is convergent. Moreover, it is convergent over the entire actual range of variation of the parameter r. Convergence of series (4) remains questionable because it is not possible to construct a generalterm.Itishoweverpossibletoevaluateaconvergence domainforspecific foil shapes.
Substituting (4) and (5) into (1) and resolving the solution into two terms (terms of the same t power are taken equal) y ields a system of singular inte gral e quations with a Cauchy-type kernel:
W = 0' a 2, ... W=1' 2' ... (6)
^oitiver^ir^j^ thio equotion into class functions wehave
>) (=-_± EI+f El^to (7)
The function fm = 1, 2, . .., AN) is foundfrom the boundary conditions via the solutions 2i";L ) ( = 0 2, ..., N-l). -etus now writethesolutions for N =13 and n = 1 for a plate when
F$= w •
0<)) =-2Tc<x; o« = {yf^d,;
Substituting (8) into (7) gives
^) = 2a|=| (9)
where m = 3 , 2,..., 13;
105 3 „ . 53,, . ++3 23,4 35 ,5 9 ,7.
- :52^<5 -
TTT U55 217, 145 ,u 19 ,3 165,4 33,5 27 </^.,<-7 dl ,8 ,9
Wm = 256 + 128'' + ~^ ^ ~^ + ~^ + ^ + ~^ + ^
W = 1227 471, _ 617£2 _ 2553 ,3 _ 889 ,4 305 ,5 415 ,6 _ 3,7 _ 275 ,s _ 12 = 1024 + 256 ^ 256 ^ 128 ^ 64 ^ + 0 ^ + 0 ^ ° ^
_ 45,9 13 „,10 ,11. 2 ^ 2 1 q
w14 = +i1943,4 + _83f,6 _
2613 263 , 5825,2 1631,3 11943,4 10773 ,5 _ 837 s6 553 s7 '14 2048 1024 ^ 512 ^ 128 ^ + 256 ^ + 128 ^ 32 1295 ,8 . 305 ,9 , 273 ,10 , „,11 , 15,12 , ,13.
-,8 + ^ ,9 + ^ ,10 + 33,11 + ,12 +
16 8 4 2
W = 36297 5617, 20883,2 27063,3 99597,4 6939,5 73617,6 16 = 32768 2048 ^ 2048 ^ + 1024 ^ + 1024 ^ 256 ^ 256 ^ 20669 ,7 + + 32835 ,8 + 5325 ,9 + 1035,10 _ 101,11 _ 945 ,12_ 91,13 _ 17,14 _,15.
128 128 16 16 8 8 2 2 W =4500439 2374493, 81825,2 295437,3 13037,4 10018325677,6
18 = 65536 + 32768 ^ + 1024 ^ + 2048 ^ + 128 ^ 512 ^ 128 ^ + +165239 ,7 + 33753 ,8 + 2049 ,9 20049,10 594,11 + 77,12 + 294,13 +187,14 + 256 32 8 32 s s s s
+60,15 +129,16 + ,17;
W = 12616275 5179823, 8336061,2 5550025,3 315945/ ,4 /45381,5 +
12616275 5179823, 8336061,2 5550025,3 3159457,4 745381 ^ '20 262144 65536 ^ 65536 ^ 32768 ^ 8192 ^ 1024 149741,13 1885,14 1151,15 2223,16 _153,17 _21£l8 _,19. 1024 4 q 2 q 8 ^ 2 q 2
_ 132266203353 317275015003, 158781624145,2 28124698217,3
W =__1--, +--,2 +--,3 +
22 131072 262144 131072 s 16384 s
112600162755,4 67212868461,5 4202590712705060269,7 2688570425,8
+--,4 +--c +--c0 +--c' +--,8 +
65536 32768 2048 2048 s 2048 s
80850475,9 41139813,10 6691095,n 5094639,12 299933,13 66759,14
+ 265 ^ + 128 ^ + 256 ^ + 256 ^ 64 ^ 16 ^
1085,15 20835 ,16 8091,17 1575^-,19 23,90
2 , + 95,19 +y, + ,21;
_ 17129979402909 1241572992573, 1889838437643,2 1355616985531,3
W =__i--, +--£2 +--P +
24 4194304 262144 524288 s 262144 s
1311925791219 ,4 396503371015 ,5 388978944203 ,6 106117200107 ,,
+--£4 +--,5 +--£6 +--, +
262144 65536 65536 32768 s
58073081641,8 116789059,9 337015355,10 94300235,11 28915279,12
+ 16384 ^ + 512 ^ + 512 ^ 512 ^ 512 ^ _
4595813,13 612715,14 1086565,15 9434469,16 13413,17 45135,18 _
64 ^ 16 ^ 16 ^ 128 ^ 16 ^ 16 ^
13185 ,19 4301 ,20 23 1 ,21 25 ,22 _ ,23. 8 q 8 q 2 q 2 q q ;
!f we do not go beyond linearized formulation of the problem, then with (4) and (9) we can find the resultant aerodynamic characteristics - the lift force and pitching momentum coefficients - using the followingformulas
C^jVW; cM^jV^s )sds. (10)
In the first boundary conditions approximation, at n = 1 we get the functionsofinfluence of the distancenanameter c(0) =n <cf Cm iit tha foem cn x-se riet:
(1) c1 2 3 3 223 4 1 1 5 39 6 163 7 5491 77 -++T = 1 + e+2e + 48 + 328 + 1668 + 648 + 256 8 + 8Ï2
' yco
3Z:X04|.^6j7O-7 i 5379225 10 14Q ^ Q0000 15 „ 7641989002175 12
"7 32738 65536 X + Q24788 8 + 2097152 8 +
37205275391907 13
+--8 1 3 + '
16777216
(11)
M) Cm + 7 7 2 7 3 15 4 15 5 29 <5 899 7
VM = -TT-= 1 + 7T£ + —£ 2 -I- — £ 3 + —2 4 + —£ 5 + 8 „ + 0 £7 +
CM C() 2 2 1(5 32 32 64 2048
7 Moo
3587 g 1923 15 9 1 0-43301 10 ]L-2962<512—86(6221 11
---£ 8 ---£ 9--£ +1--£J1 +
8 392 16384 6536 5524288
2091064690271 12 46 27590 7 6 9 55 70263 13 + 220971522 £ + 8388608 £ +
where = C0 = -na, <=C = .
Under formal application of the discussed method, the solution y given by (4)(9) is unstable [6]. It is considered that a coarse solution can be obtained taking just a few terms of the series; a further increase in the number of terms will only enhance the instability and the resultant multi-term series will have nothing to do with the true solution of Eq. (1). If however this instability is treated as a transient process, then it can be asserted that increasing the number of approximation terms provides additional information on the behavior of the true solution.
Let y(£) be an analytical function, then expression (11) can be treated as a Taylor-series expansion. The asymptotic behavior of coefficients of this expansion y = Ea„£" is determined by the type of the function singularity. So itis naturalto look for away to deduce singularity parameters from a limited sequence{a„}ofcoefficientsoftheseries [7].
The radius of convergence of the Taylor series is determined by the singularity of the function v|/(e) that is closest too the point around which expansien is performed. Farther away singularitiesmay appearto influeaue rhe behavior of the series coefficients as well. If this is the case, one should find the spectrum of singuterioks.
Thealgorithmsconsideredaboveofferasolutiontotheproblem ofy(£)synthesis.
Numerisalsoliilion of Eq. (IS.
Seteeret pepsahion (a) can bn oqived lay reaucing ts Os u aet of olgebraic squxtiens Sdiscrete sinauieriaisr mehhed, folionation mnthsd and eihorsgThe meniionqd mepteoda prs stroqgly unstable [6] ee they saa eonveatagna( qopdrftnei fgrmulae. Attempis la mjiiyve tltf reouh b— mcreaxmj* the number eri uoif s wAnp se—opiy aggravate tlo oitus—on.
Sa^r^iii(]r of singulsfily nxosation (d) goilet ten rhgutarizgnion t^itlratclu^s, ience apron es chxle of regu)arizatos can esssntiany nnhance stability of the computation scheme.
Tiie strucPuse yf Oli soPltisu x(^) = 1(1-^)/(l + £)> where v()) is a non-zero regular function for ^ = ±1, is determined by a specific behavior of the edge flow. After substitution of this solution into(1)andsomesimplemanipulationswehave
2 +,1 /r/h^H'
wlere 5) = S/A tind ® () is the right-hand side ofEq. (.1).
Withthe noclis, referencepointsandquadrature coefficientschosenbytheformulas
5,= cose,. ; Sj = coe0,. ; 0, = 3r(2i-1)/(2N/+1), (/ = 1, 2., . .., N) ; 0. = 2nj/(2N + 1);
A = 2^/^/(2=021) , //= S 2, ..., N),
wegetth/ foil c^-v^ing of algebraic equations
^NAj = ba (/, j = a, 21, ..., iVis tlib number =f nodes),
(12)
where
4= ,st, e)/1 -Sj) ; ]r>=(i-m>+ 1 ; b =-S,() )/e2-|> Ojfjjvfa, sj N ))(dC + i) ; V = i/A ; A= ( - Sj )d + 8d
Aerodynamic of m doC are dqrived fcom fosmulga ( 10). Switching to the quadrature
fogmulas gioes
Cy = 2Ji£7, ijnri CM=2rcf;YjSj-
j=c j=c
(13)
The BCF olgrrithm is f jnoTpr-ee^ul too1 to handle mtegrals unsO resoUoe sods o° olfraic equations dud to th) qutUrsturu and cabtturc formurcs
5 = Hi
k=1
(14)
(Pk - weights, ixk - noegns, R - remainder) and ltie Nednshkovsky-SkonSiO^tko IBCF method [8] ocling os rn^ilarsnalons in tdo orate ds. Thu advai^thi-see of thio tec^ericiori; Sn, 9] is Ihai itis inherently self-negulatizetr 0ecause of mvtuaT cavngllalioo of compuiateonai errori and fs oucg (d (d 1 ittle sensitive to advaesr chvnger in hltia; ceeffieknt mntnn donttlrioniag. d gooO rnnvh is vtetoined when t he BCF method eo emgkyerl to fccelerare donvargeece on find ace adtllmid of saquenaen hv theg end] r set of equations is aonidrucieV -gsilaSi motrixes arf ToeoHlatypn ire]. ^0, fon finding thso: Stanks-Schnudt transformation oic we havn
(15)
5 Sn
+1 AS ... AS^ = Sn+1
+1 ^nn. + "^n+k.-l zk+i Sn+k
Onct(15)hor Seen solve] hravsfoimation foUows t-o formula
,n =1/( Z + + %+i).
The TK n transformation is implemented using the following system
se 029 -
(16)
1 A; AJn; AJnk~l 1A+n a„+i/„+1; ...; a„+i|(„+1)1
1 An+kn A„+kln+k;...; An+kl(n+k)
where An = ASn, An=nASn, An=ASnASn+l/A2Sn for Levin's t, w-, and votrrnsfcrmations, resp ectively.
Actuady, wo ran Uont ourselves to finding only the first two components of solutions, Zj mind yf, os rest Ho components are required for analysis of the transformation spectrum.
Transformetion of nivergent sgfuences and ofriys
f0 ochitioy of Eq. (1 ) ltoa Fe em olatainad in iht form of a serieg, tlc^ oequrnce "b.^) (( = 0, 1, 2t ...g, rt a rgHe, ii drse ontqf. aummotiion o:f snch feeuti and sqoueugos is? done usrng
nonlinear trauyformhtiono. StliLii^^ ng^dientaqion SOiStsSr-e- ^ c^e"0", whese a* is an arbitrvrn
, = i
compley number holdh foe y<SS„}, thrn bn ancOoyy witd nle Fouriec-asries, the Fire summation is applieU:
Qn=rqiqsq •
n 1
For rlie series
qr-n-i + n i rrs)
¿=0 s
Sis)^^t;im^s> we osieo-yv phe toenOion oftie rindnlariSn i]L], giving rlno to diiveojfoncn of tds snri(o, on tho gois or in tne domnm of votiosien oU ite nofsmptet e. U i0o'v,(2^jsir^t gnries gee Se conver-qd ioOo pn tool^ii" or cmndii1^0]si^ltii convengene ere by intnoantinn iotess^e onhon comjsti^risoot qunctinn. Tlg Eu(eo taansfonmatian [IS] eoS)/ tl + s) 01 n f(f 0.5fl to tCe sioriei in (ll_)yields new
orrie s
k j
where Aa0=a1-iq0; O2a0 = ae-2a€ + a0 ; .. . ; Aka0= 2 €k-_,- (-1) c= ; i s tl^«; g)j?erat^or offinltq
a—
differe nee st er srddrtt, ond C+ are tie leino mlal coefficient! t^i^triti^i-i^nrtnsr llte procrdnre e = (/(i-n-Is) it is ftasibk Oo iventualld obtain a uniformapproximalion foe tine coluiional t(e extamsr of the apecifieO
— er (
onleroal h . Thesource qerie ii re Oo 0 e repqesentey in t(;e Is^itirn (= ("!) ^n0.
r=o
Aftnr groapin0 floe ttrms ^itlln tOo seme pewoe a and dropping the terms of asdnr we
fieally ob^^i-ii the fe1 lowiy apptoximnting expressions for the foil lift force and pitching momentum at ao Infodnro fuaction ofthr /2 intervalatsmallanglesofattack:
Yi " Sn
= Sn+1
yk+1 _ Sn+k _
. 7 _ 3_h IP -3 775_4 177_5 1403-6 113230 _7 715923_8 V= 1 + _ +3_2 +o-4-e5^ +-32-fe4 7f16P 595 °-60T!i^ + "252T^7 +-819tTe8-
.787O829_9 56910769_10 530324997219_„ 3092612--25699_12 ,
-I-_p9 +__10 +_P11 +__12 +
32768 65536 524288 2097152 + l8879734589793A7-13+ .
-73h72i6 _ h' -20)
_ - n_ 2 3P_3 1261_4 239 o5 959_6 00670°. 483543_8
a^M +_8if+e +1S0e +03-e +-32-° +TT8 +°8f48T8 ^-TH-T-8 +
72in8939_9 2^0332213755 _10 128639927981 _11 7737516336203
+-_9 +-8 +-811 +-812 +
16384 65536 524288 2097152
+240090786810183_13+
8388608 £ '" '
Whilo thyee losles aoo slow to converge fgr large h, they are uniformly valid on a lairger d^jonj^iLn. Therefore when nonlinear methods of convergence acc 7leration sre ^mjgjl^ncsd eve5 AhCen'r trcnpformation can yield the bert ofproximgtioni -witli am ay-ura-y nf uj to 23 significiinl figures hor A e [0.02, 0.t] T3e aa=uracr imppovss for larger /1.
Let us new wriSo on onafytieal ^^^«^sJisior^ fisr Che anquency o1 n(Tn7 == 0, 1, 32, ..., N-1) derived ftom the series v- = anin.
Conotrnctinh a hequen=o of]portial sumo enX nurstiOnting io into (-1)8okSn > 0,
k
where z3n = ^C e)0 Cf Co ehe opnraSor o0 ocntrel 0h0erence oC order b Ci are; tlie 3inominal coefficients,
= V-
wngcr '
^ftf) (((), (21) 0 i
where 1 = a0cin n ba h a^ - cec^ ; ....; bm = aman -1 cim_°an-°.
If ln (2]) 1 -h-a = 0, tlrsc (he nominvtoo of lhe fraction i+hoeil<^ 0><s expmine0fe2 zeiiics. Suppoee, qhvre is h^ = so^ri emonc the palynomiai zeroes; tleis dfheet can be readtlf CToed. When oKp is a pole, we shouldrel tp n = Oj . Comolimes oopearancv of a pote mdihatai ihe llmirtof serins convergence. Thoo fot ilie; vie series from (H) taese sat £o = fl„/m„p1, which agrees with the results obtainedbyothertechniques[3,4].
The amount of information that can be derived for a given number of approximations does not appear to be enough for a transformation to ensure the best approximation over the entire domain of variation of parameter e. We then have to approach the true solution via various transformations for each h value.
Thr nhankhsrohmirilt-Levin transformatkmi aire; bascd on rational rpproximation of the series '"^akxk. They ccmplement each other as their representation eraor ts atsociated with the Loran-series nxpanpion aflhe fuac loon
X X
f (z) = n akzk +n bkz-k . (22)
k=0 k=1
The first term in (22) refers to errors under Shanks and second one under Levin transformations. In a general case, it is necessary to have available these transformations along with the recursive techniques for their evaluation to be able to automatically monitor the situation. If it appears that the
Shanks transformation fails or is slow to converge for n -»go, i.e. Wo^. p-od e J>e, we should then switch to the Levintransformation.
If Fire summation (19) is appliedtoseries (11) and (20) foUowed b; summation for each h usmg Shanks-Schmidt and Levin's algorithms then the result we obtain will agree with that of numerical simulation. The best analytic approximations empleymg transformation of ak„ strie s (20) are obtamed with rational fractions Oi, 3; Oi, 6; o2,7; 03, oi ce3so The p°ure ^unets the mflucnee function v|/y as eualuatnsl by (11). Also shown are the results of Fire summation applied to series (20) followed by summation using the Shanks-Schmidt-Levin algorirhm. Comdututiout reu.lts basedon )his aigercthm are iuneuy good agreement with the numerical reuuSt fnr intec^rrl oquation (0) solvvd by llii collocatihofeihnique using BCF apparatus as prescribed 0; tho uven rlgsrithm. The fous-lgrm expansion v|s» from ()1) is uniformly valid over the interval h e{hte , hin"1), the quantitative result, however, being far from exact.
Conclusion
Optimal asymptotics contains nine terms, and still it rapidly deviates from the exact solution; however an approximation such as o3, 8 for h < 0,01 already gives a relative error less than 1 %.
Fig. The function of influence of h interval on the lift force: L - influence function derived from (20) using
Levin's transformationforeach h -Numerical simulation results for Eq. (l)obtainedby the collocation method using the BCF apparatus
The discussed algorithms have been implemented as computational programs for algebraic (ALFA) and integral (OMEGA) equations, ordinary (SIMP) and improper (SECOB) integrals, including the Cauchy integral (DSECOB) as well as summation programs for sequences and series (SHENKS, AYTKEN), including divergent ones (EULER).
The reported study was funded by Russian Foundation for Basic Research, Government of Krasnoyarsk Territory, Krasnoyarsk Region Science and Technology Support Fund to the research projects № № 17-48-240386p_a and 16-41-242156p_o$u_M.
References
[1] Van Dyke M. Perturbation Methods in Fluid Mechanics. M.: Mir, 1967. 310 p. (in Russian).
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