495
Tuyen tap cdng trinh Hdi nghi Cff hoc todn quoc Ky niem 30 nam Vien Cff hoc vd 30 nam Tap chi Cff hoc Hd Ndi, ngay 8-9/4/2009
Chuffng trinh tinh toan dong tren am qua miii non
Hoang Thj Bich Ngoc, Le Minh Tri Dgi hoc Bdch khoa Hd Ndi
Tom tat: Ddng trin dm qua miii non cd the gay ra sdng va thdng tach rdi miii vdi mdt miin dudi dm cue bd sau sdng va, hoac cd thi tao nen sdng va xien bam miii vd todn bd mien kich ddng a chi do chuyin dgng tren dm. Hai tinh chdt tuang tdc trin phu thude vdo sd Mach vd cimg vd gdc ndn. Logi tuang tdc khdng tgo nin mien dudi dm cue bd, nghia Id sdng va xien bam miii ndn cd tdn that nhd han nen thudng duac sic dung trong thiet ki. Bdo cdo trinh bdy kit qud chuong trinh lap trinh sd gidi phuang trinh vi phdn ddng qua mui ndn cd sdng va xien vdi hieu icng 3D lam cong dudng ddng sau mat sdng va xien, trong dd gdc ndn sdng va cimg la mdt nghiem ciia phuang trinh vi phdn duac gidi bdng phuang phdp lap.
1. Gioi thieu
Ddng tren am cd nhung tinh chat rat khac ddng dudi am do sir xuat hien cac hien tugng song va lam gian doan cac dac trung eiia chuyen ddng. Phu thuoc vao van tdc chuyen dgng va hinh hge eua mui ndn, sdng va ed the d dang tach mui hoac bam mQi. Khac vdi bai toan nem, dudng cong gidi han phan khu vuc sdng va bam mui va tach mui ddi vdi bai toan ndn khdng giai true tilp dugc tir cae phuang trinh dai sd bac cao, ma chi dugc xac djnh sau khi ed nghiem so giai phuong trinh vi phan bang phuang phap lap. Mui ndn la mot loai hinh dang khi ddng hge dugc ling dung nhilu trong Imh vuc ddng tren am do tinh chat ed the giam luc can eua nd khi chuyin ddng d tdc do tren am. Dau mui cila than may bay va cac thiet bj bay tren am khac duge thiit kl dang mui ndn. Miii ndn cung dugc ii'ng dung trong dng hut ddng ca may bay tren am de giam ton thit sdng va trong qua trinh ha tdc cua ddng vao cac may khf nhiet trong ddng ea.
Ddng qua ndn tao nen sdng va ed cudng do nhd ban so vdi ddng qua nem cimg dieu kien ddng hge va gdc nhgn, tuy nhien tinh toan ddng qua ndn lai phiic tap hon rat nhieu so vdi ddng qua nem, vi vay nhieu trudng hgp dng hut ddng ca may bay tren am duge diing mui nhgn nem thay vi mui ndn. Bao eao nay trinh bay thuat toan va xu ly sd giai phuong trinh vi phan ddng qua non, phan tich cac qua va nhan xet vl dac diim ddng tren am qua ndn.
2. Phuong phap giai
2.1. Phirffng trinh chuyin dpng Phuang trinh lien tuc:
pv sin CO = (p -+- Ap)(v + Av) sin(cD + Aco) [l - Aco cot(co - 6)] (2.1a) trong do: w la van toe tuyet ddi, u va v la cac van tdc thanh phan; p la khdi lugng rieng; 6 la gdc
ndn; co la tia ndn dugc xet.
496 Hodng Tin Bich Ngoc, Le Minh Tri
Trien khai phuong trinh (2.1a), bd thanh phan be bac cao cua (Aco), chia hai ve eho Aco va cho Aco -> 0, CO the viet:
— (pv sin co) -I- 2pu sin CO = 0 dco
Dieu kien khdng xoay:
Trong mat kinh tuyen va he tga do try, dieu kien khdng xoay duge viet:
„ , a x 5Y av ISu V ^ , du ciirlV= = 1 = 0 hay v = —
cjy ax 5r raco r dco Phuong trinh ddng lugng va eae quan he nang lugng:
wdw^-—= 0
(2.1b)
(2.2)
(2.3) Khi dan doan nhiet ra chan khdng, van toe ed
gia tri Idn nhat ('0' la trang thai ham):
" m a x , 1 6 ^ ^ *0
Y - l P o Y - 1
Y — 1
Van tdc am thanh: a" = ( w L , - u ^ - v ^ ) The cac he thiic vao phuong trinh ddng lugng (2.3) ed the viet duge:
Y + 1 1 ^,2 Y - 1 2 w ' 2
. 2 A
1- w"
Hinh 2.1. Cac thanh phan van toe
u" = (y - l ) u
. 2 \
1 - -
w max / y - 1 . 2 \
1 - - yu Y - 1 . . . 3
(2.4) u'eotg(co) -7,— u'" —u'^ cotg(co)
Phuong trinh (2.4) ggi la phuong trinh Taylor-Maeoll [1], la phuong trinh vi phan cap 2 va phi tuyen duge giai bang phuang phap lap. Cac dilu kien bien dugc xac dinh tren mat ndn
03 = 9^ tuang ung vdi v = 0 va mat sdng va o:) = p . Ci day (3 la gdc sdng va dugc xac djnh khi qua trinh lap hdi tu.
Klii van tdc w = a dugc ggi la trang thai tdi ban: a* = w* = w ^
III
Y + 1
• w .
He sd lire can hinh dang va can sdng va tren mat ndn dugc xac djnh:
' 0-5p,V,'
(2.5a)
Chuong trinh tinh todn ddng tren dm qua miii ndn 497
C„ P2SV - P . .
O.Sp.Y/
d day, pb va p2sv la ap suat tren mat ndn va ap suat sau mat song va.
He sd luc can tdng theo cae cdng thirc (2.5) sau khi biin doi cd thi vilt [1]:
(2.5b)
C, 10 7M
i 2
VMI'"^
V J z - 1 (2.6)
vo'i: z =
6 M ' s i n ' P 5-i-M'sin'P 2.2. Xie ly so
7 M ' s i n ' p - l
- 2 5M*- - ^ / 6 u / v v _
; M = = — ; M* = = ^ 6 - M * ' eosp
Viet van tdc dudi dang khdng thir nguyen U = u/w^^^; V = v/'w^,^ Chia mien khdng gian ngoai mat ndn thanh nhung mat ndn ed biide goc h. Khai trien ehudi Taylor ciia U tren tia co:
U(03) = U, + U ' , h + U", — + - + U1,' h'- (n) h"
2! " n!
vdi Ub la van tdc khdng thu' nguyen tren mat ndn.
Sai phan theo phuang vudng gdc vdi mat ndn:
u „ = u , + o.5h(u;-fu;,)
u<„.i„. = u„,.+o.5h(u;,„ + u;„,„„)
(2.7)
(2.8a) (2.8b) Phuang trinh (2.4) dugc giai bang each the (2.7) va (2.8) theo phuong phap lap, vdi gia tri du doan F,^, va gia tri hieu chinh F.^, (F la mdt ham bat ky):
F., = F +
aF^
A x ; F , „ = F , + aF dxidP^ ^
ax Ax (2.9)
i \"^/,+i)
N I U bilt gdc sdng va p, qua trinh lap dugc thuc hien theo sa do thuat toan hinh 2.2. Tuy nhien, gdc sdng va ehi dugc xac djnh sau khi da cd cac ket qua ve phan bd van tdc, vi vay gdc sdng va p la mdt nghiem ciia vdng lap ngoai (sa do thuat toan tren hinh 2.3). Gdc sdng va nhd nhit dugc gidi han bing dudng sdng Mach. Dieu kien khdng xoay tuang duang vdi
V = — = U' va dilu kien kiem tra hdi tu tirong ii'ng tai mat non Gb la van tdc ngang. Phuang dd)
phap lap nay dugc xuit phat tir gia trj du doan gdc song va, phuang trinh dirge giai tren cac sai phan tir mat sdng va din mat ndn, ndi suy kiem tra gia trj gdc non la dieu kien dirng qua trinh lap tinh toan gdc sdng va.
4 9 8 Hodng Thi Bich Ngoc, Li Minh Tri
Gan (1)=(2)
TI.k V.Ii
Wi's(Ujk)-=f(XV,Vjt)
Uj.i>^=Uj'M-l,'2(\'j'HVj,i'91(l) Wj.ii'^fCLVi^Vj.ik)
Uj-ik=I'i'H-lQ0'i'H-VH'=)h(2)
^ J + 1 ? »J.^1
Hinh 2.2. Thuat toan giai phuang trinh Taylor-Maeeoll
M . . 8, h , s
Tinh p „ i , , p5„„ , k = I
P t ^ [^sun.. p n n ]
Giai phno-ng trinh Tavlor-Ma ceo 11
N o i s u y tai &b
t \ , p
j = j + l
Hinh 2.3. Thuat toan tinh gdc song va
3. Ket qua
Tren hinh 3.1 trinh bay ket qua tinh toan va ve dudng ddng tren mat ndn sau mat song va trong ba trudng hgp. Trudng hgp a va trudng hgp b cd cimg gdc ndn 10 do vdi so Mach v6 cimg M„=2.7 va M„=2.2. Ket qua tren hinh eho thiy gdc sdng va xien trong trudng hgp M<„=2.7 la 23.7 do va ed gia tri 28.4 do vdi M„=2.2. Cac dudng ddng sau mat sdng va bj uon cong tiem can ve mat ndn. Trudng hgp b va trudng hgp c cd cung so Mach vd cimg M„=2.2 va cac goc non tirong ling la 10 do va 15 do. Kit qua gdc sdng va trong hai trudng hgp ed gia trj khac nhau la 28.4 do va 31.3 do. Sau mat sdng va, cac dudng ddng bi udn cong do hieu iing 3D trdn xoay doi xiing true.
Ket qua tren hinh 3.2a la phan bd sd Mach tuong iing vdi trudng hgp dudng dong qua non tren hinh 3.1e co so Maoh vd cimg M„=2.2; 9b=15° Doe theo eae mat ddng, sd Mach giam tir til'. Theo phuang vudng gdc vdi mat ndn, quy luat giam sd Mach Idn nhat d mat non va yeu dan ra phia ngoai. So sanh vdi trudng hgp nem cimg thdng sd gdc d mui 15° va sd Mach v6 cung
Chuang trinh tinh todn ddng trin dm qua mui ndn 499
Ma,=2.2, phan bd sd Mach tren cac mat ddng chju budc nhay dot nggt va giu- la hing sd sau mat song va (hinh 3.2b). Ddi vdi mui ndn, so Mach giam dot nggt tir M=2.2 xuing M=2, sau do giam tir tir va tiem can vl gia trj M=1.874. Trong khi do d6i vdi nem, sd Mach giam dot nggt mot lan tir M=2.2 xudng M=1.625. Gdc sdng va eiia dong qua non la P=31.3°, trong khi do gdc sdng va eiia ddng qua nem la P=41.3° Dudng ddng sau mat sdng va ciia nem cd thi thiy tren hinh 3.2b.
„ , M =2.7,9^ = 10°
0.3, " b
b ) 0^
0.4 "^1^ = 2.2, 9^ = 15 0.3
>?0.2
Hinh 3.1. Ket qua gdc sdng va xien va dudng ddng tren mat ndn sau mat sdng va a)M„=2.7;9b=10''
b) M„=2.2; 9b=10° (eiing gdc ndn vdi (a), cimg sd Mach vdi (e)) c)M<„=2.2;9b=15''
500 Hodng Thi Bich Ngoc, Li Minh Tri
1.95
a)
0 . 2 0 . 4 0 . 6x / x .
0 . 8
2.2
I 2
1.8 1.6
b) '
N E M
^^1 - - 1
1
M = 2.2,
CD 1 1 1 1 1 i 1 1 1 1 1 1 1 1 .1.-..-1
e^ = 15°
b
I • 0.4 I : 0.2
1 \ °" .^^^^^^m.
0.5 1
0.2 0 . 4 0.6 0.8
x / x .
Hinh 3.2. Phan bd so Mach sau mat sdng va a) Ndn - M„=2.2; 9b=15°; b) Nem M„=2.2; 9b=15°
Hinh 3.3 la ket qua ve he sd can sdng va va can hinh dang theo sd Mach. Trong ca hai trudng hgp goc non 10 do va gdc ndn 20 do, he,so can hinh dang giam khi tang sd Mach con he sd can sdng va tang khi tang sd Mach. Tuy nhien tdc do giam eua he sd can hinh dang Idn hon nen trong ea hai trudng hop a va b tren hinh 3.3, he sd lue can tdng giam. He sd luc can tong cua hai trudng hgp a va b tren hinh 3.3 dugc so sanh vdi cae ket qua eua Maecoll [2] eho thay do chinh xac ciia ket qua lap trinh. Ket qua so sanh nay cung cho thay trong eae trudng hgp ddng tren am qua miii ndn 10 do va 20 do, he so can nhdt rit nhd so vdi he sd can hinh dang va can song va. Ket qua tren eae hinh 3.3 va 3.4 cung cho thay, he sd can tong tang mot each ro ret khi tang goc mui non (khi tang gdc non tii' 10 do len 20 do, he sd can tang khoang gap 3 lan).
Quy luat cae thdng sd ddng tren mat ndn phu thuge vao goc sdng va. Goc song va trong trudng hgp mui nem cd the xac djnh qua mdt he thirc phu thude vao gdc nem va sd Mach v6 eiing. Ddi vdi ndn, gdc ndn sdng va phai xac djnh qua viec giai phuong trinh vi phan nhu thuat toan trinh bay trong muc 2.2. Cae thdng sd ddng phia sau mat sdng va phu thuge vao gia trj goc song va, goc nay dac trung cho muc do manh ylu cua sdng va. Hinh 3.5 trinh bay kit qua tinh toan goc sdng va phu thude vao so Mach vdi gdc ndn 10 do va 20 do. Ket qua tinh toan cua chuong trinh so sanh vdi kit qua thuc nghiem eiia Maecoll [2] ed thi thiy rit gidng nhau. Dieu nay cho phep kiem chiing do chinh xac eiia chuong trinh lap trinh.
Ket qua tdng hgp vl gia tri goc song va phu thuoc vao gdc ndn va so Mach tinh toan tir chuong trinh lap trinh dugc trinh bay tren hinh 3.6a. Kit qua tinh toan tuong irng eiia Kopal [1]
duge trinh bay tren hinh 3.6b. So sanh ha;i kit qua nay eo thi thiy dugc sir phii hgp eua hai each tinh. Tren hinh 3.7 la so sanh kit qua vl quy luat phan bd van tdc khong thir nguyen tren m?t non theo su thay doi eiia gdc non va so Mach vd eiing duge tinh tii' chuang trinh lap trinh va ket qua tinh toan eua Kopal [1].
Chuang trinh tinh todn ddng tren dm qua miii ndn 501
e = 1 0 ° - • Can song va Can hinh dang Can tong cong
o
0.05a) 0
1.5 2 0.4
2.5 3.5
e^ = 20°
4 4.5 i
• — Can song va
" - " Can hinh dang Can tong cong
0.9 0.6
O I 0.3 I
Hinh 3.3. He sd luc can sdng va va hinh dang theo sd Mach a) Gdc non 9b=10°; b) Goc ndn 9b=20°
HE S O L U C C A N - GOC NON 2 0 °
KQ tinh toan hien tai I O KQ Maecoll [2]
-@—o O - O O o o
O
HE SO L U C C A N - GOC NON 10°
— KQ tinh toan hien tai O KQ Maecoll [2]
Hinh 3.4. So sanh he sd lire can tdng tinh toan vdi ket qua eiia Macoll [2]
502 Hodng Thi Bich Ngoc, Li Minh Tri
GOC SONG VA THEO SO MACH 701
I
60 r
^ 5 0 | 401
I I 30'-
Hinh 3.5. So sanh ket qua tinh toan gdc sdng va vdi ket qua thuc nghiem [2]
a) Gdc non 9b=10° ; b) Gdc ndn 9b=20°
QUAN HE GOC SONG VA-GOC NON
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 91
KQ TT KOPAL Til
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m m
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ll
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m V/
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M S^
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A
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0 3
^
.^
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6 3 2 /3,C
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^ '
A
.-^
1,4
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2.8 -2.6
-l,B
p
10 20 30 40 50 60 70 80 90
<r= shock cone half angle • dsgrsM
Hinh 3.6. Gdc sdng va theo sd Mach va goc ndn - Ket qua lap trinh va tinh toan [1]
QUAN HE U^ - GOC NON o
Hinh 3.7. Van tdc tren mat ndn - Ket qua lap trinh va kit qua tinh toan [1]
Chuang trinh tinh todn ddng trin dm qua mui ndn ^^^
4. Nhan xet
Chuang trinh tinh toan ddng tren am qua mui ndn vdi eae kit qua duge so sanh nghiem chiing vdi cac ket qua thuc nghiem va tinh toan da cdng bd dam bao do chinh xac eiia chuong trinh da lap trinh. U'ng dung chuong trinh lap trinh tinh toan mdt sd trudng hgp cho phep rut ra mot sd nhan xet sau:
Gdc sdng va eiia ddng tren am qua mui ndn nhd ban gdc sdng va qua mui nem eiing gdc mui nhgn va sd Mach. Gdc sdng va nhd khi so Mach Idn va tang khi goc ndn tang. Ton tai mdt so Mach cue tieu va mdt goc ndn cu'c dai tuo'ng u'ng vdi hien tugng sdng va xien chuyin thanh song va thang tach khdi miii ndn.
Ddi vdi ddng tren am qua mui ndn, sd Mach tang tuong irng he so can hinh dang giam va he so can sdng va tang. Tdc do giam lire can hinh dang manh hon tdc do tang luc can song va nen lue can tdng giam khi sd Mach tang. He sd luc can tdng tang khi gdc ndn tang.
Sat sau mat sdng va, dudng ddng cong manh va tiem can ve mat ndn d xa vd eiing. Trudng sd Mach sau mat sdng va thay ddi theo ea phuong dirdng ddng va phuong vudng gdc vdi mat ndn.
Vdi do chinh xac eua chuang trinh lap trinh da dugc kiem chiing, chuang trinh ed the ii'ng dung tinh toan nghien ciru cac trudng hop ddng khong xoay tren am qua ndn vdi sd Mach vd eiing M „ = 1 . 5 - H 5 . Khoang sd Mach M„=l-^1.5 thude loai dong qua do am [3], [4], nen ii'ng dung ly thuyet ddng qua do am dam bao do chinh xac eao ban, diing ly thuyet ddng tren am trong trudng hgp nay do chinh xac khdng eao va khi sd Mach gan gia tri sd Mach don vj, tinh toan cd the khdng hdi tu. Mien sd Mach nay cQng tuang duong vdi che do sdng va tach khdi miii ndn [5].
Tai lieu tham khao
[1] A. H. Shapiro (1983). The dynamics and thermodynamics of compressible flow. Robert E. Krieger Publishing Company, New York.
[2] Edward R. C. Miles (1962). Supersonic aerodynamics. Dover Publication, USA.
[3] Hoang Thi Bich Ngoc (2007). "Influences of the compressibility on aerodynamic characteristics of profile under the transonic flow theory" Vietnam Journal of Mechanics, VAST, V. 29, No. 4, pp.
497-506.
[4] Hoang Thi Bich Ngoc, Le H6ng Chuong (2006). "Tinh toan s6 ddng qua do am bing phuong phap giai phuong trinh thi diy du" Tuyin tap Cdng trinh Hdi nghi Ca hoc todn qudc vi Ca hoc ky thuat vd Tu ddng hda, HaNoi, tr. 171-180.
[5] Hoang Thi Bi'ch Ngoc, Trin Thanh Tiing (2007). "Chuong trinh tinh toan dong qua do am qua vat thi tron xoay" Tuyin tgp Cdng trinh Hdi nghi Ca hoc todn qudc Idn thic VIII, Ha Noi, tr. 367-378.
