NUMERICAL SIMULATION OF WING-BODY JUNCTION TURBULENCE FLOW. IChm,~ Sbtp S~'icntilh" Rc.seurJ~ Centrr. ~luxi. Jiangsu)

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1 Applied Mamatics and Mechanics [English Edition, Vol 14, No6, Jun 1993) Published by SUT, Shanghai, China NUMERICAL SIMULATION OF WING-BODY JUNCTION TURBULENCE FLOW Wang Xi-liang (t:~t~) Hc Mo-qin (("I i'~,t:k) IChm,~ Sbtp S~'icntilh" RcseurJ~ Centrr ~luxi Jiangsu) (Received Feb 10 19q2: Communicated by Chien Wei-zang) lline-t~,dr I~ttl~'tion turt,uh'm'c /low is ~inmluted by using R4NS rquation and t~oumhlrv /ittrd ~~rdtmttt' tr Three or~h'r di[j~'rcntiul ~r i~ used in ('olltflttt~lltolt of tonvccliott term dttd two htvcrs turhuh'ncc mo~h,i are employed m ruh'ululitm The one-dimensional problem of motion of a rigid flying plate under explosive attack has an analytic solution only when polytropic index of detonation products equals to three In general, a numerical Key words analysis corner is required tlow wing-body In this combination, paper, however, RANSequation, by utilizing turbulence "weak" shock behavior of reflection shock tlo~,, in explosive products, and applying small parameter purterbation method, an analytic, first-order approximate solution is obtained for problem of flying I plate Introduction driven by various high explosives with polytropic indices or than but nearly equal to three Wc consider flow past acylindroid mounted on a flat surface at zero incidence to a uniform stream This simple model shows phenomena associated with that of junctions in practical configurations, such as those in wing-fuselage intersections in aerodynamics, appendage-hull junctions in hydrodynamics, and blade-hub junctions in propellers and 1 Introduction turbomachinerv The problems are complicated indeed as flow involves interacting shear layers Explosive of quite different driven flying-plate length and technique time scale, ffmds phenomena its important of vortcx use in formation study of and behavior separation of materials Thc corner under intense flow could impulsive be divided loading, into shock two synsis kinds of The diamonds, first, and boundary explosive layers welding on and both body cladding and of wing metals are The shear method flows, of estimation this is comparatively of flyor velocity easy and to solve way of without raising it consideration are questions of irregular of common vortex interest formation The second, boundary layer on body is turbulence Flow When Under wing is assumptions blunt or of one-dimensional flow comes at a plane high detonation in,:idencc, and rigid flow flying tends plate, to be separated, normal and hard to be dealt with Thc physical phenomena and flow mechanism are being studied governing flow field of detonation products behind flyor (Fig I): widely: When wing is in a good streamline'and problem becomes solvable Presently, computations of such problem are usually carried out on huge computers But thors h/ave performed same example ap on +u_~_xp personal + computers (IBM PC/AT) with an effective extrapolate formula Pressure is calculated in Poisson's 1 equation, no slip condition is used at solid boundary, three order differential scheme is employed in convection term computation, and Buldwin-Lomax model as is as introduced in turbulence flow calculation The cylindroid standing up-right "on a plat flat is 40 cm in cord length, and 5 in aspect ratethe center point of cylindroid is located 4,5m down away from leading edge of plat, and inlet flow comes at zero angle and 5m/s in speed II respectively, Grid Generation with trajectory R of reflected shock of detonation wave D as a boundary and trajectory F of flyor as anor boundary Both are unknown; position of R and state parameters It on is it necessary are governed to by transform flow field I physical of central space rarefaction (x, V, wave z) into behind a computation detonation wave space 581

2 582 Wang Xi-liang and He Mo-qin (se, r/, ~) for simplization of boundary condition The method was originally presented by JFThompson, formula of 3-D transformation are shown as following: ~"+G'+~"=/' } rt+r],,+r~ L+t,,+t, =f, =/, (21),ft, )r, and )r are predescribed grid Control functions at st'; r/, and t directions respectively When ]',, )r and /, go to zero and grid will be uniform distributed As one hope grid concentrated at (6~, r/s, ~,) point, control functions might be written in following form; /,= - asgn (~ - $,)exp [ -bl~-se, I ] /,: - asgn (r/- rh) exp [ -b1,7 -~jj ] /s= -asgn (~-~,)exp [ -hi ~-s ] (22) where The sgn(x) one-dimensional is sign function, problem a and of b are motion constants of a and rigid which flying plate value under might explosive be chosen attack from has 100 to an 1000 analytic and solution 05 to 15 only respectively when polytropic index of detonation products equals to three In general, a numerical analysis is required In this paper, however, by utilizing "weak" shock III behavior Control of Equation reflection shock and in Turbulence explosive Model products, and applying small parameter purterbation method, an analytic, first-order approximate solution is obtained for problem of flying We consider equations of motion in Cartesian coordinates (x, 11, z, t) for plate driven by various high explosives with polytropic indices or than but nearly equal to three ufisteady, Final velocities three-dimensional, of flying plate incompressible obtained agree very flow well The with Reynolds-averaged numerical results by equation computers of continu- Thus ity an and analytic momentum formula of with two mean parameters flow are of high explosive (ie detonation velocity and polytropic u+v,+w=o (31) u,++vu,+w= 1 Introduction -lp,+ OrW q ars ~ p 8y Oz -1 Orr~ Or, materials under intense v,+uv+vv,+wv=----f-#,+ impulsive loading, shock synsis of #x diamonds, -+ ez and explosive welding and (32) cladding of metals The method of estimation of flyor velocity and way of raising it are questions dr,, 8r, Under assumptions w,+uw+vw,+ww,---- of one-dimensional plane P,4 detonation Ox and I" rigid Oy flying plate, normal after governing transformation, flow field y of detonation become products behind flyor (Fig I): where ~ur +,7,, + ~ur + ~,v r +,7,v, + ~,vr + ~ewr + r/w, + ~wr = 0 (33) ap +u_~_xp + u,+ltur162 - (~Pr 1 + t/p, + ~opr + rl v,+ Uvr Fv,+ Wv r - (~,Pr rl,p,+ ~,4Pr + r, (34) w, + Uwr + Fw, + as Wwr as - (~,Pr + Vat', + GPr + rs } respectively, with trajectory R of reflected shock of detonation wave D as a boundary and trajectory F of flyor as anor boundary F -- r/u Both + r/,v are + unknown; r/w position of R and state para- (35) meters on it are governed by flow field I of central rarefaction wave behind detonation wave

3 Numerical Simulation of Wing-Body Junction Turbulence Flow 583 r, =r,=t=, ~+ Ou a ~_x ) Ov rl,_ r :/z,tt(_~_zo + 0w (s6) rz,=rar:/zoff( a_~_ I ~ it, ft and f= can be written in same form of p term, and P has been transformed according to formula P=P/P Three order up wind scheme is used in convective terms calculation f~,j,=,(ul,=,j,b -- 2ui, i,;,=, + 9u~,~,=, -- 10ut_ ~,m, ( a,,~ +2u=_t,~,=)/6A~ when f=,~,j~0 = if,,,,(_=,,,,, +,,,,,_,,,,, + -- u=_t,j,=)/sa~ when /=,~,j<o Let us The suppose one-dimensional q is a variate problem in physical of motion space, of we a can rigid get flying it's plate Laplacant under in explosive computation attack has space as an following analytic solution only when polytropic index of detonation products equals to three In general, a numerical analysis is required In this paper, however, by utilizing "weak" shock behavior of v~q reflection = (~' +~; shock +~: in )qr162 explosive (7' +,7~ products, +,7: )q,,+ and (~'- applying +~ + ~! small )qr162 parameter purterbation method, an -F analytic, 2 (~,r/,-f first-order ~rr/e-f ~,r/,) approximate q r 2 (r/~,-f solution t/r~t-f is obtained r/,~=) for q,r problem of flying plate driven by various high explosives with polytropic indices or than but nearly equal to three Final velocities of flying +2(_~,~, plate + obtained G~,+ ~;,~,)or162 agree very 2(~,,+~,,+~,,) well with numerical q: results by computers Thus an analytic formula with + (~,,+ two ~,,+~,,)q,+ parameters of high (~,,+ explosive ~,,+G,) (ie qr detonation velocity and polytropic (38) where ~, = (y,zr 1 Introduction ~, = (z,xr -zex,)/l ~, = (x,yr -xey,)/f, r/, = (z,vr -zevr I materials under intense rj, impulsive = (xr162-xr loading, shock synsis r/, = (xr162 of diamonds, -xevr and explosive I welding and (39) cladding of metals The ~, method = (Vezt-v,z:)/], of estimation of flyor velocity ~, = (z:x, and -z,xr way of raising it are questions ~, = (xev,-x,y~)/t Under assumptions of one-dimensional plane detonation and rigid flying plate, normal xr x~ xr governing flow field of detonation products ]= behind Yr Y, Vr flyor (Fig I): (310) zr z~ zr Usually in practical computation, Poisson equation ap +u_~_xp + and momentum equation are solved simultaneously instead of soiving continuity and momentum equations Poisson equation are a(ut~) vtp=vv=d OD 8 J-~(u t) -'t" O(uu) FT] as as at Ox L 8x ay - a y L T ay az j ~2 L ax F o(,,,') (szz) n 0u 8t~ 8w where respectively, ~=--,~--+--~y with trajectory -I---~-- R of reflected In present shock computations of detonation wave k-e D as and a boundary k-l turbulence and trajectory F of flyor as anor boundary Both are unknown; position of R and state paramodels meters on with are Tlear governed wall by treatments flow field are I of widely central rarefaction used as wave y behind can simulate detonation turbulence wave tr:~n~lx)rtation, D and by initial and stage all of motion of those of only flyor also; could be position done on of F super and computers state parameters But here of products a corner flow computation has been carried out on a personnel computer In calculation two layers

4 584 Wang Xi-liang and He Mo-qin turbulence model has been used that Van-Driest model was used for inner layer and Baldwin- Lomax model t'231 for outer layer where ~~ -----/zr'f pe e={e,=(ky)2[1-exp( -y/mt) ]ZlOu/Oyl eo =h~cpf,,0k[ 1at-55(Ckl,bY/ym,x)e] -t inner laver outer layer (312) (313) Aj =26v/v~ (314) Fw,k =ym,xf,,,,, (315) F~,,, =max[yldu/dy[ ~] (316) The quantity F,x is maximum value of F( r/)-----r/co/~:r/mixis vatuc of at which it occurs For 3-D flow computation B- L model has to be modified ~-~~: e, =Ccp( O O168 pf,,,ik~ (317) The one-dimensional problem of motion of a rigid flying plate under explosive attack has where an analytic solution only when polytropic index of detonation products equals to three In general, a numerical analysis is required In this paper, 2yz however, by utilizing "weak" shock behavior of reflection shock in rl= explosive yq_zq_(yz4rzz)t/t products, and applying small parameter pur- (318) terbation method, an analytic, first-order approximate solution is obtained for problem of flying plate driven by various high explosives FwAk, with = min~ polytropic r]maxfm'x indices or than but nearly equal to three (319) ~'CwkrlmxU~/ F mtx co is absolute magnitude of vorticity :o=[( aw ~y az)' 1 Introduction _ +(00 ~ ',' Oz Ox Ox"-- ay ) ] (320) Ov +( aw) ~ Van-Driest damping factor materials under intense impulsive loading, shock synsis of diamonds, and explosive welding and cladding of metals The method _~ of = estimation 1-exp(--r/d of flyor velocity [rw p~'/26p,) and way of raising it are questions (321) The Klebanoff Under intermittency assumptions of factor one-dimensional plane detonation and rigid flying plate, normal governing flow field of detonation fl=[1-1-55( products behind Ckl~ flyor )6]-1 (Fig I): (322) COp ~ Cwk, respectively IV ~"]m a X and Cklb are constants and recommended values are 16, 025, and 03 ap +u_~_xp + Boundary Condition As limitation of computer as capacity, as grids are not enough to be distributed to be far away from solid boundary where flow can be considered as undisturbed, so an effective extrapolate formula is used to determine values at outer boundary surfaces where Inlet: p, p, S, u are pressure, density, u specific predescribed entropy as and 1/7 particle power velocity functionv=w=p=0 of detonation products respectively, Outlet: with trajectory R of u~ reflected v, tu and shock p of determined detonation by wave extrapolation D as a boundary and trajectory On body: F of flyor as anor boundary u=v----w=0, Both are p unknown; by calculation position of R and state parameters on it are governed by flow field I of central rarefaction wave behind detonation wave On wing: u ~v=w=o~, p by calculation Outer boundary: u, v, w and P by extrapolation

5 Numcrical Simulation of Wing-Body Junction Turbulcncc Flow 585 Symmetric section: u, v, tu and p by symmetry b Computational domain The one-dimensional problem of motion of a rigid flying plate under explosive attack has Fig 1 Model and domain an analytic solution only when polytropic index of detonation products equals to three In general, a numerical analysis is required In this paper, however, by utilizing "weak" shock behavior of reflection shock in explosive products, and applying small parameter purterbation method, an analytic, first-order approximate solution is obtained for problem of flying plate driven by various high explosives with polytropic indices or than but nearly equal to three index) for estimation of velocity of flying a, plate x-t# is established plane 1 Introduction b x -z plane materials under intense impulsive loading, shock Fig synsis 2 Grid of diamonds, and explosive welding and cladding of metals The method of estimation of flyor velocity and way of raising it are questions Under assumptions of one-dimensional plane detonation and rigid flying plate, normal governing flow field of detonation Fig products 3 Velocity behind profile flyor (J= (Fig 1) I): ap +u_~_xp + as as ~I-~ ~t_ ~I~I- ~="~t ~- ~i-~-~--=~ =~ :- ~'-~ ~~ -- ~-~ - - -'~-=~t;i~i~ " " 111=' ~- =" " ~ t respectively, with trajectory R of reflected shock of detonation wave D as a boundary and trajectory F of flyor as anor boundary Both are unknown; position of R and state para- ~e) Z~--- ilimm - -~ -~ ~ --= = ~ ~ ~ ~-~ -= -- =:=:"~ ~ :~ ~ ::i -- -'= ~ " ~ = =~meters on it are governed by flow field I of central rarefaction wave behind detonation wave Fig 4 Velocity distribution at different x-l/ planes

6 n 586 Wang Xi-liang and He Mo-qin -'- " Z "_'- _" "- :" : 2 2" " ~ ~ o ~ m ~ I A 9 9 s s a s o - - ~ D, 0, 9 # e 9 e a n - o * 4 a L I J l~f i=25, x----0mm i----30, x mm ~ ~ ~ ~ ~ o - _ ~ - ~ ~ - ~ The one-dimensional problem of motion of a rigid flying plate under explosive attack has ~ ~ ~ ~ an analytic solution only when polytropic index of detonation products equals to three In ~ I - ~ general, a numerical analysis is required In this paper, ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ however, ~ by utilizing ~ "weak" shock Q behavior of reflection shock in explosive products, and applying small parameter pur- ~ - terbation method, an - o -analytic, - ~ first-order i i i r i I approximate 9 J solution is obtained, for o problem of flying plate driven by various high explosives with polytropic indices or than but nearly equal to three index) for estimation of i----33, velocity x= 160ram of flying plate is established i=38 xf260mm Fig 5 V, w distribution at different cross-sections 1 Introduction materials under! ~ intense impulsive loading, 2 w Tshock m synsis of diamonds, and explosive welding and cladding of metals The method of estimation of flyor velocity and way of raising it are questions a--36oa Under assumptions of one-dimensional plane detonation and rigid flying plate, normal 8 - m 2 ~ m governing flow field of detonation products behind flyor (Fig I): V Computation Result z, - 0 ap +u_~_xp + Fig 6 Pressure distribution y (J----l) =0, at different levels as as The model is shown in Fig l, and Fig 2 describe grid distribution The grid generation takes l0 minutes computation time for 20 steps, to reach accuracy that differential of coordinates between two steps is smaller than 0005 In flow field respectively, with trajectory R of reflected shock of detonation wave D as a boundary and calculation, first time step takes 20 iterations to satisfy 001 accuracy, and each iteration trajectory F of flyor as anor boundary Both are unknown; position of R and state paracosts meters 5 on minutes it are governed Each following by flow time field step I (when of central a-~001m/s rarefaction wave z ) takes behind 4-5 detonation iterations wave to keep

7 Numerical Simulation of Wing-Body Junction Turbulence Flow 587 same accuracy The symmetrical plane velocity distribution is shown in Fig 3, and horizontal planes v~locity distributions at different levels are shown in Fig 4, we can see from Figs that velocities near bow have big increments, it seems that re is a vertical vortex Four crosssections velocity distributions are demonstrated in Fig 5 that indicate re are two contra-direction vortexes located at middle and stern sections respectively Pressure distributions on wing at different levels are presented at Fig 6 VI Conclusions A program based on RANS equation, B-L turbulence model, and boundary fitted coordinates technique for 3-D turbulent flow calculation has been set up A reasonable result has been carried out by program on personnel computer which was considered impossible to finish such calculation without super computer For simulation of blunt-fin/body junction or flow comes in an angle, scheme and model should be furr modified References The one-dimensional problem of motion of a rigid flying plate under explosive attack has an analytic solution only when polytropic index of detonation products equals to three In [1] general, Stok, a numerical H W analysis and W is Hoase, required Determination In this paper, of however, length by scale utilizing in algebraic "weak" turbulence shock behavior models of for reflection Navier-Stokes shock in methods, explosive AIAA products, Journal, and 27, applying I (1989) small parameter pur- [ terbation 2 ] Hung, method, C M an and analytic, K W first-order Mac cormack, approximate Numerical solution solution is obtained of for three-dimensional problem of flying shock plate driven wave by and various boundary high layer explosives interaction, with polytropic AIAA Journal, indices or 16, 10 than (1978) but nearly equal to three [ Final 3 ] velocities Hung, C of M flying and plate P obtained G Buning agree very Simulation well with of numerical blunt-fin-induced results by computers shock-wave Thus and boundary-layer interaction, Journal of Fluid Mech, 154 (1985), [43 Fujii, K, Developing an accurate and efficient method for compressible flow simulations--an example of CFD in aerontics, Fifth International Conference on Numerical Ship Hydrodynamics 1 (1989) Introduction materials under intense impulsive loading, shock synsis of diamonds, and explosive welding and cladding of metals The method of estimation of flyor velocity and way of raising it are questions Under assumptions of one-dimensional plane detonation and rigid flying plate, normal governing flow field of detonation products behind flyor (Fig I): ap +u_~_xp + as as respectively, with trajectory R of reflected shock of detonation wave D as a boundary and trajectory F of flyor as anor boundary Both are unknown; position of R and state parameters on it are governed by flow field I of central rarefaction wave behind detonation wave