2 Analysis of the non-linear aerodynamic loads of hypersonic flow. 1 General Introduction

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1 4 TH INTERNATIONAL CONGRESS OF THE AERONAUTICAL SCIENCES PRELIMINARY STUDY OF NON-LINEAR AEROELASTIC PHENOMENA IN HYPERSONIC FLOW Zhang Wewe, Ye Zhengyn, Yang Yngnan Cllege f Aernautcs, Nrthwestern Plytechncal Unversty, X an, 717, P.R. Chna Keywrds: Hypersnc Tme-dman Aerelastcty Nn-lnearty Abstract Analyzng velcty ptental equatn, we get the surce f aerdynamc nn-lnearty at hgh Mach numbers. The nn-lnearty f the varus engneerng methds fr hypersnc aerdynamc lads s valdated. By numercal methd, the aerelastcty f hypersnc wng s smulated n tme dman. Sme nn-lnear phenmena have been fund as fllws: (1 The system desn t apply the superpstn prncple; ( The stablty f the aerelastcty s effected by the rgnal cndtns; (3 Dstrtn appears n the tme dman respnse, that s the respnse n crtcal cndtns s nt a harmnc; (4 The LCOs n hypersnc aerelastcty; (5 The nn-lnear relatns between crtcal velcty and angle f attack/ Mach numbers. 1 General Intrductn Wth the step f the manknd s explrng the unverse and exceedng the ultmate, sme new cncept arcrafts such as hyper-x prgram becme the vsn-vehcle fr the future. There s thus a great need t nvestgate the aerelastc prblem at hypersnc speed. Aerelastcty s a crssed subject that researches the nteractn f aerdynamc frce, elastc frce and nerta. Flutter s a typcal aerelastc prblem f arcraft that can cause an unstable vbratn and an nstant crash. Because f the blunt leadng edge f the hypersnc vehcle and the hgh angle f attack flght, there s a serus nnlnearty f aerdynamc lads and brngs many dffcultes n slvng hypersnc aerelastcty. Ltter research has dne fr the nn-lnear phenmena f hypersnc aerelastcty. The crash f the hypersnc vehcle X-43 may due t the falure f the cntrl surface because f flutter. The nn-lnear aerelastcty n hypersnc flw needs serus cnsderatn n a desgn prcess. Analyss f the nn-lnear aerdynamc lads f hypersnc flw Because f thn shck-layer vscus nteractn, hgh temperature and the change f the structure mdes due t the hgh temperature, the hypersnc aerelastcty s a cmplex nnlnear system. We just cnsder the aerdynamc nn-lnearty n ths paper. Analyzng velcty ptental equatn, we get the surce f aerdynamc nn-lnearty at hgh Mach numbers, and can fnd the aerdynamc lads f slender bdy at small angle f attack s stll nnlnear n hypersnc flw [1]. Cnsder the tw-dmensnal, rrtatnal, sentrpc flw, the x cmpnent velcty, u, and the y cmpnent velcty, v. u = V + u, v = v, where u and v are called the perturbatn velctes. We defne the velcty ptental Φ, and the perturbatn velcty ptental, Φ, such that, Φ = V x + Φ, where Φ Φ = u, = v Hence, Φ Φ Φ Φ Φ Φ = V +, =, =, Φ Φ Φ Φ =, = 1

2 ZHANG WEIWEI, YE ZHENGYIN Substtutng the abve defntns nt velcty ptental equatn, we get the perturbatn velcty ptental u v u [ a ( V + u ] + ( a v ( V + u v = (1 Frm energy equatn we get a V a ( V + u + v + = + ( γ 1 γ 1 Substtutn Equatn ( nt Equatn (1, we btan ( u v u γ + 1u γ 1 v u 1 M + = M[( γ ] V V V M [( u γ + 1 u 1 u v γ ] V V V γ v u u v M [ (1 + ( + ] (3 V V Ignred the rght-hand sde, we can get the lnear velcty ptental equatn. Fr the transnc flw r hypersnc flw, the rghthand sde s nt small n cmparsn wth the left hand. Ths s why the hypersnc flw s dffcult t be lnearzed even n the small perturbatn cndtn. Varus engneerng methds fr aerdynamc lads have been presented accrdng t the characterstc f the supersnc and hypersnc flw [1~4], such as Newtnan thery, pstn thery, tangent-wedge/tangent-cne methd, shck-wave and expansn-wave methd. These methds are cnnected t slve a general shaped bdy and extended t slve the unsteady aerdynamc lads. Mdfed Newtnan thery: Cp = Cp sn θ Cp max = max γ M ( p p 1 γ 1 Pstn thery: p = p (1 + Tangent-wedge/tangent-cne methd: γ + 1 γ Cp wedge = θ [ + ( + ] 4 4 ( M θ w a γ γ 1 4sn θ ( M 1snθ Cp cne = M 1snθ The frmulas f shck-wave and expansnwave methd are lttle cmplex, whch can be fund n reference [3]. Frm the cmparng wth the expermental results r Euler results shwn n Fg. 1 and Fg., we can fnd the engneerng methds has gd precsn n ther use range. Just because f the gd precsn and hgh effcency, these methds are wdely used n engneerng [,4,5]. Cp X/C (a M=1, Fg.1 Pressure ceffcent dstrbutn f a blunt bdy α = 3 Cp X/C In ths prgram, Newtnan thery, pstn thery, tangent-wedge/tangent-cne methd, shck-wave and expansn-wave methd are cnnected t slve the randm bdy s aerdynamc lads n hypersnc flw. Newtnan thery s used n hgh arflw-httng angle surface, tangent-wedge/tangent-cne methd s used n small arflw-httng angle surface, shck-wave and expansn-wave methd (pstn thery s used n the leeward. The cnnectn f Newtnan thery and tangent-wedge/tangent-cne methd use dahembark frmula [3]. The cnnected engneerng (b M=5, α = 5 Fg. Pressure ceffcent dstrbutn f a 4% thckness rund arfl 1 Euler Precse pstn, 3 Shck-expansn methd 4 tangent-wedge

3 PRELIMINARY STUDY OF NON-LINEAR AEROELASTIC PHENOMENA IN HYPERSONIC FLOW methd s used t slve supersnc r hypersnc unsteady lads n tme dman. 3 The tme-based smulatn f the aerelastcty [,6~8] The tme-based smulatn s an mprtant methd fr the engneer. It s the mst expedent t cnstruct, the mst capable f handng cmplex mdelng ssues, and the mst understandable wth an engneer s physcal ntutn. Fr example, a nn-lnear aerelastc system s ften easly slved by a tme-based apprach. Assumng the defrmatn f the wng s very small, the elastcty f the wng s lnear. We use the tradtnal mdes t descrbe the wng s scllatn, z( x, y, t = N = 1 Φ ( x, y q ( t, the N s the rders f the mdes, φ ( x, y s the th rder mde. q (t s the generalzed crdnate crrespndng φ ( x, y. Basng n the Lagrange equatn, the wng s mvement equatns can be wrtten n matrx frm: [ M ]{ q& } + [ G]{ q& } + [ K]{ q} = { F} (4 [M] s the mass matrx [G] s the structural damp matrx [K] s the stffness matrx [F] s the generalzed aerdynamc lads. F p( x, y, t φ ( x, y dxdy, the unsteady = s pressure p(x,y,t s slved by the unsteady aerdynamc slver at every tme step. In rder t slve Eq. 4 by Rnge-Kutta tme dscretzatn we ntrduce a varable {E}, T T T T {E}= [{ q },{ q& } ] = [ q 1, q, L, qn, q& 1, q&, L, q& N ], Eq. 4 becmes: [ O] [ I] [ O] { E& } = { E} + { F} [ M] [ K] [ M] [ G] [ M] (5 By the furth rder Rnge-Kutta dscretzatn, we can get the wngs pressure dstrbutn whch s needed fr the generalzed aerdynamc lads matrx at every tme steps. If there s an ntal angle f attack, the wng wll have a steady defrmatn under the steady aerdynamc lads except the dynamcal defrmatns. The steady defrmatn has n effectns t the aerelastcty fr the lnear systerm. But n the hypersnc flw, the aerdymanc lads are nn-lnear, the steady defrmatn has effects n the aerelastc prblem. In rder t cnsder the effects, the steady defrmatn can be calculated by Eq.6: [K][q] s ={F}, whch s get frm Eq.5 by remve the dynamc term. Subscrpt s means steady defrmatns. The detaled prcesses fr the supersnc/hypersnc aerelastc prblems are: (1 Calculatng the aerdynamc lads n the wng, frm the elastc equatns and aerdynamc equatns, we can get the steady defrmatn f the wng. At the base f the angle f attack, we add the defrmatn, slve the new aerdynamc lads, then slve the new steady defrmatn, tme and tme untl the steady defrmatn cnverge t a stagnatn data. Ths s the steady defrmatn at the angle f attack. ( When we get the steady defrmatn, there s a balance between the elastc frce and the aerdynamc frce. Addng an mpulse at the balance, the wng wll turn nt a dynamc respnse. By analyzng the respnse, we can get the characterstcs f the aerelastc prblem. Changng the Mach number, the angle f attack and the free stream dynamc pressure, we can get all the aerelastc characterstcs. Table 1 shws that cmputed flutter speeds are abut 1% f expermentally determned nes []. Table 1 The cmputed results cmparng wth the expermental results M Expermental velcty/ m. s -1 Cmputatnal velcty/ m. s -1 Relatve errr/% Expermental frequency/hz Cmputatnal frequency/hz Lacked Lacked %

4 ZHANG WEIWEI, YE ZHENGYIN 4 Several nn-lnear phenmena n hypersnc aerelastcty By smulatng the hypersnc wng s aerelastc respnses n tme dman, sme nn-lnear phenmena have been fund as fllws: (1 The system desn t apply the superpstn prncple. Ths example gves the cmparatve respnses f the wng s frst mde at angle f attack and 5 angle f attack at 1 Mach numbers. Frm Fg. 3 we can fnd the wng s steady defrmatn has effect n the respnse. It changes the frequences, ampltude and the phase f the respnse. The angles f attack have n effect n the fgures f the wng s respnse n lnear aerelastc system. value and abslute value f the decreasng range f M s much less than M 5 s as shwn n Fgure 4, that means V V. Ths << ( DF ( DF M = M = 5 example shws that the nn-lnearty ncreases wth the Mach numbers n supersnc flw. (3 Dstrtn appears n the tme dman respnse. That means the aerelastc respnse n crtcal cndtns s a perdc vbratn, but nt a harmnc vbratn [6]. Ths s a typcal phenmenn fr nn-lnear systems. A crtcal respnse f a wng at M 1 has been smulated. The respnses f the tw mdes are nt harmnc but perdc vbratns as shwn n Fgure 5..1 F.3. alf=5 alf=.5.1,f t Fg. 3 The effect f the angle f attack n the respnse ( The stablty f the aerelastcty s affected by the ntal cndtns. The ntal turbulence and the angel f attack nfluence the aerelastc stablty. The crtcal velctes (V at bth M and M 5 decrease wth the ncreasng f the ntal turbulence (DF, but bth the cmparatve t Fg. 5 Tme hstry f a wng s respnse at M 1 (4 The LCOs are typcal phenmena n the nn-lnear system []. Fg. 6 and Fg. 7 shw the LCOs n hypersnc aerelastcty at dfferent velctes. Frm Table we can fnd the frequences and ampltudes bth ncrease wth the velcty. 165 V(m/s V(m/s F F F3 F4 V DF DF (a M=5 (b M= Fg. 4 The effect f ntal cndtns n the flutter speed T S1 Fg. 6 The wng s respnse at V f =m/s -15 4

5 PRELIMINARY STUDY OF NON-LINEAR AEROELASTIC PHENOMENA IN HYPERSONIC FLOW.1.5 F T F F3 F4 V S1 Table The wng s nnlnear respnses Velcty m/s Fg.7 The wng s respnse at V f =3m/s Type 3 1 (5 The nn-lnear relatns between crtcal velcty and angle f attack/ Mach numbers n hypersnc aerelastc system as shwn n Fgure 8 [,9]. The crtcal velcty decreases greatly wth the ncreasng f the angle f attack at hgh M due t the nn-lnear aerdynamc lads. But fr the lw M supersnc flw, the angle f attack almst has n effect n the crtcal velcty as shwn n Fgure 8 (a. Fr lw M supersnc flw, the crtcal velcty ncreases wth the M. But fr the hgh Mach numbers (M>1,Fgure 8 (b, the crtcal velctes are almst ndependent f the M. Ths phenmenn als suts the hypersnc Mach number ndependence prncple. Nt cnsderng the effect f M n the structure, the hypersnc aerelastcty accrds wth the Mach number ndependence prncple. ω n rad/s Ampltude 15 Cnvergence 16 LCO LCO LCO LCO Dffusng V/(m/s Ma1.5 Ma Ma.5 Ma3 Ma5 Ma7 Ma1 Ma alfa(degree (a Crtcal velcty vs α at dfferent M V/(m/s degree 3 degrees 5 degrees 1 degrees 5 1 M (b Crtcal velcty vs M at dfferent α Fg.8 The relatn between crtcal velcty and M/α 5 Cnclusns Analyzng the hypersnc flw, we testfy the nn-lnearty f the aerdynamc lads at hgh Mach numbers that drectly nduces the nnlnearty f the hypersnc aerelastcty. By numercal methd, the nn-lnear aerelastcty s smulated n tme-dman and sme nnlnear phenmena have been fund as fllws: (1 The system desn t apply the superpstn prncple; ( The stablty f the aerelastcty s effected by nput and the rgnal cndtns; (3 Dstrtn appears n the tme dman respnse, that s the respnse n crtcal cndtns s nt a harmnc; (4 The LCOs n hypersnc aerelastcty; (5 The nn-lnear relatns between crtcal velcty and angle f attack / Mach numbers. Because f the dffculty f the nn-lnearty, the wrk needs further research n the future. Reference [1] Andersn. J.D Fundamentals f Aerdynamcs, MCGRAW-HILL, [], 3,1(6: [3] 1995: [4] Gupta K K, and Velker LS, CFD-based Aerelastc Analyss f The X-43 Hypersnc Flght Vehcle, AIAA paper 1-71, 1. [5] Nydck I, Fredmann P P, and zhng X L. Hypersnc Panel Flutter Studes n Curved Panels, AIAA paper ,

6 ZHANG WEIWEI, YE ZHENGYIN [6] Dwell E, Tang D. Nnlnear aerelastcty and unsteady aerdynamcs. AIAA --3. [7] Gupta K K, Velker L S. CFD-based Aerelastc Analyss f The X-43 Hypersnc Flght Vehcle. AIAA paper 1-71, 1 [8] (1: [9] 1995:54(: