Simulation of axial combustion instability development and suppression in solid rocket motors

Transcription

1 Intenational jounal of say and combustion dynamics Volume. 1 Numbe ages Simulation of axial combustion instability develoment and suession in solid ocket motos David R. Geatix Deatment of Aeosace Engineeing, Ryeson Univesity, 150 Victoia Steet, Toonto, Ontaio, Canada M5B 2K3, Received 29 th Febuay 2008; Acceted 9 th July 2008 ABSTRACT In the design of solid-oellant ocket motos, the ability to undestand and edict the exected behaviou of a given moto unde unsteady conditions is imotant. Reseach towads edicting, quantifying, and ultimately suessing undesiable stong tansient axial combustion instability symtoms necessitates a comehensive numeical model fo intenal ballistic simulation unde dynamic flow and combustion conditions. An udated numeical model incooating ecent develoments in edicting negative and ositive eosive buning, and tansient, fequency-deendent combustion esonse, in conjunction with essuedeendent and acceleation-deendent buning, is alied to the investigation of instabilityelated behaviou in a small cylindical-gain moto. Petinent key factos, like the initial essue distubance magnitude and the oellant s net suface heat elease, ae evaluated with esect to thei influence on the oduction of instability symtoms. Two taditional suession techniques, axial tansitions in gain geomety and inet aticle loading, ae in tun evaluated with esect to suessing these axial instability symtoms. NOMENCLATURE A local coe coss-sectional aea, m 2 A d downsteam coe coss-sectional aea, m 2 A u usteam coe coss-sectional aea, m 2 a l longitudinal (o lateal) acceleation, m/s 2 a n nomal acceleation, m/s 2 C de St. Robet coefficient, m/s-pa n C m aticle secific heat, J/kg-K C gas secific heat, J/kg-K C s secific heat (solid hase), J/kg-K d local coe hydaulic diamete, m E local total secific enegy of gas in coe, J/kg local total secific enegy, aticle hase in flow, J/kg E Coesondence. Phone: (416) ext. 6432; FAX: (416) ; geatix@yeson.ca

2 144 Simulation of axial combustion instability develoment and suession in solid ocket motos f fequency, Hz, o Dacy-Weisbach fiction facto f esonant combustion fequency, Hz H s net suface heat of eaction, J/kg h effective heat tansfe coefficient unde tansiation, W/m 2 -K K b bun ate limiting coefficient, s 1 K δ shea laye coefficient, m 1 k gas themal conductivity, W/m-K k s themal conductivity (solid hase), W/m-K M a magnitude of attenuation M limit magnitude (cyclic inut) m mean mass of a aticle, kg n exonent (de St. Robet s law) local gas static essue, Pa d initial ulse distubance ste essue, Pa w limit essue wave magnitude, eak-to-tough, Pa R secific gas constant, J/kg-K b instantaneous buning ate, m/s b, o efeence buning ate, m/s b, qs quasi-steady buning ate, m/s b unconstained buning ate, m/s o base buning ate, m/s T f flame temeatue, gas hase, K T i initial temeatue (solid hase), K T s buning suface temeatue, K t time incement, s u coe axial gas velocity, m/s u coe axial aticle velocity, m/s v f nominal flamefont velocity, m/s x distance fom head end, m x satial incement in axial diection, m y adial distance fom buning suface, m y satial incement in adial diection, solid hase, m y Fo Fouie limit satial incement, m α aticle mass faction of oveall coe flow α s themal diffusivity (solid hase), m 2 /s δ o efeence combustion zone thickness, m δ esultant combustion zone thickness, m κ vibation-based wall dilatation tem (1/A A/ t), s 1 ρ gas density, kg/m 3 ρ density, aticle hase in coe flow, kg/m 3 ρ s density (solid hase), kg/m 3 φ acceleation oientation angle, ad long./lateal-acceleation-based dislacement oientation angle, ad φ d

3 Intenational jounal of say and combustion dynamics Volume. 1 Numbe INTRODUCTION Ove the last fifty yeas o so, thee has been a numbe of eseach effots diected towads undestanding the hysical mechanisms, o at least the suounding factos, behind the aeaance of symtoms associated with nonlinea axial combustion instability [1 2] in solid ocket motos (SRMs). Taditionally, these symtoms ae a sustained axial essue wave esence in the combustion chambe, sometimes accomanied by a substantial ise in the base chambe essue [dc shift]. The motivation fo these ast and esent studies was and is of couse to bing this bette undestanding to bea in moe ecisely suessing, if not eliminating, these symtoms. Studies of nonlinea axial combustion instability have anged fom numeous exeimental test fiing seies on the one hand [3], and linea/nonlinea acoustic theoy modelling on the othe [2, 4 6]. Lagely, the acoustic analysis oduces fequency-based standing wave solutions fo a given chambe geomety, but tyically without some useful quantitative infomation. On occasion, eseaches have emloyed a numeical modelling aoach, to wok towads a moe comehensive quantitative undestanding of the hysics involved [7 8]. The numeical model would tyically oduce a tavelling wave solution to a limit essue wave amlitude and a coesonding small o lage dc shift in chambe essue, a time-based esult evolving fom an initial ulse distubance intoduced into the chambe flow. Available comutational owe and associated esult tunaound times commonly foced some simlifications in the given numeical model. A comehensive numeical model fo simulation of nonlinea dynamic flow and combustion conditions is ultimately essential in the quest fo the ability to edict and quantify axial combustion instability symtoms in SRMs. An effective model combines the effects of the unsteady one- o two-hase flow, the tansient combustion ocess, and the stuctual dynamics of the suounding oellant/casing stuctue. On the numeical ediction side, as vaious comonent models evolve, say fo tansient buning ate o stuctual vibation, thei incooation into an oveall intenal ballistics simulation ogam allows fo new moto fiing simulations to take lace, which in tun allows fo udated comaisons to available exeimental fiing data. Examles of ulsed moto fiing simulations comleted eviously, at inteim stages of ealie simulation model develoment, may be found in [9 12]. In the esent investigation, an udated numeical model incooating the above attibutes is used in the ediction of the unsteady nonlinea instability-elated behaviou in a cylindical-gain moto. The focus of this study will be on stong axial shock-wave-elated symtoms; two- and thee-dimensional instability symtoms of smalle magnitude, e.g., due to votex shedding obseved in segmented motos, will not be unde consideation hee. A moe ecent vesion of a tansient buning ate model (based on the Zeldovich-Novozhilov [Z-N] aoach [13 16], with secific modifications fo the simulation model as descibed in [17]) is emloyed fo the esent study. The latest vesion of this Z-N model allows fo the inclusion of a net suface heat of eaction tem ( H s ), that bette enables a match to exeimental esonse data. Additionally, the combustion module fo the simulation ogam emloys a ecently udated eosive buning model [18] (that allows fo ediction of negative eosive buning at low flow seeds, whee the oveall buning ate falls below the base essue-deendent value, in

4 146 Simulation of axial combustion instability develoment and suession in solid ocket motos Sleeve Casing Poellant Figue 1: Refeence SRM model setu, fo static test fiing in laboatoy. addition to ediction of ositive augmentation at highe flow seeds). Examle esults ae esented in this ae in ode to ovide the eade with some backgound on the sensitivities of a numbe of etinent aametes in the esent study of a small cylindical-gain SRM, including such key factos as the afoementioned H s, and the initial essue distubance magnitude d, with esect to the aeaance of axial instability symtoms. Two taditional means fo instability suession ae in tun demonstated: modifying the intenal ot geomety, and intoducing inet aticles into the coe flow. 2. NUMERICAL MODEL A simlified schematic diagam of the hysical system of an SRM laced on a static test stand is ovided in Figue 1. In this case, the cylindical-gain moto is fee to vibate adially without any extenal constaint (i.e., only constained as indicated by the thick steel static-test sleeve suounding the aluminum flightweight moto casing), while axial motion is constained to a lage degee by the thust-measuing load cell (eesented hee as a sing/dame) at the lefthand bounday. Unde nomal (nominal) quasi-equilibium oeating conditions, the intenal gas flow (o gas-aticle flow, if two-hase) moves smoothly fom the buning oellant suface though and beyond the exhaust nozzle. The equations of motion descibing the nonsteady coe flow within the SRM must be solved in conjunction with the local buning ate b of the solid oellant, and the suounding stuctue s instantaneous geometic defomation. As etains to the esent study of a small moto having a lage length-to-diamete atio, the quasi-one-dimensional hydodynamic consevation equations fo the axial gas flow ae given below: ρ + ( ρu) 1 A = ρ + α ρ + κ t x A x u 4b 4b ( 1 ) s ( )ρ d d ( ρu) A ( ρu + ) = ρ ( + κ) ρ ρ t x A x u 2 4b u al ρ d m D (1) (2)

5 Intenational jounal of say and combustion dynamics Volume. 1 Numbe ( ρe) + 1 A 4b ( ρue + u) = ( ρue + u) ( + κ)ρe t x A x d 2 4 ρ b f + ( 1 α) ρs ( f + ) ρ ( + ) d CT v ua 2 m ud Q l (3) Hee, the total secific enegy of the gas is defined fo a themally efect gas as E = /[(γ 1)ρ] + u 2 /2. The coesonding equations of motion fo an inet (non-buning) monodisese aticle hase within the axial flow may be found fom: ρ + ( ρu ) 1 A 4b 4b = ρu + αρs ( + κρ ) t x A x d d + 2 ( ρu ) ( ρu ) 1 A 2 4b = ρu ( + κ) ρ u t x A x d ρ a + ρ m D l (4) (5) ( ρ + E) ( ρue ) 1 A 4 b = ( ρ ue ) ( +κρ E t x A x d 2 4 ρ b f + αρ s ( m f + ) ρ + ( + ) d CT v ua 2 m ud Q l ) (6) Hee, the total secific enegy of a local gouing of aticles is given by E = C m T + u 2 /2, whee T is the mean temeatue of that gou. The aticles (and gas) that ente into the cental coe flow fom the buning oellant suface, at a given time ste, ae assumed to ente without any flow losses (in ast acoustic analyses, a flow-tuning loss facto has been consideed as otentially having some influence on the esulting essue wave behaviou). As outlined in [19], the viscous inteaction between the gas and aticle hases is eesented by the dag foce D, and the heat tansfe fom the coe flow to the aticles is defined by Q. In the case of dag between the gas and a eesentative sheical aticle at a given axial location, one notes that: 2 πdm D = C ρ( d u u ) u u, 8 (7) whee C d is the dag coefficient fo a shee in a steady flow with low flow tubulence (detemined as function of elative Reynolds numbe, elative flow Mach numbe, and temeatue diffeence between the aticle and the gas). In the case of heat tansfe fom the coe flow to a eesentative aticle at a given axial location, the following alies:

6 148 Simulation of axial combustion instability develoment and suession in solid ocket motos Q= πd k Nu ( T T ), m (8) whee the Nusselt numbe Nu can be found as a function of Pandtl and elative Reynolds numbe fo a shee of mean diamete d m. Longitudinal acceleation a l aeas in the gas and aticle momentum and enegy equations as a body foce contibution within a fixed Euleian efeence (fixing of x = 0 to moto head end, x ositive moving ight on stuctue as e Figue 1; acceleation of local suounding stuctue ightwad is designated ositive a l ), and may vay both satially along the length of the moto and with time. The effects of such factos as tubulence can be included though one o moe additional equations that emloy the infomation fom the bulk flow oeties aising fom the solution of the above onedimensional equations of motion. As e Figue 1, the flow system is bounded on the left-hand side by a zeo-flow condition at the head end wall, and a continuous-flow condition on the ight-hand side at the nozzle exit lane (suesonic exit flow). The incial diffeential equations themselves can be solved via a highe-ode, exlicit, finite-volume andom-choice method (RCM) aoach [9 12], a Riemann solve known fo low atificial disesion ove time of wave activity in tubes, etc. [19]. Stuctual vibation can lay a significant ole in unsteady SRM intenal ballistic behaviou, as evidenced by exeimentally obseved changes in combustion instability symtoms as allied to changes in the stuctue suounding the intenal flow (e.g., oellant gain configuation, wall thickness, mateial oeties [20 22]). In the ast it was tyical in acoustic analyses to teat the suounding stuctue as nominally igid, and assigning an imedance loss facto to the head end and/o the nozzle egions. The afoementioned exeimental obsevation of stuctue-deendent combustion instability has led to moe ecent consideation of such factos as cyclic nomal acceleation of the oellant suface, induced by the assage of tavelling intenal-flow essue waves, acting to augment the local tansient buning ate [9]. As discussed late in this ae, solid oellants, esecially those at lowe base buning ates, can exeience substantial inceases in buning unde a nomal acceleation field. In the oscillatoy situation of shock wave assage back-and-foth in the chambe, tansient acceleation eaks at the oellant suface can exceed g. The level of sohistication equied fo modelling the moto stuctue (oellant, casing, static-test sleeve, nozzle) and alicable bounday conditions (load cell on static test stand) can vay, deending on the aticula alication and moto design. Loncaic et al [11] and Montesano et al [12] emloyed a finite-element aoach towads the stuctual modeling of the given moto configuation. In the esent study, a cylindical-gain configuation allows fo a simle aoach via thick-wall theoy, as eoted in [9]. The adial defomation dynamics of the oellant/casing/sleeve ae modelled by a seies of indeendent ing elements along the length of the moto. Axial motion along the length of the stuctue is modelled via beam theoy, and bounded by the sing/dame load cell at the moto s head end. Viscous daming is alied in the adial and axial diections. Refeence stuctual oeties ae assumed fo an ammonium-echloate/hydoxyl-teminated olybutadiene (AP/HTPB) comosite oellant suounded by an aluminum casing

7 Intenational jounal of say and combustion dynamics Volume. 1 Numbe and steel sleeve. Fo geate accuacy, some oeties like the oellant/casing/sleeve assembly s natual adial fequency may be edetemined via a finite-element numeical solution (as was done hee), athe than via theoetical aoximations. With esect to tansient, fequency-deendent buning ate modeling, the Z-N solidhase enegy consevation aoach used in the esent simulation ogam may be eesented by the following time-deendent temeatue-based elationshi (see [17]): b 1 = b, qs ( T T H / C ) t s i s s 0 Tdy (9) whee b, qs is the quasi-steady buning ate (value fo buning ate as estimated fom steady-state infomation fo a given set of local flow conditions), T i is the initial oellant temeatue, and in this context, T = T(y, t) T i is the temeatue distibution in moving fom the buning oellant suface at y = 0 (and T = T s ) to that satial location in the oellant whee the temeatue eaches T i. One may note at this junctue the inclusion of a net suface heat elease tem, H s, in the calculations. This tem is commonly included as an add-on coection to the solid-hase enegy contibution in an enegyconsevation model involving solid oellants (e.g., see Eq. (15) below fo eosive buning). On a agmatic level, adjusting the value of H s ositively uwad (moe exothemic contibution) inceases the buning esonse magnitude of the oellant at a given fequency, and visa vesa fo a moe endothemic contibution, which acts to deaden the esonse [17]. The tansient heat conduction in the solid hase can be solved by an aoiate finite-diffeence scheme. One needs to take cae in setting the solid-hase satial incement y, to be in accodance with the Fouie stability limit y Fo, which is a function of the chosen time incement t [17], which in this study is on the ode of s, based on the axial node distibution fo this size of moto (head end to nozzle exit lane). The time incement itself must be coodinated between the flow and stuctual model solution systems [12]. In Eq. (9), b * is the nominal instantaneous buning ate, as edicted by that enegy-consevation equation, fo a given time ste. The actual instantaneous buning ate b may be found as a function of b * though the ate limiting equation [17], db dt = K ( ). b b b (10) The ate limiting coefficient K b effectively dams the unconstained buning ate b * when fo a finite time incement t: K < 1 b t. (11) This daming acts to constain the exchange of enegy though the buning suface inteface, allowing fo some vaiability in bette comaing to a given set of exeimental

8 150 Simulation of axial combustion instability develoment and suession in solid ocket motos esonse data, and event so-called buning-ate unaway (unstable divegence of b with time). The quasi-steady buning ate may be ascetained as a function of vaious aametes; in this study, as a function of local static essue, coe flow velocity u (eosive buning comonent), and nomal/lateal/longitudinal acceleation, such that: b, qs= + u+ a (12) The essue-based buning comonent may be found though de St. Robet s law: n = C. (13) The flow-based eosive buning comonent (negative and ositive) is established though the following exession [18]: b b = o+ δ e o (14) whee at lowe flow seeds, the negative comonent esulting fom a stetched combustion zone thickness (δ > δ o ) may cause an aeciable do in the base essuedeendent buning ate o. The stetching of the flame zone at low seed may be viewed as being the esult of a lamina-like sliding ocess of the local axial flow in the bounday laye acting to extend and cuve the ath of a eesentative aticle moving u fom the buning suface towads the flame font, such that the effective eactive length is inceased. Fom [18], in the low-seed egime, the following exession may be alied, such that the coe flow Dacy-Weisbach fiction facto f is below the limit value f lim at which oint negative eosive buning is no longe in effect: b o K δ ρu = cos tan 1 12 / [ ( [ 1 ( f/ f ) ] )]. f < f δ δ o lim ρ lim s o (15) The aamete K δ is a shea laye coefficient, whose value set at 2600 m 1, along with a value fo f lim of , oduced a good comaison to exeimental data fo vaious oellants and motos [18]. At highe flow seeds whee flow tubulence begins to become moe intense, the ositive eosive buning comonent e, established fom a convective heat feedback emise [18, 23 24], should dominate: e ht ( T) f s = ρ [ C ( T T) H ]. s s s i s (16) Fo the above case, whee the base buning ate o is a function of the othe mechanisms (essue and acceleation), one finds the velocity-based comonent of bun ate fom Eq. (14) via u = b o. At highe flow seeds, u becomes equivalent to e. The effect

9 Intenational jounal of say and combustion dynamics Volume. 1 Numbe of nomal acceleation a n esulting fom adial oellant/casing/sleeve vibation may be detemined via [25]: b C ( T T ) f s = C ( T T) H s s i s ( + G / ρ ) b a s. exc [ δ ( ρ + G )/ k] 1 o s b a (17) The above equation fo oveall buning ate is tanscendental, with the alicable solution fo b being geate than the base buning ate value of o, as the aamete G a becomes moe negative in value, as noted below. The comessive effect of nomal acceleation and the dissiative effect of steady o oscillatoy longitudinal (o lateal, if say fo a sta gain configuation) acceleation a l is stiulated though the acceleative mass flux G a : G a a n o = δ RT b f o b = o φ 0 2 cos φ. d (18) Note that the longitudinal/lateal-acceleation-based dislacement oientation angle φ d is geate than the nominal acceleation vecto oientation angle (φ; zeo when only nomal acceleation a n elative to the buning oellant suface is esent [25]). One should also note that a n is negative when acting to comess the combustion zone, and teated as zeo when diected away fom the zone (in coelating with exeimental obsevation). Fo the above case, whee the base ate o is a function of the othe flow mechanisms (essue and coe flow), one finds the acceleation-based comonent of buning ate fom Eq. (17) via a = b o. With esect to the buning suface temeatue T s, one has the otion of teating it as constant, o allowing fo its vaiation, deending on the henomenological aoach being taken fo estimating the buning ate [17]. Based on good comaisons in geneal to exeimental data as eoted in [17], a constant T s was emloyed in the Z-N henomenological combustion model, fo the esent ballistic simulations. 3. RESULTS AND DISCUSSION The chaacteistics of the efeence moto fo this study ae listed in Table 1. The moto, based in lage measue on a simila exeimental moto [9], is a small (half-mete-long) cylindical-gain design with an aluminum casing and static-test steel sleeve, with a elatively lage length-to-diamete atio. The efeence base oeating essue in the moto chambe fo the examle simulations in this ae, aound 10.5 to 11 MPa, is in coesondence with this test moto, which had been configued fo combustion instability studies at a highe chambe essue to be otentially moe suscetible to symtom develoment [9]. A tyical mean oeational essue fo a solid ocket moto utilized by a flight vehicle is moe likely somewhat lowe, say close to 7 MPa, o even lowe, to save on chambe stuctual wall thickness (and thus weight). The moto at the time of ulsing has a modeate ot-to-thoat aea atio, with a consideable oellant web thickness emaining. The esonant fundamental adial fequency of the stuctue is

10 152 Simulation of axial combustion instability develoment and suession in solid ocket motos Table 1: Refeence moto chaacteistics. Poellant gain length, L 52 cm Initial ot diamete, d i 3.6 cm Nozzle thoat diamete, d t 1.6 cm Gain/nozzle-conv. length atio, L /L c 16:1 Poellant secific heat, C s 1500 J/kg-K Poellant density, ρ s 1730 kg/m 3 Poellant themal conductivity, k s 0.4 W/m-K Poellant themal diffusivity, α s m 2 /s Poellant flame temeatue, T f 3000 K Poellant suface temeatue, T s 1000 K Poellant initial temeatue, T i 294 K Gas secific heat, C 1920 J/kg-K Secific gas constant, R 320 J/kg-K Gas themal conductivity, k 0.2 W/m-K Gas absolute viscosity, µ kg/m-s Gas secific heat atio, γ 1.2 De St. Robet exonent, n 0.35 De St. Robet coefficient, C 0.05 cm/s-(kpa) n Paticle mass faction, α 0% Poellant elastic modulus, E A 45 MPa Poellant Poisson s atio, υ A Casing inne wall adius, m 3.24 cm Casing wall thickness, h B cm Casing mateial density, ρ B 2700 kg/m 3 Casing elastic modulus, E B 80 GPa Casing mateial Poisson s atio, υ B 0.33 Sleeve wall thickness, h C 0.47 cm Sleeve mateial density, ρ C 7850 kg/m 3 Sleeve elastic modulus, E C 200 GPa Sleeve mateial Poisson s atio, υ C 0.30 Casing/o. ad. daming atio, ξ R 0.35 Casing/o. long. daming atio, ξ L 0.10 aound 15 khz at this junctue in the fiing; assage of the esulting sustained axial shock wave may oduce eak nomal acceleations on the ode of 1000 g o moe at the steel sleeve extenal wall, and on the ode of g o moe at the oellant suface [9]. The edicted combustion fequency esonse fo the AP/HTPB oellant at thee diffeent settings fo the net suface heat elease value may be viewed in Figue 2. The esulting esonse ofile fo the lowest value of H s (30000 J/kg) is tyical fo a nonaluminized comosite oellant, showing a modeate eak magnitude, while the heightened ofiles at the highe H s values would be indicative of a moto having an

11 Intenational jounal of say and combustion dynamics Volume. 1 Numbe Limit magnitude H s = J/kg H s = J/kg H s = J/kg Fequency, Hz Figue 2: Fequency esonse of efeence oellant ( b, o = 1.27 cm/s, K b = s 1, diffeing H s ) in tems of nondimensional limit magnitude. inceasingly undesiable suscetibility to instability symtoms. The geneal esonse is given in tems of the nondimensional limit magnitude, defined by [17] M = b, eak b, o l b, qs, eak b, o whee the efeence buning ate b, o in this case is the moto s aoximate mean bun ate at the oint of ulsing (1.27 cm/s), being subjected to a ±0.1 cm/s sinusoidal comonent on b, qs fom t = 0 s; the limit amlitude fo b, eak would equal b, qs, eak (1.37 cm/s) when M l is unity. The oellant s esonant combustion fequency f is set via the value of K b (20000 s 1 ) to be on the ode of 1 khz (a value within the ange of what might be exected fo this tye of comosite oellant at that base buning ate). This value fo f is in fact elatively close to the fundamental longitudinal acoustic fequency f 1L of the combustion chambe, in oviding examles late in this ae that ae close to the wost-case scenaio fo suscetibility to axial combustion instability symtoms. As a fist case of inteest, one can examine the esult fo the efeence moto when one has no fequency deendence in the buning ate model. Hee, the Z-N esonse is emoved, and as a esult, the actual buning ate will be assumed to follow the quasisteady buning equations at any given instant, without any lead o lag in value. A ulse is intoduced into the flow to initiate wave activity, in a manne coesonding to exeimental, yotechnically ulsed fiings. The ulse magnitude in the simulations may be substantially geate than one might see in actual fiings, in ode to enhance the, M l (19)

12 154 Simulation of axial combustion instability develoment and suession in solid ocket motos 20 Pessue, MPa Time, s Figue 3: Pedicted head-end essue-time ofile, efeence moto ( d = 24 atm), no Z-N esonse. otential fo tiggeing a moto into instability, in some of the simulation uns. As noted late, when a stonge Z-N buning-ate esonse is included, a lowe distubance magnitude, coesonding to yotechnic ulse stengths seen in actual test fiings, may suffice fo tiggeing the simulation model into instabilty. In the esent simulations, the initial tavelling ulse distubance s shae is set as a elatively stee-fonted ste above the local base essue, a modeate-length midsection at the ste s nominal magnitude of d, with a shallowe amed tail at the end of the ste, to etun to the base essue of the chambe; the oveall initial ulse length is on the ode of 20% of the moto s gain length. Figue 3 illustates a segment of the moto s head-end essue-time fiing ofile, showing a sustained 1L (single-wave) axial wave system afte the initial stong (24 atm) ulse distubance (a 12-atm ulse distubance was also able to delive a sustained 1L wave system, while a 6-atm distubance could not). Thee is a clealy evident dc shift o base essue ise of aound 1.5 MPa (fom a mean chambe essue of 10.5, to about 12 MPa) associated with this activity. This is imaily esulting fom the tansient acceleation-augmented buning of the local oellant with each shock wave assage, and the coesonding stuctual vibation at the oellant suface eaking aound g. Figue 4 ovides a cleae ictue of the tavelling essue wave s shae ove a given cycle a bit late into the fiing. The ost-shock-font decay is quite aid in doing to the base essue befoe the subsequent shock font aival at a late time as one begins the next cycle.

13 Intenational jounal of say and combustion dynamics Volume. 1 Numbe Pessue, MPa Time, s Figue 4: Pedicted head-end essue wave ofile, efeence moto ( d = 24 atm), no Z-N esonse. One can now incooate the Z-N fequency esonse into the buning ate model, with the usage of Eqs. (9) and (10). Fo this case, a modeate net suface heat of eaction value of J/kg (exothemic elease) is chosen, in conjunction with a K b setting of s 1. In ode to seaate out the influence of H s on the Z-N esonse behaviou, suface heat eaction tems in the quasi-steady buning comonent models will be etained at zeo (a common selection fo ast simulations, when comaing to steadystate exeimental data). Figue 5 illustates a segment of the head-end essue-time ofile, showing a sustained 1L wave system afte the initial stong (24 atm) ulse distubance. Note that a weake 12-atm ulse failed to oduce a sustained 1L system. The limit axial wave magnitude is lowe ealie in the fiing (on the ode of 3 MPa), and then tansitions late to a highe limit magnitude (aoaching 6 MPa). In egad to this tansition, given the maked diffeence in system behaviou in moving fom one quasi-equilibium solution to anothe in a shot eiod of time, one would susect that a theshold of some kind has been assed by the system, in undetaking this tend change to a new equilibium. One notes that the oellant web thickness with bunback is deceasing as the simulated fiing ogesses in time. This oduces a highe ot-tothoat aea atio when the tavelling axial essue wave iminges on the nozzle convegence section. Fom shock wave dynamics, a eflected shock wave will be stonge in magnitude subsequent to imingement on an aea contaction of a highe atio (in the limit, the eak magnitude being fo wave imingement on a closed end wall). As a esult, one might identify the inceased aea atio, ehas in assing a

14 156 Simulation of axial combustion instability develoment and suession in solid ocket motos 20 Pessue, MPa Time, s Figue 5: Pedicted head-end essue-time ofile, efeence moto (K b = s 1, H s = J/kg; d = 24 atm). cetain theshold value, as contibuting to the uwad tansition of the essue wave s stength. One may also note that the mean chambe essue is inceasing with time into the fiing, and the buning suface aea is inceasing, given the cylindical gain configuation. The tansition in wave stength may be due in at to the base essue assing above a theshold value (fo suoting a highe wave stength), o due to the inceased enegy inut of the inceased buning suface aea assing a cetain theshold. Finally, one can note that the decease in oellant web thickness would lead to a elative decease in oellant suface deflection amlitudes fo the same wave stength assing ove it, and a modeate incease in the esonant adial vibation fequency. As a esult, the net stuctual vibation effect on acceleation-elated buning in contibuting to the above tansition in limit wave stength is not clea. As one might exect, the associated dc ise is lowe ealie on io to the tansition, and moe substantial late into the fiing, with the substantially stonge axial shock wave esent late in the fiing. Figue 6 ovides a cleae ictue of the tavelling essue wave s shae ove a given cycle, at its lowe limit magnitude ealie in the fiing simulation. One can clealy see that the ost-shock decay ofile of the wave s tail is now much shallowe in comaison to that seen in Figue 4, undoubtedly due to the lagging esonse of the local buning ate. Figue 7 shows an axial essue wave ofile fom a comaable oint in an exeimental ulsed moto fiing [9]. While the dc shift is elatively simila between Figs. 6 and 7, one can see that the essue wave s limit magnitude (tough-toeak w ) is somewhat lowe in the exeimental case. In these simulation uns, the dc

15 Intenational jounal of say and combustion dynamics Volume. 1 Numbe Pessue, MPa Time, s Figue 6: Pedicted head-end essue wave ofile, efeence moto (K b = s 1, H s = J/kg, d = 24 atm), ealie in fiing. 15 Pessue, MPa Time, s Figue 7: Exeimental head-end essue wave ofile [3].

16 158 Simulation of axial combustion instability develoment and suession in solid ocket motos 25 Pessue, MPa Time, s Figue 8: Pedicted head-end essue-time ofile, efeence moto (K b = s 1, H s = J/kg, d = 2 atm). shift and the limit essue wave stength ae intimately tied to the acceleation sensitivity of the solid oellant s buning, and in tun the stuctual vibation chaacteistics of the oellant buning suface that delives the acceleation inut. The fequency-deendent esonse of the oellant buning aeas to sead the enegy of the essue wave acoss a wide length (vesus a stee shock font and a tail having a aid do-off in base essue). A somewhat lowe exeimental value fo w and a geate effective tail length (when comaing Figs. 6 and 7) might suggest that the fequency esonse of the exeimental solid oellant is not exactly in line with the assumed esonse ofile fo the simulation model. Retuning to the edictive model of the esent investigation, inceasing the oellant s exothemic suface heat elease, to J/kg (fom J/kg), esults in a much moe active moto, as evidenced in Figue 8. Also, as demonstated with the use of a much weake ulse magnitude of 2 atm, the 1L wave system can moe eadily be tiggeed and sustained. The dc shift late in the fiing is quite lage, as is the limit wave magnitude. Acceleation-augmented buning is laying a lage ole hee in these symtom magnitudes, but the heightened fequency esonse is also now a bigge influence, woking in conjunction with stuctual vibation, in the esulting ofile. The heightened fequency esonse is also allowing fo a lowe initial ulse distubance to delive a sustained wave ofile. Figue 9 illustates a 2L wave system (two tavelling shocks of comaable stength within the combusto, vesus one). This is bought about by the incease in the buning

17 Intenational jounal of say and combustion dynamics Volume. 1 Numbe Pessue, MPa Time, s Figue 9: Pedicted head-end essue wave ofile, efeence moto (K b = s 1, H s = J/kg, d = 24 atm), acceleation active. ate gain K b value fom s 1 to s 1, thus binging the combustion esonse esonant fequency f close to (but still below) the chambe s two-wave acoustic esonant fequency of aoximately 2 khz. With a limit magnitude of about 4 MPa fo each of the two axial essue waves, and with two waves assing though the system, it is not suising that the dc shift is makedly highe hee (aoximately 3.5 MPa in going fom 10.5 to about 14 MPa), as comaed to that seen in Figue 6 (aound 1.5 MPa, in going fom 10.5 to about 12 MPa),- with the inceased stuctual vibation activity (aoximately double the net acceleation inut into the buning ocess fo a 2L wave system, vesus a 1L wave system ealie). In ode to demonstate futhe the intelay between cyclic acceleation and fequency-deendent esonse, one can emove the acceleation sensitivity of the solid oellant. Figue 10 shows the edicted esult, with the same low initial ulse distubance. The dc shift is now almost entiely absent. As shown by Figue 10 and the exanded wave ofile of Figue 11 late in the fiing simulation, the essue wave s limit magnitude is significantly lowe vesus the ealie esult of Figue 8. Along with the weake magnitude, one also notes that the axial wave no longe has an aeciable shock font, unlike the evious cases. Figue 12 illustates the case of a lowe net suface heat elease value ( J/kg), binging the limit magnitude of the 1L axial wave system down to about 0.05 MPa at t = 0.26 s, fom aoximately 1.42 MPa fo a net suface heat elease value of J/kg. The wave ofile moe closely esembles a simle sinusoidal cycle, at this low stength.

18 160 Simulation of axial combustion instability develoment and suession in solid ocket motos 13 Pessue, MPa Time, s Figue 10: Pedicted head-end essue-time ofile, efeence moto (K b = s 1, H s = J/kg, d = 2 atm), acceleation nullified. 13 Pessue, MPa Time, s Figue 11: Pedicted head-end essue wave ofile, efeence moto (K b = s 1, H s = J/kg, d = 2 atm), acceleation nullified.

19 Intenational jounal of say and combustion dynamics Volume. 1 Numbe Pessue, MPa Time, s Figue 12: Pedicted head-end essue wave ofile, efeence moto (K b = s 1, H s = J/kg, d = 2 atm), acceleation nullified. One aoach towads suessing combustion instability (c.i.) symtoms is to alte the intenal gain geomety. In actice, one sometimes notes that c.i. symtoms only occu within a cetain timefame within a given moto design s fiing, suggesting that the gain shae duing that inteim eiod is somehow moe suscetible to essuewave o dc shift develoment. Convesely then, outside of that time eiod, something about the gain s shae is acting to suess symtom develoment. As an examle, in Figue 13, one can see a comaison in limiting wave stength (measued at the head end) at a late oint in the fiing with the intoduction of a oughly 2:1 coss-sectional coe aea contaction (moving left to ight) whose midoint asses though the quate-length osition (x/l = 0.25, whee L is the effective acoustic chambe length in the axial diection) on the gain of the efeence moto, educing the essue wave to about 0.14% of its efeence stength. Note that the nozzle thoat diamete was set at 1.68 cm (vs. the efeence 1.6 cm) fo the contaction case, fo a comaable chambe essue, given that the aft coe diamete stats the simulated fiing at the efeence value of 3.6 cm, while the fowad coe diamete, usteam of the gain aea tansition, stats at 5.1 cm. An attenuation ma fo the above moto is ovided in Figue 14. The attenuation may be defined by the nondimensional attenuation magnitude M a as M a, = w eak w w, eak, (20)

20 162 Simulation of axial combustion instability develoment and suession in solid ocket motos 11.0 Pessue, MPa 0.25L/2:1 contaction Time, s Figue 13: Pedicted head-end essue-time ofiles, efeence moto (K b = s 1, H s = J/kg, d = 2 atm, α = 0%), gain contaction case, acceleation nullified. 1 A u /A d = 1.5 A u /A d = 2.0 A u /A d = 2.5 Attenuation } Axial osition Figue 14: Nondimensional attenuation as function of aea contaction and gain tansition osition (x/l), efeence moto, acceleation nullified.

21 Intenational jounal of say and combustion dynamics Volume. 1 Numbe Pessue, MPa Time, s Figue 15: Pedicted head-end essue wave ofile, efeence moto (K b = s 1, H s = J/kg, d = 2 atm), acceleation nullified, 4% 10-µm Al aticle loading by mass. whee w is the limit essue wave magnitude (eak-to-tough) fo a given simulation, and w, eak is the efeence zeo-suession essue wave magnitude (1.42 MPa in this case, when acceleation effects ae nullified) at the same junctue in the fiing (in this case, 0.26 s is the efeence time). Hee, vaious M a cuves ae dislayed as a function of nondimensional axial osition (x/l) of the gain tansition (given the aea tansition has a elatively shallow sloe as equied fo a quasi-one-dimensional flow model, with the tansition length being set at 30% of the gain length, the midoint of the tansition is the efeence location), with each cuve being at a aticula aea contaction atio. The case of Figue 13 is just below the eak M a oint of the 2:1 contaction cuve in Figue 14 (i.e., aoaching full suession of sustained axial wave activity). The x-axis in essence eesents the staight-cylinde case (no suession at any axial osition). At the left-hand end of the 1.5:1 contaction cuve, the bace symbol eesents that substantial vaiability u o down fom the tend cuve, when moving futhe left, can be exected. Fo examle, fo this case, ealy in the simulated fiing fo x/l of 0.25, a low essue wave slowly aeas to, as the intenal coe exands with buning, tansition towads a new, highe wave limit beyond t = 0.26 s (the nominal efeence time fo these cuves). Imlementing a stonge initial essue distubance of 5 atm (as oosed to the efeence d of 2 atm used fo these uns) bings the highe wave limit solution to quasi-equilibium ealy on (M a = 0.51, a value substantially below that exected fom the tend cuve in Figue 14). The intoduction of noneactive sheical 10-µm-diamete aluminum aticles (4% by mass) into the flow oduces a substantial eduction in wave stength fo the staightcylinde moto of Figue 10, as illustated by the esults of Figue 15. The limit

22 164 Simulation of axial combustion instability develoment and suession in solid ocket motos 1 α = 1% α = 2% α = 3% α = 4% α = 5% Attenuation Diamete, µm Figue 16: Nondimensional attenuation as function of aticle diamete and loading, efeence moto, acceleation nullified. magnitude of the 1L wave system is doed fom 1.42 MPa down to just below 0.5 MPa (a value fo M a of 0.65) at 0.26 s into the fiing. Histoically, suession of highfequency tangential and adial essue waves in SRMs by the use of aticles in the ange of 1 to 3% loading by mass has been in geneal lagely successful. In the case of axial essue waves, the effectiveness of aticles fom 1% to ove 20% loading in suessing wave develoment has been less consistent, elative to the eviously mentioned tansvese cases. In the case of Figue 15, emembeing that acceleation as a facto has been nullified in the combustion ocess, a loading of 4% at 10 µm does aea to be quite effective in suessing (although not eliminating) axial wave develoment in this aticula moto, at this oint in its fiing. An attenuation ma fo the above moto is ovided in Figue 16. Hee, vaious M a cuves ae dislayed as a function of aluminium aticle diamete, with each cuve being at a aticula loading ecentage. The case of Figue 15 is just below the eak M a oint of the 4% loading cuve in Figue 16. The x-axis in essence eesents the 0% loading case (no suession at any aticle diamete). The same junctue in the fiing simulations is the time efeence fo each data oint. The analysis undetaken fo this ae ovides a few examles taken fom ast actice fo how one might suess, to some degee, instability symtoms within a given moto s design. By no means ae these the only techniques available to the moto designe. Ramohalli [26] ovides a good eview of a numbe of techniques that have been attemted ove the yeas.

23 Intenational jounal of say and combustion dynamics Volume. 1 Numbe CONCLUSION The imlications of such factos as stuctual vibation and fequency-deendent combustion esonse on nonlinea axial combustion instability symtom develoment have been demonstated by the examle numeical simulation esults esented in this study of a small cylindical-gain solid ocket moto. Futhemoe, the imlications of such factos as gain geomety and inet aticle loading on symtom suession have been illustated. Undoubtedly, futhe wok emains to be done in establishing a moe thoough undestanding of the vaious mechanisms involved in diving instability symtoms in these motos, in both the axial and tansvese diections. Futue analyses can include the teatment of eactive aticles, and aoiate tacking of aticles of vaying diamete in the flow. This and othe effots will allow fo moe efined and tageted techniques in suessing these instability symtoms. REFERENCES [1] Kanesky, A. L. and Colucci, S.E., Recent Occuences of Combustion Instability in Solid Rocket Motos An Oveview, Jounal of Sacecaft and Rockets, Vol. 12, No.1, Januay 1975, [2] Pice, E. W., Solid Rocket Combustion Instability An Ameican Histoical Account, in: De Luca, L., Pice, E. W. and Summefield, M., eds., Nonsteady Buning and Combustion Stability of Solid Poellants, Pogess in Astonautics & Aeonautics seies, Vol. 143, AIAA Publications, Washington DC, 1992, [3] Bownlee, W. G., Nonlinea Axial Combustion Instability in Solid Poellant Motos, AIAA Jounal, Vol. 2, No. 2, Feb. 1964, [4] Baèe, M., Intoduction to Nonsteady Buning and Combustion Instability, in: De Luca, L., Pice, E. W. and Summefield, M., eds., Nonsteady Buning and Combustion Stability of Solid Poellants, Pogess in Astonautics & Aeonautics seies, Vol. 143, AIAA Publications, Washington DC, 1992, [5] Culick, F. E. C., Pediction of the Stability of Unsteady Motions in Solid- Poellant Rocket Motos, in: De Luca, L., Pice, E. W. and Summefield, M., eds., Nonsteady Buning and Combustion Stability of Solid Poellants, Pogess in Astonautics & Aeonautics seies, Vol. 143, AIAA Publications, Washington DC, 1992, [6] Fishbach, S. R., Majdalani, J. and Flando, G. A., Acoustic Instability of the Slab Rocket Moto, Jounal of Poulsion & Powe, Vol. 23, No. 1, Jan.-Feb. 2007, [7] Kooke, D. E. and Zinn, B. T., Tiggeing Axial Instabilities in Solid Rockets: Numeical Pedictions, AIAA Pae. No , Nov. 5 7, [8] Baum, J. D. and Levine, J. N., Modeling of Nonlinea Longitudinal Instability in Solid Rocket Motos, Acta Astonautica, Vol. 13, No. 6/7, 1986, [9] Geatix, D. R. and Hais, P. G., Stuctual Vibation Consideations fo Solid Rocket Intenal Ballistics Modeling, AIAA Pae No , July 17 19, 2000.

24 166 Simulation of axial combustion instability develoment and suession in solid ocket motos [10] Geatix, D. R., Nonsteady Inteio Ballistics of Cylindical-Gain Solid Rocket Motos, Comutational Ballistics II, oceedings by WIT Pess, , Codoba, Sain, May 18-20, [11] Loncaic, S., Geatix, D. R. and Fawaz, Z., Sta-Gain Rocket Moto Nonsteady Intenal Ballistics, Aeosace Science & Technology, Vol.8, No. 1, Jan. 2004, [12] Montesano, J., Geatix, D. R., Behdinan, K. and Fawaz, Z., Stuctual Oscillation Consideations fo Solid Rocket Intenal Ballistics Modeling, AIAA Pae No , July 10 13, [13] Bewste, M. Q., Solid Poellant Combustion Resonse: Quasi-Steady (QSHOD) Theoy Develoment and Validation, in: Yang, V., Bill, T. B. and Ren, W. Z., eds., Solid Poellant Chemisty, Combustion, and Moto Inteio Ballistics, Vol. 185, Pogess in Astonautics and Aeonautics, AIAA, 2000, [14] Kooke, D. E. and Nelson, C. W., Numeical Solution of Solid Poellant Tansient Combustion, Jounal of Heat Tansfe, Vol. 101, May 1979, [15] Nelson, C. W., Resonse of Thee Tyes of Tansient Combustion Models to Gun Pessuization, Combustion & Flame, Vol. 32, 1978, [16] Kuo, K. K., Kuma, S. and Zhang, B., Tansient Buning Chaacteistics of JA2 Poellant Using Exeimentally Detemined Zel dovich Ma, AIAA Pae No , July 20 23, [17] Geatix, D. R., Tansient Buning Rate Model fo Solid Rocket Moto Intenal Ballistic Simulations, Intenational Jounal of Aeosace Engineeing, Vol. 2008, Aticle , 2008, [18] Geatix, D. R., Model fo Pediction of Negative and Positive Eosive Buning, Canadian Aeonautics & Sace Jounal, Vol. 53, No. 1, Ma. 2007, [19] Gottlieb, J. J. and Geatix, D. R., Numeical Study of the Effects of Longitudinal Acceleation on Solid Rocket Moto Intenal Ballistics, Jounal of Fluids Engineeing, Vol. 114, No. 3, Set. 1992, [20] Dotson, K. W. and Sako, B. H., An Investigation of Poulsion-Stuctue Inteaction in Solid Rocket Motos, AIAA Pae No , July 11 14, [21] Hais, P. G., Wong, F. C. and de Chamlain, A., The Influence of Stuctual Vibations on Pulse-Tiggeed Nonlinea Instability in Solid Rocket Motos: An Exeimental Study, AIAA Pae No , July 1 3, [22] Blomshield, F. S., Histoical Pesective of Combustion Instability in Motos: Case Studies, AIAA Pae No , July 8 11, [23] Lenoi, J. M. and Robillad, G., A Mathematical Model to Pedict the Effects of Eosive Buning in Solid-Poellant Rockets, 6 th Symosium (Intenational) on Combustion, Reinhold Publishing, New Yok, 1957, [24] Razdan, M. K. and Kuo, K. K., Eosive Buning of Solid Poellants, in: Kuo, K. K. and Summefield, M., eds., Fundamentals of Solid Poellant Combustion, Pogess in Astonautics & Aeonautics seies, Vol. 90, AIAA Publications, Washington DC, 1984,

25 Intenational jounal of say and combustion dynamics Volume. 1 Numbe [25] Geatix, D. R., Paametic Analysis of Combined Acceleation Effects on Solid- Poellant Combustion, Canadian Aeonautics and Sace Jounal, Vol. 40, No. 2, June 1994, [26] Ramohalli, K., Technologies and Techniques fo Instability Suession in Motos, in: De Luca, L., Pice, E. W. and Summefield, M., eds., Nonsteady Buning and Combustion Stability of Solid Poellants, Vol. 143, Pogess in Astonautics & Aeonautics seies, AIAA Publications, Washington DC, 1992,