Improvement of Two-Equation Turbulence Model with Anisotropic Eddy-Viscosity for Hybrid Rocket Research

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1 evenh Inernaonal onference on ompaonal Fld Dynamcs (IFD7), Bg Island, awa, Jly 9-, IFD7-9 Improvemen of Two-Eqaon Trblence Model wh Ansoropc Eddy-Vscosy for ybrd oce esearch M. Mro * and T. hmada ** orrespondng ahor: 79@mal.ecc.-oyo.ac.p * Deparmen of Aeronacs and Asronacs, chool of Engneerng, he Unversy of Toyo, Japan. ** Inse of pace and Asronacal cence, Japan Aerospace Eploraon Agency. Absrac: For a hybrd roce engne, has been proven by grond and flgh epermens ha he fel regresson rae can be acceleraed by swrlng necon of he odzer. I s nown a swrlng rblen flow has ansoropc nare n s rblen vscosy. In hs sdy, he obecve s o mprove he applcably of esng reamen of soropc rblen model sch as wo-eqaon, eddy vscosy, model by nrodcng ansoropc eddy-vscosy coeffcens ha can ads parclar drecon of hese wh sbsanal dervave of vorcy, n order o smlae swrlng rblen flows n hybrd roce engnes. mlaon resls for some swrlng rblen flows wh he esng model and wh an mproved model wll be compared. Keywords: wrlng Flow, Trblence Modelng, AN, ybrd oce. Bacgrond and Obecves ybrd roce proplson s one of space proplson echnqes for he ne generaon now researched acvely. As shown n Fgre, hs ype of roce consss of sold fel and lqd odzer. Typcally, acrylc or wa s sed as he sold fel, and lqd oygen s sed as he odzer. Therefore, has hgh safey becase he fel does no conan eplosves as ngredens and low envronmenal load by ehas han ha of a sold roce moor. In addon, he roce shows good characerscs of capably ha hrs modlaon le a lqd roce engne, and hgher specfc mplse han a sold roce moor. Odzer Fel Gran Fgre : Olne of a hybrd roce engne Thogh, n case a convenonal fel, sch as, TPB or PE, ec., a hybrd roce engne has lmaon ha only low fel regresson rae can be obaned. As one of mehods for enhancng fel regresson rae, a swrlng-odzer-yped hybrd roce engne ha ndces swrlng flow n he chamber was nvened. I has been clarfed by epermenal nvesgaons ha he regresson rae can be mproved by hs mehod. Also, some nmercal sdes are repored. In general, for he case ha FD s employed as a desgn ool of he combson chamber, he eynolds Averaged Naver-oes (AN) eqaons have been sed becase of her reasonable compaonal coss compared o hose of LE and DN. On hs occason, a wo-eqaon rblence

2 model, sch as - model, has been ofen seleced n order o oban rblence sresses. owever, as s well nown, ordnary lner rblence models based on he eddy-vscosy assmpon canno smlae flow feld le swrlng rblence n ppe precsely. Becase he rblence ranspor n swrlng flows s sally ansoropc whch s small n selecve drecons, a rblen model assmng soropc rblence ranspor canno descrbe swrlng flow n prncple. ence he eynolds sress models (M) solvng for each dreconal eynolds sresses have been sed for hese swrlng flow feld. Thogh, becase he varables of hese models are ncreased by nmbers of elemens of eynolds sresses, compley of codng and calclaon cos wdely ncrease n comparson wh sandard wo-eqaon rblence models. ecenly, Yoshzawa, e al. have developed he mproved wo-eqaon rblence model, whch s sable for smlang swrlng rblence flow []. Ths model s based on he sandard - model. An eddy-vscosy of hs model s redced by a coeffcen consrced wh sbsanal dervave of vorcy ndcang swrlng of mean flow. o, he rblence sresses of all drecons are redced nformly. Ahors smlaed he epermen of low velocy swrlng flow n a ppe by Mraam, e al. [] as a pror case for evalang applcably of he mproved model o he swrlng-odzer-yped hybrd roce engne []. And was confrmed ha he predcve accracy for wea swrlng flow s clearly mproved han he resl of sandard - model. Thogh, he resl ha applyng mproved model o srong swrlng flow for nsance vore-be was no favorable. I s consdered ha hese flow felds have srong ansoropc rblence. ence, ahors aemp o ha mprovemen smlaon resl of Yoshzawa model by addng nonlnear erm of rblen sress formlaon of helcy model n order o append propery of ansoropy o eddy vscosy. Obecve of hs sdy s o comprehend characersc of hs model by means of smlae esng epermenal resls and o valdae avalably. Governng Eqaons Favre averaged rblen compressble Naver-oes Eqaons are sed as governng eqaons, whch are wren n erms of mean varables as follows. Mass conservaon: Momenm conservaon: p ˆ Mean energy conservaon: E ˆ q () And where, ˆ (4) (5) p E e h (6) q T T p PrL PrT () () (7)

3 and s me, s caresan coordnae, and are sff, s densy, s velocy componen, p s pressre, E s specfc oal energy, s specfc oal enhalpy, s rblence energy, s sress ensor, T s emperare, q s hea fl, p s consan pressre specfc hea, PrL s lamnar Prandl nmber and Pr T s rblen Prandl nmber. Defnon of n mass enhalpy h e p and an eqaon of sae p T are added o he above eqaon sysem wh specfc nernal energy e and gas consan. The eddy vscosy and he rblen sran ensor are evalaed by rblence model. Trblence Models The compose me scale - model (herenafer called Yoshzawa model) bld by Yoshzawa e al. [] s shown below, whch s base rblence model of crren sdy. In case of normal wo eqaons rblence model, rblence energy cascade me scale E s ofen sed o deermne eddy vscosy. The characersc of hs compose me scale model s o se me scale of sran, me scale of vorcy and me scale of lagrangan varaon of vorcy vecor n addon o E. Formlaons of hese me scales are shown as follows. (8) (9) 4 DΩ L () D And s vorcy ensor and Ω s vorcy vecor. These me-scale-effecs are ncorporaed as correcon facor n prodcon of eddy vscosy wh convenonal formlaon sed n normal - model. Formlaons of hs rblence model are shown as follows. Trblen energy ranspor: () Trblen energy dsspaon rae ranspor: f () And where, V f () DΩ L Ω D (4)

4 (5) Where, f and f are low-eynolds nmber correcon fncons. In hs sdy, he fncons consrced by Abe e al. 6 are sed. In hs case, for he propose of seady sae analyss, Lagrange dervave s defned as follows, D D Ω Ω (6) In addon, cenral dfference scheme s sed o dscreze hs dervave and he model consans are ndcaed as follows.. V, 5.,. L, 4., 4., 5., 9. In he case of swrlng flow, hs model descrbes lamnarzaon phenomenon a cener of ppe de o sppress prodcon of eddy vscosy by means of ha hrd erm of nner roo of become domnan and s more han. The - - model bld by Yoo e, al. [4] (herenafer called helcy model) s shown below. Ths model consss of hree ranspor eqaons of rblen energy, dsspaon rae and rblen helcy ( ω ) whch descrbe swrlng of rblence n flow and rblen sress ensor s nonlnear. Trblen energy ranspor: (7) Trblen energy dsspaon rae ranspor: f Ω (8) Trblen helcy ranspor: (9) And where, Ω () 4 () The rblence model sng n hs sdy s composed by ha addng nonlnear erm of rblen sress of helcy model o ha of Yoshzawa model. Namely, eqaon of rblen sress ensor of Yoshzawa model s changed from eqaon (5) o eqaon (), and eqaon of dsspaon rae s changed from eqaon () o eqaon (9) n erms of con he effec of helcy. Where, he helcy s obaned from eqaon (9) nder he assmpon of local eqlbrm as below.

5 () Yoshzawa model s he rblence model ha has soropc eddy vscosy, b by addng nonlnear erm o rblen sress, can be consdered ha he soropc eddy vscosy s convered no ansoropc eddy vscosy as follows. Ω () Ω (4) (5) The coeffcens appear on Yoshzawa model are no changed, and ha appear on helcy model are shown as follows, whch are se wh epermenally ral calclaons. 6, 4, 4 Nmercal Mehod The governng eqaons are specally dscrezed by fne volme mehod. Nmercal fl of convecon erm s calclaed de o LAU scheme consrced by hma e al. [5] The rd order MUL scheme reglaed by Van-Albada slope lmer wh prmve varable nerpolaon s sed o oban hgher order accracy a cell bondary. Nmercal fl of vscos erm s evalaed by nd order cenral dfference scheme. nd order sages nge-ka scheme s sed o negrae me dervave erm. 5 Nmercal ep Two case of swrlng flow are seleced as analyss obecs. One case s an epermen condced by Mraam e, al. whch has reard flow n cener of he ppe and se waer as worng fld. Anoher s an epermen by Ko [6] whch ndcae srong reverse flow n cener and se ar as worng fld. The nmercal grd pon nmber for he former case s abo. mllon, and he nmber of each drecons of aal, rads and crcmferenal are 7, 5 and 8 pons respecvely, and he grd nmber for he laer case s abo.58 mllon, and he nmber of each drecons are, 7 and 8 pons respecvely. The mnmm grd lengh s abo μm a wall n each case. The velocy bondary condons of aal and crcmferenal drecon a nle are adoped velocy profles whch were measred by epermen, and rads drecon velocy se o. In case of epermen of Mraam e, al., he poson measrng velocy s a dsance of 5 mes of ppe rads from he swrler, and n case of Ko e, al., he poson s.4 mes of ppe rads. Becase he daa of velocy profle have been nondmensonalzed by bl velocy, n hs sdy, he bl velocy s se o 5 m/s. In hs case, he eynolds nmber s abo order of 5. The free oflow condon s mpored a ole bondary and he non-slp condon s se a he wall. The bondary condons for

6 and are shown below, where, sff n and w ndcae nflow and wall, respecvely. Inflow ondon: Wall ondon: n.u m (6).9. (7) n (8) w w (9) n w 6 esls and Dscsson Frs, he smlaon resls for he epermen by Mraam e, al. are shown. Fgre and Fgre ndcae radal dsrbon of aal velocy and crcmferenal velocy, respecvely, boh of hem are averaged crcmferenally and are nondmensonalzed. In boh fgres, do ndcaes he epermenal vales, dashed lne and sold lne ndcae poson of measrng secons whch are 4 and 7.8 from nle, lne color of red and ble ndcae crren model and Yoshzawa model, respecvely. Fgre : Aal velocy dsrbons Fgre : rcmferenal velocy dsrbons In Fgre, abo aal velocy dsrbon, can be seen ha he crren model smlae beer resls han ha of he Yoshzawa model n cenral and near wall regons. In cenral regon, he crren model esmaes smaller absoles of aal velocy han he resls Yoshzawa model, and n near wall regon, esmaes bgger absoles of velocy. In fgre, abo crcmferenal velocy dsrbon, dfferences beween he crren model and he Yoshzawa model are seen only n near wall regon a boh secons. In hs saon, crren model esmaes bgger absoles of velocy han he resls of Yoshzawa model. Fgre 4 shows crcmferenally averaged dsrbons of soropc eddy vscosy whch are calclaed by crren model and Yoshzawa model a secons of 4 and 7.8. In boh secons, absole vale of crren model s smaller han ha of Yoshzawa model a small regon near cener, hs relaon rn over n nermedae regon, and he vale s smaller han ha of Yoshzawa model agan a near wall. Also, he absole vales of boh models have ncreasng endency dependng on

7 dsance from he nle. Fgre 4: Isoropc eddy vscosy dsrbon Fgre 5 shows s componens of rblen sress along y as of aresan coordnae a secon 4. These componens aren averaged crcmferenally, and hese are descrbed as below:, y, z, yy, yz and zz. In hs saon,, y and z as of aresan coordnae are eqvalen o aal, radal and crcmferenal drecon, respecvely. Dsplay rle of sress ensor s canoncal ha frs of wo nde ndcaes drecon of normal vecor of face applyng sress, and second nde ndcaes drecon of sress. (a) (b) (c) (d)

8 (e) Fgre 5: omponens of rblen sress (f) From fgre 5 (a) and (c) ( and z), can be seen ha he absole vale of rblen sress by crren model s bgger han ha of Yoshzawa model n cenral regon. ence, dreconal momenm decrease as compared o he resl of Yoshzawa model, and he dfference of aal velocy dsrbon arses as cenral regon of fgre. Also, from fgre 5 (a) and (b) ( and ), can be seen ha he absole vale of rblen sress by crren model s smaller han ha of Yoshzawa model n near wall regon. ence, dreconal momenm doesn decrease as compared o he resl of Yoshzawa model, and he aal velocy dsrbon s mananed as near wall regon of fgre. Addonally, from fgre 5 (e) and (f) ( and ), can be seen ha he absole vale of rblen sress by crren model s smaller han ha of Yoshzawa model n near wall regon. ence, z dreconal momenm doesn decrease as compared o he resl of Yoshzawa model, and he dfference of crcmferenal velocy dsrbon arses as near wall regon of fgre. econd, he smlaon resls for he epermen by Ko are shown. Fgre 6 and Fgre 7 ndcae radal dsrbon of aal velocy and crcmferenal velocy, respecvely, boh of hem are averaged crcmferenally and are nondmensonalzed. The means of lnes and colors are same as fgre and fgre, and he posons of measrng secon are 4.6 and 8 from nle. yz zz y Fgre 6: Aal velocy dsrbons Fgre 7: rcmferenal velocy dsrbons In fgre 6 and 7, even dsances from nle o posons of measrng secon are longer han he cases of Mraam, can be seen ha boh of crren model and Yoshzawa model can smlae epermenal resls of Ko. Thogh, boh dsrbons have rends smlar o cases of Mraam.

9 Fgre 8: Isoropc eddy vscosy dsrbon Fgre 8 shows crcmferenally averaged dsrbons of soropc eddy vscosy whch are calclaed by crren model and Yoshzawa model a secons of 4 and 8. Unle smlaon case for he epermen of Mraam, he vale sn changed by rblence models ecep cenral regon of ppe, and dsrbons n hs regon have same rend o smlaon resls for he case of Mraam. When fgre 8 and fgre 4 are compared, can be seen ha vales of eddy vscosy n fgre 8 s several mes bgger han ha of fgre 8. ence, s hogh ha boh models canno smlae he epermenal resls de o defecon of dampng eddy vscosy. 7 mmary In hs sdy, Yoshzawa model s mproved by addng nonlnear erm of rblen sress formlaon of helcy model n order o append propery of ansoropy o eddy vscosy. In he case of smlaon for epermen of Mraam e al., whch doesn have reverse flow n cenral regon, aal velocy dsrbons are slghly mproved a cenral and near wall regon, and absole vale of crcmferenal velocy become bgger han ha of Yoshzawa model. Meanwhle, n he case of smlaon for epermen of Ko, whch have srong reverse flow n cenral regon, boh Yoshzawa model and crren model canno acheve favorable resls. eferences [] Yoshzawa, A., e al., Trblen-vscosy modelng applcable o swrlng flows, based on a compose me scale wh mean-flow helcy parally ncorporaed, Jornal of Trblence. [] Mraam, M., e al., An Epermenal dy of wrlng Flow n Ppes, Transacons of he Japan ocey of Mechancal Engneers. B, vol.4, No.46, pp.79-8, 975. [] Mooe, M, e al., Valdaon of Nmercal mlaon of wrlng Trblen Flow for ybrd oce esearch, AIAA -56 [4] Yoo, N., e al., Three-Eqaon Trblence Model wh oheren rcres Incorporaed hrogh he elcy : Applcaon o a wrlng Flow n a ragh Ppe, Transacons of he Japan ocey of Mechancal Engneers. B, vol.58, No.55, pp.74-7, 99. [5] hma, E., e al., On New mple Low-Dsspaon cheme of AUM-Famly for All peeds, AIAA Paper 9-6 [6] Ko, O., Epermenal sdy of rblen swrlng flow n a sragh ppe, Jornal of Fld Machancs, vol. 5, pp , 99