Numerical Study of Reactive Flow Past a Wedge in a Channel

Transcription

1 AIAA Numerical Study of Reactive Flow Pat a Wedge in a Channel Hui-Yuan Fan 1 and Frank K. Lu 2 Aerodynamic Reearch Center echanical and Aeropace Engineering Department The Univerity of Texa at Arlington Arlington TX USA Detonation of the flow of a combutible mixture over a wedged channel i numerically imulated. A two-dimenional time accurate finite-volume-baed method i ued to perform the computation and a five-pecie two-tep chemical reaction i aumed for a toichiometric hydrogen-air mixture. The combution channel i made of a wedged ection followed by a contant area ection. The imulation wa performed with wedge of up to 20 deg emi-angle and ach number from 2 to 6.5 with other inflow parameter fixed. Different type of flow arie depending on the wedge angle and the incoming ach number. Propagating and tanding detonation wave mode were found both of which can be further ubdivided depending on where the detonation i initiated. oreover a cae without combution wa dicovered for a narrow range of ach number for the 5 deg wedge. W I. Introduction EDGE-induced detonation i an intrinically unteady combution proce that encompae variou aerodynamic and chemical phenomena. Such a cla of problem ha received increaing attention recently becaue of it potential application in future hyperonic propulion device. For example a wedge-induced detonation wave may be ued in a ram accelerator to accelerate projectile to very high peed. 1-4 Another relevant concept that make ue of detonation wave for propulion purpoe i the oblique detonation wave engine. 5-7 The idea i to ue the thrut from the wedge-induced detonation of a fuel-air mixture. unipalli et al. 8 and Wilon et al. 9 have propoed that unteady detonation wave can be incorporated into a multimode propulion ytem for high-peed flight. Wedge-induced detonation i alo a phenomenon of fundamental interet with a rich variety of time and length cale (ee for example Ref ). Depite numerou fundamental tudie there are till a lot of poorly undertood feature. A eriou handicap in undertanding confined wedge-induced detonation i experimental difficultie. However numerical imulation of wedge-induced detonation wave have been performed by variou author recently. Figueria da Silva and Dehaie 15 revealed two different kind of overall flow configuration ariing from wedgeinduced detonation wave namely direct initiation of a detonation wave within the tagnation region or an oblique hock wave/oblique detonation wave tranition occurring at ome ditance downtream from the wedge leading edge. ot the available tudie of wedge-induced detonation wave involve long wedge that i the wedge are too long for their top corner to affect the tructure of the reaction zone (Papalexandri 16 ). Alo mot tudie idealize the flowfield to only a ingle wedge in a emi-infinite domain without conidering a confined wedge. Such tudie explore only a limited range of phenomena. Confining the detonation within a channel unveil a rich variety of phenomena that do not appear to be well explored. The preent numerical tudy intend to explore uch phenomena revealing complex wave reflection that can eriouly influence the initiation tranition and propagation wave. II. ethod Numerical imulation of confined wedge-induced detonation wave were performed to explore the phenomena found in uch flow with particular attention to the wave procee. The numerical method taken to perform the required computation i a two-dimenional time accurate finite-volume-baed method developed by Kim at al. 17 A chematic of the configuration i hown in Fig. 1. A two-dimenional channel i formed from a ymmetric wedge 1 Reearch Scientit 2 Profeor Aociate Fellow AIAA 1 American Intitute of Aeronautic and Atronautic

2 with a traight end-ection ymmetrically arranged within a traight chamber. The wedge half-angle ranged from 5 to 20 deg. An incoming toichiometric oxygen hydrogen and nitrogen mixture at 700 K and Pa flow pat the wedge with ach number ranging from 2 to 6.5. A. odel Equation The numerical technique wa previouly reported in Kim et al. 17 and only a brief overview i provided here. The timedependent two-dimenional Euler equation are ued to decribe Figure 1. Schematic of wedged channel configuration. an invicid non-heat-conducting reacting ga flow in which thermal non-equilibrium i modeled with a twotemperature model. Thee equation can be expreed in the Carteian coordinate a U F G + + = S (1) t x y where U i the vector of conerved variable F and G are the convective flux vector and S i the vector of ource term: ρ ρ u ρv w 2 ρu ρu + p ρuv 0 U = ρv F = ρuv 2 G = ρv + p S = 0 (2) ρev ρuev ρvev wv ρe ρue + pu ρve + pv 0 The ubcript = N where N i the number of pecie. The firt 2 American Intitute of Aeronautic and Atronautic N row repreent pecie continuity followed by the two momentum conervation equation for the mixture. The next row decribe the rate of change in the vibrational energy and the final row i the total energy conervation equation. The term u and v are the velocitie in the x and y direction repectively ρ = ρ i the mixture denity ρ i the denity of pecie = 1 p i the preure e v i the vibrational energy E i the total energy per unit ma of mixture w i the ma of production rate of pecie per unit volume and w v i the vibrational energy ource. The internal energy baed on the two-temperature model i aumed to comprie of an equilibium portion at the tranlational temperature T and a nonequilibium portion at the vibrational temperature T v and can be defined a e = eeq( T ) + ev ( Tv ) (3) Thee energy component can be determined with certain thermodynamic relation. The ource term for the pecie ma production rate in the chemical reaction can be written a w = N r r= 1 N ( β α )( R R ) (4) where i the molecular weight of pecie N r i the number of reaction α r and β r are the toichiometric coefficient for reactant and product repectively in the r th reaction. The forward and backward reaction rate of the r th reaction are R f r and R b r repectively. Thee rate can be determined by the Arrheniu law. The ource term of vibrational energy can be written a w + v Qv wev (5) r r = The firt term on the right hand ide Q v repreent the vibrational energy exchange rate of pecie due to the relaxation proce with tranlational energy which can be determined by the Landau-Teller formulation The econd term w repreent the amount of vibrational energy gained or lot due to production or depletion of e v pecie from chemical reaction. A finite-volume algorithm wa ued to olve thee equation numerically. The advantage of thi method i it ue of the integral form of the equation which enure conervation and which allow for the correct treatment of f r b r

3 dicontinuitie. Nonequilibrium flow involving finite-rate chemitry and thermal energy relaxation often can be very difficult to olve numerically becaue of tiffne. The preent method include a point implicit treatment of ource term to reduce the inherent tiffne of the ytem by effectively recaling all the characteritic time in the field into the ame order of magnitude. Roe flux-difference plit cheme i combined with the Runge-Kutta integration cheme for econd-order accuracy in capturing the hock wave in pace and time. In the current tudy the Roger-Chinitz 20 hydrogen-air combution mechanim of 5-pecie (N 2 O 2 H 2 H 2 O and OH) and 2-reaction (H 2 + O 2 = 2OH and 2OH + H 2 = 2H 2 O) i ued. Thi model wa developed to repreent hydrogen-air chemical kinetic with a few reaction tep a poible while till giving reaonably accurate global reult. In thi model nitrogen i counted a a colliional partner in the thermodynamic model and relaxation proce but not included in the chemical reaction model ince the maximum temperature in the hydrogen-air reaction doe not reach the diociation temperature of nitrogen. B. Configuration and Flow Condition The upper half wedged channel i taken a the computation domain. The configuration a hown in Fig. 1 comprie of a harp wedge with half angle θ from 5 through 20 deg. The harp wedge pointing uptream i 60 mm wide and i attached to an aftbody 100 mm long and of the ame width a the bae of the wedge. The entire body i placed in a channel 100 mm wide. The length of the computational domain varie from 300 through 500 mm depending on θ. Thi variation i to enure that the clearance between the wedge tip and channel inlet remain approximately contant. The incoming uperonic flow comprie a premixed toichiometric hydrogen-air mixture. The preure and temperature of the flow are fixed at p = Pa and T =700 K repectively while the incoming ach number i allowed to vary from 2 through 6.5. The matrix of tet cae i hown in Fig. 2. The figure alo how variou flow domain which will be dicued later. The different part of the computational domain are mehed with tructured grid which are not all identical. For example cell are ued for the 5-deg half-angle wedge wherea meh cell are ued for 15- deg half-angle wedge. The flow olver time tep in imulation i timecale of interet. 3 American Intitute of Aeronautic and Atronautic Figure 2. atrix of tet cae and flow domain; 1 propagating detonation wave/hock wave mode 1' propagating detonation wave/hock wave mode with wedge tip initiation 2 no combution mode 3 tanding detonation wave/hock wave mode 3' tanding detonation wave/hock wave mode with wedge tip initiation which i deemed capable of reolving the III. Reult and Dicuion Detonation initiation i accompanied by a deflagration-to-detonation tranition (DDT) unle the initiation energy i ufficiently high. 21 The preent imulation how that the ubequent flow tranition rapidly to the Chapman-Jouguet (CJ) tate ( p CJ =0.665 Pa; T CJ =3014 K; and D CJ =1927 m/). Our previou work uing the ame numerical technique ugget that a tranition phenomenon can be captured The rapid tranition in the preent imulation upport the experimental obervation by Lu and Wilon 23 that hock-induced detonation occur almot intantly and eem to indicate that a direct initiation i poible with hock-induced detonation. In the preent tudy we propoe the poibility that the impingement of an oblique hock initiate combution. If the oblique hock i weak combution occur after one or a number of hock reflection. The imulation indicate that the combution rapidly tranit to detonation. For brevity mot of the dicuion i for θ =5 and 15 deg. A. Five Deg Half-Angle Wedge Three different mode were oberved for flow pat a wedge with a 5 deg half-angle depending on the incoming ach number. When a Type 1 propagating detonation wave mode i obtained in which the initiated detonation wave alway propagate uptream. Detonation i initiated at the impingement of an oblique hock after multiple hock reflection. A number of hock reflection are needed to compre the reactive mixture to caue it to detonate. Unlike obervation of propagating detonation in a contant-ection tube where an entire detonation front form the preent reult how that detonation initiate at the impingement of an oblique hock. For example Fig. 3 how the beginning of a detonation kernel on the aftbody at t = m after ix reflection. The evolution of the kernel i hown in finer detail in Fig. 4(a). Thi figure how that the detonation initiate jut downtream of the

4 t = m t = m t = 0.43 m t = 0.43 m t = m t = m t = m t = m t=0.435m t = 0.44 m t = m t = 0.45 m t=0.44m t = 0.46 m t = 0.48 m t= 0.5 m t = m (a) =3 t = 0.55 m t = 0.28 m t= 0.6 m t = m t= 0.7 m t= 0.8 m t = m t= 0.9 m t = m t= 0.95 m t= 0.29 m Figure 3. Evolution of detonation wave for =3 andθ =5 deg (Type 1); thi and ubequent figure are preure contour plot. t= 0.3 m t= m t = 0.35 m t = 0.40 m (b) =3.5 Figure 4. Evolution of detonation wave in it initiation phae =3 and 3.5 for θ =5 deg. wedge houlder. However there are intance where the detonation i initiated on the channel wall. Thi i hown in Fig. 4(b). In ummary due to the large number of reflection required to trigger the detonation detonation occur ome ditance downtream either on the urface of the aftbody jut downtream of the wedge houlder or on the channel wall. Returning to Fig. 3 the detonation kernel grow to fill the entire width of the channel by m and i elongated a it downtream propagating front i wept by the incoming flow. However the uptream front encounter the incoming flow and propagate at a lower rate. A thi uptream detonation front propagate pat the wedge it 4 American Intitute of Aeronautic and Atronautic

5 encounter the teady oblique hock ytem. A complex λ -foot tructure form a can be een for example at t = 0.7 and 0.8 m. Eventually thi uptream detonation front propagate out of the left computational boundary normal to the incoming flow. When < < 4. 5 the imulation exhibit a teady hock ytem form without any combution occurring in the channel. In other word the incoming flow depite multiple hock reflection cannot ignite the reactant. A detailed mapping how that thi region of no combution labeled Type 2 occur for the narrow ach number range. oreover there alo appear to be a threhold in the value of θ below which ignition alo doe not occur. Thi entire region i indicated by a dahed boundary in Fig. 2. In Type 2 flow thermodynamic parameter behind the final oblique hock within the channel depite multiple reflection remain lower that the CJ tate. The flow generated by the oblique hock ytem approache a teady tate a hown for four cae in Fig. 5. When > a Type 3 tanding detonation/hock wave mode occur. The tanding wave tructure for five cae in thi region are hown in Fig. 6. For each cae the flow in the entire computational domain and an enlargement of the downtream region are hown. For example at: =4.5 the flow i tabilized at t = 0.45 m with the time to achieve tabilization decreaing with increaing ach number. The enlarged view how a normal detonation front tretching from the aftbody to the channel wall. Downtream of the normal detonation front are erie of hock and expanion reflecting from both urface. In ome cae for example at = 5.25 and 5.5 the tanding detonation wave i motly normal to the incoming flow but exhibit a λ tructure. Part of the detonation wave interact with the Prandtl-eyer expanion at the wedge houlder and i deflected downtream. Thi i particularly noticeable in Fig. 6(e). A more detailed equence of event for the (a) =3.875 t 0.38 m (b) =4 t 0.36 m (c ) =4.25 t 0.33 m (d) =4.375 t 0.31 m Figure 5. Final tanding hock wave pattern at four incoming ach number for θ = 5 deg (Type 2). =4.5 and 5 cae are hown in Fig. 7. Figure 7(a) of the =4.5 cae how ignition on the aftbody urface a hort ditance after hock reflection imilar to Type 1 the difference being that eventually the Type 3 flow tabilize into a teady tate. Figure 7(b) for =5 cae on the other hand how ignition on the channel wall. Thi proce i more complicated that that of Fig. 7(1) in that the impinging hock at t = m i till not tabilized and in moving uptream. Thi movement appear to induce the ignition and detonation zone to move forward ( t =0.165 m). Subequently a normal detonation front form and tabilize at t =0.35 m t 0.40 m 8 B. Fifteen Deg Half-Angle Wedge Simulation for θ = 15 deg included a range of ach number from 2 through 6.5. In all thee cae a detonation wave i induced on the channel or aftbody urface. Thi i different from the θ = 5 deg cae where detonation occur only on the aftbody urface if at all. The detonation/hock wave mode can alo be divided into propagating and tanding type at a ach number threhold of approximately In addition the detonation in the θ =15 deg cae can be induced at the wedge tip when > 4.5. Thu three mode (Type 1 1' and 3' can be obtained when θ =15 deg a decribed below. When 4.5 a typified in Fig. 8 the detonation i initiated downtream of (a) =4.5 t 0.45 m (c) =5 t 0.35 m (e) 5.5 t 0.25 m 8 =5.5 t 0.25 m (b) =4.75 t 0.4 m (d) 5 5 t 0 8 =5.25 t 0.28 m Figure 6. Final tanding detonation and hock wave tructure at five incoming ach number for θ = 5 deg (Type 3). 5 American Intitute of Aeronautic and Atronautic

6 t = m t = m t=0.17m t = 0.24 m t=0.13m t=0.18m t = m t=0.135m t=0.19m t = 0.25 m t=0.14m t=0.2m t = 0.26 m t=0.15m t=0.22m t = 0.28 m t=0.155m t=0.25m t = 0.35 m t=0.16m t=0.3m t = 0.45 m t=0.165m t=0.35m (a) =4.5 (b) =5 Figure 7. Evolution of detonation wave at =4.5 and 5 for θ =5 deg the wedge houlder and continuouly propagate uptream. Thi mode belong to the Type 1 mode mentioned previouly (Fig. 3). However the wave evolution in the 15 deg cae i more complex than in the 5 deg cae. The detonation initiate where an oblique hock impinge the aftbody wall a can be een at t = m in Fig. 8. The detonation wave i further trengthened by ubequent reflection on the oppoite wall followed by it uptream front to form an oblique detonation front at t = m. eanwhile the downtream front exit from the channel outlet. The etablihed oblique detonation front evolve into a normal front by t = m. At t = m another detonation wave i initiated ahead of thi uptream propagating detonation front. It i thought that the precuror detonation wave i triggered by the trengthening of the uptream oblique hock from the uptream propagating detonation front. Thi precuror detonation wave doe not completely conume the reactant thereby allowing the downtream detonation front to exit. A complex hock-detonation wave i formed which continue to propagate uptream through the wedge region to the channel inlet. While Fig. 8 how a detonation wave triggered on the aftbody at t = m the preent parametric tudy alo revealed that the detonation can be triggered from the channel wall. When = the detonation wave initiate at the tip of the wedge inducing either a propagating or a tanding mode. A propagating mode labeled Type 1' occur when 4.5 < < 5.5; otherwie the wedge induce a Type 3' tanding detonation/hock wave tructure. In thee mode the detonation initiate at the tip of the wedge that promptly develop over the whole wedge urface. The detonation front expand continuouly reflected by the oppoite channel wall to form an oblique detonation front that attache at the wedge tip. Behind the oblique detonation wave hock reflection and expanion wave can developed to produce either Type 1' or 3' flow domain. Figure 9 how the evolution of the former mode at = 5. In thi mode the downtream reflection hock wave induced by the detonation wave move uptream reulting in a new normal hock wave that overtake the exiting oblique detonation front and a a new detonation front to propagate uptream. Figure 10 how the evolution the Type 3' flow for = 6. The figure how that the downtream hock reflection induced by the detonation wave i eventually tabilized. It i worth noting that the ach reflection phenomenon can be oberved for the propagating Type 1 and 1' mode. For example in Fig. 8 and 9 when the detonation front move over the oblique hock attached at the wedge 6 American Intitute of Aeronautic and Atronautic

7 t= m t = m t=0.005m t=0.2m DW t=0.065 m DW t=0.135 m t=0.015m t=0.25m t = 0.07 m t=0.15 m t=0.025m t=0.3m t=0.075 m t=0.17 m t=0.035m t=0.35m t = 0.09 m t=0.19 m t=0.05m t=0.4m t=0.1m t=0.22 m t=0.1m t=0.5m t=0.115 m t = 0.25 m t = m t=0.3m t=0.15m t=0.6m DW t = 0.12 m t=0.37 m t=0.175m Figure 8. Evolution of detonation wave at deg (Type 1) =3 for θ = 15 Figure 9. Evolution of detonation wave at (Type 1') =5 for θ = 15 deg tip a ach reflection configuration appear. In addition the Type 1' mode in Fig.9 how the intrinic intability of a normal propagating detonation front that i uually encountered in a detonation front propagating in channel (ee t =0.6 m). C. Larger Wedge Angle and Complex Area in Fig. 2 Due to limitation of the current code the imulation the θ =20 deg cae only can be performed for < 5.25 and only Type 1 and 1' mode are obtained for thee cae. Further a i preented in Fig. 2 there i the complex area in which the detonation mode may be propagating or tanding. The imulation reult how that either a propagating or a tanding mode can occur here. We hypotheize that thi i a tranition region between the two major mode. Further work i needed to explore and define thi tranition region over a larger range of wedge angle. t=0.005m DW t = m t = m t = m t = 0.05 m t=0.1m t = 0.15 m t=0.2m t>=0.24m IV. Concluion Figure 10. Evolution of detonation wave at deg (Type 3') The wedge-induced detonation occurring in a twodimenional ymmetric wedged channel wa tudied numerically. A five-pecie and two-tep reaction mechanim wa adopted to model the thermo-chemical dynamic of the detonation procee and a time-accurate and finitevolume-baed method wa ued to numerical imulate the procee. Half-wedge angle of up to 20 deg were conidered. The incoming ach number wa allowed to vary while the other inflow aerodynamic parameter were 7 American Intitute of Aeronautic and Atronautic =6 for θ = 15

8 fixed. A qualitative undertanding wa obtained of the complex and rich detonation flow. Different flow domain arie depending on the incoming ach number and the wedge angle. Propagating and tanding detonation wave were found both of which can be further ubdivided depending on where the detonation i initiated uch a on the channel wall or along the wedge urface after multiple reflection or at the wedge tip. It wa thought that a tranitional region exit between thee two mode. Finally a no-combution cae wa dicovered for a narrow range of ach number for the 5-deg half-angle wedge. Thi cae occur between the propagating mode at a lower ach number and a tanding mode at a higher ach number. It wa thought that the geometry i uch that the hock reflection could not induce a detonation within the domain. Reference 1 Grimer. J. and Power J.. Calculation for Steady Propagation of a Generic Ram Accelerator Configuration AIAA Journal of Propulion and Power Vol. 11 No pp Lefebvre. H. and Fujiwara T. Numerical odeling of Combution Procee Induced by a Superonic Conical Blunt Body Combution and Flame Vol. 100 No pp Hertzberg A. Bruckner A. P. and Bogdanoff D. W. Ram Accelerator: A New Chemical ethod for Accelerating Projectile to Ultrahigh Velocitie AIAA Journal Vol. 26 No pp Kailaanath K. Recent Development in the Reearch on Pule Detonation Engine AIAA Journal Vol. 41 No pp Brackett D. C. and Bogdanoff D. W. Computational Invetigation of Oblique Detonation Ramjet-in-Tube Concept AIAA Journal of Propulion and Power Vol. 5 No pp Power J.. and Stewart D. S. Approximate Solution for Oblique Detonation in Hyperonic Limit AIAA Journal Vol. 30 No pp Terao K. Ihii K. Totuka T. and Ihikawa Y. An Experimental Invetigation of Hyperonic Combution for Ram Jet Engine Applying Detonation Wave AIAA Paper unipalli R. Shankar V. Wilon D. R. Kim H. Lu F. K. and Hageth P. E. A Puled Detonation Baed ultimode Engine Concept AIAA Paper Wilon D. R. Lu F. K. Kim H. and unipalli R. Analyi of a Puled Normal Detonation Wave Engine Concept AIAA Paper Thaker A. A. and Chelliah H. K. Numerical Prediction of Oblique Detonation Wave Structure Uing Detailed and Reduced Reaction echanim Combution Theory and odeling Vol. 1 No pp Grimer. J. and Power J.. Numerical prediction of oblique detonation tability boundarie Shock Wave Vol. 6 No pp Power J.. and Gonthier K. A. Reaction Zone Structure for Strong Weak Overdriven and Weak Underdriven Oblique Detonation Phyic of Fluid A Vol. 4 No pp Ohyagi S. Obara T. Nakata F. and Hohi S. A Numerical Simulation of Reflection Procee of a Detonation Wave on a Wedge Shock Wave Vol. 10 No pp Guo C. Zhang D. and Xie W. The ach Reflection of a Detonation Baed on Soot Track eaurement Combution and Flame Vol. 127 No pp Figueira da Silva L. F. and Dehaie B. Stabilization of an Oblique Detonation Wave by a Wedge: A Parametric Numerical Study Combution and Flame Vol. 121 No pp Papalexandri. A Numerical Study of Wedge-Induced Detonation Combution and Flame Vol. 120 No pp Kim H. Lu F. K. Anderon D. A. and Wilon D. R. Numerical Simulation of Detonation Proce in a Tube Computational Fluid Dynamic Journal Vol. 12 No pp illikan R. C. and White D. R. Sytematic of Vibrational Relaxation Journal of Chemical Phyic Vol. 39 No pp Vincenti W. G. and Kruger C. H. Introduction to Phyical Ga Dynamic Krieger alabar Florida Roger R. C. and Chinitz W. Uing a Global Hydrogen-Air Combution odel in Turbulent Reacting Flow Calculation AIAA Journal Vol. 21 No pp Knytauta R.and Lee J. H. S. On the Effective Energy for Direct Initiation of Detonation Combution and Flame Vol pp Fan H. Y. and Lu F. K. Comparion Study of Detonation Procee in a Variable Cro-Section Chamber and a Simple Tube AIAA Journal of Propulion and Power 2004 (to be publihed). 23 Lu F. K. and Wilon D. R. Detonation Driver for Enhancing Shock Tube Performance Shock Wave Vol. 12 No pp American Intitute of Aeronautic and Atronautic