Premixed Flame Response to Acoustic Waves in a Porous-Walled Chamber with Surface Mass Injection. Propulsion Conference and Exhibit

Transcription

1 AIAA -369 Premixed Flame Response to Acoustic Waves in a Porous-Walled Chamber with Surace Mass Injection Wen-Wei Chu and Vigor Yang The Pennsylvania State University University Park, PA 68 Propulsion Conerence and Exhibit 7 July Indianapolis, IN For permission to copy or republish, contact the copyright owner named on the irst page. For AIAA-held copyright, write to AIAA, Permissions Department, 8 Alexander Bell Drive, Suite 5, Reston, VA

2 38th AIAA/ASME/SAE/ASEE Joint Propulsion Conerence and Exhibit AIAA July, Indianapolis, Indiana Premixed Flame Response to Acoustic Waves in a Porous-Walled Chamber with Surace Mass Injection Wen-Wei Chu* and Vigor Yang The Pennsylvania State University, University Park, PA 68 and Anand B. Vyas and Joseph Majdalani Marquette University, Milwaukee, WI 5333 In order to study the internal coupling between lame dynamics and vortico-acoustic waves ormed during solid propellant combustion, a numerical simulation o an idealized rocket combustion chamber is carried out. The chamber is modeled as a rectangular enclosure along whose porous walls a laminar mixture o premixed reactants is uniormly injected. The mathematical model is based on the conservation equations in two space dimensions. Full account is taken o variable thermo-physical properties and inite-rate chemical kinetics. Boundary conditions are speciied using the method o characteristics that accommodates the transport o entropy, vorticity, and acoustic waves. The governing equations and associated boundary conditions are solved numerically using a preconditioning technique and a dual time-stepping integration procedure. For illustrative purposes, a propane-air mixture is injected through the chamber walls in an eort to emulate the lame evolution inside a solid-propellant rocket motor under laminar conditions. First, detailed lame structures under steady-state conditions are realized. Subsequently, traveling acoustic waves are imposed at the head end. Simulation results indicate that the oscillatory velocity exhibits a multi-dimensional structure caused by unsteady vorticity, pressure, and lame oscillations. Accordingly, the eects o laminar premixed lame oscillations are limited to a thin region above the burning surace. This region becomes thinner as the requency o oscillations is increased. The lowield outside the lame zone bears a striking resemblance to recent analytical solutions obtained in a geometrically similar chamber. Nomenclature a speed o sound, γ RT A pressure amplitude C p speciic heat E, F convective lux vectors Ev, F v diusive lux vectors e speciic total energy k thermal conductivity M i molecular weight o species i p pressure *Graduate Research Associate. Member AIAA. Proessor, Department o Mechanical Engineering. Fellow AIAA. Graduate Research Associate. Member AIAA. Assistant Proessor o Mechanical and Industrial Engineering. Member AIAA. Q! rate o heat release q! rate o heat release per unit mass R the ideal gas constant or reactants Re injection Reynolds number T temperature t physical time u, v velocities in x and y directions, respectively V speciic volume v b blowing velocity in a cold-low model v transverse velocity directly above the lame v w transverse velocity directly above the wall x, y Cartesian coordinates Y i mass raction o species i Greek Symbols δ thickness o the reactive lame zone. λ spatial wavelength o unsteady velocity γ ratio o speciic heats American Institute o Aeronautics and Astronautics Copyright by Vigor Yang. Published by the American Institute o Aeronautics and Astronautics, Inc., with permission.

3 µ,ν dynamic and kinematic viscosities τ c characteristic time associated with heat release luctuation ω! i mass production rate o species i Ω vorticity Superscripts acoustic luctuation steady state ^ amplitude o luctuating quantity Subscripts a acoustic mode b blowing condition or a cold low lame condition i chemical species i s entropy mode v vorticity mode w wall condition I I. Introduction N this article, the premixed laminar lame response to vortico-acoustic oscillations is investigated inside a porous-walled chamber simulating a solid rocket motor. Following the injection o a premixed low o reactants across the chamber walls, the resulting thin laminar lame that orms in the vicinity o the porous wall is used to mimic the burning zone in solid propellant rocket motors. Harmonic pressure luctuations are superimposed to represent acoustic oscillations in actual motors. This work ocuses on characterizing the thermo-acoustic boundary layer above the injection surace, which plays an important role in determining the energy exchange between unsteady combustion processes and core low dynamics. Understanding the mechanisms behind these thermo-acoustic energy exchange processes helps to improve our ability to predict combustion instabilities in rocket motors, gas turbines, and other large combustors. Combustion instabilities arise in a number o physical contexts that generally exhibit a sel-sustained coupling between unsteady chemical heat sources and chamber acoustics. Understanding their attributes is o crucial importance in the development o combustion devices such as solid rocket motors, liquid rocket engines, 3,4 gas turbines, 5 and pulsed combustors o the Helmholtz 6 and Rijke types. 7 Initially, small-amplitude acoustic disturbances arise rom the intrinsic low instabilities in the chamber. I these disturbances oscillate in avorable phase with the rate o heat addition, they are ampliied in accordance with Rayleigh s criterion. In certain applications, the growing instability may reach a level suicient to cause undesirable structural vibrations or outright ailure o the system. A thorough understanding o this energy transer mechanism requires detailed studies o a) luid dynamic processes that dictate the environment in which chemical reactions occur, and b) chemical processes that provide the energy or driving thermoacoustic instabilities. Clearly, the interaction between these two processes plays a key role in determining the dynamic behavior o unsteady motions in combustion chambers. The present paper deals with the unsteady response o a laminar lame to traveling acoustic waves in a twodimensional chamber. A simple geometry is selected in order to acilitate theoretical and numerical comparisons. Unlike other studies that have ocused on turbulent lames in similar physical settings, 8- the scope here will be limited to laminar lames only. The exploration o laminar lames will be shown to acilitate the identiication o key mechanisms that can be corroborated by theoretical solutions. The study begins in Sec. II by a description o the model and its attendant boundary conditions. The numerical scheme is briely visited in Sec. III where it is ollowed by a steady-state solution o the problem at hand. Ater describing the mean low and thermal characteristics in Sec. IV, the onset o unsteady disturbances is examined in Sec. V. Therein, the thermo-acoustic coupling due to heat release is explored at dierent requencies and spatial locations along the length o the chamber. The Rayleigh number is also invoked to explain the connection between vorticoacoustic waves and thermal oscillations. Finally, conclusions are provided in Sec. VI. II. Formulation A. Model The planar domain is shown schematically in Fig. a. The chamber is closed at the head-end, while the porous top and bottom suraces admit a premixed combustible mixture. The injection process simulates the gas-phase combustion and low development in a rocket motor loaded with a homogeneous solid propellant. A similar coniguration was experimentally studied by Sankar et al. Here we shall study the mutual coupling between low oscillations and lame dynamics, with emphasis placed on the interactions among the acoustic, vortical, and entropy waves that arise in the chamber. But irst, a numerical analysis based on the complete conservation equations and inite-rate chemical kinetics will be developed. At the outset, various undamental mechanisms, including vorticity generation and transport, lame oscillations, and acoustic motions, will be investigated. American Institute o Aeronautics and Astronautics

4 B. Equations The ormulation is based on the conservation equations o mass, momentum, energy, and species concentrations or a multi-component, chemically reacting system. Full account is taken o inite-rate chemical kinetics and variations in thermo-dynamical and transport properties. To start, the governing equations in two-dimensional Cartesian coordinates are written in the generic vector orm Q + ( E Ev) + ( F Fv) = S () t x y where x and y represent the axial and transverse coordinates, respectively. Using standard descriptors, the vector o conserved variables Q is deined by T Q = [ ρ, ρu, ρv, ρe, ρy i ] () The inviscid lux vectors ( E, F ) and the viscous lux vectors ( E v, F v ) ollow the deinitions given by Chu. 3 The thermal and mass diusion processes are approximated by Fourier s and Fick s laws, respectively. The source term S includes contributions rom chemical reactions taking place inside the combustion chamber. Within the temperature and pressure ranges o the present study, the speciic heat, viscosity, and thermal conductivity are taken to be unctions o temperature and can be approximated by ourth-order polynomials. 4 The mixture speciic heat is obtained by mass-raction weighing o each constituent species. Thermal conductivity and viscosity are calculated using Wilke s mixing rule. The binary mass diusivity D ij between species i and j is obtained using the Chapman-Enskog theory based on the Lennard-Jones intermolecular potential-energy unctions. 5 The eective mass diusion coeicient or species i in a multi-component mixture is related to the binary diusion coeicients through Wilke s ormula. 6 Since the present study is concerned with the eect o unsteady heat release and low motion on chamber aeroacoustics, an accurate estimate o the lame speed and heat release rate is essential. The single-step global reaction mechanism proposed by Westbrook and Dryer 7 or hydrocarbon/air systems is thereore adopted because o its reasonable prediction o these two parameters. The small number o species involved urther renders this model an attractive choice rom the standpoint o computational eiciency. C. Boundary Conditions The method o characteristics is used to handle the boundary conditions. Since the low is subsonic throughout the chamber, three physical conditions are speciied at the inlow boundary and only one at the H premixed C 3 H 8 air mixture H y lame zone v δ v w a).l.4l x.6l.8l L H y b).l.4l.6l.8l L x Fig. Schematic diagram o the physical model illustrating: a) the adopted space coordinates, lame zone location, and mean velocity evolution across the lame zone; and b) the non-uniorm grid extending rom the wall to the core. 3 American Institute o Aeronautics and Astronautics

5 exit. The mass low rate and temperature o the premixed gases are ixed at the porous walls, along with the no-slip condition that enorces normal injection o the reactants. At the head end, the axial velocity and pressure gradient, as well as the radial velocity gradient, are set to zero or steady-state calculations. This prevents the occurrence o numerically generated recirculation near the head end. It also coincides with the head end boundary conditions used in mathematical idealizations o non-reactive core-lows. 8 For unsteady low simulations, a periodic sinusoidal pressure oscillation is imposed at a known requency. The axial velocity and temperature luctuations are accordingly speciied rom the isentropic relations to generate a traveling acoustic wave propagating in the axial direction. Finally, low symmetry is assumed along the midsection plane. The exit boundary conditions are also treated careully to avoid the production o numerically generated, spurious relecting waves. For this purpose, the methodology described by Watson and Meyer 9 is used to speciy the low properties at the exit section. The governing equations are rearranged into our decoupled relations controlling the vorticity, entropy, and acoustic wave propagation. The complete orm o these relations are catalogued by Chu. 3 The boundary conditions are imposed implicitly to be consistent with the treatment o the interior points. III. Numerical Method The solution algorithm is based on a preconditioning technique developed to circumvent the computational diiculties arising in low Mach-number reacting lows., Temporal discretization is obtained using a second-order dual-time stepping integration, and a second-order lux dierencing technique or spatial discretization. The governing equations are treated in a ully coupled manner by means o an alternatingdirection-implicit (ADI) actorization scheme. The implicit treatment and dual-time stepping procedure guarantees a converged solution at every physical time step. This procedure also helps in capturing the transients in the lowield and its attendant chemical reactions. The numerical scheme described above is implemented to study the lame response to traveling acoustic waves in a porous chamber. The chamber has an eective aspect ratio o, being m in length ( L ) and.5 m in wall-to-core height ( H ). A propane-air mixture having an equivalence ratio o.7 is injected across the porous walls at a low rate o. kg m s, a temperature ( T w ) o 35 K, and a pressure ( P ) o bar. The low rate translates into an injection velocity ( v w ) o.6 ms and a surace Mach number ( M w ) o where the speed o sound ( a w ) is 368 m/s. Ater crossing the reactive lame zone ( y = δ ), the gas velocity ( v ) increases to a speed o.54 ms, at a Mach number ( M ) o.36 3 (see Fig. a). The computational grid is illustrated in Fig. b and consists o 6 by non-uniorm cells in the axial and transverse directions, respectively. In order to better resolve the rapid variations o low properties in the lame zone, cells are clustered more closely near the porous boundary. The smallest cell is nearly.5 mmthick just above the transpiring wall. IV. Steady-State Results Beore addressing the characteristics o the unsteady low, it is helpul to examine the simulation results or the mean lowield. Figure illustrates the distributions o mean low Mach number, vorticity, and gauge pressure under steady-state conditions. Because the coniguration is symmetric, only the lower hal o the chamber is displayed. A. Mean Velocity and Pressure Distributions As indicated in Fig., the coordinates y/ H = and correspond to the porous wall and core, respectively. Throughout, Mach-number contours are shown to exhibit the two-dimensional structure typical o an a) b) c) y/h y/h y/h x/l Fig. Contours o a) Mach number, b) gauge pressure, and c) vorticity under steady-state conditions. 4 American Institute o Aeronautics and Astronautics

6 injection-driven low. For example, the Mach number increases linearly along the core rom zero at the head end to.44 at the exit. The linear increase is consistent with cold-low predictions or a similar chamber simulating a slab rocket motor. 3 According to coldlow analysis, the mean velocity could be calculated rom uxy (, ) = vb F + ( x/ H) F (3) where v b is the blowing or ejection velocity o the gases into the chamber. The large injection mean low unction is given by 3 F = cos θ + ( π / 4) ε φcosφ sinφ ln tan φ/ {( ) ( ) [ A ]} O + cosφ φ I( φ) + ( ε ) (4) k + k k x k I( x) = x+ π ( ) j k= k + j= = x+ x + x + (5) 8 8 and the derivative is given by F = π sin θ + O( ε). The asymptotic truncation order in Eq. (4) is gauged to the reciprocal o the injection Reynolds number, 3 ε = Re = ν /( vh b ) = O( ). Here, the trigonometric π arguments θ = πy / H and φ = θ are used or brevity. Based on this laminar mean low solution, the Mach number can be determined rom M ( xy, ) = Mb cos θ + xπ sin θ + O( εm) 4 b (6) which, along the chamber midsection, becomes M ( xh, ) = π( x/ HM ) ( ) b + OεMb. Assuming that the bulk low motion outside the lame zone can be approximated by the low behavior o a non-reacting mixture (i.e., the propane-air gas products), one may set vb = v in the cold-low equations. Thus, based on Eq. (6), the local Mach number can be expected to vary linearly along the midsection plane rom zero to 3 π (/.5).36 =.43 at at end. This result is clearly consistent with the downstream value displayed in Fig. a (although not shown on the graph, the numerical simulation predicts a maximum value o.439). Had the injection speed enhancement due to chemical reactions been ignored, the Mach number prediction at the downstream end would have been signiicantly underestimated (in this case, to.8). We thus conclude that, despite the adequacy o cold-low models in approximating the gas motion outside the lame zone, the thermally adjusted speed v should be used to represent the equivalent blowing speed rather than the wall injection speed v w. Otherwise, the useulness o a cold-low model can be severely compromised. The increase in eective gas injection can be attributed to the presence o chemical reactions and accompanying volumetric dilatations that are not present in the equivalent cold-low model. In like ashion, the pressure contours in Fig. b display an almost one-dimensional distribution that is weakly sensitive to the downstream location. However, no discernible variation in the transverse direction is observed, except or the small change arising rom the volume dilatation in the near-wall lame zone. These observations may be attributed to the low injection Mach number and can be corroborated by the theoretical results obtained without chemical reactions. In act, based on Majdalani and Van Moorhem, 3 the mean-pressure varies according to pxy (, ) = P ρv ( / ) ( sin ) cos b π x H + πε θ + θ 4 (7) Along the midsection plane, it is clear that the transverse variation is maniested by the cos θ term, which is too small to be considered. On the other hand, the diminution in gauge pressure along the chamber length can be calculated to be π ρ v ( / ) 8 b x H. Unlike the cold-low simulation where all thermostatic properties are kept constant, the density in the actual simulation varies along the chamber axis. At x = L, a 3 density o.8 kg m can be used to predict a pressure drop o nearly Pa. This value concurs with the actual pressure decrease recorded in the actual simulation and shown in Fig. b (the maximum recorded pressure was 5 Pa). B. Mean Flow Vorticity To understand the sources o mean low vorticity, attention can be turned to the Crocco-Vazsonyi equation. The material derivative o vorticity is expanded into D Ω ( γ + ) = ( Ω ) u Ω ( u ) V p D t ( γ ) + ν Ω µ V Ω + 4 µ V u (8) ( ) ( ) where Ω = u and V = / ρ is the speciic volume. In the present two-dimensional simulation, the vortex stretching mechanism ( Ω ) u disappears. Vorticity is generated at the wall because o the no-slip condition that orces the parallel component o velocity to vanish along the porous surace. 8 Vorticity varies rapidly in the lame zone where both the volume dilatation Ω ( u ) and baroclinicity V P become important. Their inluence is evidenced in Fig. c by the sudden emergence o vorticity overshoot near the surace. The last two terms in Eq. (8) represent viscous damping and both appear to be secondary because o the large injection Reynolds number and the small mean low velocity gradients in the entire chamber. In the absence o viscosity, baroclinicity, and volumetric dilation outside the lame zone, the mean low vorticity 3 5 American Institute o Aeronautics and Astronautics

7 shown in Fig. c appears to be nearly conserved while driting downstream. Its magnitude increases almost linearly rom zero at the head end to 5s at the at end. To a good approximation, it can be evaluated rom 3 Ω= π ( x/ H ) v cos ( ) 54s 4 b θ + O ε vb = (9) This value concurs with the mean vorticity at the at end. Here too, the use o a non-reactive ormulation or the bulk vorticity outside the lame zone seems appropriate. C. Mean Temperature and Heat Generation To complete our coverage o lowield attributes, the mean temperature distributions are presented in Fig. 3 at various axial locations ranging rom x/ H = to 7. The proximity to the head end is purposeully targeted to ensure laminar conditions. Due to the uniormly distributed heat o reaction along the surace, the overlaid temperature curves appear to be independent o the axial location. Rather, a one-dimensional variation in the y direction is observed. The weak thermal sensitivity to the axial position can be attributed to the geometric aspect ratio and larger gradients in the transverse direction. Apparently, the downstream low convection exerts hardly any inluence on the lame structure. As shown in the graph, the temperature increases rom the wall injection value o 35 K to a maximum o 938 K just past the lame edge, at y = δ = mm. The thickness δ p is used to denote the premixed lame stando distance, which is approximately.9 mm. Due to the rapid inward convection, this maximum temperature spreads uniormly across the core. The climbing temperature segment shown in Fig. 3 is sustained by a non-zero heat input curve or Q!. It may be useul to note that the calculated lame temperature o 938 K agrees with the chemical equilibrium result o 95 K predicted by the NASA CET93 code. 4 The slight discrepancy may be due to Temperature T, K δ p δ 938 K y, mm Q, GW/m 3 Fig. 3 Mean temperature and heat release above the porous surace at several stations corresponding to: x/h =, 7,, 7. the more detailed reactions incorporated in the CET93 calculations. The injection velocity o.6 ms agrees well with the experimental value o.9 ms measured by Metghalchi and Keck. 4 This agreement urther supports the current methodology based on global chemical kinetics. The typical S-shaped temperature curve displayed in Fig. 3 can be ascribed to the expected thermal surge across the lame zone. The retention o an almostconstant core temperature is caused by the strong convective action that brings the hot products closer to the core. As demonstrated by Majdalani and Van Moorhem, 3 the transverse penetration o injectants can be signiicant, especially in planar conigurations, where the lack o curvature enables the injected mixture to draw nearer to the core beore merging into the axial stream. D. Simple Asymptotic Solution Using standard descriptors, the dierential orms o mass, momentum, and energy conservation can be expressed by ρ + ( ρu ) = () t u ρ + ( u ) u = p + µ t u () T p ρcp + ( u ) T + ( ) T = k T + Q t u! t () where ρ, p, u = ( uv, ), T, and t represent the mean low density, pressure, velocity, temperature, and time, respectively. While the heat source Q! is used to represent the net rate o exothermic chemical reactions, the remaining properties represent the speciic heat C p, viscosity µ, and thermal conductivity k. Equations () () can be simpliied based on the assumptions that: () Gradients in temperature normal to the porous walls are larger than axial gradients ( T / y # T / x). This assumption is consistent with the multidimensional simulation results. () Thermal diusion is small when compared to thermal convection and heat generation. Such a relative comparison can be accomplished using an order o magnitude analysis. (3) The velocity inside the lame zone has a negligible axial component because o the no-slip condition at the wall where the parallel x component o u vanishes. (4) The bulk velocity outside the lame zone is adequately represented by the isothermal mean low solution based on the thermally-enhanced blowing speed v. This mean low solution has been shown in 5 to provide a suiciently accurate approximation. 6 American Institute o Aeronautics and Astronautics

8 Under these underlying conditions, Eqs. () () reduce to ( ρv) = or ρ v = constant (3) y T ρvc p = Q! (4) y The decoupling o the momentum equation can be attributed to the one-dimensionality o the velocity distribution inside the lame zone. Equations (3) (4) exhibit the ollowing arrangement o boundary conditions: at y = (imposed), ρ = ρw ; v = vw ; T = Tw (5) at y = δ (deduced), ρ = ρ ; v = v ; T = T (6) While conservation o mass dictates that ρv = ρwvw inside the thin lame zone, conservation o energy cannot be used without providing inormation regarding the manner in which chemical energy is spatially released over the interval y δ. The analytical model must be based on the prescribed orm o heat released by chemical reactions. This orm needs to be determined beorehand, either experimentally, or by simulation. In the numerical model, the exothermic heat o reaction associated with propane-air combustion is considered. This leads to a typical skewed symmetric unction that exhibits a peak value inside the lame zone. Similar heat o reaction curves have been associated with other combusting hydrocarbons whose resolution has invariably demanded numerical integrations. 6-9 The heat release curve obtained by the numerical simulation can be accurately represented by a Gaussian distribution. Gaussian distributions and error unctions have been employed previously to describe both temperature and concentration proiles in several documented studies. Using single reactions and identical species diusion coeicients, Borghi and Destriau 3 have deined their sum o mass ractions via Shvab-Zeldovich unctions that can be represented by Gaussian distributions. Warnatz, Maas, and Dibble 3 have also shown that, or a species transport equation in which convective action is equally matched by diusion, the solution may be expressed in terms o error unctions. For near equi-diusion lames (NEF) that arise in the vicinity o stagnation points, Buckmaster and Ludord 3 have developed an expression or the temperature proile using error unctions as well. Their solution has been based on the coupled equations representing a balance between species diusion, thermal diusion, and convective transport o species and heat. In the current study, the heat source is represented by the Gaussian orm Q! y C + C exp[ C ( y C ) ] (7) ( ) 3 4 where the characteristic coeicients Ci, i =,,4 can be determined rom the heat o ormation o diverse species involved in the combustion o a given ueloxidizer mixture. Equation (7) can be substituted into Eq. (4) and integrated. Using Eq. (5) to speciy the single temperature boundary condition at the wall, one obtains C C3y+ C π T( y) = Tw + C3 ρvcp { er ( C C ) + er ( y C ) C } (8) In order to allow or comparisons between Eq. (8) and computations that take into account global chemical kinetics, the numerically generated heat release unction is used as an example. Using the method o leastsquares, the corresponding coeicients in Eq. (7) are ound to be 6 C = 6.34 ; 9 C =.6 ; 6 C 3 =. ; C 4 =.935. (9) In order to evaluate the temperature distribution, the density and injection velocities at the wall are taken as 3 ρ w =. kgm and v w =. ms. 3 The speciic heat used in Eq. (8) is approximated by the average speciic heat o the reactants. This is due to the lame zone being predominantly occupied by the reactants. The gas products, it has been shown, occupy the majority o the domain outside the lame region. The reactant mixture Cp ( T ) is determined using the sum o species speciic heats weighed by individual mass ractions. For propane-air reactants, we use Cp( T) = YC i p = YCHC 3 8 pch + Y i 3 8 OC po + Y NC pn () where Y i represents the mass raction o species i. The average speciic heat o the reactants is then calculated rom T Cp = Cp dt T Tw T () w where T w and T are 35 K and 938K, respectively. 3 Having determined the means to approximate the spatial evolution o the temperature, attention is turned to the thermally-enhanced gas velocity directly above the lame edge. Using mass conservation and the ideal gas law under isobaric conditions, one inds ρ T w v = vw = vw () ρ Tw Consequently, given the injection temperature and velocity at the wall, the key or determining the thermally-enhanced velocity lies in the accurate prediction o the lame temperature T. 7 American Institute o Aeronautics and Astronautics

9 Fig. 4a compares the analytical approximation or Qy! ( ), given by Eq. (7), and the numerical prediction. 3,5 Also shown in Fig. 4b are the temperature distributions obtained analytically, rom Eq. (8), and numerically, based on Chu. 3 While the analytical solution matches the simulated temperature at the wall, it closely approximates T at the lame edge (i.e., at y = δ ). Outside the lame zone, the numerical solution yields a constant temperature o 938K, while Eq. (8) predicts 935K. Coincidentally, the analytical value is closer to the lame temperature o 95K procured rom NASA CET93; 4 the latter utilizes more detailed chemical reactions than those employed by Chu. 3,5 Note that, since the analytically predicted T is close to the assumed value used in Eq. (), there is no need to re-evaluate the average speciic heat. In the absence o a priori knowledge o T, however, the ollowing algorithm may be used: () assume T ; () calculate C p rom Eq. (); () calculate T rom Eq. (8); and, (3) repeat steps until convergence. It should be noted that, outside the lame zone, the analytical solution predicts a slightly increasing temperature in the inward direction. This is contrary to the numerical simulation results which predict a constant temperature throughout the chamber core. Interestingly, the slight increase in temperature outside the lame zone is oten exhibited in laminar premixed lame proiles or methane-air mixtures and other combustibles described by Turns (c. p. 7). 6 At the outset, it appears that the simple analytical model presented here can aithully reproduce the undamental thermal trends conirmed in previous studies. 6-9 Further reinements are necessary to reduce its limitations and broaden its applicability. In like ashion, the validity o Eq. () can be veriied or the speciic problem o propane-air combustion. Whereas the velocity at the lame edge is calculated to be v =.5 ms, the approximate analysis based on Eq. () yields.4 ms. The % error in the analytical prediction may be deemed reasonable, considering its relative simplicity in comparison to the iner lame zone detail captured by computational measurements. V. Unsteady Flow Results Ater steady-state solutions are reached, periodic low oscillations are imposed at the head end in an eort to trigger acoustic waves at user-prescribed requencies. The boundary conditions along the burning surace are let unchanged with respect to those used during steady-state simulations. Following other experimental and numerical benchmark studies, 8- the impressed oscillation requency is varied rom to 8 Hz, while the amplitude o pressure luctuations ( A ) is set at two percent o the mean pressure ( 6 db). It should be noted that, according to acoustic theory, 33 the natural requencies o the simulated chamber can be estimated rom = ma /( L) ; a = γ RT = 846 m/s (3) m where γ =.65, R = 9 J kg K, and T = 938 K. Note that, due to the small thickness o the lame zone, the vast portion o the chamber cavity is illed with product gases such as CO, H O, and air. For this reason, physical properties to be used in Eq. (3) are derived rom the C 3 H 8 air products whose molecular weight averages 8.47 kg kmol. Hence, or a unit chamber length, the problem s natural requencies can be estimated rom = 43m, or m =,,, etc. m A. Unsteady Pressure and Velocity Fluctuations Figure 5 illustrates the instantaneous isobaric contours o the luctuating pressure and iso-velocity ields or = 4 Hz. This value is only 5.4% smaller than the irst undamental requency o the combustion chamber at m =. Here the luctuating variables are obtained by subtracting mean values, calculated over a complete cycle o oscillation, rom their total time- Q, GW/m 3. a) T, K b) y, mm δ non-reactive zone isothermal zone Fig. 4. Spatial distribution o a) the chemical rate o heat release Qy!( ), and b) the steady-state temperature T( y) based on computational luid dynamics ( ) and analytical solutions (- - -). Note that or y > δ, the reaction heat released is negligible. 8 American Institute o Aeronautics and Astronautics

10 dependent quantities. As a result o this siting operation, a well-deined traveling wave motion is realized. The wave motion bears a striking resemblance to the standing mode shape described by Majdalani. 34 Therein, the physical setting is identical except that a non-reacting gas is used. Nonetheless, the analytical predictions obtained under laminar conditions seem to agree quite avorably with the current results. 34 With respect to the unsteady pressure and velocity, Fig. 5 suggests the same overall mode shape by exhibiting a pressure node and a velocity anti-node midway along the chamber length at x =.5L. The near-wall combustion interactions do not seem to exert any marked inluence on the general character o the acoustic velocity, so long as the low remains dominated by longitudinal strain rates. It should be noted that the current contours are quite similar to those described analytically, based on the closed-orm expression = sin ( mπx/ L) sin ( mπa t/ L) u ( x, y, t) A a ρ ( ) ( ) Fsin mπxf / L exp( ζ)sin mπa t/ L+Φ (4) where A = Pa, and a = 846 ms, y/ H F 3 ( x )d x ζ ξ = m =, ω = π m, (5) { ln tan π ( ) sec tan } = ξ π + + θ θ. (6) The viscous parameter is 3 ξ = νω Hv (7) y, mm 9-9.L.4L x.6l.8l L 3 4 ( ) Fig. 5 Contour plots o instantaneous acoustic pressure and axial velocity luctuations at 4 Hz. ( θ ) y / H F x x π π Φ= Sr ( )d = Sr ln tan + ; (8) where the Strouhal number is, as usual, given by Sr = ωh v (9) As seen in Fig. 5, rotational eects are lited o the burning surace by approximately % due to the presence o lame interactions. In addition to the expected coupling eects between acoustic and vorticity waves near the surace, the velocity luctuations seem to exhibit an additional twodimensional correction resulting rom interactions between the vortico-acoustic wave and the non-uniorm temperature distribution across the lame zone. Away rom the surace, vortical and thermal eects quickly decay and are dominated by a simple inviscid wave structure. This is maniested in Fig. 5 by a onedimensional acoustic velocity distribution over the range.4 y/ H. The unsteady wave amplitude o approximately 3 ms can be interpreted in light o the established theory o aeroacoustics; 8 accordingly, an acoustic velocity amplitude o A/( a ρ) 3 ms be expected near the core. However, due to the thermal suppression o vorticity waves across the lame zone, a relatively small overshoot in unsteady velocity is observed near the wall. This overshoot is caused by the pairing o vorticity and acoustic waves and is much smaller than the typical % overshoot value reported in cold-low simulations. The attenuation o the vorticity wave is due, in part, to the strong volume dilatation across the lame zone. The ormation o acoustically induced shear layers are examined in Fig. 6 by plotting the transverse distributions o both amplitudes and phases o the luctuations u, v, and T at dierent requencies. The lame location is also included to acilitate the discussion. The most salient eatures are the shortening in spatial wavelength ( λ ) and the reduction in wave amplitude with requency. These observations are in agreement with ormer studies o non-reacting lows. 3,34-37 For example, based on theoretical grounds, the diminution in wavelength may be inerred rom Eq. / can (4) to be λ = vf / m. This relation clearly displays an inverse relation between λ and m. The reduction in total wave amplitude with requency is attributable to the reduced wavelength which, in turn, causes the coupling between acoustical and vortical waves to take place nearer to the lame zone. There the phase intererence with thermal waves leads to a aster attenuation o vortical strength. Outside the lame zone, the vortical presence is urther exacerbated by the increased shearing, reversal, and damping o luid particles over shorter length and time scales. 9 American Institute o Aeronautics and Astronautics

11 The observed behavior may once more be illuminated by the Crocco-Vazsonyi equation. Using similar arguments to those given in Sec. IV to explain the mean low vortical growth, the interplay between volume dilatation and baroclinicity in the lame zone can be used to explain the attenuation o unsteady vortical growth. In addition, the wavelength o the vortical motion is increased in the lame zone due to the increasing wave propagation speed (partly triggered by volumetric dilatation). At the outset, the combination o volume dilatation, baroclinicity, and increased wave propagation speed in the lame zone tend to reduce the role o vorticity near the wall, especially as compared to the case o a non-reacting low. This explains why the largest unsteady amplitudes occur outside the lame zone and later decay as the core is approached. In particular, note that, despite the presence o chemical reactions, the amplitude o v is smaller than that o u by the order o the Mach number, M O result is signiicant as it lends support to the theoretical predictions obtained rom cold-low studies o the core low. 34 u, m/s δ 5 5 a). v, m/s c) T, K e) δ p 3 = ( ). This.. 5 y, cm B. Velocity Decomposition In order to better understand the coupling between the lame response and velocity waves, it is helpul to identiy the separate contributions o pressure-driven, boundary-driven, and entropy-driven disturbances. Chu and Kovásznay s 38 mode decomposition approach can be implemented to subdivide the total velocity luctuation into three parts representing acoustical, vortical, and thermal oscillation modes. This approach has been previously employed by Flandro 8 and Majdalani, 34 who have separated the velocity ield into two parts: an irrotational, acoustical u a, and a rotational, vortical, u ν. Since the current study involves chemical reactions, the contribution rom heat luctuations must be added in the orm o a thermal, entropy-driven correction u s. The total luctuation can hence be split into u = u a + u ν + u s (3) On the one hand, the acoustic velocity u a is directly correlated with the pressure luctuation through the acoustic impedance, ρ a. This product varies rapidly in arg(u ), deg b) arg(v ), deg d) -8 9 arg(t ), K ) y, cm Fig. 6 Amplitudes and phases o unsteady velocity and temperature luctuations at x/ H = 3 requencies corresponding to: = Hz, Hz, 4 Hz, 8 Hz. and our American Institute o Aeronautics and Astronautics

12 the lame zone and is proportional to the inverse o T. On the other hand, the rotational part u ν incorporates the eects o unsteady vorticity generation and transport. The newly introduced velocity luctuation u s results rom the unsteady heat release originating in the lame zone. Based on acoustic theory, 33 this quantity can be approximated by k T s u s (3) ρcpt x where T s represents the non-isentropic temperature luctuation. Since the temperature gradient only exists in the transverse direction, the eect o entropy waves on the axial velocity luctuation can be ignored. Its inluence on the transverse velocity luctuation must be retained, however, so long as it is capable o exciting large transverse oscillations. 3 C. Unsteady Temperature Fluctuations Figures 6e illustrate the eect o requency on the amplitude and phase o temperature luctuations. The corresponding steady-state distributions o temperature and heat release rate have already been examined in Fig. 3. Pursuant to the velocity decomposition, the temperature luctuation can be obtained by superimposing the acoustic and entropy-driven contributions. The acoustic component T a obeys the isentropic relation governing the acoustic pressure. In the current study, it has an amplitude o 8 K in the core region. The non-isentropic component T s is approximately proportional to the product o entropy luctuation and mean temperature. 33 In view o Eq. (3), it can be expressed by T s q! τ c / Cp (3) where q! is the luctuation in the rate o chemical heat release, and τ c represents its associated characteristic time. As shown in Fig. 6c, the luctuating temperature response exhibits a peak directly below the line o maximum mean temperature. This peak reaches about 6 K at a requency o Hz and then drops to nearly 35 K at 8 Hz. The thermal oscillations also persist longer and thereore, penetrate deeper into the chamber core at lower oscillation requencies. This result can be attributed to the intimate coupling near the wall with the longitudinal vortico-acoustic waves whose role is to sustain these thermal oscillations. Since the vorticoacoustic depth o penetration diminishes at higher requencies, so will the depth o thermal luctuations. This eect is observed in both modulus and phase variations in Fig. 6c. So long as the vortico-acoustic waves cause the lame to lutter in the vicinity o the wall, thermal luctuations can be sustained. These luctuations are empowered by alternating high-temperature sweeps into the preheat zone that can periodically expedite the ongoing chemical combustion and attendant heat o reaction. The character o thermal luctuations near the wall can be adequately represented by the Arrhenius relation connecting reaction rates and temperatures. When suiciently removed rom the lame zone (where heat is constantly supplied), the unsteady temperature becomes dominated by the high-capacity, pressure-driven masslike response T a that always prevails in the core region. 38 Between the lame edge and the core, the entropy luctuation T s is smoothed out due to heat conduction and the thermal lywheel eect that accompanies the traveling acoustic wave. Since the lame oscillation is induced by the vorticoacoustic motion, the characteristic time τ c described in Eq. (3) may be estimated by the period o acoustic oscillations, τ c / m = L/( ma ). Hence, a lowrequency acoustic motion will lead to a largeramplitude temperature luctuation while allowing a deeper penetration o the vortical wave into the chamber core. These theoretical predictions are clearly relected in the current numerical results. D. Response to Unsteady Heat Addition The unsteady heat release rate is shown in both amplitude and phase in Fig. 7 at one axial position and our dierent requencies. For the reasons given to explain the behavior o the temperature luctuations, the amplitude and depth o penetration o the heat waves decay more rapidly at higher requencies. However, two uneven peaks in Q! are observed below and above the lame location. The value o the peak outside the lame zone is approximately 6% o the inner one, which is consistent with the experimental observation o Sankar et al. From a theoretical perspective, the two energy peaks serve as a combination o an acoustic monopole and a dipole source or driving the observed oscillations. 33 In order to explain the observed damping outside the lame zone, attention can be turned to the undamental role o heat addition in driving or suppressing the unsteady luctuations. In principle, the ampliication o an acoustic wave p due to periodic heat input supplied at a rate q! can be characterized using Rayleigh s parameter, 39 < pq! >= p ( x, t) q! ( x, t) dt (33) τ c American Institute o Aeronautics and Astronautics

13 An impressed acoustic excitation will grow when the above parameter is positive. Thus, i heat is added at the moment when the combustion mixture is locally expanding (i.e., during a avorable pressure phase), the dilatational thermal energy is positively gained insoar as it leads to a more pronounced volumetric expansion. When this occurs, the thermal wave can be considered to be in avorable phase with the acoustic motion that it tends to sustain and ampliy. Conversely, i heat is added at the moment o local gas compression (i.e., during an adverse pressure phase), the eect o thermal addition is that o opposing the volumetric contraction. Wave cancellation takes place as some o the acoustic energy is eliminated. In the problem at hand, two regions o positive and negative Rayleigh numbers can be recognized. As shown in Fig. 8, the two regions o avorable and adverse thermo-acoustic coupling are delineated by the lame stando distance. These regions are identiied by plotting the Rayleigh parameter < pq! > at x/ H = 3 and the same our requencies. In both regions, the magnitude o < pq! > diminishes at higher m, indicating a weaker thermo-acoustic coupling at higher requencies. The weaker coupling at higher requencies may explain the diminution in temperature oscillations at higher m in Fig. 6c. In particular, the negative Rayleigh number or y > δ p is indicative o a damped out region in which the role o unsteady heat addition switches to that o vortico-acoustical wave attenuation. This eature may explain the damping character o unsteady energy release in the outer lame region (y >.9 mm). VI. Conclusions This study has ocused on the interactions between acoustic waves and premixed lame dynamics in porous chambers. In the process, the complete conservation equations were numerically simulated while accommodating variations in thermo-physical properties. Finite-rate chemical kinetics are also enabled to spatially resolve the lame structure. Both mean and unsteady lame attributes are examined in various oscillatory low environments. The unsteady motions are ound to comprise contributions rom acoustical, vortical, and thermal modes that strongly depend on the requency o the impressed acoustic wave. Out o the three disturbance types, thermal oscillations are restricted to a thin region near the wall. Outside this lame zone, the spatial distribution o pressure and velocity closely resemble the theoretical predictions obtained rom cold-low solutions based on the lame-enhanced gas injection speed. In particular, the low is ound to be mainly isothermal and the Q, 8 W/m 3. a) arg(q ), deg. b) δ p y, mm Fig. 7 Amplitudes and phases o unsteady chemical heat release rate at x/ H = 3 and the same our requencies shown in Fig. 6.. Normalized <p Q >..5. δ p -.5 y, mm luctuating velocity in the transverse direction appears to be smaller than the axial component by the order o the Mach number M. Furthermore, the amplitude o all luctuating components decreases at higher requencies. This behavior applies to the acoustic boundary layer thickness as well. To explain the pairing between thermo-acoustic and vortico-acoustic waves, an energy cascade among these three separate luctuation modes is identiied. Within the lame stando distance, the thermo-acoustic response is ound to be avorable to vortico-acoustic growth. Outside this region, however, a negative Rayleigh number is conirmed, signaling an adverse thermo-acoustic coupling. As a result o the thermal oscillations induced by the vortico-acoustic wave, the rate o unsteady heat δ weakly reactive zone Fig. 8 Normalized Rayleigh parameter at / 3 x H = and the our requencies o Fig. 6. δ American Institute o Aeronautics and Astronautics

14 release is ound to exhibit two uneven peaks. These can be attributed to the Arrhenius type o temperature dependence on reaction rates. The two peaks can also be viewed as a combination o an acoustic monopole and a dipole source or driving low oscillations. Finally, the ampliication and suppression roles o thermo-acoustic coupling due to unsteady heat addition are realized by calculating local Rayleigh numbers. Reerences De Luca, L. T., Theory o Nonsteady Burning and Combustion Stability o Solid Propellants by Flame Models, Nonsteady Burning and Combustion Stability o Solid Propellants, Vol. 43, edited by L. T. De Luca, E. W. Price, and M. Summerield, AIAA Progress in Astronautics and Aeronautics, Washington, DC, 99, pp Sankar, S. V., Jagoda, J. I., and Zinn, B. T., Oscillatory Velocity Response o Premixed Flat Flames Stabilized in Axial Acoustic Fields, Combustion & Flame, Vol. 8, No. 3-4, 99, pp Culick, F. E. C., and Yang, V., Overview o Combustion Instabilities in Liquid Propellant Rocket Engines, Liquid Rocket Engine Combustion Instability, Vol. 69, edited by V. Yang and W. E. Anderson, AIAA Progress in Astronautics and Aeronautics, 995, pp Yang, V., Wicker, J., and Yoon, M. W., Acoustic Waves in Combustion Chambers, Liquid Rocket Engine Combustion Instability, Vol. 69, edited by V. Yang and W. E. Anderson, AIAA Progress in Astronautics and Aeronautics, 995, pp Lieuwen, T. C., Neumeier, Y., and Zinn, B. T., Determination o Unsteady Heat Release Distribution in Unstable Combustor rom Acoustic Pressure Measurements, Journal o Propulsion & Power, Vol. 5, No. 4, 999, pp Tang, Y. M., Waldherr, G., Jagoda, J. I., and Zinn, B. T., Heat Release Timing in a Nonpremixed Helmholtz Pulse Combustor, Combustion & Flame, Vol., No. -, 995, pp Entezam, B., Van Moorhem, W. K., and Majdalani, J., A Full-Scale Numerical Model o the Thermoacoustic Interactions inside the Rijke Tube Pulse Combustor, Journal o Numerical Heat Transer: A-Applications, Vol. 4, No. 3,, pp Apte, S., and Yang, V., Unsteady Flow Evolution in a Porous Chamber with Surace Mass Injection. Part : Free Oscillation, AIAA Journal, Vol. 39, No. 8,, pp Apte, S., and Yang, V., Unsteady Flow Evolution in a Porous Chamber with Surace Mass Injection. Part : Acoustic Excitation, AIAA Journal, Vol. 4, No.,, pp Apte, S., and Yang, V., Turbulent Flame Dynamics o Homogeneous Solid Propellant in a Rocket Motor, International Symposium on Combustion Paper 39,. Liou, T. M., Lien, W. Y., and Hwang, P. W., Transition Characteristics o Flowield in a Simulated Solid-Rocket Motor, Journal o Propulsion & Power, Vol. 4, No. 3, 998, pp Liou, T.-M., and Lien, W.-Y., Numerical Simulations o Injection-Driven Flows in a Two- Dimensional Nozzleless Solid-Rocket Motor, Journal o Propulsion & Power, Vol., No. 4, 995, pp Chu, W.-W., Dynamic Responses o Combustion to Acoustic Waves in Porous Chambers with Transpiration, Ph.D. Dissertation, The Pennsylvania State University, McBride, B. J., and Gordon, S., CET93 and CETPC: An Interim Updated Version o the NASA Lewis Computer Program or Calculating Complex Chemical Equilibria with Applications, Technical Rept. TM-4557, NASA, Reid, R. C., Prausnitz, J. M., and Poling, B. E., The Properties o Gases and Liquids, 4th ed., McGraw Hill, New York, 987, p Wilke, C. R., Diusional Properties o Multicomponent Gases, Chemical Engineering Progress, Vol. 46, 95, pp Westbrook, C. K., and Dryer, F. L., Simpliied Reaction Mechanisms or the Oxidization o Hydrocarbon Fuels in Flames, Combustion Science & Technology, Vol. 7, 98, pp Flandro, G. A., Eects o Vorticity on Rocket Combustion Stability, Journal o Propulsion and Power, Vol., No. 4, 995, pp Watson, W. R., and Myers, M. K., Inlow-Outlow Boundary Conditions or Two-Dimensional Acoustic Waves in Channels with Flow, AIAA Journal, Vol. 9, No. 9, 99, pp Tseng, I. S., and Yang, V., Combustion o a Double-Base Homogeneous Propellant in a Rocket Motor, Combustion & Flame, Vol. 96, No. 4, 994, pp Hsieh, S. Y., and Yang, V., A Preconditioning Flux-Dierencing Scheme or Chemically Reacting Flows at All Mach Numbers, International Journal o Computational Fluid Dynamics, Vol. 8, 997, pp Beam, R. M., and Warming, R. G., An Implicit Factored Scheme or the Compressible Navier-Stokes Equations, AIAA Journal, Vol. 6, No. 4, 978, pp Majdalani, J., and Van Moorhem, W. K., Laminar Cold-Flow Model or the Internal Gas Dynamics o a 3 American Institute o Aeronautics and Astronautics