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Combustion-induced pressure effects in supersonic diffusion flames
Twenty-Seventh Symposium (International) on Combustion/The Combustion Institute, 1998/pp. 2165–2171
COMBUSTION-INDUCED PRESSURE EFFECTS IN SUPERSONIC DIFFUSION FLAMES K. H. LUO1 and K. N. C. BRAY2 1Department of Engineering Queen Mary and Westfield College London E1 4NS, UK 2Department of Engineering University of Cambridge Cambridge CB2 1PZ, UK
A turbulent diffusion flame at a convective Mach number 1.2 was investigated using direct numerical simulation (DNS). The DNS employed the full time-dependent compressible Navier–Stokes equations coupled with a one-step chemical reaction governed by the Arrhenius kinetics. Detailed study of combustion-induced pressure effects on turbulence generation, conserved scalar, and stagnation enthalpy transport was conducted. Local countergradient diffusion (CGD) of a conserved scalar flux was observed for the first time in a diffusion flame where heat release was strong enough while gradient diffusion prevailed when heat release was zero or weak. The CGD occurred in spite of the absence of an externally imposed mean pressure gradient and was attributed to combustion-induced pressure fluctuations. The balance of the turbulent kinetic energy budget was strongly influenced by the pressure dilatation and the (combustioninduced) mean pressure work when heat release was strong. Both terms can be a source or a sink of turbulence, depending on the intricate interactions between turbulence and combustion. However, the temporal change in pressure ]p/]t had an insignificant influence on the stagnation enthalpy transport. A linear relation between the stagnation enthalpy and the mixture fraction was confirmed, which could lead to considerable simplification in modeling high-speed turbulent combustion.
Introduction In supersonic turbulent diffusion flames, pressure effects are expected to be significant due to the coupling of compressibility and chemical heat release. Combustion models that are traditionally developed for low-speed combustion must be subjected to careful scrutiny before being extended to high-speed combustion. For example, the pressure-related phenomenon of countergradient diffusion, which occurs in low-speed turbulent combustion [1,2], may invalidate models that are based on the gradient-diffusion assumption. Another complication is related to the combustion-generated turbulence through pressure [3,4], which in turn would affect the combustion process itself. These complicated topics have been carefully examined in two review papers by Bray et al. [5] and Bray [6] recently, which call for further studies in the areas. The principal obstacle to understanding and modeling pressure effects lies in the lack of techniques that can directly measure the relevant quantities. Another problem arises from the close but poorly understood relationship between pressure and acoustics. The employment of direct numerical simulation (DNS) marks a new era in the fundamental study of
both premixed and nonpremixed turbulent combustion. Since DNS resolves the whole spectra of time and length scales of turbulence and combustion, many previously intractable terms (e.g., pressure-velocity correlations) can be computed accurately with minimum interference of modeling. This makes it very effective in studying the combined effects of compressibility and chemical heat release. Early work in DNS of compressible reacting layers (e.g., Refs. [7,8]) usually did not go far enough to see turbulent effects, and the heat release was rather weak. More recently, pressure effects were studied in turbulent premixed combustion [9,10] with the turbulence being represented by a prescribed energy spectrum. However, the low-Mach-number approximation invoked in the DNS excluded acoustics and strong coupling between compressibility and chemical heat release. Stoukov [11] reported DNS results of a hydrogen-air flame with detailed chemistry, but the flow field was a simplified 2-D supersonic mixing layer, which like earlier work [7] could not properly address turbulence or high-Mach-number effects. The present study extends the DNS method developed in Refs. [12,13] to include finite-rate chemical reactions governed by the Arrhenius kinetic
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mechanism. The full compressible formulation enables detailed examination of pressure effects arising from the coupling of compressibility and heat release. Several fundamental issues in turbulent diffusion flames are then discussed. Direct Numerical Simulation The DNS was performed with the DSTAR code [13], in which the full 3-D time-dependent compressible governing equations were directly solved for a turbulent reacting mixing layer. A one-step reaction mFMF ` mOMO → mPMP was simulated. The reaction rate is given by the Arrhenius law, which after normalization takes the following form:
1W 2 1W 2
xT 4 Da
qYF
mF
qYO
F
3
2 exp 1Ze
mO
O
1T 1 T 24 1
1
(1)
f
It follows that the reaction rates for individual species are xF 4 1mFWFxT, xO 4 1mOWOxT, and xP 4 mPWPxT, respectively. The heat release rate is xh 4 QhxT, where Qh 4 mFWFDhof,F ` o mOWODhf,O 1 mPWPDhof,P is a heat release parameter. To start with, a nonreacting mixing layer (case CD) was simulated, in which an initially laminar flow of a Reynolds number 200 subject to linear disturbances went through transition and became turbulent at later times [12,13]. The fuel stream (the upper stream) and the oxidant stream (the lower stream) move at the same Mach number 1.2 but in opposite directions, leading to a convective Mach number Mc 4 1.2. Three reacting cases were then simulated with increasing heat release Qh 4 1 (Case HL), Qh 4 3 (Case HM), and Qh 4 6 (Case HH). Combustion was initiated at time t 4 70, when the nonreacting flow had become turbulent. Other simulation conditions were Da 4 3, Ze 4 3, and Tf 4 3. The computational box was 20 2 30 2 11.5, and the grid was 96 2 301 2 96. The grid lines were concentrated in the intense reaction zones to ensure adequate resolution throughout the simulations. More grid points could be added in each direction whenever necessary judged by the energy spectra and the value of the conserved scalar that must fall between 0 and 1. In all cases, length was normalized by the initial vorticity thickness and other quantities by their values in the upper free stream. The initial mean pressure was uniform, whose normalized value was 1/[c(Mc)2]. The Lewis number and Prandtl number were both unity. The molecular viscosity was temperature dependent (;T0.76). In all simulations, Fourier spectral methods were used in the homogeneous streamwise (x1) and spanwise (x3) directions, while a modified sixthorder compact finite-difference scheme was adopted
in the inhomogeneous lateral (x2) direction, in which characteristic nonreflecting boundary conditions were imposed. In the spanwise direction, a symmetry condition was also imposed to reduce computational cost. A third-order compact-storage RungeKutta method was employed for time advance [12,13]. Diffusion Flames in Compressible Turbulence The one-step chemical reaction with different heat release rates previously described is examined here. The combustion process was simulated from time t 4 70 to 80, during which the maximum reaction rate decreased rapidly with time. This was due to the fact that at ignition, there were large amounts of mixed fluids ready to burn. At a later stage, mixture preparation cannot keep up with the faster chemical reaction, leading to smaller and smaller reaction rate. To give a typical picture of the combustion process, the instantaneous product mass fraction in a plane cut at t 4 71 for case HH is plotted in Fig. 1a. It is shown that the flame is highly threedimensional with intense reactions concentrated in some narrow regions. These regions correspond to near-stoichiometric conditions. On average, the product is concentrated around the central layer (Fig. 1b), where mixing is more complete. Figure 2 shows the mean pressure profile at different stages of the flame. It should be recalled that in the nonreacting case, the mean pressure over the entire mixing layer is constant. The mean pressure gradient shown is purely induced by combustion. The peak mean pressure is 50% higher than that in the nonreacting flow {1/[c(Mc)2]}. Such large pressure variations, accompanied by equally significant changes in temperature and density, have a profound influence on turbulence, as shown in the next sections. Combustion-Generated Turbulence The turbulence kinetic energy (TKE) equation for full compressible flow is written as ] ˜ ] ˜ ˜ l) 4 q¯ k ` (q¯ ku ]t ]xl 1qu9u9 i l 1 u9i
]u˜i ] 1 1 qu9u9u9 i i l ]xl ]xl 2
1
2
]p ] ]u9i ` (u9r ) 1 r ]xi ]xl i li ]xl li
(2)
where on the left-hand side (L.H.S.) are the temporal (tkk) and convection (Ckk) terms, and on the right-hand side (R.H.S.) are the production (Pkk), triple correlation (Tkk), velocity-pressure gradient
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Fig. 1. Product formation in case HH. (a) Side view of the instantaneous product mass fraction field in the plane cut x3 4 L3/8 at t 4 71 (15 contours). (b) Mean mass fractions of species at t 4 80. Solid line, Y¯F; dotted line, Y¯O; dashed line, Y¯P.
Fig. 2. Combustion-induced mean pressure in case HH. Solid line, t 4 71; dotted line, t 4 75; dashed line, t 4 80.
correlation (Pkk), viscous diffusion (Dkk), and viscous dissipation (ekk) terms, respectively. Among these terms, Tkk and Dkk are purely redistributive, because they become zero after integration over x2. Only Pkk, Pkk, and ekk may generate or destroy turbulence. To see the effects of combustion on turbulence, the integrated TKE is plotted in Fig. 3 for cases CD (nonreacting) and HH (high heat release). In case CD, turbulence energy increases almost linearly with time, suggesting that some degree of self-similarity in the TKE balance has been reached. In case HH, turbulence energy increases significantly just after combustion starts at t 4 70 but then decreases dramatically at later times. It is interesting to note the differences in magnitude and ˜ caused by density effects. phase between qk ¯ ˜ and k,
Fig. 3. Temporal evolution of total turbulent kinetic en`` ˜ ergy. Solid line, *`` 1` ¯pk dx2, case CD; dotted line, *1` ˜ k˜ dx2, case CD; dashed line, *`` qk ¯ dx , case HH; dash 1` 2 ˜ dotted line, *`` 1` k dx2, case HH.
However, combustion effects on qk ¯ ˜ and k˜ are qualitatively the same. To understand these effects, the integrated production and dissipation are shown in Fig. 4a for the two cases. The production is seen to decrease with heat release almost monotonically. The viscous dissipation decreases during the main heat release period but picks up slightly as combustion becomes weak. Neither trend can explain the effects of combustion on turbulence as shown in Fig. 3. The pressure term Pkk is now examined, which can be split into the following terms Pkk 4 1u9i
]p¯ ]u9 ] ` p8 i 1 p8u9i ]xi ]xi ]xi
(3)
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Fig. 4. Temporal evolution of integrated terms in the TKE budget. (a) Solid line, *`` 1` Pkk dx2, case CD; dotted line, `` `` `` *`` 1` ekk dx2, case CD; dashed line, *1` Pkk dx2, case HH; dash dotted line, *1` ekk dx2, case HH. (b) Solid line, *1` 1 `` u9i ]p¯/]xi dx2, case CD; dotted line, *`` ¯ /]xidx2, case HH; dash dotted line, 1` p8]u9/]x i i dx2, case CD; dashed line, *1` 1u9i ]p `` *1` p8]u9/]x i idx2, case HH.
The terms on the R.H.S. of equation 3 are the mean pressure work, the pressure-dilatation, and the pressure-velocity correlation, respectively. The last term is a diffusive term that does not contribute to turbulence generation. The mean pressure work is zero in a constant density flow because in that case u9i 4 u8i 4 0. However, it was identified to be a source of combustion-generated turbulence in a turbulent premixed flame [1] and a turbulent diffusion flame with an externally imposed mean pressure gradient [14]. The pressure dilatation is still poorly understood and was typically ignored in previous analyses and calculations (e.g., Ref. [1]). Recently, Zheng and Bray [15] included a pressure-dilatation model, developed by Sarkar [16] for nonreacting flow, in the calculation of a nonpremixed supersonic hydrogenair flame but observed negligible effects. Results of the mean pressure work and pressure dilatation from the present DNS are shown in Fig. 4b. As expected, the mean pressure work is identically zero (which indirectly gives an indication of the high numerical accuracy achieved in the calculation), and the pressure-dilatation is negligibly small in the nonreacting flow (case CD). However, in the reacting flow (case HH), there is a positive mean pressure work accompanying the initial high heat release rate. This is due to the strong coupling of combustioninduced buoyancy and mean pressure gradient across the turbulent flamelets [1]. At later stages when combustion becomes weak, the induced buoyancy and the mean pressure gradient are no longer aligned positively. This causes the mean pressure work to be a sink of turbulence. The pressure dilatation also fluctuates between being a source and being a sink of turbulence, albeit at a much greater magnitude. The rise and fall in the pressure-dilatation correlate very well with the behavior of the TKE in Fig. 3, suggesting that the turbulence generation
is mainly due to the pressure-dilatation, enhanced further by a positive mean pressure work. The eventual decline of turbulence in the reacting flow relative to the nonreacting flow is attributed primarily to the decrease in Reynolds stress production by mean shear, exacerbated by a negative mean pressure work. The viscous dissipation, which is so often blamed, is not seen to be the main direct cause for the turbulence energy reduction. However, increasing viscosity due to heat release does have an adverse effect on turbulence production by mean shear. Countergradient Diffusion Countergradient diffusion (CGD) was observed in premixed combustion [1,2,9] but not in nonpremixed combustion [17]. It is generally accepted that CGD occurs when the mean pressure gradient acts differentially on the low-density burned gas and high-density unburned gas, respectively [1]. Bray [6] deemed it more appropriate to call the phenomenon “pressure gradient diffusion.” In most cases, CGD was found because there was an externally imposed mean pressure gradient acting in the opposite direction of the scalar concentration gradient [9,14]. Purely combustion-induced CGD has not been investigated sufficiently. In order to investigate the possible existence of CGD in the diffusion flame, we consider a conserved scalar defined by n 4 (sYF 1 YO ` 1)/(1 ` s), where s 4 mOWO/mFWF. It takes value 0 in the oxidant free stream and 1 in the fuel free stream, and is called a mixture fraction. In the present flame, n has positive gradient in the x2 direction, so that CGD means a positive scalar flux qn9u92. This quantity at t 4 75 is plotted for both the nonreacting and reacting cases in Fig. 5. In the nonreacting case, gradient diffusion rules. In the reacting cases, there is
COMBUSTION-INDUCED PRESSURE EFFECTS
2169
and the large negative peak of Uni in the center correlate with the countergradient diffusion region shown in Fig. 5. This is hardly surprising because Wni represents diffusion by pressure, whereas Uni represents transport by scalar gradient. It is tempting to say that Wni promotes CGD while Uni suppresses it. However, because both terms can change sign, it is difficult to make a definite conclusion. It is interesting to observe in Fig. 2 that the mean pressure gradient (and hence 1n9]p¯/]xi) in the central layer region is relatively small at this time (t 4 75). The pressure effects on scalar transport come primarily from the fluctuating pressure through 1n9]p8/]xi. Fig. 5. Conserved scalar flux qn9u92 at time t 4 75. Solid line, case HH (Qh 4 6); dotted line, case HM (Qh 4 3); dashed line, case HL (Qh 4 1); dash dotted line, case CD (Qh 4 0).
a tendency toward CGD in the center of the shear layer as the heat release increases. In case HH, CGD appears in the central layer, while gradient diffusion still dominates in other regions. To understand what is happening in the central layer, plane cuts through the conserved scalar flux field (shown in Fig. 6) and the heat release rate field (not shown) are investigated. It becomes clear that in regions of high heat release, positive qn9u92 dominates. In regions of low heat release, qn9u2 9 remains negative. The extent and duration of such local CGD are highly dependent on the local heat release rate. In fact, a case with Qh 4 12 conducted after submission of this paper shows that CGD becomes more prominent and more frequent. To further investigate the mechanisms behind CGD, the scalar flux transport equation is examined here. It is written for compressible flow in a symbolic form as ] ] qn9u9i ` ( qn9u9u i ˜ l) ]t ]xl 4 Pni ` Tni ` Pni ` Dni ` eni (4) where on the L.H.S. are the temporal and convection terms, and on the R.H.S., the production (Pni), triple correlation (Tni), scalar-pressure gradient correlation (Pni), viscous diffusion (Dni), and viscous dissipation (eni) terms, respectively. The pressure term Pni can be decomposed into two terms as Pni [ 1 n9]p/]xi [ p]n9/]xi 1 ](pn9)/]xi [ Uni ` Wni. The various terms on the R.H.S. of equation 4 for the x2 component are shown in Fig. 7 for case HH. The pressure terms Uni and Wni are dominate in scalar flux transport, although both terms have large lateral variation. Such oscillatory behavior of pressure-related terms in compressible turbulence is well documented [9,10,12,16] and is not attributed to numerical errors. The large positive peak of Wni
Pressure Effects on Stagnation Enthalpy Transport In modeling high-speed reacting flow, in which pressure effects are prominent, a question arises as to whether or not models developed for low-speed combustion can be borrowed. Bray et al. [5] suggested that by replacing the static enthalpy with the stagnation enthalpy in the energy equation, the conserved scalar presumed PDF method of low-speed combustion can be extended to high-speed combustion. Such a proposition depends crucially on a linear, algebraic relationship between the stagnation enthalpy and the mixture fraction: hs(x, t) 4 hs,O ` n(x, t)(hs,F 1 hs,O)
(5)
o where hs 4 (N a41 haYa ` uiui/2 and ha 4 hf,a ` *TTo Cp,a(T)dT. For a reacting system with unity Lewis number and Prandtl number, such a relation should hold if the pressure term ]p/]t is negligible. It was shown in Ref. [19] that equation 5 is indeed valid in a nonreacting compressible flow up to an effective Mach number 2.4. For the present reacting flow, the joint PDFs of hs and n (not shown) for all cases also exhibit a preferred direction corresponding to a linear relation between hs and n. However, the breadth of these PDFs suggests that the coupling between hs and n is to some degree weakened by the term ]p/]t. Nevertheless, the mean profiles of hs and n shown in Fig. 8 demonstrate beyond any doubt that the above linearity assumption is valid for combustion modeling purposes within the Mach number and heat release rate range tested.
Discussions and Conclusions Direct numerical simulations have been performed of reacting and nonreacting mixing layers at Mc 4 1.2 using the full compressible Navier–Stokes equations. A one-step chemical reaction governed by the Arrhenius kinetics has been simulated with various heat release rates. Combustion was found to
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HIGH SPEED COMBUSTION
Fig. 6. Plan view of the instantaneous conserved scalar flux qn9u92 in the plane cut x2 4 0 at t 4 75 for case HH. (Fifteen contours between 10.024 and `0.050: solid line, countergradient diffusion, dotted line, gradient diffusion.)
Fig. 7. Conserved scalar flux (qn9u92 ) budget at time t 4 75 for case HH. Solid line, pressure-scalar correlation (Wni); dashed line, pressure-scalar gradient (Uni); thin solid line, production (Pni); dotted line, triple correlation (Tni). Viscous diffusion (Dni) and dissipation (eni) are negligible.
generate large pressure, temperature, and density fluctuations. The analysis at the second-moment closure level was focused on combustion-induced pressure effects on turbulence generation and the transport of a conserved scalar and the stagnation enthalpy. A most significant finding was the existence of local countergradient diffusion where the heat release rate was sufficiently large. Such a phenomenon was previously observed only in premixed flames. More remarkably, the present diffusion flame had no externally imposed mean pressure gradient, so that the combustion-induced pressure fluctuations were solely responsible for the occurrence of countergradient diffusion. This fact suggests that the class of reacting flow with the presence of countergradient diffusion should be much larger than previously thought. Combustion-generated turbulence was also observed and attributed to the pressure dilatation and mean pressure work terms in the TKE budget. However, both terms can be a sink or a source of turbulence, depending on the intricate interactions between turbulence and combustion. For the Mach number and heat release rate range tested, the stagnation enthalpy and the mixture fraction showed a linear relation, due to the almost negligible influence of the temporal change in pressure ]p/]t. This implies that certain models developed for low-speed combustion in terms of the static enthalpy can be conveniently adopted for high-speed combustion using the stagnation enthalpy instead. How to incorporate all the above findings into turbulent combustion models remains a challenge.
Nomenclature Fig. 8. Linear relation between the mean stagnation enthalpy and the mean mixture fraction: solid line, case CD (Qh 4 0); dotted line, case HL (Qh 4 1); dashed line, case HM (Qh 4 3); dash dotted line, case HH (Qh 4 6).
Da, Ze, Tf h, hs, Dhof,a
Damko¨hler number, Zeldovich number, and flame temperature enthalpy, stagnation enthalpy, and standard enthalpy of formation
COMBUSTION-INDUCED PRESSURE EFFECTS
k˜ Ma Mc p Qh Re t T u Wa xi, x Ya
;
9 [u9u i i/2, turbulent kinetic energy chemical symbol for species a convective Mach number pressure a heat release parameter Reynolds number time temperature velocity molecular weight for species a Cartesian coordinates and space vector mass fraction for species a
Greek c ma n q rij x
ratio of specific heats stoichiometric coefficient for species a conserved scalar, mixture fraction density viscous stress components reaction rate
Superscripts
f 4 f¯ ` f8Reynolds average decomposition of a quantity f f 4 f˜ ` f9Favre average decomposition of a quantity f Subscripts F, O, P i, j, k, l
a
fuel, oxidant, and product tensor indexes for the three Cartesian components index for chemical species Acknowledgment
Supercomputing time from the UK EPSRC under Grant GR/K 93761 is gratefully acknowledged.
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REFERENCES 1. Bray, K. N. C., Libby, P. A., Masuya, G., and Moss, J. B., Combust. Sci. Technol. 25:127–140 (1981). 2. Moss, J. B., Combust. Sci. Technol. 22:119–129 (1980). 3. Bray, K. N. C., AGARD Conference Proceedings No. 164, Propulsion and Energitics Panel Meeting, Paris, NATO, 1975. 4. Bilger, R. W., Prog. Energy Combust. Sci. 1:87–109 (1976). 5. Bray, K. N. C., Libby, P. A., and Williams, F. A., in Turbulent Reactive Flows (P. A. Libby and F. A. Williams, eds.), Academic, New York, 1994, pp. 609–638. 6. Bray, K. N. C., Proc. R. Soc. London A 451:231–256 (1995). 7. Givi, P., Madnia, C. K., and Steinberger, C. J., Combust. Sci. Technol. 78:33–67 (1991). 8. Miller, R. S., Madnia, C. K., and Givi, P., Combust. Sci. Technol. 99:1–36 (1994). 9. Rutland, C. J. and Cant, R. S., in Proc. Summer Prog., Center for Turbulence Research, Stanford, 1994, pp. 75–94. 10. Zhang, S. W. and Rutland, C. J., Combust. Flame 102:447–461 (1995). 11. Stoukov, A., “Etude Nume´rique de La Couche de Me´lange Re´active Supersonique,” Doctoral thesis, Universite´ de Rouen, 1996. 12. Vreman, A. W., Sandham, N. D., and Luo, K. H., J. Fluid Mech. 320:235–258 (1996). 13. Luo, K. H., in ERCOFTAC Series: Direct and LargeEddy Simulation II (J.-P. Chollet, P. R. Voke, and L. Kleiser, eds.), Kluwer Academic Publishers, Dordrecht, vol. 5, 1997, pp. 167–178. 14. Sta˚rner, H. and Bilger, R. W., Combust. Sci. Technol. 21:259–276 (1980). 15. Zheng, L. L. and Bray, K. N. C., Combust. Flame 99:440–448 (1994). 16. Sarkar, S., Phys. Fluids A 4(12):2674–2682 (1992). 17. Bilger, R. W., Ann. Rev. Fluid Mech. 24:101–135 (1989). 18. Libby, P. A., and Bray, K. N. C., AIAA J. 19:205–213 (1981). 19. Luo, K. H., Bray, K. N. C., and Sandham, N. D., in 16th ICDERS, Cracow, Poland, 1997, p. 580.
COMBUSTION-INDUCED PRESSURE EFFECTS IN SUPERSONIC DIFFUSION FLAMES K. H. LUO1 and K. N. C. BRAY2 1Department of Engineering Queen Mary and Westfield College London E1 4NS, UK 2Department of Engineering University of Cambridge Cambridge CB2 1PZ, UK
A turbulent diffusion flame at a convective Mach number 1.2 was investigated using direct numerical simulation (DNS). The DNS employed the full time-dependent compressible Navier–Stokes equations coupled with a one-step chemical reaction governed by the Arrhenius kinetics. Detailed study of combustion-induced pressure effects on turbulence generation, conserved scalar, and stagnation enthalpy transport was conducted. Local countergradient diffusion (CGD) of a conserved scalar flux was observed for the first time in a diffusion flame where heat release was strong enough while gradient diffusion prevailed when heat release was zero or weak. The CGD occurred in spite of the absence of an externally imposed mean pressure gradient and was attributed to combustion-induced pressure fluctuations. The balance of the turbulent kinetic energy budget was strongly influenced by the pressure dilatation and the (combustioninduced) mean pressure work when heat release was strong. Both terms can be a source or a sink of turbulence, depending on the intricate interactions between turbulence and combustion. However, the temporal change in pressure ]p/]t had an insignificant influence on the stagnation enthalpy transport. A linear relation between the stagnation enthalpy and the mixture fraction was confirmed, which could lead to considerable simplification in modeling high-speed turbulent combustion.
Introduction In supersonic turbulent diffusion flames, pressure effects are expected to be significant due to the coupling of compressibility and chemical heat release. Combustion models that are traditionally developed for low-speed combustion must be subjected to careful scrutiny before being extended to high-speed combustion. For example, the pressure-related phenomenon of countergradient diffusion, which occurs in low-speed turbulent combustion [1,2], may invalidate models that are based on the gradient-diffusion assumption. Another complication is related to the combustion-generated turbulence through pressure [3,4], which in turn would affect the combustion process itself. These complicated topics have been carefully examined in two review papers by Bray et al. [5] and Bray [6] recently, which call for further studies in the areas. The principal obstacle to understanding and modeling pressure effects lies in the lack of techniques that can directly measure the relevant quantities. Another problem arises from the close but poorly understood relationship between pressure and acoustics. The employment of direct numerical simulation (DNS) marks a new era in the fundamental study of
both premixed and nonpremixed turbulent combustion. Since DNS resolves the whole spectra of time and length scales of turbulence and combustion, many previously intractable terms (e.g., pressure-velocity correlations) can be computed accurately with minimum interference of modeling. This makes it very effective in studying the combined effects of compressibility and chemical heat release. Early work in DNS of compressible reacting layers (e.g., Refs. [7,8]) usually did not go far enough to see turbulent effects, and the heat release was rather weak. More recently, pressure effects were studied in turbulent premixed combustion [9,10] with the turbulence being represented by a prescribed energy spectrum. However, the low-Mach-number approximation invoked in the DNS excluded acoustics and strong coupling between compressibility and chemical heat release. Stoukov [11] reported DNS results of a hydrogen-air flame with detailed chemistry, but the flow field was a simplified 2-D supersonic mixing layer, which like earlier work [7] could not properly address turbulence or high-Mach-number effects. The present study extends the DNS method developed in Refs. [12,13] to include finite-rate chemical reactions governed by the Arrhenius kinetic
2165
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HIGH SPEED COMBUSTION
mechanism. The full compressible formulation enables detailed examination of pressure effects arising from the coupling of compressibility and heat release. Several fundamental issues in turbulent diffusion flames are then discussed. Direct Numerical Simulation The DNS was performed with the DSTAR code [13], in which the full 3-D time-dependent compressible governing equations were directly solved for a turbulent reacting mixing layer. A one-step reaction mFMF ` mOMO → mPMP was simulated. The reaction rate is given by the Arrhenius law, which after normalization takes the following form:
1W 2 1W 2
xT 4 Da
qYF
mF
qYO
F
3
2 exp 1Ze
mO
O
1T 1 T 24 1
1
(1)
f
It follows that the reaction rates for individual species are xF 4 1mFWFxT, xO 4 1mOWOxT, and xP 4 mPWPxT, respectively. The heat release rate is xh 4 QhxT, where Qh 4 mFWFDhof,F ` o mOWODhf,O 1 mPWPDhof,P is a heat release parameter. To start with, a nonreacting mixing layer (case CD) was simulated, in which an initially laminar flow of a Reynolds number 200 subject to linear disturbances went through transition and became turbulent at later times [12,13]. The fuel stream (the upper stream) and the oxidant stream (the lower stream) move at the same Mach number 1.2 but in opposite directions, leading to a convective Mach number Mc 4 1.2. Three reacting cases were then simulated with increasing heat release Qh 4 1 (Case HL), Qh 4 3 (Case HM), and Qh 4 6 (Case HH). Combustion was initiated at time t 4 70, when the nonreacting flow had become turbulent. Other simulation conditions were Da 4 3, Ze 4 3, and Tf 4 3. The computational box was 20 2 30 2 11.5, and the grid was 96 2 301 2 96. The grid lines were concentrated in the intense reaction zones to ensure adequate resolution throughout the simulations. More grid points could be added in each direction whenever necessary judged by the energy spectra and the value of the conserved scalar that must fall between 0 and 1. In all cases, length was normalized by the initial vorticity thickness and other quantities by their values in the upper free stream. The initial mean pressure was uniform, whose normalized value was 1/[c(Mc)2]. The Lewis number and Prandtl number were both unity. The molecular viscosity was temperature dependent (;T0.76). In all simulations, Fourier spectral methods were used in the homogeneous streamwise (x1) and spanwise (x3) directions, while a modified sixthorder compact finite-difference scheme was adopted
in the inhomogeneous lateral (x2) direction, in which characteristic nonreflecting boundary conditions were imposed. In the spanwise direction, a symmetry condition was also imposed to reduce computational cost. A third-order compact-storage RungeKutta method was employed for time advance [12,13]. Diffusion Flames in Compressible Turbulence The one-step chemical reaction with different heat release rates previously described is examined here. The combustion process was simulated from time t 4 70 to 80, during which the maximum reaction rate decreased rapidly with time. This was due to the fact that at ignition, there were large amounts of mixed fluids ready to burn. At a later stage, mixture preparation cannot keep up with the faster chemical reaction, leading to smaller and smaller reaction rate. To give a typical picture of the combustion process, the instantaneous product mass fraction in a plane cut at t 4 71 for case HH is plotted in Fig. 1a. It is shown that the flame is highly threedimensional with intense reactions concentrated in some narrow regions. These regions correspond to near-stoichiometric conditions. On average, the product is concentrated around the central layer (Fig. 1b), where mixing is more complete. Figure 2 shows the mean pressure profile at different stages of the flame. It should be recalled that in the nonreacting case, the mean pressure over the entire mixing layer is constant. The mean pressure gradient shown is purely induced by combustion. The peak mean pressure is 50% higher than that in the nonreacting flow {1/[c(Mc)2]}. Such large pressure variations, accompanied by equally significant changes in temperature and density, have a profound influence on turbulence, as shown in the next sections. Combustion-Generated Turbulence The turbulence kinetic energy (TKE) equation for full compressible flow is written as ] ˜ ] ˜ ˜ l) 4 q¯ k ` (q¯ ku ]t ]xl 1qu9u9 i l 1 u9i
]u˜i ] 1 1 qu9u9u9 i i l ]xl ]xl 2
1
2
]p ] ]u9i ` (u9r ) 1 r ]xi ]xl i li ]xl li
(2)
where on the left-hand side (L.H.S.) are the temporal (tkk) and convection (Ckk) terms, and on the right-hand side (R.H.S.) are the production (Pkk), triple correlation (Tkk), velocity-pressure gradient
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Fig. 1. Product formation in case HH. (a) Side view of the instantaneous product mass fraction field in the plane cut x3 4 L3/8 at t 4 71 (15 contours). (b) Mean mass fractions of species at t 4 80. Solid line, Y¯F; dotted line, Y¯O; dashed line, Y¯P.
Fig. 2. Combustion-induced mean pressure in case HH. Solid line, t 4 71; dotted line, t 4 75; dashed line, t 4 80.
correlation (Pkk), viscous diffusion (Dkk), and viscous dissipation (ekk) terms, respectively. Among these terms, Tkk and Dkk are purely redistributive, because they become zero after integration over x2. Only Pkk, Pkk, and ekk may generate or destroy turbulence. To see the effects of combustion on turbulence, the integrated TKE is plotted in Fig. 3 for cases CD (nonreacting) and HH (high heat release). In case CD, turbulence energy increases almost linearly with time, suggesting that some degree of self-similarity in the TKE balance has been reached. In case HH, turbulence energy increases significantly just after combustion starts at t 4 70 but then decreases dramatically at later times. It is interesting to note the differences in magnitude and ˜ caused by density effects. phase between qk ¯ ˜ and k,
Fig. 3. Temporal evolution of total turbulent kinetic en`` ˜ ergy. Solid line, *`` 1` ¯pk dx2, case CD; dotted line, *1` ˜ k˜ dx2, case CD; dashed line, *`` qk ¯ dx , case HH; dash 1` 2 ˜ dotted line, *`` 1` k dx2, case HH.
However, combustion effects on qk ¯ ˜ and k˜ are qualitatively the same. To understand these effects, the integrated production and dissipation are shown in Fig. 4a for the two cases. The production is seen to decrease with heat release almost monotonically. The viscous dissipation decreases during the main heat release period but picks up slightly as combustion becomes weak. Neither trend can explain the effects of combustion on turbulence as shown in Fig. 3. The pressure term Pkk is now examined, which can be split into the following terms Pkk 4 1u9i
]p¯ ]u9 ] ` p8 i 1 p8u9i ]xi ]xi ]xi
(3)
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Fig. 4. Temporal evolution of integrated terms in the TKE budget. (a) Solid line, *`` 1` Pkk dx2, case CD; dotted line, `` `` `` *`` 1` ekk dx2, case CD; dashed line, *1` Pkk dx2, case HH; dash dotted line, *1` ekk dx2, case HH. (b) Solid line, *1` 1 `` u9i ]p¯/]xi dx2, case CD; dotted line, *`` ¯ /]xidx2, case HH; dash dotted line, 1` p8]u9/]x i i dx2, case CD; dashed line, *1` 1u9i ]p `` *1` p8]u9/]x i idx2, case HH.
The terms on the R.H.S. of equation 3 are the mean pressure work, the pressure-dilatation, and the pressure-velocity correlation, respectively. The last term is a diffusive term that does not contribute to turbulence generation. The mean pressure work is zero in a constant density flow because in that case u9i 4 u8i 4 0. However, it was identified to be a source of combustion-generated turbulence in a turbulent premixed flame [1] and a turbulent diffusion flame with an externally imposed mean pressure gradient [14]. The pressure dilatation is still poorly understood and was typically ignored in previous analyses and calculations (e.g., Ref. [1]). Recently, Zheng and Bray [15] included a pressure-dilatation model, developed by Sarkar [16] for nonreacting flow, in the calculation of a nonpremixed supersonic hydrogenair flame but observed negligible effects. Results of the mean pressure work and pressure dilatation from the present DNS are shown in Fig. 4b. As expected, the mean pressure work is identically zero (which indirectly gives an indication of the high numerical accuracy achieved in the calculation), and the pressure-dilatation is negligibly small in the nonreacting flow (case CD). However, in the reacting flow (case HH), there is a positive mean pressure work accompanying the initial high heat release rate. This is due to the strong coupling of combustioninduced buoyancy and mean pressure gradient across the turbulent flamelets [1]. At later stages when combustion becomes weak, the induced buoyancy and the mean pressure gradient are no longer aligned positively. This causes the mean pressure work to be a sink of turbulence. The pressure dilatation also fluctuates between being a source and being a sink of turbulence, albeit at a much greater magnitude. The rise and fall in the pressure-dilatation correlate very well with the behavior of the TKE in Fig. 3, suggesting that the turbulence generation
is mainly due to the pressure-dilatation, enhanced further by a positive mean pressure work. The eventual decline of turbulence in the reacting flow relative to the nonreacting flow is attributed primarily to the decrease in Reynolds stress production by mean shear, exacerbated by a negative mean pressure work. The viscous dissipation, which is so often blamed, is not seen to be the main direct cause for the turbulence energy reduction. However, increasing viscosity due to heat release does have an adverse effect on turbulence production by mean shear. Countergradient Diffusion Countergradient diffusion (CGD) was observed in premixed combustion [1,2,9] but not in nonpremixed combustion [17]. It is generally accepted that CGD occurs when the mean pressure gradient acts differentially on the low-density burned gas and high-density unburned gas, respectively [1]. Bray [6] deemed it more appropriate to call the phenomenon “pressure gradient diffusion.” In most cases, CGD was found because there was an externally imposed mean pressure gradient acting in the opposite direction of the scalar concentration gradient [9,14]. Purely combustion-induced CGD has not been investigated sufficiently. In order to investigate the possible existence of CGD in the diffusion flame, we consider a conserved scalar defined by n 4 (sYF 1 YO ` 1)/(1 ` s), where s 4 mOWO/mFWF. It takes value 0 in the oxidant free stream and 1 in the fuel free stream, and is called a mixture fraction. In the present flame, n has positive gradient in the x2 direction, so that CGD means a positive scalar flux qn9u92. This quantity at t 4 75 is plotted for both the nonreacting and reacting cases in Fig. 5. In the nonreacting case, gradient diffusion rules. In the reacting cases, there is
COMBUSTION-INDUCED PRESSURE EFFECTS
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and the large negative peak of Uni in the center correlate with the countergradient diffusion region shown in Fig. 5. This is hardly surprising because Wni represents diffusion by pressure, whereas Uni represents transport by scalar gradient. It is tempting to say that Wni promotes CGD while Uni suppresses it. However, because both terms can change sign, it is difficult to make a definite conclusion. It is interesting to observe in Fig. 2 that the mean pressure gradient (and hence 1n9]p¯/]xi) in the central layer region is relatively small at this time (t 4 75). The pressure effects on scalar transport come primarily from the fluctuating pressure through 1n9]p8/]xi. Fig. 5. Conserved scalar flux qn9u92 at time t 4 75. Solid line, case HH (Qh 4 6); dotted line, case HM (Qh 4 3); dashed line, case HL (Qh 4 1); dash dotted line, case CD (Qh 4 0).
a tendency toward CGD in the center of the shear layer as the heat release increases. In case HH, CGD appears in the central layer, while gradient diffusion still dominates in other regions. To understand what is happening in the central layer, plane cuts through the conserved scalar flux field (shown in Fig. 6) and the heat release rate field (not shown) are investigated. It becomes clear that in regions of high heat release, positive qn9u92 dominates. In regions of low heat release, qn9u2 9 remains negative. The extent and duration of such local CGD are highly dependent on the local heat release rate. In fact, a case with Qh 4 12 conducted after submission of this paper shows that CGD becomes more prominent and more frequent. To further investigate the mechanisms behind CGD, the scalar flux transport equation is examined here. It is written for compressible flow in a symbolic form as ] ] qn9u9i ` ( qn9u9u i ˜ l) ]t ]xl 4 Pni ` Tni ` Pni ` Dni ` eni (4) where on the L.H.S. are the temporal and convection terms, and on the R.H.S., the production (Pni), triple correlation (Tni), scalar-pressure gradient correlation (Pni), viscous diffusion (Dni), and viscous dissipation (eni) terms, respectively. The pressure term Pni can be decomposed into two terms as Pni [ 1 n9]p/]xi [ p]n9/]xi 1 ](pn9)/]xi [ Uni ` Wni. The various terms on the R.H.S. of equation 4 for the x2 component are shown in Fig. 7 for case HH. The pressure terms Uni and Wni are dominate in scalar flux transport, although both terms have large lateral variation. Such oscillatory behavior of pressure-related terms in compressible turbulence is well documented [9,10,12,16] and is not attributed to numerical errors. The large positive peak of Wni
Pressure Effects on Stagnation Enthalpy Transport In modeling high-speed reacting flow, in which pressure effects are prominent, a question arises as to whether or not models developed for low-speed combustion can be borrowed. Bray et al. [5] suggested that by replacing the static enthalpy with the stagnation enthalpy in the energy equation, the conserved scalar presumed PDF method of low-speed combustion can be extended to high-speed combustion. Such a proposition depends crucially on a linear, algebraic relationship between the stagnation enthalpy and the mixture fraction: hs(x, t) 4 hs,O ` n(x, t)(hs,F 1 hs,O)
(5)
o where hs 4 (N a41 haYa ` uiui/2 and ha 4 hf,a ` *TTo Cp,a(T)dT. For a reacting system with unity Lewis number and Prandtl number, such a relation should hold if the pressure term ]p/]t is negligible. It was shown in Ref. [19] that equation 5 is indeed valid in a nonreacting compressible flow up to an effective Mach number 2.4. For the present reacting flow, the joint PDFs of hs and n (not shown) for all cases also exhibit a preferred direction corresponding to a linear relation between hs and n. However, the breadth of these PDFs suggests that the coupling between hs and n is to some degree weakened by the term ]p/]t. Nevertheless, the mean profiles of hs and n shown in Fig. 8 demonstrate beyond any doubt that the above linearity assumption is valid for combustion modeling purposes within the Mach number and heat release rate range tested.
Discussions and Conclusions Direct numerical simulations have been performed of reacting and nonreacting mixing layers at Mc 4 1.2 using the full compressible Navier–Stokes equations. A one-step chemical reaction governed by the Arrhenius kinetics has been simulated with various heat release rates. Combustion was found to
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Fig. 6. Plan view of the instantaneous conserved scalar flux qn9u92 in the plane cut x2 4 0 at t 4 75 for case HH. (Fifteen contours between 10.024 and `0.050: solid line, countergradient diffusion, dotted line, gradient diffusion.)
Fig. 7. Conserved scalar flux (qn9u92 ) budget at time t 4 75 for case HH. Solid line, pressure-scalar correlation (Wni); dashed line, pressure-scalar gradient (Uni); thin solid line, production (Pni); dotted line, triple correlation (Tni). Viscous diffusion (Dni) and dissipation (eni) are negligible.
generate large pressure, temperature, and density fluctuations. The analysis at the second-moment closure level was focused on combustion-induced pressure effects on turbulence generation and the transport of a conserved scalar and the stagnation enthalpy. A most significant finding was the existence of local countergradient diffusion where the heat release rate was sufficiently large. Such a phenomenon was previously observed only in premixed flames. More remarkably, the present diffusion flame had no externally imposed mean pressure gradient, so that the combustion-induced pressure fluctuations were solely responsible for the occurrence of countergradient diffusion. This fact suggests that the class of reacting flow with the presence of countergradient diffusion should be much larger than previously thought. Combustion-generated turbulence was also observed and attributed to the pressure dilatation and mean pressure work terms in the TKE budget. However, both terms can be a sink or a source of turbulence, depending on the intricate interactions between turbulence and combustion. For the Mach number and heat release rate range tested, the stagnation enthalpy and the mixture fraction showed a linear relation, due to the almost negligible influence of the temporal change in pressure ]p/]t. This implies that certain models developed for low-speed combustion in terms of the static enthalpy can be conveniently adopted for high-speed combustion using the stagnation enthalpy instead. How to incorporate all the above findings into turbulent combustion models remains a challenge.
Nomenclature Fig. 8. Linear relation between the mean stagnation enthalpy and the mean mixture fraction: solid line, case CD (Qh 4 0); dotted line, case HL (Qh 4 1); dashed line, case HM (Qh 4 3); dash dotted line, case HH (Qh 4 6).
Da, Ze, Tf h, hs, Dhof,a
Damko¨hler number, Zeldovich number, and flame temperature enthalpy, stagnation enthalpy, and standard enthalpy of formation
COMBUSTION-INDUCED PRESSURE EFFECTS
k˜ Ma Mc p Qh Re t T u Wa xi, x Ya
;
9 [u9u i i/2, turbulent kinetic energy chemical symbol for species a convective Mach number pressure a heat release parameter Reynolds number time temperature velocity molecular weight for species a Cartesian coordinates and space vector mass fraction for species a
Greek c ma n q rij x
ratio of specific heats stoichiometric coefficient for species a conserved scalar, mixture fraction density viscous stress components reaction rate
Superscripts
f 4 f¯ ` f8Reynolds average decomposition of a quantity f f 4 f˜ ` f9Favre average decomposition of a quantity f Subscripts F, O, P i, j, k, l
a
fuel, oxidant, and product tensor indexes for the three Cartesian components index for chemical species Acknowledgment
Supercomputing time from the UK EPSRC under Grant GR/K 93761 is gratefully acknowledged.
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