- Email: [email protected]
progress variable model for supersonic combustion flows
JID: PROCI
ARTICLE IN PRESS
[m;July 15, 2016;11:21]
Available online at www.sciencedirect.com
Proceedings of the Combustion Institute 000 (2016) 1–10 www.elsevier.com/locate/proci
Representative interactive flamelet model and flamelet/progress variable model for supersonic combustion flows Zhenxun Gao∗, Chongwen Jiang, Chun-Hian Lee National Laboratory for Computational Fluid Dynamics, School of Aeronautic Science and Engineering, Beihang University, Xueyuan Road 37, Haidian District, Beijing 100191, China Received 2 December 2015; accepted 28 June 2016 Available online xxx
Abstract Two flamelet-type models are developed for supersonic combustion flows. First, the representative interactive flamelet (RIF) model is introduced to fully incorporate the local high-Mach-number effects into the flamelet library, and some algorithms specifically for supersonic flows are designed in the RIF model. Secondly, the flamelet/progress variable (FPV) model is also extended to supersonic combustion flows, based on a new compressible rescaling for the tabulated chemical source term of the progress variable. These two models, together with the steady flamelet (SF) model, are then applied in a supersonic combustion experimental case. Numerical results indicate that the RIF model induces very limited changes on the temperature and major species concentrations compared to the SF model, implying that the species concentrations in the flamelet library are not sensitive to the local high-Mach-number effects. Overall, the RIF model does not show obvious improvement on the simulation accuracy for the studied case. While for the FPV model, the obtained numerical results achieve much better agreements with the experimental data, including the temperature, major species concentrations and auto-ignition position. It is also found that the proposed compressible rescaling for the FPV model reasonably incorporates the high-Mach-number effects on the tabulated chemical source term of the progress variable and effectively improves the simulation accuracy of the FPV model for supersonic combustion flows. © 2016 by The Combustion Institute. Published by Elsevier Inc. Keywords: Supersonic combustion; Representative interactive flamelet model; Flamelet/progress variable model; Computational fluid dynamics
1. Introduction
∗
Corresponding author. E-mail address: [email protected], [email protected] (Z. Gao).
The numerical simulation techniques of turbulent nonpremixed combustion in supersonic flows are of great interest because of the worldwide development of scramjet engines [1]. In the framework of Reynolds-averaged Navier–Stokes
http://dx.doi.org/10.1016/j.proci.2016.06.184 1540-7489 © 2016 by The Combustion Institute. Published by Elsevier Inc.
Please cite this article as: Z. Gao et al., Representative interactive flamelet model and flamelet/progress variable model for supersonic combustion flows, Proceedings of the Combustion Institute (2016), http://dx.doi.org/10.1016/j.proci.2016.06.184
JID: PROCI
2
ARTICLE IN PRESS
[m;July 15, 2016;11:21]
Z. Gao et al. / Proceedings of the Combustion Institute 000 (2016) 1–10
(RANS) or large eddy simulation (LES), a turbulent combustion model is needed to describe the turbulent effects on the chemical reactions occurring on the unresolved scales [2,3]. Some turbulent combustion models, originally established for low-Mach-number flows, have been applied in supersonic combustion flows, such as the transport probability density function (PDF) model [4], linear eddy model [5], conditional moment closure [6], steady flamelet model [7]. Among these models, the steady flamelet (SF) model, with reasonable accuracy but low computational complexity, attracted relatively more attention in recent years [7–15]. However, due to its low-Mach-number nature, many problems need to be addressed before the SF model can be used in supersonic combustion flows. The original SF model calculates the mean temperature, as well as the species concentrations, from the so-called flamelet library, while this strategy cannot consider the strong coupling between the velocity and temperature in supersonic flows. Zheng and Bray [8] made the first attempt to construct an empirical modified equation for the mean temperature to consider the conversion from kinetic energy to internal energy. Then, Oevermann [7] proposed a strategy that only the species concentrations are tabulated from the flamelet library while the temperature is solved implicitly from the solution of the energy equation. This strategy almost became a standard method for the subsequent supersonic combustion simulations based on the SF model, including RANS [10,11] and LES [12] researches. Nonetheless, other problems still exist: several assumptions made in the derivation of the flamelet equations, used for generation of the flamelet library, would not hold in supersonic flows. First, pressure is regarded as a constant, but the spatial variation of pressure in supersonic flows could be very large. Additionally, the kinetic energy is neglected relative to the static enthalpy based on a low-Mach-number assumption and a linear relation between the static enthalpy and mixture fraction is used in the flamelet equations [9]. While in supersonic flows, the relation between the static enthalpy and mixture fraction could largely deviate from linearity due to the effects of isentropic compression/expansion, shockwave, viscous dissipation or wall heat transfer [11]. These highMach-number effects could influence the species concentrations contained in the flamelet library. Sabel’nikov et al. [9] once attempted to include part of the high-Mach-number effects into the flamelet library by introducing a quadratic function including the velocity information between the conditional averaged static enthalpy and mixture fraction, but this correction is only suitable for some simple flows and thus not used extensively. So far, most of the simulations based on the SF model for supersonic combustion flows just use the lowMach-number flamelet library, and there are no
studies fully incorporating the high-Mach-number effects into the flamelet library. In addition, supersonic combustion flows are generally featured by the auto-ignition process. However, the SF model is not able to appropriately describe an auto-ignition process, which is the motivation for Pierce and Moin [16] to develop the flamelet/progress variable (FPV) model. This FPV model has been introduced to supersonic combustion flows by Pecˇ nik et al. [13] (RANS) and Saghafian et al. [14,15] (LES). For the FPV model, not only the species concentrations, but also the chemical source terms of the progress variable need to be stored in the flamelet library, so new compressible issue comes for the source terms. In [14], Saghafian et al. proposed a compressible rescaling for the tabulated source term from the flamelet library to include the compressibility effects. However, this compressible rescaling is not universal and must be used together with their approximate method for calculating temperature from internal energy. Furthermore, the numerical cases in [13– 15] did not include experimental data about the auto-ignition, nor the temperature and species concentrations. Hence, the FPV model for supersonic combustion flows should be further developed and evaluated. This paper has two objectives. One is to present modifications based on the SF model to include the high-Mach-number effects into the flamelet library and examine the resultant influences on the simulation results. The other objective is to develop a new compressibility correction for the FPV model and evaluate its accuracy for supersonic combustion flows. These studies would be performed in the framework of RANS. In Section 2, description of the SF model is first given. Then in order to fully incorporate the high-Mach-number effects, the representative interactive flamelet (RIF) model, originally proposed by Peters et al. [17], is extended to supersonic combustion flows, for the first time, and some algorithms specifically for supersonic flows are designed. Also, the FPV model is introduced, with a new compressible rescaling for the tabulated chemical source term of the progress variable. In Section 3, an experimental case is selected to verify the developed models, utilizing the measured data of temperature, species concentrations and autoignition position. 2. Turbulent combustion models 2.1. SF model The SF model is designed to directly model the mean species concentrations and temperature [18]. By introducing the mixture fraction, simulations of the flame inner structure and turbulent mixing process are uncoupled in formalism. The flame inner structure can be obtained through solving the
Please cite this article as: Z. Gao et al., Representative interactive flamelet model and flamelet/progress variable model for supersonic combustion flows, Proceedings of the Combustion Institute (2016), http://dx.doi.org/10.1016/j.proci.2016.06.184
ARTICLE IN PRESS
JID: PROCI
Z. Gao et al. / Proceedings of the Combustion Institute 000 (2016) 1–10
[m;July 15, 2016;11:21]
3
flamelet equations that are derived based on an asymptotic analysis in the mixture fraction space, while the turbulent mixing process is described by the joint PDF of the mixture fraction and its dissipation rate. Lastly, using the PDF integration, the mean species concentrations and temperature in turbulent combustion flows can be obtained [19]. The flamelet equations for the SF model are [9] 1 ∂ 2Ys ρχ + ωs = 0, s = 1, 2, . . . , ns 2 ∂ Z2 h=
ns
Ys hs (T ) = (hF − hO )Z + hO
(1) (2)
s=1
p0 = ρRT
ns
Ys /Ms
(3)
Fig. 1. Coupling between the RANS module and flamelet module in RIF model.
s=1
p, ρ, T and h are the pressure, density, temperature and static enthalpy, respectively. Ys , ωs , hs , and Ms are the mass fraction, chemical source term, static enthalpy and molecular weight of species s. Moreover, Z and χ are the mixture fraction and its dissipation rate. The boundary conditions are imposed as Z = 0, Ys = YsO , h = hO ; Z = 1, Ys = YsF , h = hF (4) where the superscripts ‘F ’ and ‘O’ represent the variable values in the fuel and oxidizer streams, respectively. It is seen that a linear relation between h and Z is used in Eq. (2) and the pressure in Eq. (3) is assumed as constant. The value of χ corresponding to the stoichiometric Z, denoted by χ st , is supposed to be the parameter controlling the influence that the turbulent mixing field imposes on the flame inner structure [18]. With appropriate boundary conditions, by solving Eqs. (1)–(3) under different given values of χ st , Ys (Z, χ st ) and T(Z, χ st ) are obtained and then stored in the flamelet library. Then, the mean species mass fractions (superscripts ‘-’ and ‘∼’ represent Reynolds and Favrè average, respectively) are obtained from the PDF integration of the flamelet library: Y˜ s = Ys (Z, χst )P˜ (Z, χst )d χst dZ (5) while the local mean temperature is calculated implicitly, using the mean species mass fractions and the static enthalpy solved from the energy equation [7]. For the joint PDF P˜ (Z, χst )in Eq. (5), statistical independence between Z and χ st is assumed, and a β-function is used for the PDF of Z while a δ-function for χ st . Therefore, transport equations of the first- and second-order moments of Z, i.e. Z˜ 2 , need to be solved additionally [19]. and Z 2.2. RIF model As mentioned before, the assumptions of constant pressure (Eq. (3)) and linear h–Z relation
(Eq. (2)) are invalid because of the high-Machnumber effects. However, the flamelet library is generated only once before the flow simulation in the SF model, and thus the local high-Mach-number effects are hard to be incorporated. Then the logical idea is to generate a local flamelet library for each point using different pressure values and h–Z relations, which leads to the RIF model, originally proposed by Peters et al. [17] to consider the flame transient effect. Here, this RIF model is introduced to supersonic combustion flows to fully incorporate the local high-Mach-number effects in the flamelet library. The main idea of the RIF model is to solve the flamelet equations in a coupled manner with the flow equations, as shown in Fig. 1. The RANS module provides parameters of the local flowfield point to the flamelet module. The flamelet equations are solved in the flamelet module based on these parameters and the established flamlelet library is then returned to calculate the mean species mass fractions. In the flamelet library for each point, only the mixture fraction varies and the scalar dissipation rate is fixed. Because of this coupling, it is possible to establish flamelet libraries including the local high-Mach-number effects. In the flamelet module, the pressure in Eq. (3) can be directly replaced by the local mean pressure p¯ provided by the RANS module, which implies that the local pressure fluctuation is ignored according to the fact that this small pressure perturbation would not lead to obvious influence on the species concentrations in the flamelet library [14]. While for the modification of h–Z relation in Eq. (2), Peters et al. [17] still kept the linear relation but changed the boundary condition parameters hF and hO according to the evolution of the pressure using the law of adiabatic compression. This method is inappropriate for supersonic flows because the variation of static enthalpy is caused not only by the adiabatic compression/expansion, but also the shockwave, viscous dissipation or wall heat transfer. However,
Please cite this article as: Z. Gao et al., Representative interactive flamelet model and flamelet/progress variable model for supersonic combustion flows, Proceedings of the Combustion Institute (2016), http://dx.doi.org/10.1016/j.proci.2016.06.184
ARTICLE IN PRESS
JID: PROCI
4
[m;July 15, 2016;11:21]
Z. Gao et al. / Proceedings of the Combustion Institute 000 (2016) 1–10
it is impossible to construct an accurate h–Z relation only given a local mean static enthalpy h˜ and mixture fraction Z˜ from the RANS module. Here, an approximate quadratic h–Z relation, instead of the linear relation, is assumed. Three parameters need to be identified to establish a quadratic function. In addition to the two boundary conditions in Eq. (4), the third condition comes from the local values of h˜ and Z˜ by assuming that h = h˜ when Z = Z˜ . Different points have different h˜ − Z˜ values, and thus each point would use its own quadratic h – Z relation. Finally, because the local p¯ and h˜ − Z˜ , calculated from the flow governing equations by the RANS module, include the mean high-Mach-number effects on the variations of pressure and static enthalpy, the species concentrations in the flamelet library would be directly influenced by these effects. In order to reduce the computational expense of the RIF model, the flamelet equations are solved only for the region where combustion occurs, and this region can be determined by the local value of the mixture fraction. For the Cheng et al. case [20] studied in the present paper, the CPU time for every iteration of the RIF model is about 1.92 times that of the SF model. 2.3. FPV model For the SF model, the species concentrations in the flamelet library are parameterized by Z and χ st , while Pierce and Moin [16] showed that χ st cannot describe the whole so-called S-shape curve representing the solution of the flamelet equations. The FPV model was proposed in [16] to overcome this problem. A progress variable parameter , which can uniquely identify each single flame state on the whole S-shape curve, is introduced to replace χ st . is defined as the value of a progress variable C at Z = Zst, and is independent of Z. Given a Z and C, the reaction progress parameter and its corresponding flame state can be determined [21]. Hence, mean species mass fractions are calculated as Y˜ s = Ys (Z, C )P˜ (Z, C )dZdC (6) The joint PDF is written as P˜ (Z, C ) = P˜ (C|Z )P˜ (Z )
(7)
where P˜ (Z) is also given by an assumed β-function form, while the conditional PDF P˜ (C|Z ) is expressed using a δ-function [16]: (8) P˜ (C|Z ) = δ C − C| Z C| Z can be obtained by some single flamelet solution determined by the following constraint: C˜ = C| ZP˜ (Z )dZ (9)
Hence, in addition to the transport equations 2 , another one of the mean progress of Z˜ and Z ˜ variable, C , is also needed, and the mean chemical source term in its transport equation is calculated by ω¯˙ C = ω˙ C (Z, C )P˜ (Z, C )dZdC (10) where ω˙ C (Z,C) is stored in the flamelet library together with Ys (Z,C). The applications of this FPV model in supersonic combustion flows are very limited [13–15], and this model has similar problems with the SF model, that is, how to include the local high-Machnumber effects into the flamelet library. However, it is not easy to achieve this aim in the framework of the above RIF model. It can be seen in Fig. 1 that only one single flamelet solution determined by χ˜ needs to be solved for every point. If using the FPV model to replace the SF model in Fig. 1, for every point, the whole S-shape curve should be solved and then one of these flamelet solutions is chosen according to Eq. (9), which would result in unacceptable computational complexity. Hence, for the FPV model, the high-Mach-number effects are not introduced into the species mass fractions in the flamelet library. However, it is still necessary to introduce some compressible correction for the chemical source terms of the progress variable according to Saghafian et al [14]. Instead of correcting these source terms in the flamelet library, a compressible rescaling was proposed for the tabulated source term ω¯˙ C in [14] based on an approximate method for calculating temperature from internal energy. However, this rescaling is an empirical expression to imitate the Arrhenius behavior, and includes a series of parameters which can only be determined in the approximate method for calculating temperature from internal energy. So this rescaling cannot be implemented if one does not use this approximate method. Here, the idea to include the local high-Mach-number effects through rescaling the tabulated source term is adopted, but a new compressible rescaling form is proposed as follow: ¯ T˜ ω˙ test ρ, ω¯˙ C = testC f l (11) ω˙ C (ρ , T f l ) ω¯˙ C0 where ω¯˙ C0 is the tabulated source term from the flamelet library not including any high-Machnumber effect. Here, the rescaling coefficient is obtained using two testing source terms of the progress variable: ω˙ Ctest (ρ, ¯ T˜ ) and ω˙ Ctest (ρ f l , T f l ), which are calculated using the same tabulated species concentrations but different density and temperature. The former one utilizes the local mean density ρ¯ and temperature T˜ solved from RANS while the latter one is obtained based on the tabulated density ρ fl and temperature Tfl , averaged from the flamelet library using the joint
Please cite this article as: Z. Gao et al., Representative interactive flamelet model and flamelet/progress variable model for supersonic combustion flows, Proceedings of the Combustion Institute (2016), http://dx.doi.org/10.1016/j.proci.2016.06.184
JID: PROCI
ARTICLE IN PRESS Z. Gao et al. / Proceedings of the Combustion Institute 000 (2016) 1–10
[m;July 15, 2016;11:21]
5
3.1. Case description
Fig. 2. Computational and experimental ignition delay times of a diluted hydrogen/air mixture.
PDF just like the species mass fractions in Eq. (6). Note that ρ fl and Tfl only reflect the influences by the combustion, while ρ¯ and T˜ computed from the flow governing equations could also include the local high-Mach-number effects. Hence, the ratio of the two testing source terms is supposed to reflect the local high-Mach-number effects on the tabulated source term. It is believed that this rescaling is more physical and universal than the one in [14], even though some additional computation is introduced for calculating the testing source terms. 3. Numerical results and analysis The following numerical simulations are performed using a finite-difference code, named ACANS, developed by the authors. The reliability of ACANS has been validated by a variety of simulations [11,22–24]. The flow governing equations of continuity, momentum and energy, together with 2 are solved for the SF the equations of Z˜ and Z and RIF models, while another equation of C˜ is added for the FPV model. The inviscid flux vectors are discretized using the second-order AUSMDV [25] scheme and the viscous flux vectors by the central difference scheme. The implicit LU-SGS method [26] is employed for the time marching and the turbulence model is Menter’s SST model [27]. A hydrogen/air reaction mechanism including 9 species (H2 , O2 , H2 O, H, O, OH, HO2 , H2 O2 , N2 ) and 19 elementary reactions [28] is adopted here, and the species of H2 O is selected as the progress variable for the FPV model. Figure 2 compares the ignition delay times of a diluted hydrogen/air mixture calculated by the adopted reaction mechanism with the experimental data form Bhaskaran et al. [29], and good agreements can be seen.
An axisymmetric supersonic combustion experiment conducted by Cheng et al. [20] is selected to evaluate the SF, RIF and FPV models. Figure 3 shows the configuration of the experiment, where a vitiated air stream (oxygen-enriched air and hydrogen are adopted in the vitiated heater) and a hydrogen stream are injected from different nozzles and an axisymmetric flame is generated downstream. The axisymmetric governing equations are solved on a 2-D structural mesh. Table 1 gives the free stream conditions of the air and hydrogen streams, where p and T are the static pressure and temperature, respectively. Moreover, according to Cheng et al. [20], the uncertainty for the total pressure is ± 4%, while those for the gas temperature and major species concentrations are ± 5%. In the simulations, the flowfiled is initialized using the ambient air conditions: p0 = 1 atm, T0 = 288 K, U0 = 0 m/s, and thus the initializing turbulent kinetic energy and turbulent viscosity are both 0. 3.2. Results and analysis The variations of the mean pressure with x and the mean static enthalpy with mean mixture fraction along y = 0 m, obtained by the FPV model, are first shown in Fig. 4. They are used to qualitatively show that the pressure deviates from the constant value adopted in the generation of the flamelet library and the h˜ − Z˜ relation is also different from a linear relation. However, the deviations are not very large because of the parallel injection scheme in this case which would not induce strong shockwaves. Then, the temperature contours obtained by the three models are illustrated in Fig. 5. It is found that the RIF model does not lead to obvious changes on the mean temperature field by contrast with the results of the SF model, implying that the species concentrations in the flamelet library are not sensitive to the high-Mach-number effects. While the adoption of the FPV model remarkably delays the auto-ignition position compared to the other two models. Actually, experimental observation showed that the ignition occurs at around x/d = 25 [20]. Clearly, both the SF and RIF models predict the ignition too much earlier, and the inclusion of highMach-number effects in the RIF model has little effect on the auto-ignition simulation. While for the FPV model, the combustion initiates at around x/d = 23 to 24, pretty close to the observed result, which means that the FPV model can also effectively improve the auto-ignition prediction for supersonic combustion flows. Figure 6 shows the temperature distributions calculated by the three models at four streamwise locations, together with the corresponding experimental data. Compared to the SF model, the RIF
Please cite this article as: Z. Gao et al., Representative interactive flamelet model and flamelet/progress variable model for supersonic combustion flows, Proceedings of the Combustion Institute (2016), http://dx.doi.org/10.1016/j.proci.2016.06.184
ARTICLE IN PRESS
JID: PROCI
6
[m;July 15, 2016;11:21]
Z. Gao et al. / Proceedings of the Combustion Institute 000 (2016) 1–10
Fig. 3. Geometric configuration of Cheng et al. experiment.
Table 1 Free stream conditions for Cheng et al. case ( p and T are both static variables).
Air stream Hydrogen stream
M
p/kPa
T/K
YH2
YO2
YH2 O
YOH
YN2
2 1
107 112
1250 545
0 1
0.245 0
0.175 0
6.65e − 4 0
0.58 0
Fig. 4. Distributions of (a) pressure with x and (b) static enthalpy with mixture fraction along y = 0 m.
Fig. 5. Temperature contours obtained by the SF, RIF and FPV models.
Please cite this article as: Z. Gao et al., Representative interactive flamelet model and flamelet/progress variable model for supersonic combustion flows, Proceedings of the Combustion Institute (2016), http://dx.doi.org/10.1016/j.proci.2016.06.184
ARTICLE IN PRESS
JID: PROCI
[m;July 15, 2016;11:21]
Z. Gao et al. / Proceedings of the Combustion Institute 000 (2016) 1–10 3000
SF RIF FPV Experiment
2000
3000
SF RIF FPV Experiment
2500
x/d=10.8
2500
x/d=21.5
T/K
T/K
T/K
x/d=43.1
2000
2000
1500
1500
SF RIF FPV Experiment
2500
x/d=32.3
2000 1500
3000
SF RIF FPV Experiment
T/K
2500
7
1500
1000
500 -10
-5
0
y/d
5
1000
1000
1000
500
500
500
-10
10
-5
0
y/d
5
10
-10
-5
0
y/d
5
10
-10
-5
0
y/d
5
10
40
45
Fig. 6. Temperature distributions at x/d = 10.8, 21.5, 32.3 and 43.1. 1
0.6
0.6
0.4
H2
0.4
H 2O
0.2
0.2 0
SF RIF FPV Experiment
Mole fraction of H2O
Mole fraction of H2
SF RIF FPV Experiment
0.8
0
5
10
15
20
x/d
25
30
35
40
45
0
0
5
10
15
20
x/d
25
30
35
Fig. 7. Mole fraction distributions of H2 and H2 O along y = 0 m.
model induces slight decreases of the temperature peak at x/d = 10.8 and 43.1, while small increases at x/d = 21.5 and 32.3. However, these changes of temperature are all within 40 K, and some are too small to be resolved in Fig. 6. Moreover, comparisons with the measured data at x/d = 10.8 again indicate that the SF and RIF models predict the ignition earlier. While for the FPV model, the temperature achieves a much better agreement with the experimental data at x/d = 10.8. Asymmetric distribution is noticed for the experimental data at x/d = 21.5, where the maximum temperature is close to 2000 K on the positive-y side while only around 1500 K on the other side. The FPV model predicts a temperature distribution in good accordance with the measured data on the negative-y side while the results by the SF and RIF models are close to the experimental data on the positive-y side. At x/d = 32.3 and 43.1, the FPV model shows consistent results with the other two models. It is also seen in Fig. 6 that the computations obviously underestimate the jet diameter at downstream locations, while it is believed that this is not from the combustion simulations. Similar results can be found in previous simulations including RANS [30] and LES [31]. The distributions of the mole concentrations of H2 and H2 O along y = 0 m are depicted in Fig. 7. Similar to the temperature field, the RIF model leads to very limited changes on the concentrations
of H2 and H2 O, and the results obtained by the SF and RIF models show some deviations from the measured data. Due to a more accurate autoignition prediction, the FPV model achieves much better agreements with the experimental data. In order to assess the effect of the compressible rescaling on the tabulated source term in the FPV model, the variation of the rescaling coefficient (the right side of Eq. (11)) along y = 0 m is shown in Fig. 8a, and also the temperature distributions obtained by the FPV model with/without the rescaling in Fig. 8b. It is seen that the rescaling coefficient takes on some oscillations due to the influence of the wave system generated between the air and hydrogen jets. In front of the ignition (x/d ≈ 24), it can be found in Fig. 4 that the local pressure is generally larger than the value adopted in the flamelet library and meanwhile the local static enthalpy is also higher than the value calculated from a linear relation with the mixture fraction, which makes the values of the rescaling coefficient larger than 1 in Fig. 8a. This amplification of the tabulated source term remarkably advances the auto-ignition, from around x/d ≈ 30 (without rescaling) to x/d ≈ 24 (with rescaling), which can be seen in Fig. 8b. Downstream of the ignition, the rescaling coefficient becomes smaller and less than 1 in some region. Hence, the temperature with rescaling at x/d ≈ 43.1 is a little lower than that without rescaling, and closer to the
Please cite this article as: Z. Gao et al., Representative interactive flamelet model and flamelet/progress variable model for supersonic combustion flows, Proceedings of the Combustion Institute (2016), http://dx.doi.org/10.1016/j.proci.2016.06.184
JID: PROCI
8
ARTICLE IN PRESS
[m;July 15, 2016;11:21]
Z. Gao et al. / Proceedings of the Combustion Institute 000 (2016) 1–10
Fig. 8. Distributions of (a) the rescaling coefficient and (b) temperature with/without scaling along y = 0 m.
Fig. 9. Mesh refinement study for the FPV model.
measured data. Obviously, the proposed compressible rescaling reasonably incorporates the influences of the local high-Mach-number effects on the tabulated source term of the progress variable and effectively improves the simulation accuracy of the FPV model for supersonic combustion flows. Above results show that the high-Mach-number effects produce very limited changes for the RIF model but considerable influences for the FPV model. Actually, the same conclusion can be drawn in another case [32], whose results are shown in the Appendix. The reason is that the high-Machnumber effects in the RIF model only affects the species concentrations in flamelet library, but has no direct effect on the local scalar dissipation rate that mainly controls which flamelet solution in the library is used to perform the PDF integration. However, for the FPV model, the high-Mach-number effects introduced by rescaling the tabulated source term would directly influence the local values of the progress variable and then change the local flamelet state. Additionally, it is shown in [14] that in the flamelet library, the chemical source terms are much more sensitive to the
high-Mach-number effects than the concentrations of major species. Then, it is not hard to understand that the high-Mach-number effects produce more influences on the FPV model. Finally, to ensure the grid independence of the above results, a mesh refinement study is performed for the FPV model. The mesh adopted in above simulations includes 37,757 grid points, while the refined mesh has 56,667 grid points. Figure 9 compares the temperature distributions at two streamwise positions and it is seen that almost the same results are obtained by the two meshes. 4. Conclusions In order to fully incorporate the local highMach-number effects into the flamelet library for the steady flamelet (SF) model, the representative interactive flamelet (RIF) model is introduced to supersonic combustion flows. In this RIF model, the flamelet equations are solved in a speciallydesigned coupling manner with the flow governing equations, through which flamelet library including local high-Mach-number effects could be established for each point in the combustion region.
Please cite this article as: Z. Gao et al., Representative interactive flamelet model and flamelet/progress variable model for supersonic combustion flows, Proceedings of the Combustion Institute (2016), http://dx.doi.org/10.1016/j.proci.2016.06.184
JID: PROCI
ARTICLE IN PRESS Z. Gao et al. / Proceedings of the Combustion Institute 000 (2016) 1–10
[m;July 15, 2016;11:21]
9
Fig. A1. Geometric configuration of DLR experiment.
Fig. A2. Temperature distributions at x = 108 mm and 216 mm.
The application in a supersonic combustion experimental case indicates that the RIF model could induce some changes on the temperature and major species concentrations compared to the results of the SF model, but these changes are very limited. Also, the RIF model has little influence on the auto-ignition simulation. Overall, the RIF model does not show obvious improvement for the studied case, which implies that the species concentrations in the flamelet library are not sensitive to the local high-Mach-number effects. The flamelet/progress variable (FPV) model is also applied in supersonic combustion flows. A new compressible rescaling form is proposed to rescale the tabulated source term of the progress variable, so as to consider the local high-Mach-number effects. The numerical results of the same case using the FPV model achieve much better agreements with the experimental data than those of the SF and RIF models, including the temperature, major species concentrations and auto-ignition position. It is also indicated that the proposed compressible rescaling reasonably incorporates the influences of the high-Mach-number effects on the tabulated source term of the progress variable and effectively
improves the simulation accuracy of the FPV model for supersonic combustion flows. Acknowledgments This research is supported by the National Natural Science Foundation of China through the Grant 11202014 and also partially supported by the China Scholarship Council (File no.: 201406025016). Appendix Another experimental case is also simulated using the SF, RIF and FPV models for further validation, and the results are shown in this Appendix. This case is a hydrogen-fueled scramjet combustor experimentally studied by Waidmann et al. [32] in German Aerospace Center (DLR). The experimental configuration of the combustor is shown as Fig. A1, and a 2D approximation for this case can be employed according to previous studies [7,11]. Also, the free stream conditions of the air stream
Please cite this article as: Z. Gao et al., Representative interactive flamelet model and flamelet/progress variable model for supersonic combustion flows, Proceedings of the Combustion Institute (2016), http://dx.doi.org/10.1016/j.proci.2016.06.184
JID: PROCI
10
ARTICLE IN PRESS
[m;July 15, 2016;11:21]
Z. Gao et al. / Proceedings of the Combustion Institute 000 (2016) 1–10
and hydrogen jet can be found in [7,11]. Here, the temperature distributions at two streamwise positions, together with the corresponding experimental data, are shown in Fig. A2 to examine the accuracy of different models. It is seen that for this case, the RIF model also induces limited changes on the temperature results compared to the SF model, and the FPV model achieves better agreements with the measured data. Comparing the results of the FPV model with/without the compressible rescaling, it is found that the rescaling enhances the temperature at both the two positions, which makes the temperature results of the FPV model closer to the experimental data. References [1] F. Ladeinde, AIAA 2009-127 (2009). [2] D. Veynante, L. Vervisch, Prog. Energy Combust. Sci. 28 (3) (2002) 193–266. [3] H. Pitsch, Annu. Rev. Fluid Mech. 38 (2006) 453–482. [4] H. Möbus, P. Gerlinger, D. Brüggemann, Combust. Flame 132 (1–2) (2003) 3–24. [5] F. Génin, S. Menon, AIAA J. 48 (3) (2010) 526–539. [6] M. Picciani, in: Supersonic Combustion Modeling Using the Conditional Moment Closure Approach, Ph.D. thesis, Cranfield University, Cranfield, UK, 2014. [7] M. Oevermann, Aerosp. Sci. Technol. 4 (7) (2000) 463–480. [8] L.L. Zheng, K.N.C. Bray, Combust. Flame 99 (2) (1994) 440–448. [9] V. Sabel’nikov, B. Deshaies, L.F. Figueira da Silva, Combust. Flame 114 (3–4) (1998) 577–584. [10] Z. Fan, W. Liu, M. Sun, Z. Wang, F. Zhuang, W. Luo, Sci. China-Technol. Sci. 55 (1) (2012) 193–205. [11] Z. Gao, J. Wang, C. Jiang, C.H. Lee, Combust. Theory Model. 18 (6) (2014) 652–691.
[12] M. Berglund, C. Fureby, Proc. Combust. Inst. 31 (2) (2007) 2497–2504. [13] R. Pecˇ nik, V.E. Terrapon, F. Ham, G. Iaccarino, H. Pitsch, AIAA J. 50 (8) (2012) 1717–1732. [14] A. Saghafian, V.E. Terrapon, H. Pitsch, Combust. Flame 162 (3) (2015) 652–667. [15] A. Saghafian, L. Shunn, D.A. Philips, F. Ham, Proc. Combust. Inst. 35 (2) (2015) 2163–2172. [16] C.D. Pierce, P. Moin, J. Fluid Mech. 504 (2004) 73–97. [17] H. Pitsch, H. Barths, N. Peters, SAE-962057 (1996). [18] N. Peters, Prog. Energy Combust. Sci. 10 (3) (1984) 319–339. [19] N. Peters, Turbulent Combustion, Cambridge University Press, 2000. [20] T.S. Cheng, J.A. Wehrmeyer, R.W. Pitz, O. Jarrett Jr., G.B. Northam, Combust. Flame 99 (1) (1994) 157–173. [21] M. Ihme, C.M. Cha, H. Pitsch, Proc. Combust. Inst. 30 (1) (2005) 793–800. [22] Z. Gao, C. Jiang, S. Pan, C.H. Lee, AIAA J. 53 (7) (2015) 1949–1968. [23] Z. Gao, C. Jiang, C.H. Lee, Appl. Therm. Eng. 92 (2016) 62–70. [24] Z. Gao, C. Jiang, C.H. Lee, Sci. China-Technol. Sci. 56 (10) (2013) 2501–2515. [25] Y. Wada, M.S. Liou, SIAM J. Sci. Comput. 18 (3) (1997) 633–657. [26] S. Yoon, A. Jameson, AIAA J. 26 (9) (1988) 1025–1026. [27] F.R. Menter, AIAA J. 32 (8) (1994) 1598–1605. [28] T. Poinsot, D. Veynant, Theoretical and Numerical Combustion, 2nd ed., R.T. Edwards Inc., Philadelphia, USA, 2005, p. 10. [29] K.A. Bhaskaran, M.C. Gupta, T. Just, Combust. Flame 21 (1) (1973) 45–48. [30] R.A. Baurle, S.S. Girimaji, Combust. Flame 134 (1–2) (2003) 131–148. [31] P. Boivin, A. Dauptain, C. Jiménez, B. Cuenot, Combust. Flame 159 (4) (2012) 1779–1790. [32] W. Waidmann, F. Alff, M. Böhm, U. Brummund, W. Clauβ, M. Oschwald, Space Technol 15 (6) (1995) 421–429.
Please cite this article as: Z. Gao et al., Representative interactive flamelet model and flamelet/progress variable model for supersonic combustion flows, Proceedings of the Combustion Institute (2016), http://dx.doi.org/10.1016/j.proci.2016.06.184
ARTICLE IN PRESS
[m;July 15, 2016;11:21]
Available online at www.sciencedirect.com
Proceedings of the Combustion Institute 000 (2016) 1–10 www.elsevier.com/locate/proci
Representative interactive flamelet model and flamelet/progress variable model for supersonic combustion flows Zhenxun Gao∗, Chongwen Jiang, Chun-Hian Lee National Laboratory for Computational Fluid Dynamics, School of Aeronautic Science and Engineering, Beihang University, Xueyuan Road 37, Haidian District, Beijing 100191, China Received 2 December 2015; accepted 28 June 2016 Available online xxx
Abstract Two flamelet-type models are developed for supersonic combustion flows. First, the representative interactive flamelet (RIF) model is introduced to fully incorporate the local high-Mach-number effects into the flamelet library, and some algorithms specifically for supersonic flows are designed in the RIF model. Secondly, the flamelet/progress variable (FPV) model is also extended to supersonic combustion flows, based on a new compressible rescaling for the tabulated chemical source term of the progress variable. These two models, together with the steady flamelet (SF) model, are then applied in a supersonic combustion experimental case. Numerical results indicate that the RIF model induces very limited changes on the temperature and major species concentrations compared to the SF model, implying that the species concentrations in the flamelet library are not sensitive to the local high-Mach-number effects. Overall, the RIF model does not show obvious improvement on the simulation accuracy for the studied case. While for the FPV model, the obtained numerical results achieve much better agreements with the experimental data, including the temperature, major species concentrations and auto-ignition position. It is also found that the proposed compressible rescaling for the FPV model reasonably incorporates the high-Mach-number effects on the tabulated chemical source term of the progress variable and effectively improves the simulation accuracy of the FPV model for supersonic combustion flows. © 2016 by The Combustion Institute. Published by Elsevier Inc. Keywords: Supersonic combustion; Representative interactive flamelet model; Flamelet/progress variable model; Computational fluid dynamics
1. Introduction
∗
Corresponding author. E-mail address: [email protected], [email protected] (Z. Gao).
The numerical simulation techniques of turbulent nonpremixed combustion in supersonic flows are of great interest because of the worldwide development of scramjet engines [1]. In the framework of Reynolds-averaged Navier–Stokes
http://dx.doi.org/10.1016/j.proci.2016.06.184 1540-7489 © 2016 by The Combustion Institute. Published by Elsevier Inc.
Please cite this article as: Z. Gao et al., Representative interactive flamelet model and flamelet/progress variable model for supersonic combustion flows, Proceedings of the Combustion Institute (2016), http://dx.doi.org/10.1016/j.proci.2016.06.184
JID: PROCI
2
ARTICLE IN PRESS
[m;July 15, 2016;11:21]
Z. Gao et al. / Proceedings of the Combustion Institute 000 (2016) 1–10
(RANS) or large eddy simulation (LES), a turbulent combustion model is needed to describe the turbulent effects on the chemical reactions occurring on the unresolved scales [2,3]. Some turbulent combustion models, originally established for low-Mach-number flows, have been applied in supersonic combustion flows, such as the transport probability density function (PDF) model [4], linear eddy model [5], conditional moment closure [6], steady flamelet model [7]. Among these models, the steady flamelet (SF) model, with reasonable accuracy but low computational complexity, attracted relatively more attention in recent years [7–15]. However, due to its low-Mach-number nature, many problems need to be addressed before the SF model can be used in supersonic combustion flows. The original SF model calculates the mean temperature, as well as the species concentrations, from the so-called flamelet library, while this strategy cannot consider the strong coupling between the velocity and temperature in supersonic flows. Zheng and Bray [8] made the first attempt to construct an empirical modified equation for the mean temperature to consider the conversion from kinetic energy to internal energy. Then, Oevermann [7] proposed a strategy that only the species concentrations are tabulated from the flamelet library while the temperature is solved implicitly from the solution of the energy equation. This strategy almost became a standard method for the subsequent supersonic combustion simulations based on the SF model, including RANS [10,11] and LES [12] researches. Nonetheless, other problems still exist: several assumptions made in the derivation of the flamelet equations, used for generation of the flamelet library, would not hold in supersonic flows. First, pressure is regarded as a constant, but the spatial variation of pressure in supersonic flows could be very large. Additionally, the kinetic energy is neglected relative to the static enthalpy based on a low-Mach-number assumption and a linear relation between the static enthalpy and mixture fraction is used in the flamelet equations [9]. While in supersonic flows, the relation between the static enthalpy and mixture fraction could largely deviate from linearity due to the effects of isentropic compression/expansion, shockwave, viscous dissipation or wall heat transfer [11]. These highMach-number effects could influence the species concentrations contained in the flamelet library. Sabel’nikov et al. [9] once attempted to include part of the high-Mach-number effects into the flamelet library by introducing a quadratic function including the velocity information between the conditional averaged static enthalpy and mixture fraction, but this correction is only suitable for some simple flows and thus not used extensively. So far, most of the simulations based on the SF model for supersonic combustion flows just use the lowMach-number flamelet library, and there are no
studies fully incorporating the high-Mach-number effects into the flamelet library. In addition, supersonic combustion flows are generally featured by the auto-ignition process. However, the SF model is not able to appropriately describe an auto-ignition process, which is the motivation for Pierce and Moin [16] to develop the flamelet/progress variable (FPV) model. This FPV model has been introduced to supersonic combustion flows by Pecˇ nik et al. [13] (RANS) and Saghafian et al. [14,15] (LES). For the FPV model, not only the species concentrations, but also the chemical source terms of the progress variable need to be stored in the flamelet library, so new compressible issue comes for the source terms. In [14], Saghafian et al. proposed a compressible rescaling for the tabulated source term from the flamelet library to include the compressibility effects. However, this compressible rescaling is not universal and must be used together with their approximate method for calculating temperature from internal energy. Furthermore, the numerical cases in [13– 15] did not include experimental data about the auto-ignition, nor the temperature and species concentrations. Hence, the FPV model for supersonic combustion flows should be further developed and evaluated. This paper has two objectives. One is to present modifications based on the SF model to include the high-Mach-number effects into the flamelet library and examine the resultant influences on the simulation results. The other objective is to develop a new compressibility correction for the FPV model and evaluate its accuracy for supersonic combustion flows. These studies would be performed in the framework of RANS. In Section 2, description of the SF model is first given. Then in order to fully incorporate the high-Mach-number effects, the representative interactive flamelet (RIF) model, originally proposed by Peters et al. [17], is extended to supersonic combustion flows, for the first time, and some algorithms specifically for supersonic flows are designed. Also, the FPV model is introduced, with a new compressible rescaling for the tabulated chemical source term of the progress variable. In Section 3, an experimental case is selected to verify the developed models, utilizing the measured data of temperature, species concentrations and autoignition position. 2. Turbulent combustion models 2.1. SF model The SF model is designed to directly model the mean species concentrations and temperature [18]. By introducing the mixture fraction, simulations of the flame inner structure and turbulent mixing process are uncoupled in formalism. The flame inner structure can be obtained through solving the
Please cite this article as: Z. Gao et al., Representative interactive flamelet model and flamelet/progress variable model for supersonic combustion flows, Proceedings of the Combustion Institute (2016), http://dx.doi.org/10.1016/j.proci.2016.06.184
ARTICLE IN PRESS
JID: PROCI
Z. Gao et al. / Proceedings of the Combustion Institute 000 (2016) 1–10
[m;July 15, 2016;11:21]
3
flamelet equations that are derived based on an asymptotic analysis in the mixture fraction space, while the turbulent mixing process is described by the joint PDF of the mixture fraction and its dissipation rate. Lastly, using the PDF integration, the mean species concentrations and temperature in turbulent combustion flows can be obtained [19]. The flamelet equations for the SF model are [9] 1 ∂ 2Ys ρχ + ωs = 0, s = 1, 2, . . . , ns 2 ∂ Z2 h=
ns
Ys hs (T ) = (hF − hO )Z + hO
(1) (2)
s=1
p0 = ρRT
ns
Ys /Ms
(3)
Fig. 1. Coupling between the RANS module and flamelet module in RIF model.
s=1
p, ρ, T and h are the pressure, density, temperature and static enthalpy, respectively. Ys , ωs , hs , and Ms are the mass fraction, chemical source term, static enthalpy and molecular weight of species s. Moreover, Z and χ are the mixture fraction and its dissipation rate. The boundary conditions are imposed as Z = 0, Ys = YsO , h = hO ; Z = 1, Ys = YsF , h = hF (4) where the superscripts ‘F ’ and ‘O’ represent the variable values in the fuel and oxidizer streams, respectively. It is seen that a linear relation between h and Z is used in Eq. (2) and the pressure in Eq. (3) is assumed as constant. The value of χ corresponding to the stoichiometric Z, denoted by χ st , is supposed to be the parameter controlling the influence that the turbulent mixing field imposes on the flame inner structure [18]. With appropriate boundary conditions, by solving Eqs. (1)–(3) under different given values of χ st , Ys (Z, χ st ) and T(Z, χ st ) are obtained and then stored in the flamelet library. Then, the mean species mass fractions (superscripts ‘-’ and ‘∼’ represent Reynolds and Favrè average, respectively) are obtained from the PDF integration of the flamelet library: Y˜ s = Ys (Z, χst )P˜ (Z, χst )d χst dZ (5) while the local mean temperature is calculated implicitly, using the mean species mass fractions and the static enthalpy solved from the energy equation [7]. For the joint PDF P˜ (Z, χst )in Eq. (5), statistical independence between Z and χ st is assumed, and a β-function is used for the PDF of Z while a δ-function for χ st . Therefore, transport equations of the first- and second-order moments of Z, i.e. Z˜ 2 , need to be solved additionally [19]. and Z 2.2. RIF model As mentioned before, the assumptions of constant pressure (Eq. (3)) and linear h–Z relation
(Eq. (2)) are invalid because of the high-Machnumber effects. However, the flamelet library is generated only once before the flow simulation in the SF model, and thus the local high-Mach-number effects are hard to be incorporated. Then the logical idea is to generate a local flamelet library for each point using different pressure values and h–Z relations, which leads to the RIF model, originally proposed by Peters et al. [17] to consider the flame transient effect. Here, this RIF model is introduced to supersonic combustion flows to fully incorporate the local high-Mach-number effects in the flamelet library. The main idea of the RIF model is to solve the flamelet equations in a coupled manner with the flow equations, as shown in Fig. 1. The RANS module provides parameters of the local flowfield point to the flamelet module. The flamelet equations are solved in the flamelet module based on these parameters and the established flamlelet library is then returned to calculate the mean species mass fractions. In the flamelet library for each point, only the mixture fraction varies and the scalar dissipation rate is fixed. Because of this coupling, it is possible to establish flamelet libraries including the local high-Mach-number effects. In the flamelet module, the pressure in Eq. (3) can be directly replaced by the local mean pressure p¯ provided by the RANS module, which implies that the local pressure fluctuation is ignored according to the fact that this small pressure perturbation would not lead to obvious influence on the species concentrations in the flamelet library [14]. While for the modification of h–Z relation in Eq. (2), Peters et al. [17] still kept the linear relation but changed the boundary condition parameters hF and hO according to the evolution of the pressure using the law of adiabatic compression. This method is inappropriate for supersonic flows because the variation of static enthalpy is caused not only by the adiabatic compression/expansion, but also the shockwave, viscous dissipation or wall heat transfer. However,
Please cite this article as: Z. Gao et al., Representative interactive flamelet model and flamelet/progress variable model for supersonic combustion flows, Proceedings of the Combustion Institute (2016), http://dx.doi.org/10.1016/j.proci.2016.06.184
ARTICLE IN PRESS
JID: PROCI
4
[m;July 15, 2016;11:21]
Z. Gao et al. / Proceedings of the Combustion Institute 000 (2016) 1–10
it is impossible to construct an accurate h–Z relation only given a local mean static enthalpy h˜ and mixture fraction Z˜ from the RANS module. Here, an approximate quadratic h–Z relation, instead of the linear relation, is assumed. Three parameters need to be identified to establish a quadratic function. In addition to the two boundary conditions in Eq. (4), the third condition comes from the local values of h˜ and Z˜ by assuming that h = h˜ when Z = Z˜ . Different points have different h˜ − Z˜ values, and thus each point would use its own quadratic h – Z relation. Finally, because the local p¯ and h˜ − Z˜ , calculated from the flow governing equations by the RANS module, include the mean high-Mach-number effects on the variations of pressure and static enthalpy, the species concentrations in the flamelet library would be directly influenced by these effects. In order to reduce the computational expense of the RIF model, the flamelet equations are solved only for the region where combustion occurs, and this region can be determined by the local value of the mixture fraction. For the Cheng et al. case [20] studied in the present paper, the CPU time for every iteration of the RIF model is about 1.92 times that of the SF model. 2.3. FPV model For the SF model, the species concentrations in the flamelet library are parameterized by Z and χ st , while Pierce and Moin [16] showed that χ st cannot describe the whole so-called S-shape curve representing the solution of the flamelet equations. The FPV model was proposed in [16] to overcome this problem. A progress variable parameter , which can uniquely identify each single flame state on the whole S-shape curve, is introduced to replace χ st . is defined as the value of a progress variable C at Z = Zst, and is independent of Z. Given a Z and C, the reaction progress parameter and its corresponding flame state can be determined [21]. Hence, mean species mass fractions are calculated as Y˜ s = Ys (Z, C )P˜ (Z, C )dZdC (6) The joint PDF is written as P˜ (Z, C ) = P˜ (C|Z )P˜ (Z )
(7)
where P˜ (Z) is also given by an assumed β-function form, while the conditional PDF P˜ (C|Z ) is expressed using a δ-function [16]: (8) P˜ (C|Z ) = δ C − C| Z C| Z can be obtained by some single flamelet solution determined by the following constraint: C˜ = C| ZP˜ (Z )dZ (9)
Hence, in addition to the transport equations 2 , another one of the mean progress of Z˜ and Z ˜ variable, C , is also needed, and the mean chemical source term in its transport equation is calculated by ω¯˙ C = ω˙ C (Z, C )P˜ (Z, C )dZdC (10) where ω˙ C (Z,C) is stored in the flamelet library together with Ys (Z,C). The applications of this FPV model in supersonic combustion flows are very limited [13–15], and this model has similar problems with the SF model, that is, how to include the local high-Machnumber effects into the flamelet library. However, it is not easy to achieve this aim in the framework of the above RIF model. It can be seen in Fig. 1 that only one single flamelet solution determined by χ˜ needs to be solved for every point. If using the FPV model to replace the SF model in Fig. 1, for every point, the whole S-shape curve should be solved and then one of these flamelet solutions is chosen according to Eq. (9), which would result in unacceptable computational complexity. Hence, for the FPV model, the high-Mach-number effects are not introduced into the species mass fractions in the flamelet library. However, it is still necessary to introduce some compressible correction for the chemical source terms of the progress variable according to Saghafian et al [14]. Instead of correcting these source terms in the flamelet library, a compressible rescaling was proposed for the tabulated source term ω¯˙ C in [14] based on an approximate method for calculating temperature from internal energy. However, this rescaling is an empirical expression to imitate the Arrhenius behavior, and includes a series of parameters which can only be determined in the approximate method for calculating temperature from internal energy. So this rescaling cannot be implemented if one does not use this approximate method. Here, the idea to include the local high-Mach-number effects through rescaling the tabulated source term is adopted, but a new compressible rescaling form is proposed as follow: ¯ T˜ ω˙ test ρ, ω¯˙ C = testC f l (11) ω˙ C (ρ , T f l ) ω¯˙ C0 where ω¯˙ C0 is the tabulated source term from the flamelet library not including any high-Machnumber effect. Here, the rescaling coefficient is obtained using two testing source terms of the progress variable: ω˙ Ctest (ρ, ¯ T˜ ) and ω˙ Ctest (ρ f l , T f l ), which are calculated using the same tabulated species concentrations but different density and temperature. The former one utilizes the local mean density ρ¯ and temperature T˜ solved from RANS while the latter one is obtained based on the tabulated density ρ fl and temperature Tfl , averaged from the flamelet library using the joint
Please cite this article as: Z. Gao et al., Representative interactive flamelet model and flamelet/progress variable model for supersonic combustion flows, Proceedings of the Combustion Institute (2016), http://dx.doi.org/10.1016/j.proci.2016.06.184
JID: PROCI
ARTICLE IN PRESS Z. Gao et al. / Proceedings of the Combustion Institute 000 (2016) 1–10
[m;July 15, 2016;11:21]
5
3.1. Case description
Fig. 2. Computational and experimental ignition delay times of a diluted hydrogen/air mixture.
PDF just like the species mass fractions in Eq. (6). Note that ρ fl and Tfl only reflect the influences by the combustion, while ρ¯ and T˜ computed from the flow governing equations could also include the local high-Mach-number effects. Hence, the ratio of the two testing source terms is supposed to reflect the local high-Mach-number effects on the tabulated source term. It is believed that this rescaling is more physical and universal than the one in [14], even though some additional computation is introduced for calculating the testing source terms. 3. Numerical results and analysis The following numerical simulations are performed using a finite-difference code, named ACANS, developed by the authors. The reliability of ACANS has been validated by a variety of simulations [11,22–24]. The flow governing equations of continuity, momentum and energy, together with 2 are solved for the SF the equations of Z˜ and Z and RIF models, while another equation of C˜ is added for the FPV model. The inviscid flux vectors are discretized using the second-order AUSMDV [25] scheme and the viscous flux vectors by the central difference scheme. The implicit LU-SGS method [26] is employed for the time marching and the turbulence model is Menter’s SST model [27]. A hydrogen/air reaction mechanism including 9 species (H2 , O2 , H2 O, H, O, OH, HO2 , H2 O2 , N2 ) and 19 elementary reactions [28] is adopted here, and the species of H2 O is selected as the progress variable for the FPV model. Figure 2 compares the ignition delay times of a diluted hydrogen/air mixture calculated by the adopted reaction mechanism with the experimental data form Bhaskaran et al. [29], and good agreements can be seen.
An axisymmetric supersonic combustion experiment conducted by Cheng et al. [20] is selected to evaluate the SF, RIF and FPV models. Figure 3 shows the configuration of the experiment, where a vitiated air stream (oxygen-enriched air and hydrogen are adopted in the vitiated heater) and a hydrogen stream are injected from different nozzles and an axisymmetric flame is generated downstream. The axisymmetric governing equations are solved on a 2-D structural mesh. Table 1 gives the free stream conditions of the air and hydrogen streams, where p and T are the static pressure and temperature, respectively. Moreover, according to Cheng et al. [20], the uncertainty for the total pressure is ± 4%, while those for the gas temperature and major species concentrations are ± 5%. In the simulations, the flowfiled is initialized using the ambient air conditions: p0 = 1 atm, T0 = 288 K, U0 = 0 m/s, and thus the initializing turbulent kinetic energy and turbulent viscosity are both 0. 3.2. Results and analysis The variations of the mean pressure with x and the mean static enthalpy with mean mixture fraction along y = 0 m, obtained by the FPV model, are first shown in Fig. 4. They are used to qualitatively show that the pressure deviates from the constant value adopted in the generation of the flamelet library and the h˜ − Z˜ relation is also different from a linear relation. However, the deviations are not very large because of the parallel injection scheme in this case which would not induce strong shockwaves. Then, the temperature contours obtained by the three models are illustrated in Fig. 5. It is found that the RIF model does not lead to obvious changes on the mean temperature field by contrast with the results of the SF model, implying that the species concentrations in the flamelet library are not sensitive to the high-Mach-number effects. While the adoption of the FPV model remarkably delays the auto-ignition position compared to the other two models. Actually, experimental observation showed that the ignition occurs at around x/d = 25 [20]. Clearly, both the SF and RIF models predict the ignition too much earlier, and the inclusion of highMach-number effects in the RIF model has little effect on the auto-ignition simulation. While for the FPV model, the combustion initiates at around x/d = 23 to 24, pretty close to the observed result, which means that the FPV model can also effectively improve the auto-ignition prediction for supersonic combustion flows. Figure 6 shows the temperature distributions calculated by the three models at four streamwise locations, together with the corresponding experimental data. Compared to the SF model, the RIF
Please cite this article as: Z. Gao et al., Representative interactive flamelet model and flamelet/progress variable model for supersonic combustion flows, Proceedings of the Combustion Institute (2016), http://dx.doi.org/10.1016/j.proci.2016.06.184
ARTICLE IN PRESS
JID: PROCI
6
[m;July 15, 2016;11:21]
Z. Gao et al. / Proceedings of the Combustion Institute 000 (2016) 1–10
Fig. 3. Geometric configuration of Cheng et al. experiment.
Table 1 Free stream conditions for Cheng et al. case ( p and T are both static variables).
Air stream Hydrogen stream
M
p/kPa
T/K
YH2
YO2
YH2 O
YOH
YN2
2 1
107 112
1250 545
0 1
0.245 0
0.175 0
6.65e − 4 0
0.58 0
Fig. 4. Distributions of (a) pressure with x and (b) static enthalpy with mixture fraction along y = 0 m.
Fig. 5. Temperature contours obtained by the SF, RIF and FPV models.
Please cite this article as: Z. Gao et al., Representative interactive flamelet model and flamelet/progress variable model for supersonic combustion flows, Proceedings of the Combustion Institute (2016), http://dx.doi.org/10.1016/j.proci.2016.06.184
ARTICLE IN PRESS
JID: PROCI
[m;July 15, 2016;11:21]
Z. Gao et al. / Proceedings of the Combustion Institute 000 (2016) 1–10 3000
SF RIF FPV Experiment
2000
3000
SF RIF FPV Experiment
2500
x/d=10.8
2500
x/d=21.5
T/K
T/K
T/K
x/d=43.1
2000
2000
1500
1500
SF RIF FPV Experiment
2500
x/d=32.3
2000 1500
3000
SF RIF FPV Experiment
T/K
2500
7
1500
1000
500 -10
-5
0
y/d
5
1000
1000
1000
500
500
500
-10
10
-5
0
y/d
5
10
-10
-5
0
y/d
5
10
-10
-5
0
y/d
5
10
40
45
Fig. 6. Temperature distributions at x/d = 10.8, 21.5, 32.3 and 43.1. 1
0.6
0.6
0.4
H2
0.4
H 2O
0.2
0.2 0
SF RIF FPV Experiment
Mole fraction of H2O
Mole fraction of H2
SF RIF FPV Experiment
0.8
0
5
10
15
20
x/d
25
30
35
40
45
0
0
5
10
15
20
x/d
25
30
35
Fig. 7. Mole fraction distributions of H2 and H2 O along y = 0 m.
model induces slight decreases of the temperature peak at x/d = 10.8 and 43.1, while small increases at x/d = 21.5 and 32.3. However, these changes of temperature are all within 40 K, and some are too small to be resolved in Fig. 6. Moreover, comparisons with the measured data at x/d = 10.8 again indicate that the SF and RIF models predict the ignition earlier. While for the FPV model, the temperature achieves a much better agreement with the experimental data at x/d = 10.8. Asymmetric distribution is noticed for the experimental data at x/d = 21.5, where the maximum temperature is close to 2000 K on the positive-y side while only around 1500 K on the other side. The FPV model predicts a temperature distribution in good accordance with the measured data on the negative-y side while the results by the SF and RIF models are close to the experimental data on the positive-y side. At x/d = 32.3 and 43.1, the FPV model shows consistent results with the other two models. It is also seen in Fig. 6 that the computations obviously underestimate the jet diameter at downstream locations, while it is believed that this is not from the combustion simulations. Similar results can be found in previous simulations including RANS [30] and LES [31]. The distributions of the mole concentrations of H2 and H2 O along y = 0 m are depicted in Fig. 7. Similar to the temperature field, the RIF model leads to very limited changes on the concentrations
of H2 and H2 O, and the results obtained by the SF and RIF models show some deviations from the measured data. Due to a more accurate autoignition prediction, the FPV model achieves much better agreements with the experimental data. In order to assess the effect of the compressible rescaling on the tabulated source term in the FPV model, the variation of the rescaling coefficient (the right side of Eq. (11)) along y = 0 m is shown in Fig. 8a, and also the temperature distributions obtained by the FPV model with/without the rescaling in Fig. 8b. It is seen that the rescaling coefficient takes on some oscillations due to the influence of the wave system generated between the air and hydrogen jets. In front of the ignition (x/d ≈ 24), it can be found in Fig. 4 that the local pressure is generally larger than the value adopted in the flamelet library and meanwhile the local static enthalpy is also higher than the value calculated from a linear relation with the mixture fraction, which makes the values of the rescaling coefficient larger than 1 in Fig. 8a. This amplification of the tabulated source term remarkably advances the auto-ignition, from around x/d ≈ 30 (without rescaling) to x/d ≈ 24 (with rescaling), which can be seen in Fig. 8b. Downstream of the ignition, the rescaling coefficient becomes smaller and less than 1 in some region. Hence, the temperature with rescaling at x/d ≈ 43.1 is a little lower than that without rescaling, and closer to the
Please cite this article as: Z. Gao et al., Representative interactive flamelet model and flamelet/progress variable model for supersonic combustion flows, Proceedings of the Combustion Institute (2016), http://dx.doi.org/10.1016/j.proci.2016.06.184
JID: PROCI
8
ARTICLE IN PRESS
[m;July 15, 2016;11:21]
Z. Gao et al. / Proceedings of the Combustion Institute 000 (2016) 1–10
Fig. 8. Distributions of (a) the rescaling coefficient and (b) temperature with/without scaling along y = 0 m.
Fig. 9. Mesh refinement study for the FPV model.
measured data. Obviously, the proposed compressible rescaling reasonably incorporates the influences of the local high-Mach-number effects on the tabulated source term of the progress variable and effectively improves the simulation accuracy of the FPV model for supersonic combustion flows. Above results show that the high-Mach-number effects produce very limited changes for the RIF model but considerable influences for the FPV model. Actually, the same conclusion can be drawn in another case [32], whose results are shown in the Appendix. The reason is that the high-Machnumber effects in the RIF model only affects the species concentrations in flamelet library, but has no direct effect on the local scalar dissipation rate that mainly controls which flamelet solution in the library is used to perform the PDF integration. However, for the FPV model, the high-Mach-number effects introduced by rescaling the tabulated source term would directly influence the local values of the progress variable and then change the local flamelet state. Additionally, it is shown in [14] that in the flamelet library, the chemical source terms are much more sensitive to the
high-Mach-number effects than the concentrations of major species. Then, it is not hard to understand that the high-Mach-number effects produce more influences on the FPV model. Finally, to ensure the grid independence of the above results, a mesh refinement study is performed for the FPV model. The mesh adopted in above simulations includes 37,757 grid points, while the refined mesh has 56,667 grid points. Figure 9 compares the temperature distributions at two streamwise positions and it is seen that almost the same results are obtained by the two meshes. 4. Conclusions In order to fully incorporate the local highMach-number effects into the flamelet library for the steady flamelet (SF) model, the representative interactive flamelet (RIF) model is introduced to supersonic combustion flows. In this RIF model, the flamelet equations are solved in a speciallydesigned coupling manner with the flow governing equations, through which flamelet library including local high-Mach-number effects could be established for each point in the combustion region.
Please cite this article as: Z. Gao et al., Representative interactive flamelet model and flamelet/progress variable model for supersonic combustion flows, Proceedings of the Combustion Institute (2016), http://dx.doi.org/10.1016/j.proci.2016.06.184
JID: PROCI
ARTICLE IN PRESS Z. Gao et al. / Proceedings of the Combustion Institute 000 (2016) 1–10
[m;July 15, 2016;11:21]
9
Fig. A1. Geometric configuration of DLR experiment.
Fig. A2. Temperature distributions at x = 108 mm and 216 mm.
The application in a supersonic combustion experimental case indicates that the RIF model could induce some changes on the temperature and major species concentrations compared to the results of the SF model, but these changes are very limited. Also, the RIF model has little influence on the auto-ignition simulation. Overall, the RIF model does not show obvious improvement for the studied case, which implies that the species concentrations in the flamelet library are not sensitive to the local high-Mach-number effects. The flamelet/progress variable (FPV) model is also applied in supersonic combustion flows. A new compressible rescaling form is proposed to rescale the tabulated source term of the progress variable, so as to consider the local high-Mach-number effects. The numerical results of the same case using the FPV model achieve much better agreements with the experimental data than those of the SF and RIF models, including the temperature, major species concentrations and auto-ignition position. It is also indicated that the proposed compressible rescaling reasonably incorporates the influences of the high-Mach-number effects on the tabulated source term of the progress variable and effectively
improves the simulation accuracy of the FPV model for supersonic combustion flows. Acknowledgments This research is supported by the National Natural Science Foundation of China through the Grant 11202014 and also partially supported by the China Scholarship Council (File no.: 201406025016). Appendix Another experimental case is also simulated using the SF, RIF and FPV models for further validation, and the results are shown in this Appendix. This case is a hydrogen-fueled scramjet combustor experimentally studied by Waidmann et al. [32] in German Aerospace Center (DLR). The experimental configuration of the combustor is shown as Fig. A1, and a 2D approximation for this case can be employed according to previous studies [7,11]. Also, the free stream conditions of the air stream
Please cite this article as: Z. Gao et al., Representative interactive flamelet model and flamelet/progress variable model for supersonic combustion flows, Proceedings of the Combustion Institute (2016), http://dx.doi.org/10.1016/j.proci.2016.06.184
JID: PROCI
10
ARTICLE IN PRESS
[m;July 15, 2016;11:21]
Z. Gao et al. / Proceedings of the Combustion Institute 000 (2016) 1–10
and hydrogen jet can be found in [7,11]. Here, the temperature distributions at two streamwise positions, together with the corresponding experimental data, are shown in Fig. A2 to examine the accuracy of different models. It is seen that for this case, the RIF model also induces limited changes on the temperature results compared to the SF model, and the FPV model achieves better agreements with the measured data. Comparing the results of the FPV model with/without the compressible rescaling, it is found that the rescaling enhances the temperature at both the two positions, which makes the temperature results of the FPV model closer to the experimental data. References [1] F. Ladeinde, AIAA 2009-127 (2009). [2] D. Veynante, L. Vervisch, Prog. Energy Combust. Sci. 28 (3) (2002) 193–266. [3] H. Pitsch, Annu. Rev. Fluid Mech. 38 (2006) 453–482. [4] H. Möbus, P. Gerlinger, D. Brüggemann, Combust. Flame 132 (1–2) (2003) 3–24. [5] F. Génin, S. Menon, AIAA J. 48 (3) (2010) 526–539. [6] M. Picciani, in: Supersonic Combustion Modeling Using the Conditional Moment Closure Approach, Ph.D. thesis, Cranfield University, Cranfield, UK, 2014. [7] M. Oevermann, Aerosp. Sci. Technol. 4 (7) (2000) 463–480. [8] L.L. Zheng, K.N.C. Bray, Combust. Flame 99 (2) (1994) 440–448. [9] V. Sabel’nikov, B. Deshaies, L.F. Figueira da Silva, Combust. Flame 114 (3–4) (1998) 577–584. [10] Z. Fan, W. Liu, M. Sun, Z. Wang, F. Zhuang, W. Luo, Sci. China-Technol. Sci. 55 (1) (2012) 193–205. [11] Z. Gao, J. Wang, C. Jiang, C.H. Lee, Combust. Theory Model. 18 (6) (2014) 652–691.
[12] M. Berglund, C. Fureby, Proc. Combust. Inst. 31 (2) (2007) 2497–2504. [13] R. Pecˇ nik, V.E. Terrapon, F. Ham, G. Iaccarino, H. Pitsch, AIAA J. 50 (8) (2012) 1717–1732. [14] A. Saghafian, V.E. Terrapon, H. Pitsch, Combust. Flame 162 (3) (2015) 652–667. [15] A. Saghafian, L. Shunn, D.A. Philips, F. Ham, Proc. Combust. Inst. 35 (2) (2015) 2163–2172. [16] C.D. Pierce, P. Moin, J. Fluid Mech. 504 (2004) 73–97. [17] H. Pitsch, H. Barths, N. Peters, SAE-962057 (1996). [18] N. Peters, Prog. Energy Combust. Sci. 10 (3) (1984) 319–339. [19] N. Peters, Turbulent Combustion, Cambridge University Press, 2000. [20] T.S. Cheng, J.A. Wehrmeyer, R.W. Pitz, O. Jarrett Jr., G.B. Northam, Combust. Flame 99 (1) (1994) 157–173. [21] M. Ihme, C.M. Cha, H. Pitsch, Proc. Combust. Inst. 30 (1) (2005) 793–800. [22] Z. Gao, C. Jiang, S. Pan, C.H. Lee, AIAA J. 53 (7) (2015) 1949–1968. [23] Z. Gao, C. Jiang, C.H. Lee, Appl. Therm. Eng. 92 (2016) 62–70. [24] Z. Gao, C. Jiang, C.H. Lee, Sci. China-Technol. Sci. 56 (10) (2013) 2501–2515. [25] Y. Wada, M.S. Liou, SIAM J. Sci. Comput. 18 (3) (1997) 633–657. [26] S. Yoon, A. Jameson, AIAA J. 26 (9) (1988) 1025–1026. [27] F.R. Menter, AIAA J. 32 (8) (1994) 1598–1605. [28] T. Poinsot, D. Veynant, Theoretical and Numerical Combustion, 2nd ed., R.T. Edwards Inc., Philadelphia, USA, 2005, p. 10. [29] K.A. Bhaskaran, M.C. Gupta, T. Just, Combust. Flame 21 (1) (1973) 45–48. [30] R.A. Baurle, S.S. Girimaji, Combust. Flame 134 (1–2) (2003) 131–148. [31] P. Boivin, A. Dauptain, C. Jiménez, B. Cuenot, Combust. Flame 159 (4) (2012) 1779–1790. [32] W. Waidmann, F. Alff, M. Böhm, U. Brummund, W. Clauβ, M. Oschwald, Space Technol 15 (6) (1995) 421–429.
Please cite this article as: Z. Gao et al., Representative interactive flamelet model and flamelet/progress variable model for supersonic combustion flows, Proceedings of the Combustion Institute (2016), http://dx.doi.org/10.1016/j.proci.2016.06.184