Laminar near-wall combustion: Analysis of tabulated chemistry simulations by means of detailed kinetics

International Journal of Heat and Fluid Flow 70 (2018) 259–270

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International Journal of Heat and Fluid Flow journal homepage: www.elsevier.com/locate/ijhff

Laminar near-wall combustion: Analysis of tabulated chemistry simulations by means of detailed kinetics

T

⁎

Sebastian Ganter ,a, Christina Straßackerb, Guido Kuennea, Thorsten Meiera, Arne Heinricha, Ulrich Maasb, Johannes Janickaa a b

Institute of Energy and Power Plant Technology, TU Darmstadt, Jovanka-Bontschits-Straße 2, Darmstadt 64287, Germany Institue of Technical Thermodinamics, Karlsruhe University (TH), Kaiserstrasse 12, Karlsruhe 76128, Germany

A R T I C LE I N FO

A B S T R A C T

Keywords: Premixed Methane FGM REDIM Flame–wall-interaction Side-wall-quenching

Chemistry pre-tabulation is well suited to include information of detailed reaction kinetics at reasonable computational costs to allow for simulations of realistic devices. In order to evaluate its accuracy in the near wall region, a study with numerical simulations of flame–wall-interaction is performed in this work. A laminar sidewall-quenching scenario is considered to judge on the prediction of the global flame behavior as well as on local species formation of practical relevance. The configuration considered represents a subsection of a side-wallquenching burner introduced recently with the purpose of numerical validation in mind. The measured temperature and carbon-monoxide concentration are used to identify deficiencies of the tabulated chemistry approach. Furthermore, detailed chemistry simulations are carried out to identify the root cause for those deficiencies. The corresponding analysis is based on the transformation of the species transport equation into the composition space where the physical significance of the scalar dissipation rates provides clear indications regarding the pre-tabulation assumptions. The evaluation of individual terms allows to quantify the interaction of flamelets in the near-wall region where diffusive fluxes cause a departure from the presumed manifold. Based on this analysis, improvements are then suggested. First, as a proof of concept, the direct tabulation of the thermochemical states obtained by the detailed chemistry simulation is applied to evaluate whether the reduction to controlling variables is in general possible in such a physical scenario. Second, as an alternative way of pretabulating, the reaction-diffusion-manifold (REDIM) approach is then adopted. By building the REDIM based on a gradient estimate from a computationally inexpensive transient one-dimensional flame–wall-interaction simulation it is possible to obtain realistic dissipation rates without a-priory knowledge. By this approximation a significant gain in prediction is achieved when compared to the original tabulation.

1. Introduction In many technical combustion systems, the reaction zone approaches the enclosing walls rendering the flame–wall-interaction (FWI) an important area of research. In these devices, the flame can be in direct contact with a relatively cold wall, which can have an influence on the efficiency of the combustor as well as on the pollutant formation (Poinsot and Veynante, 2005). The phenomena become of increasing importance with e.g. downsizing concepts for internal combustion engines (ICE) (Dreizler and Böhm, 2015) or lean burn technologies within aero engines (Lazik et al., 2008). Within scientific research as well as industrial developments, the numerical simulation represents a powerful tool of increasing importance. However, current capabilities enforce a trade-off regarding the accuracy, computational effort and the problem size considered. ⁎

Besides the well-known turbulent closures required within computational fluid dynamics (CFD) of complex geometries, the chemistry treatment causes severe restrictions. At this, the application of a detailed reaction mechanism represents the most accurate one but is limited to simple domains since it requires to solve for all relevant species usually interacting in hundreds of elementary reactions. Accordingly, this approach has been exclusively applied to FWI in generic configurations to obtain a fundamental insight and understanding of the underlying processes. Numerous simulations covered the transient one-dimensional head-on-quenching (HOQ) scenario, which mimics the situation found in ICE where the flame approaches the wall with a parallel alignment. Valuable knowledge has been gained by investigating processes like unburnt hydrocarbon formation, heat fluxes, surface reactions, and the relevance of different transport phenomena depending on parameters like the fuel employed or the wall

Corresponding author. E-mail address: [email protected] (S. Ganter).

https://doi.org/10.1016/j.ijheatfluidflow.2018.02.015 Received 27 October 2017; Received in revised form 26 February 2018; Accepted 26 February 2018 0142-727X/ © 2018 Elsevier Inc. All rights reserved.

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the near wall evolution of the temperature and CO concentration as a reference. The first simulations of this configuration have been carried out by Heinrich et al. (2017a) by means of a tabulation approach. They obtained satisfactory results regarding the flame shape and temperature distributions but significant errors considering the CO prediction. On the other hand, Ganter et al. (2017), who simulated only a sub-domain of the burner covering the near-wall region which allowed for the application of detailed chemistry, were able to match the measurements very closely. Accordingly, the simulation with pre-tabulation fails to predict species like CO due to the chemistry approximation, which we will detail in this work. We consider the most obvious choice of premixed tabulations based on a progress variable (Y ) and the enthalpy (h) to account for the heat losses. Specifically the objectives are:

temperature (Ezekoye et al., 1992; Wichman and Bruneaux, 1995; Popp et al., 1996; Popp and Baum, 1997; Hasse et al., 2000). Likewise, similar studies have been carried out for the steady two-dimensional sidewall-quenching (SWQ) (Andrae et al., 2002; 2008). Utilizing significant computational resources, Gruber et al. (2010, 2012, 2015) investigated the FWI in a turbulent channel flow using direct numerical simulations (DNS) and a detailed hydrogen reaction mechanism. The studies provided insight into the flame-turbulence regimes in the near wall region and the mechanism of boundary layer flashback. To enable the simulation of more complex geometries or even real devices, the chemistry has to be approximated in order to significantly lower the computational effort by reducing the number of variables computed and the spatial and temporal resolution requirements. A common approach is to use reduced mechanisms. They can be derived as a subset of the full mechanism by identifying the rate-determining reactions. Besides such mathematical reductions, often even just single step (or very few steps) are used to represent the chemistry. Due to the large loss of information, it is necessary to adjust their parameters to meet certain physical properties like the flame speed or its temperature. An alternative to reduced mechanisms is to use lookup tables. Such a lookup table contains all required thermo-chemical states parameterized based on only a few controlling variables that have to be solved for in the CFD. By pre-computing this chemistry database by means of the full reaction mechanism it enables to include valuable information of the underlying detailed chemistry into the CFD. However, obviously, with the type of detailed chemistry simulation conducted in advance, assumptions have to be made. If the actual physical situation differs from the pre-tabulation scenario, errors are introduced. The most basic example of pre-tabulation is the parameterization onto the mixture fraction f for the prediction of non-premixed flames (see e.g. Burke and Schumann, 1928; Zel’dovich, 1949; Williams, 1985; Peters, 2000). If the strain rate in the real flow is different from the detailed chemistry simulation employed for the pre-tabulation, errors in the burnt gas temperature and species arise. This can be accounted for by introducing the strain by means of the scalar dissipation rate of the mixture fraction as an additional table dimension (Peters, 1984, 1986). If premixing takes place the approach fundamentally fails due to its intrinsic mixed is burnt assumption and it is required to employ premixed flames for the pre-tabulation by employing a reaction progress variable Y (Gicquel et al., 2000; van Oijen and de Goey, 2000). This approach in turn can be extended for inhomogeneous systems by adding the mixture fraction as a second controlling variable (see e.g. Oijen, 2002; Ketelheun et al., 2009). However, again errors are introduced based on the dissipation rate of the latter. Accordingly, pretabulation can introduce errors as exemplified by the scalar dissipation rate or completely fail when the selected controlling variables cannot even approximate the physical situation as required by the mixed is burnt assumption. Hence, for an accurate prediction, the required apriori knowledge and table dimensions can represent a significant challenge for its application. This is demonstrated by the multidimensional flamelet generated manifold approach of Nguyen et al. (2009) which requires a five-dimensional table for a physical situation that can be described by only two variables, i.e. besides the two variables the individual scalar dissipation rates as well as their cross scalar dissipation rate is required for the full description. Building and employing such a table is rather difficult and therefore generally less universal tables are utilized and certain errors are accepted. In this work, we want to analyze chemistry tabulation approaches in the context of FWI. In contrast to the above-mentioned scenario where the alignment of concentration gradients determines the suitability and accuracy of the approach, within FWI the thermal boundary layer can cause a departure from the pre-computed states. For the analysis we consider a laminar methane-air SWQ configuration. The burner has been investigated experimentally by Jainski et al. (2017b) providing

• Conduct simulations using tabulation as well as detailed chemistry •

to show the root cause of the deficiencies. This is done by considering the processes in composition space where departures from the exact equation can be quantified by means of the scalar dissipation rates. Introduce an alternative approach that accounts for the enthalpy fluxes in the pre-tabulation process to improve the prediction quality for the FWI without increasing neither the tabulation nor the simulation effort noticeably.

In the following, Section 2 outlines the configuration considered followed by Section 3 providing information on the numerics including the detailed and tabulated chemistry. Section 4 then contains the results and the error analysis of the tabulation. Finally, in Section 5, improved tabulation techniques are suggested and evaluated. At the end a summary is given.

2. Configuration The configuration that is analyzed in this work is sketched in Fig. 1. A stoichiometric methane-air mixture issues from the nozzle at ambient conditions (p = 1 atm, T = 300 K) and passes a rod (∅ 1 mm) where a V-flame stabilizes. The nozzle flow Reynolds number is about 5000. The left flame branch approaches the wall at an angle of approximately 10° where the SWQ takes place. The wall temperature slightly varies along the water-cooled wall and is estimated to be between 300 and 350 K based on thermocouple measurements beneath the wall surface (Jainski et al., 2017a; 2017b). The burner has a thermal power of approximately 9.3 kW. As shown by Ganter et al. (2017), the relevant processes may be analyzed considering a reduced two-dimensional sub-domain (gray in Fig. 1) which enables the use of detailed chemistry simulations. The sub-domain, which covers about 80% of the considered flame branch, is magnified on the right of Fig. 1. Based on the experimental estimation, the wall is assumed to have a constant temperature of 300 K. Catalytic effects were not included since they are negligible at these wall temperatures (Popp and Baum, 1997). The reduction of the 3D experimental configuration implies the usage of a generic parabolic inflow velocity profile, which is visualized in Fig. 1 by white arrows on blue ground. The maximum velocity of the fresh mixture is 1.7 m/s. Since no rod is included in the 2D sub-domain, the flame is stabilized by injecting hot exhaust gas under equilibrium conditions (T = 2202 K) in a 0.5 mm wide section of the inlet as marked with red color in Fig. 1. The velocity of the hot gas was set to 3.81 m/s to compensate partially for the difference in density of the fresh and the burned inlet gases. Zerogradients boundary conditions were applied at all outlets for velocity, enthalpy and species, which result in the freely developed velocity profiles given in Fig. 1.

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u [m/s] 5 4 3 2 1

T(K)

Outlet

2200

2000

1800

19 mm PIV + OHLIF

1600

WALL

20 mm

CO + T d c b a

Outlet

1400

1200

1000

ROD 10 mm

800

16 mm

23.5 mm

20 mm

30 mm

PIV + OHLIF

Fig. 1. SWQ-Burner and measurement regions with the numerical sub-domain magnified on the right. The latter shows the simulated temperature distribution, the prescribed inlet velocity profile of the fresh mixture at 300 K (blue with white arrows) and burned gas at 2202 K (red with white arrows). Also shown is the flame shape by an isoline of the CO2 source term (black) and streamlines (black dashed). The resulting velocities at the outlets are represented by blue arrows. The four black horizontal lines near the flame tip denote the extraction lines for the CO and temperature data. Thereby the positions (a), (b), (c) and (d) are defined by the vertical offsets to the flame–wallquenching point of −550, −50, 450 and 950 μ m, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

600

15 mm 400

40 mm

v [m/s] 6 mm

1 2 3

T (K)

YCO ( - )

2100

-500

-500

0.04

-1000

0.03

-1500

0.02

1500

-1500

1200

-2000

900

-2500

600

h ( kJ/kg )

h ( kJ/kg )

1800

-1000

-2000 0.01

-2500

300

0

0.05

0.1

0

YCO2 ( - )

0.05

0.1

YCO2 ( - )

Fig. 2. Illustration of the FGM table in the h-YCO2 space colored with the temperature (left) and the CO mass fraction (right). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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0.05

2200

position d

1600

YCO ( − )

T (K)

0.04

1000

0.03 0.02 0.01

400

0 0.05

2200

position c

1600

YCO ( − )

T (K)

0.04

1000

0.03 0.02 0.01

400

0 0.05

2200

position b

1600

YCO ( − )

T (K)

0.04

1000

0.03 0.02 0.01

400

0 0.05

2200

position a

1600

YCO ( − )

T (K)

0.04

Exp DC FGM

1000

0.03 0.02 0.01

400

0 0

1

2

3

4

0

1

y ( mm )

2

3

y ( mm )

4

400

1000

1600

2200

T (K)

Fig. 3. CO mass fraction profiles (left), temperature profiles (center) and their evolution in state space (right) for the positions a–d as marked in Fig. 1. The gray dots mark the individual measured values and the gray curve (right) marks the adiabatic detailed chemistry solution.

3. Numerical methods and modeling 3.1. The finite volume code

non-regular cells. Boundedness of scalar quantities is assured by the total variation diminishing scheme suggested by Zhou et al. (1995). The code uses a cell centered variable arrangement on a non-staggered grid with selective interpolation of the mass fluxes as proposed by Rhie and Chow (1983). The time integration is done with an explicit three-stage Runge–Kutta scheme (Houwen, 1972) combined with a pressure correction procedure to satisfy continuity. The flow field is described by the conservation equation for mass and momentum neglecting volume forces:

FASTEST

All simulations are performed with the academic CFD code FASTEST. The incompressible, variable density finite volume code uses a blockstructured boundary fitted hexahedral mesh. The spatial interpolation of the velocity is based on the Taylor series expansion of Lehnhäuser and Schäfer (2002) to maintain second order accuracy on 262

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approaches. Ganter et al. (2017) found that this assumption has only a minor influence on the results of the side-wall-quenching simulation and is not responsible for the major deficiencies of the Flamelet Generated Manifold (FGM) simulation results mentioned above. Accordingly, the transport equation for the mass fraction Yk of species k reads:

CO,cond.

0

0.5

1

∂ρYk ∂ ∂ ⎛ ∂Yk ⎞ (ρui Yk ) = ρD + ω˙ k + ∂t ∂x i ∂x i ⎝ ∂x i ⎠ ⎜

6

4 3

xq ( mm )

2 position d position c position b position a

0

(3)

where a constant average molar mass throughout the reaction is assumed which is well fulfilled for methane/air combustion (Somers, 1994). The transport coefficients (thermal conductivity λ, dynamic viscosity μ and species diffusion D) are computed according to the local conditions (Smooke and Giovangigli, 1991; Meier et al., 2013; Vreman et al., 2008). The chemical source term ω˙ k of the species k is defined as sum of the contribution of all elementary reactions Nr (Poinsot and Veynante, 2005). These contributions depend on the concentrations ci of the species i and the molar stoichiometric coefficients νij′ and νij″ of species i of reaction j for the forward reaction and the backward reaction, respectively. Furthermore, the contributions depend on the stoichiometric coefficient νkj of species j of reaction k. The forward reaction constant kf is modeled using the modified Arrhenius law (Laidler, 1996) whereas the backward reaction constant kb is calculated using the equilibrium constant Kc which depends on the Gibbs free energy change of reaction and the atmospheric pressure. With that, the chemical source term may be formulated as:

5

1

⎟

Nr

ω˙ k = Wk

Ns

ν′

Ns

ν″

∑ νkj ⎛⎜k f,j ∏ ci ji − kb,j ∏ ci ji⎞⎟ j=1

⎝

i=1

i=1

⎠

whereby the GRI mechanism (Smith et al., 1999) is used in this study. The transport equation for the enthalpy hs is solved in its sensible formulation:

-1

∂ρh s ∂ ∂ ⎛ λ ∂h s ⎞ ˙T + (ρui hs) = ⎜ ⎟ + ω ∂t ∂x i ∂x i ⎝ c p ∂x i ⎠

-2

hs = h −

∑ Δh f0k Yk

(6)

k

-4 0

1

2

ω˙ T = − ∑ Δh f0k ω˙ k

3

Further description of the detailed chemistry implementation can be found in Ganter et al. (2017), Meier et al. (2013) and Meier (2017).

Fig. 4. Relative error of the CO mass fraction based on FGM and DC simulation results according to Eq. (10).

∂ρui ∂ρ = 0 + ∂x i ∂t

+

∂uj ⎞ ∂p ∂ ∂ ⎛ ⎛ ∂ui 2 ∂uk ⎞ (ρuj ui ) = ρν ⎜ + ρν δji − ⎟ − ∂x i ∂x i ⎜ ⎝ ∂x j ∂x i ⎠ 3 ∂xk ⎟ ∂x j ⎝ ⎠

(7)

k

y ( mm )

∂t

(5)

where the sensible enthalpy hs and its source term ω˙ T can be formulated by the enthalpy h, the standard enthalpies of formation Δh f0k of all species, their mass fractions Yk and their source terms ω˙ k as:

-3

∂ρuj

(4)

3.3. Description of the flamelet-generated-manifold For the tabulated chemistry simulation, the FGM approach is applied as detailed by van Oijen and de Goey (2000) and Ketelheun et al. (2013, 2009). Beside the reaction progress variable, the enthalpy is used to account for the heat losses to the wall. The transport equations for the progress variable Y which was chosen to be the mass fraction of CO2 and the enthalpy as sum of sensible and formation enthalpy read:

(1)

(2)

where ρ, ui, ν and p denote the density, velocity, kinematic viscosity and pressure, respectively.

∂ρY ∂ ∂ ⎛ ∂Y ⎞ (ρui Y ) = ρD + ω˙ Y , + ∂t ∂x i ∂x i ⎝ ∂x i ⎠

(8)

∂ρh ∂ ∂ ⎛ λ ∂h ⎞ + (ρui h) = ⎜ ⎟. ∂t ∂x i ∂x i ⎝ c p ∂x i ⎠

(9)

⎜

3.2. Description of detailed chemistry The SWQ configuration is simulated using tabulated as well as detailed chemistry (DC). The latter involves solving transport equations for all species that are considered depending on the reaction mechanism applied. For the detailed chemistry simulation, unity Lewis number was assumed in order to match the assumptions made within the tabulation

⎟

The transport coefficients for the FGM simulation are calculated as for the detailed chemistry. The FGM-table-generation is based on a series of 1D flamelet calculations performed with the detailed chemistry solver CHEM1D 263

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Fig. 5. Evaluation of the individual terms of the transport equation in state space (12): term I (

), term II (

), term III (

), term IV (

), sum of all terms (

).

studies of Jainski et al. (2017b). The variation of the enthalpy level is realized by means of the burner stabilized flame configuration (van Oijen and de Goey, 2000; Ketelheun et al., 2013). The chemical states of the described calculation are then mapped onto the controlling variables. Besides density, viscosity and the source of CO2 which feed back to the transport equations, various additional scalars of interest are tabulated in order to compare them with experimental data.

(Somers, 1994) using unity Lewis number assumption. The laminar premixed and stoichiometric methane-air flamelets are isenthalpic i.e. the enthalpy does not vary along the spatial coordinate, however, their enthalpy level is lowered flamelet by flamelet. Fig. 2 shows some of these flamelets in the CO2-enthalpy-space being the table access parameters. As one can see, the flamelets are horizontal isenthalpic lines which do not interact in the initial table generation process. They are then combined to form the two-dimensional table. The color gives the temperature (left) that rises for increasing reaction progress and reduces with the enthalpy level. On the right of the figure, the CO mass fraction is visualized which was also measured in the experimental

4. Results and error analysis As mentioned, the two-dimensional SWQ configuration was 264

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YCO ( - )

h ( kJ/kg )

-500

0.04

-1000

0.03

-1500

0.02

-2000

0.01

-2500 0

0.05

0.1

0

0.05

YCO2 ( - )

0.1

YCO2 ( - )

Fig. 6. Mass fraction of CO in the state space for the FGM table (left) and the detailed chemistry based side-wall-quenching simulation (right).

∂2Yk ∂2Yk ∂ 2Y ∂Yk + 2χh,CO2 + χh 2k + ω˙ k − 0 = χCO2 ω˙ CO2 2 ∂ ∂ ∂ ∂ Y h h YCO2 ∂YCO CO 2   2          

simulated using detailed chemistry as well as FGM. The results of the simulation regarding CO mass fraction and temperature are shown in Fig. 3 along the wall normal lines (a–d) as indicated in Fig. 1. Beside the temperature (left column) and the CO2 mass fraction profile (central column), a representation in state space is added in the right column which provides an expanded view of the near-wall region. As previously done by Ganter et al. (2017) the results are compared with experimental data of Jainski et al. (2017b). Summarizing the findings given by Ganter et al. (2017), the FGM and the detailed chemistry capture temperature at all positions and also are in good agreement regarding CO at position (a). For position (b–d), however, the detailed chemistry performs significantly better compared with the FGM results with respect to the CO mass fraction. The remaining discrepancy between the detailed chemistry simulation and the measurements is related to unavoidable uncertainties in the boundary conditions, modeling (e.g. diffusivities Ganter et al., 2017) and experiments (Jainski et al., 2017b). However, as mentioned above, these cause only a slight deviation being not related to the systematic errors seen in the FGM simulation. Therefore, to enable a clear error identification, the detailed chemistry simulation is taken as the reference in the following. First the spatial extent of the deficiency is identified by the relative error of CO, which is depicted in Fig. 4. To avoid an influence of spatial deviations between the FGM and the detailed chemistry simulations the relative error is calculated based on YCO2 − h− conditioned YCO values according to:

ϵCO,cond. =

FGM DC YCO − YCO DC YCO

h,CO2

term III

term IV

(12) where

∂YCO2 ∂YCO2 ⎞ χCO2 = ρD ⎛ ⎝ ∂x i ∂x i ⎠ ∂h ∂h ⎞ χh = ρD ⎛ ∂ ⎝ x i ∂x i ⎠ ⎜

⎜

⎟

⎟

∂YCO2 ∂h ⎞ χh,CO2 = ρD ⎛ ⎝ ∂x i ∂x i ⎠ ⎜

⎟

(13)

denote the scalar dissipation rate of the YCO2 field, the enthalpy field and the cross scalar dissipation rate, respectively. From Eq. (12) it can be seen that even if the chemical state for this quenching scenario can be described by the two variables YCO2 and h the process in state space, however, is actually depending on five parameters. Namely the controlling variable themselves plus the three scalar dissipation rates. In order to quantify the magnitude of the terms in Eq. (12) and hence to identify the main source of error, each term is evaluated. Therefore, the CO2 mass fraction and enthalpy gradients, which occur in the scalar dissipation rates, are calculated by determining the bicubic interpolation function on the regular spatial grid followed by an evaluation of its analytical derivative. The second derivatives of the considered species Yk with respect to the state space coordinates are computed by locally fitting a two-dimensional polynomial of fourth degree whose second derivatives are also obtained analytically. Both, the interpolation function and the fitted function are generated using the MATLAB Curve Fitting Toolbox (MathWorks Incorporation, 2006). This evaluation procedure turned out to be a robust approach to face numerical inaccuracies that may be amplified by differentiation using lower order approaches. The remaining inaccuracies of this procedure are sufficiently small and allow for reasonable interpretation of the evaluated terms. The terms are plotted in Fig. 5 for the detailed chemistry and the FGM simulation. The results for the SWQT and the REDIM will be discussed later. As the transport equation (12) states, the sum of the four terms should vanish. Considering the sum, which is marked in black in Fig. 5, this holds for the detailed chemistry results for all positions. This finding confirms the general validity of the assumption expressed by Eq. (11). While position (a) is dominated by terms I and IV, at positions (b)–(d) terms II and III gain in significance. The FGM results meet Eq. (12) for position (a), where the flame is almost adiabatic. An increasing non-zero sum, however, is obtained for the positions (b) and (c) which only reduces again at position (d). Comparing the result of the FGM and the detailed chemistry at position (b) it becomes obvious that the main error arises from term III whose absolute value is considerably over-predicted. This finding can be explained by comparing the CO distribution in the state space for the detailed

(10)

That is, the relative error of CO is formed using CO values belonging to the same pair of YCO2 and h for both simulations. Fig. 4 shows the spatial distribution of the relative error revealing that it is restricted to the region where the FWI takes place. Considering the region, where the flame is adiabatic, i.e. below position (a), the CO prediction of the simulations agree perfectly. Approaching the region of heat loss, the error increases. Further downstream reducing values are found along with decreasing wall heat fluxes. Reviewing Fig. 3, the region of FWI presents a rise in CO whose physical cause could be explained in Ganter et al. (2017) by means of detailed chemistry. Since the FGM does not capture this, a prediction of CO within real combustion devices requiring chemistry reduction, is not possible. Hence, this study focuses on the origin of the tabulation specific error and presents improved tabulation approaches. Therefore, the species transport equation (3) is transformed using the assumption that is made within the FGM table generation, namely that the species composition Yk is only depending on the mass fraction YCO2 and enthalpy h which reads

Yk = Yk (YCO2, h)

term II

term I

(11)

For the considered steady configuration this leads to the species transport equation in state space: 265

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0.05

2200

position d

1600

YCO ( − )

T (K)

0.04

1000

0.03 0.02 0.01

400

0 0.05

2200

position c

1600

YCO ( − )

T (K)

0.04

1000

0.03 0.02 0.01

400

0 0.05

2200

position b

1600

YCO ( − )

T (K)

0.04

1000

0.03 0.02 0.01

400

0 0.05

2200

position a

1600

YCO ( − )

T (K)

0.04

DC FGM REDIM SWQT

1000

0.03 0.02 0.01

400

0 0

1

2

3

4

0

1

y ( mm )

2

3

y ( mm )

4

400

1000

1600

2200

T (K)

Fig. 7. CO mass fraction profiles (left), temperature profiles (center) and their evolution in state space (right) for the positions a–d as marked in Fig. 1.

term. The preliminary conclusion from these findings is, that scalar dissipation rates cannot be neglected during the table generation process when these have significant influence on the process within the configuration to be simulated.

chemistry and FGM as depicted in Fig. 6. As mentioned above, the scalar dissipation rate in enthalpy direction is implicitly assumed to be zero during the FGM table generation process, which inhibits CO diffusion in the enthalpy direction. Steep gradients of CO in the enthalpy direction are the consequence, which can be seen in Fig. 6 (left). By contrast, Fig. 6 (right) shows the CO for the detailed chemistry simulation with inherently correct scalar dissipation rates, which leads to smoothened CO gradients in enthalpy direction. Evaluating term III by means of the actual scalar dissipation rate χh of the side-wall-quenching simulation, which significantly differs from zero, together with the deficiently steepened CO leads to the over-prediction of the considered

5. Improved chemistry tabulation methods The following section aims first to clarify weather tabulation with two variables can capture the detailed chemistry results in principal. Therefor the detailed chemistry results are tabulated directly which represents a costly brute force method. Second, a more sophisticated 266

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Fig. 8. Relative error profile of temperature (left), CO mass fraction (center) and their evolution in state space (right) for the positions a–d as marked in Fig. 1.

simulation. The actual tabulation is done by a simple mapping of the chemical states on the controlling variables (YCO2 and h). The corresponding chemistry table will be referred to as Side-Wall-QuenchingTable (SWQT). Performing a SWQ simulation using this chemistry table leads to the CO and temperature results in physical space as well as in state space shown in Fig. 7. As can be seen, the SWQT results show an almost perfect congruence with the original detailed chemistry results for both quantities at all positions, which proves that tabulation in general may reach a high degree of prediction accuracy if the

and computationally inexpensive chemistry reduction method is applied and assessed, which represents a reasonable trade-off between prediction accuracy and computational costs.

5.1. Direct tabulation of the DC results The direct tabulation of the detailed chemistry simulation results provides a consequent method that inherently implies the correct scalar dissipation rates, since these could develop freely during the 267

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h ( kJ/kg )

time. The time scales of relatively slow chemical processes overlap with the physical processes like molecular transport and they are both accounted for in the context of the REDIM. Therefore, the REDIM is a reduced kinetic model that accounts for both, chemical kinetics and diffusion processes. The evolution equation of a general reacting system is given by a system of partial differential equations for the state vector Ψ = (h, p , w1/ M1, …, wn/ Mn ) where h represents the specific enthalpy, p the pressure and wi/Mi the specific mole fraction consisting of mass fraction wi and the molar mass Mi of the species i. According to the REDIM method the system state vector can be redefined as Ψ = Ψ (θ ) with the parametrization vector θ = (θ1, θ2, …, θm) that represents the local coordinates of the m-dimensional manifold (Bykov and Maas, 2007). In order to generate the manifold, the REDIM evolution equation (Maas and Bykov, 2011; Bykov and Maas, 2007) is integrated until a stationary state is reached which gives the REDIM-table. As mentioned before, a spatial gradient estimate ∇Ψ which is obtained from a one-dimensional head-on-quenching simulation is required during the integration. The state variables enthalpy and the species CO2 are used to parametrize for the estimation of gradients (similar as in Steinhilber and Maas, 2013). For the modeling of the molecular transport equal diffusivities and unity Lewis number are assumed which was justified before in Section 3.2. Moreover, Dirichlet boundary conditions are applied at the boundary of the REDIM. It should be mentioned, that these assumptions are made for simplicity and the use of more sophisticated boundary conditions (Neagos et al., 2017; Straßacker et al., 2018) and transport models (Maas and Bykov, 2011; Straßacker et al., 2018) is possible. As is the case with FGM, the computed REDIM table contains the source terms and the transport properties. In order to implement the reduced model equation in physical variables, the REDIM is re-parametrized in terms of CO2 and enthalpy and the source term as well as the diffusion coefficients are appropriately reformulated. This enables using the same controlling variable as for the FGM and the SWQT tables. A qualitative comparison of the FGM and REDIM table with detailed chemistry results is given in Fig. 9 by means of the CO mass fraction isolines in the controlling variable space. As already indicated in Fig. 6, in comparison to the FGM, the detailed chemistry reflects the physical diffusion process in the enthalpy direction leading to rather vertically aligned isolines. This behavior is now also visible in the chemistry table constructed with the REDIM technique with a moderate over-estimation of CO in the near-wall region (i.e. at low enthalpies). As we will further discuss below, this is likely caused by constructing the REDIM using gradients of a HOQ simulation which are steeper than in the laminar SWQ flame considered here. This gradient being represented by scalar dissipation rate in Eq. (12) then also results in a non-zero sum right hand side as visible in Fig. 5 (fourth column) for positions (b) and (c). To judge on the performance of the REDIM in physical space, results have been added to Figs. 7 and 8. Within an overall assessment the REDIM results in Fig. 7 do not achieve the prediction quality of the SWQT but comes considerably closer compared with the FGM result. Comparing the relative errors of CO (Fig. 8) reveals the REDIM to be more accurate than the FGM. The largest errors occur directly at the wall for all positions. The error of the REDIM is about a quarter at position (a), less than an eighth at positions (b) and (c) and about a sixth at position (d) compared to the FGM. Considering the SWQT simulation it can be seen that the error is vanishingly small except for the near wall region at position (a) where interpolation errors are presumed to become visible as mentioned above. Here, the REDIM actually seems closer to the detailed chemistry but a detailed analysis (not shown) revealed that it benefits from an error compensation resulting from the predicted flame position on the one hand and its shape on the other hand while the SWQT only has a slight error in the former. So far the results showed, that by incorporating approximate gradients (scalar dissipation rates) into the tabulation procedure significant improvements are possible. Especially for minor species like CO this is necessary where our FGM table fails while it was still reasonable

0.0 3 0.0 8 2 0.0 8 1 0.0 8 08

YC

O

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-1000

-2000

FGM DC REDIM

-3000 0

0.05

0.1

YCO2 ( - ) Fig. 9. Isolines of the CO mass fraction for the FGM and the REDIM table compared with detailed chemistry results. Slight shading marks out the DC and the REDIM manifold. Dark shading marks out the FGM manifold.

assumptions made are reasonable. Fig. 8 gives the relative error based on the detailed chemistry results defined by:

ϵCO =

DC YCO − YCO DC YCO

(14)

Only a slight deviation is observed for CO at position (a). However, this error is only present at this position where gradients are large while the absolute CO concentration is low. This amplifies the relative deviation by only marginal differences between the simulations in the predicted flame position. Those in turn could be caused by interpolation errors due to a locally insufficient resolution within the tabulation procedure. Furthermore, Fig. 5 (third column) demonstrates that the SWQT simulation results fulfill the transport equation (12). However, since this chemistry reduction procedure is based on a costly two-dimensional detailed chemistry simulation there is no overall reduction of computing time. 5.2. REDIM for near-wall combustion Accounting for realistic scalar dissipation rates at low computational effort is the objective of the reaction-diffusion-manifold (REDIM) approach, which is developed by Bykov and Maas (2007). Thereby the advantage of the REDIM approach arises from the fact that rough estimations for scalar dissipation rates already lead to a significantly improved chemistry table compared to the FGM table outlined above. However, it remains to obtain these estimations which are unknown in advance for the configuration of interest. Since this is the general case when considering complex geometries that require the use of chemistry tabulation, one has to consider a simple configuration which resembles the relevant physics as close as possible. Accordingly, in this work the estimations base on a one-dimensional HOQ simulation performed with detailed chemistry, which demands only a fraction of computational effort compared with the two-dimensional simulation. In order to meet the SWQ conditions, this HOQ simulation is performed with a stoichiometric mixture at atmospheric pressure. Likewise the temperature of the wall and the fresh gas was set to 300 K. The gradients of temperature and species, and therewith the scalar dissipation rates, however, are in the same order of magnitude compared with the SWQ configuration. The REDIM method is based on the assumption that the fast and slow chemical processes of a reacting system can be decoupled (Bykov and Maas, 2007). The fast chemical processes can be assumed to be quasi-stationary because they are completed in a short period of 268

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the detailed chemistry results within the side-wall-quenching simulation appears to be a reasonable trade-off.

for global quantities like the temperature. Regarding the generality of these findings, some considerations shall be added. As the contributions in Fig. 5 and the comparison of the corresponding results to the SWQT in Figs. 7 and 8 show, the remaining error to the actual physics is caused by the estimation of the scalar dissipation rates. These vary in between configurations and obviously in turbulent near-wall flows found in most practical applications. Specifically, Gruber et al. (2010) observed for the hydrogen flame, that the heat flux in the turbulent flame can reach up to 150% of the HOQ value. For the turbulent operation of a methane flame similar to the one considered here, Heinrich et al. (2017b) observed an increase of about 70%.1 These heat fluxes indicate a similar variation in the near-wall gradients which in turn will cause different concentrations. It is not possible to completely quantify this based on our data, however some estimations can be made. The HOQ gradients used for the REDIM are about 30% larger than the ones of the laminar SWQ configuration (not shown). As visible at position (b) in Fig. 8, this leads to an overestimation of CO by about 12%. This quantification was also confirmed by an analysis in state space (not shown for brevity). Considering the above mentioned turbulent cases, the SWQ will show up to approx. 50% larger gradients than the HOQ compared to the 30% lower values in the laminar case, i.e. the HOQ gradients are in between the laminar case considered here and the turbulent case. Accordingly, with the REDIM table, underestimations of the CO concentration will occur in regions of high gradients in the turbulent case. A quantification is difficult. Based on the above it can be estimated to be of about 20% but that does not account for an expected non-linear behavior. Accordingly, we can conclude that also in different situations, an inclusion of reasonable gradients into the tabulation will significantly improve the result. However, according to theoretical results, an improvement is also possible by increasing the number of controlling variables. It has been shown (Bykov and Maas, 2007) that the accuracy requirements for the gradient estimates reduces considerably with increasing dimension of the reduced model. Its practical implementation, however, can be difficult considering Eq. (12) must be satisfied at all locations.

Acknowledgments Financial support by Deutsche Forschungsgemeinschaft (DFG) through Grants SFB/TRR 150 and in the framework of the Excellence Initiative, Darmstadt Graduate School of Excellence Energy Science and Engineering (GSC 1070) is gratefully acknowledged. All computations were performed on the Lichtenberg High Performance Computer of TU Darmstadt. References Andrae, J., Rnbom, P.B., Edsberg, L., Eriksson, L.-E., 2002. A numerical study of side wall quenching with propane/air flames. Proc. Combust. Inst. 29 (1), 789–795. Andrae, J.C.G., Brinck, T., Kalghatgi, G.T., 2008. Hcci experiments with toluene reference fuels modeled by a semidetailed chemical kinetic model. Combust. Flame 155 (4), 696–712. Burke, S.P., Schumann, T.E.W., 1928. Diffusion flames. Ind. Eng. Chem. 20 (10), 998–1005. Bykov, V., Maas, U., 2007. The extension of the ILDM concept to reaction diffusion manifolds. Combust. Theor. Model. 11 (6), 839–862. Dreizler, A., Böhm, B., 2015. Advanced laser diagnostics for an improved understanding of premixed flame–wall interactions. Proc. Combust. Inst. 35 (1), 37–64. Ezekoye, O., Greif, R., Sawyer, R., 1992. Increased surface temperature effects on wall heat transfer during unsteady flame quenching. Symp. (Int.) Combust. 24 (1), 1465–1472. Ganter, S., Heinrich, A., Meier, T., Kuenne, G., Jainski, C., Rißmann, M.C., Dreizler, A., Janicka, J., 2017. Numerical analysis of laminar methane–air side-wall-quenching. Combust. Flame 186, 299–310. Gicquel, O., Darabiha, N., Thevenin, D., 2000. Laminar premixed hydrogen/air counterflow flame simulations using flame prolongation of ildm with differential diffusion. Proc. Combust. Inst. 28 (2), 1901–1908. Gruber, A., Chen, J.H., Valiev, D., Law, C.K., 2012. Direct numerical simulation of premixed flame boundary layer flashback in turbulent channel flow. J. Fluid Mech. 709, 516–542. Gruber, A., Kerstein, A.R., Valiev, D., Law, C.K., Kolla, H., Chen, J.H., 2015. Modeling of mean flame shape during premixed flame flashback in turbulent boundary layers. Proc. Combust. Inst. 35 (2), 1485–1492. Gruber, A., Sankaran, R., Hawkes, E.R., Chen, J.H., 2010. Turbulent flame–wall interaction: a direct numerical simulation study. J. Fluid Mech. 658, 5–32. Hasse, C., Bollig, M., Peters, N., Dwyer, H., 2000. Quenching of laminar isooctane flames at cold walls. Combust. Flame 122 (1), 117–129. Heinrich, A., Ganter, S., Kuenne, G., Jainski, C., Dreizler, A., Janicka, J., 2017. 3D numerical simulation of a laminar experimental SWQ burner with tabulated chemistry. Flow Turbul. Combust. 100 (2), 535–559. Heinrich, A., Ries, F., Kuenne, G., Ganter, S., Hasse, C., Sadiki, A., Janicka, J., 2017. Large eddy simulation with tabulated chemistry of an experimental sidewall quenching burner. Heat Fluid Flow. (Unpublished result, this issue). Houwen, P.J., 1972. Explicit Runge–Kutta formulas with increased stability boundaries. Numer. Math. 20 (2), 149–164. Jainski, C., Rißmann, M., Böhm, B., Dreizler, A., 2017. Experimental investigation of flame surface density and mean reaction rate during flame–wall interaction. Proc. Combust. Inst. 36 (2), 1827–1834. Jainski, C., Rißmann, M., Böhm, B., Janicka, J., Dreizler, A., 2017. Sidewall quenching of atmospheric laminar premixed flames studied by laser-based diagnostics. Combust. Flame 183, 271–282. Ketelheun, A., Kuenne, G., Janicka, J., 2013. Heat transfer modeling in the context of large eddy simulation of premixed combustion with tabulated chemistry. Flow Turbul. Combust. 91 (4), 867–893. Ketelheun, A., Olbricht, C., Hahn, F., Janicka, J., 2009. Premixed generated manifolds for the computation of technical combustion systems. Proceedings of ASME Turbo Expo Conference. pp. 695–705. Laidler, K.J., 1996. A glossary of terms used in chemical kinetics, including reaction dynamics (IUPAC recommendations 1996). Pure Appl. Chem. 68 (1), 149–192. Lazik, W., Doerr, T., Bake, S., Bank, R.v.D., Rackwitz, L., 2008. Development of lean-burn low-nox combustion technology at Rolly-Royce Deutschland. Proceedings of ASME Turbo Expo Conference Proceedings, 797–807. Lehnhäuser, T., Schäfer, M., 2002. Improved linear interpolation practice for finite-volume schemes on complex grids. Numer. Methods Fluids 38 (7), 625–645. Maas, U., Bykov, V., 2011. The extension of the reaction diffusion manifold concept to systems with detailed transport models. Proc. Combust. Inst. 33 (1), 1253–1259. MathWorks Incorporation, 2006. Curve Fitting Toolbox 1: User’s Guide. MathWorks. Meier, T., 2017. Entwicklung Numerischer Methoden zur Anwendung Detaillierter Chemie in Komplexen Verbrennungssystemen. Technische Universität Darmstadt Ph.D. thesis. Meier, T., Kuenne, G., Ketelheun, A., Janicka, J., 2013. Numerische Abbildung Von Verbrennungsprozessen Mit Hilfe Detaillierter Und Tabellierter Chemie. VDI Berichte. VDI Verlag GmbH, Duisburg, Germany, pp. 643–652. Neagos, A., Bykov, V., Maas, U., 2017. Adaptive hierarchical construction of reaction

6. Conclusion This work considered the modeling of flame–wall-interaction with respect to the prediction of carbon monoxide and temperature within the flame quenching region. Different chemistry tabulation approaches were evaluated by means of the laminar side-wall burner configuration which has also been investigated experimentally. The analyses of the FGM simulation results by means of the species transport equation in state space showed that unsuitable assumptions regarding scalar dissipation rates within the FGM tabulation approach leads to a systematic prediction deficiency. These assumptions are implicitly made during the table generation procedure by using one dimensional flamelets that present no enthalpy gradients which, however, actually have a significant influence of the simulated SWQ configuration. Furthermore, it was demonstrated that generating a table directly from detailed chemistry simulation results and performing a reduced chemistry simulation provides good agreement with the original detailed chemistry results. Hence it could be shown that a chemistry table using two controlling variables leading to accurate results exists in principle. Analyzing the simulation results indicates that the scalar dissipation rates of this tabulation method are inherently correct since they develop freely within the detailed chemistry simulation. As an improved tabulation we introduced the usage of a REDIM table based on one-dimensional detailed chemistry HOQ results for this SWQ scenario. This presents a significant reduction of tabulation effort compared to the direct tabulation method using the two-dimensional detailed chemistry simulation. The good agreement with only moderate deviations from 1

There might be inaccuracies due to modeling in this quantity.

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