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Estimation of 3D flame surface density and global fuel consumption rate from 2D PLIF images of turbulent premixed flame
Combustion and Flame 162 (2015) 2087–2097
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Combustion and Flame j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m b u s t fl a m e
Estimation of 3D flame surface density and global fuel consumption rate from 2D PLIF images of turbulent premixed flame Meng Zhang a, Jinhua Wang a,⇑, Wu Jin a, Zuohua Huang a,⇑, Hideaki Kobayashi b, Lin Ma c a
State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China Institute of Fluid Science, Tohoku University, Sendai, Miyagi 980-8577, Japan c Department of Aerospace and Ocean Engineering, Virginia Tech, Blacksburg, VA 24061, USA b
a r t i c l e
i n f o
Article history: Received 13 June 2014 Received in revised form 3 January 2015 Accepted 4 January 2015 Available online 23 February 2015 Keywords: Turbulent premixed flame OH-PLIF 3D flame surface density Flame stretch factor
a b s t r a c t In premixed turbulent combustion, flame surface density (FSD) is a key parameter and can be used to estimate the system reaction rates. Even though laser diagnostic technics (Mie scattering or OH/CH-PLIF) on flame front provided very useful information of flame front wrinkles, the measurement is limited to a plane on which wrinkles in the third direction is unavailable. In this study, the estimation of 3D FSD (R) and the global fuel consumption rate (W) from planar measurements of 2D FSD (R2D) was conducted on a Bunsen-type burner fueled with methane/air mixture at the equivalence ratio of 0.9. Assuming symmetry of the mean flow, five different models designated as Method 1 to Method 5 (M1 to M5) based on different additional assumptions were utilized. M1 connected the 2D to 3D FSD with a typical value of 0.69. M2 and M3 were based on isotropic flame front normal vector distribution and identical / and h distribution which designate the direction angle of front normal in 3D space and the measurement plane. M4 and M5 assumed that normal vector fluctuation intensity of transverse direction was similar with x or y direction on the measurement plane, respectively. W was also obtained by integrating the R within the flame domain and the flame stretch factor, I0 was evaluated based on fractal analysis of the 2D measurements. For all methods, the results are satisfying. Results of M1 indicate that a typical direction cosine value of 0.69 is valid for the turbulent Bunsen flame in this study and the satisfied W estimation under higher turbulent intensities is provided. Results of M2 are relatively rough for overestimating W by about 40% under most conditions because of its intrinsic deficiency of the 1=hcos /is evaluation. M3 based on the assumed identical cosine value of mean direction angle of 3D and 2D flame front presented by / and h gave rather good estimation as M4. M4 and M5 provide the best evaluation of W, absolute error within 17% except low turbulence conditions (u0 /SL 0.2 and 0.4) of M4, by the normal vector fluctuation analysis. 2D data, as expected, underestimates W. Better W can be obtained considering the flame stretch factor, I0. Ó 2015 The Combustion Institute. Published by Elsevier Inc. All rights reserved.
1. Introduction Under the flamelet framework for turbulent premixed combustion [1], flame surface density (FSD) is defined as the flamelet surface area per unit volume. The FSD is a well-developed approach for turbulent premixed combustion modeling and can be solved or estimated on the basis of Reynolds averaged Navier Stokes theory [2–4], algebraic approach [5,6] which can be expressed by algebraic equations and spectral or fractal analysis [7]. As an evolution of the FSD, other quantities such as turbulent burning velocity [8,9], curvature [10,11], and strain-rate [12] have received considerable ⇑ Corresponding authors. Fax: +86 29 82668789. E-mail addresses: [email protected] (J. Wang), [email protected]. edu.cn (Z. Huang).
research interests in the combustion community. The statistics of flame surface density has been extensively studied both numerically [4,13] and experimentally [6,14–16], providing estimation models of various terms of its transport equation [17]. Extensive work has been conducted on FSD by detecting the flame front location and structure mainly using planar laser induced florescence (PLIF) [6,18,19] or Rayleigh scattering from the small and vaporizing droplets seeded in the flame [20,21]. These 2D techniques show great talent on combustion measurements and provide a lot of valuable information to derive FSD, as summarized in Driscoll’s review paper [22]. However, such 2D measurements only provide information on a single plane while the turbulent flames are inherently threedimensional (3D) and 2D measurements are unable to fully resolve the flame wrinkles in 3D. Instantaneous 3D techniques are ultimately needed to resolve the 3D flame wrinkles. However, at
http://dx.doi.org/10.1016/j.combustflame.2015.01.007 0010-2180/Ó 2015 The Combustion Institute. Published by Elsevier Inc. All rights reserved.
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present, complicated experimental apparatus are required and experimental difficulties will be encountered in performing the direct 3D measurements. Despite the challenges and difficulties, several 3D techniques are being developed to directly obtain 3D measurements in turbulent flames, including techniques that are based on two crossed planes [23], several parallel planes [24], and tomographic approaches [25–27]. Also, some direct 3D measurements of direction cosine and several items in the transport R-equation (i. e. propagation and curvature terms) have been obtained by using ‘‘liquid flames’’ propagating in near-isotropic turbulence, which can be closely matching the assumptions made by flamelet models of gaseous flames [28,29]. While these 3D techniques are still in development, it is important to analyze and understand the various assumptions invoked in the interpretation of 2D planar measurements. For instance, the determination of 2D FSD from OH-PLIF images is typically based on the assumption that the instantaneous corrugated flame front is not wrinkled in the transverse or tangential direction normal to the measuring plane [11]. It is important to quantify the accuracy and applicable range of correcting R2D (denotes the 2D FSD measured by planar techniques) to obtain statistics of FSD in 3D (R). The importance of such analysis also extends beyond the measurement of FSD itself. For example, once R is estimated, the global fuel consumption rates (W), another critical concept in turbulent combustion models, can be obtained by integrating the local burning rate over the available flame surface. Deschamps et al. [15] proposed an algebraic formulation for the correction of R2D, and verified the formulation by the cosine value of the averaged orientation angle in the measured plane with a Bunsen burner and an optical engine. In recent works, Halter et al. [30] analyzed the R obtained from the planar images with two models, and compared the total reaction rate estimated from the integration of R to the value supplied from the inlet fuel flow rate. They found that R and R2D had significant differences when the normal component fluctuation in the third direction in the wrinkled flame fronts was taken into account. The global fuel burning rate obtained by integrating R had an uncertainty of 50%. However, the stretch effect not considered when integrating R with SL (un-stretched laminar flame speed) in their work. Daniele et al. [31] deduced the mean flamelet speed (or local stretched laminar flame speed), SL,k, from the angle fractal method and performed a quantitative analysis of W. Veynante et al. [32] calibrated four methods including Halter’s R estimation methods by DNS simulation and showed that R2D underestimated true R values by about 12–16%, but with no additional experimental validation. Based on the past work, the overall objective of this paper is to perform an experimental study of the relationship between the mean statistics of R2D and R. The specific objectives are to perform controlled experiments to examine the following assumptions invoked in Veynante’s methods [32]: (1) The flow is statistically 2D, (2) the measurement plane is a plane of symmetry for the mean flow, either in the translational or rotational sense, and (3) the flame front movements in the third direction are similar to that of the measurement plane. Five models are proposed and analyzed to evaluate the R by considering unknown flame front wrinkling parameters embedded in the third dimension from known quantities. Global fuel consumption rate was then obtained subsequently. Flame stretch effect on the local laminar flame speeds due to flow strain or front curvature was taken into account to correct the global fuel consumption rate. The rest of the paper is organized as follows. Section 2 describes the experimental setup and the flame conditions tested in this study. Section 3 introduces the formulations of the correction methods. Section 4 presents the results and our observations. Lastly, Section 5 concludes the paper.
2. Experimental details and image processing method 2.1. Experiment setup and the measurement The experimental setup is schematically shown and has been detailed in Zhang et al. [33]. CH4/air mixture at equivalence ratio of 0.9 and various turbulent intensities was tested. The components and laminar flame parameters of the tested mixture are shown in Tables 1 and 2. A brief summary is provided here to facilitate the discussion. The experimental apparatus included the air supply system, the turbulent Bunsen burner, and the OH-PLIF system. Figure 1 shows the standard turbulent Bunsen burner with an axially-symmetric copper nozzle with the diameter of 20 mm and water cooling. An impinging plate and two pieces of sintering metals were used to enhance the mixing of fuel and air and also to rectify the flow. Three kinds of perforated plates with different hole arrangements and diameters, shown in Fig. 2, were installed 40 mm upstream of the nozzle outlet to generate the turbulence. A H2 diffusion flame around the nozzle outlet served as a pilot flame to stabilize the turbulent flame. Different turbulence intensities (normalized by the unstretched laminar burning velocity) had been investigated and the performed experimental condition matrix was shown in Table 3. Turbulence intensities were measured using a constanttemperature hot-wire anemometer (Dantec, Streamline 90N) at the center and 1 mm above the burner outlet. The measurements were performed at a sampling rate of 300 kHz for duration of 5 s. The turbulence intensity was then calculated under the assumption of Taylor’s hypothesis, isotropy of the turbulence, and its linearity with respect to the velocity at the burner exit as demonstrated in our previous research [33]. Figure 3 illustrates the position of the measurement plane relative to the flame and the 3D coordinate system used in current study. The instantaneous flame front structure was imaged by OH-PLIF, at an excitation wavelength of 282.769 nm. The laser system consists of an Nd:YAG laser (Quanta-Ray Pro-190), wavelength of 355 nm, 10 Hz with 10 ns pulse duration and a dye laser (Sirah PRSC-G-3000) with a frequency doubler to excite the Q1(8) line of the A2R X2P(1,0) bands of OH at a the excitation wavelength. The OH fluorescence was detected by an ICCD camera through a UV lens (Nikon Rayfact PF 10545MF-UV) with intensified Relay Optics (LaVision VC08-0094) and OH bandpass filter (LaVision VZ08–0222). The OH images were focused onto the CCD chip with a resolution of 800 600 pixels, with each pixel corresponding to 136 lm in the measurement plane. 2.2. Image processing method The image processing method is illustrated in Fig. 4. First, the OH-PLIF images were filtered to remove the pixel noise by applying a 5 5 median filter and then were binarized based on the histogram of the intensity of the images, as Fig. 4a. Then, the flame
Table 1 The components of inlet gas tested in this study. Components
/
XCH4
XN2
XO2
CH4/air
0.85
0.082
0.725
0.193
Table 2 Parameters of burner outlet and the fuel mixture. SL (m/s)
S0 (m2)
q0 (kg/m3)
Ma
LM
Leeff
dT (mm)
0.3097
0.000314
1.1118
3.81
0.25
0.96
0.066
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fit the continuous flame front curve for the subsequent calculation. A third-order smoothing scheme was used to remove the digitization noise from the binarization process as shown in Fig. 4c. The smooth flame front was then discretized with an interval of 1.0 mm, which was verified to be sufficient larger than laminar flame thickness and smaller than inner cutoff scale of all the flames in present study. Flame normal vector and flame front local angle were then calculated as Fig. 4d shown. Furthermore, fractal analysis was applied to 100 flame images including fractal inner cutoffs (ei), outer cutoffs (e0), and the fractal dimension (D2) to evaluate the stretch factor (I0). Detailed information of the fractal analysis will be discussed in later section.
Main flame Pilot flame H2 Cooling water
3. Theoretical derivation and 3D estimation models
Perforated plate 3.1. Coordinate system and definition of R
Sintering metall
Impinging plate
MainFlow Fig. 1. The structure of turbulent premixed Bunsen burner.
front contours were extracted as shown in Fig. 4b and flame brush was obtained. Third, 2D flame surface density fields were obtained by boxing method with an interrogation box of 11 11 pixel [18,34]. 450 images were utilized for the 2D processing from which the 3D estimation parameter was evaluated. More than 1500 points of each OH-PLIF image were tracked which was proved sufficient to
Figure 3 shows the coordinate system to clarify the wrinkled flame surface and the measurement plane. The x axis was defined along the direction of the flow through the center of the jet. The following derivations are based on the fact that the turbulent Bunsen flame generated in the present experimental configuration was statistically axially symmetric. The corresponding position of oxy displayed in Bunsen burner and 3D coordinates was also shown in Fig. 3, resulting oxy a plane of symmetry. The OH-PLIF therefore measures a 2D wrinkled flame front defined by the intersection between the 3D instantaneous flame and the measurement plane, as illustrated in Fig. 3b. The corrugation transverse to measurement plane is unavailable. The unit normal vector of the 3D instantaneous flame front and its projection on the measurement plane are denoted as n and nP, respectively. n(x,y) is the 2D flame front unit normal vector. The relationship of n, nP and n(x,y) was clarified in Fig. 3b. Under the model of turbulent premixed combustion, the 3D FSD, R, and 2D FSD, R2D or R(x,y) (R(x,y) indicates the flame surface density measured on xoy plane in this study) of the iso-c⁄ surface are estimated from the conditional gradient of c according to Pope [35] using Eqs. (1) and (2) as below:
Rðc Þ ¼ jrcjdðc c Þ ¼ ðjrcjjc ¼ c ÞPðc Þ
ð1Þ
Fig. 2. The structure and diameters of the perforated plates used in this study.
Table 3 Inlet gas velocity and volume flow rate tested in this study. Plate no.
u0 (m/s)
Uave (m/s)
SL (cm/s)
a
0.062 0.124
1.97 3.19
30.97 30.97
b
0.232 0.279
2.18 2.92
c
0.325 0.387 0.449
2.67 3.53 4.39
u0 /Uave (%)
u0 /SL
Q (l/min)
3.15 3.89
0.20 0.40
37.12 60.15
30.97 30.97
10.64 9.55
0.75 0.90
41.12 54.97
30.97 30.97 30.97
12.17 10.96 10.23
1.05 1.25 1.45
50.38 66.55 82.72
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Rðx;yÞ ðc Þ ¼ jrcjðx;yÞ dðc c Þ ¼ ðjrcjðx;yÞ jc ¼ c ÞPðc Þ
ð2Þ
where c represents the progress variable which is 0 in the unburned mixture and 1 in the products, jrcj the absolute value of the spatial gradient of c, dðc c Þ the instantaneous flame front position, and Q the statistical average of the measured quantity of Q. The term ðjrcjjc ¼ c Þ represents the conditional average of jrcj for the isoc⁄ and Pðc Þ represents the probability to find c ¼ c at the given location. The subscript (x, y) indicates the value in the measurement plane. As shown in Fig. 3b, / represents the direction angle of the 3D instantaneous flame front on the measurement plane, which cannot be measured by the 2D PLIF technique. While, h is the direction angle of measured 2D flame front, which value at any point is available. As jrcjðx;yÞ is the 2D measured value of jrcj in the measurement plane, the relationship between these two local gradients can be established mathematically according to Fig. 3b:
jrcjðx;yÞ ¼ cos /jrcj
ð3Þ
Combining Eqs. (1)–(3) yields:
Rðx;yÞ ðc Þ ¼ ðcos/jrcjdðc c ÞÞ ¼ hcos /is Rðc Þ
ð4Þ
where hcos /is is the average of the cos / on the iso-c⁄ surface defined as below:
hcos /is ¼
ðcos /jrcjdðc c ÞÞ ðjrcjdðc c ÞÞ
¼
ðcos /jrcjjc ¼ c Þ ðjrcjjc ¼ c Þ
The derivations above relate the 2D and 3D flame surface densities. Specifically, Eq. (4) relates the 2D and 3D flame surface densities though the surface averaged cosine value of the local iso-c⁄ direction angle. This average value can either be measured experimentally [23] or deduced from the known quantities under additional assumptions [17,30,32]. Details about the estimation of this value are introduced in the following section. 3.2. Different models for evaluating hcos /is 3.2.1. Identical cosine value of mean / and h along flame symmetry axis Deschamps et al. [15] conducted the surface density measurements in both a Bunsen burner and an optical engine and proposed a correction method for R2D (R(x,y)) distribution along progress and cos and variable, c. Their method implied that cos / h (/ h represent the mean value of h and /) along the burner axis were identical since the flame was axisymmetric. Their results showed that for the Bunsen flames studied, the mean direction angle cosine, cos h, of the flame front had a typical value of 0.69. Thus, R can be estimated by Eq. (6):
R ¼ Rðx;yÞ =0:69
ð6Þ
Fig. 3. The coordinates in this study.
Instantaneous flame front
SL,k
x U0
y
A0
(a)
(b)
ð5Þ
(c)
(d)
Fig. 4. Image processing procedures to determine the normal vector of flame front.
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3.2.2. Isotropic flame front normal vector distribution and identical / and h distribution The assumption that distribution profiles of / and h are identical, while cos h is measureable and R can be easily evaluated by Eq. (7):
R¼
Rðx;yÞ hcos /is
Rðx;yÞ hcos his
Rðx;yÞ
ð7Þ
hnðx;yÞ is
This assumption was supported by both computational [32] and experimental [23] works. And cautions should be taken according to our definition of / and h in Fig. 3b, p=2 6 / 6 p=2 and p 6 h 6 p. As a result, cos h can be negative while cos / is always positive. Another model is derived from the isotropic normal vector of the flame front surface or the isotropic and independent distribution of / and h. Hawkes et al. [36] made the isotropic assumption for approximating the scalar dissipation rates form low-dimensional measurements, and their further studies [17,37] extended to 3D flame surface densities. Let p(/) and p(h) denote the probability density function (PDF) of / and h. Then, according to our definitions, the probability of a 3D surface element corresponding to the normal vector n is:
pð/ÞpðhÞd/dh ¼ cos /d/dh=S
ð8Þ
2D
M1
M2
The right hand side of Eq. (8) is also the proportion of an elementary surface area and the surface of a unity radius sphere. Combining the property of the PDF and the assumption of the independent distribution of / and h yields:
Z p Z p=2 p
Z p
pð/ÞpðhÞd/dh ¼
p=2
Z p Z p=2 p
cos /d/dh=S ¼ 1
ð9Þ
p=2
pðhÞdh ¼ 1
ð10Þ
p
Combining (8)–(10) to integrate p(/) within the / domain, the following relationships are then obtained:
pð/Þ ¼
cos / 2
and hcos /is ¼
ð11Þ Z p=2
cos /pð/Þd/ ¼
p
p=2
4
ð12Þ
Thus, R is estimated by:
R
4Rðx;yÞ
ð13Þ
p
M3
M4
M5
u’/S L
0.2
0.8
1.45
Fig. 5. Comparison of flame surface density (mm1) of 2D and M1–M5 at u0 /SL 0.2, 0.8 and 1.45.
M. Zhang et al. / Combustion and Flame 162 (2015) 2087–2097
3.5
or two components of normal vector, n or n(x,y) can be further decomposed into their mean and fluctuating components:
3.0
ni ¼ hni is þ mi with hmi is ¼ 0
2.5 2.0 1.5 1.0
Down stream location x/D
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where hni is is the mean value and mi represents the fluctuation. The projection of n on the measurement plane np and 2D normal vector n(x,y) are collinear and connected with cos /. So are those components of np and n(x,y) on xoy plane:
ni ¼ nðx;yÞi cos /
0.0 50
100
150
hnx nx is þ hny ny is þ hnz nz is
200
¼ hnx is hnx is þ hmx mx is þ hny is hny is þ hmy my is þ hmz mz is ¼ 1
Flame surface density (mm2/mm)
ð16Þ
(a)
Eqs. (15) and (16) are exact equality. The following approximations were introduced in this work to connect those values with 2D information derived from (15):
Flame surface density (mm2/mm)
300 2D M1 M2 M3 M4 M5
250 200
hni is hcos /is hniðx;yÞ is
ð17Þ
hni ni is hcos2 /is hnðx;yÞi nðx;yÞi is
ð18Þ
150
hmi mi is hðni hni is Þðni hni is Þis hcos2 /is hnðx;yÞi nðx;yÞi is hcos /i2s hnðx;yÞi is hnðx;yÞi is
100
hcos2 /is hcos /i2s
50 0 0.0
0.5
1.0
1.5
2.0
2.5
Down stream location x/D
ð21Þ 2D M1 M2 M3 M4 M5
400 350 300 250
Substituting (17)–(20) into (16) and combining the result with (21) yields:
hcos /i2s þ hmz mz is ¼ 1
150 100 50 0 1.0
1.5
2.0
2.5
3.0
ð22Þ
Eq. (22) links the fluctuation component of the transverse direction
200
0.5
ð20Þ
In the measurement plane, the normal vector of the 2D instantaneous flame front satisfies the following relation:
450
0.0
ð19Þ
hnðx;yÞx is hnðx;yÞx is þ hmðx;yÞx mðx;yÞx is þ hnðx;yÞy is hnðx;yÞy is þ hmðx;yÞy mðx;yÞy is ¼ 1
(b) Flame surface density (mm2/mm)
ð15Þ
ni and nðx;yÞi denote the components on xoy plane of 3D and 2D normal vector respectively. By definition, unit normal vector is written as (16) with Eq. (14) and hnz is ¼ 0 based on the measurement plane being a symmetric plane:
0.5
0
ð14Þ
3.5
with hcos /i2s . This equation also indicates that the true flame surface density in the measurement plane differs from the value measured by 2D techniques because of third direction wrinkles off the measurement plane. Furthermore, Eq. (22) is an approximation and additional assumptions are needed since 2D techniques do not provide any values out of the measured plane. Here, we assumed that the normal vector fluctuation intensity of transverse direction is identical with x or y direction on the measurement plane, respectively:
Down stream location x/D
(c) Fig. 6. Evolution of the flame surface density with x/D, x is the downstream location and D is the exit diameter of the burner, a: indication of Eq. (28); b: u0 /SL 0.20; c: u0 /SL 1.45.
hmz mz is hmx mx is
ð23-aÞ
or
hmz mz is hmy my is
ð23-bÞ
Combining Eq. (23) with Eqs. (19) and (20) yields: 3.2.3. Analysis of instantaneous flame front normal vector fluctuations Halter et al. [30] analyzed the 3D local flame surface density evaluation based on normal vectors of the flame wrinkles and a fixed value at various conditions and depicted the components distribution. Veynante et al. [32] validated Halter’s model by DNS results and proposed a new model upon normal vector fluctuations corresponding to Halter’s model. According to Cant et al. [38], three
hmz mz is hcos /i2s hmðx;yÞx mðx;yÞx is
ð24Þ
or
hmz mz is hcos /i2s hmðx;yÞy mðx;yÞy is
ð25Þ
Based on the additional assumptions reflected in Eq. (24) or (25), hcos /is is then obtained with Eq. (22). Eq. (4) then becomes:
M. Zhang et al. / Combustion and Flame 162 (2015) 2087–2097
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Fig. 7. Fluctuation and mean components of the normal vector for a, b, c and d: u0 /SL 0.20; e, f, g and h: u0 /SL 1.45.
R¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ hmðx;yÞx mðx;yÞx is Rðx;yÞ
ð26Þ
R¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ hmðx;yÞy mðx;yÞy is Rðx;yÞ
ð27Þ
Up to now, five different models can be developed based on the above derivations to infer 3D properties based on 2D measurements. These models are encapsulated in Eqs. (6), (7), (13), (26) and (27), respectively, and are designated as Method1 to Method5 (M1 to M5) in the later sections. 4. Results and discussions 4.1. 3D estimation from five models The typical instantaneous flame front measured OH-PLIF diagnostics exhibits wrinkled structures, an inherent feature of turbulent premixed flames [34]. Figure 5 shows the processed 2D flame surface density and the 3D estimations on the measurement plane for u0 /SL 0.20, 0.80 and 1.45. The 2D flame surface density was obtained using the method detailed in our previous publication [34]. The 3D flame surface densities estimated from the above models were larger than the 2D image processed values, indicating the underestimation of 2D flame surface density and this was also verified by Fig. 6. This discrepancy was due to the limitation of 2D diagnostics, because they do not provide information to the wrinkles on the transverse direction, either via translation as in a slot burner or via rotation as in the axially-symmetric Bunsen flame used here. While, this wrinkles in transverse direction are the
turbulent flame natures. Though M1 and M3 were deduced from different models under different assumptions, 3D results obtained from these methods were different from the 2D values by a common proportional constant. This result implies that these models have all considered the fact that the wrinkles off the measurement plane have an equal effect on the flame surface density within the whole flame zone (flame brush), i.e., the flame tip and the two flame roots near the burner exit. Since wrinkles are often observed in the flame tip and the flat pattern in roots where the flame is still roughly two-dimensional, this effect could lead to an overestimation at flame roots and an underestimation at flame tip. However, as shown in the next section, M1 and M3 exhibit different level of accuracy when applied to calculate the overall fuel consumption. In contrast to M1 and M3, M2 as well as M4 and M5 used the local 2D to 3D coefficient deduced from the local wrinkle properties in the measurement plane and yield different local values within the flame zone. Therefore, M2, M4 and M5 were expected to provide more accurate estimation than M1 and M3, because they spatially resolve the local 2D flame front wrinkles to estimate the 2D to 3D FSD factor. Figure 6a displays the flame surface density evolution which indicated flame surface in the unit length along the burner centerline in mainstream direction. And this flame surface which solely depends on the downstream location, x, is defined as Eq. (28):
RðxÞ ¼
Z Z z
Rdydz
ð28Þ
y
It can be clearly seen from Fig. 6b and c that all results of 3D estimation were larger than that of 2D values. The discrepancy between
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and uniform values when compared with hnðx;yÞy is under both conditions, indicating that hnðx;yÞy is is more suitable to represent flame front wrinkles than hnðx;yÞx is at flame root. Figure 7c, d, g and h show the map of fluctuations, i.e., hmðx;yÞx mðx;yÞx is and hmðx;yÞy mðx;yÞy is . Values of hmðx;yÞy mðx;yÞy is are low at flame root and both wings while hmðx;yÞx mðx;yÞx is exhibits larger values within the flame zone. Radial profiles 10 mm and 25 mm above the burner exit are depicted in Fig. 8 which also verified this. The hypothesis of constant hcos /is seems to be more validated for higher turbulence intensity which has higher Reynolds number and more developed turbulence. Figures 7 and 8 imply that (1) the dimension of wrinkles in the y direction is different from that in the x direction, and (2) this flame was not completely isotropic. Once the flame surface density was evaluated through models M1 to M5, the global fuel consumption rate, W, of the reactive system can be calculated as discussed immediately next.
1.0 10mms 10mm s 25mm s 25mm s
Mean components
0.8 0.6 0.4 0.2 0.0 -0.2
-10
0
-5
5
10
Radial Position (mm)
(a) 4.2. Global mass consumption rate tests 1.0 10mms 10mm s 25mm s 25mm s
Mean components
0.8 0.6
The global fuel consumption rate, W, can be obtained by integrating the mean flame surface density R over the flame brush [22,39].
W¼
Z
þ1
0
0.4
0.0 -0.2 -5
0
5
þ1
0
Rq0 I0 SL pxdxdy
ð29Þ
where I0 is the stretch factor, SL is un-stretched laminar burning velocity and q0 represents the density of unburnt gas. In this work, the laminar burning velocities and the unburnt gas densities were calculated by the PREMIX code [40] of CHEMKIN II [41] with the GRI 3.0 mechanism [42]. These data are summarized in Table 2. In this work, W was calculated as the fuel consumption rate through the flame surface within the flame brush and it is therefore identical to the fuel supply from the burner outlet. As a result, the ratio (R) between W and the fuel supply rate q0 U 0 S0 is expected to be unity as shown below.
0.2
-10
Z
10
Radial Position (mm)
(b)
R þ1 R þ1
Fig. 8. Radical profiles of the fluctuations components, hmðx;yÞi mðx;yÞi is 10 mm and 25 mm above the burner exit, a: u0 /SL 0.20; b: u0 /SL 1.45.
R¼
2D and 3D models was different along the downstream location and was more obvious at higher turbulence intensity. M1 overestimates the contribution of out of plane flame wrinkles at flame root due to the flat patterns, especially at lower turbulence intensity. For M2, M4 and M5, the proportion of R3D/R2D varies with x and this was caused by the relative contribution from the base and tip regions as analyzed earlier in this section. M4 and M5 were deduced from normal to flame front and the fluctuation component. The mean components of the normal vectors are presented in Fig. 7a, b, e and f for u0 /SL 0.20 and 1.45. Note that hnðx;yÞx is and hnðx;yÞy is are not unique within the flame zone, while the values are both virtually symmetric. hnðx;yÞy is is larger at flame root than flame tip. That means the flame front is more wrinkled at flame tip. The map of hnðx;yÞx is shows relatively smaller
where U0 and A0 are the mean flow velocity and cross section area at burner exit as depicted in Fig. 4a. If the stretch factor is not considered (i.e., I0 1), the value of R can be directly obtained. Table 4 lists the ratios obtained from the 2D measurements using the five different models at u0 /SL 0.2, 0.8 and 1.45 considering I0 1. At relatively high turbulence intensities (e.g., u0 /SL 0.8 and 1.45), the errors from M4 and M5 were within 10%. The maximum error from M1 through M5 was below 60%. Figure 9 illustrates the variation of this ratio with respect to turbulence intensity using different models. Under all turbulence conditions, M5 generated the best estimation of the global fuel consumption rate. M3 and M4 also provided good estimation, especially at higher turbulent intensities. The results from M4 and M5 revealed that the estimation of the fluctuation in the transverse direction represented by
0
0
RI0 SL q0 pxdxdy q 0 U 0 A0
ð30Þ
Table 4 Integrated global fuel consumption rate and the ratio R; u0 /SL is the turbulence intensity and Q is the volume flow rate of fresh gas at the burner outlet. Model
Deschamps et al. Isotropic n Hawkes et al. Halter et al. Veynante et al.
hcos /is
2D 0.69 hnðx;yÞy is p/4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1= 1 þ hmðx;yÞx mðx;yÞx is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1= 1 þ hmðx;yÞy mðx;yÞy is
u0 /SL, 0.2; Q, 37.12 R þ1 R þ1 SL 0 RI0 prdxdy 0
R
u0 /SL, 0.8; Q, 54.97 R þ1 R þ1 SL 0 RI0 prdxdy 0
R
u0 /SL, 1.45; Q, 82.72 R þ1 R þ1 SL 0 RI0 prdxdy 0
R
38.22 48.34 54.11 51.59 47.71
1.03 1.30 1.45 1.39 1.29
49.32 62.80 71.42 80.23 60.01
0.90 1.14 1.30 1.45 1.09
69.98 89.01 101.41 117.04 85.18
0.85 1.08 1.23 1.41 1.03
41.58
1.12
56.46
1.03
80.93
0.98
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2.0
2.5
2D M1 M2
1.8
CH 4
M3 M4 M5
2.4 2.3
1.4
2.2
R
log L
1.6
1.2
(0.30, 2.355)
1-D2 =slope
2.1 (1.85, 1.973)
1.0
2.0
0.8
1.9
0.6 0.0
1.8 -2
0.4
0.8
1.2
1.6
u'/S L ≈ 1.45
log ε0
log εi -1
1
2
3
log r
u'/SL Fig. 9. Ratio between the integrated global burning rate and the inlet fresh gas for various turbulent intensities I0 1.
=0.5 Fitting Curve
=0.5
Fig. 11. Fractal analysis of the instantaneous flame front.
when considering thermodiffusional effect of stretched flame. Since the flame front can be viewed as the laminar flamelet, for the turbulent Bunsen flame, this local propagation speed, SL,k, is the averaged value over time and space and is expected to be linear with respect to stretch. Hence, the stretch factor (I0) of turbulent flame represents the ensemble sum of the competing effects of positive and negative stretch over time and space. It can be anticipated that the more reliable W estimation would be prospective if I0 can be evaluated properly. 4.3. Stretch factor, I0 The analysis of the stretch factor starts from the mass balance at the burner exit. At the inlet section, applying the mass balance to the average and instantaneous flame front as Fig. 4a yields:
q0 A0 U 0 ¼ q0 Aav ST
ð32Þ
The turbulent mass consumption speed, ST is:
ST ¼
(a)
(b)
Fig. 10. Position of hci = 0.5 overlaid onto the flame surface density map and the average flame fitting curve.
hmðx;yÞx mðx;yÞx is and hmðx;yÞy mðx;yÞy is was reliable. Results obtained from M1 indicated that a typical direction cosine value of 0.69 is valid for the turbulent Bunsen flame in this study. M2 overestimated W by about 40% under all conditions because of the intrinsic deficiency of the 1=hcos /is estimation. The defect of M2 resulting from the rough identical distribution of / and h even though the model resolve the local flame front wrinkles. For the 2D FSD data, as expected, it underestimates the flame surface density as well as the global fuel consumption rate. Results from these models are in good agreement when the error margins are considered. However, the stretch effect characterized by the Markstein number, Ma, was not taken into account. For a laminar flame, I0 relates the stretch rate to the stretched laminar flame speed [22], SL,k in the following way:
SL;k K ¼ I0 ¼ 1 Ma 2 SL ðSL =a0 Þ
ð31Þ
where K represents the stretch rate of flame, a0 is the diffusivity of the mixture. Obviously, I0 is not always equal to unity especially
A0 U0 Aav
ð33Þ
where Aav is the averaged or the expected surface of the turbulent flame. In this study, the contours of hci = 0.5 is applied and fitted by quadratic least square method to obtain Aav, as shown in Fig. 10. There are also other ways to determine the contours [43]. To derive the ST from the ensemble average of the instantaneous flame, the normalized ST/SL,k, by the local laminar flame speed as shown below:
ST Ain;j ¼ SL;k Aav
ð34Þ
where Ain,j is the jth instantaneous flame front surface area and ðÞ represents the statistical average. Evaluation of Ain,j/Aav can be performed by fractal analysis for the averaged and each individual instantaneous image [44,45]. According to Gouldin et al. [46]:
Ain;j ¼ Aav
D2 1
e0 ei
ð35Þ
where ei, e0, and D2 denote the inner cutoff, outer cutoff, and 2D fractal dimension of each instantaneous flame image, respectively. Figure 11 shows the typical fractal characteristics of the caliper and flame front length. The stretch factor, I0 with the flamelet concept can be deduced by combining (30)–(32). That would be:
SL;k I0 ¼ ¼ SL
A0 U 0 =Aav ðe0 =ei ÞD2 1 SL
! ð36Þ
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5. Conclusions
1.6
I0
1.4
I0
1.2 1.0 0.8 0.6 0.4 0.6
0.8
1.0
1.2
1.4
1.6
u'/S L Fig. 12. Variation of stretch factor, I0, with turbulence intensity.
2.0
2D M1 M2
1.8
M3 M4 M5
1.4 1.2
R
I0corrected
1.6
1.0 0.8 0.6 0.6
0.8
1.0
1.2
1.4
1.6
u'/S L
Flame surface density which is defined as flamelet surface area per unit volume is a key parameter in turbulent combustion models. Turbulent combustion is inherently three-dimensional, and consequently the flame surface density cannot be directly measured by planar 2D laser diagnostics. Five models were evaluated to estimate 3D flame surface density under the common assumption that the measurement plane is a symmetry plane of the mean flow and the additional assumptions. The global fuel consumption rate, W, was obtained by integrating R and the flame stretch factor was evaluated based on fractal analysis of the 2D images. This work examined the various assumptions typically invoked in each model for the estimation of 3D FSD and W, and quantified the error from each model. We expect these results to clarify the error and applicability of these models, and to provide valuable guidance for the processing of 2D planar measurements of turbulent flame properties. Under all turbulence intensities tested in this work, the errors in the results obtained from M1 to M5 were within 57%. Results obtained from M1 indicated that a typical direction cosine value of 0.69 is valid for the turbulent Bunsen flame in this study, and that M1 provided satisfactory estimation of global fuel consumption rate under high turbulent intensities. Results obtained from M2 overestimated W by about 40% under most conditions because of the intrinsic deficiency of the 1=hcos /is evaluation, i.e., the nonidentical cosine value of mean / and h and non-ideal distribution profile of / and h. M3 provides better estimation as well as M4. Results obtained from M4 and M5 provided the best evaluation of the global fuel consumption rate by estimating the normal vector fluctuation in the transverse direction assuming that they are equivalent to those in the horizontal and vertical directions in the measurement plane. For 2D data, as expected, they underestimate flame surface density and W. Lastly, the flame stretch factor, I0, can also be deduced from the fractal analysis of the turbulent flame front, which helps to improve the estimations of the global fuel consumption rate.
Fig. 13. I0 corrected ratio, RI0 corrected , with turbulent intensities.
Acknowledgments Table 5 The standard derivation against unity of ratios between global fuel consumption rate and fuel supply rate.
I0 = 1 I0 corrected
M1
M2
M3
M4
M5
0.1486 0.1373
0.2998 0.2924
0.4476 0.4419
0.1111 0.0957
0.0501 0.0331
This study is supported by National Natural Science Foundation of China (Nos. 51376004, 91441203) and the National Basic Research Program of China (2013CB228406). The Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry is also appreciated. References
This method has been used by Daniele et al. [31]. In this work, 100 flame front images were used for the fractal analysis. Spatial resolution of the image (0.136 mm/pixel) is adequate for this analysis according to Wyngaard [47]. The stretch factor, I0 with turbulent intensities and the error bar representing the standard deviation are depicted in Fig. 12 removing low turbulence intensity (u0 /SL 0.2 and 0.4) for the large uncertainty of fractal characteristics. It can be seen that I0 are around unity, indicating the nearly neutral diffusional effects of CH4/air flame (with / = 0.7–1.0) with respect to stretch [48]. Figure 13 shows the corrected ratios of the global fuel consumption rate to the fuel supply rate, RI0 corrected . Comparing to Fig. 9, it is clear that RI0 corrected are closer to unity giving better estimation on W. This is also verified by the data shown in Table 5, which lists the standard deviation of R against unity with and without the stretch effects.
[1] N. Peters, Proc. Combust. Inst. 21 (1) (1986) 1231–1250. [2] D. Veynante, L. Vervisch, Prog. Energy Combust. Sci. 28 (3) (2002) 193–266. [3] D. Veynante, J. Piana, J.M. Duclos, C. Martel, Proc. Combust. Inst. 26 (1) (1996) 413–420. [4] R. Knikker, D. Veynante, C. Meneveau, Phys. Fluids 16 (11) (2004) L91–L94. [5] K.N.C. Bray, Proc. R. Soc. London Ser. 431 (1882) (1990) 315–335. [6] G.G. Lee, K.Y. Huh, H. Kobayashi, Combust. Flame 122 (1–2) (2000) 43–57. [7] M. Wirth, N. Peters, Proc. Combust. Inst. 24 (1) (1992) 493–501. [8] S. Daniele, P. Jansohn, J. Mantzaras, K. Boulouchos, Proc. Combust. Inst. 33 (2) (2011) 2937–2944. [9] H. Kobayashi, T. Tamura, K. Maruta, T. Niioka, F.A. Williams, Proc. Combust. Inst. 26 (1996) 389–396. [10] R. Sankaran, E.R. Hawkes, J.H. Chen, T. Lu, C.K. Law, Proc. Combust. Inst. 31 (1) (2007) 1291–1298. [11] J. Wang, M. Zhang, Z. Huang, T. Kudo, H. Kobayashi, Combust. Flame 160 (11) (2013) 2434–2441. [12] C. Mounaïm-Rousselle, I. Gökalp, Proc. Combust. Inst. 25 (1) (1994) 1199– 1205. [13] J.B. Bell, M.S. Day, J.F. Grcar, M.J. Lijewski, J.F. Driscoll, S.A. Filatyev, Proc. Combust. Inst. 31 (1) (2007) 1299–1307. [14] J. Hult, S. Gashi, N. Chakraborty, M. Klein, K.W. Jenkins, S. Cant, C.F. Kaminski, Proc. Combust. Inst. 31 (2007) 1319–1326.
M. Zhang et al. / Combustion and Flame 162 (2015) 2087–2097 [15] B.M. Deschamps, G.J. Smallwood, J. Prieur, D.R. Snelling, Ö.L. Gülder, Proc. Combust. Inst. 26 (1) (1996) 427–435. [16] B. Deschamps, A. Boukhalfa, C. Chauveau, I. Gökalp, I.G. Shepherd, R.K. Cheng, Proc. Combust. Inst. 24 (1) (1992) 469–475. [17] E.R. Hawkes, R. Sankaran, J.H. Chen, Proc. Combust. Inst. 33 (1) (2011) 1447– 1454. [18] M. Zhang, J. Wang, Y. Xie, Z. Wei, W. Jin, Z. Huang, H. Kobayashi, Exp. Therm. Fluid Sci. 52 (2014) 288–296. [19] S.A. Filatyev, J.F. Driscoll, C.D. Carter, J.M. Donbar, Combust. Flame 141 (1–2) (2005) 1–21. [20] F. Halter, C. Chauveau, I. Gökalp, Int. J. Hydrogen Energy 32 (13) (2007) 2585– 2592. [21] H.S. Guo, B. Tayebi, C. Galizzi, D. Escudie, Int. J. Hydrogen Energy 35 (20) (2010) 11342–11348. [22] J.F. Driscoll, Prog. Energy Combust. Sci. 34 (1) (2008) 91–134. [23] A.N. Karpetis, R.S. Barlow, Proc. Combust. Inst. 30 (1) (2005) 665–672. [24] J. Hult, A. Omrane, J. Nygren, C. Kaminski, B. Axelsson, R. Collin, P.E. Bengtsson, M. Aldén, Exp. Fluids 33 (2) (2002) 265–269. [25] W. Cai, X. Li, F. Li, L. Ma, Opt. Express 21 (6) (2013) 7050–7064. [26] X. Li, L. Ma, Opt. Express 22 (4) (2014) 4768–4778. [27] T.D. Upton, D.D. Verhoeven, D.E. Hudgins, Exp. Fluids 50 (1) (2011) 125–134. [28] S.S. Shy, R.H. Jang, W.K. I, K.L. Gee, Proc. Combust. Inst. 26 (1) (1996) 283–289. [29] S.S. Shy, W.K. I, E.I. Lee, T.S. Yang, Combust. Flame 118 (4) (1999) 606–618. [30] F. Halter, C. Chauveau, I. Gokalp, D. Veynante, Combust. Flame 156 (3) (2009) 657–664. [31] S. Daniele, J. Mantzaras, P. Jansohn, A. Denisov, K. Boulouchos, J. Fluid Mech. 724 (2013) 36–68. [32] D. Veynante, G. Lodato, P. Domingo, L. Vervisch, E. Hawkes, Exp. Fluids 49 (1) (2010) 267–278.
2097
[33] M. Zhang, J. Wang, Y. Xie, W. Jin, Z. Wei, Z. Huang, H. Kobayashi, Int. J. Hydrogen Energy 38 (26) (2013) 11421–11428. [34] M. Zhang, J. Wang, J. Wu, Z. Wei, Z. Huang, H. Kobayashi, Int. J. Hydrogen Energy 39 (10) (2014) 5176–5185. [35] S.B. Pope, Int. J. Eng. Sci. 26 (5) (1988) 445–469. [36] E.R. Hawkes, R. Sankaran, J.H. Chen, S.A. Kaiser, J.H. Frank, Proc. Combust. Inst. 32 (1) (2009) 1455–1463. [37] N. Chakraborty, E.R. Hawkes, Phys. Fluids 23 (6) (2011) 1–19 (1994–present). [38] R. Cant, S. Pope, K. Bray, Proc. Combust. Inst. (1991) 809–815. [39] I.G. Shepherd, Proc. Combust. Inst. 26 (1) (1996) 373–379. [40] R.J. Kee, J.F. Grcar, M. Smooke, J. Miller, E. Meeks, PREMIX: A Fortran Program for Modeling Steady Laminar One-Dimensional Premixed Flames, Report SAND85-8240, Sandia National Laboratories, 1985. [41] R.J. Kee, F.M. Rupley, J.A. Miller, Chemkin-II: A Fortran Chemical Kinetics Package for the Analysis of Gas-Phase Chemical Kinetics, Report SAND-898009, Sandia National Laboratories, 1989. [42] P. Gregory, D.M. Smith, M. Golden, N.W. Frenklach, B. Moriarty, M. Eiteneer, C.T. Goldenberg, R.K. Bowman, S. Hanson, W.C. Song, J. Gardiner, V.V. Lissianski, Z. Qin.
Contents lists available at ScienceDirect
Combustion and Flame j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m b u s t fl a m e
Estimation of 3D flame surface density and global fuel consumption rate from 2D PLIF images of turbulent premixed flame Meng Zhang a, Jinhua Wang a,⇑, Wu Jin a, Zuohua Huang a,⇑, Hideaki Kobayashi b, Lin Ma c a
State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China Institute of Fluid Science, Tohoku University, Sendai, Miyagi 980-8577, Japan c Department of Aerospace and Ocean Engineering, Virginia Tech, Blacksburg, VA 24061, USA b
a r t i c l e
i n f o
Article history: Received 13 June 2014 Received in revised form 3 January 2015 Accepted 4 January 2015 Available online 23 February 2015 Keywords: Turbulent premixed flame OH-PLIF 3D flame surface density Flame stretch factor
a b s t r a c t In premixed turbulent combustion, flame surface density (FSD) is a key parameter and can be used to estimate the system reaction rates. Even though laser diagnostic technics (Mie scattering or OH/CH-PLIF) on flame front provided very useful information of flame front wrinkles, the measurement is limited to a plane on which wrinkles in the third direction is unavailable. In this study, the estimation of 3D FSD (R) and the global fuel consumption rate (W) from planar measurements of 2D FSD (R2D) was conducted on a Bunsen-type burner fueled with methane/air mixture at the equivalence ratio of 0.9. Assuming symmetry of the mean flow, five different models designated as Method 1 to Method 5 (M1 to M5) based on different additional assumptions were utilized. M1 connected the 2D to 3D FSD with a typical value of 0.69. M2 and M3 were based on isotropic flame front normal vector distribution and identical / and h distribution which designate the direction angle of front normal in 3D space and the measurement plane. M4 and M5 assumed that normal vector fluctuation intensity of transverse direction was similar with x or y direction on the measurement plane, respectively. W was also obtained by integrating the R within the flame domain and the flame stretch factor, I0 was evaluated based on fractal analysis of the 2D measurements. For all methods, the results are satisfying. Results of M1 indicate that a typical direction cosine value of 0.69 is valid for the turbulent Bunsen flame in this study and the satisfied W estimation under higher turbulent intensities is provided. Results of M2 are relatively rough for overestimating W by about 40% under most conditions because of its intrinsic deficiency of the 1=hcos /is evaluation. M3 based on the assumed identical cosine value of mean direction angle of 3D and 2D flame front presented by / and h gave rather good estimation as M4. M4 and M5 provide the best evaluation of W, absolute error within 17% except low turbulence conditions (u0 /SL 0.2 and 0.4) of M4, by the normal vector fluctuation analysis. 2D data, as expected, underestimates W. Better W can be obtained considering the flame stretch factor, I0. Ó 2015 The Combustion Institute. Published by Elsevier Inc. All rights reserved.
1. Introduction Under the flamelet framework for turbulent premixed combustion [1], flame surface density (FSD) is defined as the flamelet surface area per unit volume. The FSD is a well-developed approach for turbulent premixed combustion modeling and can be solved or estimated on the basis of Reynolds averaged Navier Stokes theory [2–4], algebraic approach [5,6] which can be expressed by algebraic equations and spectral or fractal analysis [7]. As an evolution of the FSD, other quantities such as turbulent burning velocity [8,9], curvature [10,11], and strain-rate [12] have received considerable ⇑ Corresponding authors. Fax: +86 29 82668789. E-mail addresses: [email protected] (J. Wang), [email protected]. edu.cn (Z. Huang).
research interests in the combustion community. The statistics of flame surface density has been extensively studied both numerically [4,13] and experimentally [6,14–16], providing estimation models of various terms of its transport equation [17]. Extensive work has been conducted on FSD by detecting the flame front location and structure mainly using planar laser induced florescence (PLIF) [6,18,19] or Rayleigh scattering from the small and vaporizing droplets seeded in the flame [20,21]. These 2D techniques show great talent on combustion measurements and provide a lot of valuable information to derive FSD, as summarized in Driscoll’s review paper [22]. However, such 2D measurements only provide information on a single plane while the turbulent flames are inherently threedimensional (3D) and 2D measurements are unable to fully resolve the flame wrinkles in 3D. Instantaneous 3D techniques are ultimately needed to resolve the 3D flame wrinkles. However, at
http://dx.doi.org/10.1016/j.combustflame.2015.01.007 0010-2180/Ó 2015 The Combustion Institute. Published by Elsevier Inc. All rights reserved.
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present, complicated experimental apparatus are required and experimental difficulties will be encountered in performing the direct 3D measurements. Despite the challenges and difficulties, several 3D techniques are being developed to directly obtain 3D measurements in turbulent flames, including techniques that are based on two crossed planes [23], several parallel planes [24], and tomographic approaches [25–27]. Also, some direct 3D measurements of direction cosine and several items in the transport R-equation (i. e. propagation and curvature terms) have been obtained by using ‘‘liquid flames’’ propagating in near-isotropic turbulence, which can be closely matching the assumptions made by flamelet models of gaseous flames [28,29]. While these 3D techniques are still in development, it is important to analyze and understand the various assumptions invoked in the interpretation of 2D planar measurements. For instance, the determination of 2D FSD from OH-PLIF images is typically based on the assumption that the instantaneous corrugated flame front is not wrinkled in the transverse or tangential direction normal to the measuring plane [11]. It is important to quantify the accuracy and applicable range of correcting R2D (denotes the 2D FSD measured by planar techniques) to obtain statistics of FSD in 3D (R). The importance of such analysis also extends beyond the measurement of FSD itself. For example, once R is estimated, the global fuel consumption rates (W), another critical concept in turbulent combustion models, can be obtained by integrating the local burning rate over the available flame surface. Deschamps et al. [15] proposed an algebraic formulation for the correction of R2D, and verified the formulation by the cosine value of the averaged orientation angle in the measured plane with a Bunsen burner and an optical engine. In recent works, Halter et al. [30] analyzed the R obtained from the planar images with two models, and compared the total reaction rate estimated from the integration of R to the value supplied from the inlet fuel flow rate. They found that R and R2D had significant differences when the normal component fluctuation in the third direction in the wrinkled flame fronts was taken into account. The global fuel burning rate obtained by integrating R had an uncertainty of 50%. However, the stretch effect not considered when integrating R with SL (un-stretched laminar flame speed) in their work. Daniele et al. [31] deduced the mean flamelet speed (or local stretched laminar flame speed), SL,k, from the angle fractal method and performed a quantitative analysis of W. Veynante et al. [32] calibrated four methods including Halter’s R estimation methods by DNS simulation and showed that R2D underestimated true R values by about 12–16%, but with no additional experimental validation. Based on the past work, the overall objective of this paper is to perform an experimental study of the relationship between the mean statistics of R2D and R. The specific objectives are to perform controlled experiments to examine the following assumptions invoked in Veynante’s methods [32]: (1) The flow is statistically 2D, (2) the measurement plane is a plane of symmetry for the mean flow, either in the translational or rotational sense, and (3) the flame front movements in the third direction are similar to that of the measurement plane. Five models are proposed and analyzed to evaluate the R by considering unknown flame front wrinkling parameters embedded in the third dimension from known quantities. Global fuel consumption rate was then obtained subsequently. Flame stretch effect on the local laminar flame speeds due to flow strain or front curvature was taken into account to correct the global fuel consumption rate. The rest of the paper is organized as follows. Section 2 describes the experimental setup and the flame conditions tested in this study. Section 3 introduces the formulations of the correction methods. Section 4 presents the results and our observations. Lastly, Section 5 concludes the paper.
2. Experimental details and image processing method 2.1. Experiment setup and the measurement The experimental setup is schematically shown and has been detailed in Zhang et al. [33]. CH4/air mixture at equivalence ratio of 0.9 and various turbulent intensities was tested. The components and laminar flame parameters of the tested mixture are shown in Tables 1 and 2. A brief summary is provided here to facilitate the discussion. The experimental apparatus included the air supply system, the turbulent Bunsen burner, and the OH-PLIF system. Figure 1 shows the standard turbulent Bunsen burner with an axially-symmetric copper nozzle with the diameter of 20 mm and water cooling. An impinging plate and two pieces of sintering metals were used to enhance the mixing of fuel and air and also to rectify the flow. Three kinds of perforated plates with different hole arrangements and diameters, shown in Fig. 2, were installed 40 mm upstream of the nozzle outlet to generate the turbulence. A H2 diffusion flame around the nozzle outlet served as a pilot flame to stabilize the turbulent flame. Different turbulence intensities (normalized by the unstretched laminar burning velocity) had been investigated and the performed experimental condition matrix was shown in Table 3. Turbulence intensities were measured using a constanttemperature hot-wire anemometer (Dantec, Streamline 90N) at the center and 1 mm above the burner outlet. The measurements were performed at a sampling rate of 300 kHz for duration of 5 s. The turbulence intensity was then calculated under the assumption of Taylor’s hypothesis, isotropy of the turbulence, and its linearity with respect to the velocity at the burner exit as demonstrated in our previous research [33]. Figure 3 illustrates the position of the measurement plane relative to the flame and the 3D coordinate system used in current study. The instantaneous flame front structure was imaged by OH-PLIF, at an excitation wavelength of 282.769 nm. The laser system consists of an Nd:YAG laser (Quanta-Ray Pro-190), wavelength of 355 nm, 10 Hz with 10 ns pulse duration and a dye laser (Sirah PRSC-G-3000) with a frequency doubler to excite the Q1(8) line of the A2R X2P(1,0) bands of OH at a the excitation wavelength. The OH fluorescence was detected by an ICCD camera through a UV lens (Nikon Rayfact PF 10545MF-UV) with intensified Relay Optics (LaVision VC08-0094) and OH bandpass filter (LaVision VZ08–0222). The OH images were focused onto the CCD chip with a resolution of 800 600 pixels, with each pixel corresponding to 136 lm in the measurement plane. 2.2. Image processing method The image processing method is illustrated in Fig. 4. First, the OH-PLIF images were filtered to remove the pixel noise by applying a 5 5 median filter and then were binarized based on the histogram of the intensity of the images, as Fig. 4a. Then, the flame
Table 1 The components of inlet gas tested in this study. Components
/
XCH4
XN2
XO2
CH4/air
0.85
0.082
0.725
0.193
Table 2 Parameters of burner outlet and the fuel mixture. SL (m/s)
S0 (m2)
q0 (kg/m3)
Ma
LM
Leeff
dT (mm)
0.3097
0.000314
1.1118
3.81
0.25
0.96
0.066
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fit the continuous flame front curve for the subsequent calculation. A third-order smoothing scheme was used to remove the digitization noise from the binarization process as shown in Fig. 4c. The smooth flame front was then discretized with an interval of 1.0 mm, which was verified to be sufficient larger than laminar flame thickness and smaller than inner cutoff scale of all the flames in present study. Flame normal vector and flame front local angle were then calculated as Fig. 4d shown. Furthermore, fractal analysis was applied to 100 flame images including fractal inner cutoffs (ei), outer cutoffs (e0), and the fractal dimension (D2) to evaluate the stretch factor (I0). Detailed information of the fractal analysis will be discussed in later section.
Main flame Pilot flame H2 Cooling water
3. Theoretical derivation and 3D estimation models
Perforated plate 3.1. Coordinate system and definition of R
Sintering metall
Impinging plate
MainFlow Fig. 1. The structure of turbulent premixed Bunsen burner.
front contours were extracted as shown in Fig. 4b and flame brush was obtained. Third, 2D flame surface density fields were obtained by boxing method with an interrogation box of 11 11 pixel [18,34]. 450 images were utilized for the 2D processing from which the 3D estimation parameter was evaluated. More than 1500 points of each OH-PLIF image were tracked which was proved sufficient to
Figure 3 shows the coordinate system to clarify the wrinkled flame surface and the measurement plane. The x axis was defined along the direction of the flow through the center of the jet. The following derivations are based on the fact that the turbulent Bunsen flame generated in the present experimental configuration was statistically axially symmetric. The corresponding position of oxy displayed in Bunsen burner and 3D coordinates was also shown in Fig. 3, resulting oxy a plane of symmetry. The OH-PLIF therefore measures a 2D wrinkled flame front defined by the intersection between the 3D instantaneous flame and the measurement plane, as illustrated in Fig. 3b. The corrugation transverse to measurement plane is unavailable. The unit normal vector of the 3D instantaneous flame front and its projection on the measurement plane are denoted as n and nP, respectively. n(x,y) is the 2D flame front unit normal vector. The relationship of n, nP and n(x,y) was clarified in Fig. 3b. Under the model of turbulent premixed combustion, the 3D FSD, R, and 2D FSD, R2D or R(x,y) (R(x,y) indicates the flame surface density measured on xoy plane in this study) of the iso-c⁄ surface are estimated from the conditional gradient of c according to Pope [35] using Eqs. (1) and (2) as below:
Rðc Þ ¼ jrcjdðc c Þ ¼ ðjrcjjc ¼ c ÞPðc Þ
ð1Þ
Fig. 2. The structure and diameters of the perforated plates used in this study.
Table 3 Inlet gas velocity and volume flow rate tested in this study. Plate no.
u0 (m/s)
Uave (m/s)
SL (cm/s)
a
0.062 0.124
1.97 3.19
30.97 30.97
b
0.232 0.279
2.18 2.92
c
0.325 0.387 0.449
2.67 3.53 4.39
u0 /Uave (%)
u0 /SL
Q (l/min)
3.15 3.89
0.20 0.40
37.12 60.15
30.97 30.97
10.64 9.55
0.75 0.90
41.12 54.97
30.97 30.97 30.97
12.17 10.96 10.23
1.05 1.25 1.45
50.38 66.55 82.72
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Rðx;yÞ ðc Þ ¼ jrcjðx;yÞ dðc c Þ ¼ ðjrcjðx;yÞ jc ¼ c ÞPðc Þ
ð2Þ
where c represents the progress variable which is 0 in the unburned mixture and 1 in the products, jrcj the absolute value of the spatial gradient of c, dðc c Þ the instantaneous flame front position, and Q the statistical average of the measured quantity of Q. The term ðjrcjjc ¼ c Þ represents the conditional average of jrcj for the isoc⁄ and Pðc Þ represents the probability to find c ¼ c at the given location. The subscript (x, y) indicates the value in the measurement plane. As shown in Fig. 3b, / represents the direction angle of the 3D instantaneous flame front on the measurement plane, which cannot be measured by the 2D PLIF technique. While, h is the direction angle of measured 2D flame front, which value at any point is available. As jrcjðx;yÞ is the 2D measured value of jrcj in the measurement plane, the relationship between these two local gradients can be established mathematically according to Fig. 3b:
jrcjðx;yÞ ¼ cos /jrcj
ð3Þ
Combining Eqs. (1)–(3) yields:
Rðx;yÞ ðc Þ ¼ ðcos/jrcjdðc c ÞÞ ¼ hcos /is Rðc Þ
ð4Þ
where hcos /is is the average of the cos / on the iso-c⁄ surface defined as below:
hcos /is ¼
ðcos /jrcjdðc c ÞÞ ðjrcjdðc c ÞÞ
¼
ðcos /jrcjjc ¼ c Þ ðjrcjjc ¼ c Þ
The derivations above relate the 2D and 3D flame surface densities. Specifically, Eq. (4) relates the 2D and 3D flame surface densities though the surface averaged cosine value of the local iso-c⁄ direction angle. This average value can either be measured experimentally [23] or deduced from the known quantities under additional assumptions [17,30,32]. Details about the estimation of this value are introduced in the following section. 3.2. Different models for evaluating hcos /is 3.2.1. Identical cosine value of mean / and h along flame symmetry axis Deschamps et al. [15] conducted the surface density measurements in both a Bunsen burner and an optical engine and proposed a correction method for R2D (R(x,y)) distribution along progress and cos and variable, c. Their method implied that cos / h (/ h represent the mean value of h and /) along the burner axis were identical since the flame was axisymmetric. Their results showed that for the Bunsen flames studied, the mean direction angle cosine, cos h, of the flame front had a typical value of 0.69. Thus, R can be estimated by Eq. (6):
R ¼ Rðx;yÞ =0:69
ð6Þ
Fig. 3. The coordinates in this study.
Instantaneous flame front
SL,k
x U0
y
A0
(a)
(b)
ð5Þ
(c)
(d)
Fig. 4. Image processing procedures to determine the normal vector of flame front.
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3.2.2. Isotropic flame front normal vector distribution and identical / and h distribution The assumption that distribution profiles of / and h are identical, while cos h is measureable and R can be easily evaluated by Eq. (7):
R¼
Rðx;yÞ hcos /is
Rðx;yÞ hcos his
Rðx;yÞ
ð7Þ
hnðx;yÞ is
This assumption was supported by both computational [32] and experimental [23] works. And cautions should be taken according to our definition of / and h in Fig. 3b, p=2 6 / 6 p=2 and p 6 h 6 p. As a result, cos h can be negative while cos / is always positive. Another model is derived from the isotropic normal vector of the flame front surface or the isotropic and independent distribution of / and h. Hawkes et al. [36] made the isotropic assumption for approximating the scalar dissipation rates form low-dimensional measurements, and their further studies [17,37] extended to 3D flame surface densities. Let p(/) and p(h) denote the probability density function (PDF) of / and h. Then, according to our definitions, the probability of a 3D surface element corresponding to the normal vector n is:
pð/ÞpðhÞd/dh ¼ cos /d/dh=S
ð8Þ
2D
M1
M2
The right hand side of Eq. (8) is also the proportion of an elementary surface area and the surface of a unity radius sphere. Combining the property of the PDF and the assumption of the independent distribution of / and h yields:
Z p Z p=2 p
Z p
pð/ÞpðhÞd/dh ¼
p=2
Z p Z p=2 p
cos /d/dh=S ¼ 1
ð9Þ
p=2
pðhÞdh ¼ 1
ð10Þ
p
Combining (8)–(10) to integrate p(/) within the / domain, the following relationships are then obtained:
pð/Þ ¼
cos / 2
and hcos /is ¼
ð11Þ Z p=2
cos /pð/Þd/ ¼
p
p=2
4
ð12Þ
Thus, R is estimated by:
R
4Rðx;yÞ
ð13Þ
p
M3
M4
M5
u’/S L
0.2
0.8
1.45
Fig. 5. Comparison of flame surface density (mm1) of 2D and M1–M5 at u0 /SL 0.2, 0.8 and 1.45.
M. Zhang et al. / Combustion and Flame 162 (2015) 2087–2097
3.5
or two components of normal vector, n or n(x,y) can be further decomposed into their mean and fluctuating components:
3.0
ni ¼ hni is þ mi with hmi is ¼ 0
2.5 2.0 1.5 1.0
Down stream location x/D
2092
where hni is is the mean value and mi represents the fluctuation. The projection of n on the measurement plane np and 2D normal vector n(x,y) are collinear and connected with cos /. So are those components of np and n(x,y) on xoy plane:
ni ¼ nðx;yÞi cos /
0.0 50
100
150
hnx nx is þ hny ny is þ hnz nz is
200
¼ hnx is hnx is þ hmx mx is þ hny is hny is þ hmy my is þ hmz mz is ¼ 1
Flame surface density (mm2/mm)
ð16Þ
(a)
Eqs. (15) and (16) are exact equality. The following approximations were introduced in this work to connect those values with 2D information derived from (15):
Flame surface density (mm2/mm)
300 2D M1 M2 M3 M4 M5
250 200
hni is hcos /is hniðx;yÞ is
ð17Þ
hni ni is hcos2 /is hnðx;yÞi nðx;yÞi is
ð18Þ
150
hmi mi is hðni hni is Þðni hni is Þis hcos2 /is hnðx;yÞi nðx;yÞi is hcos /i2s hnðx;yÞi is hnðx;yÞi is
100
hcos2 /is hcos /i2s
50 0 0.0
0.5
1.0
1.5
2.0
2.5
Down stream location x/D
ð21Þ 2D M1 M2 M3 M4 M5
400 350 300 250
Substituting (17)–(20) into (16) and combining the result with (21) yields:
hcos /i2s þ hmz mz is ¼ 1
150 100 50 0 1.0
1.5
2.0
2.5
3.0
ð22Þ
Eq. (22) links the fluctuation component of the transverse direction
200
0.5
ð20Þ
In the measurement plane, the normal vector of the 2D instantaneous flame front satisfies the following relation:
450
0.0
ð19Þ
hnðx;yÞx is hnðx;yÞx is þ hmðx;yÞx mðx;yÞx is þ hnðx;yÞy is hnðx;yÞy is þ hmðx;yÞy mðx;yÞy is ¼ 1
(b) Flame surface density (mm2/mm)
ð15Þ
ni and nðx;yÞi denote the components on xoy plane of 3D and 2D normal vector respectively. By definition, unit normal vector is written as (16) with Eq. (14) and hnz is ¼ 0 based on the measurement plane being a symmetric plane:
0.5
0
ð14Þ
3.5
with hcos /i2s . This equation also indicates that the true flame surface density in the measurement plane differs from the value measured by 2D techniques because of third direction wrinkles off the measurement plane. Furthermore, Eq. (22) is an approximation and additional assumptions are needed since 2D techniques do not provide any values out of the measured plane. Here, we assumed that the normal vector fluctuation intensity of transverse direction is identical with x or y direction on the measurement plane, respectively:
Down stream location x/D
(c) Fig. 6. Evolution of the flame surface density with x/D, x is the downstream location and D is the exit diameter of the burner, a: indication of Eq. (28); b: u0 /SL 0.20; c: u0 /SL 1.45.
hmz mz is hmx mx is
ð23-aÞ
or
hmz mz is hmy my is
ð23-bÞ
Combining Eq. (23) with Eqs. (19) and (20) yields: 3.2.3. Analysis of instantaneous flame front normal vector fluctuations Halter et al. [30] analyzed the 3D local flame surface density evaluation based on normal vectors of the flame wrinkles and a fixed value at various conditions and depicted the components distribution. Veynante et al. [32] validated Halter’s model by DNS results and proposed a new model upon normal vector fluctuations corresponding to Halter’s model. According to Cant et al. [38], three
hmz mz is hcos /i2s hmðx;yÞx mðx;yÞx is
ð24Þ
or
hmz mz is hcos /i2s hmðx;yÞy mðx;yÞy is
ð25Þ
Based on the additional assumptions reflected in Eq. (24) or (25), hcos /is is then obtained with Eq. (22). Eq. (4) then becomes:
M. Zhang et al. / Combustion and Flame 162 (2015) 2087–2097
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Fig. 7. Fluctuation and mean components of the normal vector for a, b, c and d: u0 /SL 0.20; e, f, g and h: u0 /SL 1.45.
R¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ hmðx;yÞx mðx;yÞx is Rðx;yÞ
ð26Þ
R¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ hmðx;yÞy mðx;yÞy is Rðx;yÞ
ð27Þ
Up to now, five different models can be developed based on the above derivations to infer 3D properties based on 2D measurements. These models are encapsulated in Eqs. (6), (7), (13), (26) and (27), respectively, and are designated as Method1 to Method5 (M1 to M5) in the later sections. 4. Results and discussions 4.1. 3D estimation from five models The typical instantaneous flame front measured OH-PLIF diagnostics exhibits wrinkled structures, an inherent feature of turbulent premixed flames [34]. Figure 5 shows the processed 2D flame surface density and the 3D estimations on the measurement plane for u0 /SL 0.20, 0.80 and 1.45. The 2D flame surface density was obtained using the method detailed in our previous publication [34]. The 3D flame surface densities estimated from the above models were larger than the 2D image processed values, indicating the underestimation of 2D flame surface density and this was also verified by Fig. 6. This discrepancy was due to the limitation of 2D diagnostics, because they do not provide information to the wrinkles on the transverse direction, either via translation as in a slot burner or via rotation as in the axially-symmetric Bunsen flame used here. While, this wrinkles in transverse direction are the
turbulent flame natures. Though M1 and M3 were deduced from different models under different assumptions, 3D results obtained from these methods were different from the 2D values by a common proportional constant. This result implies that these models have all considered the fact that the wrinkles off the measurement plane have an equal effect on the flame surface density within the whole flame zone (flame brush), i.e., the flame tip and the two flame roots near the burner exit. Since wrinkles are often observed in the flame tip and the flat pattern in roots where the flame is still roughly two-dimensional, this effect could lead to an overestimation at flame roots and an underestimation at flame tip. However, as shown in the next section, M1 and M3 exhibit different level of accuracy when applied to calculate the overall fuel consumption. In contrast to M1 and M3, M2 as well as M4 and M5 used the local 2D to 3D coefficient deduced from the local wrinkle properties in the measurement plane and yield different local values within the flame zone. Therefore, M2, M4 and M5 were expected to provide more accurate estimation than M1 and M3, because they spatially resolve the local 2D flame front wrinkles to estimate the 2D to 3D FSD factor. Figure 6a displays the flame surface density evolution which indicated flame surface in the unit length along the burner centerline in mainstream direction. And this flame surface which solely depends on the downstream location, x, is defined as Eq. (28):
RðxÞ ¼
Z Z z
Rdydz
ð28Þ
y
It can be clearly seen from Fig. 6b and c that all results of 3D estimation were larger than that of 2D values. The discrepancy between
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and uniform values when compared with hnðx;yÞy is under both conditions, indicating that hnðx;yÞy is is more suitable to represent flame front wrinkles than hnðx;yÞx is at flame root. Figure 7c, d, g and h show the map of fluctuations, i.e., hmðx;yÞx mðx;yÞx is and hmðx;yÞy mðx;yÞy is . Values of hmðx;yÞy mðx;yÞy is are low at flame root and both wings while hmðx;yÞx mðx;yÞx is exhibits larger values within the flame zone. Radial profiles 10 mm and 25 mm above the burner exit are depicted in Fig. 8 which also verified this. The hypothesis of constant hcos /is seems to be more validated for higher turbulence intensity which has higher Reynolds number and more developed turbulence. Figures 7 and 8 imply that (1) the dimension of wrinkles in the y direction is different from that in the x direction, and (2) this flame was not completely isotropic. Once the flame surface density was evaluated through models M1 to M5, the global fuel consumption rate, W, of the reactive system can be calculated as discussed immediately next.
1.0 10mm
Mean components
0.8 0.6 0.4 0.2 0.0 -0.2
-10
0
-5
5
10
Radial Position (mm)
(a) 4.2. Global mass consumption rate tests 1.0 10mm
Mean components
0.8 0.6
The global fuel consumption rate, W, can be obtained by integrating the mean flame surface density R over the flame brush [22,39].
W¼
Z
þ1
0
0.4
0.0 -0.2 -5
0
5
þ1
0
Rq0 I0 SL pxdxdy
ð29Þ
where I0 is the stretch factor, SL is un-stretched laminar burning velocity and q0 represents the density of unburnt gas. In this work, the laminar burning velocities and the unburnt gas densities were calculated by the PREMIX code [40] of CHEMKIN II [41] with the GRI 3.0 mechanism [42]. These data are summarized in Table 2. In this work, W was calculated as the fuel consumption rate through the flame surface within the flame brush and it is therefore identical to the fuel supply from the burner outlet. As a result, the ratio (R) between W and the fuel supply rate q0 U 0 S0 is expected to be unity as shown below.
0.2
-10
Z
10
Radial Position (mm)
(b)
R þ1 R þ1
Fig. 8. Radical profiles of the fluctuations components, hmðx;yÞi mðx;yÞi is 10 mm and 25 mm above the burner exit, a: u0 /SL 0.20; b: u0 /SL 1.45.
R¼
2D and 3D models was different along the downstream location and was more obvious at higher turbulence intensity. M1 overestimates the contribution of out of plane flame wrinkles at flame root due to the flat patterns, especially at lower turbulence intensity. For M2, M4 and M5, the proportion of R3D/R2D varies with x and this was caused by the relative contribution from the base and tip regions as analyzed earlier in this section. M4 and M5 were deduced from normal to flame front and the fluctuation component. The mean components of the normal vectors are presented in Fig. 7a, b, e and f for u0 /SL 0.20 and 1.45. Note that hnðx;yÞx is and hnðx;yÞy is are not unique within the flame zone, while the values are both virtually symmetric. hnðx;yÞy is is larger at flame root than flame tip. That means the flame front is more wrinkled at flame tip. The map of hnðx;yÞx is shows relatively smaller
where U0 and A0 are the mean flow velocity and cross section area at burner exit as depicted in Fig. 4a. If the stretch factor is not considered (i.e., I0 1), the value of R can be directly obtained. Table 4 lists the ratios obtained from the 2D measurements using the five different models at u0 /SL 0.2, 0.8 and 1.45 considering I0 1. At relatively high turbulence intensities (e.g., u0 /SL 0.8 and 1.45), the errors from M4 and M5 were within 10%. The maximum error from M1 through M5 was below 60%. Figure 9 illustrates the variation of this ratio with respect to turbulence intensity using different models. Under all turbulence conditions, M5 generated the best estimation of the global fuel consumption rate. M3 and M4 also provided good estimation, especially at higher turbulent intensities. The results from M4 and M5 revealed that the estimation of the fluctuation in the transverse direction represented by
0
0
RI0 SL q0 pxdxdy q 0 U 0 A0
ð30Þ
Table 4 Integrated global fuel consumption rate and the ratio R; u0 /SL is the turbulence intensity and Q is the volume flow rate of fresh gas at the burner outlet. Model
Deschamps et al. Isotropic n Hawkes et al. Halter et al. Veynante et al.
hcos /is
2D 0.69 hnðx;yÞy is p/4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1= 1 þ hmðx;yÞx mðx;yÞx is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1= 1 þ hmðx;yÞy mðx;yÞy is
u0 /SL, 0.2; Q, 37.12 R þ1 R þ1 SL 0 RI0 prdxdy 0
R
u0 /SL, 0.8; Q, 54.97 R þ1 R þ1 SL 0 RI0 prdxdy 0
R
u0 /SL, 1.45; Q, 82.72 R þ1 R þ1 SL 0 RI0 prdxdy 0
R
38.22 48.34 54.11 51.59 47.71
1.03 1.30 1.45 1.39 1.29
49.32 62.80 71.42 80.23 60.01
0.90 1.14 1.30 1.45 1.09
69.98 89.01 101.41 117.04 85.18
0.85 1.08 1.23 1.41 1.03
41.58
1.12
56.46
1.03
80.93
0.98
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2.0
2.5
2D M1 M2
1.8
CH 4
M3 M4 M5
2.4 2.3
1.4
2.2
R
log L
1.6
1.2
(0.30, 2.355)
1-D2 =slope
2.1 (1.85, 1.973)
1.0
2.0
0.8
1.9
0.6 0.0
1.8 -2
0.4
0.8
1.2
1.6
u'/S L ≈ 1.45
log ε0
log εi -1
1
2
3
log r
u'/SL Fig. 9. Ratio between the integrated global burning rate and the inlet fresh gas for various turbulent intensities I0 1.
Fig. 11. Fractal analysis of the instantaneous flame front.
when considering thermodiffusional effect of stretched flame. Since the flame front can be viewed as the laminar flamelet, for the turbulent Bunsen flame, this local propagation speed, SL,k, is the averaged value over time and space and is expected to be linear with respect to stretch. Hence, the stretch factor (I0) of turbulent flame represents the ensemble sum of the competing effects of positive and negative stretch over time and space. It can be anticipated that the more reliable W estimation would be prospective if I0 can be evaluated properly. 4.3. Stretch factor, I0 The analysis of the stretch factor starts from the mass balance at the burner exit. At the inlet section, applying the mass balance to the average and instantaneous flame front as Fig. 4a yields:
q0 A0 U 0 ¼ q0 Aav ST
ð32Þ
The turbulent mass consumption speed, ST is:
ST ¼
(a)
(b)
Fig. 10. Position of hci = 0.5 overlaid onto the flame surface density map and the average flame fitting curve.
hmðx;yÞx mðx;yÞx is and hmðx;yÞy mðx;yÞy is was reliable. Results obtained from M1 indicated that a typical direction cosine value of 0.69 is valid for the turbulent Bunsen flame in this study. M2 overestimated W by about 40% under all conditions because of the intrinsic deficiency of the 1=hcos /is estimation. The defect of M2 resulting from the rough identical distribution of / and h even though the model resolve the local flame front wrinkles. For the 2D FSD data, as expected, it underestimates the flame surface density as well as the global fuel consumption rate. Results from these models are in good agreement when the error margins are considered. However, the stretch effect characterized by the Markstein number, Ma, was not taken into account. For a laminar flame, I0 relates the stretch rate to the stretched laminar flame speed [22], SL,k in the following way:
SL;k K ¼ I0 ¼ 1 Ma 2 SL ðSL =a0 Þ
ð31Þ
where K represents the stretch rate of flame, a0 is the diffusivity of the mixture. Obviously, I0 is not always equal to unity especially
A0 U0 Aav
ð33Þ
where Aav is the averaged or the expected surface of the turbulent flame. In this study, the contours of hci = 0.5 is applied and fitted by quadratic least square method to obtain Aav, as shown in Fig. 10. There are also other ways to determine the contours [43]. To derive the ST from the ensemble average of the instantaneous flame, the normalized ST/SL,k, by the local laminar flame speed as shown below:
ST Ain;j ¼ SL;k Aav
ð34Þ
where Ain,j is the jth instantaneous flame front surface area and ðÞ represents the statistical average. Evaluation of Ain,j/Aav can be performed by fractal analysis for the averaged and each individual instantaneous image [44,45]. According to Gouldin et al. [46]:
Ain;j ¼ Aav
D2 1
e0 ei
ð35Þ
where ei, e0, and D2 denote the inner cutoff, outer cutoff, and 2D fractal dimension of each instantaneous flame image, respectively. Figure 11 shows the typical fractal characteristics of the caliper and flame front length. The stretch factor, I0 with the flamelet concept can be deduced by combining (30)–(32). That would be:
SL;k I0 ¼ ¼ SL
A0 U 0 =Aav ðe0 =ei ÞD2 1 SL
! ð36Þ
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5. Conclusions
1.6
I0
1.4
I0
1.2 1.0 0.8 0.6 0.4 0.6
0.8
1.0
1.2
1.4
1.6
u'/S L Fig. 12. Variation of stretch factor, I0, with turbulence intensity.
2.0
2D M1 M2
1.8
M3 M4 M5
1.4 1.2
R
I0corrected
1.6
1.0 0.8 0.6 0.6
0.8
1.0
1.2
1.4
1.6
u'/S L
Flame surface density which is defined as flamelet surface area per unit volume is a key parameter in turbulent combustion models. Turbulent combustion is inherently three-dimensional, and consequently the flame surface density cannot be directly measured by planar 2D laser diagnostics. Five models were evaluated to estimate 3D flame surface density under the common assumption that the measurement plane is a symmetry plane of the mean flow and the additional assumptions. The global fuel consumption rate, W, was obtained by integrating R and the flame stretch factor was evaluated based on fractal analysis of the 2D images. This work examined the various assumptions typically invoked in each model for the estimation of 3D FSD and W, and quantified the error from each model. We expect these results to clarify the error and applicability of these models, and to provide valuable guidance for the processing of 2D planar measurements of turbulent flame properties. Under all turbulence intensities tested in this work, the errors in the results obtained from M1 to M5 were within 57%. Results obtained from M1 indicated that a typical direction cosine value of 0.69 is valid for the turbulent Bunsen flame in this study, and that M1 provided satisfactory estimation of global fuel consumption rate under high turbulent intensities. Results obtained from M2 overestimated W by about 40% under most conditions because of the intrinsic deficiency of the 1=hcos /is evaluation, i.e., the nonidentical cosine value of mean / and h and non-ideal distribution profile of / and h. M3 provides better estimation as well as M4. Results obtained from M4 and M5 provided the best evaluation of the global fuel consumption rate by estimating the normal vector fluctuation in the transverse direction assuming that they are equivalent to those in the horizontal and vertical directions in the measurement plane. For 2D data, as expected, they underestimate flame surface density and W. Lastly, the flame stretch factor, I0, can also be deduced from the fractal analysis of the turbulent flame front, which helps to improve the estimations of the global fuel consumption rate.
Fig. 13. I0 corrected ratio, RI0 corrected , with turbulent intensities.
Acknowledgments Table 5 The standard derivation against unity of ratios between global fuel consumption rate and fuel supply rate.
I0 = 1 I0 corrected
M1
M2
M3
M4
M5
0.1486 0.1373
0.2998 0.2924
0.4476 0.4419
0.1111 0.0957
0.0501 0.0331
This study is supported by National Natural Science Foundation of China (Nos. 51376004, 91441203) and the National Basic Research Program of China (2013CB228406). The Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry is also appreciated. References
This method has been used by Daniele et al. [31]. In this work, 100 flame front images were used for the fractal analysis. Spatial resolution of the image (0.136 mm/pixel) is adequate for this analysis according to Wyngaard [47]. The stretch factor, I0 with turbulent intensities and the error bar representing the standard deviation are depicted in Fig. 12 removing low turbulence intensity (u0 /SL 0.2 and 0.4) for the large uncertainty of fractal characteristics. It can be seen that I0 are around unity, indicating the nearly neutral diffusional effects of CH4/air flame (with / = 0.7–1.0) with respect to stretch [48]. Figure 13 shows the corrected ratios of the global fuel consumption rate to the fuel supply rate, RI0 corrected . Comparing to Fig. 9, it is clear that RI0 corrected are closer to unity giving better estimation on W. This is also verified by the data shown in Table 5, which lists the standard deviation of R against unity with and without the stretch effects.
[1] N. Peters, Proc. Combust. Inst. 21 (1) (1986) 1231–1250. [2] D. Veynante, L. Vervisch, Prog. Energy Combust. Sci. 28 (3) (2002) 193–266. [3] D. Veynante, J. Piana, J.M. Duclos, C. Martel, Proc. Combust. Inst. 26 (1) (1996) 413–420. [4] R. Knikker, D. Veynante, C. Meneveau, Phys. Fluids 16 (11) (2004) L91–L94. [5] K.N.C. Bray, Proc. R. Soc. London Ser. 431 (1882) (1990) 315–335. [6] G.G. Lee, K.Y. Huh, H. Kobayashi, Combust. Flame 122 (1–2) (2000) 43–57. [7] M. Wirth, N. Peters, Proc. Combust. Inst. 24 (1) (1992) 493–501. [8] S. Daniele, P. Jansohn, J. Mantzaras, K. Boulouchos, Proc. Combust. Inst. 33 (2) (2011) 2937–2944. [9] H. Kobayashi, T. Tamura, K. Maruta, T. Niioka, F.A. Williams, Proc. Combust. Inst. 26 (1996) 389–396. [10] R. Sankaran, E.R. Hawkes, J.H. Chen, T. Lu, C.K. Law, Proc. Combust. Inst. 31 (1) (2007) 1291–1298. [11] J. Wang, M. Zhang, Z. Huang, T. Kudo, H. Kobayashi, Combust. Flame 160 (11) (2013) 2434–2441. [12] C. Mounaïm-Rousselle, I. Gökalp, Proc. Combust. Inst. 25 (1) (1994) 1199– 1205. [13] J.B. Bell, M.S. Day, J.F. Grcar, M.J. Lijewski, J.F. Driscoll, S.A. Filatyev, Proc. Combust. Inst. 31 (1) (2007) 1299–1307. [14] J. Hult, S. Gashi, N. Chakraborty, M. Klein, K.W. Jenkins, S. Cant, C.F. Kaminski, Proc. Combust. Inst. 31 (2007) 1319–1326.
M. Zhang et al. / Combustion and Flame 162 (2015) 2087–2097 [15] B.M. Deschamps, G.J. Smallwood, J. Prieur, D.R. Snelling, Ö.L. Gülder, Proc. Combust. Inst. 26 (1) (1996) 427–435. [16] B. Deschamps, A. Boukhalfa, C. Chauveau, I. Gökalp, I.G. Shepherd, R.K. Cheng, Proc. Combust. Inst. 24 (1) (1992) 469–475. [17] E.R. Hawkes, R. Sankaran, J.H. Chen, Proc. Combust. Inst. 33 (1) (2011) 1447– 1454. [18] M. Zhang, J. Wang, Y. Xie, Z. Wei, W. Jin, Z. Huang, H. Kobayashi, Exp. Therm. Fluid Sci. 52 (2014) 288–296. [19] S.A. Filatyev, J.F. Driscoll, C.D. Carter, J.M. Donbar, Combust. Flame 141 (1–2) (2005) 1–21. [20] F. Halter, C. Chauveau, I. Gökalp, Int. J. Hydrogen Energy 32 (13) (2007) 2585– 2592. [21] H.S. Guo, B. Tayebi, C. Galizzi, D. Escudie, Int. J. Hydrogen Energy 35 (20) (2010) 11342–11348. [22] J.F. Driscoll, Prog. Energy Combust. Sci. 34 (1) (2008) 91–134. [23] A.N. Karpetis, R.S. Barlow, Proc. Combust. Inst. 30 (1) (2005) 665–672. [24] J. Hult, A. Omrane, J. Nygren, C. Kaminski, B. Axelsson, R. Collin, P.E. Bengtsson, M. Aldén, Exp. Fluids 33 (2) (2002) 265–269. [25] W. Cai, X. Li, F. Li, L. Ma, Opt. Express 21 (6) (2013) 7050–7064. [26] X. Li, L. Ma, Opt. Express 22 (4) (2014) 4768–4778. [27] T.D. Upton, D.D. Verhoeven, D.E. Hudgins, Exp. Fluids 50 (1) (2011) 125–134. [28] S.S. Shy, R.H. Jang, W.K. I, K.L. Gee, Proc. Combust. Inst. 26 (1) (1996) 283–289. [29] S.S. Shy, W.K. I, E.I. Lee, T.S. Yang, Combust. Flame 118 (4) (1999) 606–618. [30] F. Halter, C. Chauveau, I. Gokalp, D. Veynante, Combust. Flame 156 (3) (2009) 657–664. [31] S. Daniele, J. Mantzaras, P. Jansohn, A. Denisov, K. Boulouchos, J. Fluid Mech. 724 (2013) 36–68. [32] D. Veynante, G. Lodato, P. Domingo, L. Vervisch, E. Hawkes, Exp. Fluids 49 (1) (2010) 267–278.
2097
[33] M. Zhang, J. Wang, Y. Xie, W. Jin, Z. Wei, Z. Huang, H. Kobayashi, Int. J. Hydrogen Energy 38 (26) (2013) 11421–11428. [34] M. Zhang, J. Wang, J. Wu, Z. Wei, Z. Huang, H. Kobayashi, Int. J. Hydrogen Energy 39 (10) (2014) 5176–5185. [35] S.B. Pope, Int. J. Eng. Sci. 26 (5) (1988) 445–469. [36] E.R. Hawkes, R. Sankaran, J.H. Chen, S.A. Kaiser, J.H. Frank, Proc. Combust. Inst. 32 (1) (2009) 1455–1463. [37] N. Chakraborty, E.R. Hawkes, Phys. Fluids 23 (6) (2011) 1–19 (1994–present). [38] R. Cant, S. Pope, K. Bray, Proc. Combust. Inst. (1991) 809–815. [39] I.G. Shepherd, Proc. Combust. Inst. 26 (1) (1996) 373–379. [40] R.J. Kee, J.F. Grcar, M. Smooke, J. Miller, E. Meeks, PREMIX: A Fortran Program for Modeling Steady Laminar One-Dimensional Premixed Flames, Report SAND85-8240, Sandia National Laboratories, 1985. [41] R.J. Kee, F.M. Rupley, J.A. Miller, Chemkin-II: A Fortran Chemical Kinetics Package for the Analysis of Gas-Phase Chemical Kinetics, Report SAND-898009, Sandia National Laboratories, 1989. [42] P. Gregory, D.M. Smith, M. Golden, N.W. Frenklach, B. Moriarty, M. Eiteneer, C.T. Goldenberg, R.K. Bowman, S. Hanson, W.C. Song, J. Gardiner, V.V. Lissianski, Z. Qin.