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Fractals and turbulent premixed engine flames
C O M B U S T I O N A N D F L A M E 77: 295-310 (1989)
295
Fractais and Turbulent Premixed Engine Flames J. MANTZARAS, P. G. FELTON, and F. V. BRACCO Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544
The fractal nature of premixed turbulent flames in an internal combustion engine is examined. A sheet of laser light, approximately 200 #m thick, is shone through the cylinder of a single-cylinder ported internal combustion engine. The homogeneous charge of propane and air is seeded with submicron TiO2 pa~cles and the scattered light is collected through a quartz window in the engine head and is imaged on a I00 × 100 diode array camera. The number density of the Ti02 particles scales with the gas density so that a two-dimensional map of reactants and products is obtained. A field of view 2 x 2 cm in the center of the cylinder is examined and fractal analysis is performed on the front separating reactants from products. Results are presented for two equivalence ratios and three engine speeds, corresponding to different laminar flame speeds Sn, laminar flame thicknesses c5~, turbulent intensities u ' , and Kolmogorov scales 71. The examined flames were found to exhibit fractal character within a range of length scales as low as 200 /~m and as high as 4.5 mm, which is twice the measured lateral integral length scale in this engine configuration. At stoichiometric conditions, the fractal dimension of the flame surface is found to be statistically different for engine speeds of 300, 1200, and 2400 rpm (u'/S~ = 0.5, 2, and 4). It increases with increasing u'/St. At lean conditions (4~ = 0.59), when u '/S~ ~, 1, the fractal dimension does not change with engine speed. For 4 _ u'/St < 50 and 0.1 < ~/6~ _< 1 (estimated ranges), the fractal dimension is 2.36 _+ 3%. A turbulent flame speed model based on fractal analysis is also briefly examined.
INTRODUCTION The structure of turbulent premixed flames has received considerable theoretical and experimental study in recent years. A review by Clavin [1] summarizes current understanding of the dynamics of premixed flame fronts in laminar and turbulent flows. Even though significant progress has been made, our understanding of the turbulent flame structure is still limited, particularly under conditions of turbulence with characteristic length and time scales comparable with the laminar flame thickness and the residence time within the laminar flame. Recent experiments employing laser techniques have provided two-dimensional images of turbulent flame fronts [2-5] in various combustion configurations with controllable turbulent quantities. In particular, Ref. 2-4 examined the structure of turbulent premixed flames in internal combustion engines using two-dimensional laser imaging techniques. The experiments provided direct inCopyright (© 1989 by The Combustion Institute Published by Elsevier Science Publishing Co., Inc. 655 Avenue of the Americas, New York, NY 10010
formation on the turbulent flame structure. However, the quantification of this information and its relation to fundamental turbulent and combustion quantities, such as turbulent intensities and scales and turbulent flame speeds, was just initiated. A principal obstacle to this quantification is that the studies show that the flame front surface can be both wrinkled and highly fragmented and therefore difficult to characterize using classical geometrical concepts. The analysis of rough surfaces or curves has followed an independent mathematical route, which is known as fractal analysis, largely advanced by Mandelbrot [6, 7]. Mandelbrot [8] proposed that scalar isosurfaces in turbulent flows could be described as fractal surfaces and derived 2 two possible values for the fractal dimension; 23 1 for Gauss-Kolmogorov turbulence and 2~ for Gauss-Bergers turbulence. A variety of turbulent flows have since been observed to show fractal behavior, ranging from jet boundaries in free shear flows [9] to storm clouds [10]. If the
0010-2180/89/$03.50
296 structure of the premixed flame surface is determined by the turbulent flow, then the flame surface should also be fractal. Gouldin [11] applied fractals to model turbulent premixed flame speeds and North and Santavicca [5] found that turbulent flames in a flow reactor exhibit fractal character; they examined conditions of low values of the ratio of turbulence intensity u' to laminar flame speed Sj (0.2 < u'/Sj < 0.5). In Ref. 11 the fractal dimension of the turbulent flame surface was hypothesized because direct experimental information at values of u'/Sj > 1 was lacking. In this article, the terminology of fractal analysis, the apparatus and experimental procedure, and the analysis of the flame images are described first. The discussion of the results is then divided into two distinct parts. In the first part we simply consider the question of whether our flame fronts exhibit fractal character. The geometrical properties of the flame front are important in themselves and are not limited by the validity of various theories. In the second part we relate our findings to current theories. Although all of the lean data and some of the stoichiometric data presented here are new, the experimental technique and much of the image processing are described fully in Refs. 4 and 12, which the reader may wish to consult for the discussion of important details.
THEORY OF FRACTALS The application of fractals as a mathematical tool for the analysis of rough surfaces or curves has received increasing interest in recent years. Fractals are objects that display self-similarity over a wide range of scales. That is, a rough curve is a fractal when, as Mandelbrot [6] states, "the specific mechanisms that brought about both small and large details of the curve are geometrically identical except for scale." Consider a rough curve on a plane. Enscribe it by a polygon of N equal sides of length e and measure how the length of the curve changes with the resolution e of the side of the polygon. If the curve is irregular, one cannot in general specify the manner in which the length will increase with increasing resolution e. However, if the curve has
J. MANTZARAS ET AL. similarity (in the statistical sense) at all levels of resolution e, its measured length L will increase according to a power law: L oc ~I-D2.
(1)
The exponent D2 is called the fractal dimension of the curve and is l < D2 < 2. The subscript 2 indicates that the curve is embedded in a twodimensional (2-D) space. The fractal dimension /92 is a measure of the fragmentation and roughness of the curve; the rougher and/or more fragmented the curve, the larger the exponent D2. For smooth curves the fractal dimension 192 reduces to unity, which is the classical Euclidean dimension. An expression similar to Eq. 1 applies also to a rough surface. The measured area A of a surface of a three-dimensional (3-D) fractal object depends on the size ~2 of the square used to resolve it: A oc ~2-D3.
(2)
The exponent D3 is the fractal dimension of the surface, and 2 < /93 < 3. The subscript 3 indicates that the surface is embedded in a 3-D space. It is difficult to measure the fractal dimension of the surface of a 3-D fractal object because it requires the knowledge of its structure in 3-D space; few experimental techniques can provide 3D information on surfaces of interest with adequate resolution. To overcome this difficulty, a plane intersection the 3-D object is considered and the fractal dimension /92 of the plane-object intersection is determined. Then, it is argued [7] that D3 = D2 + 1, provided that the surface of the object is isotropic, i.e., the orientation of the plane of intersection is irrelevant. The main thrust for the application of fractals in the field of turbulence and combustion came when Mandelbrot [8] suggested that surfaces of constant properties of passive scalars in homogeneous and isotropic turbulent flows possess fractal character. These surfaces should exhibit fractal behavior within a range of scales. The range should be limited at the low end by the Kolmogorov scale. A measure of such a surface with higher resolution than the Kolmogorov scale cannot reveal more surface structure, because surface convolution of
FRACTALS AND TURBULENT PREMIXED ENGINE FLAMES finer scale does not exist. At the high end, the fractal character should be bounded by the large scales of the flow, which are comparable with the integral length scale of turbulence. At larger Reynolds numbers the range of scales over which fractal behavior applies becomes larger. Figure 1 shows how the measured area is expected to change as a function of the resolving scale for a surface that appears to be smooth at large measurement scales. According to Eq. 2, the plot of log (A) versus log (~) will be a straight line with slope 2 - D3 in the range of scales where fractal behavior applies. Fractal dimensions have been measured experimentally. Sreenivasan and Meneveau [9] measured the fractal dimension of the interface between the turbulent and the laminar parts of jets and found values of D3 between 2.3 and 2.4. Lovejoy [10] reported the fractal dimension of storm cloud surfaces to be 2.37. In Ref. 5 the fractal dimension of premixed turbulent flame surfaces had values between 2.05 and 2.14. The application of fractals to turbulent combustion flame surfaces has some complications; the laminar flame front is not a constant-property surface of a passive scalar in the context defined in Ref. 8. Implications of this point will be discussed in the section on Inner Cutoff and Turbulent Flame Speed. APPARATUS PROCEDURE
AND EXPERIMENTAL
A ported single-cylinder, homogeneous-charge engine, with a high-speed Waukesha CFR-48 crankcase was used. The operating parameters are given in Table 1. The six intake ports were directed 30 degrees from the cylinder radius to produce swirl, and upward at 30 degrees to improve scavenging efficiency. The engine was water cooled, with cooling passages in the cylinder head, around the cylinder and around the exhaust manifold. The piston was of an elongated design, which reduced the amount of oil that reached the combustion chamber. Three self-lubricating, bronze-impregnated Teflon rings were used, which allowed the engine to be run with minimal cylinder lubrication, thus greatly reducing window
297
LogA
- 3
r
,, ~i
Log~
%
Fig. 1. The fractal behaviorof a surface with inner and outer cutoffs.
fouling. The engine was provided with a quartz ring between the cylinder and cylinder head, the dimensions of which were 8.23 cm i.d., 12.7 cm o.d. and 1.91 cm thickness. This ring was surrounded by an aluminum blast shield (0.32 cm thick) to contain the quartz in the event of window failure. A 5 mm-high slot in the shield provided a window for the laser sheet to pass through the combustion chamber. Optical access to the combustion chamber from above was provided by a 9.53 cm-diameter, 3.81 cm-thick quartz window mounted in the cylinder head. Sketches of the cylinder, the piston and cylinder-head arrangement are shown in Fig. 2. Because two of the three surfaces of the combustion chamber are made of quartz, it was necessary to relocate the spark plug (a single Champion 304-396 type plug with a spark gap of 1 mm) in the top of the piston. A flow chart of the experimental setup is shown
TABLE 1
Engine Parameters Engine Bore Stroke Compression ratio TDC swirl number Clearance height at TDC Timing of intake ports Timing of exhaust ports Connecting rod length
Ported 82.6 mm 114.3 mm 8 4
13 mm + 126 degrees + 113 degrees 253.8 m
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J. MANTZARAS ET AL.
in Fig. 3. Air was supplied to the engine by an air compressor at 0.7 MPa. The gas was then passed through an air filter, a Hastings mass flow meter (of two available in parallel) and a metering valve. The throttled gas was routed through a surge tank, a straight, 1.8 m-long section of 3.8 cm pipe, 0.9 m of 5.08 cm pipe, and then through a 30 degree upward turn into the intake manifold via 0.2 m of 5.08 cm pipe. A perforated plate was installed approximately 0.6 m downstream of the surge tank to break up large scale turbulence and to improve the mixing of air, fuel, and seed. The air flow rate used corresponded to a scavenging ratio of 100% at all engine speeds. After the flow meter, part of the compressed air was sent to a seeding tank containing titanium tetrachloride (TiCI4) vapor through a pressure regulator, a drying cartridge, and control valves. The mixture of the dry air and titanium tetrachloride vapor then flowed through a 0.953 cm-i.d, stainless-steel pipe to the intake pipe where it was injected 0.2 m downstream of the propane injection. The TiCI4 vapor then
CYLINDER HEAD
n [)01
4
I
CYLINDER
I
PISTON
Fig. 2. Schematicof the portedcylinder,elongatedpiston, and cylinder head assembly.
reacted with water vapor in the humid air to form submicron titanium dioxide smoke. Observations of the light scattering in the absence of flames showed the seed distribution to be uniform across the cylinder cross section. The propane was supplied by three 100 lb cylinders of commercial grade propane (96%) used in parallel. The propane was passed through a filter, a Hastings mass flow meter, two Brooks mass flow meters, and a solenoid operated shutoff valve. The flow rate of the propane was metered with a precision value and then injected into the intake pipe just upstream of the perforated plate. Different mass flow meters in the air and propane systems were used for different ranges of flow rates, enabling reasonably accurate measurements to be made over broad ranges. The equivalence ratios used were 1.0 and 0.59 _+ 3%. Firing occurred every 4, 8, and 14 cycles at 300, 1200, and 2400 rpm engine speeds respectively, so that negligible residual products were present during combustion. The engine crankangle and speed were monitored using an optical shaft encoder that sent pulses to an electronic control module at 1 degree intervals. The output of the electronic control module was used to trigger the laser and the camera. The laser used to illuminate the seed was a frequency-doubled (532 nm) Quanta-Ray DCR1A Nd:YAG operated at five pulses per second, thus limiting the experiment to one image per engine cycle. The laser beam was focused into a sheet by a 150 mm focal length cylindrical lens and a 2 m focal length biconvex lens. The thickness of the resulting laser sheet was approximately 200 #m throughout the chamber. The laser pulse lasted less than 10 ns so that the motion of the flame front was frozen. The intensity of the scattered light changes from a high value in the reactants to a low one in the products, corresponding to the change in gas density and hence seed number density. Because the pressure is virtually uniform inside the engine (the flow Mach number if less than 0.1), the large temperature difference between burned and unburned regions results in a large density change. The 2-D flow visualization optics are shown in Fig. 3. The light scattered by the seeding particles was collected from above through the cylinder head
FRACTALS AND TURBULENT PREMIXED ENGINE FLAMES
299
kI~SOR LENS FILTER
LENSES
~ SHAFT ENCOOER
AEC.~ATOfl A-~OTAMETEA
i i TM, Fig. 3. Flow chart of the test rig.
window. A Reticon MC-521 camera (100 x 100 pixels), interfaced via a Reticon RS-521 controller, was used to acquire the images. The camera was mounted directly above the engine on an XY-Z traverse and used a 50 mm, ft.4 lens. The field of view of the camera could be adjusted by changing the height of the camera above the engine and by the use of extension tubes between the lens and the camera. This meant that fields of view from 1 x 1 cm to 9 x 9 cm (the full chamber) could be used. For the images presented here, the central 2 × 2 cm portion of the chamber was used; thus each pixel covers 0.2 x 0.2 mm and also the images are at different (but known) times after spark for different conditions. This field of view was chosen so that the pixel size equals the smallest resolvable scale, which in this experiment is set by the laser sheet thickness (0.2 mm). The laser thickness was measured by shining the laser sheet light onto a white target, imaging with the camera, and digitizing its profile with
high resolution (50 #m per pixel); assuming a Gaussian profile, the 1/e drop point of its peak intensity was used to define its thickness. The location of the field of view with respect to the intake and the spark plug is shown in Fig. 4. The output of the camera was digitized by a Tecmar Labmaster A/D converter and up to 70 frames were stored in the video memory of a Number Nine video adapter board and displayed in real time on the high-resolution (1024 × 768) color monitor of an IBM PC AT compatible computer. Using software developed in the Princeton Engine Laboratory [12] these images could be processed, displayed in false color, and then stored in the mass storage of the computer. ANALYSIS OF THE IMAGES The digitized image from the diode array was corrected for dark noise by subtracting the image obtained while the laser sheet was blocked and the
300
J. MANTZARAS ET AL.
SWIRL
l/
DIRECTION HT
Fig. 4. Schematic of the spark plug, intake, and field of view location.
engine was not firing. It was then corrected for systematic nonuniformities in pixel sensitivity, laser sheet profde, and collection optics by normalizing with a reference image taken under nonfiring conditions [2]. The gray-scale intensity of each pixel could vary from the value of 0 to the value of 255 units, as determined by the 8-bit resolution of the A/D converter. The intensity histograms of the processed flame images show a
strong bimodal distribution. This is because in the flame images with high contrast the transition between products (low intensity) and reactants (high intensity) generally occurs rapidly, i.e., within a couple of pixels. Therefore only a small fraction of the pixels are due to the intermediate flame zone intensity. This is shown in Fig. 5b, where the intensity level is plotted across a line cut through the flame of Fig. 5a; the vertical scale in
~I
•
•
0
°
.A'-',:'...
Cc)
255
INTENSITY
(b)
....:-;'."
Cz."""
Fig. 5. (a) Two-color flame image; white is products and black is reactants. (b) Light intensity plot along the line indicated in Fig. 5a showing separation between low intensity scattering (products) and high intensity scattering (reactants). (c) Intensity histogram of the whole image showing separation between the reactant and product zones.
FRACTALS AND TURBULENT PREMIXED ENGINE FLAMES
301
TABLE 2
Conditions and Results CASE 1 2 3 4 5 6
¢
1.0 1.0 1.0 0.59 0.59 0.59
rpm
Os
O~
2 - D2
tr~
D3
u' /S~
~ (/zm)
~/~1
300 1200 2400 300 1200 2400
15 15 15 30 30 30
9 7 0 8 5 5
0.88 0.71 0.63 0.67 0.64 0.64
0.037 0.059 0.066 0.041 0.067 0.052
2.12 2.29 2.37 2.33 2.36 2.36
0.5 2.0 4.0 2-7 8-28 16-54
30-150 9-50 6-30 30-150 9-50 6-30
3.0 1.5 1.0 0.9 0.5 0.3
Fig. 5b is arbitrary. The intensity histogram of the flame image is also shown in Fig. 5c. In order to determine the flame front location, a two-color image was created using a thresholding technique. A threshold intensity was chosen that was at the valley of the bimodal intensity distribution, and a pixel with an intensity smaller than this threshold is defined to be in the burned side and with an intensity greater than this threshold in the unburned side. Thus a black and white image is produced. The threshold level was chosen so that it accurately represents the flame front location as determined by the intensity cross sections of the type shown in Fig. 5. The threshold level chosen was found to have no influence on the fractal analysis. For the fractal study, flames at three engine speeds and two equivalence ratios ~b were
used. Table 2 shows the conditions used; 0s is the spark timing and 0 i the image timing in crankangle degrees BTDC. Typical two-color 2 × 2 cm flame images corresponding to the above conditions are shown in Figs. 6 and 7. For each condition five images are shown, with each image corresponding to a different firing cycle. The products are represented in white and the reactants in black. These images show that at higher speeds or at lower equivalence ratios the flame front becomes more contorted. Once the borderline (flame front) between reactants and products has been defined via the thresholding technique, its fractal dimension/)2 is calculated. The method followed here is a modified algorithm of that described in the section on fractal theory; it is mentioned in Mandelbrot [6]
Fig. 6. Stoichiometricflame images at various engine speeds. Top row 300 rpm, middle row 1200 rpm, bottom row 2400 rpm. White is products and black is reactants.
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Fig. 7. Lean flame images (~b = 0.59) at various engine speeds. Top row 300 rpm, middle row 1200 rpm, bottom row 2400 rpm. White is products and black is reactants.
and was employed by Sreenivasan and Menevau [9]. This method is particularly suited for the analysis of fragmented images. The algorithm is summarized below. One considers the area of reactants and products that is within a distance e from the flame front. This area is 2eL, where L is the length of the flame front. Substituting for L from Eq. 1, it is seen that this area is proportional to 62-02. The plot of the logarithm of the area versus log (e) has slope 2 - D2 in the region of 6 where fractal behavior applies. The quantity 2 - D2 is called the codimension. The implementation of this method involves the following steps (see also Ref. 9): A circle of radius 6 is drawn around each pixel of the image (reactants as well as products), and it is determined whether this circle intersects the flame front. Let N(e) be the number of pixels whose circles intersect the flame front for a fixed radius e and repeat the process for different e, thus determining the variation of N(e) with 6. Because N(6) represents the area of the strip with width 2E about the flame front (when the unit area is taken to be the pixel area), the previous discussion yields N(6) cx ~2-D2.
2) in that, if the fractal domain is limited, at scales outside the fractal domain (~i > 6 or 6 > Co) the slope does not become zero, as in Fig. 1. For values of e sufficiently larger than eo and sufficiently smaller than ei the slope becomes one. This results from Eq. 3 because in these regions the fractal dimension D2 reduces to the Euclidean dimension of one. For values of e close to ei or 6o the slope changes in an unpredictable manner. Thus, ei and eo are defined by the values of E at which the log [N(e)] versus log (e) curve ceases to be linear. A plot of Eq. 3 is shown in Fig. 8, for a 10 4
N(E) I0 3
(3) 10 2
The log-log plot of Eq. 3 has slope 2 - /92 and hence the fractal dimension is easily calculated. The ideal form of this plot differs from Fig. 1 (Eq.
I0
102
E
Fig. 8. Test run for the fractal behavior of a square geometrical object.
FRACTALS AND TURBULENT PREMIXED ENGINE FLAMES test run where the digitized image was a square of 30 × 30 pixels in the center of the 100 × 100 pixel image and the radius e was increased in integer multiples of the pixel size. The theoretical slope of this regular object should be one, and a value of 0.993 is obtained. For a regular object such as this, no cutoffs should be observed. The outer cutoff shown in Fig. 8 is due to the finite size of the space surrounding the square object. Thus there are some practical limitations in the use of the above algorithm to determine 6i and Co. They include the following. To detect similarity at small scales and to define the inner cutoff, ei, one requires resolution of the digitized image significantly smaller than ~i (in Ref. 9 a factor of at least two is stated). In our case the best resolution dictated by the experiment was, as mentioned in the previous section, 0.2 mm. It is thus expected that, if the physical scales of the flame front are smaller than the pixel size no inner cutoff will be revealed. To determine properly eo, one requires many of the large scales in the finite-size digitized image. To compensate for this effect, 25 images at each condition were analyzed and then ensemble averaging was performed; thus, statistically, the largescale effects were properly accounted for. In addition, as is shown in the next section, the integral length scale (and hence the outer cutoff point, Co) was sufficiently smaller than the 2 cm size of the image. Another possible problem encountered is that the areas of reactants and products can influence the outer cutoff end. If the areas are very different, the flame front is near the border of the image and hence the strip 2eL is confined to a smaller region. Then, if the distance between the flame front and the border of the image is smaller than the natural outer cutoff, the turning point at the high-end cutoff will be affected by it. We reduced this problem by considering only images with approximately equal reactant and product areas (see Figs. 6 and 7). Thus, after the fractal analysis is performed on each of the 25 images for each of the six cases, the mean and standard deviation of the fraetal dimension DE are calculated and the existence of cutoffs is examined. The fractal dimension of the flame
303
surface D3 is then calculated from the relation D3 =D2+ t.
RESULTS AND DISCUSSION This section is divided into two clearly separated parts. In the first part we show that the boundary separating reactants from products unequivocally exhibits fractal behavior. This conclusion is independent of any theory of turbulent combustion. In the second part, we relate the fractal dimension to various theoretical concepts of turbulent combustion.
Fractal Dimension The fractal analysis described in the previous section was applied to the six conditions of Table 2. Plots of log IN(e)] versus log (e) are shown in Fig. 9. Each plot is typical of the 25 flame fronts 104
N(e)
o° / g 0 O
00 / o °
io3
oD D
O
~=l.0
o D
0
D 300 RPI,I
D
o
1200 RPM
0
2400 1~14
0
2xlO 2 I04
N(e)
q~ 0 )0
ooo
¢Do oo o o o
10 3
O ¢
O 300 RPH
oo
o 1200 RP~t
0 2400 RPM
I0
I0 2
E
Fig. 9. Fractal plots of flame images, a. Stoichiometric flames
at three engine speeds; the scale~is in units of 0.2 mm. b. ~an flamesat three engine speeds; the scale c is in units of 0.2 mm.
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J. MANTZARAS ET AL. I0 5
(c) I0 4
N(E) .0 10 3
-: 10 2
,
,
,
9 cm
, ,,,,=
,
,
,
q , ....
I0 E
10 2
Fig. 9. c. F u l l f i ¢ l d f l a m e at lean c o n d i t i o n s s h o w i n g the u p p e r c u t o f f ; the scale e is i n units o f 0 . 9 ram. T h e insert s h o w s the f l a m e i m a g e ; w h i t e is products and b l a c k is reactants.
analyzed in each case. It is seen that there exists fractal behavior over a significant range of scales. A straight line was fitted through the points where fractal behavior applies, and its slope (2 - D2) was calculated. This process was repeated for each image. Figure 10 shows the pdfs of the slopes (2 - D2) obtained from the analysis of the 25 images. The mean slopes, 2 - /)2, and standard deviations, os, of the slopes for the six conditions are given in Table 2. The variation of the slope at fixed conditions is not only due to the finite size ot the image as previously stated, but also to the fact that each cycle is slightly different from all others in an engine. It should be noted that the standard error of the mean gradient is always 2 % or less. Figure 9a shows the fractal plots for the stoichiometric condition and the three engine speeds. The relative standard deviations of the slopes at qb = 1.0 are 4% for 300 rpm and about 10% at 1200 and 2400 rpm. There is a mean slope separation of about three standard deviations between conditions at 300 and 1200 rpm and two standard deviations between 1200 and 2400 rpm. Thus at stoichiometric conditions, the flame fronts at different engine speeds have statistically different fractal dimensions. The slopes of the lean conditions shown in Fig. 9b, on the other hand,
(k = 1.0 300 RPM .... ----
1200 RPM 2400RPM .
,
r] rj
r--
!
o
r-u i L
i
0.4
0.6
-J
"1
v1 "-i I[ I
, 0.8 SLOPE ( 2 - O 2)
•
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1.0
• = (/.59 300 RPM - ~ - 1200RPM - - - - - 2400RPM(b) r - ~F--liti- -
o 0.4
i
i 0.6 0.8 SLOPE ( 2-D 2)
1.0
Fig. 10. Probability density function of the slopes of fractal plots at two equivalence ratios and three engine speeds.
FRACTALS AND TURBULENT PREMIXED ENGINE FLAMES exhibit in general smaller variation with engine speed. At engine speeds of 1200 and 2400 rpm the slopes are statistically the same. It is also seen that at 300 and 1200 rpm the leaner flames have smaller slopes than the stoichiometric ones. We shall comment on these variations in the next section. The fractal dimension 1)3 of the flame surface can be calculated, assuming isotropy, from /)3 = /)2 + 1. Table 2 gives the fractal dimensions of the conditions examined. As explained earlier, at scales large compared to the integral length scale we expect to see a transition from the fractal gradient of the fractal region to a gradient of one. That this is not observed in Figs. 9a and 9b may be due to the finite size of the image that influences the gradient before scales are reached that are large compared to the integral length scale. In order to test this we analyzed some images taken at a lower resolution (9 × 9 cm); in these images a complete section through the flame is seen and the flame did not intersect the image boundary. This allowed the fractal analysis to be performed in a 300 × 300 pixel square with the flame front occupying the center portion of the square, thus eliminating the edge effects. The fractal plot of one of these images (Fig. 9c) clearly shows a transition from a gradient of 0.66, typical of the fractal plots at higher resolution, to a gradient of one, expected for a Euclidean object. The first deviation from the initial gradient occurs at a scale of about 5 to 6 pixels (4.5-5.4 mm), this value taken to be the outer cutoff scale, Co. In order to determine the outer cutoff scale it is therefore necessary to have an image that has flame front boundaries sufficiently far from the image boundaries. The ensemble lateral integral length scale was measured in an identical engine at 600 rpm using a two-point LDV method [13] and its value is about 2 mm around TDC (Fig. 11). The longitudinal integral length scale is twice the lateral one for homogeneous, isotropic turbulence and in our engine there exists some degree of homogeneity and isotropy around TDC [14]. Thus the outer cutoff is observed to be approximately the longitudinal integral length scale. No inner cutoff was observed with our resolution of 0.2 ram, as exemplified in Fig. 9.
305
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........UnSll01)the(~!
_$ ~, 3£ :=
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zo
I.C 600 rpm
TDC i
310
330
i 550
370
390
CRANKANGLE
Fig. l l . Ensemble latcr~ in~gral|engthsc~e m600rpm,
13.
~om Ref.
In the following subsections, we now consider the following theoretical topics: the regimes of premixed turbulent combustion of our data; the possibility that the wrinkled flame front may be a constant property scalar surface of the turbulent field; the question of whether we should have observed an inner cutoff; and some correlations of the turbulent flame speed that include fractal information.
Regimes of Turbulent Combustion For any theoretical consideration, it is first necessary to assess the regime(s) of turbulent combustion of our data. Figure 12 [15] is a plot of the Damkohler number Da^ versus the turbulence Reynolds number RA [16]. The Damkohler number is the ratio of the characteristic turnover time, zt, of an eddy of the size of the integral length scale, A, to the characteristic transit time through the laminar flame, 7"1: DaA--
7"t
--
7"1
A
Sl
(4)
U ~ t~1 '
where Sl is the laminar flame speed, u' the turbulence intensity, and ~l the laminar flame thickness. The turbulence Reynolds number is u'A RA =
(5)
P
Relevant parameters appearing in this plot are the
J. MANTZARAS ET AL.
306
/ Z /
/-~
.... / "!~ /
REACTION SHEETS
'
1
DISTRIBUTEDN 10.4 R/EACTIONS "
/~2
~ ~ IO 4
.
"
~ 10 8
RA
Fig. 12. Regimesof turbulent combustion. The rectangleis an estimate of the regimes applicable to internal combustion engines [16].
ratio of the Kolmogorov scale, ~/, to the laminar flame thickness, and the ratios A/(~1 and u'/Sl. Some understanding of the structure of turbulent flames exists in two regimes; the regimes of reaction sheets and of distributed reactions. In the regime of reaction sheets, turbulence does not affect chemistry and the residence time in the laminar flame is much smaller than the characteristic turnover time of the smallest eddies. In this regime, the propagating flame fronts are just wrinkled and convoluted by the turbulence. In the distributed reaction regime, chemical reactions proceed together with, or after, turbulent mixing, and the concept of laminar flame does not apply. The lower bound of the reaction sheet regime can be set at ~///tl = 1 (the quantity 7/2/8]2 is a measure of the stretch that the flame is subjected to in a turbulent flow) and M/it = 1 is the upper bound of the distributed reaction regime shown by the heavy lines in Figure 12. In the intermediate regions, 7/ < 8~ < A, the structure of the turbulent flame has not yet been identified. The reaction sheet regime is divided in two subregimes. For u'/S1 < 1,
turbulence is weak and a single continuous reaction sheet can be identified; at higher turbulent intensities, adjacent sheets may collide and cut off pockets of reactants, forming multiply-connected reaction sheets. The line u'/Sl = 1 may be taken as the boundary between these subregimes. The rectangle in Fig. 12 indicates the combustion regimes for various engine operating conditions as estimated in Ref. 16. It is seen that most of the operating conditions in an engine are expected to fall in the reaction sheet regime. For our stoichiometric conditions, and using the correlations of Metghalchi and Keck [17], the laminar flame speed is estimated to be 1.25 m/s + 5%. For our lean conditions, Sz would be estimated to be between 0.09 and 0.31 m/s by extrapolating the results of [17]. For an identical engine the turbulence intensity was measured [18] using LDV. Figure 13 shows the turbulence intensity near TDC at various engine locations and engine speeds under nonfiring conditions. The turbulence intensities at the times at which the flame images were taken are estimated from Fig. 13 with an uncertainty of + 7 % (Table 2). The laminar flame thickness was estimated in Ref. 16 to be between 10 and 50 #m at stoichiometric conditions. For the lean conditions it was estimated to be between 50 and 200 #m, the upper bound being set by the measurements of Ref. 4. The estimated values of 71are given in Table 2. It is seen that the ratio ~/8] is expected to be of order
10 E
PORTED SWIRL
~
z _z r..) Z
RPM 2400
::
:~i?il;;
]8oo
TDC 0 --45
I
--30
J
I
--15
J
[
O
i
I
15
30
CRANKANGLE Fig. 13. Turbulenceintensities around TDC for three engine speeds, at various engine locations from [18].
FRACTALS AND TURBULENT PREMIXED ENGINE FLAMES
307
and we expect the structure of the laminar flame itself to be altered by turbulence. In any case, the results of Fig. 14 are in good agreement with those of Ref. 5, where D3 was found to increase with increasing u'/SI from the value of 2.05 at u"/$1 0.2 to the value of 2.14 at u'/S1 ~ 0.5.
2.6
2.4
D3
Inner Cutoff
2.2
0
qb ~ 1.0
o ¢~= 0 59
2.C I0 -I
......................... I u'IS L
I0
F i g . 14. Plot o f f r a c t a l d i m e n s i o n D 3 v e r s u s
I0 :~
u'/S~.
one for the stoichiometric cases and smaller than order one for the lean cases. Thus our stoichiometric flames are expected to fall within the multiply-connected reaction sheet regime, i.e., to be composed of laminar flames whose structure is unaffected by turbulence, whereas our lean flames are expected to fall within the intermediate regime so that the structure of their laminar flames is expected to be affecled by turbulence.
Constant-Property Scalar Surface We can now address the question as to whether the wrinkled flame fronts may act as a constantproperty scalar surface of the turbulent field. Figure 14 shows a plot of/)3 versus u '/$1, with the vertical error bars representing the standard deviation of /)3. The fractal dimension D3 increases with increasing u'/Sl for u'/S~ < 4 and seems to reach a constant value at 2.36 _+ 3% for u'/S1 > 4. The limit value o f / ) 3 happens to coincide with those measured by Sreenivasan and Meneveau in turbulent jets [9] and by Lovejoy in storm clouds [10]. Thus, as far as the fractal dimension is concerned, the laminar flame surfaces of our lean turbulent flames act as if they were constant-property scalar surfaces of the turbulent field. If confirmed, this behavior must result from a rather special balance of events since for our lean flames we have estimated 7//~ < 1
In general the laminar flame front is expected to be smoother than that implied by the unperturbed turbulent field [19]. The smoothing of the flame surface is predominant in the small scales whereas the larger scales tend to remain unaffected [11]. This suggests that the inner cutoff ei tends to shift to higher values than the Kolmogorov scale but also that the fractal behavior of turbulence should be recovered for sufficiently large values of e. For ~//6j > 1 and u'/S1 ,~ 1, Gouldin [11] suggests that the effect of the laminar flame propagation is predominant in determining the roughness of the flame surface, so that the inner cutoff ei approaches the outer cutoff ~o. For ~]/Sj > 1 and high values of u'/SI, turbulence becomes important in determining the roughness of the flame surface even though the effect of $1 persists in smoothing the small scales. In this case the inner cutoff increases but the fractal dimension of turbulence should still prevail at large values of ~ (our values of D3 at ~b = 1 and 300, 1200 rpm suggest that wrinkles of all sizes are affected by combustion for ~1/6~ ~ 1, u'/S~ 1). Peters [20] argued that in the multiplyconnected sheet regime the inner cutoff shifts from ~/to another scale, the Gibson scale Lg:
Lg = A( S1/u')3.
(6)
The underlying assumption in Eq. 6 is that the laminar flame propagation smoothens turbulent eddies with circumferential velocity less or equal to SI. The size of the turbulent eddy that has circumferential velocity equal to $1 is the Gibson scale. When ~//~i1 > 1 and u ' / S l ~, 1 the effect of turbulence is predominant and the inner cutoff should become that of the turbulent field, i.e., the Kolmogorov scale [11].
308
J. MANTZARAS ET AL.
For our data ~/ is always smaller than the 200 a m resolution of our measurements (see Table 2) and we estimate the Gibson scale to be 250 #m at 1200 rpm and 32 #m at 2400 rpm, using u'/Sl of Table 2 and A = 2 mm for ~b = 0.59. Thus, current estimates suggest that the inner cutoff is smaller than the resolution of our technique and could not be observed.
20
Turbulent Flame Speeds The following method to estimate the turbulent flame speeds based on fractals was suggested by Gouldin [11, 21] for ~/~1 > 1 and u'/Sj ~, 1. He states that the effect of turbulence on the flame surface is to increase the ensemble-average flame area by the ratio of the areas of the inner and outer cutoff Ai/Ao (Fig. 1). Then,
St~S1= Ai/Ao,
where St is the turbulent flame speed. Since ~//~l be the Kolmogorov scale ~/, and substituting Eq. 2 in Eq. 7, one obtains [11] ( A / 7 / ) 03 -2.
u'/S~ Fig. 15. Turbulent flame speeds in engines. Solid bars are results from Ref. 4 and open bars (shifted horizontally for clarity) are fractal predictions.
(7)
and u'/St are large, the inner cutoff is expected to
St~St =
I
(8)
Equation 8 was applied to cases 2 and 3 of Table 2, which come closer (~/~l = 1 and u'/Si > 1) to satisfying the conditions necessary for its validity. A was replaced by the measured cutoff Eo = 4.5 mm and/93 was taken from Table 2, assuming isotropic flame surfaces. (Note that even if eo is underestimated because of the effects of the finite image size this will not have a large effect on St/S~ due to the small value of D3 - 2. For example, an unlikely underestimate of eo by a factor of two will lead to an underestimate of St/Sj by less than 30%.) The results are shown by the open bars in Fig. 15. The solid bars are estimates of St~S1 from Ref. 4. There St~S1 was calculated by measuring the turbulent flame front length directly from flame images and dividing it with a reference laminar flame length that was taken to be either the perimeter of a circle with the same burned area or the flame front length of the ensemble-average flame front. Then, assuming isotropy of wrinkling in the direction perpendicular to the plane of the
image, the ratio of turbulent to laminar flame surface areas was calculated by squaring the flame length ratio since the fractal nature of the flame front was not known at that time. The fractal analysis of this paper is seen to give St~S1 lower than in Ref. 4 by some 35% at u'/Sl = 2 and 30% at u'/Sl = 4. Since both Ref. 4 and this work measure the flame length ratio of the flame images obtained with the same apparatus, we can reconcile the differences in the flame length ratios and hence in St~S1 as follows. In Ref. 4 the length of the turbulent flame front L r was measured using a yardstick E equal to 0.2 mm. The length of the corresponding laminar flame front LR was taken to be that of a circle of same area with the flame area, or that of a reference ensemble flame measured with the same yardstick of 0.2 mm. If we want the corresponding lengths measured with yardstick 7, then
LT,I=LT~(rl/~.)I-D2,
(9) (10)
LR,~ = LR~ = LR.
In Eq. 10, D2 = 1 because the reference laminar front is Euclidean. Thus
LT,JLR = [LT /LR](~I/¢)
I -D2.
(11)
FRACTALS AND TURBULENT PREMIXED ENGINE FLAMES The quantity in the brackets was measured by Ref. 4, and according to Gouldin [11], L.r,~/LR is equal to St/SI (for a fractal isotropic surface Eqs. 1 and 2 show that length ratios are equal to area ratios). In Eq. 11 the ratio LT,~/LR can be obtained using values of ~/from Table 2 and the bracketed term from Ref. 4, and it represents an adjustment of the estimates of Ref. 4 that considers the fractal behavior of the flame fronts. The fractal predictions (Eq. 8) and the adjusted estimates of Ref. 4 (Eq. 11) are plotted in Fig. 16. It is seen that the fractal predictions fall within the adjusted estimates of Ref. 4. The solid lines in Fig. 16 are correlations proposed by various researchers [4, 22, 23, 24]. Although the two magnitudes predicted by Eq. 8 are acceptable, the trends implied by Eq. 8 are not acceptable for engine applications. Thus, for D3 = 2.33, A/rl o: RA°.75, and A and ~ constant, it gives St/S] - ( u ' A / v ) °-25, where the exponent of u' is much too small for engine combustion. An exponent of at least 0.7 is more realistic. If, instead of using the Kolmogorov scale as the inner cutoff, one uses the Gibson scale, then Eq. 11 is replaced by (12)
LTg/LR = [LTe/LR](Lg/e) 1-/92,
2O
/
Q
oquation 8
•
equation
equation I t
5t/S;
~/__ .
\
-
~
L2 ~ 7
/
IO ~
% ....
; ....
i ....
i ....
i ....
5
u'/sL
Fig. 16. Turbulent flame speeds obtained with fractal analysis aM correlations suggested by various authors (O, A shifted horizontally for clarity).
309
and Eq. 7 gives St/Sl = (u' /S1) 3(0~- 2).
(13)
Adjusting the estimates of Ref. 4 with Eq. 12 and taking D3 from Table 2, one finds that the turbulent flame speed predictions fall at the lower end of the predictions of both Eq. 8 and 11, as shown in Fig. 16. For D3 = 2.33, Eq. 13 gives St - u ' , which is acceptable for fully-developed turbulent engine flames. For/23 = 2.23 and with a coefficient of 3.5, Eq. 13 would become equal to Klimov's correlation [24] for u ' / S l ~, 1 (Fig. 16). But, D3 = 2.23 is smaller than we have measured. Thus more work is needed to relate fractal analysis to turbulent flame speed even in the simplest of cases in which ~/5~ >- 1 and u ' / S ] ~. 1.
CONCLUSIONS The structure of the turbulent flame fronts in an internal combustion engine was shown to be fractal in nature for scales as small as 200 t~m and as large as twice the lateral integral length scale, for six engine conditions corresponding to different turbulent intensities and equivalence ratios. At stoichiometric conditions the fractal dimension of the flame surfaces was found to be statsfically different for engine speeds of 300, 1200, and 2400 rpm and to increase with engine speed and therefore u'/S1, which ranges from 0.5 to 4.0. At lean conditions (~b = 0.59) where u ' / S l ~,. 1 the fractal dimension did not change with engine speed. For 50 > u' /S1 > 4 and 0.1 < rl/6~ <- 1 (estimated ranges), the fractal dimension was 2.36% + 3% for both stoichiometric and lean conditions. This is in good agreement with the fractal dimension determined for other turbulent flows [9, 10], which is about 2.35. Thus, as far as the fractal dimension is concerned, the laminar flame surfaces of our lean turbulent flames act as if they were constant-property scalar surfaces of the turbulent field. At lower values of u ' / S t the flame propagation becomes important in determining the wrinkling of the flame front, although the flame front is still a fractal with reduced fractal dimension.
310 No inner cutoff was observed at any condition with the resolution of 0.2 mm of this experiment. The outer cutoff occurred at about 4.5 mm, which is twice the measured ensemble lateral integral length scale for this engine. A turbulent flame speed model based on fractals was examined for two of our stoichiometric cases that appear to satisfy the conditions of the model. For the two cases, St/Sl had also been determined by a different technique. The values given by the fractal model for St/SI are not too far from the range of values obtained by other researchers. But the conclusion is reached that more work is needed to relate fractal analysis to turbulent flame speed even in the simplest of cases in which 7//~1 -< 1 and u'/S~ ~ 1.
J. MANTZARAS ET AL.
6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
16.
Support for this work was provided by the Department o f Energy, Office o f Energy Utilization Research, Energy Conservation and Utilization Technologies Program (Contract DEAS-O4-86AL33209), General Motors Corp., Ford Motor Corp., and Cummins Engine Co.
17. 18. 19. 20.
REFERENCES I. 2.
Clavin, P., Prog. Ener. Combust. Sci. 11:1-57 0985). zur Loye, A. O., Santavicca, D. A., and Bracco, F. V., Preliminary study of flame structure in an internal combustion engine using 2-D flow visualization, International Symposium on Diagnostics and Modelling of Combustion in Reciprocating Engines (COMODIA 85), Tokyo, Japan 1985. 3. Baritaud, T., and Green, R., SAE Paper No. 860025, 1986. 4. zur Loye, A. O., and Bracco, F. V., SAE Paper No. 870454, 1987. 5. North, G. L., and Santavicca, D. A., Fractal analysis of premixed turbulent flame structure, presented at the
21. 22.
23. 24.
Eastern States Combustion Institute Meeting, Puerto Rico 1986, and personal communication. Mandelbrot, B. B., Fractals, W.H. Freeman, San Francisco, 1977. Mandelbrot, B. B., The Fractal Geometry of Nature, W.H. Freeman, San Francisco, 1982. Mandelbrot, B. B., J. Fluid Mech. 72:401--416 (1975). Sreenivasan, K. R., and Meneveau, C., J. FluidMech. 173:357-386 (1986). Lovejoy, S., Science 216:185-187 (1982). Gouldin, F. C., Combust. Flame 68:249-266 (1987). zur Loye, A. O., Ph.D. thesis No. 1768-T, Princeton University, February 1987. Fraser, R. A., Felton, P. G., Santavicca, D. A., and Bracco, F. V., SAE paper No. 860021, 1986. Hall, M. J., and Bracco, F. V., SAE paper No. 870453, 1987. Bray, K. N. C., in Turbulent Reacting Flows (P. A. Libby and F. A. Williams, Eds.), Springer-Verlag, Berlin, 1980. Abraham, J., Williams, F. A., and Bracco, F. V., SAE paper No. 850345, 1985. Metghalchi, M., and Keck, J. C., Combust. Flame 38:143-154 (1980). Liou, T., Hall, M., Santavicca, D. A., and Bracco, F. V., SAE paper No. 8403775, 1984. Williams, F. A., Combustion Theory, Benjamin-Cummiags, Menlo Park, CA, 1985. Peters, N., Twenty-First Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1986, pp. 1231-1250. Gouldin, F. C., U.S. Army Research Office Report No. DAAG29-82-K-0187, Cornell University, 1987. Mattavi, J. N. Groff, E. G., and Matekunas, F. A., Turbulence, flame motion and combustion chamber geometry--Their interactions in a lean-combustion engine, Proceedings of the I. Mech. E. Conf. on Fuel Economy and Emissions of Lean Burn Engines, 1979. Witze, P. O. and Mendes-Lopes, J. M. C., SAE Paper No. 861531, 1986. Klimov, A. M., Prog. Astronaut. Aeronaut., 88:133146 (1983).
Received 5 January 1988; revised 22 September 1988
295
Fractais and Turbulent Premixed Engine Flames J. MANTZARAS, P. G. FELTON, and F. V. BRACCO Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544
The fractal nature of premixed turbulent flames in an internal combustion engine is examined. A sheet of laser light, approximately 200 #m thick, is shone through the cylinder of a single-cylinder ported internal combustion engine. The homogeneous charge of propane and air is seeded with submicron TiO2 pa~cles and the scattered light is collected through a quartz window in the engine head and is imaged on a I00 × 100 diode array camera. The number density of the Ti02 particles scales with the gas density so that a two-dimensional map of reactants and products is obtained. A field of view 2 x 2 cm in the center of the cylinder is examined and fractal analysis is performed on the front separating reactants from products. Results are presented for two equivalence ratios and three engine speeds, corresponding to different laminar flame speeds Sn, laminar flame thicknesses c5~, turbulent intensities u ' , and Kolmogorov scales 71. The examined flames were found to exhibit fractal character within a range of length scales as low as 200 /~m and as high as 4.5 mm, which is twice the measured lateral integral length scale in this engine configuration. At stoichiometric conditions, the fractal dimension of the flame surface is found to be statistically different for engine speeds of 300, 1200, and 2400 rpm (u'/S~ = 0.5, 2, and 4). It increases with increasing u'/St. At lean conditions (4~ = 0.59), when u '/S~ ~, 1, the fractal dimension does not change with engine speed. For 4 _ u'/St < 50 and 0.1 < ~/6~ _< 1 (estimated ranges), the fractal dimension is 2.36 _+ 3%. A turbulent flame speed model based on fractal analysis is also briefly examined.
INTRODUCTION The structure of turbulent premixed flames has received considerable theoretical and experimental study in recent years. A review by Clavin [1] summarizes current understanding of the dynamics of premixed flame fronts in laminar and turbulent flows. Even though significant progress has been made, our understanding of the turbulent flame structure is still limited, particularly under conditions of turbulence with characteristic length and time scales comparable with the laminar flame thickness and the residence time within the laminar flame. Recent experiments employing laser techniques have provided two-dimensional images of turbulent flame fronts [2-5] in various combustion configurations with controllable turbulent quantities. In particular, Ref. 2-4 examined the structure of turbulent premixed flames in internal combustion engines using two-dimensional laser imaging techniques. The experiments provided direct inCopyright (© 1989 by The Combustion Institute Published by Elsevier Science Publishing Co., Inc. 655 Avenue of the Americas, New York, NY 10010
formation on the turbulent flame structure. However, the quantification of this information and its relation to fundamental turbulent and combustion quantities, such as turbulent intensities and scales and turbulent flame speeds, was just initiated. A principal obstacle to this quantification is that the studies show that the flame front surface can be both wrinkled and highly fragmented and therefore difficult to characterize using classical geometrical concepts. The analysis of rough surfaces or curves has followed an independent mathematical route, which is known as fractal analysis, largely advanced by Mandelbrot [6, 7]. Mandelbrot [8] proposed that scalar isosurfaces in turbulent flows could be described as fractal surfaces and derived 2 two possible values for the fractal dimension; 23 1 for Gauss-Kolmogorov turbulence and 2~ for Gauss-Bergers turbulence. A variety of turbulent flows have since been observed to show fractal behavior, ranging from jet boundaries in free shear flows [9] to storm clouds [10]. If the
0010-2180/89/$03.50
296 structure of the premixed flame surface is determined by the turbulent flow, then the flame surface should also be fractal. Gouldin [11] applied fractals to model turbulent premixed flame speeds and North and Santavicca [5] found that turbulent flames in a flow reactor exhibit fractal character; they examined conditions of low values of the ratio of turbulence intensity u' to laminar flame speed Sj (0.2 < u'/Sj < 0.5). In Ref. 11 the fractal dimension of the turbulent flame surface was hypothesized because direct experimental information at values of u'/Sj > 1 was lacking. In this article, the terminology of fractal analysis, the apparatus and experimental procedure, and the analysis of the flame images are described first. The discussion of the results is then divided into two distinct parts. In the first part we simply consider the question of whether our flame fronts exhibit fractal character. The geometrical properties of the flame front are important in themselves and are not limited by the validity of various theories. In the second part we relate our findings to current theories. Although all of the lean data and some of the stoichiometric data presented here are new, the experimental technique and much of the image processing are described fully in Refs. 4 and 12, which the reader may wish to consult for the discussion of important details.
THEORY OF FRACTALS The application of fractals as a mathematical tool for the analysis of rough surfaces or curves has received increasing interest in recent years. Fractals are objects that display self-similarity over a wide range of scales. That is, a rough curve is a fractal when, as Mandelbrot [6] states, "the specific mechanisms that brought about both small and large details of the curve are geometrically identical except for scale." Consider a rough curve on a plane. Enscribe it by a polygon of N equal sides of length e and measure how the length of the curve changes with the resolution e of the side of the polygon. If the curve is irregular, one cannot in general specify the manner in which the length will increase with increasing resolution e. However, if the curve has
J. MANTZARAS ET AL. similarity (in the statistical sense) at all levels of resolution e, its measured length L will increase according to a power law: L oc ~I-D2.
(1)
The exponent D2 is called the fractal dimension of the curve and is l < D2 < 2. The subscript 2 indicates that the curve is embedded in a twodimensional (2-D) space. The fractal dimension /92 is a measure of the fragmentation and roughness of the curve; the rougher and/or more fragmented the curve, the larger the exponent D2. For smooth curves the fractal dimension 192 reduces to unity, which is the classical Euclidean dimension. An expression similar to Eq. 1 applies also to a rough surface. The measured area A of a surface of a three-dimensional (3-D) fractal object depends on the size ~2 of the square used to resolve it: A oc ~2-D3.
(2)
The exponent D3 is the fractal dimension of the surface, and 2 < /93 < 3. The subscript 3 indicates that the surface is embedded in a 3-D space. It is difficult to measure the fractal dimension of the surface of a 3-D fractal object because it requires the knowledge of its structure in 3-D space; few experimental techniques can provide 3D information on surfaces of interest with adequate resolution. To overcome this difficulty, a plane intersection the 3-D object is considered and the fractal dimension /92 of the plane-object intersection is determined. Then, it is argued [7] that D3 = D2 + 1, provided that the surface of the object is isotropic, i.e., the orientation of the plane of intersection is irrelevant. The main thrust for the application of fractals in the field of turbulence and combustion came when Mandelbrot [8] suggested that surfaces of constant properties of passive scalars in homogeneous and isotropic turbulent flows possess fractal character. These surfaces should exhibit fractal behavior within a range of scales. The range should be limited at the low end by the Kolmogorov scale. A measure of such a surface with higher resolution than the Kolmogorov scale cannot reveal more surface structure, because surface convolution of
FRACTALS AND TURBULENT PREMIXED ENGINE FLAMES finer scale does not exist. At the high end, the fractal character should be bounded by the large scales of the flow, which are comparable with the integral length scale of turbulence. At larger Reynolds numbers the range of scales over which fractal behavior applies becomes larger. Figure 1 shows how the measured area is expected to change as a function of the resolving scale for a surface that appears to be smooth at large measurement scales. According to Eq. 2, the plot of log (A) versus log (~) will be a straight line with slope 2 - D3 in the range of scales where fractal behavior applies. Fractal dimensions have been measured experimentally. Sreenivasan and Meneveau [9] measured the fractal dimension of the interface between the turbulent and the laminar parts of jets and found values of D3 between 2.3 and 2.4. Lovejoy [10] reported the fractal dimension of storm cloud surfaces to be 2.37. In Ref. 5 the fractal dimension of premixed turbulent flame surfaces had values between 2.05 and 2.14. The application of fractals to turbulent combustion flame surfaces has some complications; the laminar flame front is not a constant-property surface of a passive scalar in the context defined in Ref. 8. Implications of this point will be discussed in the section on Inner Cutoff and Turbulent Flame Speed. APPARATUS PROCEDURE
AND EXPERIMENTAL
A ported single-cylinder, homogeneous-charge engine, with a high-speed Waukesha CFR-48 crankcase was used. The operating parameters are given in Table 1. The six intake ports were directed 30 degrees from the cylinder radius to produce swirl, and upward at 30 degrees to improve scavenging efficiency. The engine was water cooled, with cooling passages in the cylinder head, around the cylinder and around the exhaust manifold. The piston was of an elongated design, which reduced the amount of oil that reached the combustion chamber. Three self-lubricating, bronze-impregnated Teflon rings were used, which allowed the engine to be run with minimal cylinder lubrication, thus greatly reducing window
297
LogA
- 3
r
,, ~i
Log~
%
Fig. 1. The fractal behaviorof a surface with inner and outer cutoffs.
fouling. The engine was provided with a quartz ring between the cylinder and cylinder head, the dimensions of which were 8.23 cm i.d., 12.7 cm o.d. and 1.91 cm thickness. This ring was surrounded by an aluminum blast shield (0.32 cm thick) to contain the quartz in the event of window failure. A 5 mm-high slot in the shield provided a window for the laser sheet to pass through the combustion chamber. Optical access to the combustion chamber from above was provided by a 9.53 cm-diameter, 3.81 cm-thick quartz window mounted in the cylinder head. Sketches of the cylinder, the piston and cylinder-head arrangement are shown in Fig. 2. Because two of the three surfaces of the combustion chamber are made of quartz, it was necessary to relocate the spark plug (a single Champion 304-396 type plug with a spark gap of 1 mm) in the top of the piston. A flow chart of the experimental setup is shown
TABLE 1
Engine Parameters Engine Bore Stroke Compression ratio TDC swirl number Clearance height at TDC Timing of intake ports Timing of exhaust ports Connecting rod length
Ported 82.6 mm 114.3 mm 8 4
13 mm + 126 degrees + 113 degrees 253.8 m
298
J. MANTZARAS ET AL.
in Fig. 3. Air was supplied to the engine by an air compressor at 0.7 MPa. The gas was then passed through an air filter, a Hastings mass flow meter (of two available in parallel) and a metering valve. The throttled gas was routed through a surge tank, a straight, 1.8 m-long section of 3.8 cm pipe, 0.9 m of 5.08 cm pipe, and then through a 30 degree upward turn into the intake manifold via 0.2 m of 5.08 cm pipe. A perforated plate was installed approximately 0.6 m downstream of the surge tank to break up large scale turbulence and to improve the mixing of air, fuel, and seed. The air flow rate used corresponded to a scavenging ratio of 100% at all engine speeds. After the flow meter, part of the compressed air was sent to a seeding tank containing titanium tetrachloride (TiCI4) vapor through a pressure regulator, a drying cartridge, and control valves. The mixture of the dry air and titanium tetrachloride vapor then flowed through a 0.953 cm-i.d, stainless-steel pipe to the intake pipe where it was injected 0.2 m downstream of the propane injection. The TiCI4 vapor then
CYLINDER HEAD
n [)01
4
I
CYLINDER
I
PISTON
Fig. 2. Schematicof the portedcylinder,elongatedpiston, and cylinder head assembly.
reacted with water vapor in the humid air to form submicron titanium dioxide smoke. Observations of the light scattering in the absence of flames showed the seed distribution to be uniform across the cylinder cross section. The propane was supplied by three 100 lb cylinders of commercial grade propane (96%) used in parallel. The propane was passed through a filter, a Hastings mass flow meter, two Brooks mass flow meters, and a solenoid operated shutoff valve. The flow rate of the propane was metered with a precision value and then injected into the intake pipe just upstream of the perforated plate. Different mass flow meters in the air and propane systems were used for different ranges of flow rates, enabling reasonably accurate measurements to be made over broad ranges. The equivalence ratios used were 1.0 and 0.59 _+ 3%. Firing occurred every 4, 8, and 14 cycles at 300, 1200, and 2400 rpm engine speeds respectively, so that negligible residual products were present during combustion. The engine crankangle and speed were monitored using an optical shaft encoder that sent pulses to an electronic control module at 1 degree intervals. The output of the electronic control module was used to trigger the laser and the camera. The laser used to illuminate the seed was a frequency-doubled (532 nm) Quanta-Ray DCR1A Nd:YAG operated at five pulses per second, thus limiting the experiment to one image per engine cycle. The laser beam was focused into a sheet by a 150 mm focal length cylindrical lens and a 2 m focal length biconvex lens. The thickness of the resulting laser sheet was approximately 200 #m throughout the chamber. The laser pulse lasted less than 10 ns so that the motion of the flame front was frozen. The intensity of the scattered light changes from a high value in the reactants to a low one in the products, corresponding to the change in gas density and hence seed number density. Because the pressure is virtually uniform inside the engine (the flow Mach number if less than 0.1), the large temperature difference between burned and unburned regions results in a large density change. The 2-D flow visualization optics are shown in Fig. 3. The light scattered by the seeding particles was collected from above through the cylinder head
FRACTALS AND TURBULENT PREMIXED ENGINE FLAMES
299
kI~SOR LENS FILTER
LENSES
~ SHAFT ENCOOER
AEC.~ATOfl A-~OTAMETEA
i i TM, Fig. 3. Flow chart of the test rig.
window. A Reticon MC-521 camera (100 x 100 pixels), interfaced via a Reticon RS-521 controller, was used to acquire the images. The camera was mounted directly above the engine on an XY-Z traverse and used a 50 mm, ft.4 lens. The field of view of the camera could be adjusted by changing the height of the camera above the engine and by the use of extension tubes between the lens and the camera. This meant that fields of view from 1 x 1 cm to 9 x 9 cm (the full chamber) could be used. For the images presented here, the central 2 × 2 cm portion of the chamber was used; thus each pixel covers 0.2 x 0.2 mm and also the images are at different (but known) times after spark for different conditions. This field of view was chosen so that the pixel size equals the smallest resolvable scale, which in this experiment is set by the laser sheet thickness (0.2 mm). The laser thickness was measured by shining the laser sheet light onto a white target, imaging with the camera, and digitizing its profile with
high resolution (50 #m per pixel); assuming a Gaussian profile, the 1/e drop point of its peak intensity was used to define its thickness. The location of the field of view with respect to the intake and the spark plug is shown in Fig. 4. The output of the camera was digitized by a Tecmar Labmaster A/D converter and up to 70 frames were stored in the video memory of a Number Nine video adapter board and displayed in real time on the high-resolution (1024 × 768) color monitor of an IBM PC AT compatible computer. Using software developed in the Princeton Engine Laboratory [12] these images could be processed, displayed in false color, and then stored in the mass storage of the computer. ANALYSIS OF THE IMAGES The digitized image from the diode array was corrected for dark noise by subtracting the image obtained while the laser sheet was blocked and the
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J. MANTZARAS ET AL.
SWIRL
l/
DIRECTION HT
Fig. 4. Schematic of the spark plug, intake, and field of view location.
engine was not firing. It was then corrected for systematic nonuniformities in pixel sensitivity, laser sheet profde, and collection optics by normalizing with a reference image taken under nonfiring conditions [2]. The gray-scale intensity of each pixel could vary from the value of 0 to the value of 255 units, as determined by the 8-bit resolution of the A/D converter. The intensity histograms of the processed flame images show a
strong bimodal distribution. This is because in the flame images with high contrast the transition between products (low intensity) and reactants (high intensity) generally occurs rapidly, i.e., within a couple of pixels. Therefore only a small fraction of the pixels are due to the intermediate flame zone intensity. This is shown in Fig. 5b, where the intensity level is plotted across a line cut through the flame of Fig. 5a; the vertical scale in
~I
•
•
0
°
.A'-',:'...
Cc)
255
INTENSITY
(b)
....:-;'."
Cz."""
Fig. 5. (a) Two-color flame image; white is products and black is reactants. (b) Light intensity plot along the line indicated in Fig. 5a showing separation between low intensity scattering (products) and high intensity scattering (reactants). (c) Intensity histogram of the whole image showing separation between the reactant and product zones.
FRACTALS AND TURBULENT PREMIXED ENGINE FLAMES
301
TABLE 2
Conditions and Results CASE 1 2 3 4 5 6
¢
1.0 1.0 1.0 0.59 0.59 0.59
rpm
Os
O~
2 - D2
tr~
D3
u' /S~
~ (/zm)
~/~1
300 1200 2400 300 1200 2400
15 15 15 30 30 30
9 7 0 8 5 5
0.88 0.71 0.63 0.67 0.64 0.64
0.037 0.059 0.066 0.041 0.067 0.052
2.12 2.29 2.37 2.33 2.36 2.36
0.5 2.0 4.0 2-7 8-28 16-54
30-150 9-50 6-30 30-150 9-50 6-30
3.0 1.5 1.0 0.9 0.5 0.3
Fig. 5b is arbitrary. The intensity histogram of the flame image is also shown in Fig. 5c. In order to determine the flame front location, a two-color image was created using a thresholding technique. A threshold intensity was chosen that was at the valley of the bimodal intensity distribution, and a pixel with an intensity smaller than this threshold is defined to be in the burned side and with an intensity greater than this threshold in the unburned side. Thus a black and white image is produced. The threshold level was chosen so that it accurately represents the flame front location as determined by the intensity cross sections of the type shown in Fig. 5. The threshold level chosen was found to have no influence on the fractal analysis. For the fractal study, flames at three engine speeds and two equivalence ratios ~b were
used. Table 2 shows the conditions used; 0s is the spark timing and 0 i the image timing in crankangle degrees BTDC. Typical two-color 2 × 2 cm flame images corresponding to the above conditions are shown in Figs. 6 and 7. For each condition five images are shown, with each image corresponding to a different firing cycle. The products are represented in white and the reactants in black. These images show that at higher speeds or at lower equivalence ratios the flame front becomes more contorted. Once the borderline (flame front) between reactants and products has been defined via the thresholding technique, its fractal dimension/)2 is calculated. The method followed here is a modified algorithm of that described in the section on fractal theory; it is mentioned in Mandelbrot [6]
Fig. 6. Stoichiometricflame images at various engine speeds. Top row 300 rpm, middle row 1200 rpm, bottom row 2400 rpm. White is products and black is reactants.
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J. MANTZARAS ET AL.
Fig. 7. Lean flame images (~b = 0.59) at various engine speeds. Top row 300 rpm, middle row 1200 rpm, bottom row 2400 rpm. White is products and black is reactants.
and was employed by Sreenivasan and Menevau [9]. This method is particularly suited for the analysis of fragmented images. The algorithm is summarized below. One considers the area of reactants and products that is within a distance e from the flame front. This area is 2eL, where L is the length of the flame front. Substituting for L from Eq. 1, it is seen that this area is proportional to 62-02. The plot of the logarithm of the area versus log (e) has slope 2 - D2 in the region of 6 where fractal behavior applies. The quantity 2 - D2 is called the codimension. The implementation of this method involves the following steps (see also Ref. 9): A circle of radius 6 is drawn around each pixel of the image (reactants as well as products), and it is determined whether this circle intersects the flame front. Let N(e) be the number of pixels whose circles intersect the flame front for a fixed radius e and repeat the process for different e, thus determining the variation of N(e) with 6. Because N(6) represents the area of the strip with width 2E about the flame front (when the unit area is taken to be the pixel area), the previous discussion yields N(6) cx ~2-D2.
2) in that, if the fractal domain is limited, at scales outside the fractal domain (~i > 6 or 6 > Co) the slope does not become zero, as in Fig. 1. For values of e sufficiently larger than eo and sufficiently smaller than ei the slope becomes one. This results from Eq. 3 because in these regions the fractal dimension D2 reduces to the Euclidean dimension of one. For values of e close to ei or 6o the slope changes in an unpredictable manner. Thus, ei and eo are defined by the values of E at which the log [N(e)] versus log (e) curve ceases to be linear. A plot of Eq. 3 is shown in Fig. 8, for a 10 4
N(E) I0 3
(3) 10 2
The log-log plot of Eq. 3 has slope 2 - /92 and hence the fractal dimension is easily calculated. The ideal form of this plot differs from Fig. 1 (Eq.
I0
102
E
Fig. 8. Test run for the fractal behavior of a square geometrical object.
FRACTALS AND TURBULENT PREMIXED ENGINE FLAMES test run where the digitized image was a square of 30 × 30 pixels in the center of the 100 × 100 pixel image and the radius e was increased in integer multiples of the pixel size. The theoretical slope of this regular object should be one, and a value of 0.993 is obtained. For a regular object such as this, no cutoffs should be observed. The outer cutoff shown in Fig. 8 is due to the finite size of the space surrounding the square object. Thus there are some practical limitations in the use of the above algorithm to determine 6i and Co. They include the following. To detect similarity at small scales and to define the inner cutoff, ei, one requires resolution of the digitized image significantly smaller than ~i (in Ref. 9 a factor of at least two is stated). In our case the best resolution dictated by the experiment was, as mentioned in the previous section, 0.2 mm. It is thus expected that, if the physical scales of the flame front are smaller than the pixel size no inner cutoff will be revealed. To determine properly eo, one requires many of the large scales in the finite-size digitized image. To compensate for this effect, 25 images at each condition were analyzed and then ensemble averaging was performed; thus, statistically, the largescale effects were properly accounted for. In addition, as is shown in the next section, the integral length scale (and hence the outer cutoff point, Co) was sufficiently smaller than the 2 cm size of the image. Another possible problem encountered is that the areas of reactants and products can influence the outer cutoff end. If the areas are very different, the flame front is near the border of the image and hence the strip 2eL is confined to a smaller region. Then, if the distance between the flame front and the border of the image is smaller than the natural outer cutoff, the turning point at the high-end cutoff will be affected by it. We reduced this problem by considering only images with approximately equal reactant and product areas (see Figs. 6 and 7). Thus, after the fractal analysis is performed on each of the 25 images for each of the six cases, the mean and standard deviation of the fraetal dimension DE are calculated and the existence of cutoffs is examined. The fractal dimension of the flame
303
surface D3 is then calculated from the relation D3 =D2+ t.
RESULTS AND DISCUSSION This section is divided into two clearly separated parts. In the first part we show that the boundary separating reactants from products unequivocally exhibits fractal behavior. This conclusion is independent of any theory of turbulent combustion. In the second part, we relate the fractal dimension to various theoretical concepts of turbulent combustion.
Fractal Dimension The fractal analysis described in the previous section was applied to the six conditions of Table 2. Plots of log IN(e)] versus log (e) are shown in Fig. 9. Each plot is typical of the 25 flame fronts 104
N(e)
o° / g 0 O
00 / o °
io3
oD D
O
~=l.0
o D
0
D 300 RPI,I
D
o
1200 RPM
0
2400 1~14
0
2xlO 2 I04
N(e)
q~ 0 )0
ooo
¢Do oo o o o
10 3
O ¢
O 300 RPH
oo
o 1200 RP~t
0 2400 RPM
I0
I0 2
E
Fig. 9. Fractal plots of flame images, a. Stoichiometric flames
at three engine speeds; the scale~is in units of 0.2 mm. b. ~an flamesat three engine speeds; the scale c is in units of 0.2 mm.
304
J. MANTZARAS ET AL. I0 5
(c) I0 4
N(E) .0 10 3
-: 10 2
,
,
,
9 cm
, ,,,,=
,
,
,
q , ....
I0 E
10 2
Fig. 9. c. F u l l f i ¢ l d f l a m e at lean c o n d i t i o n s s h o w i n g the u p p e r c u t o f f ; the scale e is i n units o f 0 . 9 ram. T h e insert s h o w s the f l a m e i m a g e ; w h i t e is products and b l a c k is reactants.
analyzed in each case. It is seen that there exists fractal behavior over a significant range of scales. A straight line was fitted through the points where fractal behavior applies, and its slope (2 - D2) was calculated. This process was repeated for each image. Figure 10 shows the pdfs of the slopes (2 - D2) obtained from the analysis of the 25 images. The mean slopes, 2 - /)2, and standard deviations, os, of the slopes for the six conditions are given in Table 2. The variation of the slope at fixed conditions is not only due to the finite size ot the image as previously stated, but also to the fact that each cycle is slightly different from all others in an engine. It should be noted that the standard error of the mean gradient is always 2 % or less. Figure 9a shows the fractal plots for the stoichiometric condition and the three engine speeds. The relative standard deviations of the slopes at qb = 1.0 are 4% for 300 rpm and about 10% at 1200 and 2400 rpm. There is a mean slope separation of about three standard deviations between conditions at 300 and 1200 rpm and two standard deviations between 1200 and 2400 rpm. Thus at stoichiometric conditions, the flame fronts at different engine speeds have statistically different fractal dimensions. The slopes of the lean conditions shown in Fig. 9b, on the other hand,
(k = 1.0 300 RPM .... ----
1200 RPM 2400RPM .
,
r] rj
r--
!
o
r-u i L
i
0.4
0.6
-J
"1
v1 "-i I[ I
, 0.8 SLOPE ( 2 - O 2)
•
I
h
1.0
• = (/.59 300 RPM - ~ - 1200RPM - - - - - 2400RPM(b) r - ~F--liti- -
o 0.4
i
i 0.6 0.8 SLOPE ( 2-D 2)
1.0
Fig. 10. Probability density function of the slopes of fractal plots at two equivalence ratios and three engine speeds.
FRACTALS AND TURBULENT PREMIXED ENGINE FLAMES exhibit in general smaller variation with engine speed. At engine speeds of 1200 and 2400 rpm the slopes are statistically the same. It is also seen that at 300 and 1200 rpm the leaner flames have smaller slopes than the stoichiometric ones. We shall comment on these variations in the next section. The fractal dimension 1)3 of the flame surface can be calculated, assuming isotropy, from /)3 = /)2 + 1. Table 2 gives the fractal dimensions of the conditions examined. As explained earlier, at scales large compared to the integral length scale we expect to see a transition from the fractal gradient of the fractal region to a gradient of one. That this is not observed in Figs. 9a and 9b may be due to the finite size of the image that influences the gradient before scales are reached that are large compared to the integral length scale. In order to test this we analyzed some images taken at a lower resolution (9 × 9 cm); in these images a complete section through the flame is seen and the flame did not intersect the image boundary. This allowed the fractal analysis to be performed in a 300 × 300 pixel square with the flame front occupying the center portion of the square, thus eliminating the edge effects. The fractal plot of one of these images (Fig. 9c) clearly shows a transition from a gradient of 0.66, typical of the fractal plots at higher resolution, to a gradient of one, expected for a Euclidean object. The first deviation from the initial gradient occurs at a scale of about 5 to 6 pixels (4.5-5.4 mm), this value taken to be the outer cutoff scale, Co. In order to determine the outer cutoff scale it is therefore necessary to have an image that has flame front boundaries sufficiently far from the image boundaries. The ensemble lateral integral length scale was measured in an identical engine at 600 rpm using a two-point LDV method [13] and its value is about 2 mm around TDC (Fig. 11). The longitudinal integral length scale is twice the lateral one for homogeneous, isotropic turbulence and in our engine there exists some degree of homogeneity and isotropy around TDC [14]. Thus the outer cutoff is observed to be approximately the longitudinal integral length scale. No inner cutoff was observed with our resolution of 0.2 ram, as exemplified in Fig. 9.
305
I
4.01
--
imootheci
........UnSll01)the(~!
_$ ~, 3£ :=
i.-- 2.C
zo
I.C 600 rpm
TDC i
310
330
i 550
370
390
CRANKANGLE
Fig. l l . Ensemble latcr~ in~gral|engthsc~e m600rpm,
13.
~om Ref.
In the following subsections, we now consider the following theoretical topics: the regimes of premixed turbulent combustion of our data; the possibility that the wrinkled flame front may be a constant property scalar surface of the turbulent field; the question of whether we should have observed an inner cutoff; and some correlations of the turbulent flame speed that include fractal information.
Regimes of Turbulent Combustion For any theoretical consideration, it is first necessary to assess the regime(s) of turbulent combustion of our data. Figure 12 [15] is a plot of the Damkohler number Da^ versus the turbulence Reynolds number RA [16]. The Damkohler number is the ratio of the characteristic turnover time, zt, of an eddy of the size of the integral length scale, A, to the characteristic transit time through the laminar flame, 7"1: DaA--
7"t
--
7"1
A
Sl
(4)
U ~ t~1 '
where Sl is the laminar flame speed, u' the turbulence intensity, and ~l the laminar flame thickness. The turbulence Reynolds number is u'A RA =
(5)
P
Relevant parameters appearing in this plot are the
J. MANTZARAS ET AL.
306
/ Z /
/-~
.... / "!~ /
REACTION SHEETS
'
1
DISTRIBUTEDN 10.4 R/EACTIONS "
/~2
~ ~ IO 4
.
"
~ 10 8
RA
Fig. 12. Regimesof turbulent combustion. The rectangleis an estimate of the regimes applicable to internal combustion engines [16].
ratio of the Kolmogorov scale, ~/, to the laminar flame thickness, and the ratios A/(~1 and u'/Sl. Some understanding of the structure of turbulent flames exists in two regimes; the regimes of reaction sheets and of distributed reactions. In the regime of reaction sheets, turbulence does not affect chemistry and the residence time in the laminar flame is much smaller than the characteristic turnover time of the smallest eddies. In this regime, the propagating flame fronts are just wrinkled and convoluted by the turbulence. In the distributed reaction regime, chemical reactions proceed together with, or after, turbulent mixing, and the concept of laminar flame does not apply. The lower bound of the reaction sheet regime can be set at ~///tl = 1 (the quantity 7/2/8]2 is a measure of the stretch that the flame is subjected to in a turbulent flow) and M/it = 1 is the upper bound of the distributed reaction regime shown by the heavy lines in Figure 12. In the intermediate regions, 7/ < 8~ < A, the structure of the turbulent flame has not yet been identified. The reaction sheet regime is divided in two subregimes. For u'/S1 < 1,
turbulence is weak and a single continuous reaction sheet can be identified; at higher turbulent intensities, adjacent sheets may collide and cut off pockets of reactants, forming multiply-connected reaction sheets. The line u'/Sl = 1 may be taken as the boundary between these subregimes. The rectangle in Fig. 12 indicates the combustion regimes for various engine operating conditions as estimated in Ref. 16. It is seen that most of the operating conditions in an engine are expected to fall in the reaction sheet regime. For our stoichiometric conditions, and using the correlations of Metghalchi and Keck [17], the laminar flame speed is estimated to be 1.25 m/s + 5%. For our lean conditions, Sz would be estimated to be between 0.09 and 0.31 m/s by extrapolating the results of [17]. For an identical engine the turbulence intensity was measured [18] using LDV. Figure 13 shows the turbulence intensity near TDC at various engine locations and engine speeds under nonfiring conditions. The turbulence intensities at the times at which the flame images were taken are estimated from Fig. 13 with an uncertainty of + 7 % (Table 2). The laminar flame thickness was estimated in Ref. 16 to be between 10 and 50 #m at stoichiometric conditions. For the lean conditions it was estimated to be between 50 and 200 #m, the upper bound being set by the measurements of Ref. 4. The estimated values of 71are given in Table 2. It is seen that the ratio ~/8] is expected to be of order
10 E
PORTED SWIRL
~
z _z r..) Z
RPM 2400
::
:~i?il;;
]8oo
TDC 0 --45
I
--30
J
I
--15
J
[
O
i
I
15
30
CRANKANGLE Fig. 13. Turbulenceintensities around TDC for three engine speeds, at various engine locations from [18].
FRACTALS AND TURBULENT PREMIXED ENGINE FLAMES
307
and we expect the structure of the laminar flame itself to be altered by turbulence. In any case, the results of Fig. 14 are in good agreement with those of Ref. 5, where D3 was found to increase with increasing u'/SI from the value of 2.05 at u"/$1 0.2 to the value of 2.14 at u'/S1 ~ 0.5.
2.6
2.4
D3
Inner Cutoff
2.2
0
qb ~ 1.0
o ¢~= 0 59
2.C I0 -I
......................... I u'IS L
I0
F i g . 14. Plot o f f r a c t a l d i m e n s i o n D 3 v e r s u s
I0 :~
u'/S~.
one for the stoichiometric cases and smaller than order one for the lean cases. Thus our stoichiometric flames are expected to fall within the multiply-connected reaction sheet regime, i.e., to be composed of laminar flames whose structure is unaffected by turbulence, whereas our lean flames are expected to fall within the intermediate regime so that the structure of their laminar flames is expected to be affecled by turbulence.
Constant-Property Scalar Surface We can now address the question as to whether the wrinkled flame fronts may act as a constantproperty scalar surface of the turbulent field. Figure 14 shows a plot of/)3 versus u '/$1, with the vertical error bars representing the standard deviation of /)3. The fractal dimension D3 increases with increasing u'/Sl for u'/S~ < 4 and seems to reach a constant value at 2.36 _+ 3% for u'/S1 > 4. The limit value o f / ) 3 happens to coincide with those measured by Sreenivasan and Meneveau in turbulent jets [9] and by Lovejoy in storm clouds [10]. Thus, as far as the fractal dimension is concerned, the laminar flame surfaces of our lean turbulent flames act as if they were constant-property scalar surfaces of the turbulent field. If confirmed, this behavior must result from a rather special balance of events since for our lean flames we have estimated 7//~ < 1
In general the laminar flame front is expected to be smoother than that implied by the unperturbed turbulent field [19]. The smoothing of the flame surface is predominant in the small scales whereas the larger scales tend to remain unaffected [11]. This suggests that the inner cutoff ei tends to shift to higher values than the Kolmogorov scale but also that the fractal behavior of turbulence should be recovered for sufficiently large values of e. For ~//6j > 1 and u'/S1 ,~ 1, Gouldin [11] suggests that the effect of the laminar flame propagation is predominant in determining the roughness of the flame surface, so that the inner cutoff ei approaches the outer cutoff ~o. For ~]/Sj > 1 and high values of u'/SI, turbulence becomes important in determining the roughness of the flame surface even though the effect of $1 persists in smoothing the small scales. In this case the inner cutoff increases but the fractal dimension of turbulence should still prevail at large values of ~ (our values of D3 at ~b = 1 and 300, 1200 rpm suggest that wrinkles of all sizes are affected by combustion for ~1/6~ ~ 1, u'/S~ 1). Peters [20] argued that in the multiplyconnected sheet regime the inner cutoff shifts from ~/to another scale, the Gibson scale Lg:
Lg = A( S1/u')3.
(6)
The underlying assumption in Eq. 6 is that the laminar flame propagation smoothens turbulent eddies with circumferential velocity less or equal to SI. The size of the turbulent eddy that has circumferential velocity equal to $1 is the Gibson scale. When ~//~i1 > 1 and u ' / S l ~, 1 the effect of turbulence is predominant and the inner cutoff should become that of the turbulent field, i.e., the Kolmogorov scale [11].
308
J. MANTZARAS ET AL.
For our data ~/ is always smaller than the 200 a m resolution of our measurements (see Table 2) and we estimate the Gibson scale to be 250 #m at 1200 rpm and 32 #m at 2400 rpm, using u'/Sl of Table 2 and A = 2 mm for ~b = 0.59. Thus, current estimates suggest that the inner cutoff is smaller than the resolution of our technique and could not be observed.
20
Turbulent Flame Speeds The following method to estimate the turbulent flame speeds based on fractals was suggested by Gouldin [11, 21] for ~/~1 > 1 and u'/Sj ~, 1. He states that the effect of turbulence on the flame surface is to increase the ensemble-average flame area by the ratio of the areas of the inner and outer cutoff Ai/Ao (Fig. 1). Then,
St~S1= Ai/Ao,
where St is the turbulent flame speed. Since ~//~l be the Kolmogorov scale ~/, and substituting Eq. 2 in Eq. 7, one obtains [11] ( A / 7 / ) 03 -2.
u'/S~ Fig. 15. Turbulent flame speeds in engines. Solid bars are results from Ref. 4 and open bars (shifted horizontally for clarity) are fractal predictions.
(7)
and u'/St are large, the inner cutoff is expected to
St~St =
I
(8)
Equation 8 was applied to cases 2 and 3 of Table 2, which come closer (~/~l = 1 and u'/Si > 1) to satisfying the conditions necessary for its validity. A was replaced by the measured cutoff Eo = 4.5 mm and/93 was taken from Table 2, assuming isotropic flame surfaces. (Note that even if eo is underestimated because of the effects of the finite image size this will not have a large effect on St/S~ due to the small value of D3 - 2. For example, an unlikely underestimate of eo by a factor of two will lead to an underestimate of St/Sj by less than 30%.) The results are shown by the open bars in Fig. 15. The solid bars are estimates of St~S1 from Ref. 4. There St~S1 was calculated by measuring the turbulent flame front length directly from flame images and dividing it with a reference laminar flame length that was taken to be either the perimeter of a circle with the same burned area or the flame front length of the ensemble-average flame front. Then, assuming isotropy of wrinkling in the direction perpendicular to the plane of the
image, the ratio of turbulent to laminar flame surface areas was calculated by squaring the flame length ratio since the fractal nature of the flame front was not known at that time. The fractal analysis of this paper is seen to give St~S1 lower than in Ref. 4 by some 35% at u'/Sl = 2 and 30% at u'/Sl = 4. Since both Ref. 4 and this work measure the flame length ratio of the flame images obtained with the same apparatus, we can reconcile the differences in the flame length ratios and hence in St~S1 as follows. In Ref. 4 the length of the turbulent flame front L r was measured using a yardstick E equal to 0.2 mm. The length of the corresponding laminar flame front LR was taken to be that of a circle of same area with the flame area, or that of a reference ensemble flame measured with the same yardstick of 0.2 mm. If we want the corresponding lengths measured with yardstick 7, then
LT,I=LT~(rl/~.)I-D2,
(9) (10)
LR,~ = LR~ = LR.
In Eq. 10, D2 = 1 because the reference laminar front is Euclidean. Thus
LT,JLR = [LT /LR](~I/¢)
I -D2.
(11)
FRACTALS AND TURBULENT PREMIXED ENGINE FLAMES The quantity in the brackets was measured by Ref. 4, and according to Gouldin [11], L.r,~/LR is equal to St/SI (for a fractal isotropic surface Eqs. 1 and 2 show that length ratios are equal to area ratios). In Eq. 11 the ratio LT,~/LR can be obtained using values of ~/from Table 2 and the bracketed term from Ref. 4, and it represents an adjustment of the estimates of Ref. 4 that considers the fractal behavior of the flame fronts. The fractal predictions (Eq. 8) and the adjusted estimates of Ref. 4 (Eq. 11) are plotted in Fig. 16. It is seen that the fractal predictions fall within the adjusted estimates of Ref. 4. The solid lines in Fig. 16 are correlations proposed by various researchers [4, 22, 23, 24]. Although the two magnitudes predicted by Eq. 8 are acceptable, the trends implied by Eq. 8 are not acceptable for engine applications. Thus, for D3 = 2.33, A/rl o: RA°.75, and A and ~ constant, it gives St/S] - ( u ' A / v ) °-25, where the exponent of u' is much too small for engine combustion. An exponent of at least 0.7 is more realistic. If, instead of using the Kolmogorov scale as the inner cutoff, one uses the Gibson scale, then Eq. 11 is replaced by (12)
LTg/LR = [LTe/LR](Lg/e) 1-/92,
2O
/
Q
oquation 8
•
equation
equation I t
5t/S;
~/__ .
\
-
~
L2 ~ 7
/
IO ~
% ....
; ....
i ....
i ....
i ....
5
u'/sL
Fig. 16. Turbulent flame speeds obtained with fractal analysis aM correlations suggested by various authors (O, A shifted horizontally for clarity).
309
and Eq. 7 gives St/Sl = (u' /S1) 3(0~- 2).
(13)
Adjusting the estimates of Ref. 4 with Eq. 12 and taking D3 from Table 2, one finds that the turbulent flame speed predictions fall at the lower end of the predictions of both Eq. 8 and 11, as shown in Fig. 16. For D3 = 2.33, Eq. 13 gives St - u ' , which is acceptable for fully-developed turbulent engine flames. For/23 = 2.23 and with a coefficient of 3.5, Eq. 13 would become equal to Klimov's correlation [24] for u ' / S l ~, 1 (Fig. 16). But, D3 = 2.23 is smaller than we have measured. Thus more work is needed to relate fractal analysis to turbulent flame speed even in the simplest of cases in which ~/5~ >- 1 and u ' / S ] ~. 1.
CONCLUSIONS The structure of the turbulent flame fronts in an internal combustion engine was shown to be fractal in nature for scales as small as 200 t~m and as large as twice the lateral integral length scale, for six engine conditions corresponding to different turbulent intensities and equivalence ratios. At stoichiometric conditions the fractal dimension of the flame surfaces was found to be statsfically different for engine speeds of 300, 1200, and 2400 rpm and to increase with engine speed and therefore u'/S1, which ranges from 0.5 to 4.0. At lean conditions (~b = 0.59) where u ' / S l ~,. 1 the fractal dimension did not change with engine speed. For 50 > u' /S1 > 4 and 0.1 < rl/6~ <- 1 (estimated ranges), the fractal dimension was 2.36% + 3% for both stoichiometric and lean conditions. This is in good agreement with the fractal dimension determined for other turbulent flows [9, 10], which is about 2.35. Thus, as far as the fractal dimension is concerned, the laminar flame surfaces of our lean turbulent flames act as if they were constant-property scalar surfaces of the turbulent field. At lower values of u ' / S t the flame propagation becomes important in determining the wrinkling of the flame front, although the flame front is still a fractal with reduced fractal dimension.
310 No inner cutoff was observed at any condition with the resolution of 0.2 mm of this experiment. The outer cutoff occurred at about 4.5 mm, which is twice the measured ensemble lateral integral length scale for this engine. A turbulent flame speed model based on fractals was examined for two of our stoichiometric cases that appear to satisfy the conditions of the model. For the two cases, St/Sl had also been determined by a different technique. The values given by the fractal model for St/SI are not too far from the range of values obtained by other researchers. But the conclusion is reached that more work is needed to relate fractal analysis to turbulent flame speed even in the simplest of cases in which 7//~1 -< 1 and u'/S~ ~ 1.
J. MANTZARAS ET AL.
6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
16.
Support for this work was provided by the Department o f Energy, Office o f Energy Utilization Research, Energy Conservation and Utilization Technologies Program (Contract DEAS-O4-86AL33209), General Motors Corp., Ford Motor Corp., and Cummins Engine Co.
17. 18. 19. 20.
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Received 5 January 1988; revised 22 September 1988