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Dissipation element analysis of scalar ﬁeld in turbulent jet ﬂow A.M. Soliman a, Mohy S. Mansour b, Norbert Peters c, Mohamed H. Morsy d,⇑ a

Faculty of Petroleum and Mining Engineering, Suez Cannal University, Suez, Egypt Faculty of Engineering, Beni Suef University, Beni Suef, Egypt c Institute of Combustion Technology, RWTH Aachen, Germany d Faculty of Engineering, King Saud University, Riyadh, Saudi Arabia b

a r t i c l e

i n f o

Article history: Received 20 April 2011 Received in revised form 1 September 2011 Accepted 6 October 2011 Available online 13 October 2011 Keywords: Dissipation element Turbulent ﬂow Rayleigh scattering

a b s t r a c t For better understanding of turbulence, the geometry of turbulent structures in turbulent jet ﬂow should be analyzed. The aim of the present work was to experimentally verify the dissipation element theory on highly resolved two-dimensional measurements turbulent jets using Rayleigh scattering technique. The statistical analysis of the characteristic parameters of dissipation elements; namely the linear length connecting the extremal points and the absolute value of the scalar difference at these points, respectively was also investigated. Rayleigh scattering was used to topographically produce 2D images of turbulent mixing to obtain the concentration distribution of two gases in a turbulent shear ﬂow. The scalar ﬁeld obtained was subdivided into numerous ﬁnite size regions. In each of these regions local extremal points of the ﬂuctuating scalar are determined via gradient trajectory method. Gradient trajectories starting from any point in the scalar ﬁeld /(x, y) in the directions of ascending and descending scalar gradients will always reach a minimum and a maximum point where r/ = 0. The dissipation element has two extremal points (one maximal and one minimal) and two saddle points at the boundaries. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction The dynamics of the ﬁne scales is important to turbulence theory as well as to the development and validation of subscale models. The mixing of two or more ﬂuid components at the molecular level in turbulent shear ﬂows plays a dominant role in a wide range of practical problems, especially when chemical reactions occur between the constituents. Combustion requires fuel and oxidizer to be mixed at the molecular level. Molecular mixing of fuel and oxidizer takes place at the interface between eddies. For these reasons, numerous experiments and simulations of ﬁne scale turbulence have been conducted. Much experimental effort has thus been invested on mapping scalar ﬁelds in three dimensions, usually by reconstructing them from several planar cuts taken in quick succession in time. Experimental measurement of the scalar dissipation imply to the difﬁcult task of obtaining the instantaneous spatial derivative of the mixture fraction with sufﬁcient resolution [1]. An experimental study done by Dahm et al. [2] gave an assessment of the accuracy of Taylor’s hypothesis in approximating the magnitude of the scalar dissipation rate from such one or twodimensional measurements of the scalar gradients. There have been many recent experimental studies to quantify scalar dissipation in turbulent axisymmetric and planar jets [2–6]. ⇑ Corresponding author. Address: Department of Mechanical Engineering, College of Engineering, King Saud University, Riyadh 11421, Saudi Arabia. E-mail address: [email protected] (M.H. Morsy). 0894-1777/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.expthermﬂusci.2011.10.003

There have been some other attempts to deﬁne the geometrical elements that one intuitively believes to represent eddies of different size in turbulent ﬂows [7–9]. Townsend [7] suggested that turbulent motion is essentially a random distribution of vortex sheets and tubes. More recently, Wary and Hunt [8] subdivided the threedimensional vertical ﬁeld obtained from direct numerical simulations (DNS) into four types of space-ﬁlling regions, classifying them tentatively by characteristic values of the second invariant of the velocity derivative tensor as well as the pressure. By using DNS, Peters and Wang [9] set the basis for prediction methods of scalar ﬁelds that incorporate information from length scales below the integral scales. They found that the dissipation element has elongated shape with nearly constant diameter of few Kolmogorov length scale and a variable length that has the mean of a Taylor length scale. They parameterize the geometry of these elements by linear distance between their extremal points and their scalar structure by absolute value of scalar difference at these points. To better understand turbulent ﬂow, it is useful to study and analyze dissipation elements with highly resolved measurements in turbulent jet ﬂow. The objective of the present work is to experimentally verify the dissipation element theory on highly resolved 2D measurements of turbulent jet ﬂow using Rayleigh scattering technique. The statistical analysis of the characteristic parameters of dissipation elements; namely the linear length connecting the extremal points and the absolute value of the scalar difference at these points, respectively is also investigated.

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Nomenclature b1/2 D d E(k) K k l lo Re Sc U0 Ucenter

width of the jet where the mean velocity has dropped to half its centerline value (m) diffusivity (m2/s) diameter of jet (m) energy spectrum (m3/s2) turbulent kinetic energy (m2/s2) wave number (m1) linear distance between extreme point (m) integral length scale (m) Reynolds number Schmidt number characteristic velocity at characteristic length lo (m/s) RMS center line speed (m/s)

u0 x

ﬂuctuating turbulence velocity in x direction distance from the jet, assuming the virtual origin of jet equal zero

Greek symbols m kinematic viscosity (m2/s) /(x, y) passive scalar ﬁeld g Kolmogorov length scale (m) kT Taylor length scale (m) Batchelor length scale (m) kB e rate of kinetic energy dissipation (m2/s3) v the scalar dissipation rate ((concentration)2/s)

2. Deﬁnition of dissipation element Richardson [10] introduced that the larger eddies are unstable and break up transferring their energy to somewhat smaller eddies undergo a similar break-up process, and transfer their energy to yet smaller eddies. This energy cascade, in which energy is transferred to successively smaller and smaller eddies, until it is ﬁnally dissipated by viscous effects. Eddies are statistical entities and any statistical evaluation in terms of the distribution function, requires a clear deﬁnition of the quantity to be sampled. Therefore, it is necessary to divide the art of ﬂow to regions of ﬁnite size. The regions should not be arbitrary, which is the case of numerical mesh, but should follow the structures of the ﬂow itself. Fig. 1 shows the 2D scalar ﬁeld with minimum and maximum points distributed randomly. The iso-scalar lines around the extremal points and geodetic lines that connect two maxima or minima and pass through a saddle points are evident. Starting from given initial points, trajectories move along the normal directions of iso-scalar lines, shown as thin lines. Each bold solid edge line, in most of the cases, connects two maximal points

Fig. 2. Schematic of apparatus, mixing ﬁeld and laser sheet (red box indicates the location of the window).

Fig. 1. Schematic sketch of a 2D scalar ﬁeld including the trajectory from an initial point to the minimum and maximum points (red boundary represents the dissipation element).

Fig. 3. Experimental conﬁguration for high-resolution Rayleigh imaging.

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Fig. 4. CCD camera, image intensiﬁer and mirror conﬁguration.

Table 1 Experimental conditions. Speed of helium (m/s)

x/ d

Re

Ucenter (m/s)

Fluctuating speed (m/s)

Half width of the jet b1/2 (mm)

Integral length scale (mm)

The rate of kinetic energy dissipation (m2/s)

Komogrov length scale (mm)

Taylor length scale (mm)

300 400 500 300 400 500 300 400 500 300 400 500

5 5 5 10 10 10 15 15 15 20 20 20

5.32E+03 7.09E+03 8.86E+03 5.32E+03 7.09E+03 8.86E+03 5.32E+03 7.09E+03 8.86E+03 5.32E+03 7.09E+03 8.86E+03

360 480 600 180 240 300 120 160 200 87 116 145

36 48 60 18 24 30 12 160 20 87 11.6 14.5

1.122 1.122 1.122 2.244 2.244 2.244 3.366 3.366 3.366 4.488 4.488 4.488

0.2244 0.2244 0.2244 0.4488 0.4488 0.4488 0.6732 0.6732 0.6732 0.8976 0.8976 0.8976

733626.3 1738966 3396418 2566845 6084373 11883541 12994652 30802139 60160428 2.08E+08 4.93E+08 9.63E+08

0.009792 0.007891 0.006675 0.019583 0.015783 0.013351 0.029375 0.023674 0.020026 0.040175 0.032378 0.027389

0.107719 0.093287 0.083439 0.215438 0.186575 0.166877 0.323157 0.279862 0.25036 0.438241 0.379528 0.33946

and one saddle point to determine the basic topology of each dissipation element. Usually the overall distribution of 2D dissipation elements will be appeared like a deformed and twisted soft net, where each element has two extremal points (one maximal and one minimal) and two saddle points at the boundaries.

3.1. Deﬁnitions of dissipation length scale

3. Experimental technique and resolution

is the kinematic viscosity. Kolmogorov scales are the smallest scales

The integral scales are the largest scales in the ﬂow ﬁeld, lo = k3/2/e, which is comparable to the overall scales exerted from boundaries, where k is the turbulent kinetic energy deﬁned as 2 i 2 k ¼ 32 u0 ; e is the energy dissipation deﬁned as e ¼ m @u and m @xi 1

The ability to measure several scalars at once to resolve measurements in three dimensions and to make time-correlated measurements allows new aspects of combustion theory to be probed and indeed may help the development of novel, more accurate models. Experimentally turbulent combustion is just as challenging as it is for the theorist. The spatial and temporal resolution requirements, the need for multiple excitation and detection systems, and the complex procedures for the extraction of quantitative data are reasons why the number of multi-scalar as well as multidimensional measurements have been few in the past.

(g ¼ ðm3 =heiÞ4 ) at which random ﬂuctuations from turbulence are 13 pﬃﬃﬃﬃﬃﬃ 3=2 dissipated. The Taylor length scale kT ¼ 10 g2=3 k =e is the

Table 2 Measurements with resolution 26.7 lm/pix. U

x/d

g (lm)

k (lm)

g/resolution

Resolution

300 300 300 300 400 400 400 400 500 500 500 500

5 10 15 20 5 10 15 20 5 10 15 20

9.792 19.583 29.375 40.175 7.891 15.783 23.674 32.378 6.675 13.351 20.026 27.389

107.719 215.438 323.157 438.241 93.287 186.575 279.862 379.528 83.439 166.877 250.316 339.46

0.37 0.73 1.1 1.5 0.29 0.59 0.88 1.21 0.25 0.502 0.75 1.02

2.7g 1.3g 0.9g 0.6g 3.3g 1.6g 1.1g 0.8g 4g 1.9g 1.3g 0.96g

Fig. 5. Averaged scalar power spectra for the present data.

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Fig. 6. The average (left) and RMS (right) of images for a speed of 300 m/s at: (a) x/d = 20, (b) x/d = 15, (c) x/d = 10 and (d) x/d = 5.

intermediate length scale between the integral length scale and Kolmogrov length scale. According to Batchelor [11], the smallest scalar structure occurs at Batchelor scale kB ¼ g Sc1=2 , where Sc is the Schmidt number that equal to m/D where D is the diffusivity. The present study thus

aims to identify the physical characteristics of the ﬁne structure of scalar ﬁeld for a gas mixture. An important issue that arises in experimental studies of the dynamics of ﬁne scales is the resolution required to capture these scales. However, some disagreement exists in the literature over

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Fig. 7. Proﬁle of the average and RMS of helium concentration at horizontal line at a speed of 300 m/s at: (a) x/d = 20, (b) x/d = 15, (c) x/d = 10 and (d) x/d = 5.

the true ﬁnest dissipative length scale. For example, Antonia and Mi [12] conducted a dissipation measurements using cold wires separated by a known distance and concluded that a resolution of 3g is apparently sufﬁcient to capture the dissipation structure.

Buch and Dahm [3] computed a PDF of scalar dissipative structure thickness and observed that the width of the thinnest scalar dissipation layers in gas-phase jet is approximately equal to 3g. A consensus remains to be reached concerning the resolution necessary

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for the correct measurement of the scalar dissipation rate, v, but a resolution of g should be sufﬁcient for both / and v [13]. The objective of the current study is to investigate the structure of the dissipative scales in a gas-phase turbulent jet and to determine the requirements for full resolution of the true ﬁnest scales. In this work, a high resolution of about 0.6g with a wide range from 0.6g to 4g is achieved. These experiments were based on high-resolution two-dimensional imaging laser Rayleigh scattering in the self-similar far ﬁeld of an axisymmetric co-ﬂowing turbulent jet in air. 3.2. Apparatus Simple open jet geometry was selected for the present work. Helium was selected for the main jet surrounded by a co-ﬂow of dry air. The air and helium concentrations were measured using highly resolved two-dimensional Rayleigh scattering technique. The helium-air gases were selected due to the large difference between the Rayleigh cross-section of helium as compared to air (the ratio between the helium Rayleigh cross-section to that of air is about 0.014). The experimental technique and the test-rig are described below. The nozzle consists of two concentric tubes; the inner and outer tubes have an inside diameters of 2.2 mm and 50 mm, respectively. The helium ﬂows through the inner tube while the air ﬂows through the outer tube. The outer co-ﬂow air velocity was ﬁxed at 1.5 m/s, while the helium ﬂows with variable velocities which were selected to be 300, 400 and 500 m/s. The measurements were collected at ﬁxed locations from the nozzle exit of each helium jet velocity. The window of measurements was selected to be 8 mm 8 mm. This allows full coverage of the length scales of the present jets. The mass ﬂow rates of air are controlled through a mass ﬂow controller for air (0–40 kg/h). Helium ﬂow rate is controlled through a volume ﬂow controller for N2 (0–250 L/min) which is calibrated to measure He. Fig. 2 shows schematic diagram for mixing ﬁeld and laser sheet. Fig. 3 illustrates the Rayleigh technique used at the combustion institute (ITV-Aachen University) laboratory. The laser used is a tunable KrF excimer laser (Lambda Physik model EMG 150 MSC). It produces tunable UV laser beam at 248 nm with pulse energy of 250 mJ and rectangular cross section of 4 mm 22 mm. The laser pulse width is 22 ns. One spherical lens with a focal length of f1 = 156 mm and one cylindrical lens with a focal length of f2 = 310 mm, separated by a distance of 70 mm, were used to form a laser sheet with an average

thickness of 110 lm and a height of about 12 mm at the measuring section. The laser sheet thickness is 100 lm at the image center and increases to about 120 lm at the edge. The laser sheet passes vertically above the jet center, as shown in Fig. 2. The Rayleigh scattering images were collected at right angle using a special dichroic mirror that reﬂects the Rayleigh signal at 248 nm and passes the rest of the wavelength. The mirror is located with inclination of 45°, as shown in Fig. 3. The Rayleigh images were then collected using intensiﬁed CCD (charge coupled device) ﬂow master camera with 1280 1024 pixel array and 6.7 lm 6.7 lm pixel size. The scan area for the spatial resolution of 26.7 pix/lm is 8.01 mm 8.01 mm and that for the spatial resolution of 8.6 pix/lm is 4.3 mm 4.3 mm. As shown in Fig. 4, the image intensiﬁer is connected with CCD camera which has a minimum useful photocathode and phosphor green screen of 1280 1024 square pixels with intensiﬁer control via RS-232. The magniﬁcation factor has been adjusted to suit the spatial resolution required for the present jets. 3.3. Flow parameters and operating conditions As described above, the ﬂow ﬁeld investigated here is a simple turbulent jet surrounded by co-ﬂowing air at low velocity of 1.5 m/ s. The jet nozzle has a diameter of 2.2 mm and the diameter of outer tube is 50 mm. The integral length scale is calculated as follows [14]:

lo ¼ 0:2b1=2

ð1Þ

where b1/2 is the half width of the jet as determined by the point where the mean velocity has dropped to half its centerline value that calculated as follows:

b1=2 0:102 x

ð2Þ

where x is the distance from the jet, assume the virtual origin of jet equal zero [14]. The root mean square (RMS) of center line speed is calculated as follows:

U center ¼ ð5:8d=xÞU 0

ð3Þ

where d is the diameter of jet. The ﬂuctuating speed is calculated as follows:

u0 ¼ 0:1U center The rate of kinetic energy dissipation is deﬁned as:

Fig. 8. The trajectory line between extreme points. The minimum points are denoted by blue circle and the maximum points are denoted by red circle.

ð4Þ

A.M. Soliman et al. / Experimental Thermal and Fluid Science 37 (2012) 57–64

63

Fig. 9. PDFs of linear distance between minimum and maximum points for a resolution of 26.7 lm at speeds of: (a) 300 m/s, (b) 400 m/s and (c) 500 m/s.

3

e ¼ u0 =lo

ð5Þ

The Kolmogorov length scale is deﬁned as: 3

1 4

g ¼ ðm =heiÞ

ð6Þ

and the Taylor length scale is deﬁned as:

kT ¼

pﬃﬃﬃﬃﬃﬃ 1 10 g2=3 ðK 3=2 =eÞ3

ð7Þ

where k is the turbulent kinetic energy which is deﬁned as [15]:

K¼

3 02 u 2

ð8Þ

The kinematic viscosity of helium is m ¼ 1:24 104 m2 =s [15] and the diffusivity of helium in air at this condition is D 0:72 104 m2 =s [16] while the Schmidt number is Sc ¼ m=D 1:7. Experimental conditions for the planar jet measurement are tabulated in Table 1.

4. Results and discussion The set of measurements has a spatial resolution of 26.7 pix/lm and are tabulated in Table 2. Five hundred images were collected for each condition at each location. This allows small statistical errors in the present work. The two most important characteristic scales in the cascade process are the integral scales and the dissipative (Kolmogorov) scales. Between these two scales lies the inertial range, where the viscous effect can be neglected so that energy will be transported without remarkable loss. Fig. 5 shows the scalar power spectra for the full set of data and shows the evidence for an inertial range. This evidence of the inertial range provides a support for the generality, especially to higher Reynolds numbers, of the results for the structural properties of the small-scale turbulent mixing presented in this work. This produces the conﬁdence in the accuracy of measurements. Previously, it was

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found that spectral theories for turbulent velocity have direct analogs with turbulent scalar ﬁelds [3]. For Schmidt numbers near to unity as in the present measurements, the power spectrum of scalar ﬂuctuations should have a similar form to that of the velocity ﬂuctuations. In particular, for sufﬁciently high Reynolds numbers, one can expect that there must be an inertial range of wave numbers 5=3 kw for which the scalar spectrum has a kw dependence. For higher values of k, The energy spectrum, E(k), falls off more rapidly as the energy of the scalar ﬂuctuation is dissipated. Fig. 6 shows the average and RMS of images at a speed of 300 m/s and x/d = 20, 15, 10, and 5. For the conserved scalar ﬁeld /(x, t) of each image, pure blue denotes the lowest range of air (ambient ﬂuid) beginning at minimum value, with colors ranging from blue to red identifying linearly increasing scalar values. Pure red1 denotes the lowest of scalar values and highest value for pure air. Although not shown, similar results have been obtained for speeds of 400 m/s and 500 m/s. Fig. 7 shows the proﬁle of the average and RMS of helium concentration at horizontal line (at the middle of image) for a speed of 300 m/s. It can be seen that the concentration of helium decays on each side of jet centerline in horizontal direction with increase in radius. The maximum concentration of helium is increased with the decrease in the radial distance and shifts closer to the nozzle. It is seen that radial proﬁles are Gaussian as expected. Similar results have been also obtained for speeds of 400 m/s and 500 m/s. The topology of dissipation elements in 2D space is shown in Fig. 8. It shows the scalar ﬁeld of mixture fraction / and the minimum points denoted by blue circle and maximum points denoted by red circle. The connection between the minimum and maximum points is represented by yellow lines which called trajectories. The spatial PDFs of linear distance between two extremal points, l, for various values of x/d are computed from 500 images each of which has 500 500 pixels in resolution. The PDFs of l for three speeds (V = 300, 400, 500 m/s) and four upstream locations (x/d = 5, 10, 15, 20) are shown in Fig. 9. The peaks of PDFs are at l equal to approximately 100 lm (the values of l are ranged from 11.98g to 1.98g or ranged from 0.958kT to 0.182kT). These results agree well with that obtained by Peters and Wang [9] using DNS. It can be seen that the PDFs show a steep rise at the origin and an exponential decay for large l. It is clear that the PDFs follow nearly F distribution and that a hump is formed at about 160 lm which can be attributed to the image noise. 5. Concluding remarks The results presented from this study offer detailed views of the ﬁne–scale structure of conserved scalar mixing in turbulent ﬂows. These imaging of measurements have provided highly resolved data on the conserved scalar and scalar gradient ﬁelds. Furthermore, they

1 For interpretation of color in Figs. 1, 2, 6 and 8, the reader is referred to the web version of this article.

give a physical picture of the ﬁne-scale structure of scalar mixing process with a resolution of few Kolmogorov length scale (0.6g – 4g). The peaks of PDFs are at approximately 100 lm (the values of l are ranged from 11.98g to 1.98g or ranged from 0.958kT to 0.182kT). These PDFs show a steep rise at the origin and an exponential decay for large l. It is clear that the PDFs are nearly F distribution and found that a hump at about 160 lm, this hump due to the image noise. The gradient trajectory method that introduced here is able to identify ﬁnite size regions in a turbulent scalar ﬁeld without arbitrariness. The dissipation element theory is veriﬁed and the linear length scale distribution function agrees with the theoretical one derived by Wang [16]. The measurement of length scales and dissipation structures are important for developing accurate combustion models and determining necessary detector resolutions for experimental studies of turbulent ﬂames. References [1] C.H. Gibson, Kolmogorov similarity hypotheses for scalar ﬁelds: sampling intermittent turbulent mixing in the ocean and galaxy, Proc. Roy. Soc. Lond. A 434 (1991) 149–164. [2] W.J.A. Dahm, K. Southerland, K.A. Buch, Four-dimensional measurements of the ﬁne scale structures of Sc 1 molecular mixing in turbulent ﬂows, A-Fluid 3 (5) (1991) 1115–1127. [3] K.A. Buch, W.J.A. Dahm, Experimental study of the ﬁne-scale structure of conserved scalar mixing in turbulent shear ﬂows: Part 2. Sc 1, J. Fluid Mech. 364 (1998) 1–29. [4] K.A. Buch, W.J.A. Dahm, Experimental study of the ﬁne-scale structure of conserved scalar mixing in turbulent shear ﬂows: Part 2. Sc 1, J. Fluid Mech. 317 (1996) 21–71. [5] M.S. Tsurikov, N.T. Clemens, The structure of dissipative scales in axisymmetric turbulent gas-phase jets, AIAA 2002-0164. [6] L.K. Su, N.T. Clemens, The structure of ﬁne-scale scalar mixing in gas-phase planar turbulent jets, J. Fluid Mech. 488 (2003) 1–29. [7] A. Townsend, On the ﬁne structure of turbulence, Proc. Roy. Soc. Lond. A 208 (1951) 534–542. [8] A. Wary, J. Hunt, Algorithms for classiﬁcation of turbulent structures, in: Lipo, A. Tsinobar (Eds.), Topological Fluid Mechanics, Cambridge University Press, 1990, pp. 45–104. [9] N. Peters, L. Wang, The length scale distribution function of the distance between extreme points in passive scalar turbulence, J. Fluid Mech. 554 (2006) 457–475. [10] L.F. Richardson, Weather Prediction by Numerical Process, Cambridge University Press, 1922. [11] G.K. Batchelor, Small-scale variation of convicted quantities like temperature in turbulent ﬂuid: Part 1. General discussion and the case of small conductivity, J. Fluid Mech. 5 (1959) 113–133. [12] R.A. Antonia, J. Mi, Temperature dissipation in a turbulent round jet, J. Fluid Mech. 250 (1993) 531–551. [13] W.K. George, H.J. Hussein, Locally axisymmetric turbulence, J. Fluid Mech. 233 (1991) 1–23. [14] S.B. Pope, Turbulent Flows, Cambridge University Press, 2000. [15] H. Petersen, The Properties of Helium, Danish Atomic Energy Commission Risö Report No. 224, 1970. [16] L. Wang, Geometrical Description of Homogeneous Shear Turbulence Using Dissipation Element Analysis, PhD Thesis, Aachen University, Germany, 2008.