The results of calculating the averaged concentration of methane in the flow field of a turbulent axisymmetric jet on the basis of experimental data o...

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HEAT AND MASS TRANSFER IN COMBUSTION PROCESSES TALBOT METHOD OF INVESTIGATION OF THE CONCENTRATION DISTRIBUTION OF METHANE IN A TURBULENT AXISYMMETRIC JET P. P. Khramtsov, O. G. Penyazkov, I. N. Shatan, and I. A. Shikh

UDC 533.9.082.5; 537.523.2

The results of calculating the averaged concentration of methane in the flow field of a turbulent axisymmetric jet on the basis of experimental data obtained by the Talbot method have been given. Keywords: Talbot effect, turbulent jet, refractive index. Introduction. The recent years have seen considerable progress made in the field of studying the processes of turbulent mixing and combustion. In experimental investigations, use is made of diagnostic methods based on the laserinduced fluorescence of molecules, Rayleigh or Raman radiation scattering on molecules, and others [1−5]. These methods enable one to measure a large number of parameters with a sufficient temporal and spatial resolution; however there are a number of difficulties associated with the small value of measured signals or with the selection of suitable fluorescent impurities. In the present work, we diagnose turbulent flow on the basis of an analysis of the characteristics of the wave front of probing radiation that has traversed the object under study. To record the change in the wave-front structure we use the phenomenon of self-reproduction of the amplitude structure of coherent radiation with a periodically modulated wave front, which is known as the Talbot effect [6]. The coherent monochromatic radiation, on traversal of the Talbot grating and the object under study, is recorded by the matrix of a digital camera which is installed in a self-reproduction plane. The distance between the grating and self-reproduction planes is determined by the radiation wavelength and the structural period. The presence of the optical inhomogeneity produces a change in the refractive index and a corresponding distortion of the wave front of the probing radiation which is recorded in the self-reproduction plane. Relative displacements in the image due to this distortion characterize the deflection of the light beams from the initial direction. From the measured angles of deflection of the light beams, we can find the distribution of the refractive index and calculate, from these data, the characteristics of the object under study. The method of investigation of optical inhomogeneities, which is underlain by the Talbot effect, has been developed in [7, 8]. The efficiency of application of this method to finding the concentration distribution of helium in an axisymmetric jet and in a binary mixture in the case of two-dimensional jet flow past a cylinder has been shown in [9, 10]. Automatic processing of the full Talbot image makes it possible to obtain turbulence characteristics throughout the flow field, as exemplified by the investigation of the averaged and pulsation parameters of turbulence for a two-dimensional submerged jet [11, 12] and during the study of the statistical characteristics of density pulsations in the boundary layer behind the shockwave front in a shock tube [12, 13]. In the present work, the method of averaged Talbot images has been used for diagnostics of a turbulent methane jet flowing out into an air space. The distribution of the averaged refractive index in the flow field of the jet was restored on the basis of experimentally measured angles of deflection of the light beams; thereafter, the methane concentration was calculated from the data obtained. Experimental Setup and Measurement Procedure. The object of investigation was an isothermal free methane jet flowing out of an axisymmetric nozzle into an air space. The nozzle with a diameter of the outlet cross section of 4 mm, manufactured in the form of a cylinder of diameter 28 mm and length 100 mm with the upper end pointed at an angle of 45o, A.V. Luikov Heat and Mass Transfer Institute, National Academy of Sciences of Belarus, 15 P. Brovka Str., Minsk, 220072, Belarus; email: [email protected] Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 86, No. 2, pp. 247–255, March– April, 2013. Original article submitted August 29, 2012. 0062-0125/13/8602-0259 ©2013 Springer Science+Business Media New York

259

Fig. 1. Model of the experimental stand for investigation of turbulent flows.

Fig. 2. Scheme of the Talbot grating. was installed on a moving table for accurate positioning of the volume observed in the flow field of the jet. Commercial-purity methane (with a volume fraction of >99.9%) was used for the experiments. The initial velocity of outflow of methane at exit from the nozzle was u = 14 m/s (Re = uD/ν = 3900), which corresponds to a developed turbulent flow. An experimental setup for investigation of turbulent flows using the Talbot effect is shown in Fig. 1. The radiation of a gas laser with a wavelength of 514.5 nm was used as the probing radiation. The laser radiation was converted, with a collimator, to a homogeneous wide-aperture light beam of diameter 0.2 m with an approximately plane wave front. The Talbot matrix represented a reflecting grating with a system of equidistant holes of identical diameter (Fig. 2). It was arranged in the region of the laser beam immediately behind the collimator normally to the vector of propagation of the laser radiation and possessed the following parameters: optical density 4, period of the grating p = 1000 μm, and hole diameter 200 μm. For coherent radiation of the selected wavelength, the distance to the first self-reproduction plane was L = p2/λ = 1.944 m. The specimen was manufactured from a quartz plate coated with chromium. The repetition frequency of the period was ±2 μm. The investigated turbulent jet of methane was at a certain distance from the Talbot grating, which did not exceed the distance to the first self-reproduction plane. A light-sensitive matrix of a Nikon-D700 camera with a physical dimension of 24 × 36 mm and a number of active pixels of 12.1 Mp was placed in the corresponding zone of reproduction depending on the required sensitivity of the method. In an effort to match the apertures of the image of the Talbot matrix and the light-sensitive matrix of the camera, an aperture converter was installed immediately ahead of the camera. The probing radiation was recorded for the reference Talbot image and for the Talbot image of the methane jet. In the absence of the object under study, the field of radiation intensity in the self-reproduction plane represents a regular structure 260

Fig. 3. Differential Talbot image of the turbulent axisymmetric methane jet. of the set of light spots that corresponds to the structure of holes in the Talbot grating. The reference Talbot image recorded in this case determines the initial intensity field. Changes in the refractive index in the flow field of methane induce distortions of the wave front which lead to a displacement of the light spots in the Talbot image of the jet. Measurement of the value of displacement of each light spot enables us to determine the angles of deflection of light beams. Figure 3 gives the differential Talbot image of the methane jet under study, obtained for an exposure time of 10 s. Computational Methods for Determination of the Characteristics of the Axisymmetric Flow. Let us guide the symmetry axis of the turbulent jet vertically upward along the coordinate axis y and perpendicularly to the optional axis of the setup z (Fig. 4). For optical inhomogeneities of axial symmetry, the relationship between the deflection of the light beams and the distribution of the refractive index is expressed by the equations [14, 15]

2 εx = n0

R

∫ x

∂n ∂r

x r2 − x2

2 dr , ε y = n0

R

∫ x

∂n ∂y

r r2 − x2

dr ,

(1)

where εx and εy are the projections of the angle of deflection of the light beams along the corresponding coordinate axes. It is assumed that the gradient angles of the refractive index is equal to zero for a radius r exceeding a certain finite radius R = R(y). Equations (1) represent a special form of the Abelian integral equation of the first kind. As a result of the replacement of the variables r = R 2 − t and x = R 2 − s , they take the form s

ϕ( s ) =

∫

f (t )

0

dt . s−t

(2)

1 ∂n nε For the first equation of (1), the function f(t) corresponds to the function f(r) = , and the function φ(s) – φ(x) = 0 x . r ∂r x For Eq. (2), we know of the analytical solution

1 d f (t ) = π dt

t

∫ 0

ϕ( s )ds , t−s

(3)

⎞ ⎟. x 2 − r 2 ⎠⎟

(4)

which, for the first equation of (1), is of the form R n0 d ⎛⎜ ∂n = ∂r π dr ⎜ ∫r ⎝

ε x dx

261

Fig. 4. Coordinate system for mathematical modeling of the process of diagnostics of the jet. One usually does not use directly the analytical solution of the Abelian equation in integrating experimental data because of the presence of the derivative of the experimental function [16, 17]. Integration of the radial component for the gradient of the refractive index yields [14, 15] r

n( r ) − n( R ) =

∫

R

∂n dr = ∂r

r

∫

R

⎛n ⎜ 0 ⎜π ⎝

R

∫ r

⎞ ⎟ dr = − n0 π x 2 − r 2 ⎟⎠ ε x dx

R

∫ r

ε x dx x2 − r2

.

(5)

The given equation, in contrast to (4), contains no derivative of the experimental function. The singularity at the point x = r is easily integrated numerically, if we take into account that near the point x = r = 0, the function of experimental εx data must take on a value of zero. In diagnosing by the method of Talbot images, experimental data represent a discrete set of the parameters εx(x) ≡ ε kx dependent on the period of the Talbot grating. Numerical integration of Eq. (5) can be based on interpolation of the experimental data by the Lagrange polynomial L2(x) of the second degree on each segment [xk, xk+2] [14]. In the present work, evaluation of the integral of (5) is based on interpolation of the εx data by a cubic spline. In this case the integral of (5) will be equal to the sum of the integrals on each segment of the corresponding polynomial of the third degree. The distribution (restored from formula (5)) of the averaged refractive index of the medium in the case of the isothermal jet may be used for calculation of the averaged concentration of methane in the flow field. The possibility of the concentration being determined further derives from the additivity property of the refractions of the components forming the mixture. Since the refractive indices of gaseous substances differ little from unity, the initial system of equations can be written as [14] n −1 =

1 ( N1a1 + N 2 a 2 ) , 2

N1 + N 2 = N ,

(6)

(7)

where the first equation follows from the Lorentz−Lorenz relation for a gaseous mixture, whereas the second represents the sum of particles in a unit volume. The concentration of the components Xi in this case is proportional to the change in the medium’s refractive index and can be expressed, for methane, by the equation X CH 4 =

262

n − n0 . nCH 4 − n0

(8)

Fig. 5. Three-dimensional distributions of the averaged deflection angles in the flow field of the axisymmetric methane jet: a) 〈εx〉; b) 〈εy〉.

Fig. 6. Distributions of the averaged deflection angles 〈εx〉 (a) and 〈εy〉 (b) in different cross sections of the axisymmetric methane jet: 1) y = 10D, 2) 15D, and 3) 20D. Results of Experimental Investigation of the Processes of Turbulent Mixing in the Axisymmetric Methane Jet. Figure 5 gives three-dimensional distributions (resulting from the processing of experimental Talbot images) of the averaged deflection angles 〈εx〉 and 〈εy〉 in the axisymmetric methane jet flowing out into an air space. Figure 6 shows the distributions of the deflection angles for cross sections of the jet. Different signs of deflections point to the fact that the displacement of disturbed individual elements of the Talbot image from the initial undisturbed ones occurred both right and left along the x axis and up and down along the y axis. The maximum values of the deflection angles along the x axis do not exceed 3 ⋅10 −4 rad ( 60′′ ) in magnitude; along the y axis, they are no higher than 2.5 ⋅10 −5 rad ( 5′′ ). On the symmetry axis, the values of the two projections of the deflection angles are equal to zero. Figure 7 gives the distribution of the averaged refractive index in the axisymmetric methane jet, that has been obtained from the distribution of the deflection angles. Results for the averaged methane concentration with a maximum value on the jet axis, which has been calculated from these data, are presented in Fig. 8. A reduction in the methane concentration downstream of the jet is due to the turbulent mixing of methane and air. Figure 9 shows the reciprocal concentration of methane along the jet axis as a function of the dimensionless distance to the nozzle exit section y/D. The straight line drawn through the experimental data is a result of approximation by a linear function. To construct the approximation we have used the data whose abscissa satisfied the condition y/D ≥ 7. The functional dependence of the concentration along the jet axis on the distance to the nozzle exit section is expressed by the formula [2, 5]

( X CH 4 )0 ( X CH 4 )C

=

C1 ( y − y01 ) , D

(9)

263

Fig. 7. Three-dimensional radial distribution of the refractive index throughout the flow field of the axisymmetric jet (a) and for the jet cross sections (b): 1) y = 10D, 2) 15D, and 3) 20D.

Fig. 8. Distributions of the averaged concentration of methane in the axisymmetric jet throughout the flow field (a) and for the jet cross sections (b): 1) y = 10D, 2) 15D, and 3) 20D.

where ( X CH 4 )0 = 1 is the concentration of methane at exit from the nozzle, C1 is the constant determining the slope of the straight line, and y01 is the position of the virtual jet source. For the data given in the present work, the indicated parameters are C1 = 0.16 and y01 = 1.5D. The opening angle of the jet is characterized with the aid of r1/2, i.e., of the radial distance at which the concentration is halved compared to the value on the jet axis. At a sufficient distance from the nozzle’s edge, the value of r1/2 can be approximated by a function proportional to the distance from the virtual source y02. This dependence is expressed by the formula [2] r1 2 = C2 ( y − y02 ) ,

(10)

where C2 is the parameter characterizing the opening angle of the jet, and y02 is the position of the virtual source corresponding to the opening angle of the jet. Figure 10 gives r1/2 values which have been calculated from the experimental data obtained in the present work and as a result of their approximation by the function (10) in the domain y/D ≥ 7. A comparison of the results of measurement of the

264

Fig. 9. Inverse concentration of methane along the jet axis vs. distance to the nozzle exit section. Fig. 10. Dependence of r1/2 on the distance to the nozzle exit section.

Fig. 11. Radial distributions of the averaged concentration of methane: 1) data of the present work for the cross section y = 22.7D; 2) data of [2]. methane concentration on the basis of the method of averaged Talbot images to the results obtained in other investigations was based on the approximations of (9) and (10). We selected [2] and [5] in which methane-air flows have also been investigated. In [2], a study was made of the outflow of a methane jet into a slowly moving concurrent air flow, whereas in [5], the methane jet flowed out into a quiescent air space. The value of the Reynolds number for the jet investigated in the present work does not differ, in practice, from the value given in [2] but noticeably differs from its value in [5]. The method employed in [2] for measurements of the concentration of the components of a binary mixture is based on the Rayleigh scattering of laser radiation. In [5], the methane jet was investigated using the method based on Raman scattering. Table 1 gives the value of the parameters in Eqs. (9) and (10) for the data measured in the present work and published in [2] and [5]. In comparing the values of C1 given in the table, we see that the disagreement between these data amounts to as much as 10%. It should be taken into account that this coefficient is sensitive to the selection of the approximation domain y/D. Therefore, it is likely that on selection of the corresponding region, the disagreement between the data will decrease. The values of the parameter C2 for the approximation of (9) are in good agreement with the results of other authors. From Table 1, it is clear that there is a strong similarity of the results, despite the fact that the Reynolds number in [5] is four times larger, whereas in [2], the jet flows out into a concurrent flow. The differences in virtual sources point to dissimilar initial conditions of jet outflow in all the three cases. Figure 11 shows the radial concentration distribution of methane for the cross section 22.7D, obtained in the present work, and the data presented in Fig. 10 in [2]. The cross sections are selected so that the distance to the virtual source y02 is the same. As is clear from the figure, the differences in the data are minor despite the absence of a concurrent air flow. 265

TABLE 1. Coefficients of Eqs. (9) and (10) for Experimental Data Source Experimental results

Re 3900

C1 0.16

y01 1.5D

Data from [2]

4130

0.17

−3.7D

Data from [5]

16,000

0.18

1.7D

C2 0.107 0.108 0.105 0.097

y02 1.95D 0 −4.6D 0

On the basis of the good agreement between the characteristics of the concentration field that have been measured in the present work and have been published in [2] and [5] and of the absence of experimental proofs pointing to the significant difference in the values of the parameters measured, we can infer that the presented data provide a reliable description of the concentration field in the axisymmetric jet. Conclusions. In this work, we have presented the results of application of the method of Talbot images to determination of the concentration field in the binary mixture of an isothermal turbulent flow. It has been shown that the procedure ensures measurements of the field of deflection angles of probing radiation with a high spatial resolution. From the time-averaged values of the deflection angles, we have restored the distribution of the refractive index in an axisymmetric turbulent jet. Taking into account that the change in the refractive index in the binary isothermal jet is directly proportional to the change in the quantitative composition of the mixture, we have calculated the averaged concentration of methane along the jet flow. A comparison of the presented data to the results of concentration measurements, obtained by other methods, has been made. The possibility of measuring the spatial distribution of the averaged concentration of the mixture components makes this method highly promising for investigation of the processes of mixing. The use of high-speed image recording together with the presented procedure will enable one to investigate turbulent processes with a high temporal resolution. In the future, it is planned to use the method of Talbot images for measurement of the dispersion dependence of the refractive index in the flow field of a reactive methane jet.

NOTATION a, molecular polarizability, m3; D, diameter of the nozzle outlet cross section, m; F, focal length of the objective, mm; L, self-reproduction distance, m; n, refractive index of the medium; N, number of particles in a unit volume, m −3 ; p, period of the grating, m; r, radius in a cylindrical coordinate system, m; R, radius of an optical inhomogeneity, m; u, velocity of flow of the methane jet out of the nozzle, m/s; X, relative concentration of the component of the mixture; x, y, and z, Cartesian coordinate system, m; ε, angle of deflection of light, angular seconds; λ, probing-radiation wavelength, nm; ν, kinematic viscosity, m2/s. Subscripts: 0, undisturbed medium; k, reproduction-zone number.

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