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Figure 4: Theoretical n profiles compared with experimental data.

Figure 4 Theoretical n profiles compared with experimental data.

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‘ isotropic turbulence, the thickness ‘kK’ is defined for the ‘Kolmogorov sublayer’ using the same variables of the viscous sub-layer. When mass transfer occurs, an ‘outer-diffusive sublayer’ is defined, which inserts the Schmidt number Sc in the quantification of the mean value of the thickness ‘O’. Finally, very close to the surface, the ‘Batchelor sublayer’ is defined with thickness ‘B’ , representing the mean distance that is more affected by characteristics of the smallest eddies in mass transfer. This last sublayer uses the same parameters of the outer-diffusive sublayer for its quantification. In the sketch of Fig. 1, it is assumed that Sc>1.
‘ isotropic turbulence, the thickness ‘kK’ is defined for the ‘Kolmogorov sublayer’ using the same variables of the viscous sub-layer. When mass transfer occurs, an ‘outer-diffusive sublayer’ is defined, which inserts the Schmidt number Sc in the quantification of the mean value of the thickness ‘O’. Finally, very close to the surface, the ‘Batchelor sublayer’ is defined with thickness ‘B’ , representing the mean distance that is more affected by characteristics of the smallest eddies in mass transfer. This last sublayer uses the same parameters of the outer-diffusive sublayer for its quantification. In the sketch of Fig. 1, it is assumed that Sc>1.
Figure 2: Mean statistical properties obtained from series of instantaneous measured fields. bi-modal random signals and corrective functions. In the 1D case of scalar transport four such functions are needed. Thus, as mentioned by Refs. [12] and [13], also four equations are needed. By rearranging terms, two of the parameters were linked, remaining three to be solved through only three equations. By considering one of the parameters constant (the reduction function ‘a’) it was possible to reduce the problem substantially and establish one equation for the so-called partition function ‘n’. In fact, the reduction function varies with z, the distance to the surface, and profiles of ‘a’ obtained from experimental data may be found in Refs. [14] and [8], for example (see Fig. 3). For stationary turbulence and constant a the equation for n was theoretically solved ({15] and [16]), allowing checking the behavior of the mean concentration profile. Equations (1) and (2) are the 1D mass conservation equation and the nondimensional form of the turbulent mas flux for the RSW method, respectively.
Figure 2: Mean statistical properties obtained from series of instantaneous measured fields. bi-modal random signals and corrective functions. In the 1D case of scalar transport four such functions are needed. Thus, as mentioned by Refs. [12] and [13], also four equations are needed. By rearranging terms, two of the parameters were linked, remaining three to be solved through only three equations. By considering one of the parameters constant (the reduction function ‘a’) it was possible to reduce the problem substantially and establish one equation for the so-called partition function ‘n’. In fact, the reduction function varies with z, the distance to the surface, and profiles of ‘a’ obtained from experimental data may be found in Refs. [14] and [8], for example (see Fig. 3). For stationary turbulence and constant a the equation for n was theoretically solved ({15] and [16]), allowing checking the behavior of the mean concentration profile. Equations (1) and (2) are the 1D mass conservation equation and the nondimensional form of the turbulent mas flux for the RSW method, respectively.
ion’, which quantifies the coherence between concentration and velocity fluctuations; ¥@” is he RMS of «, the fluctuation of the vertical velocity (ms), being ¥@** the normalizec ntensity of turbulence and A=1-a replaces the normalized reduction function. c* and z* are he normalized concentration fluctuation and the vertical dimension, respectively, the last vith origin at the surface and positive downwards; D is the molecular diffusivity of the gas ir he liquid (ms‘?), E is the concentration boundary layer thickness (m), given by the positior vhere n=0.01; JJ* = w*c * is the normalized turbulent mass flux. C s and C, are the concen: rations of gas at the surface and the bulk liquid, respectively (kg.m"3). The integration of eqr 1) furnishes the mass flux that crosses the medium, j*= -dn/dz*+IJ*, nondimensional anc onstant along z* (integration constant). Equation (1) shows that if dn/dz* has a critical point maximum or minimum) in the physical domain, then /J* also has a critical point at the same osition. By assuming a minimum value of zero for //*, the integration constant assumes the value j*=-dn/dz* | .,_,. On the other hand, considering the structure shown in Figs | and 2.
ion’, which quantifies the coherence between concentration and velocity fluctuations; ¥@” is he RMS of «, the fluctuation of the vertical velocity (ms), being ¥@** the normalizec ntensity of turbulence and A=1-a replaces the normalized reduction function. c* and z* are he normalized concentration fluctuation and the vertical dimension, respectively, the last vith origin at the surface and positive downwards; D is the molecular diffusivity of the gas ir he liquid (ms‘?), E is the concentration boundary layer thickness (m), given by the positior vhere n=0.01; JJ* = w*c * is the normalized turbulent mass flux. C s and C, are the concen: rations of gas at the surface and the bulk liquid, respectively (kg.m"3). The integration of eqr 1) furnishes the mass flux that crosses the medium, j*= -dn/dz*+IJ*, nondimensional anc onstant along z* (integration constant). Equation (1) shows that if dn/dz* has a critical point maximum or minimum) in the physical domain, then /J* also has a critical point at the same osition. By assuming a minimum value of zero for //*, the integration constant assumes the value j*=-dn/dz* | .,_,. On the other hand, considering the structure shown in Figs | and 2.
K is a nondimensional constant coefficient. Equation (4) was integrated for several values of 0,, or A, the last being a measure of the relative influence between molecular diffusivity and turbulence in the transfer phenomenon, or, equivalently, a measure of agitation. In the present study the values 0,=2 (or A=0.43099) and 6,=4 (or A=0.30007) are considered, which allow integrating eqn (4) obtaining eqns (5a) and (5b), respectively:
K is a nondimensional constant coefficient. Equation (4) was integrated for several values of 0,, or A, the last being a measure of the relative influence between molecular diffusivity and turbulence in the transfer phenomenon, or, equivalently, a measure of agitation. In the present study the values 0,=2 (or A=0.43099) and 6,=4 (or A=0.30007) are considered, which allow integrating eqn (4) obtaining eqns (5a) and (5b), respectively:
Figure 3: Experimental data of a varying along z*, and the two constant values used here.
Figure 3: Experimental data of a varying along z*, and the two constant values used here.
Figure 5: (a) n and concentration RMS profiles; (b) Higher order central moment profiles. The full lines of Fig. 5a were already presented in Fig. 4. Because 0, (or A) are measures of the agitation, these curves show that different agitation conditions produce different normalized pro- files. For 7=2, eqn (6) furnishes the RMS function, given by:
Figure 5: (a) n and concentration RMS profiles; (b) Higher order central moment profiles. The full lines of Fig. 5a were already presented in Fig. 4. Because 0, (or A) are measures of the agitation, these curves show that different agitation conditions produce different normalized pro- files. For 7=2, eqn (6) furnishes the RMS function, given by:
Figure 6: Experimental and theoretical concentration RMS profiles. Data of [5, 19, 20]. Equations (7a) and (7b) show that the solutions for constant ‘a’ produce RMS profiles that are paralel to the over axis at z*=0. That is, the derivative of the RMS function tends to inifinite at this point. Figure 6 shows experimental and theoretical RMS profiles, the last obtained with eqns (Sa), (5b) and (7). In this figure the horizontal axis follows, for convenience to reproduce the data, the normalization adopted by [5] as z+=z/d, being 6 the position where the concentration difference C(z)-C, atains 36.7% of the total difference C,-C,. The shape of the theoretical solutions follow the experimental data. Maximum values and peak positions are in the range of the observed values, and eqn (5a) follows more closely the observed data. The
Figure 6: Experimental and theoretical concentration RMS profiles. Data of [5, 19, 20]. Equations (7a) and (7b) show that the solutions for constant ‘a’ produce RMS profiles that are paralel to the over axis at z*=0. That is, the derivative of the RMS function tends to inifinite at this point. Figure 6 shows experimental and theoretical RMS profiles, the last obtained with eqns (Sa), (5b) and (7). In this figure the horizontal axis follows, for convenience to reproduce the data, the normalization adopted by [5] as z+=z/d, being 6 the position where the concentration difference C(z)-C, atains 36.7% of the total difference C,-C,. The shape of the theoretical solutions follow the experimental data. Maximum values and peak positions are in the range of the observed values, and eqn (5a) follows more closely the observed data. The
Figure 7: Diffusive flux dn/dz*; turbulent flux @*c *, and total constant flux j*.
Figure 7: Diffusive flux dn/dz*; turbulent flux @*c *, and total constant flux j*.
Figure 8: Normalized theoretical turbulent fluxes from indicated equations, and envelope ot measured values showing the decreasing of the turbulent flux for z*>~1. PE Se ee) Mn: Te Ne: ee a ee Figure 8 shows the envelope of experimental data of w*c * for several agitation conditions obtained by Ref. [5], together with eqns (9a), (9b) and (9c). The horizontal axis follows the same definition used in Fig. 6. The envelope shows possible positive, null, and negative val- ues of w*c *close to z*=0. The negative range of w*c* would imply, when using eqn (2), also negative values of the RMS velocity, a situation not observed in the experimental data. The growing region of the experimental data for z*<~1 (dark grey) is followed more closely
Figure 8: Normalized theoretical turbulent fluxes from indicated equations, and envelope ot measured values showing the decreasing of the turbulent flux for z*>~1. PE Se ee) Mn: Te Ne: ee a ee Figure 8 shows the envelope of experimental data of w*c * for several agitation conditions obtained by Ref. [5], together with eqns (9a), (9b) and (9c). The horizontal axis follows the same definition used in Fig. 6. The envelope shows possible positive, null, and negative val- ues of w*c *close to z*=0. The negative range of w*c* would imply, when using eqn (2), also negative values of the RMS velocity, a situation not observed in the experimental data. The growing region of the experimental data for z*<~1 (dark grey) is followed more closely
Figure 9: (a) Inner and outer sides of the boundary layer; (b) Overall /2 slope trend. ‘onsidering here only the case of n profiles without inflexion points (derivatives without ritical points), eqn (11) can be easily calculated because eqns (5a) and (7a) furnishy cw ; nd eqn (9a) furnishes @*c *. For the mentioned equations, Fig. 9 shows the ¥@*’ profile btained from the RSW method, evidently applied to the space of the concentration boundary ayer, together with experimental data from the literature, obtained as nearer as possible to the nterface, but not entering the boundary layer. The measured values were obtained from the eferences [1] and [20]. In order to consider similar experimental conditions of both sources, he data having similar reference turbulence velocities at the surface were used. The reference
Figure 9: (a) Inner and outer sides of the boundary layer; (b) Overall /2 slope trend. ‘onsidering here only the case of n profiles without inflexion points (derivatives without ritical points), eqn (11) can be easily calculated because eqns (5a) and (7a) furnishy cw ; nd eqn (9a) furnishes @*c *. For the mentioned equations, Fig. 9 shows the ¥@*’ profile btained from the RSW method, evidently applied to the space of the concentration boundary ayer, together with experimental data from the literature, obtained as nearer as possible to the nterface, but not entering the boundary layer. The measured values were obtained from the eferences [1] and [20]. In order to consider similar experimental conditions of both sources, he data having similar reference turbulence velocities at the surface were used. The reference
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Materials ScienceTurbulence statisticsConcentration fluctuationsRandom Square Waves
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