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Direction of scalar transport in turbulent channel flow
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10.1063/1.3657825
/content/aip/journal/pof2/23/11/10.1063/1.3657825
http://aip.metastore.ingenta.com/content/aip/journal/pof2/23/11/10.1063/1.3657825

Figures

Image of FIG. 1.
FIG. 1.

Material cross-correlation coefficients plotted as a function of time for different Pr, in cases of forwards and backwards dispersion of markers captured and correlated in the viscous sub-layer of Poisueille channel flow: (a) R uv , y = 5; (b) R vu , y = 5; (c) R uw , y = 5; (d) R wx , y = 5; (e) R vw , y = 5; and (f) R wv , y = 5. In (b)–(f), in order to clearly present the results, the curves for Pr > 6 are all represented by the curve for Pr = 1000.

Image of FIG. 2.
FIG. 2.

Material cross-correlation coefficients plotted as a function of time for different Pr, in cases of forwards and backwards dispersion for markers captured and correlated in the transition region of Poiseuille channel flow: (a) R uv , y = 37; (b) R vu , y = 37. In order to clearly present the results, the curves for Pr > 6 are all represented by the curve for Pr = 1000. Also, since the material cross-correlation coefficients obtained from correlations with the spanwise velocities are zero, they are not presented for the transition regions of Poiseuille channel flow.

Image of FIG. 3.
FIG. 3.

Material cross-correlation coefficients plotted as a function of time for different Pr, in cases of forwards and backwards dispersion for markers captured and correlated in the logarithmic region of Poiseuille channel flow: (a) R uv , y = 75; (b) R vu , y = 75. In order to clearly present the results, the curves for Pr > 6 are all represented by the curve for Pr = 1000. Also, since the material cross-correlation coefficients obtained from correlations with the spanwise velocities are zero, they are not presented for the logarithmic regions of Poiseuille channel flow.

Image of FIG. 4.
FIG. 4.

Material cross-correlation coefficients plotted as a function of time for different Pr, in cases of forwards and backwards dispersion for markers captured and correlated in the center of the channel for Poiseuille channel flow: (a) R uv , y = 150; (b) R vu , y = 150. In order to clearly present the results, the curves for Pr > 6 are all represented by the curve for Pr = 1000. Also, since the material cross-correlation coefficients obtained from correlations with the spanwise velocities are zero, they are not presented for the center of the Poiseuille channel flow.

Image of FIG. 5.
FIG. 5.

Material cross-correlation coefficients plotted as a function of time for different Pr, in cases of forwards and backwards dispersion of markers captured and correlated in the viscous sub-layer of plane Couette flow: (a) R uv , y = 5; (b) R vu , y = 5. In (b), to clearly present the results, the curves for Pr > 6 are all represented by the curve for Pr = 500. Also, since the material cross-correlation coefficients obtained from correlations with the spanwise velocities are zero, they are not presented for the viscous region of plane Couette flow.

Image of FIG. 6.
FIG. 6.

Material cross-correlation coefficients plotted as a function of time for different Pr, in cases of forwards and backwards dispersion of markers captured and correlated in the transition region of plane Couette flow: (a) R uv , y = 37; (b) R vu , y = 37. In order to clearly present the results, the curves for Pr > 6 are all represented by the curve for Pr = 500. Also, since the material cross-correlation coefficients obtained from correlations with the spanwise velocities are zero, they are not presented for the transition region of plane Couette flow.

Image of FIG. 7.
FIG. 7.

Material cross-correlation coefficients plotted as a function of time for different Pr, in cases of forwards and backwards dispersion of markers captured and correlated in the logarithmic region of plane Couette flow: (a) R uv , y = 75; (b) R vu , y = 75. In order to clearly present the results, the curves for Pr > 6 are all represented by the curve for Pr = 500. Also, since the material cross-correlation coefficients obtained from correlations with the spanwise velocities are zero, they are not presented for the logarithmic region of plane Couette flow.

Image of FIG. 8.
FIG. 8.

Highest eigenvalues obtained from the correlation coefficient matrix for both forwards and backwards dispersion plotted as a function of time for different Pr in case of Poiseuille channel flow: (a) y = 5; (b) y = 37; (c) y = 75; and (d) y = 150. In order to present the plot with clarity, the curves for Pr > 6 are all represented by the curve for Pr = 1000.

Image of FIG. 9.
FIG. 9.

Highest eigenvalues obtained from the correlation coefficient matrix for both forwards and backwards dispersion plotted as a function of time for different Pr in case of plane Couette flow: (a) y = 5; (b) y = 37; and (c) y = 75. In order to present the plot with clarity, the curves for Pr > 6 are all represented by the curve for Pr = 500.

Image of FIG. 10.
FIG. 10.

Orientation of the eigenvectors corresponding to the highest eigenvalues plotted in three dimensions in a domain comparable to the computational box, not to exact scale, as a function of time for a Pr = 0.1 in all four regions of Poiseuille channel flow: (a) forwards dispersion; (b) backwards dispersion.

Image of FIG. 11.
FIG. 11.

Orientation of the eigenvectors corresponding to the highest eigenvalues plotted in three dimensions in a domain comparable to the computational box, not to exact scale, as a function of time for a Pr = 0.1 in all three regions of plane Couette flow: (a) forwards dispersion; (b) backwards dispersion.

Image of FIG. 12.
FIG. 12.

Representation of the different angles made by the primary eigenvector makes with the normal of the three different planes in our current study.

Image of FIG. 13.
FIG. 13.

Schematic of the suggested analogy between optics and turbulent backwards and forwards dispersion. The angle of incidence of light in medium 1 (θ1) is similar to the angle that the direction of backwards dispersion of heat makes with the normal of the plane (presented also as θ1 in the right panel), while the angle of refraction in medium 2 (θ2) is comparable to that of the forwards dispersion with the normal of the plane (presented as θ2 in the right panel of the figure).

Image of FIG. 14.
FIG. 14.

Direction of the eigenvector corresponding to the highest eigenvalue obtained for markers captured and correlated in the viscous sub-layer with forwards and backwards dispersion plotted as a function of time, for the case of different Pr in Poiseuille channel flow: (a) angle with the xy plane; (b) angle with the yz plane; and (c) angle with the zx plane. In order to present the plot with clarity, the curves for Pr > 6 are all represented by the curve for Pr = 1000.

Image of FIG. 15.
FIG. 15.

Direction of the eigenvector corresponding to the highest eigenvalue obtained for markers captured and correlated in the transition region with forwards and backwards dispersion plotted as a function of time, for the case of different Pr in Poiseuille channel flow: (a) angle with the xy plane; (b) angle with the yz plane; and (c) angle with the zx plane. In order to present the plot with clarity, the curves for Pr > 6 are all represented by the curve for Pr = 1000.

Image of FIG. 16.
FIG. 16.

Direction of the eigenvector corresponding to the highest eigenvalue obtained for markers captured and correlated in the log-layer with forwards and backwards dispersion plotted as a function of time, for the case of different Pr in Poiseuille channel flow: (a) angle with the xy plane; (b) angle with the yz plane; (c) angle with the zx plane. In order to present the plot with clarity, the curves for Pr > 6 are all represented by the curve for Pr = 1000.

Image of FIG. 17.
FIG. 17.

Direction of the eigenvector corresponding to the highest eigenvalue obtained for markers captured and correlated in the viscous sub-layer with forwards and backwards dispersion plotted as a function of time, for the case of different Pr in plane Couette flow: (a) angle with the xy plane; (b) angle with the yz plane; and (c) angle with the zx plane. In order to present the plot with clarity, the curves for Pr > 6 are all represented by the curve for Pr = 500.

Image of FIG. 18.
FIG. 18.

Direction of the eigenvector corresponding to the highest eigenvalue obtained for markers captured and correlated in the transition region with forwards and backwards dispersion plotted as a function of time, for the case of different Pr in plane Couette flow: (a) angle with the xy plane; (b) angle with the yz plane; and (c) angle with the zx plane. In order to present the plot with clarity, the curves for Pr > 6 are all represented by the curve for Pr = 500.

Image of FIG. 19.
FIG. 19.

Direction of the eigenvector corresponding to the highest eigenvalue obtained for markers captured and correlated in the log-layer with forwards and backwards dispersion plotted as a function of time, for the case of different Pr in plane Couette flow: (a) angle with the xy plane; (b) angle with the yz plane; (c) angle with the zx plane. In order to present the plot with clarity, the curves for Pr > 6 are all represented by the curve for Pr = 500.

Image of FIG. 20.
FIG. 20.

Spectrum of the material autocorrelation coefficient R vv in case of forwards and backwards dispersion of markers captured and correlated in the viscous sub-layer for a low and a high Pr: (a) Poiseuille channel flow; (b) plane Couette flow. The lines marked “Analytical” show the spectrum of the material autocorrelation coefficient of R vv  = exp(−t y )

Image of FIG. 21.
FIG. 21.

Spectrum of the material autocorrelation coefficient R vv in case of forwards and backwards dispersion of markers captured and correlated in the log-layer for a low and a high Pr: (a) Poiseuille channel flow; (b) plane Couette flow. The lines marked “Analytical” show the spectrum of the material autocorrelation coefficient of R vv  = exp(−t y ).

Tables

Generic image for table
Table I.

Lagrangian material time scale in the vertical direction presented for the cases of different Pr, at different regions of Poiseuille channel flow, for both forwards and backwards turbulent dispersion.

Generic image for table
Table II.

Lagrangian material time scale in the vertical direction presented for the cases of different Pr, at different regions of plane Couette flow, for both forwards and backwards turbulent dispersion.

Generic image for table
Table III.

Angles of inclinations of the eigenvector directions corresponding to the highest eigenvalue with the normal to the different planes, similar to the angles in optics, at the vertical Lagrangian material scales for different regions of Poiseuille channel flow for both forwards and backwards dispersion, with changes in Pr.

Generic image for table
Table IV.

Angles of inclinations of the eigenvector directions corresponding to the highest eigenvalue with the normal to the different planes, similar to the angles in optics, at the vertical Lagrangian material scales for different regions of plane Couette flow for both forwards and backwards dispersion, with changes in Pr.

Generic image for table
Table V.

Measure of the turbulent dispersive ratio (forwards dispersive index to the backwards dispersive index), obtained from the ratio of sine of the angle of backwards dispersion primary eigenvector to the sine of the angle of forwards dispersion primary eigenvector with the three different planes for different Pr, at various regions of the Poiseuille channel flow.

Generic image for table
Table VI.

Measure of the turbulent dispersive ratio (forwards dispersive index to the backwards dispersive index), obtained from the ratio of sine of the angle of backwards dispersion primary eigenvector to the sine of the angle of forwards dispersion primary eigenvector with the three different planes for different Pr, at various regions of the plane Couette flow.

Generic image for table
Table VII.

Measure of the slopes of the Lagrangian scalar spectrum at the intermediate frequency range obtained from the forwards and backwards auto-correlation coefficient in the x, y, z directions, for different Pr, at various regions of the Poiseuille channel flow.

Generic image for table
Table VIII.

Measure of the slopes of the Lagrangian scalar spectrum at the intermediate frequency range obtained from the forwards and backwards auto-correlation coefficient in the x, y, z directions, for different Pr, at various regions of plane Couette flow.

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/content/aip/journal/pof2/23/11/10.1063/1.3657825
2011-11-11
2014-04-18
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Direction of scalar transport in turbulent channel flow
http://aip.metastore.ingenta.com/content/aip/journal/pof2/23/11/10.1063/1.3657825
10.1063/1.3657825
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