^{1}and Dimitrios V. Papavassiliou

^{1,a)}

### Abstract

The concept of reverse diffusion, introduced by Corrsin to describe the motion of particles as they move towards a location in the flow field, is fundamental to the understanding of mixing. In this work, direct numerical simulations in conjunction with the tracking of scalar markers are utilized in infinitely long channels to study the principal direction of transport of heat (or mass) for both forwards and backwards single particle dispersion. The viscous sub-layer, the transition region (between the viscous sub-layer and the logarithmic region), and the logarithmic region of a Poiseuille flow and a plane Couette flow channel are studied. Fluctuating velocities of scalar markers captured in these regions are used to obtain the full autocorrelation coefficient tensor forwards and backwards with time. The highest eigenvalue of the velocity correlation coefficient tensor quantifies the highest amount of turbulentheat transport, while the corresponding eigenvector points to the main direction of transport. Different Prandtl number, *Pr*, fluids are simulated for the two types of flow. It is found that the highest eigenvalues are higher in the case of backwards dispersion compared to the case of forwards dispersion for any *Pr*, in both flow cases. The principal direction for backwards and forwards dispersion is different than for forwards dispersion, for all *Pr*, and in all flow regions for both flows. Fluids with lower *Pr* behave different than the higher *Pr* fluids because of increased molecular diffusion effects. The current study also establishes an interesting analogy of turbulentdispersion to optics defining the turbulent dispersive ratio, a parameter that can be used to identify the differences in the direction of turbulentheat transport between forwards and backwards dispersion. A spectral analysis of the auto-correlation coefficient for both forwards and backwards dispersion shows a universal behavior with slope of −1 at intermediate frequencies.

The authors are grateful for the financial support provided by NSF under CBET-0651180. The computational support offered by the TeraGrid under TRAC TG-CTS090025 is also acknowledged. The work utilized the SGI Altix system (Cobalt), Intel 64 Dell Cluster (Abe), and Intel 64 Tesla Linux Cluster (Lincoln) at NCSA at the University of Illinois. The authors would also like to acknowledge the University of Oklahoma Center for Supercomputing Education and Research (OSCER) for computer support.

I. INTRODUCTION

II. FORWARDS AND BACKWARDS DISPERSION BACKGROUND

III. NUMERICAL METHODOLOGY

A. Direct numerical simulation

B. Lagrangian scalar tracking

C. Simulation procedure

IV. RESULTS AND DISCUSSION

A. Cross-correlation coefficients for Poiseuille flow

B. Cross-correlation coefficients for plane Couette flow

C. Direction of heat transport

1. Largest eigenvalues

2. Principal directions of heat transfer

3. Lagrangian scalar spectrum

V. CONCLUSIONS

### Key Topics

- Poiseuille flow
- 75.0
- Eigenvalues
- 63.0
- Couette flows
- 61.0
- Dispersion
- 38.0
- Viscosity
- 37.0

## Figures

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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

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

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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*/τ_{ y })

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*/τ_{ y })

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*/τ_{ y }).

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*/τ_{ y }).

## Tables

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.

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.

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.

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.

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*.

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*.

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*.

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*.

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.

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.

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.

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.

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.

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.

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.

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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