^{1}and Stephen B. Pope

^{1}

### Abstract

Probability density function (PDF) calculations are reported for the dispersion from line sources in decaying grid turbulence. The calculations are performed using a modified form of the interaction by exchange with the conditional mean (IECM) mixing model. These flows pose a significant challenge to statistical models because the scalar length scale (of the initial plume) is much smaller than the turbulence integral scale. Consequently, this necessitates incorporating the effects of molecular diffusion in order to model laboratory experiments. Previously, Sawford [Flow Turb. Combust.72, 133 (2004)] performed PDF calculations in conjunction with the IECM mixing model,modeling the effects of molecular diffusion as a random walk in physical space and using a mixing time scale empirically fit to the experimental data of Warhaft [J. Fluid Mech.144, 363 (1984)]. The resulting transport equation for the scalar variance contains a spurious production term. In the present work, the effects of molecular diffusion are instead modeled by adding a conditional mean scalar drift term, thus avoiding the spurious production of scalar variance. A laminar wake model is used to obtain an analytic expression for the mixing time scale at small times, and this is used as part of a general specification of the mixing time scale. Based on this modeling, PDF calculations are performed, and comparison is made primarily with the experimental data of Warhaft on single and multiple line sources and with the previous calculations of Sawford. A heated mandoline is also considered with comparison to the experimental data of Warhaft and Lumley [J. Fluid Mech.88, 659 (1978)]. This establishes the validity of the proposed model and the significant effect of molecular diffusion on the decay of scalar fluctuations. The following are the significant predictions of the model. For the line source, the effect of the source size is limited to early times and can be completely accounted for by simple transformations. The peak centerline ratio of the rms to the mean of the scalar increases with the Reynolds number (approximately as ), whereas this ratio tends to a constant (approximately 0.4) at large times independent of . In addition, the model yields a universal long-time decay exponent for the temperature variance.

This paper is dedicated to John Kim on the occasion of his 60th birthday. S.V. would like to thank Professor Z. Warhaft and Brian Sawford for valuable comments and also Prasad Bhave and Haifeng Wang for insightful discussions during the course of this work. This research is supported by the Department of Energy under Grant No. DE-FG02-90ER. This research was conducted using the resources of the Cornell University Center for Advanced Computing, which receives funding from Cornell University, New York State, the National Science Foundation, and other leading public agencies, foundations, and corporations.

I. INTRODUCTION

II. EXPERIMENTAL DETAILS

III. MODELING

A. Turbulence

B. Mixing model

1. IECM mixing model

2. Modified IECM mixing model

C. Laminar thermal wake modeling

D. Mixing rate

1. IECM model

2. Modified IECM model

E. Summary of the model

IV. IMPLEMENTATION

V. RESULTS AND DISCUSSION

A. A single line source

B. A pair of line sources

C. An array of line sources

D. The heated mandoline

E. The effect of the choices of and

F. Effect of the Reynolds number and source size

1. Dependence on the Reynolds number

2. Dependence on normalized source size

VI. CONCLUSIONS

### Key Topics

- Turbulence simulations
- 48.0
- Turbulent flows
- 48.0
- Diffusion
- 30.0
- Scalar field theory
- 17.0
- Langevin equation
- 15.0

## Figures

Sketch of the experimental setup showing the wind tunnel. The source (dot) is at a distance from the turbulence generating grid.

Sketch of the experimental setup showing the wind tunnel. The source (dot) is at a distance from the turbulence generating grid.

Comparison of the centerline intensity of fluctuations obtained using the laminar thermal wake model: (dot-dashed line) and (solid line); Warhaft (Ref. 7) data (▼) and Warhaft (Ref. 7) data (▲); Sawford’s (Ref. 23) model calculations (dashed line) plotted against flight time from the source for source position and source size .

Comparison of the centerline intensity of fluctuations obtained using the laminar thermal wake model: (dot-dashed line) and (solid line); Warhaft (Ref. 7) data (▼) and Warhaft (Ref. 7) data (▲); Sawford’s (Ref. 23) model calculations (dashed line) plotted against flight time from the source for source position and source size .

Comparison of mixing rate definitions with flight time from the source: Modified mixing model (dashed line); IECM model (thick solid line), (thin solid line); Sawford’s (Ref. 23) empirical mixing rate (dot-dashed line).

Comparison of mixing rate definitions with flight time from the source: Modified mixing model (dashed line); IECM model (thick solid line), (thin solid line); Sawford’s (Ref. 23) empirical mixing rate (dot-dashed line).

Width of the mean scalar profile normalized by the turbulence length scale at the source against normalized flight time from the source for source position ; from Eq. (11) (solid line) and from the present model calculations (◼).

Width of the mean scalar profile normalized by the turbulence length scale at the source against normalized flight time from the source for source position ; from Eq. (11) (solid line) and from the present model calculations (◼).

Comparison of the centerline intensity of fluctuations, , plotted against flight time from the source. Warhaft (Ref. 7) data: , (◼), , (▼), , (▲), and , (◆). Sawford’s (Ref. 23) calculations using the mixing rate given by Eq. (44): , (thin solid line) and , (thin dashed line). Present calculations: , (thick solid line), , (thick dashed line), and , (dotted line).

Comparison of the centerline intensity of fluctuations, , plotted against flight time from the source. Warhaft (Ref. 7) data: , (◼), , (▼), , (▲), and , (◆). Sawford’s (Ref. 23) calculations using the mixing rate given by Eq. (44): , (thin solid line) and , (thin dashed line). Present calculations: , (thick solid line), , (thick dashed line), and , (dotted line).

Comparison of IECM model calculations with the mixing rate given by Eq. (17) with the model calculations done with Eq. (50) showing the centerline intensity of fluctuations, , against flight time from the source. IECM model calculations using mixing rate given by Eq. (17): , (thick dot-dashed line). Present calculations: , (thick dashed line) and , (thick solid line). Warhaft (Ref. 7) data: , (◼), , (▼), and , (▲).

Comparison of IECM model calculations with the mixing rate given by Eq. (17) with the model calculations done with Eq. (50) showing the centerline intensity of fluctuations, , against flight time from the source. IECM model calculations using mixing rate given by Eq. (17): , (thick dot-dashed line). Present calculations: , (thick dashed line) and , (thick solid line). Warhaft (Ref. 7) data: , (◼), , (▼), and , (▲).

Radial profiles of rms scalar normalized by its centerline value at . Warhaft (Ref. 7) data: (▲), (◼), (●), and (◆). Present calculations: (dotted line), (dot-dashed line), (solid line), and (dashed line).

Radial profiles of rms scalar normalized by its centerline value at . Warhaft (Ref. 7) data: (▲), (◼), (●), and (◆). Present calculations: (dotted line), (dot-dashed line), (solid line), and (dashed line).

Integral measure of the scalar variance in nondimensional form against flight time from the source. Present calculations (solid line); Warhaft (Ref. 7) data (●). The source of size is at .

Integral measure of the scalar variance in nondimensional form against flight time from the source. Present calculations (solid line); Warhaft (Ref. 7) data (●). The source of size is at .

Higher moments on the centerline against flight time from the source: Present calculations (solid line); Sawford (Ref. 23) IECM calculations (dashed line); Sawford and Tivendale (Ref. 23) data (●): (a) skewness and (b) kurtosis .

Higher moments on the centerline against flight time from the source: Present calculations (solid line); Sawford (Ref. 23) IECM calculations (dashed line); Sawford and Tivendale (Ref. 23) data (●): (a) skewness and (b) kurtosis .

Radial profiles of higher-order moments measured at varying distances from the source. Present calculations (solid line); Sawford (Ref. 23) IECM calculations (dashed line); Sawford and Tivendale (Ref. 23) data (●): (a) skewness at , (b) skewness at , (c) skewness at , (d) kurtosis at , (e) kurtosis at , and (f) kurtosis at .

Radial profiles of higher-order moments measured at varying distances from the source. Present calculations (solid line); Sawford (Ref. 23) IECM calculations (dashed line); Sawford and Tivendale (Ref. 23) data (●): (a) skewness at , (b) skewness at , (c) skewness at , (d) kurtosis at , (e) kurtosis at , and (f) kurtosis at .

Evolution of the centerline cross-correlation coefficient for various source spacings, . The sources are placed at a distance of from the turbulence generating grid: Warhaft (Ref. 7) data (●); Sawford (Ref. 23) model calculations (dot-dashed line); present calculations (solid line).

Evolution of the centerline cross-correlation coefficient for various source spacings, . The sources are placed at a distance of from the turbulence generating grid: Warhaft (Ref. 7) data (●); Sawford (Ref. 23) model calculations (dot-dashed line); present calculations (solid line).

Radial profiles of rms scalar normalized by their respective centerline values when the sources are positioned at from the turbulence grid for different spacings between the sources, : (a) and , (b) and , and (c) and ; present model calculations (solid line); Warhaft (Ref. 7) data: (●), (◼), and (◆).

Radial profiles of rms scalar normalized by their respective centerline values when the sources are positioned at from the turbulence grid for different spacings between the sources, : (a) and , (b) and , and (c) and ; present model calculations (solid line); Warhaft (Ref. 7) data: (●), (◼), and (◆).

Radial profiles of the cross-correlation coefficient between sources 1 and 2 for different spacings between the two sources, . The sources are positioned at from the turbulence generating grid: (a) , (b) , (c) , and (d) . Present model calculations (solid line). Warhaft (Ref. 7) data: (●), (▲), (◼), (▼), and (◆).

Radial profiles of the cross-correlation coefficient between sources 1 and 2 for different spacings between the two sources, . The sources are positioned at from the turbulence generating grid: (a) , (b) , (c) , and (d) . Present model calculations (solid line). Warhaft (Ref. 7) data: (●), (▲), (◼), (▼), and (◆).

(a) Radial profiles of rms scalar corresponding to each of the four sources in an array, normalized by their respective centerline values at ; (b) radial profiles of rms scalar corresponding to ; (c) radial profiles of rms scalar corresponding to ; (d) radial profiles of rms scalar corresponding to ; (e) radial profiles of rms scalar corresponding to all the four sources. The radial profiles in (b)–(e) are normalized by the mean centerline value obtained from (a). Present model calculations (solid line); Warhaft (Ref. 7) data (●).

(a) Radial profiles of rms scalar corresponding to each of the four sources in an array, normalized by their respective centerline values at ; (b) radial profiles of rms scalar corresponding to ; (c) radial profiles of rms scalar corresponding to ; (d) radial profiles of rms scalar corresponding to ; (e) radial profiles of rms scalar corresponding to all the four sources. The radial profiles in (b)–(e) are normalized by the mean centerline value obtained from (a). Present model calculations (solid line); Warhaft (Ref. 7) data (●).

Radial profiles of the cross-correlation coefficient between pairs of sources at . Diffusion behind an array of four sources is considered. The sources are positioned at from the turbulence grid: (a) , sources 2 and 3; (b) , sources 2 and 4; (c) , sources 1 and 4. Present model calculations (solid line); Warhaft (Ref. 7) data (●).

Radial profiles of the cross-correlation coefficient between pairs of sources at . Diffusion behind an array of four sources is considered. The sources are positioned at from the turbulence grid: (a) , sources 2 and 3; (b) , sources 2 and 4; (c) , sources 1 and 4. Present model calculations (solid line); Warhaft (Ref. 7) data (●).

Experimental data of decay of normalized scalar fluctuations, , downstream of a heated mandoline from the turbulence generating grid. Relevant parameters are listed in Table III: and (●), and (◼), and and (◆).

Experimental data of decay of normalized scalar fluctuations, , downstream of a heated mandoline from the turbulence generating grid. Relevant parameters are listed in Table III: and (●), and (◼), and and (◆).

Decay of normalized scalar fluctuations, , downstream of a heated mandoline from the turbulence generating grid. Experimental data: and (●), and (◼), and and (◆). Present model calculations are denoted by lines: and (solid line), and (dashed line), and and (dot-dashed line).

Decay of normalized scalar fluctuations, , downstream of a heated mandoline from the turbulence generating grid. Experimental data: and (●), and (◼), and and (◆). Present model calculations are denoted by lines: and (solid line), and (dashed line), and and (dot-dashed line).

Decay of normalized scalar fluctuations, , against flight time from the source. Experimental data: and (●), and (◼), and and (◆). Present model calculations are denoted by lines: and (solid line), and (dashed line), and and (dot-dashed line). A dashed line of slope is shown for reference.

Decay of normalized scalar fluctuations, , against flight time from the source. Experimental data: and (●), and (◼), and and (◆). Present model calculations are denoted by lines: and (solid line), and (dashed line), and and (dot-dashed line). A dashed line of slope is shown for reference.

Effect of model coefficients and on scalar fluctuations. (a) Maximum centerline intensity of fluctuations, , against different placements of the source with respect to the turbulence grid, . (b) Centerline intensity of fluctuations, , against where . Symbols are from present calculations for different combinations of and : and (●), and (◼), and (▲), and (◆), and (▼), and and (). Solid horizontal lines correspond to the experimental data.

Effect of model coefficients and on scalar fluctuations. (a) Maximum centerline intensity of fluctuations, , against different placements of the source with respect to the turbulence grid, . (b) Centerline intensity of fluctuations, , against where . Symbols are from present calculations for different combinations of and : and (●), and (◼), and (▲), and (◆), and (▼), and and (). Solid horizontal lines correspond to the experimental data.

Correlation coefficient between a pair of line sources at plotted for different source separations, , for various combinations of and . Symbols are from present calculations for different combinations of and : and (●), and (◼), and (▲), and (◆), and (▼), and and (). Solid horizontal lines correspond to the experimental data.

Correlation coefficient between a pair of line sources at plotted for different source separations, , for various combinations of and . Symbols are from present calculations for different combinations of and : and (●), and (◼), and (▲), and (◆), and (▼), and and (). Solid horizontal lines correspond to the experimental data.

Centerline intensity of fluctuations, , vs flight time from the source for and different values of : (solid line), (dashed line), and (dotted line).

Centerline intensity of fluctuations, , vs flight time from the source for and different values of : (solid line), (dashed line), and (dotted line).

Mean plume width normalized by the turbulence integral scale at the source, , against flight time from the source for and different values of : (solid line), (dashed line), and (dotted line).

Mean plume width normalized by the turbulence integral scale at the source, , against flight time from the source for and different values of : (solid line), (dashed line), and (dotted line).

Integral of scalar variance, , normalized by against flight time from the source for and different values of : (solid line), (dashed line), and (dotted line).

Integral of scalar variance, , normalized by against flight time from the source for and different values of : (solid line), (dashed line), and (dotted line).

Skewness and kurtosis against flight time from the source for and different values of : (solid line), (dashed line), and (dotted line).

Skewness and kurtosis against flight time from the source for and different values of : (solid line), (dashed line), and (dotted line).

Maximum centerline intensity of fluctuation against for . The solid lines indicate 95% confidence intervals. Dashed line of slope 1/3 is shown for reference.

Maximum centerline intensity of fluctuation against for . The solid lines indicate 95% confidence intervals. Dashed line of slope 1/3 is shown for reference.

Estimate of the centerline intensity of fluctuation as against for . The lines indicate 95% confidence intervals.

Estimate of the centerline intensity of fluctuation as against for . The lines indicate 95% confidence intervals.

Maximum centerline intensity of fluctuation against for : (●) and (◼). The lines indicate 95% confidence intervals.

Maximum centerline intensity of fluctuation against for : (●) and (◼). The lines indicate 95% confidence intervals.

Estimate of the centerline intensity of fluctuation as against for : (●) and (◼). The lines indicate 95% confidence intervals.

Estimate of the centerline intensity of fluctuation as against for : (●) and (◼). The lines indicate 95% confidence intervals.

Centerline intensity of fluctuations, , vs flight time from the source at for different values of : (solid line), (dashed line), and (dotted line).

Centerline intensity of fluctuations, , vs flight time from the source at for different values of : (solid line), (dashed line), and (dotted line).

Mean plume width normalized by the turbulence integral scale at the source, , against flight time from the source at for different values of : (solid line), (dashed line), and (dotted line).

Mean plume width normalized by the turbulence integral scale at the source, , against flight time from the source at for different values of : (solid line), (dashed line), and (dotted line).

Integral of scalar variance, , normalized by against flight time from the source at for different values of : (solid line), (dashed line), and (dotted line).

Integral of scalar variance, , normalized by against flight time from the source at for different values of : (solid line), (dashed line), and (dotted line).

Skewness and kurtosis against flight time from the source at for different values of : (solid line), (dashed line), and (dotted line).

Skewness and kurtosis against flight time from the source at for different values of : (solid line), (dashed line), and (dotted line).

Normalized mean plume width minus the effect of the source plotted against flight time from the source at for different values of : (solid line), (dashed line), and (dotted line). (The lines are indistinguishable.)

Normalized mean plume width minus the effect of the source plotted against flight time from the source at for different values of : (solid line), (dashed line), and (dotted line). (The lines are indistinguishable.)

Centerline intensity of fluctuations, , vs time, , at for different values of : (solid line), (dashed line), and (dotted line). (The lines are indistinguishable.)

Centerline intensity of fluctuations, , vs time, , at for different values of : (solid line), (dashed line), and (dotted line). (The lines are indistinguishable.)

Maximum centerline intensity of fluctuation against for . The lines indicate 95% confidence intervals.

Maximum centerline intensity of fluctuation against for . The lines indicate 95% confidence intervals.

Estimate of the centerline intensity of fluctuation as against for . The lines indicate 95% confidence intervals.

## Tables

Parameters in the laboratory measurements for diffusion behind a single line source in grid turbulence (Ref. 7), effective source size , mesh spacing , position of the source with respect to the grid, , mean speed , velocity standard deviation at one mesh length from the grid, , velocity variance decay exponent , and molecular diffusivity .

Parameters in the laboratory measurements for diffusion behind a single line source in grid turbulence (Ref. 7), effective source size , mesh spacing , position of the source with respect to the grid, , mean speed , velocity standard deviation at one mesh length from the grid, , velocity variance decay exponent , and molecular diffusivity .

Characteristics of the velocity field corresponding to the parameters in Table I; effective source size , source position relative to the grid, , Kolmogorov length scale , turbulence length scale , integral scale Reynolds number, , and Taylor scale Reynolds number at the source.

Characteristics of the velocity field corresponding to the parameters in Table I; effective source size , source position relative to the grid, , Kolmogorov length scale , turbulence length scale , integral scale Reynolds number, , and Taylor scale Reynolds number at the source.

Parameters in the laboratory measurements of Warhaft and Lumley (Ref. 6). Definitions are given in Table I.

Parameters in the laboratory measurements of Warhaft and Lumley (Ref. 6). Definitions are given in Table I.

Parameters corresponding to the cases performed in Sec. V F. Velocity variance at the source location (isotropic turbulence), , Velocity variance decay exponent , turbulence mesh spacing , mean speed , source size , Taylor scale Reynolds number at the source location, , and ratio of source to turbulence integral scale at the source location, .

Parameters corresponding to the cases performed in Sec. V F. Velocity variance at the source location (isotropic turbulence), , Velocity variance decay exponent , turbulence mesh spacing , mean speed , source size , Taylor scale Reynolds number at the source location, , and ratio of source to turbulence integral scale at the source location, .

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