^{1,a)}

### Abstract

Probability density function (PDF) methods are an established tool applied for the simulation of turbulentmixing and turbulentreactive flows.Mixingmodels are required to close the molecular diffusion term in the PDF transportequation. From the nature of molecular diffusion, several requirements or design criteria can be derived for mixingmodels. All current models have certain shortcomings with respect to these requirements. A new mixingmodel is presented which fully satisfies almost all requirements. It conserves the mean of an inert scalar, reduces its scalar variance, and relaxes closely to a Gaussian scalar PDF. Multiple inert scalars without differential diffusion effects evolve independently and are kept bounded within their allowable region. Mixing is conditional on the velocity and particle scalar trajectories are continuous in time leading to a model that is local in a weak sense. Validation tests show that the model can reproduce differential diffusion effects and mixing rate dependencies due to variable initial scalar length scales or Reynolds and Schmidt number variations.

Many constructive and detailed suggestions of two anonymous reviewers are gratefully acknowledged. The author is also very thankful to Carla Meyer-Massetti for her help during the preparation of this article.

I. INTRODUCTION

II. FORMULATION

A. PDF method

B. Mixingmodel

III. VALIDATION MIXINGMODEL REQUIREMENTS

A. Conservation of mass

B. Decay of variance

C. Boundedness

D. Independence and linearity

E. Localness

F. Conditioning on velocity

G. Dependence on initial scalar length scales and Re and Sc numbers

IV. VALIDATION WITH DNS DATA

A. Three-stream problem

1. Numerical methods

2. Results

B. Mean scalar gradient test

1. Numerical methods

2. Results

C. Multiscalar test

V. CONCLUSIONS

### Key Topics

- Turbulence simulations
- 50.0
- Diffusion
- 27.0
- Turbulent flows
- 17.0
- Trajectory models
- 15.0
- Numerical modeling
- 13.0

## Figures

Simulation of single-scalar mixing with a mean scalar gradient. Scalar trajectories (a) of particle (solid line), the corresponding drift particles (dashed line) and (dashed-dotted line), and (b) the associated drift particle lifetimes.

Simulation of single-scalar mixing with a mean scalar gradient. Scalar trajectories (a) of particle (solid line), the corresponding drift particles (dashed line) and (dashed-dotted line), and (b) the associated drift particle lifetimes.

The first two columns show density plots of planar slices through the initial scalar fields and . Contour plots of the corresponding joint scalar PDFs are shown in the third column. The ten contour levels range from 0.01 to 1 (black) and represent the PDF normalized by its maximum value. The rows correspond to different length scale pairs listed in Table II.

The first two columns show density plots of planar slices through the initial scalar fields and . Contour plots of the corresponding joint scalar PDFs are shown in the third column. The ten contour levels range from 0.01 to 1 (black) and represent the PDF normalized by its maximum value. The rows correspond to different length scale pairs listed in Table II.

Temporal evolution of the mechanical-to-scalar time scale ratio based on Fig. 14 of the DNS study (Ref. 19). and of case C and of case D (solid line); and of case A, of case D, and of case E (dashed line); of case E (dashed-dotted line); and of case B (dotted line).

Temporal evolution of the mechanical-to-scalar time scale ratio based on Fig. 14 of the DNS study (Ref. 19). and of case C and of case D (solid line); and of case A, of case D, and of case E (dashed line); of case E (dashed-dotted line); and of case B (dotted line).

Temporal evolution of the standard deviation of the first scalar (solid lines) and the second scalar (dashed lines) for the test cases (a) A to (e) E. The thick lines represent the mixing model results and the thin lines the DNS predictions (Fig. 12 in Ref. 19). The dotted lines correspond to a decay given by Eq. (5) with .

Temporal evolution of the standard deviation of the first scalar (solid lines) and the second scalar (dashed lines) for the test cases (a) A to (e) E. The thick lines represent the mixing model results and the thin lines the DNS predictions (Fig. 12 in Ref. 19). The dotted lines correspond to a decay given by Eq. (5) with .

Temporal evolution of the skewness (curves with values ) and the flatness (curves with values ) of the first scalar (solid lines) and the second scalar (dashed lines) for the test cases (a) A to (e) E. The thick lines represent the mixing model results and the thin lines the DNS predictions (Fig. 11 in Ref. 19).

Temporal evolution of the skewness (curves with values ) and the flatness (curves with values ) of the first scalar (solid lines) and the second scalar (dashed lines) for the test cases (a) A to (e) E. The thick lines represent the mixing model results and the thin lines the DNS predictions (Fig. 11 in Ref. 19).

In the first column the normalized joint scalar PDF and in the second column standard normal PDFs (solid lines), the normalized marginal PDFs (dashed lines) and (dots) are plotted. Rows (a), (d), and (e) correspond to cases A, D, and E.

In the first column the normalized joint scalar PDF and in the second column standard normal PDFs (solid lines), the normalized marginal PDFs (dashed lines) and (dots) are plotted. Rows (a), (d), and (e) correspond to cases A, D, and E.

Scalar flatness (solid line) in the three-stream problem as a function of the mixing model parameter . With the values for (dashed line) the correct scalar variance decay is resulting.

Scalar flatness (solid line) in the three-stream problem as a function of the mixing model parameter . With the values for (dashed line) the correct scalar variance decay is resulting.

Evolution of the joint scalar PDF for case A; (a) DNS, (b) mixing model; the ten contour levels range from 0.01 to 1 (black) and represent the PDF normalized by its maximum value.

Evolution of the joint scalar PDF for case A; (a) DNS, (b) mixing model; the ten contour levels range from 0.01 to 1 (black) and represent the PDF normalized by its maximum value.

Marginal PDFs for case A; (a) , (b) ; DNS (thin lines), mixing model (thick lines).

Marginal PDFs for case A; (a) , (b) ; DNS (thin lines), mixing model (thick lines).

Evolution of the conditional average of the scalar diffusion rate for case A; (a) DNS, (b) mixing model; the ten contour levels range from 0.01 to 1 (black) and represent normalized by its maximum value; the streamlines are parallel to .

Evolution of the conditional average of the scalar diffusion rate for case A; (a) DNS, (b) mixing model; the ten contour levels range from 0.01 to 1 (black) and represent normalized by its maximum value; the streamlines are parallel to .

Evolution of the joint scalar PDF for case B; same as Fig. 8.

Evolution of the joint scalar PDF for case B; same as Fig. 8.

Evolution of the conditional average of the scalar diffusion rate for case B; same as Fig. 10.

Evolution of the conditional average of the scalar diffusion rate for case B; same as Fig. 10.

(a) Evolution of the joint scalar PDF and (b) the conditional average of the scalar diffusion rate for case C; the ten contour levels range from 0.01 to 1 (black) and represent in (a) the PDF normalized by its maximum value and in (b) normalized by its maximum value; the streamlines in (b) are parallel to .

(a) Evolution of the joint scalar PDF and (b) the conditional average of the scalar diffusion rate for case C; the ten contour levels range from 0.01 to 1 (black) and represent in (a) the PDF normalized by its maximum value and in (b) normalized by its maximum value; the streamlines in (b) are parallel to .

Evolution of the joint scalar PDF for case D; same as Fig. 8.

Evolution of the joint scalar PDF for case D; same as Fig. 8.

Evolution of the conditional average of the scalar diffusion rate for case D; same as Fig. 10.

Evolution of the conditional average of the scalar diffusion rate for case D; same as Fig. 10.

Evolution of the conditional average of the scalar diffusion rate for case E; same as Fig. 10.

Evolution of the conditional average of the scalar diffusion rate for case E; same as Fig. 10.

Evolution of the joint scalar PDF for case E; same as Fig. 8.

Evolution of the joint scalar PDF for case E; same as Fig. 8.

Nondimensional scalar variance resulting from the new mixing model as function of the integer parameter and the number of particles .

Nondimensional scalar variance resulting from the new mixing model as function of the integer parameter and the number of particles .

Scalar flatness resulting from the new mixing model as function of the integer parameter and the number of particles .

Scalar flatness resulting from the new mixing model as function of the integer parameter and the number of particles .

Conditional mean of the scalar diffusion rate as a function of the velocity sample space coordinate and for different values of the integer parameter .

Conditional mean of the scalar diffusion rate as a function of the velocity sample space coordinate and for different values of the integer parameter .

Correlation coefficient between the scalar diffusion rate and the velocity as function of the integer parameter and the number of particles .

Correlation coefficient between the scalar diffusion rate and the velocity as function of the integer parameter and the number of particles .

Nondimensional scalar variance resulting from the modified Curl mixing model as function of the integer parameter and the number of particles .

Nondimensional scalar variance resulting from the modified Curl mixing model as function of the integer parameter and the number of particles .

Scalar flatness resulting from the modified Curl mixing model as function of the integer parameter and the number of particles .

Scalar flatness resulting from the modified Curl mixing model as function of the integer parameter and the number of particles .

Nondimensional scalar variance (solid line), scalar flatness (dashed line), correlation coefficient (dashed-dotted line), and (dotted line) as a function of . Lines with and without symbols correspond to (no velocity conditioning) and , respectively.

Nondimensional scalar variance (solid line), scalar flatness (dashed line), correlation coefficient (dashed-dotted line), and (dotted line) as a function of . Lines with and without symbols correspond to (no velocity conditioning) and , respectively.

Conditional mean of the scalar diffusion rate as function of the velocity sample space coordinate and different ratios.

Conditional mean of the scalar diffusion rate as function of the velocity sample space coordinate and different ratios.

Scalar flatness as function of the number of (a) particles and (b) scalars .

Scalar flatness as function of the number of (a) particles and (b) scalars .

In (a), the normalized joint scalar PDF that is resulting in the mean scalar gradient test case with and is plotted. In (b), the corresponding marginal PDFs (dashed line) and (dots) are compared to a standard normal PDF (solid line).

In (a), the normalized joint scalar PDF that is resulting in the mean scalar gradient test case with and is plotted. In (b), the corresponding marginal PDFs (dashed line) and (dots) are compared to a standard normal PDF (solid line).

Computational cost (CPU-time) as function of the number of (a) particles and (b) scalars ; (solid line) mixing model performance (dotted line) linear increase.

Computational cost (CPU-time) as function of the number of (a) particles and (b) scalars ; (solid line) mixing model performance (dotted line) linear increase.

## Tables

Mixing model assessment based on the requirements listed by Subramaniam and Pope (Ref. 12) and Fox (Ref. 17). The symbols ●, ○, and × refer to requirements that are completely, partly, and not fulfilled, respectively. The accuracy of a mixing model that includes initial scalar length scale dependencies, and Re, Sc or Pr number effects depends on the corresponding mixing time scale model.

Mixing model assessment based on the requirements listed by Subramaniam and Pope (Ref. 12) and Fox (Ref. 17). The symbols ●, ○, and × refer to requirements that are completely, partly, and not fulfilled, respectively. The accuracy of a mixing model that includes initial scalar length scale dependencies, and Re, Sc or Pr number effects depends on the corresponding mixing time scale model.

Scalar field parameters of the different DNS cases in Ref. 19.

Scalar field parameters of the different DNS cases in Ref. 19.

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