^{1,a)}and Rajdeep Deb

^{2}

### Abstract

Simulations of spatially inhomogeneous turbulent mixing in decaying grid turbulence with a joint velocity–concentration probability density function (PDF) method were conducted. The inert mixing scenario involves three streams with different compositions. The mixing model of Meyer [“A new particle interaction mixing model for turbulent dispersion and turbulent reactive flows,” Phys. Fluids **22**(3), 035103 (2010)], the interaction by exchange with the mean (IEM) model and its velocity-conditional variant, i.e., the IECM model, were applied. For reference, the direct numerical simulation data provided by Sawford and de Bruyn Kops [“Direct numerical simulation and lagrangianmodeling of joint scalar statistics in ternary mixing,” Phys. Fluids **20**(9), 095106 (2008)] was used. It was found that velocity conditioning is essential to obtain accurate concentration PDF predictions. Moreover, the model of Meyer provides significantly better results compared to the IECM model at comparable computational expense.

I. INTRODUCTION

II. FORMULATION

A. PDF method

B. Mixing models

III. NUMERICAL SIMULATIONS

A. Test case

B. Numerical solution

IV. RESULTS

V. CONCLUSIONS

### Key Topics

- Turbulence simulations
- 25.0
- Turbulent flows
- 22.0
- Lagrangian mechanics
- 13.0
- Diffusion
- 10.0
- Langevin equation
- 5.0

##### B01F3/00

## Figures

Distributions of concentrations ϕ_{1} and ϕ_{2} in the DNS study of Sawford and de Bruyn Kops^{2} at time *t* = 0. The concentration field statistics are inhomogeneous in *y*-direction only and change over time *t*.

Distributions of concentrations ϕ_{1} and ϕ_{2} in the DNS study of Sawford and de Bruyn Kops^{2} at time *t* = 0. The concentration field statistics are inhomogeneous in *y*-direction only and change over time *t*.

Decay of (a) turbulent kinetic energy *k* and (b) dissipation rate ɛ as a function of time *t*. Results of (solid lines) the PDF method and (dots) DNS-based power laws (15) and (16) from Ref. 2 are plotted.

Decay of (a) turbulent kinetic energy *k* and (b) dissipation rate ɛ as a function of time *t*. Results of (solid lines) the PDF method and (dots) DNS-based power laws (15) and (16) from Ref. 2 are plotted.

Velocity-conditional concentration means (a) 〈ϕ_{1}|*v*, *y*, *t*〉 and (b) 〈ϕ_{2}|*v*, *y*, *t*〉 for different times *t* and velocities (solid lines) *v* = +*u* ^{′}(*t*) and (dashed lines) *v* = −*u* ^{′}(*t*). Results of (dots) the PDF method with the mixing model (11) and (lines) analytical expression (20) from Ref. 2 are plotted.

Velocity-conditional concentration means (a) 〈ϕ_{1}|*v*, *y*, *t*〉 and (b) 〈ϕ_{2}|*v*, *y*, *t*〉 for different times *t* and velocities (solid lines) *v* = +*u* ^{′}(*t*) and (dashed lines) *v* = −*u* ^{′}(*t*). Results of (dots) the PDF method with the mixing model (11) and (lines) analytical expression (20) from Ref. 2 are plotted.

Concentration standard deviations of ((a) and (c)) ϕ_{1} and ((b) and (d)) ϕ_{2} for different times (solid lines) *t*/*T* _{0} = 1, (dashed lines) 2, and (dotted lines) 4. Results of ((a) and (b)) the second simulation with mixing model (11) and ((c) and (d)) IEM model (7) are plotted. For reference, (dots) the DNS results from Ref. 2 are provided. The agreement of the first simulation with model (11) and with the IECM mixing model (10) is almost identical to (a) and (b).

Concentration standard deviations of ((a) and (c)) ϕ_{1} and ((b) and (d)) ϕ_{2} for different times (solid lines) *t*/*T* _{0} = 1, (dashed lines) 2, and (dotted lines) 4. Results of ((a) and (b)) the second simulation with mixing model (11) and ((c) and (d)) IEM model (7) are plotted. For reference, (dots) the DNS results from Ref. 2 are provided. The agreement of the first simulation with model (11) and with the IECM mixing model (10) is almost identical to (a) and (b).

Concentration covariance for different times (solid lines) *t*/*T* _{0} = 1, (dashed lines) 2, and (dotted lines) 4. Results of (a) the second simulation with mixing model (11) and (b) IEM model (7) are plotted. For reference, (dots) the DNS results from Ref. 2 are provided. The agreement of the first simulation with model (11) and with the IECM mixing model (10) is almost identical to (a).

Concentration covariance for different times (solid lines) *t*/*T* _{0} = 1, (dashed lines) 2, and (dotted lines) 4. Results of (a) the second simulation with mixing model (11) and (b) IEM model (7) are plotted. For reference, (dots) the DNS results from Ref. 2 are provided. The agreement of the first simulation with model (11) and with the IECM mixing model (10) is almost identical to (a).

Logarithm of the joint concentration PDF *g* _{ϕ}(**ψ**; *y*, *t*) at location *y* = 0 and time *t* = 4*T* _{0}. Provided are (a) the DNS results,^{2} (b) the first and (c) second simulations with mixing model (11), and (d) the IEM and (e) IECM model results.

Logarithm of the joint concentration PDF *g* _{ϕ}(**ψ**; *y*, *t*) at location *y* = 0 and time *t* = 4*T* _{0}. Provided are (a) the DNS results,^{2} (b) the first and (c) second simulations with mixing model (11), and (d) the IEM and (e) IECM model results.

Conditional mean diffusion rate distribution of the first concentration at location *y* = 0 and time *t* = 4*T* _{0}. Provided are (a) the DNS results,^{2} (b) the first and (c) second simulations with mixing model (11), and (d) the IEM and (e) IECM model results.

Conditional mean diffusion rate distribution of the first concentration at location *y* = 0 and time *t* = 4*T* _{0}. Provided are (a) the DNS results,^{2} (b) the first and (c) second simulations with mixing model (11), and (d) the IEM and (e) IECM model results.

Conditional mean diffusion rate distribution of the second concentration at location *y* = 0 and time *t* = 4*T* _{0}. Provided are (a) the DNS results,^{2} (b) the first and (c) second simulations with mixing model (11), and (d) the IEM and (e) IECM model results.

Conditional mean diffusion rate distribution of the second concentration at location *y* = 0 and time *t* = 4*T* _{0}. Provided are (a) the DNS results,^{2} (b) the first and (c) second simulations with mixing model (11), and (d) the IEM and (e) IECM model results.

Conditional mean velocity 〈*u*|**ψ**, *y*, *t*〉/*u* ^{′}(*t*) at location *y* = 0 and time *t* = 4*T* _{0}. Provided are (a) the DNS results,^{2} (b) the first and (c) second simulations with mixing model (11), and (d) the IEM and (e) IECM model results.

Conditional mean velocity 〈*u*|**ψ**, *y*, *t*〉/*u* ^{′}(*t*) at location *y* = 0 and time *t* = 4*T* _{0}. Provided are (a) the DNS results,^{2} (b) the first and (c) second simulations with mixing model (11), and (d) the IEM and (e) IECM model results.

Joint concentration PDF *g* _{ϕ}(**ψ**; *y*, *t*) at location *y* = 0 and times *t* = 4*T* _{0} and 8*T* _{0}. Compared are model results from the IEM, IECM, and the second simulation with the generalized model (GM). White corresponds to zero probability density.

Joint concentration PDF *g* _{ϕ}(**ψ**; *y*, *t*) at location *y* = 0 and times *t* = 4*T* _{0} and 8*T* _{0}. Compared are model results from the IEM, IECM, and the second simulation with the generalized model (GM). White corresponds to zero probability density.

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