# An Overview of the Lagrangian Dispersion Modeling of Heavy Particles in Homogeneous Isotropic Turbulence and Considerations on Related LES Simulations

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

#### 2.1. Taylor’s Turbulent Diffusion Theory and Batchelor’s Generalization

#### 2.2. Toward Turbulent Dispersion

#### 2.2.1. Scales of Turbulent Motion

- the Kolmogorov micro-length scale $\eta ={({\nu}^{3}/\epsilon )}^{1/4}\propto {({\nu}^{3}L/{U}^{3})}^{1/4}$,
- the Kolmogorov time scale ${\tau}_{\eta}={(\nu /\epsilon )}^{1/2}$,

#### 2.2.2. Equation of Motion

_{p}, its motion is given to a good approximation by Newton’s equations:

_{p}is the particle Reynolds number,

_{p}, the nonlinear factor f and depends on the fluid velocity ${u}_{i}^{p}(t)={u}_{i}({\overrightarrow{X}}_{p}(t),t)$ at the particle position ${\overrightarrow{X}}_{p}(t)$.

- the Stokes number $St=S{t}_{mE}$, based on the Eulerian moving scale ${T}_{mE}$ is a measure of the relative importance of the particle inertia; it characterizes the particle’s response to the turbulent fluid velocity fluctuations;
- the drift parameter $\gamma $, which is a dimensionless drift velocity related to the turbulence level, given by$$\gamma =\frac{{v}_{ch}}{{u}_{0}}=\frac{g{\tau}_{s}}{{u}_{0}}\hspace{1em}\mathrm{with}\hspace{1em}{v}_{ch}=(\rho -1)\cdot {\tau}_{s}g/(\rho \cdot f)\approx {\tau}_{s}g;$$
- the drag correction factor $f$, a function that increases with Reynolds number (fluid–particle drift velocity or particle size).

#### Note on Stokes Numbers

- $S{t}_{\eta}={\tau}_{p}/{\tau}_{\eta}$ based on the Kolmogorov microscale,
- $S{t}_{mE}={\tau}_{p}/{T}_{mE}$ based on the Eulerian moving macroscale,
- $S{t}_{E}={\tau}_{p}/{T}_{E}$ based on the classical Eulerian macroscale,
- $S{t}_{L}={\tau}_{p}/{T}_{L}$ based on the Lagrangian time scale,
- $S{t}_{MF}={\tau}_{p}/{T}_{MF}$ based on a time scale of the mean flow ${T}_{MF}=L/U$ indicating if the particles follow the mean fluid flow.

#### 2.2.3. Qualitative Analysis of Turbulent Dispersion

#### 2.2.4. Short Review of Analytical Approaches

- –
- –
- since the terminal velocity is a measure of the inertia, the particle does not completely follow the high-frequency fluctuations of the turbulent fluid velocity; thus, Yudine did not separate inertia and gravity effects;
- –
- if it has an appreciable settling velocity, a particle will fall from one eddy to another, whereas a fluid point will remain in the same eddy throughout the lifetime of the eddy; this is one of the first papers mentioning “overshooting”.

- for long time diffusion ${S}_{C}(t\to \infty )=\frac{1}{F}={\left(1+{\left(\frac{{v}_{ch}}{{v}_{p}^{\prime}}\right)}^{2}\right)}^{-1/2}$.

- the ratio of the fluctuating velocity variances is$${\left(\frac{v\prime}{{u}_{0}}\right)}^{2}=\frac{{T}_{L11}^{f}}{{T}_{L11}^{f}+{\tau}_{s}}=\frac{1}{1+{\tau}_{s}/{T}_{L11}^{f}}=\frac{1}{1+{S}_{t11}^{}}.$$

## 3. Numerical Methods

#### 3.1. Eddy Interaction Models (EIM–DRW)

#### 3.1.1. General Description

- the particle (P) leaves the fluid eddy (F) for entering into a new one when the eddy (F) lifetime ${\tau}^{*}$ is elapsed (condition 1),
- or when the distance between the particle and the eddy center does exceed the eddy length ${\lambda}^{*}$ a radial dimension (condition 2).

#### 3.1.2. Historical Background

- a first estimate of the fluid velocity with the criteria $y<\mathsf{\Delta}t/{T}_{L}^{f}$,
- a spatial step that calculates the fluid–particle velocity ${u}_{f}^{p}({X}_{p}(t),t)$ at time $t$ at ${X}_{p}(t)$ with a correlation function:$${R}_{E}^{}(r)=\frac{\overline{{u}_{f}^{p}{u}_{f}}}{\overline{{u}_{f}(t){u}_{f}(t)}},$$
- the integration of the equation of motion with ${u}_{f}^{p}({X}_{p}(t),t)$ and $v\hspace{0.17em}(t)$ to obtain the new discrete particle velocity.

#### 3.2. Random Walk Models (RWM-CWM)

#### 3.2.1. Classical Approach

#### 3.2.2. Two-Step Space–Time Approach

_{T}, some other factors can influence the simulations and justify discrepancies, the number of computed trajectories, the choice of the random number generator or the time step $\mathsf{\Delta}t$. Lain and Grillo [134] simulated the Wells and Stock experiment [16] and a jet flow [135].

#### 3.3. CRW–Matrix Methods

_{k}with a Gaussian probability density function:

#### 3.4. Numerical Context

^{3}and a diameter d in the range of 10 to 500 microns. The spheres are released from a point source and disperse in a homogeneous isotropic stationary turbulence (HIST), the fluid flow being air under standard conditions (density ${\mathsf{\rho}}_{\mathrm{F}}$ = 1.275 kg/m

^{3}and kinematic viscosity ${\mathsf{\nu}}_{\mathrm{F}}$ = 1.36 × 10

^{−5}m/s

^{2}) without gravity and a gravity field. The scale of the turbulent fluid is ${\mathrm{u}}_{0}$ = 0.131 m/s and the Lagrangian integral time scale is set to ${T}_{L}$ = 0.091 s. The experimental data by Sato and Yamamoto [174] suggested that for a grid turbulence ${T}_{L}=0.3-0.6\hspace{0.17em}{T}_{mE}$ where ${T}_{mE}$ is the moving Eulerian integral time scale. Burnage and Huilier [175] performed a diffusion experiment with small droplets released isokinetically from a tube in a grid decaying turbulence, using LDA and a light diffusion technique to measure the turbulence and concentration field and found ${T}_{L}=0.5\hspace{0.17em}-\hspace{0.17em}0.95\hspace{0.17em}\hspace{0.17em}{T}_{mE}$ [13]. In the present study, we suppose that ${T}_{L}/{T}_{mE}=0.356$ (Appendix A). As a consequence, in the present simulations, the Stokes number, $St={\tau}_{s}/{T}_{mE}$, will range from 0.007 to 4.4, ${\tau}_{s}$ being the Stokesian particle relaxation time (details are summarized in Table 2). All the statistics that will characterize the particulate dispersion, namely, turbulent energies (variances) ${v}_{ii}^{2}$, Lagrangian particle integral time scales ${T}_{p,ii}$ and long time particle dispersion coefficients ${D}_{P,ii}(\infty )={v}_{ii}^{2}\cdot {T}_{p,ii}$, are calculated for 50,000 trajectories for each particle size (Stokes number), a number of trajectories sufficient to ensure statistical stability and convergence.

^{−4}s to 10

^{−3}s with particle relaxation times from 1.2 × 10

^{−3}s (d = 20 µm) to 0.8 s (d = 500 µm).

^{2}varies from 0.02 for the smallest particles to 15 for the largest. The long time dispersion characteristics are calculated for 4 s trajectories, a time that largely exceeds by a factor of 5 to 10 the particle relaxation time of the heavier particles and the Lagrangian scales. Besides other tests on nonstationary forces (added mass, Basset term), lift forces confirmed that statistically there is no influence on long time dispersion of heavy particles [93], in agreement with other results in the literature.

## 4. Numerical Results

#### 4.1. EIM–DRW Results

- (1)
- the distance between the eddy center and the heavy particle position does not exceed the random eddy length (no overshooting condition),
- (2)
- the interaction time of the particle does not exceed the random eddy lifetime.

#### 4.1.1. Classical Dispersion Without Gravity (g = 0)

#### 4.1.2. Modified EIM with Wang and Stock Correction (No Gravity)

#### 4.1.3. Dispersion under Gravity Effect

#### 4.1.4. Conclusion on EIM–DRW Monte-Carlo Methods

_{L}). Major modifications were brought to Monte-Carlo approaches by Huang et al. [179], Graham [95,96,97], Launay et al. [176,177], among others, and some eddy–particle interaction models tried to better take into account the space–time characteristics of the effective fluid turbulence controlling the particle motion, to better simulate the so-called “true turbulence experienced by the particle’.”

#### 4.2. Markovian Methods

- The Markovian model developed by a two-step space–time approach (Section 3.2.2) with the following equations and the method proposed by Lu et al. [126,127,128,129,130]:$${{u}^{\prime}}_{i}^{p}(t+\mathsf{\Delta}t)={R}_{F,ii}^{P}(\mathsf{\Delta}t,\mathsf{\Delta}s){{u}^{\prime}}_{i}^{p}(t)+{u}_{0}{\xi}_{ti}\sqrt{1-{[{R}_{F,ii}^{P}(\mathsf{\Delta}t,\mathsf{\Delta}s)]}^{2}},$$$${R}_{F,ii}^{P}(\mathsf{\Delta}t,\mathsf{\Delta}s)=\mathrm{exp}\left(-\frac{\mathsf{\Delta}t}{{T}_{L,ii}}\right)\mathrm{exp}\left(-\frac{\mathsf{\Delta}s}{{L}_{E,ii}}\right)$$
- A more classical Markovian model with a Langevin equation based on the correlation functions proposed by Wang and Stock (Equation (A2)):$${{u}^{\prime}}_{i}^{p}(t+\mathsf{\Delta}t)={R}_{F,ii}^{P}(\mathsf{\Delta}t){{u}^{\prime}}_{i}^{p}(t)+{u}_{0}{\xi}_{ti}\sqrt{1-{[{R}_{F,ii}^{P}(\mathsf{\Delta}t)]}^{2}}.$$

#### 4.2.1. CRW–Lu Model

- Dispersion without gravity (g = 0)

- RWM–Lu with gravity effects (g ≠ 0)

#### 4.2.2. Langevin-WS Model

#### 4.2.3. Conclusions on Markovian Models

#### 4.3. Matrix Method and Related Considerations

#### 4.3.1. Matrix Method without Gravity (g = 0)

#### 4.3.2. Matrix Method with Gravity (g ≠ 0)

## 5. Discussion and Present Trends of Modeling Approaches

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

- variances of the particle turbulent velocity in the i-th direction:$$\begin{array}{l}{v}_{11}^{2}={v}_{22}^{2}={u}_{0}^{2}\hspace{0.17em}\left(\frac{1}{1+S{t}_{T}\sqrt{1+{\gamma}^{2}{m}_{T}^{2}}}-\frac{0.5\gamma \hspace{0.17em}{m}_{T}S{t}_{T}}{{\left(1+S{t}_{T}\sqrt{1+{\gamma}^{2}{m}_{T}^{2}}\right)}^{2}}\right)\\ {v}_{33}^{2}=\frac{{u}_{0}^{2}}{1+S{t}_{T}\sqrt{1+{\gamma}^{2}{m}_{T}^{2}}}\end{array}$$$$\mathrm{with}St=\frac{{\tau}_{s}}{{T}_{mE}};\gamma =\frac{{v}_{ch}}{{u}_{0}};m=\frac{{T}_{mE}{u}_{0}}{{L}_{f}},S{t}_{T}=\frac{{\tau}_{s}}{{T}_{F}^{P}};{m}_{T}=\frac{{T}_{F}^{P}\hspace{0.17em}{u}_{0}}{{L}_{f}}=m\frac{{T}_{F}^{P}(St)}{{T}_{mE}};$$
- long time coefficient of the heavy particle in the i-th direction:$$\begin{array}{l}{D}_{P,11}(\infty )={D}_{P,22}(\infty )={u}_{0}^{2}{T}_{F}^{P}\hspace{0.17em}\left(\frac{\sqrt{1+{m}_{T}^{2}{\gamma}^{2}}-0.5{m}_{T}\gamma}{1+{m}_{T}^{2}{\gamma}^{2}}\right)\\ {D}_{P,33}(\infty )={u}_{0}^{2}{T}_{F}^{P}\left(\frac{1}{\sqrt{1+{m}_{T}^{2}{\gamma}^{2}}}\right)\end{array}$$
- particle Lagrangian integral time scale in the i-th direction:$$\begin{array}{l}{T}_{P,11}={T}_{P,22}=\frac{{D}_{P,11}(\infty )}{{v}_{11}^{2}}={T}_{F}^{P}(St)\frac{\left(1+\sqrt{1+{\gamma}^{2}{m}_{T}^{2}}-0.5\gamma \hspace{0.17em}{m}_{T}\right){\left(1+S{t}_{T}\sqrt{1+{\gamma}^{2}{m}_{T}^{2}}\right)}^{2}}{\left(1+{m}_{T}^{2}{\gamma}^{2}\right)\left(1+S{t}_{T}\sqrt{1+{\gamma}^{2}{m}_{T}^{2}}-0.5\gamma \hspace{0.17em}{m}_{T}S{t}_{T}\right)}\\ {T}_{P,33}=\frac{{D}_{P,33}(\infty )}{{v}_{33}^{2}}={T}_{F}^{P}(St)\frac{1+S{t}_{T}\sqrt{1+{\gamma}^{2}{m}_{T}^{2}}}{\sqrt{1+{\gamma}^{2}{m}_{T}^{2}}}\end{array}$$

- variances of the particle turbulent velocity in the i-th direction (zero drift velocity):$$\frac{{v}_{11}^{2}}{{u}_{0}^{2}}=\frac{{v}_{22}^{2}}{{u}_{0}^{2}}=\frac{{v}_{33}^{2}}{{u}_{0}^{2}}=\frac{1}{\left(1+\frac{{\tau}_{s}}{{T}_{F}^{P}(St)}\right)};$$
- particle Lagrangian integral time scale in the i-th direction (zero drift velocity):$${T}_{P,11}={T}_{P,22}={T}_{P,33}={\tau}_{s}+{T}_{F}^{P}\left(St\right);$$
- long time coefficient of the heavy particle in the i-th direction (zero drift velocity):$${D}_{P,11}(\infty )={D}_{P,22}(\infty )={D}_{P,33}(\infty )={u}_{0}^{2}{T}_{F}^{P}\left(St\right).$$

_{s}is large.

## Nomenclature

${d}_{p}$ | Particle diameter |

${D}_{ij}^{f}$ | Fluid turbulent diffusion coefficient |

${D}_{ij}^{p}$ | Particle turbulent diffusion coefficient |

${E}_{Lij}^{f}(\omega )$ | Lagrangian turbulent fluid energy spectrum |

$f$ | Non-Stokesian drag correction factor |

$f(r)$ | Longitudinal Eulerian space correlation function |

$g(r)$ | Lateral Eulerian space correlation function |

$k$ | Turbulent kinetic energy |

${l}_{e}$ | Typical eddy length scale |

${L}_{Eij}$ | Eulerian integral length scale (i and j directions) |

${L}_{f}$ | Size of the largest eddies |

m | Structure parameter of turbulence |

${\mathrm{Re}}_{p}$ | Particle Reynolds number |

${R}_{Eij}^{f}(\tau )$ | Eulerian fluid correlation function related to i,j directions |

${R}_{Lij}^{f}(\tau )$ | Lagrangian fluid correlation function related to i,j directions |

${R}_{Lij}^{p}(\tau )$ | Lagrangian particle correlation function related to i,j directions |

$St$ | Stokes number |

${T}_{Lij}^{f}{}_{}$ | Lagrangian integral fluid time scale related to i,j directions |

${T}_{Lij}^{p}{}_{}$ | Lagrangian integral particle time scale related to i,j directions |

${T}_{0}$ | Eddy turnover time |

${T}_{E}$ | Eulerian integral time scale |

${T}_{mE}$ | Moving Eulerian time |

${T}_{CTE}$ | Characteristic time for the crossing-trajectories effect |

${T}_{Max}$ | Interaction time as defined by Graham [ ] |

${u}_{f}(t)$ | Turbulent fluid velocity |

${u}_{f}^{p}(t)$ | Turbulent fluid velocity at the particle position |

${u}_{0}$ | Root mean-square fluid velocity |

${u}_{f}(t)$ | Turbulent fluid velocity |

$\overline{{u}^{2}}$ | Fluid velocity variance |

$\overline{{v}^{2}}$ | Particle velocity variance |

$\overline{{v}_{i}^{2}}$ | Particle velocity variance |

${v}_{ch}$ | Mean particle-fluid drift velocity |

${\overrightarrow{X}}_{p}^{}(t)$ | Particle position |

$\overline{{y}_{2}^{2}(t)}$ | Mean lateral square fluid point displacement |

Greek symbols | |

$\eta $ | Kolmogorov length scale |

$\epsilon $ | Rate of dissipation of turbulence kinetic energy |

$\hspace{0.17em}{\mathsf{\Lambda}}_{i}=\sqrt{\overline{{u}_{i}^{2}}}{T}_{Lii}$ | Lagrangian integral length scale |

${\mu}_{F}$ | Fluid dynamic viscosity |

${\nu}_{F}$, $\nu $ | kinematic viscosity of the fluid |

${\rho}_{F}$ | Fluid density |

${\rho}_{p}$ | Particle density |

$\rho $ | Relative density ${\rho}_{p}/{\rho}_{F}$ |

${\tau}_{E}$ | Eulerian microscale (Taylor) |

${\tau}_{L}$ | Lagrangian microscale (Taylor) |

${\tau}_{\eta}$ | Kolmogorov time scale |

${\tau}_{p}$ | Non-Stokesian relaxation time |

${\tau}_{s}$ | Stokesian relaxation time |

Superscript/subscript | |

f | For fluid properties |

p | For particle properties |

## Abbreviations

EIM | Eddy Interaction Model |

IE | Inertial effect |

RWM | Random Walk Model |

CE | Continuity effect |

CTE | Crossing Interaction effect |

HIST | Homogeneous Isotropic Stationary Turbulence |

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**Figure 3.**Matrices A & B used to calculate the turbulent fluid velocity seen by the particle along its path.

Case | ${\mathit{\tau}}_{\mathit{p}}>{\mathit{\tau}}_{\mathit{L}}$ | ${\mathit{d}}_{\mathit{p}}>{\mathit{\eta}}_{\mathit{K}}\hspace{0.17em}\hspace{0.17em}$ | Dispersion Regime |
---|---|---|---|

1 | yes | yes | Large structure dispersion with high-frequency cut-off and turbulence modification, inertia and CTE-CE if drift |

2 | no | yes | Large structure dispersion without high-frequency cut-off, turbulence modification, CTE-CE |

3 | yes | no | Small and large structure influence with damped particle response to high-frequency fluctuationsInertia effect (IE) and CTE-CE if drift |

4 | no | no | Turbulent diffusion |

${\mathit{d}}_{\mathit{p}}(\mathsf{\mu}\mathbf{m})$ | ${\mathit{v}}_{\mathit{c}\mathit{h}}(\mathbf{m}/\mathbf{s})$ | ${\mathit{\tau}}_{\mathit{p}}(\mathbf{ms})$ | ${\mathit{\tau}}_{\mathit{s}}=\mathit{f}{\mathit{\tau}}_{\mathit{p}}\hspace{0.17em}(\mathbf{ms})$ | ${\mathit{S}}_{\mathit{t}\mathit{s}}={\mathit{\tau}}_{\mathit{s}}/{\mathit{T}}_{\mathit{m}\mathit{E}}$ | $\begin{array}{l}{\mathit{S}}_{\mathit{t}\mathit{p}}={\mathit{\tau}}_{\mathit{p}}/{\mathit{T}}_{\mathit{m}\mathit{E}}\\ ={\mathit{\tau}}_{\mathit{s}}/(\mathit{f}{\mathit{T}}_{\mathit{m}\mathit{E}})\end{array}$ | ${\mathit{v}}_{\mathit{c}\mathit{h}}/{\mathit{u}}^{\prime}$ | ${\mathbf{Re}}_{\mathit{p}}$ |
---|---|---|---|---|---|---|---|

10 | 0.003 | 0.31 | 0.31 | 0.0017 | 0.0017 | 0.023 | 0.002 |

20 | 0.012 | 1.23 | 1.24 | 0.0068 | 0.0068 | 0.093 | 0.017 |

50 | 0.07 | 7.35 | 7.75 | 0.042 | 0.0404 | 0.55 | 0.25 |

70 | 0.135 | 13.7 | 15.2 | 0.084 | 0.0753 | 1.3 | 0.65 |

100 | 0.25 | 25 | 31 | 0.170 | 0.137 | 1.95 | 1.7 |

120 | 0.34 | 34 | 44.64 | 0.245 | 0.187 | 2.57 | 2.8 |

150 | 0.47 | 48 | 69.75 | 0.38 | 0.26 | 3.62 | 4.9 |

200 | 0.710 | 72 | 124 | 0.68 | 0.39 | 5.42 | 9.8 |

250 | 0.944 | 96 | 194 | 1.06 | 0.52 | 7.21 | 16.3 |

400 | 1.61 | 164 | 496 | 2.72 | 0.90 | 12.3 | 47 |

500 | 2.02 | 205 | 775 | 4.26 | 1.13 | 15.4 | 74 |

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Huilier, D.G.F.
An Overview of the Lagrangian Dispersion Modeling of Heavy Particles in Homogeneous Isotropic Turbulence and Considerations on Related LES Simulations. *Fluids* **2021**, *6*, 145.
https://doi.org/10.3390/fluids6040145

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Huilier DGF.
An Overview of the Lagrangian Dispersion Modeling of Heavy Particles in Homogeneous Isotropic Turbulence and Considerations on Related LES Simulations. *Fluids*. 2021; 6(4):145.
https://doi.org/10.3390/fluids6040145

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Huilier, Daniel G. F.
2021. "An Overview of the Lagrangian Dispersion Modeling of Heavy Particles in Homogeneous Isotropic Turbulence and Considerations on Related LES Simulations" *Fluids* 6, no. 4: 145.
https://doi.org/10.3390/fluids6040145