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Two-Fluid RANS Modelling of Turbulence Created by a Vertically Falling/Moving Particle Cloud

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Abstract

In particle-laden flows, a turbulent field can be produced in the carrier phase by the movement of the particle/spray cloud. In this study, the intensity and the integral length scale of the particle-induced turbulence are studied using a simple mechanistic model with comparison to experimental data and numerical simulations for large-scale numerical applications. The experimental results of DynAsp are investigated with numerical simulation results. Out of the spray nozzle, two regions can be distinguished for the spray dynamics: an inertial zone and an equilibrium zone. It is found that the initial injection velocity of the cloud has little effect on the terminal slip-velocity of the particles in the equilibrium zone far from the injection region. The turbulent kinetic energy is closely related to the particle slip-velocity and shows a maximal value when particles reach their terminal velocity inside the equilibrium zone. The integral length scale depends mainly on three parameters: particle slip-velocity, particle size and volume fraction. Combined with the terminal slip-velocity correlation, the reduced-order mechanistic model can give a reasonable estimation of the turbulent kinetic energy as well as the integral length scale of the particle-laden flow in large-scale configurations.

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Acknowledgements

The authors gratefully acknowledge the financial support from Electricité de France (EDF) within the framework of the Generation II & III reactor research program.

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Appendices

Appendix

A Mechanistic Model of Kenning et al. Kenning (1996)

For simplicity, the distribution of the dispersed phase is considered as uniform and the fluctuations induced in the carrier phase by the particles are assumed to be isotropic, even though the fluctuation in the streamwise direction is almost twice the fluctuation in the transverse direction Parthasarathy and Faeth (1990).

Since both the carrier flow and the dispersed phase exhibit fluctuating behavior, the relative fluctuations are used to investigate the turbulent energy production and dissipation due to the presence of particles. The relative particle velocity fluctuations were described from the spherical-particle motion equation as:

$$\begin{aligned} \frac{dv'_p}{dt} = \frac{1}{\tau _{p}}(u'-v'_p)+\frac{1}{2}\frac{\rho _f}{\rho _p}\frac{d(u'-v'_p)}{dt}, \end{aligned}$$
(20)

where \(\rho _f\), \(\rho _p\) denote the density of the carrier fluid and the dispersed particle, respectively and \(u'\) and \(v'_p\) are the fluid and the particle velocity fluctuations respectively, \(\tau _{p}\) is the response time of the particle.

Assuming that the fluctuating velocity components of the fluid and the particle velocity behave as:

$$\begin{aligned} u' = u_0\,e^{i\omega t} ,\,\,\, v'_p = u_0\,{\mathscr {A}} \,e^{i\omega t + \phi }, \end{aligned}$$
(21)

where \(u_0\), \({\mathscr {A}}u_0\) are the amplitudes of the fluid and the particle velocity fluctuations, respectively and \(\omega \) is the characteristic frequency of the fluid defined as:

$$\begin{aligned} \omega = \frac{v_{rel}}{\lambda }, \end{aligned}$$
(22)

where \(v_{rel}\) and \(\lambda \) denote the relative velocity between the two phases and the mean inter-particle distance of the dispersed phase, respectively.

By introducing (21) into Equation (20), we can have:

$$\begin{aligned} {\mathscr {A}} cos\phi - S_t\,sin\phi \,{\mathscr {A}} \left( 1+\frac{1}{2}\frac{\rho _f}{\rho _p}\right) = 1, \nonumber \\ {\mathscr {A}} sin\phi + S_t\,cos\phi \,{\mathscr {A}} \left( 1+\frac{1}{2}\frac{\rho _f}{\rho _p}\right) = S_t\left( \frac{1}{2}\frac{\rho _f}{\rho _p}\right) , \end{aligned}$$
(23)

where \(S_t= \omega \tau _{p}\) is the Stokes number.

From Equation (23), we can obtain \(\phi \) and \({\mathscr {A}}\) as:

$$\begin{aligned} \phi= & {} arctan\left( \frac{-2S_t}{S_t^2\frac{\rho _f}{\rho _p}\left( 1 + \frac{1}{2}\frac{\rho _f}{\rho _p}\right) + 2} \right) , \end{aligned}$$
(24)
$$\begin{aligned} {\mathscr {A}}\,= & {} \sqrt{\frac{1+tan^2\phi }{\left[ 1-S_t\,tan\phi \left( 1+\frac{1}{2}\frac{\rho _f}{\rho _p}\right) \right] ^2}}, \end{aligned}$$
(25)

Kenning et al. Kenning (1996) propose a simple expression of the fluctuation amplitude \({\mathscr {A}}\) using the Stokes number, such as:

$$\begin{aligned} {\mathscr {A}} = \sqrt{\frac{S_t^2\frac{\rho _f^2}{\rho _p^2} + 4}{4S_t^2 + 4S_t^2\frac{\rho _f}{\rho _p}+ S_t^2\frac{\rho _f^2}{\rho _p^2} + 4}}. \end{aligned}$$
(26)

We can notice from Equation (26) that \({\mathscr {A}}\) is smaller than unity, which indicates that the particles oscillation magnitude is smaller than that of the fluid fluctuations.

1.1 A.1 Turbulence generation by particles

Considering the main flow direction, the kinetic energy transfer rate from particles to fluid per unit particle mass due to the velocity difference can be estimated by:

$$\begin{aligned} P_p = \frac{(u-v_p)^2}{\tau _{p}}, \end{aligned}$$
(27)

where u and \(v_p\) are fluid and particle instantaneous velocity, respectively. \(\tau _{p}\) is the mean particle response time. The velocities can be divided into mean and fluctuating parts as:

$$\begin{aligned} u = {\overline{u}} + u', \nonumber \\ v_p = {\overline{v}}_p + v'_p, \end{aligned}$$
(28)

Considering the expression of Equation (21), we can calculate the averaged energy production rate as:

$$\begin{aligned} {\overline{P}}_{p,1}= \frac{1}{2}\left[ \frac{2\,({\overline{u}} - {\overline{v}}_p)^2 + {\mathscr {A}}^2\,u_0^2 - 2\,u_0^2\,{\mathscr {A}} \,cos\phi + u_0^2}{\tau _{p}} \right] , \end{aligned}$$
(29)

Even though the kinetic energy transfer is mainly due to the velocity in the main flow direction, the fluctuation of the fluid and the particles are basically three-dimensional. Thus, the turbulent energy production from the particle to the fluid should be three-dimensional, leading to a more general formulation of the turbulent production term:

$$\begin{aligned} {\overline{P}}_p= \frac{1}{2}\left[ \frac{2({\overline{u}} - {\overline{v}}_p)^2 + 3{\mathscr {A}}^2\,u_0^2 - 6\,u_0^2\,{\mathscr {A}}\,cos\phi + 3u_0^2}{\tau _{p}} \right] . \end{aligned}$$
(30)

1.2 A.2 Energy redistributing to particles

Particle fluctuations are mainly due to the fluid flow fluctuations. The presence of dispersed particles dissipates part of the turbulent energy of the carrier phase. The dissipation rate per unit mass of particles is derived from the following particle equation:

$$\begin{aligned} \varepsilon _p = \frac{d(\frac{1}{2} {v'}_p^2)}{dt} = \left[ \frac{1}{\tau _{p}}(u'-v'_p)+\frac{1}{2}\frac{\rho _f}{\rho _p}\frac{d(u'-v'_p)}{dt} \right] v'_p, \end{aligned}$$
(31)

Using Equation (21), a mean dissipation rate over a complete oscillation period results in:

$$\begin{aligned} {\overline{\varepsilon }}_p = 3\times u_0^2\,{\mathscr {A}} \left[ \frac{2cos\phi - 2\,{\mathscr {A}} + S_t\,sin\phi \frac{\rho _f}{\rho _p}}{4\tau _{p}} \right] . \end{aligned}$$
(32)

where the factor 3 indicates that the dissipation of the fluctuations account for three dimensional effect, similar to the turbulence generation.

1.3 A.3 Viscous flow dissipation

The presence of particles in the carrier phase will not generate only turbulence, but also modify the viscous dissipation rate of the fluid. The rate of turbulent dissipation proposed by Kenning is:

$$\begin{aligned} \varepsilon = \frac{k_t^{3/2}}{L_h}, \end{aligned}$$
(33)

where \(L_h\) is the hybrid length scale which combines the inherent integral length scale \(L_i\) and the mean inter-particle distance of the dispersed particles \(\lambda \).

$$\begin{aligned} L_h = \frac{2}{\frac{1}{\lambda } + \frac{1}{L_i}} = \frac{2L_i\lambda }{L_i + \lambda }. \end{aligned}$$
(34)

The factor 2 comes from the harmonic average of these two length scales.

Combining (30) and (32), the turbulent kinetic energy rate can be expressed as:

$$\begin{aligned} \frac{dk_t}{dt} = ({\overline{P}}_p - {\overline{\varepsilon }}_p)\left( \frac{\alpha _p}{1-\alpha _p}\right) \left( \frac{\rho _p}{\rho _f}\right) + (P_i - \varepsilon ). \end{aligned}$$
(35)

where \(\alpha _p\) denotes the volume fraction of the dispersed phase, and \(P_i\) is the inherent turbulence in the carrier fluid for no-stagnant initial conditions.

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Gai, G., Kudriakov, S., Thomine, O. et al. Two-Fluid RANS Modelling of Turbulence Created by a Vertically Falling/Moving Particle Cloud. Flow Turbulence Combust 108, 819–842 (2022). https://doi.org/10.1007/s10494-021-00295-6

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