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22.2.2 Turbulent Dispersion of Particles

The dispersion of particles due to turbulence in the fluid phase can be predicted using the stochastic tracking model or the particle cloud model (see Section  22.2.2). The stochastic tracking (random walk) model includes the effect of instantaneous turbulent velocity fluctuations on the particle trajectories through the use of stochastic methods (see Section  22.2.2). The particle cloud model tracks the statistical evolution of a cloud of particles about a mean trajectory (see Section  22.2.2). The concentration of particles within the cloud is represented by a Gaussian probability density function (PDF) about the mean trajectory. For stochastic tracking a model is available to account for the generation or dissipation of turbulence in the continuous phase (see Section  22.9.1).

figure   

Turbulent dispersion of particles cannot be included if the Spalart-Allmaras turbulence model is used.



Stochastic Tracking


When the flow is turbulent, FLUENT will predict the trajectories of particles using the mean fluid phase velocity, $\overline{u}$, in the trajectory equations (Equation  22.2-1). Optionally, you can include the instantaneous value of the fluctuating gas flow velocity,


 u = \overline{u} + u' (22.2-20)

to predict the dispersion of the particles due to turbulence.

In the stochastic tracking approach, FLUENT predicts the turbulent dispersion of particles by integrating the trajectory equations for individual particles, using the instantaneous fluid velocity, $\overline{u} + u^{'}(t)$, along the particle path during the integration. By computing the trajectory in this manner for a sufficient number of representative particles (termed the "number of tries''), the random effects of turbulence on the particle dispersion may be accounted for.

FLUENT uses a stochastic method (random walk model) to determine the instantaneous gas velocity. In the discrete random walk (DRW) model, the fluctuating velocity components are discrete piecewise constant functions of time. Their random value is kept constant over an interval of time given by the characteristic lifetime of the eddies.

The DRW model may give nonphysical results in strongly nonhomogeneous diffusion-dominated flows, where small particles should become uniformly distributed. Instead, the DRW will show a tendency for such particles to concentrate in low-turbulence regions of the flow.

The Integral Time

Prediction of particle dispersion makes use of the concept of the integral time scale, $T$, which describes the time spent in turbulent motion along the particle path, $ds$:


 T = \int_0^{\infty} \frac{ {u_p}'(t) {u_p}'(t+s) }{ \overline{{u_p}^{'2}}} ds (22.2-21)

The integral time is proportional to the particle dispersion rate, as larger values indicate more turbulent motion in the flow. It can be shown that the particle diffusivity is given by $\overline{{u_i}'{u_j}'} T $.

For small "tracer'' particles that move with the fluid (zero drift velocity), the integral time becomes the fluid Lagrangian integral time, $T_L$. This time scale can be approximated as


 T_L = C_L \frac{k}{\epsilon} (22.2-22)

where $C_L$ is to be determined as it is not well known. By matching the diffusivity of tracer particles, $\overline{{u_i}'{u_j}'} T_L $, to the scalar diffusion rate predicted by the turbulence model, $\nu_t / \sigma $, one can obtain


 T_L \approx 0.15 \frac{k}{\epsilon} (22.2-23)

for the $k$- $\epsilon$ model and its variants, and


 T_L \approx 0.30 \frac{k}{\epsilon} (22.2-24)

when the Reynolds stress model (RSM) is used [ 75]. For the $k$- $\omega$ models, substitute $\omega = \epsilon/k$ into Equation  22.2-22. The LES model uses the equivalent LES time scales.

The Discrete Random Walk Model

In the discrete random walk (DRW) model, or "eddy lifetime'' model, the interaction of a particle with a succession of discrete stylized fluid phase turbulent eddies is simulated. Each eddy is characterized by

The values of $u'$, $v'$, and $w'$ that prevail during the lifetime of the turbulent eddy are sampled by assuming that they obey a Gaussian probability distribution, so that


 u^{'} = \zeta \sqrt{\overline{u^{' 2}}} (22.2-25)

where $\zeta$ is a normally distributed random number, and the remainder of the right-hand side is the local RMS value of the velocity fluctuations. Since the kinetic energy of turbulence is known at each point in the flow, these values of the RMS fluctuating components can be defined (assuming isotropy) as


 \sqrt{\overline{u^{' 2}}} = \sqrt{\overline{v^{'2}}} = \sqrt{\overline{w^{'2}}} = \sqrt{2k/3} (22.2-26)

for the $k$- $\epsilon$ model, the $k$- $\omega$ model, and their variants. When the RSM is used, nonisotropy of the stresses is included in the derivation of the velocity fluctuations:


$\displaystyle u'$ $\textstyle =$ $\displaystyle \zeta \sqrt{\overline{u^{'2}}}$ (22.2-27)
$\displaystyle v'$ $\textstyle =$ $\displaystyle \zeta \sqrt{\overline{v^{'2}}}$ (22.2-28)
$\displaystyle w'$ $\textstyle =$ $\displaystyle \zeta \sqrt{\overline{w^{'2}}}$ (22.2-29)

when viewed in a reference frame in which the second moment of the turbulence is diagonal [ 413]. For the LES model, the velocity fluctuations are equivalent in all directions. See Section  12.9.4 for details.

The characteristic lifetime of the eddy is defined either as a constant:


 \tau_e = 2 T_L (22.2-30)

where $T_L$ is given by Equation  22.2-22 in general (Equation  22.2-23 by default), or as a random variation about $T_L$:


 \tau_e = - T_L \log (r) (22.2-31)

where $r$ is a uniform random number between 0 and 1 and $T_L$ is given by Equation  22.2-23. The option of random calculation of $\tau_e$ yields a more realistic description of the correlation function.

The particle eddy crossing time is defined as


 t_{\rm cross} = -\tau \ln \left[1- \left(\frac{L_e}{\tau\vert u-u_p\vert}\right)\right] (22.2-32)

where $\tau$ is the particle relaxation time, $L_e$ is the eddy length scale, and $\vert u - u_p\vert$ is the magnitude of the relative velocity.

The particle is assumed to interact with the fluid phase eddy over the smaller of the eddy lifetime and the eddy crossing time. When this time is reached, a new value of the instantaneous velocity is obtained by applying a new value of $\zeta$ in Equation  22.2-25.

Using the DRW Model

The only inputs required for the DRW model are the value for the integral time-scale constant, $C_L$ (see Equations  22.2-22 and 22.2-30) and the choice of the method used for the prediction of the eddy lifetime. You can choose to use either a constant value or a random value by selecting the appropriate option in the Set Injection Properties panel for each injection, as described in Section  22.12.5.

figure   

Turbulent dispersion of particles cannot be included if the Spalart-Allmaras turbulence model is used.

Stochastic Staggering of Particles

In order to obtain a better representation of an injector, the particles can be staggered either spatially or temporally. When particles are staggered spatially, FLUENT randomly samples from the region in which the spray is specified (e.g., the sheet thickness in the pressure-swirl atomizer) so that as the calculation progresses, trajectories will originate from the entire region. This allows the entire geometry specified in the atomizer to be sampled while specifying fewer streams in the input panel, thus decreasing computational expense.

When injecting particles in a transient calculation using relatively large time steps in relation to the spray event, the particles can clump together in discrete bunches. The clumps do not look physically realistic, though FLUENT calculates the trajectory for each particle as it passes through a cell and the coupling to the gas phase is properly accounted for. To obtain a statistically smoother representation of the spray, the particles can be staggered in time. During the first time step, the particle is tracked for a random percentage of its initial step. This results in a sample of the initial volume swept out by the particle during the first time step and a smoother, more uniform spatial distribution at longer time intervals.

The menu for staggering is available in the text user interface, under

define/models/dpm/options/particle-staggering.

The "staggering factor'' in the TUI is a constant which multiplies the random sample. The staggering factor controls the percentage of the initial time step that will be sampled. For example, if the staggering factor is 0.5, then the parcels in the injection will be tracked between half and all of their full initial time step. If the staggering factor is 0.1, then the parcels will be tracked between ninety percent and all of their initial time step. If the staggering factor is set to 0.9, the parcels will be tracked between ten percent and all of their initial time step. This allows the user to control the amount of smoothing between injections.

The default values for the options in the TUI are no temporal staggering and a temporal staggering factor of 1.0. The temporal staggering factor is inactive until the flag for temporal staggering is turned on.



Particle Cloud Tracking


Particle dispersion due to turbulent fluctuations can also be modeled with the particle cloud model [ 28, 29, 159, 217]. The turbulent dispersion of particles about a mean trajectory is calculated using statistical methods. The concentration of particles about the mean trajectory is represented by a Gaussian probability density function (PDF) whose variance is based on the degree of particle dispersion due to turbulent fluctuations. The mean trajectory is obtained by solving the ensemble-averaged equations of motion for all particles represented by the cloud (see Section  22.2.2).

The cloud enters the domain either as a point source or with an initial diameter. The cloud expands due to turbulent dispersion as it is transported through the domain until it exits. As mentioned before, the distribution of particles in the cloud is defined by a probability density function (PDF) based on the position in the cloud relative to the cloud center. The value of the PDF represents the probability of finding particles represented by that cloud with residence time $t$ at location $x_i$ in the flow field. The average particle number density can be obtained by weighting the total flow rate of particles represented by that cloud, $\dot{m}$, as


 \langle n(x_i) \rangle = \dot{m} P(x_i,t) (22.2-33)

The PDFs for particle position are assumed to be multivariate Gaussian. These are completely described by their mean, $\mu_i$, and variance, ${\sigma_i}^2$, and are of the form


 P(x_i,t) = \frac{1}{(2 \pi)^{3/2} \displaystyle{\prod_{i=1}^{3} \sigma_{i}}} e^{-s/2} (22.2-34)

where


 s = \sum_{i=1}^{3} \left( \frac{x_i - \mu_i}{\sigma_i}\right)^2 (22.2-35)

The mean of the PDF, or the center of the cloud, at a given time represents the most likely location of the particles in the cloud. The mean location is obtained by integrating a particle velocity as defined by an equation of motion for the cloud of particles:


 \mu_i(t) \equiv \langle x_i (t) \rangle = \int_{0}^{t} \langle V_i(t_1) \rangle dt_1 + \langle x_i (0) \rangle (22.2-36)

The equations of motion are constructed using an ensemble average.

The radius of the particle cloud is based on the variance of the PDF. The variance, $\sigma^{2}_{i} (t)$, of the PDF can be expressed in terms of two particle turbulence statistical quantities:


 \sigma^{2}_{i} (t) = 2 \int_{0}^{t} \langle u'^{2}_{p,i}(t_2) \rangle \int_{0}^{t_2} R_{p, ii} (t_2,t_1) dt_1 dt_2 (22.2-37)

where $\langle {u^{'2}_{p,i}} \rangle$ are the mean square velocity fluctuations, and $R_{p, ij}(t_2,t_1)$ is the particle velocity correlation function:


 R_{p, ij} (t_2,t_1) = \frac{\langle u'_{p,i}(t_2) u'_{p,j}(t... ...\langle u'^2_{p,i}(t_2) u'^2_{p,j}(t_2) \rangle \right]^{1/2}} (22.2-38)

By using the substitution $ \tau = \vert t_2-t_1 \vert $, and the fact that


 R_{p, ij} (t_2,t_1) = R_{p, ij} (t_4,t_3) (22.2-39)

whenever $\vert t_2-t_1\vert=\vert t_4-t_3\vert$, we can write

 \sigma^{2}_{i} (t) = 2 \int_{0}^{t} \langle u'^{2}_{p,i}(t_2) \rangle \int_{0}^{t_2} R_{p, ii} (\tau) d\tau dt_2 (22.2-40)

Note that cross correlations in the definition of the variance ( $R_{p, ij}, i \neq j $) have been neglected.

The form of the particle velocity correlation function used determines the particle dispersion in the cloud model. FLUENT uses a correlation function first proposed by Wang [ 389], and used by Jain [ 159]. When the gravity vector is aligned with the $z$-coordinate direction, $R_{ij}$ takes the form:


$\displaystyle R_{p, 11}$ $\textstyle =$ $\displaystyle \frac{u'^{2}}{\theta} e^{-(\tau/\tau_a)} \;\; {\rm St}_T \left( B - 0.5 m_T \gamma \frac{{\rm St}^{2}_{T} B^2 + 1}{\theta} \right)$  
  $\textstyle +$ $\displaystyle \frac{u'^{2}}{\theta} e^{-(\tau B/T)} \left(-1 + \frac{m_T {\rm St}^{2}_{T} \gamma B} {\theta} + 0.5 m_T \gamma \frac{\tau}{T} \right)$  
      (22.2-41)
$\displaystyle R_{p, 22}$ $\textstyle =$ $\displaystyle R_{p, 11}$ (22.2-42)
$\displaystyle R_{p, 33}$ $\textstyle =$ $\displaystyle \frac{u'^{2} {\rm St}_{T} B}{\theta} e^{-(\tau/\tau_a)} - \frac{u'^{2}}{\theta} e^{-(\tau B/T)}$ (22.2-43)

where $B = \sqrt{1+ m^{2}_{T} \gamma^2}$ and $\tau_a$ is the aerodynamic response time of the particle:


 \tau_a = \frac{\rho_p d^{2}_{p}}{18 \mu} (22.2-44)

and


$\displaystyle T$ $\textstyle =$ $\displaystyle \frac{m_T T_{mE}}{m}$ (22.2-45)
$\displaystyle T_{fE}$ $\textstyle =$ $\displaystyle \frac{C_{\mu}^{3/4} k^{3/2}}{\epsilon (\frac{2}{3} k)^{1/2}}$ (22.2-46)
$\displaystyle \gamma$ $\textstyle =$ $\displaystyle \frac{\tau_a g}{u'}$ (22.2-47)
$\displaystyle {\rm St}$ $\textstyle =$ $\displaystyle \frac{\tau_a}{T_{mE}}$ (22.2-48)
$\displaystyle {\rm St}_{T}$ $\textstyle =$ $\displaystyle \frac{\tau_a}{T}$ (22.2-49)
$\displaystyle \theta$ $\textstyle =$ $\displaystyle {\rm St}_{T}^{2} (1 + m_{T}^{2} \gamma^2) - 1$ (22.2-50)
$\displaystyle m$ $\textstyle =$ $\displaystyle \frac{\bar{u}}{u'}$ (22.2-51)
$\displaystyle T_{mE}$ $\textstyle =$ $\displaystyle T_{fE} \frac{\bar{u}}{u'}$ (22.2-52)
$\displaystyle m_T$ $\textstyle =$ $\displaystyle m \left[ 1 - \frac{G(m)}{(1+ {\rm St})^{0.4(1+0.01 {\rm St})}} \right]$ (22.2-53)
$\displaystyle G(m)$ $\textstyle =$ $\displaystyle \frac{2}{\sqrt{\pi}} \int_0^\infty \frac{e^{-y^2} \; dy } {\left(... ... erf}(y)y-1+e^{-y^2} \right) \right)^{5/2}} \; \; \; \; \; \; \; \; \; \; \; \;$ (22.2-54)

Using this correlation function, the variance is integrated over the life of the cloud. At any given time, the cloud radius is set to three standard deviations in the coordinate directions. The cloud radius is limited to three standard deviations since at least $99.2 \%$ of the area under a Gaussian PDF is accounted for at this distance. Once the cells within the cloud are established, the fluid properties are ensemble-averaged for the mean trajectory, and the mean path is integrated in time. This is done with a weighting factor defined as


 W(x_i , t) \equiv \frac{ \displaystyle{\int_{V_{\rm cell}} P... ..., t) dV}} { \displaystyle{\int_{V_{\rm cloud}} P(x_i , t) dV}} (22.2-55)

If coupled calculations are performed, sources are distributed to the cells in the cloud based on the same weighting factors.

Using the Cloud Model

The only inputs required for the cloud model are the values of the minimum and maximum cloud diameters. The cloud model is enabled in the Set Injection Properties panel for each injection, as described in Section  22.12.5.

figure   

The cloud model is not available for unsteady particle tracking, or in parallel, when using the message passing option for the particles.


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