Turbulent Dispersion

A particle in turbulent flow experiences a randomly-varying velocity field to which it responds according to its inertia. This behavior is modeled by a stochastic approach that includes the effect of instantaneous velocity fluctuations on the particle.

Following Gosman and Ioannides [661], a particle is assumed to pass through a sequence of turbulent eddies as it traverses a turbulent flow field. Here, an eddy is a local disturbance to the Reynolds-averaged velocity field. The particle remains in the eddy until either the eddy time-scale $τ_{e}$ is exceeded, or the separation between the particle and the eddy exceeds the length scale of the eddy, $l_{e}$ . In practice, an eddy transit time, $τ_{c}$ serves in place of the latter.

The particle experiences an instantaneous fluid velocity $v$ in each eddy, which is

Figure 1. EQUATION_DISPLAY

v = \overline{v} + v ′

(2999)

where $\overline{v}$ is the local Reynolds-averaged velocity and $v ′$ is the eddy velocity fluctuation, unique to each particle. The latter is a normal (Gaussian) deviate with zero mean value and a standard deviation that comes from the eddy velocity scale

Figure 2. EQUATION_DISPLAY

u_{e} = \frac{l_{t}}{τ_{t}} \sqrt{\frac{2}{3}}

(3000)

The turbulence model provides the length and time-scales of the turbulence, $l_{t}$ and $τ_{t}$ . For example, their ratio is $\sqrt{k}$ with a K-Epsilon or K-Omega turbulence model.

Once generated, a single realization of $v ′$ continues to apply to a particle until its eddy interaction time is exceeded:

Figure 3. EQUATION_DISPLAY

τ_{I} = min (τ_{e}, τ_{c})

(3001)

Thus, $v$ as given by Eqn. (2999) is used to evaluate the particle slip velocity and feeds through to other models, for example, using the particle Reynolds number, Eqn. (2966).

The eddy time-scale measures the lifetime of the eddy: the maximum interval over which a single realization of $v ′$ remains valid. It can be related to the diffusion of a passive scalar according to the underlying turbulence model, giving

Figure 4. EQUATION_DISPLAY

τ_{e} = \frac{2 μ_{t}}{ρ u_{e}^{2}}

(3002)

Particles can interact with the eddy for less than the eddy time-scale when they have non-zero slip velocity, which precludes massless particles by definition. With non-zero slip velocity, it is possible for the particle to “cross” the eddy, and hence escape it. The eddy transit time is defined only for material particles when the drag force is activated, in which case it is estimated from

Figure 5. EQUATION_DISPLAY

τ_{c} = {\begin{matrix} \infty & τ_{v} \leq \frac{l_{e}}{| v_{s} |} \\ - τ_{v} \ln (1 - \frac{l_{e}}{τ_{v} | v_{s} |}) & τ_{v} > \frac{l_{e}}{| v_{s} |} \end{matrix}

(3003)

where $τ_{v}$ is the momentum relaxation time-scale, Eqn. (2964), and $l_{e}$ is the eddy length scale $u_{e} τ_{e}$ .