Phenomenology of turbulent mixing

As seen in Chapter 2 for turbulent flow, the length-scale information needed to describe a homogeneous scalar field is contained in the scalar energy spectrum E, (k, t), which we will look at in some detail in Section 3.2. However, in order to gain valuable intuition into the essential physics of scalar mixing, we will look first at the relevant length scales of a turbulent scalar field, and we develop a simple phenomenological model valid for fully developed, statistically stationary turbulent flow. Readers interested in the detailed structure of the scalar fields in turbulent flow should have a look at the remarkable experimental data reported in Dahm et al. (1991), Buch and Dahm (1996) and Buch and Dahm (1998). 

---

Kosaly, G. 1986. Theoretical remarks on a phenomenological model of turbulent mixing. Combustion Science Technology 49 227-34. 

As the alternative, a phenomenological description of turbulent mixing gives good results for many situations. An apparent diffusivity is defined so that a diffusion-type equation may be used, and the magnitude of this parameter is then found from experiment. The dispersion models lend themselves to relatively simple mathematical formulations, analogous to the classical methods for heat conduction and diffusion. 

Besides these stochastic interpretations, deterministic interpretations are presently developed GRAY [12], KUMPINSKY and EPSTEIN [13], propose systemic approaches, commonly used in chemical engineering several ideal reactors are coupled by conservative flows with expandable coefficients, so that by-passes or dead zones may be taken into account. NICOLIS and FRISCH 14] use a quasi-Semenov equation in the limit of large diffusion coefficients and obtain a renormalization of k , DEWEL et al. [15] use a phenomenological theory of turbulent mixing to study surface effects produced by the feed of the reactor. 

The decreased overall density of the mixing layer with combustion increases the dimensions of the large vortices and reduces the rate of entrainment of fluids into the mixing layer [13]. Thus it is appropriate to modify the simple phenomenological approach that led to Eq. (6.31) to account for turbulent diffusion by replacing the molecular diffusivity with a turbulent eddy dif-fusivity. Consequently, the turbulent form of Eq. (6.38) becomes... 

In turbulent flow, mixing is to a large extent controlled by the mrbulence. Consequently, an understanding of turbulence per se is necessary before we can analyze transport phenomena. Recalhng our phenomenological description from Section 2-3.1, turbulence is three dimensional, dynamic, and mnltiscaled, even in its most ideal form. In a stirred tank, the picture is further complicated by... 

The difference between this equation for turbulent flow and the Navier-Stokes equation for laminar flow is the Reynolds stress/turbulent stress term —pujuj appears in the equation of motion for turbulent flow. This equation of motion for turbulent flow involves non-linear terms, and it is impossible to be solved analytically. In order to solve the equation in the same way as the Navier-Stokes equation, the Reynolds stress or fluctuating velocity must be known or calculated. Two methods have been adopted to avoid this problem—phenomenological method and statistical method. In the phenomenological method, the Reynolds stress is considered to be proportional to the average velocity gradient and the proportional coefficient is considered to be turbulent viscosity or mixing length ... 

In solid-liquid mixing design problems, the main features to be determined are the flow patterns in the vessel, the impeller power draw, and the solid concentration profile versus the solid concentration. In principle, they could be readily obtained by resorting to the CFD (computational fluid dynamics) resolution of the appropriate multiphase fluid mechanics equations. Historically, simplified methods have first been proposed in the literature, which do not use numerical intensive computation. The most common approach is the dispersion-sedimentation phenomenological model. It postulates equilibrium between the particle flux due to sedimentation and the particle flux resuspended by the turbulent diffusion created by the rotating impeller. 

Possible influences of nonequilibrium cross-diffusion effects on the mixing process were investigated by means of direct numerical simulations (DNS) of mass fraction fluctuations in stationary isotropic turbulence for binary mixtures under supercritical conditions (26,27). The authors have shown that after some time, the initially perfectly mixed species become segregated owing to the presence of temperature and pressure fluctuations and the resulting Soret mass cross-diffusion fluxes Jj and /f, induced by temperature and pressure gradients. Based on DNS results (26,27), we propose a phenomenological model that predicts the rate of production of the concentration variance as... 

---