**5. Comparison between different SGS scalar flux models and the FDF method**

In previous works [98, 99], we simulated a turbulent round jet (*Re* = 10,000) discharging diluted rhodamine B in a co‐flowing stream of water (*Sc* = 2000), using large eddy simulation. The flow configuration is detailed by Antoine et al. [100]. Three different models for the unknown subgrid‐scale (SGS) scalar flux term were used for closing the filtered scalar balance equa‐ tion: eddy diffusivity model with both constant turbulent Schmidt number (*ScT* = 0.7) and dynamically calculated turbulent Schmidt number, and the dynamic anisotropy model. In addition, the filtered density function (FDF) method was implemented. The interaction by exchange with the mean (IEM) mixing model was used for closing the conditional diffusion term in the transported FDF equation. The FDF transport equation was solved using a Monte Carlo method.

**Figure 1.** Radial mean concentration distribution across the jet. Experimental data (*symbols*) from Ref. [100]: + = 70; ● = 80; × = 90. Simulation results (*lines*) from Refs. [98, 99]: *Solid line*: eddy; *dashed line*: dynamic; *dotted line*: anisotropy; *dashed*‐*dotted line*: FDF.

**Figure 2.** Radial mean (a) and fluctuation (b) concentration distribution across the jet. Experimental data (*symbols*) from Ref. [100]: + = 70; ● = 80; × = 90. Simulation results (*lines*) from Refs. [98, 99]: *Solid line*: eddy; *dashed line*: dynamic; *dotted line*: anisotropy; *dashed*‐*dotted line*: FDF.

**Figures 1** and **2** compare the simulated radial mean and fluctuation distribution of the scalar concentration [98, 99], normalized by the centerline value (*η = r/x*, where *r* is the radial coordinate and *x* is the downstream distance from the nozzle exit), with the experimental data [101].

Having in mind the limitations of the eddy diffusivity model, it is noted in **Figure 1** that the mean concentration distribution is predicted reasonably well. On the other hand, **Figure 2** reveals that the three SGS scalar flux models fail to reproduce the concentration fluctuations at higher values of the nondimensional radial coordinate (*η*). Although the LES simulation results of the concentration distribution in radial direction are similar for the tested SGS scalar flux models in **Figures 1** and **2**, the dynamic anisotropy model provides a better perform‐ ance. In LES, a mesh grid refinement would improve the simulation results. Mejía et al. [98] also showed that if the mesh grid is further refined, the dynamic eddy diffusivity improved its predictive capabilities.

**Figure 3.** Streamwise velocity‐concentration correlation across the jet. Experimental data (*symbols*) from Antoine et al. [100]: + = 70; ● = 80; × = 90. Simulation results (*lines*) from Refs. [98, 99]: *Solid line*: eddy; *dashed line*: dynamic; *dotted line*: anisotropy; *dashed*‐*dotted line*: FDF.

**Figure 4.** Radial velocity‐concentration correlation across the jet. Experimental data (*symbols*) from Antoine et al. [100]: + = 70; ● = 80; × = 90. Simulation results (*lines*) from Refs. [98, 99]: *Solid line*: eddy; *dashed line*: dynamic; *dotted line*: anisotropy; *dashed*‐*dotted line*: FDF.

The simulation results of the FDF approach are in good agreement with the mean and fluctuations concentration distribution for all regions of the jet. The radial distributions of the axial and radial scalar fluxes are presented, respectively, in **Figures 3** and **4**.

**Figures 3** and **4** show that the dynamic eddy diffusivity and anisotropy models reproduce the behavior of the scalar fluxes for low and intermediate values of the radial coordinates.

The agreement of the simulations of the FDF method (**Figures 3** and **4**) with the experimen‐ tal data is good, and much betterthan that obtained with LES simulations using advanced SGS scalar flux models, such as the anisotropy model (see, e.g., **Figures 1** and **2** ). The LES/FDF can capture strong intermittency effects that take place on the surroundings of the superviscous layer formed between the jet and the co‐flowing streams, leading toward anisotropy of the scalar field. Contrary to the SGS scalarflux models evaluated in [98, 99], the FDF equation does not spread the scalar away from the superviscous layer, as seen in the predicted *rms* and scalar fluxes distributions.

The computational costs of the dynamic and anisotropy models were 1.2 and 1.7 times that of the eddy diffusivity model, respectively. When a higher level model has to be considered, it is usual that more computational resource is required. From an engineering point of view, the dynamic model captures the most important mixing characteristics at a rather low computa‐ tional cost. The computational time of the FDF, compared to the one required by using the eddy diffusivity model is approximately 1.9 times higher though [101].
