Systematic approach for modeling a nonlocal eddy diffusivity

This study considers advective and diffusive transport of passive scalar fields by spatially varying incompressible flows. Prior studies have shown that the eddy diffusivities governing the mean-field transport in such systems can generally be nonlocal in space and time. While for many flows nonlocal eddy diffusivities are more accurate than commonly used Boussinesq eddy diffusivities, nonlocal eddy diffusivities are often computationally cost prohibitive to obtain and difficult to implement in practice. We develop a systematic and more cost-effective approach for modeling nonlocal eddy diffusivities using matched moment inverse operators. These operators are constructed using only a few leading-order moments of the exact nonlocal eddy diffusivity kernel, which can be easily computed using the inverse macroscopic forcing method [Mani and Park, Phys. Rev. Fluids 6, 054607 (2021)]. The resulting reduced-order models for the mean fields that incorporate the modeled eddy diffusivities often improve Boussinesq-limit models because they capture leading-order nonlocal effects. But more importantly, these models can be expressed as partial differential equations that are readily solvable using existing computational fluid dynamics capabilities rather than as integropartial differential equations.

Figure 1

Example of a nonlocal eddy diffusivity. (a) Full nonlocal eddy diffusivity kernel, $D(x_1,y_1)$, and (b) cross sections of $D(x_1,y_1)$ at various $x_1$ locations.

Figure 2

(a) Example eddy diffusivity kernel corresponding to the Boussinesq model in Eq. (26). (b) Example eddy diffusivity kernel corresponding to the explicit model in Eq. (27) with $D^0=1/2, D^{1_s}=0$, and $D^{2_s}=1/32$ using a second-order central difference scheme and $\mathrm{\Delta }{y}_1=0.1$.

Figure 3

(a) Example temporally nonlocal eddy diffusivity kernel in Eq. (28) with $\alpha =\beta =1$. (b) Example spatially nonlocal eddy diffusivity kernel in Eq. (31) with $\alpha =\beta =1$.

Figure 4

(a) The exact spatiotemporally nonlocal eddy diffusivity, $D(y_1-x_1,\tau -t)$, for the homogeneous model problem in Sec. 3a. (b) The modeled eddy diffusivity using MMI.

Figure 5

For the inhomogeneous model problem with periodic boundaries in Sec. 4a: (a) MMI coefficients for Eq. (37). (b) The eddy diffusivity kernel from the MMI model closely approximates the exact eddy diffusivity kernel as shown for various $x_1$ locations.

Figure 6

(a) The velocity profile for the homogeneous, parallel flow ($u_1=cos\left(x_2\right), u_2=0$). (b) An initial condition corresponding to the release of a narrow band of passive scalar in the center of the domain, $c(t=0)=exp(-x_1^2/0.025)$. (c) The dispersed scalar field, $c(x_1,x_2,t)$, and $x_2$-averaged field, $\overline{c}(x_1,t)$, at time $t_1=0.5$. (d) $c(x_1,x_2,t)$ and $\overline{c}(x_1,t)$ at a later time, $t_2=4$.

Figure 7

Model comparison of the averaged field, $\overline{c}(x_1,t)$, using the initial condition $c(t=0)=exp(-x_1^2/0.025)$. The spatiotemporal MMI model in Eq. (52) closely captures the spread of the averaged field, whereas the leading-order Taylor model and higher-order Taylor model overpredict the spread of the averaged field.

Figure 8

Model comparison of leading-order Taylor model and higher-order Taylor model at (a) early time, $t=0.5$, and (b) late time, $t=4$.

Figure 9

Model comparison of the temporal MMI model and spatiotemporal MMI model at (a) early time, $t=0.5$, and (b) late time, $t=4$.

Figure 10

Comparison of the spatially nonlocal eddy diffusivities (equivalent to the spatiotemporally nonlocal eddy diffusivities integrated over $\tau$) (a) in physical space and (b) in Fourier space.

Figure 11

MMI model and MFM-inspired model comparison at (a) early time, $t=0.5$, and (b) late time, $t=4$.

Figure 12

(a) Streamlines of the velocity field in Eq. (57). (b) Contour plot of $c(x_1,x_2)$ from DNS.

Figure 13

Model comparison for the inhomogeneous problem with periodic boundary conditions. The MMI model closely matches the DNS solution.

Figure 14

(a) Streamlines of the velocity field in Eq. (63). (b) Contour plot of $c(x_1,x_2)$ from DNS.

Figure 15

(a) MMI coefficients for Eq. (61) for the wall-bounded inhomogeneous problem. (b) The exact and modeled nonlocal eddy diffusivity shown for various $x_1$ approaching the wall ($x_1=-2.922,-2.796,-2.670,-2.545$).

Figure 16

Model comparison for the wall-bounded inhomogeneous problem. The MMI model closely matches the DNS solution.

Figure 17

(a) MMI coefficients using the regularization in Eq. (64). (b) Model comparison for the wall-bounded inhomogeneous problem.

Figure 18

(a) Error comparison for the leading-order Taylor model and the spatiotemporal MMI model. For large $L$, the models follow the expected $1/L^2$ and $1/L^4$ scaling, respectively. (b) Model comparison for $t=12$. In this limit, all models are expected to perform well; however, the spatiotemporal MMI model still outperforms both the leading-order Taylor model and the higher-order Taylor model.

Figure 19

The nonlocal eddy diffusivity of the simple fractional-order model shown in Fourier space for various choices of $\alpha$ compared with the exact nonlocal eddy diffusivity and the MMI-modeled eddy diffusivity.

Figure 20

Fractional-order model comparison for the homogeneous problem in Sec. 3a at early time, $t=0.5$, and late time, $t=4$.

Figure 21

(a) Coefficients of the alternative MMI formulation in Eq. (G1) for the wall-bounded inhomogeneous problem in Sec. 4b. (b) Model comparison for the wall-bounded inhomogeneous problem.

Figure 22

(a) Coefficients for the alternative MMI formulation with coefficient regularization and $\sigma =0.01{\varepsilon }^2$. (b) Model comparison for the wall-bounded inhomogeneous problem in Sec. 4b.