Dependence of scalar mixing on initial conditions in turbulent channel flow

Turbulent scalar mixing occurs within a wide variety of natural and engineering flows. Predicting and controlling the scalar concentration(s) within these flows can yield immediate benefits to numerous applications across many fields. To that end, a better understanding of the effects of scalar-field initial conditions on the evolution(s) of scalar fields is required to either promote or delay the rate at which mixing occurs. Direct numerical simulations are employed herein to simulate the evolution in time of the hydrodynamic and (passive) scalar fields within a fully developed turbulent channel flow. The effects of the scalar field initial conditions are studied by analyzing the evolution of the scalar field subject to three different initial conditions with interfaces oriented normal to the streamwise, wall-normal, and transverse directions. Particular emphasis is placed on the scalar variance and dissipation rate budgets, including the evolutions of their constituent terms. The fastest mixing occurs for the initial condition in which the interface is aligned normal to the mean velocity vector. The rapid mixing in this case is associated with higher rates of production and destruction of the scalar dissipation, as well as strong advection and stretching of the interface by the mean flow. In addition to better mixing arising from the stronger turbulence near the wall, enhanced mixing is correlated with having the edge of an interface along a channel wall, such that a large distortion of the initial interface arises from the combined effects of the no-slip condition at the channel walls with the advection of the interface by the mean flow in the region between the walls. To maximize this effect, it is recommended that scalar interfaces be aligned normal to the mean velocity vector to promote mixing within internal flows.

Figure 1

The three scalar-field initial conditions studied herein. (a) ${\mathrm{IC}}_x$; (b) ${\mathrm{IC}}_y$; (c) ${\mathrm{IC}}_z$. (Imagery generated using VAPOR.)

Figure 2

Mean streamwise component of the velocity field (a) and the rms values of the three velocity field components (b) and comparisons with the work of Refs. [57, 62].

Figure 3

Concentration distributions generated at $\tau =7$ by the action of the same turbulent channel flow on the three scalar-field initial conditions. (a) ${\mathrm{IC}}_x$; (b) ${\mathrm{IC}}_y$; (c) ${\mathrm{IC}}_z$ (Imagery generated using VAPOR.)

Figure 4

Time evolutions of the volume-averaged scalar variance (a) and the volume-averaged scalar dissipation rate (b) for the three scalar fields resulting from the three initial conditions.

Figure 5

Evolution in time of the space and time averages of all six terms in the scalar variance budget in which statistical moments ($\langle ·\rangle$) are assessed in planes parallel to the initial location of the interface in the initial condition. (a) ${\mathrm{IC}}_x$ with $\frac{1}{L_x}\frac{1}{\tau }{\int }_0^{L_x}{\int }_0^{\tau }\mathrm{Term}d{\tau }^{\prime }dx$, (b) ${\mathrm{IC}}_y$ with $\frac{1}{2h}\frac{1}{\tau }{\int }_0^{2h}{\int }_0^{\tau }\mathrm{Term}d{\tau }^{\prime }dy$, and (c) ${\mathrm{IC}}_z$ with $\frac{1}{L_z}\frac{1}{\tau }{\int }_0^{L_z}{\int }_0^{\tau }\mathrm{Term}d{\tau }^{\prime }dz$. Note the different scales of the vertical axes.

Figure 6

Evolution in time of the space and time averages of all nine terms in the scalar dissipation rate budget in which statistical moments ($\langle ·\rangle$) are assessed in planes parallel to the initial location of the interface in the initial condition. (a) ${\mathrm{IC}}_x$ with $\frac{1}{L_x}\frac{1}{\tau }{\int }_0^{L_x}{\int }_0^{\tau }\mathrm{Term}d{\tau }^{\prime }dx$, (b) ${\mathrm{IC}}_y$ with $\frac{1}{2h}\frac{1}{\tau }{\int }_0^{2h}{\int }_0^{\tau }\mathrm{Term}d{\tau }^{\prime }dy$, and (c) ${\mathrm{IC}}_z$ with $\frac{1}{L_z}\frac{1}{\tau }{\int }_0^{L_z}{\int }_0^{\tau }\mathrm{Term}d{\tau }^{\prime }dz$. Note the different scales of the vertical axes.

Figure 7

Evolution in time of the time-averages of all nine terms in the scalar dissipation rate budget in which statistical moments ($\langle ·\rangle$) are assessed on the $y\text{−}z$ plane at $x/L_x=0.50$, i.e., $\frac{1}{\tau }{\int }_0^{\tau }\mathrm{Term}d{\tau }^{\prime }$, where all averages ($\langle ·\rangle$) in Term are given by $\langle ·\rangle ={\langle ·\rangle }_{y,z}$.

Figure 8

Evolution in time of the time averages of all nine terms in the scalar dissipation rate budget in which statistical moments ($\langle ·\rangle$) are assessed on the $x\text{−}z$ plane at $y/h=1$, i.e., $\frac{1}{\tau }{\int }_0^{\tau }\mathrm{Term}d{\tau }^{\prime }$, where all averages ($\langle ·\rangle$) in Term are given by $\langle ·\rangle ={\langle ·\rangle }_{x,z}$.

Figure 9

Evolution in time of the time averages of all nine terms in the scalar dissipation rate budget in which statistical moments ($\langle ·\rangle$) are assessed on the $x\text{−}y$ plane at $z/L_z=0.50$, i.e., $\frac{1}{\tau }{\int }_0^{\tau }\mathrm{Term}d{\tau }^{\prime }$, where all averages ($\langle ·\rangle$) in Term are given by $\langle ·\rangle ={\langle ·\rangle }_{x,y}$.

Figure 10

Evolution in time of the time averages of all nine terms in the scalar dissipation rate budget in which statistical moments ($\langle ·\rangle$) are assessed on the midplane corresponding to the initial location of the interface in the initial condition: $\frac{1}{\tau }{\int }_0^{\tau }\mathrm{Term}d{\tau }^{\prime }$. (a) ${\mathrm{IC}}_x$ with statistical moments evaluated over the $y\text{−}z$ plane at $x/L_x=0.50$, (b) ${\mathrm{IC}}_y$ with statistical moments evaluated over the $x\text{−}z$ plane at $y/h=1$, and (c) ${\mathrm{IC}}_z$ with statistical moments evaluated over the $x\text{−}y$ plane at $z/L_z=0.50$.

Figure 11

Evolution in time of the time-averages of all nine terms in the scalar dissipation rate budget in which statistical moments ($\langle ·\rangle$) are assessed in planes parallel to the initial location of the interface in the initial condition but away from the midplane: $\frac{1}{\tau }{\int }_0^{\tau }\mathrm{Term}d{\tau }^{\prime }$. (a) ${\mathrm{IC}}_x$ with statistical moments evaluated over the $y\text{−}z$ plane at $x/L_x=0.25$, (b) ${\mathrm{IC}}_y$ with statistical moments evaluated over the $x\text{−}z$ plane at $y/h=0.5$, and (c) ${\mathrm{IC}}_z$ with statistical moments evaluated over the $x\text{−}y$ plane at $z/L_z=0.25$.