Influence of spanwise rotation and scalar boundary conditions on passive scalar transport in turbulent channel flow

Direct numerical simulations of passive scalar transport in turbulent channel flow subject to spanwise rotation are carried out with two different boundary conditions for the scalar. In the first case the scalar transport is driven by an assigned scalar difference at the walls and in the second case by a constant mean streamwise scalar gradient. The Reynolds number $\text{Re}=U_{b}h/\nu$ is fixed at $14000$ and the rotation number $\text{Ro}=2\mathrm{\Omega }h/U_b$ is varied from 0 to 0.75, where $U_b$ is the mean bulk velocity, $h$ half the channel gap width, and $\mathrm{\Omega }$ the rotation rate. This work is a continuation of Brethouwer [J. Fluid Mech. 844, 297 (2018)] to further study the influence of rotation and also the influence of scalar boundary conditions on scalar transport in channel flow. Mean scalar profiles and other scalar statistics differ in the two cases with different boundary conditions but are similar in the near-wall region in terms of local wall units. The conclusion of Brethouwer that the Reynolds analogy for scalar-momentum transfer does not apply to rotating channel flow is independent of scalar boundary conditions. Rotation influences the turbulent scalar flux differently than the Reynolds shear stress and strongly reduces the turbulent Prandtl number on the unstable channel side, irrespective of the scalar boundary conditions. Scalar structures are larger than the turbulence structures in rotating channel flow, in contrast to nonrotating channel flow where these are similar.

Figure 1

Profiles of (a) $U/U_b$ and (b) $v^{+}$ at $\text{Ro}=0$ (black solid), $\text{Ro}=0.15$ (black dashed), $\text{Ro}=0.3$ (black dotted), $\text{Ro}=0.45$ (red solid), and $\text{Ro}=0.75$ (red dashed).

Figure 2

Visualizations of the scalar field in an $x\text{−}z$ plane in case 2 on the unstable channel side at (a) $\text{Ro}=0.15$ and $y=-0.5$ and (b) $\text{Ro}=0.75$ and $y=-0.9$. A dark color corresponds to a low scalar value. Visualizations of the instantaneous scalar gradients in case 2 on the stable channel side at the wall at $y=1$ at (c) $\text{Ro}=0.3$ and (d) $\text{Ro}=0.75$. A dark color corresponds to large scalar gradients. The flow is from left to right.

Figure 3

Time series at $\text{Ro}=0.75$ of ${\tau }_w^s$ (black dashed), ${\theta }_1^{\prime }$ (blue solid), $Q_{w1}^s$ (black solid), and $Q_{w2}^s$ (red solid).

Figure 4

Profiles of (a) ${\mathrm{\Theta }}_1$ and (b) ${\mathrm{\Theta }}_2/\left(\text{Pr}\text{Re}\right)$. Lines as in Fig. 1. (c) ${\tau }_w^u/{\tau }_w$ (red dashed), ${\tau }_w^s/{\tau }_w$ (blue dashed), $Q_{w2}^u/\left(\text{Pr}\text{Re}\right)$ (red solid),and $Q_{w2}^s/\left(\text{Pr}\text{Re}\right)$ (blue solid) as a function of Ro.

Figure 5

(a) Profiles of ${\mathrm{\Theta }}_1^{+}$ (thin lines) and ${\mathrm{\Theta }}_2^{*}$ (thick lines) on the unstable channel side and (b) profiles of ${\mathrm{\Theta }}_2^{*}$ on the stable (thin lines) and unstable channel sides (thick lines). Lines as in Fig. 1. The dashed green and blue lines are explained in the main text.

Figure 6

(a) Profiles of ${\theta }_2^{+}$ and (b), (c) near-wall profiles of ${\theta }_1^{+}$ (thin lines) and ${\theta }_2^{*}$ (thick lines) on the unstable and stable channel side, respectively. Lines as in Fig. 1.

Figure 7

(a) Profiles of ${\overline{u\theta }}_2^{+}$ and (b), (c) near-wall profiles of ${\overline{u\theta }}_1^{+}$ (thin lines) and ${\overline{u\theta }}_2^{*}$ (thick lines) on the unstable and stable channel side, respectively. (c) Profiles of $-{\overline{u\theta }}_1^{+}$ and ${\overline{u\theta }}_2^{*}$ on the stable channel side. Lines as in Fig. 1.

Figure 8

Profiles of (a) $-{\overline{v\theta }}_2^{+}$ and (b) ${\rho }_{v\theta 2}$. The blue lines show the sum of the first two terms on the right-hand side of Eq. (4). Lines as in Fig. 1.

Figure 9

Profiles of (a) ${\text{Pr}}_t$ and (b) ${\text{Pr}}_t$ on the unstable channel side. Lines as in Fig. 1. The thin lines are for case 1 and the thick lines for case 2.

Figure 10

(a) ${\text{Nu}}_1$ (red line) and ${\text{Nu}}_2$ (blue line), and (b) $2{\text{St}}_2/C_f$ as a function of Ro.

Figure 11

Premultiplied one-dimensional [(a), (c), (e)] spanwise spectra as a function of ${\lambda }_z^{+}$ and [(b), (d), (f)] streamwise spectra as a function of ${\lambda }_x^{+}$ at $\text{Ro}=0$ [(a), (b)], $\text{Ro}=0.3$ [(c), (d)], and $\text{Ro}=0.75$ [(e), (f)]. Shown are spectra at $y^{+}\approx 11$ (dotted lines), $y^{+}\approx 95$ (dashed lines), and $y=-0.56$ (solid lines) shifted in the vertical with an offset. Also spectra (dash-dotted lines) at $y=0$, 0.42, and 0.24 for $R=0$, 0.3, and 0.75, respectively, are shown. The red lines show $k_i{E}_K\left(k_i\right)$ and the green and blue lines $k_i{E}_{\theta \theta }\left(k_i\right)$ for case 1 and 2, respectively.

Figure 12

$C_{\epsilon 1}$ (black solid); $C_{\epsilon \theta }^1$ (solid); $C_{\epsilon \theta }^2$ (dashed). Red: ${\theta }_1$; blue: ${\theta }_2$. (a) $\text{Ro}=0$, (b) $\text{Ro}=0.3$, and (c) $\text{Ro}=0.75$.