Properties of the scalar variance transport equation in turbulent channel flow

The asymptotic scaling structure of the total scalar variance equation is investigated for fully developed turbulent channel flow subjected to uniform scalar generation. The total scalar variance balance has a four-layer structure similar to that of the total kinetic energy balance, as previously investigated by Zhou and Klewicki [Phys. Rev. Fluids 1, 044408 (2016)]. Direct numerical simulation data are used to quantify the leading balance structure. These data cover the friction Reynolds number up to ${\delta }^{+}=4088$ and Prandtl number ranging between $\text{Pr}=0.2$ and 1.0. Of the layers empirically characterized, the inner-normalized width of the third layer is analytically verified to be ${\delta }^{+}-\sqrt{{\delta }^{+}/\text{Pr}}$. This result agrees closely with the empirical observations. Consistent with previous observations, the Kármán constant, $k_{\theta }$, for the mean scalar profile for $\text{Pr}=1$ is shown to be greater than the Kármán constant, $k$, for the mean velocity profile. Unlike previous studies, the present problem formation yields identical mean equations and boundary conditions for the scalar and velocity, and this allows unambiguous comparisons regarding the noted differences between $k$ and $k_{\theta }$. Results from the mean transport equations and streamwise velocity and scalar variance budget equations, as well as the relevant correlation coefficient profiles, are used to clarify the source of the differences between $k$ and $k_{\theta }$. Through the present theory, the results reported herein connect the statistical structure of the scalar and velocity fields to the mean profile slopes.

Figure 1

(a) Ratio of the sum of the molecular transport (MT) and dissipation (D) terms to the gradient production/turbulent diffusion term (GPTD). Inset: Four-layer structure specifically presented by $\square ,{\delta }^{+}=995,\text{Pr}=0.20,{\delta }^{+}/\text{Pr}=4975$. Vertical dashed-dotted line denotes the external bound of layer i. Vertical dashed line denotes the external bound of layer ii. Vertical solid line denotes the external bound of layer iii. (b) Ratio of the sum of the molecular transport (MT) and dissipation (D) terms to the gradient production/turbulent diffusion term (GPTD) versus $y^{+}\sqrt{\text{Pr}{\delta }^{+}}$. DNS data are from Pirozzoli et al. [18]: $△,{\delta }^{+}=548,\text{Pr}=1,{\delta }^{+}/\text{Pr}=548$; $▽,{\delta }^{+}=548,\text{Pr}=0.71,{\delta }^{+}/\text{Pr}=772$; $◃,{\delta }^{+}=995,\text{Pr}=1,{\delta }^{+}/\text{Pr}=995$; $▹,{\delta }^{+}=995,\text{Pr}=0.71,{\delta }^{+}/\text{Pr}=1401$; $\times ,{\delta }^{+}=2017,\text{Pr}=1,{\delta }^{+}/\text{Pr}=2017$; $·,{\delta }^{+}=548,\text{Pr}=0.20,{\delta }^{+}/\text{Pr}=2740$; $\diamond ,{\delta }^{+}=2017,\text{Pr}=0.71,{\delta }^{+}/\text{Pr}=2841$; $\circ ,{\delta }^{+}=4088,\text{Pr}=1,{\delta }^{+}/\text{Pr}=4088$; $\square ,{\delta }^{+}=995,\text{Pr}=0.20,{\delta }^{+}/\text{Pr}=4975$; $☆,{\delta }^{+}=4088,\text{Pr}=0.71,{\delta }^{+}/\text{Pr}=5758$; $+,{\delta }^{+}=2017,\text{Pr}=0.20,{\delta }^{+}/\text{Pr}=10085$; $✴,{\delta }^{+}=4088,\text{Pr}=0.20,{\delta }^{+}/\text{Pr}=20440$.

Figure 2

(a) Inner-normalized width of layer iii for $\text{Pr}=0.20$. $·,{\delta }^{+}=548,\text{Pr}=0.20,{\delta }^{+}/\text{Pr}=2740$; $\square ,{\delta }^{+}=995,\text{Pr}=0.20,{\delta }^{+}/\text{Pr}=4975$; $+,{\delta }^{+}=2017,\text{Pr}=0.20,{\delta }^{+}/\text{Pr}=10085$; $✴,{\delta }^{+}=4088,\text{Pr}=0.20,{\delta }^{+}/\text{Pr}=20440$. Curve fit is given by $0.29({\delta }^{+}-\sqrt{{\delta }^{+}/\text{Pr}})$. (b) Inner-normalized width of layer iii for $\text{Pr}=0.71$. $▽,{\delta }^{+}=548,\text{Pr}=0.71,{\delta }^{+}/\text{Pr}=772$; $▹,{\delta }^{+}=995,\text{Pr}=0.71,{\delta }^{+}/\text{Pr}=1401$; $\diamond ,{\delta }^{+}=2017,\text{Pr}=0.71,{\delta }^{+}/\text{Pr}=2841$; $☆,{\delta }^{+}=4088,\text{Pr}=0.71,{\delta }^{+}/\text{Pr}=5758$. Curve fit is given by $0.25({\delta }^{+}-\sqrt{{\delta }^{+}/\text{Pr}})$. (c) Inner-normalized width of layer iii for $\text{Pr}=1$. $△,{\delta }^{+}=548,\text{Pr}=1,{\delta }^{+}/\text{Pr}=548$; $◃,{\delta }^{+}=995,\text{Pr}=1,{\delta }^{+}/\text{Pr}=995$; $\times ,{\delta }^{+}=2017,\text{Pr}=1,{\delta }^{+}/\text{Pr}=2017$; $\circ ,{\delta }^{+}=4088,\text{Pr}=1,{\delta }^{+}/\text{Pr}=4088$. Curve fit is given by $0.21({\delta }^{+}-\sqrt{{\delta }^{+}/\text{Pr}})$.

Figure 3

(a) Ratio of the turbulent diffusion to the gradient production part at ${\delta }^{+}=4088,\text{Pr}=0.20,{\delta }^{+}/\text{Pr}=20440$. The vertical dashed-dotted line denotes the external bound of layer ii. The vertical solid line denotes the external bound of layer iii. (b) Ratio of the mean dissipation (MD) to the turbulent dissipation (TD). The vertical dashed-dotted line denotes the external bound of layer ii at ${\delta }^{+}=548,\text{Pr}=1,{\delta }^{+}/\text{Pr}=548$. The vertical dashed line denotes the external bound of layer ii at ${\delta }^{+}=548,\text{Pr}=0.71,{\delta }^{+}/\text{Pr}=772$. The vertical solid line denotes the external bound of layer ii at ${\delta }^{+}=548,\text{Pr}=0.20,{\delta }^{+}/\text{Pr}=2740$.

Figure 4

(a) Profiles of terms PGT $(△)$, GP $(▽)$, TD $(◃)$, and SG $(▹)$ across layers iii and iv at ${\delta }^{+}=4088,\text{Pr}=1,{\delta }^{+}/\text{Pr}=4088$. The vertical dashed-dotted line denotes the external bound of layer ii. The vertical solid line denotes the external bound of layer iii. (b) Ratios of terms PGT/SG $(◃)$ and GP/TD at ${\delta }^{+}=4088,\text{Pr}=1,{\delta }^{+}/\text{Pr}=4088$. The vertical dashed-dotted line denotes the external bound of layer ii. The vertical solid line denotes the external bound of layer iii.

Figure 5

(a) Profiles of terms on the right of Eq. (28) at ${\delta }^{+}=4088$. Profiles of terms on the right-hand side of Eq. (29) at ${\delta }^{+}=4088$ and $\text{Pr}=1$. Solid line, $\langle v^{+}\partial u^{+}/\partial y^{+}\rangle$; dotted line, $\langle v^{+}\partial {\theta }^{+}/\partial y^{+}\rangle$; dashed line, $\langle u^{+}\partial v^{+}/\partial y^{+}\rangle$; dashed-dotted line, $\langle {\theta }^{+}\partial v^{+}/\partial y^{+}\rangle$. (b) Closeup plot of (a) across inertial (nondiffusive) domain. The vertical dashed line denotes $y^{+}=2.6\sqrt{{\delta }^{+}}$. The vertical dashed-dotted line denotes $y^{+}=0.3{\delta }^{+}$.

Figure 6

(a) Profiles of terms in Eq. (31) across inertial domain at ${\delta }^{+}=4088$. $△$, turbulent diffusion; $▽$, viscous diffusion; $◃$, production; $▹$, dissipation; $\square$, pressure strain. The vertical dashed line denotes the external bound of layer III for mean momentum balance. The vertical dashed-dotted line denotes $y^{+}=0.3{\delta }^{+}$. (b) Profiles of terms in Eq. (32) across nondiffusive domain at ${\delta }^{+}=4088$ and $\text{Pr}=1$. $△$, turbulent diffusion; $▽$, molecular transport; $◃$, gradient production; $▹$, dissipation. The vertical dashed line denotes the external bound of layer III for mean scalar balance. The vertical dashed-dotted line denotes $y^{+}=0.3{\delta }^{+}$.

Figure 7

(a) Streamwise velocity and scalar variances. Dotted line, ${〈u^2〉}^{+},{\delta }^{+}=548$; Dashed line, ${〈u^2〉}^{+},{\delta }^{+}=995$; dashed-dotted line, ${〈u^2〉}^{+},{\delta }^{+}=2017$; solid line, ${〈u^2〉}^{+},{\delta }^{+}=4088$; $△,{〈{\theta }^2〉}^{+},{\delta }^{+}=548,\text{Pr}=1$; $▽,{〈{\theta }^2〉}^{+},{\delta }^{+}=995,\text{Pr}=1$; $◃,{〈{\theta }^2〉}^{+},{\delta }^{+}=2017,\text{Pr}=1$; $▹,{〈{\theta }^2〉}^{+},{\delta }^{+}=4088,\text{Pr}=1$. (b) Reynolds stress and turbulent flux. Inset: Solid line, $T^{+},{\delta }^{+}=4088$; $▹,T_{\theta }^{+},{\delta }^{+}=4088,\text{Pr}=1$.

Figure 8

(a) Profiles of correlation coefficient. Dotted line, $-C_{uv},{\delta }^{+}=548$; dashed line, $-C_{uv},{\delta }^{+}=995$; dashed-dotted line, $-C_{uv},{\delta }^{+}=2017$; solid line, $-C_{uv},{\delta }^{+}=4088$; $△,-C_{v\theta },{\delta }^{+}=548,\text{Pr}=1$; $▽,-C_{v\theta },{\delta }^{+}=995,\text{Pr}=1$; $◃,-C_{v\theta },{\delta }^{+}=2017,\text{Pr}=1$; $▹,-C_{v\theta },{\delta }^{+}=4088,\text{Pr}=1$. (b) Profiles of correlation coefficient. Solid line, $-C_{uv},{\delta }^{+}=4088$; $▹,-C_{v\theta },{\delta }^{+}=4088,\text{Pr}=1$. The vertical dashed line denotes the external bound of layer III for mean scalar balance. The vertical dashed-dotted line denotes $y^{+}=0.3{\delta }^{+}$.