Multiscaling analysis of buoyancy-driven turbulence in a differentially heated vertical channel

This paper investigates the scaling properties of the mean momentum balance (MMB) equation and the mean thermal energy balance (MHB) equation for buoyancy-driven turbulent flow and heat transfer in a differentially heated vertical channel (DHVC). Based on the characteristics of force balance, a three-layer structure is developed for the mean momentum balance equation. In Layer I, a viscous inner layer adjacent to the wall, the force balance is between the viscous force and the buoyancy force. In Layer III, the outer layer, the force balance is between the Reynolds shear force and the buoyancy force. A multiscaling analysis of the MMB equation is developed for the inner and outer layers. In the outer layer, a proper length scale is the channel half width $\delta$, a proper velocity scale is the maximum mean streamwise velocity $U_{\max }$, and a proper scale for the Reynolds shear stress is $u_{\tau }{U}_{\max }$ where $u_{\tau }$ is the friction velocity. In the viscous inner layer, a proper length scale is found to be $\frac{\nu }{u_{\tau }}\frac{U_{\max }}{u_{\tau }}$, where $\nu$ is the kinematic viscosity. The structure for the MHB equation can also be divided into three layers based on the characteristics of the diffusional and turbulent heat flux. The thickness of thermal inner layer is found to be $O\left(\frac{k}{u_{\tau }}\sqrt{\frac{U_{\max }}{u_{\tau }}}\right)$, where $k$ is the thermal diffusivity. A multiscaling analysis of the MHB equation is developed for the inner and outer layers. The outer-scaling of the MHB equation in a DHVC is similar to passive scalar transport in forced convection, where the channel half width $\delta$ is a proper length scale, friction temperature ${\theta }_{\tau }$ is a proper temperature scale, and $u_{\tau }{\theta }_{\tau }$ is a proper scale for turbulent heat flux. The inner-scaling for the thermal inner layer in a DHVC, however, is distinctly different from that in forced convection. The thermal inner length scale in a DHVC is found to be $\frac{k}{u_{\tau }}\sqrt{\frac{U_{\max }}{u_{\tau }}}$. The multiscaling analysis of the MMB and MHB equations agree well with direct numerical simulation data of DHVC.

Figure 1

Sketch of a differentially heated vertical channel (DHVC). $\mathrm{\Theta }\stackrel{\mathrm{def}}{=}{T}_{\mathrm{hot}}-T$ is the mean transformed temperature. Under ideal conditions, the mean flow is antisymmetric with respect to the centerline of the channel.

Figure 2

(a) Three forces in the MMB equation: viscous force $F_{\mathrm{visc}}=\nu \frac{{\partial }^{2}U}{\partial z^2}$, Reynolds shear force $F_{\mathrm{turb}}=\frac{\partial R_{\mathrm{wu}}}{\partial z}$, and buoyancy force $F_{\mathrm{buoy}}=g\alpha ({\mathrm{\Theta }}_{\mathrm{ctr}}-\mathrm{\Theta })$, based on the DNS of Kiš at $\text{Gr}=4.4\times {10}^5$. (b) Ratio between buoyancy force and Reynolds shear force: $\frac{F_{\mathrm{buoy}}}{F_{\mathrm{turb}}}$ at three Grashof numbers. Data are from DNS of Kiš [3].

Figure 3

(a) Layer ${\mathrm{I}}_{\mathrm{m}}$ thickness $l_{{}_{{\mathrm{I}}_{\mathrm{m}}}}$ normalized by channel half width $\delta$: $\frac{l_{{}_{{\mathrm{I}}_{\mathrm{m}}}}}{\delta }$. (b) Layer ${\mathrm{I}}_{\mathrm{m}}$ thickness $l_{{}_{{\mathrm{I}}_{\mathrm{m}}}}$ normalized by $\nu /u_{\tau }$: $\frac{l_{{}_{{\mathrm{I}}_{\mathrm{m}}}}}{\nu /u_{\tau }}$. Black solid symbols are data of $\frac{U_{\max }}{u_{\tau }}$. DNS data are from three independent studies: Versteegh [1] (red), Kiš [3] (green), and Ng [4] (blue). These three DNS datasets are used in the other figures, with the same color coding.

Figure 4

Extent of Layer ${\mathrm{I}}_{\mathrm{m}}$ for the MMB equation in a DHVC, presented as $\frac{l_{{}_{{\mathrm{I}}_{\mathrm{m}}}}^{+}}{U_{\max }^{+}}$.

Figure 5

(a) Ratio of thermal displacement thickness to channel half width versus Grashof number. (b) Grashof number dependence of the factor $\frac{\text{Gr}}{{\text{Re}}_{\tau }^2}\frac{{\delta }_{\theta d}}{\delta }-1$ (open symbols). Solid black symbols represent data of $U_{\max }^{+}$. Dashed line represents the laminar result at low Grashof numbers.

Figure 6

Scaling of the maximum values of the kinematic Reynolds shear stress $|R_{\mathrm{wu}}{|}_{\text{max}}$ by $40\left(g\alpha \right|f_w{\left|k\right)}^{1/2}, \left(g\alpha \right|f_w{\left|\delta \right)}^{2/3}$, and $u_{\tau }{U}_{\max }$. The dashed black line denotes a constant of 0.52.

Figure 7

(a) Outer-scaling for mean streamwise velocity $\frac{U}{U_{\max }}$ versus $\eta$. The exact laminar solution is represented by the solid green curve. (b) Outer-scaling for the kinematic Reynolds shear stress $\frac{R_{\mathrm{wu}}}{u_{\tau }{U}_{\max }}$ versus $\eta$. DNS data are from Kiš [3].

Figure 8

Inner-scaling of the mean streamwise velocity: $\frac{U^{+}}{U_{\max }^{+}}$ versus $\frac{z^{+}}{U_{\max }^{+}}$. (a) On linear-linear axes. (b) On log-linear axes to better show the near-wall region. DNS data are from Kiš [3].

Figure 9

Inner-scaling of the Reynolds shear stress: $R_{\mathrm{wu}}^{+}$ versus $\frac{z^{+}}{U_{\max }^{+}}$. (a) On linear-linear axes. (b) On log-linear axes to better show the near-wall region.

Figure 10

(a) Diffusional term (filled symbols) and turbulent term (open symbols) in the MHB Eq. (3b) at two Grashof numbers. Peak $\frac{\partial R_{\mathrm{w}\theta }^{+}}{\partial \eta }$ location moves closer to the wall with increasing Grashof number. (b) Diffusional heat flux (filled symbols) and turbulent heat flux (open symbols) at two Grashof numbers. Crossover location of $R_{\mathrm{w}\theta }^{+}$ and $\frac{1}{{\text{Pe}}_{\tau }}\frac{\partial {\mathrm{\Theta }}^{+}}{\partial \eta }$ also moves closer to the wall with increasing Grashof number.

Figure 11

Extent of the thermal inner layer for the MHB equation, based on the maximum $\frac{\partial R_{\mathrm{w}\theta }}{\partial z}$ location, and the crossover location of $k\frac{\partial \mathrm{\Theta }}{\partial z}$ and $R_{\mathrm{w}\theta }$. (a) Normalized by the channel half width $\delta$. (b) Normalized by $\frac{k}{u_{\tau }}$. The black symbols represent data of $\sqrt{U_{\max }^{+}}$.

Figure 12

Peak $\frac{\partial R_{\mathrm{w}\theta }}{\partial z}$ location and the crossover point of $k\frac{\partial \mathrm{\Theta }}{\partial z}$ and $R_{\mathrm{w}\theta }$, normalized as $\frac{z}{\sqrt{U_{\max }^{+}}k/u_{\tau }}=\frac{\text{Pr}{z}^{+}}{\sqrt{U_{\max }^{+}}}$.

Figure 13

Scaling of the peak turbulent heat flux $R_{\mathrm{w}\theta }{|}_{\text{max}}$.

Figure 14

(a) Outer-scaling for mean temperature deficit $\frac{{\mathrm{\Theta }}_{\mathrm{ctr}}-\mathrm{\Theta }}{{\theta }_{\tau }}={\mathrm{\Theta }}_{\mathrm{ctr}}^{+}-{\mathrm{\Theta }}^{+}$ versus $\eta$. (b) Outer-scaling of the turbulent heat flux $R_{\mathrm{w}\theta }^{+}=\frac{R_{\mathrm{w}\theta }}{u_{\tau }{\theta }_{\tau }}$ versus $\eta$.

Figure 15

Inner-scaled mean temperature $\frac{{\mathrm{\Theta }}^{+}}{\sqrt{U_{\max }^{+}}}$ versus $\frac{\mathrm{Pr}{z}^{+}}{\sqrt{U_{\max }^{+}}}$. (a) On linear-linear axes. (b) On log-linear axes.

Figure 16

Inner-scaled turbulent heat flux $R_{\mathrm{w}\theta }^{+}=\frac{-\langle \mathrm{w}\theta \rangle }{u_{\tau }{\theta }_{\tau }}$ versus $\frac{\mathrm{Pr}{z}^{+}}{\sqrt{U_{\max }^{+}}}$. (a) On linear-linear axes. (b) On log-linear axes.