Scaling of the production of turbulent kinetic energy and temperature variance in a differentially heated vertical channel

This paper investigates the scaling of turbulent kinetic energy (TKE) and temperature variance production in a differentially heated vertical channel (DHVC). In a DHVC, TKE is produced by two distinctively different mechanisms: buoyancy production and shear production. In the present work, identity equations are derived for the global integrals of shear-produced TKE and temperature variance production. The derived identity equations agree well direct numerical simulation (DNS) data. At sufficiently high Rayleigh number the global integral of the shear-produced TKE is found, based on the DNS data, to scale as ${\int }_0^{\delta }{R}_{\mathrm{wu}}\frac{dU}{dz}dz\approx 0.385u_{\tau }{U}_{\max }^2$. Here $z$ is the wall-normal direction, $\delta$ is the channel half-width, $U$ is the mean streamwise velocity in the $x$ direction, $U_{\max }$ is the maximum mean streamwise velocity, and $u_{\tau }$ is the friction velocity. $R_{\mathrm{wu}}=-\langle \mathrm{wu}\rangle$ is the Reynolds shear stress, where $\mathrm{w}$ is the velocity fluctuation in the $z$ direction, $\mathrm{u}$ is the velocity fluctuation in the $x$ direction, and angle brackets $\langle \rangle$ denote averaging operation. The global integral of the buoyancy-produced TKE at sufficiently high Grashof number is found to scale as ${\int }_0^{\delta }g\alpha R_{\mathrm{u}\theta }dz\approx u_{\tau }^2{U}_{\max }$ where $g$ is the gravitational acceleration, $\alpha$ is the thermal expansion coefficient, and $R_{\mathrm{u}\theta }=-\langle \mathrm{u}\theta \rangle$ is the covariance of the streamwise velocity fluctuation $u$ and the temperature fluctuation $\theta$. The global integrals of temperature variance production and temperature dissipation ${\epsilon }_{\theta }$ are found to grow with the Grashof number in a logarithmic-like fashion as ${\int }_0^{\delta }{R}_{\mathrm{w}\theta }\frac{d\mathrm{\Theta }}{dz}dz={\int }_0^{\delta }{\epsilon }_{{}_{\theta }}dz\approx \left[0.5\ln \left(\mathrm{Gr}\right)-1.8\right]u_{\tau }{\theta }_{\tau }^2$ where $\mathrm{\Theta }$ is the mean transformed temperature, $R_{\mathrm{w}\theta }=-\langle \mathrm{w}\theta \rangle$ is the wall-normal turbulent transport of heat, ${\theta }_{\tau }$ is the friction temperature, and $\mathrm{Gr}$ is the Grashof number. Based on the characteristics of the flux Richardson number, a four-layer structure is proposed for the TKE budget equation in a 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 about the channel center line.

Figure 2

(a) DNS data to evaluate TKE identity Eq. (8a). DNS data of Versteegh are triangles $▹$, DNS data of Kiš are circles $\circ$, and DNS data of Ng are stars $\star$. (b) DNS data to evaluate the temperature variance production identity Eq. (8b). Black $\times$ symbols denote the the magnitude of ${\int }_0^1{\epsilon }_{{}_{\theta }}^{+}d\eta$. The dissipation data are from DNS of Kiš [3] only.

Figure 3

(a) Flux Richardson number versus $\eta =z/\delta$. (b) Flux Richardson number multiplied by $U_{\max }^{+}=\frac{U_{\max }}{u_{\tau }}$. Insets on the top right show the region away from the wall. Data are from DNS of Kiš [3].

Figure 4

(a) Ratio (filled symbols) between the global integrals of buoyancy-produced TKE and the shear-produced TKE as a function of Grashof number. Open symbols are data of $\frac{1}{U_{\max }^{+}}$. (b) Ratio of global integrals weighted by $U_{\max }^{+}$.