Model for the structure function constant for index of refraction fluctuations in Rayleigh-Bénard turbulence

A model for the structure function constant associated with the index of refraction fluctuations in Rayleigh-Bénard turbulence is developed. The model is based upon the following assumptions: (1) the turbulence is homogeneous and isotropic at or near the midplane, (2) the rate of production is in balance with the rate of dissipation, (3) an inertial region exists, and (4) estimates for the rate of dissipation of temperature fluctuations and of turbulent kinetic energy can be made by assuming that the large-scale turbulence is dissipated in one eddy turnover time. From these assumptions, the dependence of the structure function on the geometry, heat flux, and the properties of the fluid is obtained. The model predicts that the normalized structure function constant is independent of the Rayleigh number. To verify the model, numerical simulations of Rayleigh-Bénard turbulence were performed using two different approaches: an in-house code based on a pseudospectral method, and a finite volume code which employs a model for the smallest scales of the turbulence. The model was found to agree with the results of the simulations, thereby lending support for the assumptions underlying the theory.

Figure 1

Nusselt number ($\mathrm{Nu}=QL/k\mathrm{\Delta }T$) versus time obtained from the DNS. Arrow indicates decreasing $\mathrm{\Delta }T$.

Figure 2

Three-dimensional visualization of the temperature field, $\theta =T-T_0\left({}^{\circ }\mathrm{C}\right)$, obtained from a statistically steady state flow from the DNS for which $\mathrm{\Delta }T=20{}^{\circ }\mathrm{C}$ and $\mathrm{Ra}=2.126\times {10}^6$ at $t=180\mathrm{s}$. Hot and cold fluid associated with cusplike regions are rising and falling from bottom and top boundaries respectively. Axes are denoted in meters and the arrow indicates the direction of gravity.

Figure 3

Nusselt number vs Rayleigh number. DNS: $■$ LES: $▼$. The Nusselt number was obtained from the heat flux at the top and bottom walls of the domain during a time period in which the flow was statistically steady. A least squares curve fit of these results gives $\mathrm{Nu}=0.186\mathrm{R}{\mathrm{a}}^{0.274}$.

Figure 4

Three-dimensional snapshot of the index of refraction field, $n_0=n-1$. This image was generated from the thermal field shown in Fig. 2.

Figure 5

(a) Visualization of the index of refraction field, $n_0=n-1$, in the horizontal ($x-z)$ plane near the top surface ($y=0.045\mathrm{m}$) obtained from the snapshot shown in Fig. 4. Note the spider-web-like structure. (b) Visualization of the index of refraction field in the horizontal ($x-z)$ plane at the center ($y=0.0\mathrm{m}$) of the domain.

Figure 6

Temperature statistics obtained from the DNS. (a) Mean temperature $\overline{\theta }$ versus vertical distance $y$ for five temperature differences $\mathrm{\Delta }T.$ The bottom wall of the domain is at $y=-0.05\mathrm{m}.$ (b) Root-mean-square temperature ${\theta }_{\mathrm{rms}}$ made nondimensional by the convective thermal scale ${\theta }^{*}$.

Figure 7

Root-mean-square velocities, $u_{\mathrm{rms}}, v_{\mathrm{rms}}, w_{\mathrm{rms}}$, obtained from the DNS. These have been made nondimensional by the convective velocity scale $w^{*}$.

Figure 8

(a) Root-mean-square index of refraction, ${\eta }_{\mathrm{rms}}$, and (b) ${\eta }_{\mathrm{rms}}/{\theta }^{*}$. These results were obtained from the DNS.

Figure 9

Dependence of thermal structure function constant $C_T^2$ on the Nusselt number (a) and Rayleigh number (b). Theory given by $C_T^2={\left(\frac{Q}{\rho c_{p}L}\right)}^{4/3}{\left(\beta g\right)}^{-2/3}$ using the $Q$ from DNS ($▼$) and LES ($♦$). $C_T^2$ from DNS ($▲$) and LES ($■$). Dependence of the thermal structure function constant made nondimensional by the theory ($C_T^2/\chi$) on Nusselt number (c) and Rayleigh number (d) for DNS ($\times$) and LES ($⚫$). Dimensions of $C_T^2$ are ${}^{\circ }{\text{C}}^2{\text{m}}^{-2/3}$.

Figure 10

Dependence of the index of refraction structure function constant, $C_n^2=K^2{C}_T^2$ where $K=-C\frac{p_0}{T_0^2}$, on heat flux $Q \left(\mathrm{W}/{\mathrm{m}}^2\right)$. Simulation results: DNS ($▲$), LES ($■$). Theory ($*$) given by $C_n^2=\overline{\gamma }{K}^2{\left(\frac{Q}{\rho c_{p}L}\right)}^{4/3}{\left(\beta g\right)}^{-2/3}$. Units for $C_n^2$ are ${\mathrm{m}}^{-2/3}$.