Effect of polymer additives on heat transport and large-scale circulation in turbulent Rayleigh-Bénard convection

The present paper presents direct numerical simulations of Rayleigh-Bénard convection (RBC) in an enclosed cell filled with the polymer solution in order to investigate the viscoelastic effect on the characteristics of heat transport and large-scale circulation (LSC) of RBC. To overcome the difficulties in numerically solving a high Weissenberg number (Wi) problem of viscoelastic fluid flow with strong elastic effect, the log-conformation reformulation method was implemented. Numerical results showed that the addition of polymers reduced the heat flux and the amount of heat transfer reduction (HTR) behaves nonmonotonically, which firstly increases but then decreases with Wi. The maximum HTR reaches around $8.7\%$ at the critical Wi. The nonmonotonic behavior of HTR as a function of Wi was then corroborated with the modifications of the period of LSC and turbulent energy as well as viscous boundary layer thickness. Finally, a standard turbulent kinetic energy (TKE) budget analysis was done for the whole domain, the boundary layer region, and the bulk region. It showed that the role change of elastic stress contributions to TKE is mainly responsible for this nonmonotonic behavior of HTR.

Figure 1

Schematic of the computational model.

Figure 2

(a) Ratio of time-averaged Nu(Wi) to Nu(0) as a function of Wi. The value of Nu(0) is 24.68 at $\mathrm{Ra}={10}^8$ and $\mathrm{Pr}=0.7$. The solid line is a spline fitting to the data. (b) The ratio of standard deviation ${\sigma }_{\mathrm{Nu}}\left(\mathrm{Wi}\right)$ to Nu(Wi) as a function of Wi. The value of ${\sigma }_{\mathrm{Nu}}\left(0\right)$ is 2.36.

Figure 3

Autocorrelation functions of the fluctuating velocity measured at convection cell center.

Figure 4

Oscillation period $t_T$ (Wi) normalized by $t_T$ (0) as a function of Wi. The value of $t_T$ (0) is $6.25H/u_c$ at $\mathrm{Ra}={10}^8$ and $\mathrm{Pr}=0.7$. The solid line is a spline fitting to the data.

Figure 5

LSC kinetic energy profile over the vertical centerline as a function of Wi. (a) Cases at Wi's smaller than $\mathrm{W}{\mathrm{i}}_{c2}$, and (b) cases at Wi's larger than $\mathrm{W}{\mathrm{i}}_{c2}$. The insets are enlarged plots for kinetic energy near the wall and the peak position, respectively.

Figure 6

Time-averaged streamline (a) Newtonian fluid case (red dashed line) and the polymer solution case at $\mathrm{Wi}=0.5$ (solid line), and (b) polymer solution cases at $\mathrm{Wi}=1.5$ (red dashed line) and $\mathrm{Wi}=10$ (solid line), respectively.

Figure 7

Comparison between the time-averaged (a) temperature and (b) velocity profiles over the vertical centerline and PB profile (solid line) near the bottom plate obtained for Newtonian fluid (triangles) and polymer solution (circles) cases. The inclined straight lines are linear fittings to the mean temperature and velocity profiles. The distances are normalized by ${\lambda }_v$ and ${\lambda }_{\mathrm{th}}$, respectively.

Figure 8

Viscous BL thickness and thermal BL thickness as a function of Wi at $\mathrm{Ra}={10}^8$ and $\mathrm{Pr}=0.7$.

Figure 9

The turbulent kinetic energy budget as a function of Wi: (a) over the whole domain; (b) over the BL region; (c) over the bulk region.$F_{\mathrm{KF}}$, $P_{\mathrm{KF}}$, $V_{\mathrm{KF}}$, and $G_{\mathrm{KF}}$ represent the convective heat transport term, inertial production term, viscous dissipation term, and elastic term of the TKE budget equation, respectively.

Figure 10

Comparison of our results with the benchmark.