Enhanced enstrophy generation for turbulent convection in low-Prandtl-number fluids

Abstract

Turbulent convection is often present in liquids with a kinematic viscosity much smaller than the diffusivity of the temperature. Here we reveal why these convection flows obey a much stronger level of fluid turbulence than those in which kinematic viscosity and thermal diffusivity are the same; i.e., the Prandtl number Pr is unity. We compare turbulent convection in air at Pr=0.7 and in liquid mercury at Pr=0.021. In this comparison the Prandtl number at constant Grashof number Gr is varied, rather than at constant Rayleigh number Ra as usually done. Our simulations demonstrate that the turbulent Kolmogorov-like cascade is extended both at the large- and small-scale ends with decreasing Pr. The kinetic energy injection into the flow takes place over the whole cascade range. In contrast to convection in air, the kinetic energy injection rate is particularly enhanced for liquid mercury for all scales larger than the characteristic width of thermal plumes. As a consequence, mean values and fluctuations of the local strain rates are increased, which in turn results in significantly enhanced enstrophy production by vortex stretching. The normalized distributions of enstrophy production in the bulk and the ratio of the principal strain rates are found to agree for both Prs. Despite the different energy injection mechanisms, the principal strain rates also agree with those in homogeneous isotropic turbulence conducted at the same Reynolds numbers as for the convection flows. Our results have thus interesting implications for small-scale turbulence modeling of liquid metal convection in astrophysical and technological applications.

Keywords: direct numerical simulation; liquid metals; thermal convection; vorticity generation.

Fig. 1.

Comparison of two turbulent convection runs. (A, Left) Vertical slice cut of temperature; (A, Right) corresponding velocity magnitude. Data are for run RB1 at $\text{Pr}=0.7$. (B) Sketch of the Prandtl–Rayleigh-number plane illustrating our parameter variation between runs RB1 and RB2 (more details in Table 1). Both runs are at the same Grashof number. The dotted gray lines denote the variations at constant Rayleigh (horizontal lines) and Prandtl (vertical lines) numbers, respectively. (C) Same as A, but for run RB2 at $\text{Pr}=0.021$.

Fig. 2.

(A–D) Statistical analysis in subvolumes of the cell. Triangles are for RB2 and squares for RB1. (A) Bolgiano and Kolmogorov scales as a function of the volume fraction $V_j/V$. Additionally, we plot the Corrsin length ${\eta }_{\text{C},j}={\eta }_{\text{K},j}{\text{Pr}}^{-3/4}$ for RB2. (B) Ratio of both scales. (C) Mean kinetic energy dissipation rates. (D) Root-mean-square value of temperature T and vertical velocity $u_z\left(=w\right)$. The exact size of the subvolumes $V_j$ is listed in SI Text.

Fig. 3.

(A) Time-averaged mixed-increment moment $\langle {\mathrm{\Delta }}_{r}w{\mathrm{\Delta }}_{r}T\rangle$ as a function of $r/{\delta }_T$, where ${\delta }_T$ is the thermal boundary layer thickness. Also plotted are scales ${\eta }_{\text{K}}$ and $L_{\text{B}}$ and the integral length scale $L_{\text{int}}=1/\langle w^2\rangle {\displaystyle \int \langle w\left(z+r_z\right)w\left(z\right)\rangle d{r}_z}$. They are indicated as vertical lines, red for RB1 and blue for RB2. B, Inset shows a plume and explains the difference between $r_{\perp }$ and $r_{\mathrm{\parallel }}$ directions. (B) Correlation coefficient $C\left(r\right)$ as given in Eq. 10. The horizontal double-headed arrows indicate the extension of the Kolmogorov-like cascade range, red for RB1 and blue for RB2. All data are obtained in $V_1$.

Fig. S1.

(A) Time-averaged mixed-increment function $\langle {\mathrm{\Delta }}_{r}w{\mathrm{\Delta }}_{r}T\rangle$ as a function of $r/{\delta }_T$, where ${\delta }_T$ is the thermal boundary layer thickness. Also plotted are scales ${\eta }_{\text{K}}$ and $L_{\text{B}}$. They are indicated as vertical solid lines, red for RB1 and blue for RB2. (B) Correlation coefficient $C\left(r\right)$ as given in Eq. 9 of the main text. Data are obtained in $V_2$.

Fig. S2.

(A) Time-averaged mixed-increment function $\langle {\mathrm{\Delta }}_{r}w{\mathrm{\Delta }}_{r}T\rangle$ as a function of $r/{\delta }_T$, where ${\delta }_T$ is the thermal boundary layer thickness. Also plotted are scales ${\eta }_{\text{K}}$ and $L_{\text{B}}$. They are indicated as vertical solid lines, red for RB1 and blue for RB2. (B) Correlation coefficient $C\left(r\right)$ as given in Eq. 9 of the main text. Data are obtained in $V_3$.

Fig. S3.

Probability density function of the enstrophy production due to vortex stretching. Data correspond to Fig. 5 A and C, but are conducted in subvolume $V_5$. (A) Run RB1. (B) Run RB2.

Fig. 4.

(A–D) Probability density functions of the principal rates of strain for the four flows. In all four runs the principal rates are given in units of the inverse large-scale eddy turnover time $T_L^{-1}=\langle \mathit{ϵ}\rangle /k^2$ with $k^2$ being the turbulent kinetic energy $k^2=\langle u_i^2\rangle /2$. The mean values in the convection cases are determined over the volume $V_4$ and with respect to time. (A) Run RB1; (B) Run HI1; (C) Run RB2; (D) Run HI2.

Fig. 5.

Enstrophy production for different Prandtl number flows. (A and C) Normalized probability density function (PDF) of the production due to vortex stretching, $P_v={\omega }_i{S}_{ij}{\omega }_j$. We compare data for the full volume (denoted cell) and a subvolume in the bulk of the cell (denoted bulk, which equals $V_4$). For comparison we also display the PDFs of the corresponding isotropic box turbulence runs. (B and D) Plane-time averaged vertical profiles of the enstrophy production due to vortex stretching $\langle P_v\left(z\right)\rangle$ (B) and due to the temperature gradient $\langle P_T\left(z\right)\rangle$ (D) in the vicinity of the heating plate. The production term $\langle P_T\left(z\right)\rangle$ is negative in the vicinity of the walls for both cases. The distance from the wall is given in units of the corresponding thermal boundary layer thickness ${\delta }_T$. The viscous boundary layer thicknesses ${\lambda }_v$ are evaluated from slopes of gradients at the isothermal walls (35) and indicated by solid vertical lines. The dashed line in B shows the global maximum of $\langle P_T\left(z\right)\rangle$ for the low-$\text{Pr}$ run. (E) Root-mean-square values of $P_v$ obtained for the whole cell as a function of the time that is normalized with respect to the total integration time. (F) The sketch explains the connection between enstrophy consumption, $P_T<0$, and the detachment of a line-like plume in a simple one-dimensional picture.