Effect of heat-flux boundary conditions on the Rayleigh-Bénard instability in a rarefied gas

We consider the effect of heat-flux boundary conditions, replacing the previously studied isothermal wall conditions, on the Rayleigh-Bénard instability in a rarefied gas. The problem is investigated in the limit of small Knudsen numbers, by means of a linear stability analysis of a slip flow model, and the direct simulation Monte Carlo method. In the latter, a noniterative algorithm is applied to implement the heat-flux conditions. The results delineate the instability domain in the parameters plane of the Knudsen number (Kn), Froude number (Fr), and walls reference temperature ratio. The heat-flux conditions result in a significant destabilizing effect, extending instability to larger Knudsen numbers. At large Fr, the Boussinesq limit is recovered, and transition to instability is governed by a critical value of the Rayleigh number. With decreasing Fr, gas compressibility becomes dominant, confining the convection layer to the vicinity of the upper cold wall. Asymptotic analysis of the low-Fr limit is carried out, to highlight the impact of difference in thermal conditions. The Monte Carlo scheme is applied to investigate system instability at supercritical states, where walls heat-flux conditions lead to elevated shear stresses. Nonmonotonic variations in the walls shear stress with Kn are observed and discussed.

Figure 1

Projection of the neutral surface $\omega (k;\text{Fr},\text{Kn})=0$ over the $(\text{Fr},\text{Kn})$ plane at $R_T=0.1$. The solid line marks the neutral curve in the walls fixed heat-flux problem (along most of which $k\approx 0$; cf. the bold solid line in Fig. 2), and the dashed line shows the counterpart result in the isothermal walls setup (along most of which $k\approx 3.12$; cf. the thin dashed line in Fig. 2). The empty and filled circles denote states where the pure conduction solution is stable or unstable according to DSMC calculations, respectively, in the walls constant heat-flux problem. The red dash-dotted line marks the constant ${\text{Ra}}_m=720$ large-Fr asymptote [see Eq. (23)], and the asterisk denotes the value $\text{Fr}={\text{Fr}}_{\mathrm{cr}}\approx 0.39$, the minimal necessary Fr for instability at $\text{Kn}\to 0$ [see Eq. (25)].

Figure 2

Projection of the neutral surface over the $(\text{Fr},k)$ plane for $R_T=0.1$. The bold solid and thin dashed curves show projections of the neutral curves in Fig. 1 for the constant heat-flux and isothermal walls setups, respectively, where the cross and circle mark the respective peak $(\overline{\text{Fr}},\overline{\text{Kn}})$ locations. The colored contours show the neutral curves at the indicated values of Kn in the constant heat-flux problem.

Figure 3

Effects of the (a) temperature ratio $R_T$ and (b) perturbation wave number $k$ on the instability domain in the constant heat-flux problem. In Fig. 3, the numbers correspond to the values of $R_T$, the dashed lines mark the constant $R{a}_m$ asymptotes in each case, and the dash-dotted curve joins the locations of the neutral curve peaks $(\overline{\text{Fr}},\overline{\text{Kn}})$ at different $R_T$. In Fig. 3, $R_T=0.1$, and the numbers indicate the values of $k$ corresponding to each curve.

Figure 4

The critical tangential velocity (a), (c) and temperature (b), (d) eigenfunctions at the onset of instability for $R_T=0.1$: comparison between walls constant heat-flux (solid lines) and isothermal boundaries (dashed curves) setups at $(\text{Fr},\text{Kn})=(\overline{\text{Fr}},\overline{\text{Kn}})$ (a), (b) and $(\text{Fr},\text{Kn})=({\text{Fr}}_{\text{min}},0.005)$ (c), (d).

Figure 5

(a) The neutral curve in the $(\tilde{l},\tilde{a})$ plane for $R_T=0.5$. The dashed and dash-dotted lines show the dispersion relation Eq. (37) in the limit $\delta \to 0$ in the constant wall heat-flux and fixed wall temperature problems, respectively, and the solid blue lines correspond to the numerically calculated neutral curves at the indicated values of $\delta$ in the former case. (b) The normalized temperature eigenfunctions in the constant wall heat-flux setup for $R_T=0.5$ and $({\tilde{l}}_{\text{cr}},{\tilde{a}}_{\text{max}})$ combinations. The solid and dashed curves represent the slip-flow numerical ($T^{\left(1\right)}$) and asymptotic ($T_0^{\left(1\right)}$) solutions at the indicated values of $\delta$, respectively.

Figure 6

The effect of thermal boundary conditions on the (a) temperature and (b) vertical velocity perturbation eigenfunctions in the compressible limit. In each part, the solid and dashed curves present the normalized eigenfunctions for the walls constant heat-flux and fixed temperature problems, respectively, at the indicated values of $\delta$.

Figure 7

DSMC-calculated flow fields (arrows) and temperature perturbations (colormaps) for walls constant heat-flux (a,c,e) and fixed temperature (b,d,f) setups at $R_T=0.1,\text{Fr}=2.7$ and $\text{Kn}=0.04$ (a,b), $\text{Kn}=0.025$ (c,d), and $\text{Kn}=0.01$ (e,f). In each part, ${\left|\mathbf{u}\right|}^{\max }$ denotes the maximum value of calculated velocity amplitude.

Figure 8

Variation with Kn of the DSMC-calculated walls shear stress amplitude at $R_T=0.1$ and $\text{Fr}=2.7$. Black and blue curves present results for walls constant heat-flux and isothermal setups, respectively, with solid and dashed lines showing shear-stress amplitudes at the hot and cold walls, respectively.