Fig. 1. (a) Two-parameter theoretical temperature profile distribution for different parameters a, the black solid line is the Prandtl-Blasius predictions; (b) Pr = 4.3, Ra = 10⁹, one and two-parameter temperature profile fitting.

Fig. 2. Distribution of control parameter with Ra number.

Fig. 3. Temperature distribution close to the bottom plate with different Ra number for Pr = 0.7.

Fig. 4. The time-averaged temperature fields and streamlines in two typical RB convection, the temperature is coded in color, and the arrow indicates the flow direction. The red streamline represents the length of the large-scale circulation path^[20]: (a) Ra = 1 × 10⁸; (b) Ra = 1 × 10⁹.

Fig. 5. Distribution of fitting parameter a with Ra number: (a) Pr = 4.3; (b) Pr = 20.0.

Fig. 6. Distribution of control parameter a with Ra number.

Fig. 7. The variation of large scale circulation path length, Nu number, and parameter a with Ra number, among which the red hollow dots and blue hollow triangles are calculated by Bao Yun^[21], etc. The black dotted line is the multiple predicted by the GL theory, the red line is the characteristic Ra number position: (a) Pr = 0.7; (b) Pr = 4.3; (c) Pr = 20.