Abstract
Results on the Prandtl–Blasius-type kinetic and thermal boundary layer (BL) thicknesses in turbulent Rayleigh–Bénard (RB) convection in a broad range of Prandtl numbers are presented. By solving the laminar Prandtl–Blasius BL equations, we calculate the ratio between the thermal and kinetic BL thicknesses, which depends on the Prandtl number only. It is approximated as for and as for , with . Comparison of the Prandtl–Blasius velocity BL thickness with that evaluated in the direct numerical simulations by Stevens et al (2010 J. Fluid Mech.643 495) shows very good agreement between them. Based on the Prandtl–Blasius-type considerations, we derive a lower-bound estimate for the minimum number of computational mesh nodes required to conduct accurate numerical simulations of moderately high (BL-dominated) turbulent RB convection, in the thermal and kinetic BLs close to the bottom and top plates. It is shown that the number of required nodes within each BL depends on and and grows with the Rayleigh number not slower than . This estimate is in excellent agreement with empirical results, which were based on the convergence of the Nusselt number in numerical simulations.
Highlights
This confirms that the Prandtl–Blasius boundary layer (BL) theory is the relevant theory to describe the BL dynamics in RB convection for not too large Res
We find that this minimum number of nodes in the thermal BLs is
We found that neither the position of the maximum rms velocity fluctuations nor the position of the horizontal velocity maximum reflects the slope velocity BL thickness, many studies use these as criteria to determine the BL thickness
Summary
Solving numerically equation (5) with the boundary conditions (7) for any fixed Prandtl number, one obtains the temperature profile with respect to the similarity variable ξ (see figure 1(b)). An approximation of the ratio between the thermal and kinetic BL thicknesses in the transition region 3 × 10−4 Pr 3 is obtained by applying a least square fit to the numerical solutions of the Prandtl–Blasius equations (4)–(7). As seen, this relation is a good fit of the full solution in the transition regime As seen in figure 2, this relation is a good fit of the full solution in the transition regime
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