Horizontal structures of velocity and temperature boundary layers in two-dimensional numerical turbulent Rayleigh-Bénard convection

Abstract

We investigate the structures of the near-plate velocity and temperature profiles at different horizontal positions along the conducting bottom (and top) plate of a Rayleigh-Bénard convection cell, using two-dimensional (2D) numerical data obtained at the Rayleigh number Ra= 10 8 and the Prandtl number Pr= 4.4 of an Oberbeck-Boussinesq flow with constant material parameters. The results show that most of the time, and for both velocity and temperature, the instantaneous profiles scaled by the dynamical frame method [Q. Zhou and K.-Q. Xia, Phys. Rev. Lett. 104, 104301 (2010)] agree well with the classical Prandtl-Blasius laminar boundary layer (BL) profiles. Therefore, when averaging in the dynamical reference frames, which fluctuate with the respective instantaneous kinematic and thermal BL thicknesses, the obtained mean velocity and temperature profiles are also of Prandtl-Blasius type for nearly all horizontal positions. We further show that in certain situations the traditional definitions based on the time-averaged profiles can lead to unphysical BL thicknesses, while the dynamical method also in such cases can provide a well-defined BL thickness for both the kinematic and the thermal BLs.

Thermal boundary layer profiles in turbulent Rayleigh-Bénard convection in a cylindrical sample

“…However, a drawback of this method is that we cannot determine whether the LSC is passing the cell center at every time instance. As is shown by Zhou et al [21,23,24], the agreement between the dynamically rescaled profile and the Prandtl-Blasius-Pohlhausen profile deteriorates when one considers a position where the LSC does not pass. Similar effects can influence our results, however preventing this is very difficult as it would require an arbitrary criterion to determine the presence of the LSC at a certain location.…”

“…Even then, we obtain that the mean bulk temperature obtained far away from the thermal BL is below θ * . As discussed by Zhou et al [23,24], we attribute this to the emissions of thermal plumes. They propose to use the temperature at some position outside of the thermal BL, such as z/λ th = 3 or z * th = 3, as the asymptotic value for the BL, where λ th is the slope BL thickness obtained in the fixed reference frame and z * th (t) ≡ z/δ th (t) is the rescaled vertical distance from the plate.…”

“…This led to the study of the BL structures in dynamical reference frames, which fluctuate with the instantaneous BL thicknesses [21]. Zhou et al [21] used PIV data to show that the velocity profiles obtained in the dynamical reference frame are very close to the laminar Prandtl-Blasius profile, and numerical data [23,24] to show that a good agreement with the temperature profile of a laminar, zero-pressure gradient boundary layer according to Pohlhausen [23,24], which we will call the "Pohlhausen" profile [25] in the remainder of this paper, is obtained. In addition, they showed that most of the time the rescaled instantaneous velocity and temperature profiles are also in agreement with the theoretical prediction.…”

“…We determined the mean temperature profile in the dynamical reference frame by defining the instantaneous thermal BL thickness δ th (t) as the intersection between the temperature gradient at the plate and the temperature of the first extremum value in the temperature profile [23,24]. Even then, we obtain that the mean bulk temperature obtained far away from the thermal BL is below θ * .…”

“…In Fig. 1, we show a direct comparison between the various temperature profiles, rescaled in the described way: the fixed reference frame profiles θ (x,z)/θ (x,z = 3λ th ), the dynamical local profiles θ * (z * th )/θ * (z * th = 3), where θ * indicates the temperature in the dynamical reference frame defined as θ * (x,z * th ) ≡ θ (x,z = z * th δ th (x,t),t) [21,23,24], and the Pohlhausen temperature profile. Figures 1(a) and 1(b) show that for Ra = 10 8 and Pr = 6.4, the mean dynamically rescaled profiles are very close to the Pohlhausen temperature profile for both = 1 and 1/2 samples.…”