Rayleigh–Bénard convection
6.6 Boundary layer characteristics
The analysis of theRa-dependence of BL characteristics includes the different quantities involved.
Therefore the levels and slopes as well as the resulting BL thicknesses and their ratio are discussed.
6.6.1 Slopes and levels
As mentioned before the slopes of the temperature and wind velocity profiles are worth evaluating not only with respect to the BL thicknesses but because they are meaningful in themselves, as well. In the case of the temperature field the slopes at the wall are proportional to the local heat flux orthogonal to the wall. In the case of the bottom plate the slopes are thus proportional to the vertical local heat fluxΩz.
In figure 6.16a the local heat flux along the wind direction is depicted in terms ofNu. It is observed that the heat flux normalized withNualmost collapses for all Rayleigh numbers considered in the intervalJ={r|r∈[−0.7,0.5]}. In particular the maxima of the p.d.f.s of this local heat flux in the intervalJoccur for all Rayleigh numbers at 0.8 as shown in figure 6.16b.
OutsideJ and close to the vertical wall the heat flux reflects some interesting features due to the secondary rolls as displayed in figure 6.16a. From this we conclude that the vortices in the corners (here close tor=0.8) are also essential drivers of the heat. The comparison of the horizontal normalized heat flux profiles for differentRain figure 6.16a also shows that even though the largestΩzvalues are reached within this corner region for all theRa, it becomes less important for increasing Ra.
Contrary to this another structure becomes more important for the heat flux with increasing Ra. It was shown in figure 6.12a that in the remaining corners (here close tor=−0.9) a smaller vortex structure develops. Owing to the rising plumes close to the vertical wall, the flow detaches from the bottom plate. For higherRathis effect becomes so strong that a separation bubble and
(a)
FIGURE6.17: (a) Time-averaged wind velocity leveluLalonglLSCfor differentRawith a polynomial fit forr∈[−0.7,0.5]; (b)p.d.f. for the wind velocity level forr∈[−0.7,0.5],Pr= 0.786, Γ=1.
therefore a vortex structure develops, which induces motion against the direction of the wind close to the bottom plate and thus a positiveτw.
This argument is also supported by the levels presented in figure 6.17a which for larger Rareflect positive mean velocities in this corner region close tor=−0.9. WithinJ, where the wind is dominant, the mean velocity levels can be approximated by a parabola for all Ra considered. In figure 6.17a the fit with a second-order polynomial for r ∈J results in uL(r) =0.277(r+0.104)2−0.189. This means that the wind along the plate has a varying magnitude and reaches its maximum atr=−0.104. The existence of a varying wind velocity has been also observed by Zhouet al.[170] in two-dimensional DNS and similarly by Kaczorowski et al.[70] in DNS in a cube.
The observedRa-independence of the dimensionless wind velocity magnitude is also visible in the p.d.f.s (see figure 6.17b). They almost coincide for all investigatedRaand are symmetric arounduL≈ −0.2. Similar results were obtained in experiments. For example du Puitset al.
[103] found a mean velocity magnitude of|uL| ≈0.18 in the centre of their cylindrical cell for Ra=1.12×1011, Pr=0.7 andΓ=1.13, while Sunet al.[140] obtained|uL| ≈0.1 in their quasi-two-dimensional experiments forRa=3.43×109,Pr=4.3 andΓ=1.04. The deviation of our results from the latter might be an effect of the considerably different Prandtl numbers.
Besides the velocity levels the temperature levelsTLalso reveal an interestingRa-dependence.
Their time averaged distributions and p.d.f.s are presented in figure 6.18. As shown by [169], the fluctuating temperature level has an effect on the p.d.f. of the thermal BL thickness. We have argued that levels below zero are reached due to a global temperature minimum within the intervalI (recall figure 6.11). From figure 6.18 it is concluded that the radial distribution of the time averaged levels as well as the p.d.f. are shifted to higher levels, i.e. closer to the arithmetic mean temperature zero, with increasingRa. Thus for higherRathe case of a global temperature minimum which deviates from zero is less probable. If the temperature level would be chosen as
(a)
FIGURE6.18:Temperature levelTLalonglLSCfor differentRa, (a) time-averaged and (b) p.d.f. for r∈[−0.7,0.5],Pr=0.786, Γ=1.
FIGURE6.19:Thermal boundary layer thicknessδT in units ofH/2Nu, (a) time-averaged and (b) p.d.f.
forr∈[−0.7,0.5],Pr=0.786, Γ=1.
the arithmetic mean temperature the later obtained BL thicknesses were closer to our results in the case of largerRa.
6.6.2 Boundary layer thickness
Finally we compare the BL thicknesses for differentRa, starting with the thermal BL in figure 6.19(a), where the time-averaged BL thickness alonglLSCis presented in units of its approxima-tionH/2Nu. We observe a nearly linear andRa-independent increase of this normalised thermal BL thickness along the wind, i. e. withinJ, for decreasingr. Since the PDF ofδT/(H/2Nu) presented in figure 6.19(b)shows an almostRa-independent behaviour the normalisation with Nucompensates perfectly the decreasing BL thickness with growingRa. Thus the found PDF is universal (for the investigated tuples(Ra,Pr,Γ)) and the meanδT shows aRascaling
hδTit,r∼1/Nu∼Ra−0.285±0.003. (6.8)
(a)
FIGURE6.20:(a) p.d.f. of viscous boundary layer thicknessδuforr∈[−0.7,0.5]and (b) fit of most probableδuversusRa,Pr=0.786,Γ=1.
Finally we compare the BL thicknesses for differentRa, starting with the thermal BL in fig-ure 6.19a, where the time averaged BL thickness alonglLSC is presented in units ofH/2Nu. We observe a nearly linear andRa-independent increase of this normalized thermal BL thickness along the wind, i.e. withinJ, for decreasingr. Since the p.d.f. ofδT/(H/2Nu)presented in figure 6.19b shows an almostRa-independent behaviour the normalization withNucompensates perfectly the decreasing BL thickness with growingRa. Thus the p.d.f. found is universal (for the investigated triplets(Ra,Pr,Γ)) and the meanδT shows anRascaling
hδTit,r∼1/Nu∼Ra−0.285±0.003
. (6.9)
As one can see from figure 6.19b, the BL thickness is distributed almost symmetrically around 85% of its approximated value if a logarithmic scale is used.
In the case of the viscous BL the rescaling of the BL thickness with an approximated value is more complicated since no proper value exists. In figure 6.20a the p.d.f. of the viscous BL thicknessδureflects a Gaussian-like behaviour if scaled logarithmically. If the (not rescaled) p.d.f.s with logarithmic scale are fitted with a Gaussian curve, the most probable BL thickness δucu can be evaluated. Thisδucu shows aRa-dependence (shown in figure 6.20b)
δuc∼Ra−0.227±0.010, (6.10)
whilehδuit,r∈J scales as
hδuit,r∈J ∼Ra−0.238±0.009
(6.11) and is larger thanδucas expected from the p.d.f.
Thus the exponent in the scaling of the viscous BL thickness withRais slightly smaller than that in the scaling of the thermal BL thickness, which lets us conclude that their ratioδT/δuis slightly decreasing with increasingRa. Using the most probable valueδucfor the rescaling, we
(a)
FIGURE6.21:(a) p.d.f. of viscous boundary layer thicknessδuscaled with the most probable value δucforr∈[−0.7,0.5]and (b) time-averaged viscous BL thickness in units of its mean value,Pr=0.786, Γ=1. dashed lines correspond to the prediction by Prandtl-Blasius (PB) theory [121].
obtain a similar behaviour for all treatedRa(see figure 6.21a) as in the case of the thermal BL and as reported by Zhou & Xia [173]. Like in the case ofδT the mean thickness of the viscous BL increases almost linearly alonglLSC (see figure 6.21b).
Besides the individual BL thicknesses also their ratioδT/δuis of interest. In figure 6.22 the mean ratioδT/δuover lLSC and its p.d.f. are shown for differentRa. We observe that along the plate the ratio remains almost constant withinJ but slightly decreases with increasingRa.
This is exactly what is expected from the previous results. Indeed, the time averaged thermal and viscous BL thicknesses develop almost linearly along the plate, therefore their ratio was expected to be constant withinJ. Furthermore, as the individual BL thicknesses scale withRain a different way, anRa-dependent ratio is obtained. The p.d.f.s in figure 6.22b reflect that there is a qualitative change from the lower to the higherRacases. This might by due to a change in the dynamics of the systems aroundRa=108as was proposed in the theory by Grossmann & Lohse [51] and supported by theNuvs.Radependence presented in figure 6.2.