eurm10 msam10 119–126

# Prandtl-Blasius temperature and velocity boundary layer profiles in
turbulent Rayleigh-Bénard convection

Quan ZHOU1 Richard J. A. M. STEVENS2 Kazuyasu SUGIYAMA2,3 Siegfried GROSSMANN4
Detlef LOHSE2 Ke-Qing XIA5 1Shanghai Institute of Applied Mathematics and
Mechanics, Shanghai University, Shanghai 200072, China
2Physics of Fluids Group, Department of Science and Technology, J. M. Burgers
Centre for Fluid Dynamics, and Impact-Institute, University of Twente, 7500 AE
Enschede, The Netherlands
3 Department of Mechanical Engineering, School of Engineering, The University
of Tokyo, Tokyo, Japan
4Fachbereich Physik, Philipps-Universität Marburg, D-35032 Marburg, Germany
5Department of Physics, The Chinese University of Hong Kong, Shatin, Hong
Kong, China

(1996; ?? and in revised form ??)

###### Abstract

The shape of velocity and temperature profiles near the horizontal conducting
plates in turbulent Rayleigh-Bénard convection are studied numerically and
experimentally over the Rayleigh number range $10^{8}\lesssim Ra\lesssim
3\times 10^{11}$ and the Prandtl number range $0.7\lesssim Pr\lesssim 5.4$.
The results show that both the temperature and velocity profiles well agree
with the classical Prandtl-Blasius laminar boundary-layer profiles, if they
are re-sampled in the respective dynamical reference frames that fluctuate
with the instantaneous thermal and velocity boundary-layer thicknesses.

###### keywords:

Rayleigh-Bénard Convection, kinematic and thermal boundary layers, Prandtl-
Blasius boundary layer theory, turbulent thermal convection

††volume: 538

##  1 Introduction

The turbulent motion in a fluid layer sandwiched by two parallel plates and
heated from below, i.e. Rayleigh-Bénard (RB) convection, has become a fruitful
paradigm for understanding the physical nature of a wide range of complicated
convection problems occurring in nature and in engineering problems (Siggia
1994; Ahlers, Lohse $\&$ Grossmann 2009; Lohse $\&$ Xia 2010). A key issue in
the study of turbulent RB system is to understand how heat is transported
upwards by turbulent flow across the fluid layer. It is measured in terms of
the Nusselt number $Nu$, defined as $Nu=J/(\kappa\Delta/H)$, which depends on
the turbulent intensity and the fluid properties. These are characterized,
respectively, by the Rayleigh number $Ra$ and the Prandtl number $Pr$, namely
$Ra=\alpha gH^{3}\Delta/\nu\kappa$ and $Pr=\nu/\kappa$. Here $J$ is the
temperature current density across the fluid layer with a height $H$ and with
an applied temperature difference $\Delta$, $g$ the gravitational
acceleration, and $\alpha$, $\nu$, and $\kappa$ are, respectively, the thermal
expansion coefficient, kinematic viscosity, and thermal diffusivity of the
convecting fluid, for which the Oberbeck-Boussinesq approximation is
considered as valid. As heat transport is controlled by viscosity and thermal
diffusion in the immediate vicinity of the solid boundaries, $Nu$ is
intimately related to the physics of the boundary layers.

In thermal convective turbulent flow two types of boundary layers (BL) exist
near the top and bottom plates, both of which are generated and stabilized by
the viscous shear of the large-scale mean flow: One is the kinematic boundary
layer and the other is the thermal boundary layer. The two layers are not
isolated but are coupled dynamically to each other. They both play an
essential role in turbulent thermal convection, especially for the global heat
flux across the fluid layer. Almost all theories proposed to predict the
relation between $Nu$ and the control parameters $Ra$ and $Pr$ are based on
some kind of assumptions for the BLs, such as the stability assumption of the
thermal BL from the early marginal stability theory Malkus (1954), the
turbulent-BL assumption from the theories of Shraiman & Siggia (1990) and
Siggia (1994) and of Dubrulle (2001, 2002), and the Prandtl-Blasius laminar-BL
assumption of the Grossmann $\&$ Lohse (GL) theory Grossmann & Lohse (2000,
2001, 2002, 2004). Because of the complicated nature of the problem, different
theories based on different assumptions for the BL may yield the same
predictions for the global quantities, such as the $Nu$-$Ra$ scaling relation
Castaing et al. (1989); Shraiman & Siggia (1990). Therefore, direct
characterization of the BL properties is essential for the differences between
and the testing of the various theoretical models and will also provide
insight into the physical nature of turbulent heat transfer in RB system.

In the GL theory, the kinetic energy and thermal dissipation rates have been
decomposed into boundary layer and bulk contributions. Scaling wise and in a
time averaged sense a laminar Prandtl-Blasius boundary layer has been assumed.
This theory can successfully describe and predict the Nusselt and the Reynolds
number dependences on $Ra$ and $Pr$ (see e.g. the recent review in Ahlers et
al., 2009). As the Prandtl-Blasius laminar BL is a key ingredient of the GL
theory, it is important to make direct experimental verification of it. We
note that also the (experimentally verified) calculation of the mean
temperature in the bulk in both liquid and gaseous non-Oberbeck-Boussinesq RB
flows Ahlers et al. (2006, 2007, 2008) is based on the Prandtl-Blasius theory.

In a recent high-resolution measurement of the properties of the velocity
boundary layer, Sun, Cheung $\&$ Xia (2008) have found that, despite the
intermittent emission of plumes, the Prandtl-Blasius-type laminar boundary
layer description is indeed a good approximation, in a time-averaged sense,
both in terms of its scaling and its various dynamical properties. However,
because of the intermittent emissions of thermal plumes from the BLs, the
detailed dynamics of both kinematic and thermal BLs in turbulent RB flow are
much more complicated. On the one hand, direct comparison of experimental
velocity (du Puits, Resagk $\&$ Thess 2007) and numerical temperature
Shishkina & Thess (2009) profiles with theoretical predictions has shown that
both the classical Prandtl-Blasius laminar BL profile and the empirical
turbulent logarithmic profile are not good approximations for the time-
averaged velocity and temperature profiles. Furthermore, Sugiyama et al.
(2009) from two-dimensional (2D) and Stevens, Verzicco $\&$ Lohse (2010) from
three-dimensional (3D) numerical simulations found that the deviation of the
BL profile from the Prandtl-Blasius profile increases from the plate’s center
towards the sidewalls, due to the rising (falling) plumes near the sidewalls.
On the other hand, Qiu & Xia (1998) have found near the sidewall and Sun et
al. (2008) near the bottom plate that the velocity BL obeys the scaling law of
the Prandtl-Blasius laminar BL, i.e., its width scales as $\lambda_{v}/H\sim
Re^{-0.5}$, where $\lambda_{v}$ is the kinematic BL thickness, defined as the
distance from the wall at which the extrapolation of the linear part of the
local mean horizontal velocity profile $u(z)=\langle u_{x}(z,t)\rangle$, with
$z$ being the vertical distance from the bottom plate and
$\langle\cdots\rangle$ being the time average at the plate center, meets the
horizontal line passing through the maximum horizontal velocity
$[u(z)]_{max}$, and $Re$ is the Reynolds number based on $[u(z)]_{max}$. These
papers highlight the need to study the nature of the BL profiles, both
velocity and temperature, in turbulent thermal RB convection.

Considerable progress on this issue has recently been achieved by Zhou & Xia
(2009) who have experimentally studied the velocity BL for water ($Pr=4.3$)
with particle image velocimetry (PIV). They found that, since the dynamics
above and below the range of the boundary layer is different, a time-average
at a fixed height $z$ above the plate with respect to the laboratory (or
container) frame will sample a mixed dynamics, one pertaining to the BL range
and the other one pertaining to the bulk, because the measurement position
will be sometimes inside and sometime outside of the fluctuating width of the
boundary layer. To make a clean separation between the two types of dynamics,
Zhou & Xia (2009) studied the BL quantities in a time-dependent frame that
fluctuates with the instantaneous BL thickness itself. Within this dynamical
frame, they found that the mean velocity profile well agrees with the
theoretical Prandtl-Blasius laminar BL profile. In figure 1 we show the
essence of the results, again for the velocity boundary layer but for somewhat
larger $Pr$, now $Pr=5.4$. (For details of the experiment and the apparatus
used, please see Xia, Sun $\&$ Zhou 2003; Zhou $\&$ Xia 2009). Also here the
method of using the time dependent frame works as good as for the $Pr=4.3$
case of Zhou & Xia (2009). While at the large $Ra=1.8\times 10^{11}$ the time
and space averaged velocity profile (triangles) already considerably deviates
from the Prandtl-Blasius profile (solid line), the dynamically rescaled
profile (circles) perfectly agrees with the Prandtl-Blasius profile. Thus a
dynamical algorithm has been established to directly characterize the BL
properties in turbulent RB systems, which is mathematically well-defined and
requires no adjustable parameters.

Figure 1: Comparison between the spatial $x$-interval and time averaged
velocity profiles $u(z)$ (triangles), the dynamically rescaled velocity
profile $u^{*}(z^{*}_{v})$ (circles – for the notation we refer to section 3),
and the Prandtl-Blasius velocity profile (solid line) near the bottom plate
obtained experimentally at $Ra=1.8\times 10^{11}$ and $Pr=5.4$ (working fluid
water).

The questions which immediately arise are: (i) Does this dynamical rescaling
method also work for the temperature field, giving good agreement with the
(Prandtl number dependent) Prandtl-Blasius temperature profile? (ii) And does
the method also work for lower $Pr$, where the velocity field is more
turbulent? Both these questions cannot be answered with the current Hong Kong
experiments, as PIV only provides the velocity field and not the temperature
field, and as PIV has not yet been established in gaseous RB, i.e., at low
$Pr$ number Rayleigh-Bénard flows.

In the present paper we will answer these two questions with the help of
direct numerical simulations (DNS). To avoid the complications of oscillations
and rotations of the large scale convection roll plane and as the Prandtl-
Blasius theory is a 2D theory anyhow we will restrict ourselves to the 2D
simulations of Sugiyama et al. (2009). Our results will show that Zhou & Xia
(2009)’s idea of using time-dependent coordinates to disentangle the mixed
dynamics of BL and bulk works excellently also for the temperature field and
also for low $Pr$ flow. I.e., if dynamically rescaled, both velocity and
temperature BL profiles can be brought into excellent agreement with the
theoretical Prandtl-Blasius BL predictions, for both larger and lower $Pr$.

##  2 DNS of the 2D Oberbeck-Boussinesq equations

The numerical method has been explained in detail in Sugiyama et al. (2009).
In a nutshell, the Oberbeck-Boussinesq equations with no-slip velocity
boundary conditions at all four walls are solved for a 2D RB cell with a
fourth-order finite-difference scheme. The aspect ratio is $\Gamma\equiv
D/L=1.0$, the Rayleigh number $Ra=10^{8}-10^{9}$, and the Prandtl number
either $Pr=4.3$ (water) or $Pr=0.7$ (gas). Sugiyama et al. (2009) have
provided a detailed code validation.

As the governing equations are strictly Oberbeck-Boussinesq, there exists a
top-bottom symmetry. We therefore discuss only the velocity and temperature
profiles near the bottom plate. For the temperature profiles, we introduce the
non-dimensional temperature $\Theta(z,t)$, defined as

$\Theta(z,t)=\frac{\theta^{bot}-\theta(z,t)}{\Delta/2},$ (1)

where $\theta^{bot}$ is the temperature of the bottom plate. In this
definition, $\Theta(H)=2$ and $\Theta(0)=0$ are the temperatures for the top
and bottom plates, respectively, and $\Theta(H/2)=1$ is the mean bulk
temperature.

##  3 Dynamical BL rescaling

Figure 2: Examples of (a) an instantaneous horizontal velocity profile
$u(z,t)$ and (b) a normalized instantaneous temperature profile $\Theta(z,t)$,
averaged over $0.475<x/D<0.525$. The DNS data are obtained at $Ra=10^{9}$ and
$Pr=0.7$. (c) and (d) show enlarged portions of the velocity and temperature
profiles near the bottom plate, respectively. The two tilted dashed lines are
linear fits to the linear parts of the velocity and temperature profiles near
the plate and the two horizontal dashed lines mark the instantaneous maximum
horizontal velocity and the bulk temperature $\Theta=1$, respectively. The
distances of there crossing points from the plate define the instantaneous BL
thicknesses $\delta_{v,th}^{bot}(t)$. The instantaneous profiles are not top-
down symmetric, the time averaged ones are. Within our present statistical
error our data are consistent with zero thermal gradient in the bulk.

The idea of the Zhou & Xia (2009) method is to construct a dynamical frame
that fluctuates with the local instantaneous BL thickness. To do this, first
the instantaneous kinematic and thermal BL thicknesses are determined using
the algorithm introduced by Zhou & Xia (2009). To reduce data scatter, the
horizontal velocity and temperature profiles at each discrete time $t$,
$u(z,t)$ and $\Theta(z,t)$, are obtained by averaging the velocity and
temperature fields along the $x$-direction (horizontal) over the range
$0.475<x/D<0.525$. Figures 2(a) and (b) show examples of $u(z,t)$ and
$\Theta(z,t)$ versus the normalized height $z/H$, respectively, of the DNS
data obtained at $Ra=10^{9}$ and $Pr=0.7$. Both $u(z,t)$ and $\Theta(z,t)$
rise very quickly from 0 to either the instantaneous maximum velocity or to
the bulk temperature within very thin layers above the bottom plate. While
after reaching its maximum value, $u(z,t)$ slowly decreases in the bulk region
of the closed convection cell, $\Theta(z,t)$ reaches and stays nearly constant
at the bulk temperature $\Theta=1$. To see the velocity and the temperature in
the vicinity of plates more resolved, we plot the enlarged near-plate parts of
the $u(z,t)$ and $\Theta(z,t)$ profiles in figures 2 (c) and (d). One sees
that both profiles enjoy a linear portion near the plate. The instantaneous
velocity BL thickness $\delta_{v}(t)$ is then defined as the distance from the
plate at which the extrapolation of the linear part of the velocity profile
meets the horizontal line passing through the instantaneous maximum horizontal
velocity, and the instantaneous thermal BL thickness $\delta_{th}(t)$ is
obtained as the distance from the plate at which the extrapolation of the
linear part of the temperature profile crosses the horizontal line passing
through the bulk temperature. The arrows in figures 2(c) and (d) illustrate
how to determine $\delta_{v}(t)$ and $\delta_{th}(t)$ as the crossing point
distances.

With these measured $\delta_{v}(t)$ and $\delta_{th}(t)$, we can now construct
the local dynamical BL frames at the plate’s center. The time-dependent
rescaled distances $z^{*}_{v}(t)$ and $z^{*}_{th}(t)$ from the plate in terms
of $\delta_{v}(t)$ and $\delta_{th}(t)$, respectively, are defined as

$z^{*}_{v}(t)\equiv z/\delta_{v}(t)\mbox{\ \ and\ \ }z^{*}_{th}(t)\equiv
z/\delta_{th}(t).$ (2)

The dynamically time averaged mean velocity and temperature profiles
$u^{*}(z^{*}_{v})$ and $\Theta^{*}(z^{*}_{th})$ in the dynamical BL frames are
then obtained by averaging over all values of $u(z,t)$ and $\Theta(z,t)$ that
were measured at different discrete times $t$ but at the same relative
positions $z^{*}_{v}$ and $z^{*}_{th}$, respectively, i.e.,

$u^{*}(z^{*}_{v})\equiv\langle u(z,t)|z=z^{*}_{v}\delta_{v}(t)\rangle\mbox{\ \
and\ \
}\Theta^{*}(z^{*}_{th})\equiv\langle\Theta(z,t)|z=z^{*}_{th}\delta_{th}(t)\rangle.$
(3)

Figure 3: Comparison among (a) velocity profiles: dynamical frame based
$u^{*}(z^{*}_{v})$ (circles), laboratory frame based $u(z)$ (triangles), and
the Prandtl-Blasius velocity profile (solid line), and (b) the corresponding
temperature profiles: $\Theta^{*}(z^{*}_{th})$ (circles), $\Theta(z)$
(triangles), and the Prandtl-Blasius temperature profile (solid line) near the
bottom plate. All results obtained numerically at $Ra=10^{8}$ and $Pr=4.3$.
The inset of (b) shows enlarged portions of the profiles around the thermal
boundary layers’ mergers to the bulk.

We first discuss our results from the simulation performed at $Pr=4.3$, the
Prandtl number corresponding to water at 40 ∘C. Figure 3(a) shows the
$u^{*}(z^{*}_{v})$ profile (circles), normalized by its maximum value
$[u^{*}(z^{*}_{v})]_{max}$, obtained at $Ra=10^{8}$. For comparison, we also
plot in the figure the time-averaged horizontal velocity profile $u(z)$
($=\langle u(z,t)\rangle$) (triangles), obtained from the same simulation. The
solid line represents the Prandtl-Blasius velocity BL profile, the initial
slope of which is matched to that of the measured profiles (cf. Ahlers et al.,
2006). For the range $z^{*}_{v}\lesssim 2$ the $u^{*}(z^{*}_{v})$ profile
obtained in the dynamical frame agrees well with the Prandtl-Blasius profile,
while the time-averaged $u(z)$ profile obtained in the laboratory frame
obviously is much lower than the Prandtl-Blasius profile in the region around
a few kinematic BL widths. - Note that for $z^{*}_{v}\gtrsim 2$ the
$u^{*}(z^{*}_{v})$ profile deviates gradually from the Prandtl-Blasius profile
because $u^{*}(z^{*}_{v})$ decreases in the bulk region of the closed
convection system down to 0 in the center and then changes sign. The Prandtl-
Blasius profile, instead, describes the situation of an asymptotically
constant, nonzero flow velocity. - These DNS results are similar to those
found experimentally in a rectangular cell Zhou & Xia (2009).

Figure 3(b) shows a direct comparison among the temperature profiles obtained
from the same simulation: the dynamical frame based $\Theta^{*}(z^{*}_{th})$
(circles), the laboratory frame time-averaged temperature profile $\Theta(z)$
($=\langle\Theta(z,t)\rangle$) (triangles), and the Prandtl-Blasius
temperature profile. At first glance both the $\Theta^{*}(z^{*}_{th})$ and
$\Theta(z)$ profiles are consistent with the Prandtl-Blasius thermal profile.
However, looking more carefully at the region around the thermal BL to bulk
merger (the inset of figure 3(b)), one notes that the $\Theta^{*}(z^{*}_{th})$
profile obtained in the dynamical frame is significantly closer to the
Prandtl-Blasius profile than the time-averaged $\Theta(z)$ profile obtained in
the laboratory frame, indicating that the dynamical frame idea of Zhou & Xia
(2009) works also for the thermal BL. Taken together, figures 3(a) and (b)
illustrate that both the kinematic and the thermal BLs in turbulent RB
convection are of Prandtl-Blasius type, which is a key assumption of the GL
theory Grossmann & Lohse (2000, 2001, 2002, 2004), and the dynamical frame
idea of Zhou & Xia (2009) can achieve a clean separation for both temperature
and velocity fields between their BL and bulk dynamics.

Figure 4: Comparison between (a) velocity profiles: dynamical
$u^{*}(z^{*}_{v})$ (circles), laboratory $u(z)$ (triangles), and the Prandtl-
Blasius laminar velocity profile (solid line), and (b) temperature profiles:
dynamical $\Theta^{*}(z^{*}_{th})$ (circles), laboratory $\Theta(z)$
(triangles), and the Prandtl-Blasius laminar temperature profile (solid line)
near the bottom plate, all obtained numerically at $Ra=10^{9}$ and $Pr=0.7$,
representative for gases.

We next turn to the simulation performed at $Pr=0.7$, a Prandtl number typical
for gases, which is relevant in all atmospheric processes and many technical
applications. Figures 4(a) and (b) show direct comparison between the
temperature and velocity profiles, respectively, at $Ra=10^{9}$. Again, around
the BL-bulk merger range the laboratory frame time-averaged profiles are found
to be obviously lower than the Prandtl-Blasius profile. This once more
indicates that the time-averaged BL quantities obtained in the laboratory
frame are contaminated by the mixed dynamics inside and outside the
fluctuating BLs. On the other hand, within the dynamical frame, both
$u^{*}(z^{*}_{v})$ and $\Theta^{*}(z^{*}_{th})$ are found to agree pretty well
with the Prandtl-Blasius laminar BL profiles, indicating that the dynamical
frame idea works also for the turbulent RB system with working fluids whose
Prandtl numbers are of the same order as those for gases.

##  4 Shape factors of the velocity and temperature profiles

Let us now quantitatively compare the differences between the Prandtl-Blasius
profile and the profiles obtained from both simulations and experiments for
various $Ra$ and various $Pr$. The shapes of the velocity and temperature
(thermal) profiles, labeled by $i=v$ or $i=th$, can be characterized
quantitatively by their shape factors $H_{i}$, defined as Schlichting &
Gersten (2004),

$H_{i}=\frac{\lambda^{d}_{i}}{\lambda^{m}_{i}}~{},~{}~{}~{}i=v,th.$ (4)

$\lambda^{d}_{i}$ and $\lambda^{m}_{i}$ denote, respectively, the displacement
and the momentum thicknesses of the profile, namely,

$\lambda^{d}_{i}=\int_{0}^{\infty}\\{1-\frac{Y(z)}{[Y(z)]_{max}}\\}dz\mbox{\ \
and\ \
}\lambda^{m}_{i}=\int_{0}^{\infty}\\{1-\frac{Y(z)}{[Y(z)]_{max}}\\}\\{\frac{Y(z)}{[Y(z)]_{max}}\\}dz.$
(5)

Here $Y(z)=u(z)$ is the velocity profile if $i=v$ and $Y(z)=\Theta(z)$ the
thermal profile if $i=th$. The deviation of these profiles from the Prandtl-
Blasius profile is then measured by

$\delta H_{i}=H_{i}-H^{PB}_{i},$ (6)

where $H^{PB}_{i}$ is the shape factor for the respective Prandtl-Blasius
laminar BL profile. If a given profile exactly matches the Prandtl-Blasius
profile, $\delta H_{i}$ is zero. Note that the Prandtl-Blasius velocity
profile shape factor $H^{PB}_{v}=2.59$ is independent of $Pr$, while the
thermal Prandtl-Blasius BL profile shape factor $H^{PB}_{th}$ varies with
$Pr$.

Figure 5: (a) The shape factors for the thermal (solid line) and velocity
(dashed line) Prandtl-Blasius BL profiles as function of $Pr$. The asymptotic
value $H^{PB}_{th}(Pr\gg 1)$ is $2.61676...$ and $H^{PB}_{v}=2.59$. Both
Prandtl-Blasius BL profiles for the velocity and for the temperature coincide
for $Pr=1$. (b) The thermal Prandtl-Blasius BL profiles for three (four) $Pr$
numbers and the reference linear and exponential profiles, see (7); in the
figure’s resolution the thermal profile for $Pr=100$ is indistinguishable from
that of $Pr=1$. Note that the shape factor of the thermal Prandtl-Blasius BL
profile decreases with decreasing $Pr$ due to the slower approach to its
asymptotic level $1$.

Figure 5(a) shows the shape factors $H_{i}(Pr)$ of the thermal and the
velocity Prandtl-Blasius BL profiles as functions of $Pr$ and figure 5(b)
shows the corresponding thermal profiles as functions of $z^{*}_{th}$ for
three different $Pr$. Note that the Prandtl-Blasius velocity BL profile is
identical to the thermal one for $Pr=1$. The two figures show that the thermal
shape factor $H_{th}^{PB}$ decreases with decreasing $Pr$. We attribute this
to the decrease of the temperature profiles in the BL range and the
corresponding increase of the tails for lower $Pr$. Thus we expect that the
slower approach to the asymptotic height $1$ of the thermal profiles in the
laboratory frame in figures 3 and 4 should lead to a negative deviation of
their $H_{th}$’s from the respective Prandtl-Blasius values, cf. figure 6. In
contrast, a positive $\delta H_{i}$ is obtained if the profile runs to its
asymptotic level faster than the Prandtl-Blasius profile. To see this more
clearly, we have plotted in figure 5(b) also two extreme cases, the linear and
the exponential profiles. Using (5) one calculates the shape factor $3$ for
the linear profile $\Theta^{*}(z^{*}_{th})=\mbox{min}(1,z^{*}_{th})$ and the
shape factor $2$ for the exponential one
$\Theta^{*}(z^{*}_{th})=1-\mbox{exp}(-z^{*}_{th})$. The $H$-decreasing effect
by lowering the profile can also be demonstrated by analyzing some profiles
analytically. Using a combination of exponential profiles,

$\Theta^{*}(z^{*}_{th})=0.5(1-\mbox{exp}(-(1-n)(z^{*}_{th}))+0.5(1-\mbox{exp}(-(1+n)z^{*}_{th})),$
(7)

with $0\leq n<1$ one can evaluate, using (5), that the shape factor for small
$n$ is

$H(n)\approx H(n=0)-n^{2}=2-n^{2}.$ (8)

As is shown in figure 5(b) the profile for $n>0$ is below the profile for
$n=0$. This analytical example again reflects what we found as the
characteristic difference between the laboratory frame profiles as compared to
the dynamical frame profiles.

Figure 6: The $Ra$-dependence of the deviations of the profile shape factors
from the respective Prandtl-Blasius shape factors. (a) Laboratory frame
$\delta H_{v}$ (open symbols) and dynamical frame $\delta H^{*}_{v}$ (solid
symbols); (b) laboratory frame $\delta H_{th}$ (open symbols) and dynamical
frame $\delta H^{*}_{th}$ (solid symbols); all from simulations performed at
$Pr=0.7$ (circles), $Pr=4.3$ (triangles), and from experiments at $Pr=5.4$
(squares).

Figure 6(a) shows the velocity shape factor deviations $\delta H_{v}$ (open
symbols) and $\delta H^{*}_{v}$ (solid symbols) as obtained from simulations
at $Pr=0.7$ (circles) and $Pr=4.3$ (triangles) as well as from experiments at
$Pr=5.4$ (squares). Here, $\delta H_{v}$ is calculated with the time averaged
profile $u(z)$ in the laboratory frame, while $\delta H^{*}_{v}$ is calculated
with the dynamical, time-dependent frame profile $u^{*}(z^{*}_{v})$. The
laboratoty frame based deviations turn out to be definitely smaller than zero.
In contrast, the shape factor deviations $\delta H^{*}_{v}$ for the dynamical
frame profiles obviously are much closer to zero. A similar result is found
for the thermal BLs: Figure 6(b) shows $\delta H_{th}$ (open symbols) and
$\delta H^{*}_{th}$ (solid symbols), versus $Ra$, for the same $Pr$ number
simulations. Again $\delta H^{*}_{th}$ is nearly zero, whereas $\delta H_{th}$
is significantly off. Thus these quantitative deviation measures again
indicate that the algorithm using the dynamical coordinates can effectively
disentangle the mixed dynamics inside and outside the fluctuating BLs.

##  5 Conclusions

In summary, we have studied the velocity and temperature BL profiles in
turbulent RB convection both numerically and experimentally. We extended
previous results to different Prandtl numbers and in particular to thermal
BLs. The results show that both the velocity and the temperature BLs (at least
in the plates’ center region) are of laminar Prandtl-Blasius type in the co-
moving dynamical frame in turbulent thermal convection for the parameter
ranges studied. However, the fluctuations of the BL widths, induced by the
fluctuations of the large-scale mean flow and the emissions of thermal plumes,
cause measuring probes at fixed heights above the plate to sample a mixed
dynamics, one pertaining to the BL range and the other one pertaining to the
bulk. This is the reason why the time-averaged velocity and temperature
profiles measured in previous work in fixed laboratory (RB cell) frames
deviate from the Prandtl-Blasius profiles. To disentangle that mixed dynamics,
we constructed a dynamical BL frame that fluctuates with the instantaneous BL
thicknesses. Within this dynamical frame, both velocity and temperature
profiles are very well consistent with the classical Prandtl-Blasius laminar
BL profiles, both for lower and larger $Pr$ (from $0.7$ to $5.4$). We have
thus validated the idea and algorithm of using dynamical coordinates over a
range of $Pr$ and $Ra$ for both kinematic and thermal BLs and have shown that
the Prandtl-Blasius laminar BL profile is a valid description for the BLs of
both velocity and temperature in turbulent thermal convection. Laminar
Prandtl-Blasius BL theory in turbulent RB thermal convection has thus turned
out to indeed be valid not only scaling wise, but also in the time average as
seen from the dynamical frame, co-moving with the local, instantaneous BL
widths.

###### Acknowledgements.

We gratefully acknowledge support of this work by the Natural Science
Foundation of Shanghai (No. 09ZR1411200), “Chen Guang” project (No.
09CG41)(Q.Z.), by the Research Grants Council of Hong Kong SAR (Nos.
CUHK403806 and 403807) (K.Q. X), and by the research programme of FOM, which
is financially supported by NWO (R.J.A.M.S. and D.L.).

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