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

Abstract

The shapes of the velocity and temperature profiles near the horizontal conducting plates' centre regions in turbulent Rayleigh–Bénard convection are studied numerically and experimentally over the Rayleigh number range 108 ≲ Ra ≲ 3 × 1011 and the Prandtl number range 0.7 ≲ Pr ≲ 5.4. The results show that both the temperature and velocity profiles agree well with the classical Prandtl–Blasius (PB) 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. The study further shows that the PB boundary layer in turbulent thermal convection not only holds in a time-averaged sense, but is most of the time also valid in an instantaneous sense.

Full text

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arXiv:1002.1339v1 [physics.flu-dyn] 6 Feb 2010
Under consideration for publication in J. Fluid Mech.
1
Prandtl-Blasius temperature and velocity
boundary layer profiles in turbulent
Rayleigh-B´ enard convection
Quan ZHOU1, Richard J. A. M. STEVENS2, Kazuyasu
SUGIYAMA2,3, Siegfried GROSSMANN4, Detlef LOHSE2,
AND 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
3Department of Mechanical Engineering, School of Engineering, The University of Tokyo,
Tokyo, Japan
4Fachbereich Physik, Philipps-Universit¨ at Marburg, D-35032 Marburg, Germany
5Department of Physics, The Chinese University of Hong Kong, Shatin, Hong Kong, China
(Received ?? and in revised form ??)
The shape of velocity and temperature profiles near the horizontal conducting plates in
turbulent Rayleigh-B´ enard convection are studied numerically and experimentally over
the Rayleigh number range 108? Ra ? 3 × 1011and the Prandtl number range 0.7 ?
Pr ? 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.
Key Words: Rayleigh-B´ enard Convection, kinematic and thermal boundary layers,
Prandtl-Blasius boundary layer theory, turbulent thermal convection
1. Introduction
The turbulent motion in a fluid layer sandwiched by two parallel plates and heated
from below, i.e. Rayleigh-B´ enard (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/(κ∆/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 = αgH3∆/νκ and Pr =
ν/κ. Here J is the temperature current density across the fluid layer with a height H and
with an applied temperature difference ∆, g the gravitational acceleration, and α, ν, and
κ 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

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2
Q. Zhou et al.
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 decom-
posed 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 suc-
cessfully 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 ve-
locity (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 side-
walls, 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 λv/H ∼ Re−0.5, where λvis 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) = ?ux(z,t)?, with z being the vertical distance from the bottom plate and
?···? being the time average at the plate center, meets the horizontal line passing through

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Prandtl-Blasius temperature and velocity BL profiles in turbulent RBC
u*(z*υ )/[u*(z*υ )]max , u(z)/[u(z)]max
z
υ
3
z*, z/λth, η
z
/ ,
0123
0.0
0.2
0.4
0.6
0.8
1.0
u*(z*υ )
u(z)
PB profile

bot
υ
λ
*

Figure 1. Comparison between the spatial x-interval and time averaged velocity profiles u(z)
(triangles), the dynamically rescaled velocity profile u∗(z∗
section 3), and the Prandtl-Blasius velocity profile (solid line) near the bottom plate obtained
experimentally at Ra = 1.8 × 1011and Pr = 5.4 (working fluid water).
v) (circles – for the notation we refer to
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 × 1011the 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.
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-B´ enard flows.

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4
Q. Zhou et al.
In the present paper we will answer these two questions with the help of direct numer-
ical 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 Γ ≡ D/L = 1.0, the Rayleigh number Ra = 108− 109, 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 bot-
tom plate. For the temperature profiles, we introduce the non-dimensional temperature
Θ(z,t), defined as
Θ(z,t) =θbot− θ(z,t)
∆/2
, (2.1)
where θbotis the temperature of the bottom plate. In this definition, Θ(H) = 2 and
Θ(0) = 0 are the temperatures for the top and bottom plates, respectively, and Θ(H/2) =
1 is the mean bulk temperature.
3. Dynamical BL rescaling
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 Θ(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 Θ(z,t) versus the normalized
height z/H, respectively, of the DNS data obtained at Ra = 109and Pr = 0.7. Both
u(z,t) and Θ(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, Θ(z,t) reaches and stays nearly constant at the bulk temperature Θ = 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 Θ(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 δv(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 δth(t) is obtained as the distance from the plate at which the extrapolation

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Prandtl-Blasius temperature and velocity BL profiles in turbulent RBC
5
z/H
0.000.020.04
u(z, t) (m/s)
0.00
0.05
0.10
z/H
0.00.2 0.40.6 0.8 1.0
u(z, t) (m/s)
−0.10
−0.05
0.00
0.05
0.10
z/H
0.0 0.20.4 0.6 0.81.0
Θ (z,t)
0.0
0.5
1.0
1.5
2.0
z/H
0.000.02 0.04
Θ (z,t)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
(a)
(b)
(c)(d)

Ht
bot
/ )(
υ δ
Ht
bot
th
/ )(
δ
Figure 2. Examples of (a) an instantaneous horizontal velocity profile u(z,t) and (b) a normal-
ized instantaneous temperature profile Θ(z,t), averaged over 0.475 < x/D < 0.525. The DNS
data are obtained at Ra = 109and 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 Θ = 1, respectively. The distances of there crossing points from the plate define the
instantaneous BL thicknesses δbot
the time averaged ones are. Within our present statistical error our data are consistent with
zero thermal gradient in the bulk.
v,th(t). The instantaneous profiles are not top-down symmetric,
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
δv(t) and δth(t) as the crossing point distances.
With these measured δv(t) and δth(t), we can now construct the local dynamical BL
frames at the plate’s center. The time-dependent rescaled distances z∗
the plate in terms of δv(t) and δth(t), respectively, are defined as
v(t) and z∗
th(t) from
z∗
v(t) ≡ z/δv(t) and z∗
th(t) ≡ z/δth(t). (3.1)
The dynamically time averagedmean velocity and temperature profiles u∗(z∗
in the dynamical BL frames are then obtained by averaging over all values of u(z,t) and
Θ(z,t) that were measured at different discrete times t but at the same relative positions
z∗
vand z∗
th, respectively, i.e.,
v) and Θ∗(z∗
th)
u∗(z∗
v) ≡ ?u(z,t)|z = z∗
vδv(t)? and Θ∗(z∗
th) ≡ ?Θ(z,t)|z = z∗
thδth(t)?. (3.2)
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∗
normalized by its maximum value [u∗(z∗
v) profile (circles),
v)]max, obtained at Ra = 108. For comparison,

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6
Q. Zhou et al.

z*, z/λth, η
z
υ
/ ,
0123
u*(z*υ )/[u*(z*υ )]max , u(z)/[u(z)]max
0.0
0.2
0.4
0.6
0.8
1.0
u*(z*υ )
u(z)
PB profile
z*, z/λth, η
/ ,
0123
Θ *(z*th
), Θ (z)
0.0
0.2
0.4
0.6
0.8
1.0
1.01.52.0 2.5
0.9
1.0
(a)(b)

bot
υ
z
λ
*

bot
thth zz
λ
*

Figure 3. Comparison among (a) velocity profiles: dynamical frame based u∗(z∗
laboratory frame based u(z) (triangles), and the Prandtl-Blasius velocity profile (solid line),
and (b) the corresponding temperature profiles: Θ∗(z∗
Prandtl-Blasius temperature profile (solid line) near the bottom plate. All results obtained
numerically at Ra = 108and Pr = 4.3. The inset of (b) shows enlarged portions of the profiles
around the thermal boundary layers’ mergers to the bulk.
v) (circles),
th) (circles), Θ(z) (triangles), and the
we also plot in the figure the time-averaged horizontal velocity profile u(z) (= ?u(z,t)?)
(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∗
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∗
the u∗(z∗
v) profile deviates gradually from the Prandtl-Blasius profile because u∗(z∗
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 Θ∗(z∗
time-averaged temperature profile Θ(z) (= ?Θ(z,t)?) (triangles), and the Prandtl-Blasius
temperature profile. At first glance both the Θ∗(z∗
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 Θ∗(z∗
obtained in the dynamical frame is significantly closer to the Prandtl-Blasius profile
than the time-averaged Θ(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.
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 pro-
files, respectively, at Ra = 109. Again, around the BL-bulk merger range the laboratory
frame time-averaged profiles are found to be obviously lower than the Prandtl-Blasius
v? 2 the u∗(z∗
v) profile obtained in the
v? 2
v)
th) (circles), the laboratory frame
th) and Θ(z) profiles are consistent with
th) profile

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Prandtl-Blasius temperature and velocity BL profiles in turbulent RBC
7

z*, z/λth, η
/ ,
012

3
Θ *(z*th
), Θ (z)
0.0
0.2
0.4
0.6
0.8
1.0
Θ *(z*th
Θ (z)
PB profile
)
z*, z/λv, η
z
υ
/ ,
012

3
u*(z*υ )/[u*(z*υ )]max , u(z)/[u(z)]max
0.0
0.2
0.4
0.6
0.8
1.0
u*(z*υ )
u(z)
PB profile
(a)(b)

bot
υ
z
λ
*
bot
thth zz
λ
*
Figure 4. Comparison between (a) velocity profiles: dynamical u∗(z∗
(triangles), and the Prandtl-Blasius laminar velocity profile (solid line), and (b) temperature
profiles: dynamical Θ∗(z∗
nar temperature profile (solid line) near the bottom plate, all obtained numerically at Ra = 109
and Pr = 0.7, representative for gases.
v) (circles), laboratory u(z)
th) (circles), laboratory Θ(z) (triangles), and the Prandtl-Blasius lami-
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 fluctu-
ating BLs. On the other hand, within the dynamical frame, both u∗(z∗
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.
v) and Θ∗(z∗
th) are
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 Hi, defined as
(Schlichting & Gersten 2004),
Hi=λd
i
λm
i
,i = v,th. (4.1)
λd
profile, namely,
iand λm
i denote, respectively, the displacement and the momentum thicknesses of the
λd
i=
?∞
0
{1 −
Y (z)
[Y (z)]max}dz and λm
i=
?∞
0
{1 −
Y (z)
[Y (z)]max}{
Y (z)
[Y (z)]max}dz.(4.2)
Here Y (z) = u(z) is the velocity profile if i = v and Y (z) = Θ(z) the thermal profile if
i = th. The deviation of these profiles from the Prandtl-Blasius profile is then measured
by
δHi= Hi− HPB
where HPB
i
is the shape factor for the respective Prandtl-Blasius laminar BL profile. If
a given profile exactly matches the Prandtl-Blasius profile, δHi is zero. Note that the
Prandtl-Blasius velocity profile shape factor HPB
thermal Prandtl-Blasius BL profile shape factor HPB
Figure 5(a) shows the shape factors Hi(Pr) of the thermal and the velocity Prandtl-
Blasius BL profiles as functions of Pr and figure 5(b) shows the corresponding thermal
i
,(4.3)
v
= 2.59 is independent of Pr, while the
varies with Pr.
th

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8
Q. Zhou et al.
Pr
10−2
10−1
100
101
102
Hυ
PB
, Hth
PB
2.45
2.50
2.55
2.60
2.65
Hυ
Hth
PB
PB
z*th
0123
Θ *(z*th
)
0.0
0.2
0.4
0.6
0.8
1.0
Pr = 0.01 (Hth
Pr = 0.1 (Hth
Pr = 1 (Hth
Eq. (4.4) (n = 0) (H = 2)
Eq. (4.4) (n = 0.3) (H = 1.91)
Linear (H = 3)
PB = 2.45)
PB = 2.52)
PB = 2.59)
(a)(b)

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 HPB
and HPB
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 (4.4); 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.
th(Pr ≫ 1) is 2.61676...
profiles as functions of z∗
BL profile is identical to the thermal one for Pr = 1. The two figures show that the ther-
mal shape factor HPB
th
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 nega-
tive deviation of their Hth’s from the respective Prandtl-Blasius values, cf. figure 6. In
contrast, a positive δHiis 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 (4.2) one calculates the
shape factor 3 for the linear profile Θ∗(z∗
th) = min(1,z∗
exponential one Θ∗(z∗
th) = 1−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,
thfor three different Pr. Note that the Prandtl-Blasius velocity
th) and the shape factor 2 for the
Θ∗(z∗
th) = 0.5(1 − exp(−(1 − n)(z∗
th)) + 0.5(1 − exp(−(1 + n)z∗
th)),(4.4)
with 0 ? n < 1 one can evaluate, using (4.2), that the shape factor for small n is
H(n) ≈ H(n = 0) − n2= 2 − n2.(4.5)
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(a) shows the velocity shape factor deviations δHv (open symbols) and δH∗
(solid symbols) as obtained from simulations at Pr = 0.7 (circles) and Pr = 4.3 (trian-
gles) as well as from experiments at Pr = 5.4 (squares). Here, δHv is calculated with
the time averaged profile u(z) in the laboratory frame, while δH∗
dynamical, time-dependent frame profile u∗(z∗
turn out to be definitely smaller than zero. In contrast, the shape factor deviations δH∗
for the dynamical frame profiles obviously are much closer to zero. A similar result is
found for the thermal BLs: Figure 6(b) shows δHth (open symbols) and δH∗
v
vis calculated with the
v). The laboratoty frame based deviations
v
th(solid

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Prandtl-Blasius temperature and velocity BL profiles in turbulent RBC
9
Ra
108
109
1010
1011
δH*υ , δHυ
−1.0
−0.5
0.0
0.5
Ra
108
109
δH*th , δHth
−0.6
−0.4
−0.2
0.0
0.2
(a)(b)
linear profile
exponential profile

Figure 6. The Ra-dependence of the deviations of the profile shape factors from the respective
Prandtl-Blasius shape factors. (a) Laboratory frame δHv (open symbols) and dynamical frame
δH∗
(solid symbols); all from simulations performed at Pr = 0.7 (circles), Pr = 4.3 (triangles), and
from experiments at Pr = 5.4 (squares).
v (solid symbols); (b) laboratory frame δHth (open symbols) and dynamical frame δH∗
th
symbols), versus Ra, for the same Pr number simulations. Again δH∗
whereas δHthis significantly off. Thus these quantitative deviation measures again indi-
cate that the algorithm using the dynamical coordinates can effectively disentangle the
mixed dynamics inside and outside the fluctuating BLs.
this nearly zero,
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.
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.

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Q. Zhou et al.
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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