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# Viscous boundary layer properties in turbulent thermal convection in a
cylindrical cell: the effect of cell tilting

PINGWEIANDKE-QINGXIA  Department of Physics, The Chinese University of Hong
Kong, Shatin, China

(?; revised ?; accepted ?. - To be entered by editorial office)

###### Abstract

We report an experimental study of the properties of the velocity boundary
layer in turbulent Rayleigh-Bénard convection in a cylindrical cell. The
measurements were made at Rayleigh numbers $Ra$ in the range $2.8\times
10^{8}<Ra<5.6\times 10^{9}$ and were conducted with the convection cell tilted
with an angle $\theta$ relative to gravity, at $\theta=0.5^{o}$, $1.0^{o}$,
$2.0^{o}$, and $3.4^{o}$, respectively. The fluid was water with Prandtl
number $Pr=5.3$.

It is found that at small tilt angles ($\theta\leq 1^{o}$), the measured
viscous boundary layer thickness $\delta_{v}$ scales with the Reynolds number
$Re$ with an exponent close to that for a Prandtl-Blasius laminar boundary
layer, i.e. $\delta_{v}\sim Re^{-0.46\pm 0.03}$. For larger tilt angles, the
scaling exponent of $\delta_{v}$ with $Re$ decreases with $\theta$. The
normalized mean horizontal velocity profiles measured at the same tilt angle
but with different $Ra$ are found to have an invariant shape. But for
different tilt angles, the shape of the normalized profiles is different.

It is also found that the Reynolds number $Re$ based on the maximum mean
horizontal velocity scales with $Ra$ as $Re\sim Ra^{0.43}$ and the Reynolds
number $Re_{\sigma}$ based on the maximum rms velocity scales with $Ra$ as
$Re_{\sigma}\sim Ra^{0.55}$, with both exponents do not seem to depend on the
tilt angle $\theta$.

Several wall quantities are also measured directly and their dependency on
$Re$ are found to agree well with those predicted for a classical laminar
boundary layer. These are the wall shear stress $\tau$ ($\sim Re^{1.46}$), the
viscous sublayer $\delta_{w}$ ($\sim Re^{0.75}$), the friction velocity
$u_{\tau}$ ($\sim Re^{-0.86}$) and the skin friction coefficient $c_{f}$
($\sim Re^{-0.46}$). Again, all these near-wall quantities do not seem to
depend on the tilt angle.

We also examined the dynamical scaling method proposed bys Zhou and Xia [Phys.
Rev. Lett. 104, 104301 (2010)] and found that in both the laboratory and the
dynamical frames the mean velocity profiles show deviations from the
theoretical Prandtl-Blasius profile, with the deviations increase with $Ra$.
But profiles obtained from dynamical scaling in general have better agreement
with the theoretical profile. It is also found that the effectiveness of this
method appears to be independent of $Ra$.

##  1 Introduction

###  1.1 Rayleigh-Bénard convection

Rayleigh-Bénard (RB) convection, which is a fluid layer heated from below and
cooled from the top, is an idealized model to study turbulent flows involving
heat transport and has attracted much attention during the past few decades
(Siggia, 1994; Kadanoff, 2001; Ahlers, Grossmann & Lohse, 2009; Lohse & Xia,
2010). The system is characterized by two control parameters: the Rayleigh
number $Ra$, Prandtl number $Pr$, which are defined as

$Ra=\frac{\alpha g\Delta TH^{3}}{\nu\kappa},$ (1)

and

$Pr=\frac{\nu}{\kappa},$ (2)

respectively. Here $\alpha$ is the thermal expansion coefficient, $g$ the
gravitational acceleration, $\Delta T$ the temperature difference between the
bottom and the top plates, $H$ the height of the fluid layer between the
plates, $\nu$ the kinematic viscosity, and $\kappa$ the thermal diffusivity of
the convecting fluid. In addition, the aspect ratio $\Gamma=D/H$ ($D$ is the
lateral dimension of the system) also plays an important role in the
structures and dynamics of the flow.

In a fully developed Rayleigh-Bénard turbulent flow, most of the imposed
temperature difference is localized in the thermal boundary layers near the
surface of the top and bottom plates, within which heat is transported via
conduction (Wu & Libcharber, 1991; Belmonte, Tilgner & Libchaber, 1994; Lui &
Xia, 1998). The velocity field has the same character: velocity gradient is
localized in a thin layer near the plates, which is called viscous boundary
layer. Turbulent flow in the central region of the RB cell is approximately
homogenous and isotropic(Zhou, Sun & Xia, 2008; Ni, Huang & Xia, 2011a, 2012).
As the top and bottom boundary layers contribute the main resistance to heat
transfer through the cell and thus dominantly determine the Nusselt number,
they deserve special attention. Indeed, nearly all theories in RB convection
are in essence boundary-layer (BL) theories. For example, a turbulent BL was
assumed in the early marginal stability theory (Malkus, 1954) and also in the
models by Shraiman & Siggia (1990) and Siggia (1994) and by Dubrulle (2001,
2002). On the other hand, a Prandtl-Blasius (PB) type laminar BL was
assumption in the Grossmann & Lohse (GL) theory (Grossmann & Lohse, 2000,
2001, 2002, 2004). Therefore, direct characterization of the BL properties is
essential for testing and differentiating the various theoretical models, and
will also provide insight into the physical nature of turbulent heat transfer.

###  1.2 Boundary layer measurements in turbulent thermal convection

One of the earlier measurements of temperature and also velocity profiles in
turbulent RB convection was taken by Tilgner, Belmonte & Libchaber (1993) in
water ($Pr=6.6$) at the fixed $Ra=1.1\times 10^{9}$ and at a fixed lateral
position. Belmonte, Tilgner & Libchaber (1993) extended these measurements
over the range $5\times 10^{5}\leq Ra\leq 10^{11}$ in compressed gas (air) at
room temperature (Pr=0.7), but still at fixed lateral position. Lui & Xia
(1998) measured the mean temperature profiles at various horizontal positions
on the lower plate of a cylindrical convection cell, the result shows that the
thermal layer thickness $\delta_{th}$ varied over the plate for the same $Ra$,
and the thinnest BL is closed to the center of the plate. Wang & Xia (2003)
found similar results for a cubic cell. du Puits et al. (2007b) measured high-
resolution temperature profiles in Rayleigh-Bénard convection near the top
plate of a cylindrical container with air ($Pr=0.7$) as the working fluid.
Their result shows that the thermal BL thickness $\delta_{th}\sim Ra^{-0.25}$
in the cell with $\Gamma=1.13$. Sun, Cheung & Xia (2008) found that the
thermal boundary layer thickness scales with $Ra^{-0.32}$ in a rectangular
cell, at $Pr=4.3$ and $Ra$ ranging from $10^{8}$ to $10^{10}$.

For the velocity measurement, the methods for determining the velocity
profiles near the solid walls of the cell are developed in recent years. Since
strong temperature fluctuations exist in Rayleigh-Bénard convection, the well-
established hot-wire anemometry could not be applied to this system. For the
viscous boundary layer, the large temperature fluctuations make conventional
laser Doppler velocimetry ineffective because the temperature fluctuations
cause fluctuations in the refractive index of the fluid that in turn make it
difficult to steadily focus two laser beams to cross each other in the fluid
(Xia, Xin & Tong, 1995). Tilgner et al. (1993) introduced an electrochemical
labeling method and measured the velocity profile and boundary layer thickness
near the top plate of a cubic cell filled with water, but only at a single
value of $Ra$. In a later study, Belmonte, Tilgner & Libchaber (1993, 1994)
developed an indirect method — the correspondence between the peak position of
the cutoff frequency profile of the temperature power spectrum and the peak
position of the velocity — to infer the viscous boundary boundary layer
thickness in gaseous convection. This method has subsequently been used to
infer the viscous layer in thermal convection in mercury (Naert, Segawa &
Sano, 1997). It may just be that the method works in certain situations, but
there is no theoretical basis for it. Xin, Xia & Tong (1996), using a novel
light-scattering technique developed by Xia et al. (1995), conducted the first
direct systematic measurement of velocity profiles in RB convection in a
cylindrical cell as a function of $Ra$. They found $\delta_{\upsilon}\sim
Ra^{-0.16}$ from the velocity profile above the center of the lower plate. Qiu
& Xia (1998a, b) extended these measurements to convection in cubic cells,
finding the same scaling exponent $-0.16$ at the bottom plate, but at the
sidewall a different result $\delta_{\upsilon}\sim Ra^{-0.26}$. Using various
organic liquids, Lam et al. (2002) explored the $Pr$ dependence, finding
$\delta_{\upsilon}\sim Pr^{0.24}Ra^{-0.16}$. With the measured $Ra-Re$ scaling
relationship $Re\sim Ra^{0.5}$ obtained in these studies (in fact, it was the
Peclet number $Pe=\upsilon L/\kappa$ rather than $Re$ in some of these
studies; please see Sun & Xia (2005) for more detailed discussions), the above
results imply a scaling relation $\delta_{\upsilon}\sim Re^{-0.32}$. In recent
years, the technique of particle image velocimetry (PIV) has been introduced
to the experimental study of thermal convection (Xia, Sun & Zhou, 2003; Sun,
Xi & Xia, 2005a, b; Sun & Xia, 2005). Sun et al. (2008) further applied the
PIV technique to study the viscous BL in a rectangular cell. Their results
show that $\delta_{\upsilon}\sim Ra^{-0.27}$ and $\delta_{\upsilon}\sim
Re^{-0.5}$, which showed that the viscous BL in thermal turbulence has the
same $Re$-scaling as a Prandtl-Blasius laminar BL. This result validates the
laminar BL assumption made in the GL model in a scaling sense. Thus it appears
there is a discrepancy in the measured scaling exponent of $\delta_{\upsilon}$
with respect to $Ra$ ($Re$) between those obtained in cylindrical and cubic
cells and that obtained in rectangular cells. In Sun et al. (2008) it was
argued that because of the more complicated flow dynamics of the large-scale
circulation (LSC), such as the azimuthal motion in the cylindrical cell (Sun
et al., 2005b; Brown et al., 2005; Xi et al., 2006) and the secondary flows in
the cubic cell (Qiu & Xia, 1998a), the shear flow near the plates is less
steady as compared to that in the rectangular cell which is more close to
quasi-two-dimension (quasi-2D). As the viscous boundary is created by the
shear of the LSC, this may plausibly change the BL properties, resulting in a
different exponent. However, the above argument has not been substantiated
experimentally. Part of the motivation of the present work is to determine how
the three-dimensional LSC dynamics will affect the BL properties. It is known
that titling the cell by a small angle will “lock” the LSC in a fixed
azimuthal plane in the sense that it will limit the range of the LSC’s
azimuthal meandering (Sun et al., 2005a) and reduce its azimuthal oscillation
amplitude near the top and bottom plates of the cell (Ahlers, Brown &
Nikolaenko, 2006). In the present work, we present measurements of BL
properties in a cylindrical cell with the cell titled over a range of angles.
For small titling angles, we measure a boundary layer under a more steady
shear comparing to the “leveled” case when the LSC can freely meander in the
azimuthal direction but presumably the BL is otherwise unperturbed under such
a small titling angle ($\leq 1^{o}$). We also examine how the BL scaling
exponent and other BL properties behave when the titling angle becomes not so
small ($\geq 1^{o}$), which would amount to a perturbation to the BLs.
Boundary layers play such an important role in turbulent thermal convection,
it is therefore important to examine how BLs respond to external
perturbations. Understanding the stability or instability of BLs is also
relevant to the search for the so-called ultimate state of thermal convection,
as the transition from the “classical state” to the ultimate one is
essentially an instability transition of the BL from being laminar to being
turbulent.

In addition to the scaling of the BL thickness, the shape of the velocity
profile near the top and bottom plates has attracted a lot of attention
recently. Although the BL has been found scaling wise to be of Prandtl-Blasius
type (at least in the quasi-2D case), the time-averaged velocity profiles are
found to differ with the theoretically predicted one (du Puits et al., 2007a;
Sun et al., 2008), especially for the region around the thermal BL. Recently,
Zhou & Xia (2010) have proposed a dynamic scaling method that shows that the
mean velocity profile measured in the laboratory frame can be brought into
coincidence with the theoretical Prandtl-Blasius laminar BL profile, if it is
resampled relative to the time-dependent frame that fluctuates with the
instantaneous BL thickness. This method was tested initially for the case of
velocity profile in turbulent convection in a quasi-2D rectangular cell with
water as working fluid ($Pr=4.3$). In a follow-up study using two-dimensional
DNS data, Zhou et al. (2010) found that the method is also valid for thermal
boundary layers and for the case of $Pr=0.7$ as well. More recently, these
authors further shown, again using numerical data, that the method works also
in other positions in the horizontal plate other than the central axis (Zhou
et al., 2011) and in three-dimension (3D) cylindrical cell for moderate values
of $Ra$ (Stevens et al., 2012). However, Scheel, Kim & White (2012) and Shi,
Emran & Schumacher (2012), both using numerical approaches, have found that
dynamic scaling works less well in the 3D cylindrical geometry than in the
quasi-2D case. However, the method has not been tested experimentally so far
in a 3D system. Here we would like to examine the dynamical scaling method
using the experimentally obtained instantaneous velocity profiles in our
three-dimensional cylindrical cell.

###  1.3 Organization of the paper

The remainder of this paper is organized as follows. We give detailed
descriptions of the experimental setup and measurement instrumentation in §2
and present and analyze experimental results in §3, which are divided into six
subsections. In §3.1 we present the measured temperature profiles and
corresponding position-dependent fluid properties, which will be used to
calculate the viscous and Reynolds stresses. In §3.2, the measured velocity
profiles and their characterizations are presented. In §3.3, the scaling
properties, with both $Ra$ and $Re$, of the thickness $\delta_{v}$ obtained
from the mean velocity profiles and $\delta_{\sigma}$ obtained form r.m.s.
velocity profiles, are presented and discussed. We also discuss the influence
of the cell tilting angle $\theta$ on the boundary layer scaling. In §3.4
statistical properties (r.m.s. and skewness) of the velocity field in the
boundary layer region are discussed. In §3.5 we present results of the viscous
and Reynolds shear stresses distributions in the boundary layer, and discuss
the scaling of the wall quantities. In §3.6 we test the dynamic scaling method
with respect to the measured instantaneous velocity profiles. We summarize our
findings and conclude in §4.

##  2 Experimental apparatus

###  2.1 Convection cell

The measurements were made in a cylindrical Rayleigh-Bénard convection cell,
which has been described in detail previously (Zhou & Xia, 2002; Sun et al.,
2005b; Ni et al., 2011b). Here we give only its essential features. The top
and bottom conducting plates are made of pure copper with a thin layer of
nickel to avoid oxidation. The sidewall is made of Plexiglas. To avoid
distortions in the images viewed by the camera, a square-shaped jacket is
fitted around the sidewall of the convection cell. As shown in figure 1(b),
the jacket is filled with water. The diameter and height of the cell is
$D=19.6$ cm and $H=18.6$ cm, respectively. The aspect ratio $\Gamma=D/H$ is
thus close to $1$. Two (three) thermistors are embedded in the top (bottom)
plate. The top plate temperature is maintained constant by a refrigerated
circulator (Polyscience Model 9702) that has a temperature stability of
$0.01^{o}C$. A NiChrome wire (26 Gauge, Aerocon Systems) surrounded by
fiberglass sleeving and Teflon tape is distributed inside the grooves carved
under the bottom plate. The wire is connected with five DC power supplies (GE
Model GPS-3030) in series to provide constant and uniform heating. During the
measurement, the whole cell is placed in a homemade thermostat box that is
kept at the same temperature ($30^{o}C$) as that of the fluid at the centre of
the cell. During the experiment the cell was tilted by an angle $\theta$ such
that the circulation plane of the LSC was parallel to the image plane of the
camera (the $x$-$z$ plane, see figure 1).

###  2.2 PIV measurement

The application of PIV to thermal turbulence has been described in detail in
several previous publications (Xia et al., 2003; Sun et al., 2005b, 2008).
Here we only provide details concerning the particular features of the present
experiment. The PIV system consists of one CCD camera with $2048\times 2048$
pixels, a dual pulse Nd-YAG laser with $135$ mJ per pulse, a synchronizer and
software. As the cell was titled, both the CCD and the laser light-sheet were
titled accordingly with the same angle. A 105 mm focal-length macro lens was
attached to the CCD to achieve a measuring area with size varying from
$18\times 18$ mm2 to $30\times 30$ mm2. Each 2D velocity vector is calculated
from a subwindow (32 pixels $\times$ 32 pixels) that has 50$\%$ overlap with
its neighboring subwindows, so each vector corresponds to a region of 16
pixels $\times$ 16 pixels and each velocity map contains $127\times 127$
velocity vectors in the $x$-$z$ plane (see figure 1). This corresponds to
spatial resolutions of about $0.135\times 0.135$ mm2 to $0.236\times 0.236$
mm2 for velocities $u$ and $w$ measured in the horizontal $x$ and vertical $z$
directions, respectively. For the measurement at $\theta=3.4^{o}$ particles
with diameter $2~{}\mu$m were used, while particles with diameter of
$10~{}\mu$m were used for measurements with other three tilted angles. For
each run, typically about 25200 image pairs were acquired with frame rate of
$2$ Hz.

Figure 1:  Sketch of the convection cell and the Cartesian coordinates used in
temperature and velocity measurements. (a) side view of the setup, and (b) the
top view.

Figure 2: (a) A profile of mean temperature $\langle T\rangle$ measured at
$Ra=6.8\times 10^{8}$ and $\theta=3.4^{o}$. (b) A profile of dynamic viscosity
$\mu$ obtained from the mean temperature profile in (a).

##  3 Results and discussion

PIV measurements were made at four values of the titling angle $\theta$
$=0.5^{o}$, $1^{o}$, $2^{o}$, and $3.4^{o}$. For each $\theta$, measurements
over a range of $Ra$ were made. Table 1 lists the parameters ($\theta$, $Ra$
and $Pr$) of each measurement, which typically lasted for about 3.5 hours. As
already mentioned, titling the cell by a small angle has the effect of
“locking” the LSC’s circulation plane at a fixed azimuthal angle (in reality
it restricts the angular range of the LSC’s azimuthal meandering). Thus,
measurements made with small $\theta$ are aimed at studying BL properties
under more steady shear, but the BL itself is assumed to be unperturbed
otherwise. For large values of $\theta$ we wish to examine how the BL responds
to relatively large perturbations.

$\theta$ | Ra | Pr | $U_{max}$ | $\delta_{v}$ | $\theta$ | Ra | Pr | $U_{max}$ | $\delta_{v}$
---|---|---|---|---|---|---|---|---|---
$(deg.)$ |  |  | $(mm/s)$ | $(mm)$ | $(deg.)$ |  |  | $(mm/s)$ | $(mm)$
$0.5$ | $5.77\times 10^{8}$ | 5.45 | 4.77 | 2.80 | $2.0$ | $1.34\times 10^{9}$ | 5.42 | 7.47 | 2.57
| $2.79\times 10^{8}$ | 5.41 | 3.57 | 3.40 |  | $5.85\times 10^{8}$ | 5.42 | 5.21 | 3.25
| $1.55\times 10^{9}$ | 5.39 | 7.38 | 2.44 |  | $5.53\times 10^{9}$ | 5.32 | 13.49 | 1.97
| $2.93\times 10^{9}$ | 5.39 | 9.11 | 2.13 | $3.4$ | $2.74\times 10^{8}$ | 5.67 | 4.02 | 4.54
| $4.26\times 10^{9}$ | 5.38 | 10.88 | 2.06 |  | $4.19\times 10^{8}$ | 5.66 | 4.70 | 4.23
$1.0$ | $1.68\times 10^{9}$ | 5.45 | 7.53 | 3.46 |  | $6.69\times 10^{8}$ | 5.66 | 6.00 | 3.58
| $3.19\times 10^{9}$ | 5.39 | 9.96 | 2.87 |  | $1.38\times 10^{9}$ | 5.68 | 7.59 | 2.70
| $5.54\times 10^{9}$ | 5.42 | 11.60 | 2.66 |  | $2.19\times 10^{9}$ | 5.64 | 9.40 | 2.50
| $9.46\times 10^{8}$ | 5.45 | 5.72 | 3.82 |  | $2.66\times 10^{9}$ | 5.66 | 10.20 | 2.11
| $2.40\times 10^{8}$ | 5.42 | 3.06 | 5.07 |  | $9.89\times 10^{8}$ | 5.66 | 6.47 | 3.00
| $6.00\times 10^{8}$ | 5.43 | 4.71 | 3.94 |  | $3.20\times 10^{9}$ | 5.65 | 10.84 | 2.14
$2.0$ | $2.76\times 10^{8}$ | 5.41 | 3.76 | 4.12 |  | $4.28\times 10^{9}$ | 5.63 | 12.65 | 1.89
| $2.78\times 10^{9}$ | 5.42 | 10.09 | 2.15 |  | $5.19\times 10^{9}$ | 5.56 | 13.49 | 1.84
Table 1: Control parameters of the experiment: cell tilt angle $\theta$, the
Rayleigh number $Ra$ and the Prandtl number $Pr$; and the measured maximum
horizontal velocity $U_{max}$ and viscous boundary layer thickness
$\delta_{v}$. The data are listed in chronological order.

###  3.1 Temperature profile and fluid properties

The local values of fluid properties are needed in calculating the viscous and
Reynolds shear stresses, which requires measurement of the local temperature.
Temperature profiles for the leveled case have been measured systematically by
Lui & Xia (1998) in a similar cylindrical cell. To check whether titling the
cell by a relatively large angle will change the temperature profile, we
measured one mean temperature along the central axis ($x=y=0$) of the cell at
a titling angle $\theta=3.4^{o}$ ($Ra=6.8\times 10^{8}$). The result is shown
in figure 2(a) and the dynamic viscosity corresponding to the local
temperature is shown in figure 2(b). As these results are similar to those
obtained in previous studies by Lui & Xia (1998) and Sun et al. (2008), we
will use results from those studies at similar $Ra$ in the calculations of
Reynolds stress (Sec. 3.5) and other wall quantities that require position
dependent viscosity (density).

###  3.2 Velocity profiles and the Reynolds number scaling

Figure 3:  Coarse-grained vector maps of the instantaneous (a) and time-
averaged (b) velocity field measured near the center of the bottom plate
($Ra=4.2\times 10^{8}$ with $\theta=3.4^{o}$), the velocity scale bar is in
unit of $mm/s$.

Figure 4:  Time-averaged horizontal velocity profiles measured at tilt angles
$\theta=0.5^{o}$ (a), $1.0^{o}$ (b), $2.0^{o}$ (c) and $3.4^{o}$ (d). In each
plot the corresponding value of $Ra$ decreases from top to bottom (see Table 1
for exact values).

Figure 5:  Profiles normalized by their respective maximum velocity
$U_{max}(Ra)$ and the corresponding viscous boundary layer thickness
$\delta_{v}(Ra)$ with tilt angles $\theta=0.5^{o}$ (a), $1.0^{o}$ (b),
$2.0^{o}$ (c) and $3.4^{o}$ (d). The solid line in each plot represents the
theoretical Prandtl-Blasius profile.

Figure 6:  Normalized profiles measured at different tilt angles $\theta$ but
with approximately the same value of $Ra$. Profiles in (a) have a nominal
value of $Ra=5\times 10^{8}$ and in (b) have a nominal value of $Ra=1.5\times
10^{9}$. In both figures the symbols are: inverted triangles
($\theta=0.5^{o}$); squares ($\theta=1.0^{o}$); triangles ($\theta=2.0^{o}$);
and circles ($\theta=3.4^{o}$). The solid line in each plot represents the
theoretical Prandtl-Blasius profile.

Figure 7:  (a) $Re$ based on the maximum horizontal velocity $U_{max}$, and
(b) $Re_{\sigma}$ based on the maximum velocity fluctuation $\sigma_{max}$ as
a function of $Ra$ for different tilting angles. Inverted triangles:
$\theta=0.5^{o}$; squares: $1^{o}$; triangles: $2^{o}$; circles: $3.4^{o}$.
The dashed lines in (a) represent power-law fits to the individual data sets
all with a scaling exponent $-0.43$ (see text for the fitting results). The
solid line in (b) is a power law fit to all data sets in the plot, which gives
$Re_{\sigma}=0.007Ra^{0.55\pm 0.01}$.

Figure 3 (a) shows an example of measured instantaneous velocity map and (b)
time-averaged velocity field taken over a period of $3.5$ h (corresponding to
$25200$ velocity frames), with the cell tilted at $\theta=3.4^{o}$ and at
$Ra=4.2\times 10^{8}$. In the present measurement, $x$ spans from $-8.75$ mm
to $8.75$ mm, and $z$ spans from $0$ to $17.5$ mm. From the velocity scale in
figure 3 (a) and (b), it is seen that there exist velocity bursts with values
much larger than the maximum velocity in the time-averaged velocity field. It
is found that velocity maps measured at other tilt angles have similar
features. As the mean velocity and the velocity fluctuations do not exhibit
any obvious dependence on the horizontal position $x$ over the small range of
the measurement, the quantities presented below are based on values averaged
along the $x$-direction over the width of the measuring area.

Figure 4 plots the velocity profiles for different tilt angles and various
values of $Ra$, which shows that the shapes of the profiles are rather similar
at this level of detail. Figure 5 plots normalized profiles in which $U(z)$ is
normalized by the maximum horizontal velocity $U_{max}(Ra)$ (for ease of
reference the values of $U_{max}$ are also listed in Table 1) and the distance
$z$ from the wall by the viscous boundary layer thickness $\delta_{v}(Ra)$ (to
be defined below). The figure shows that up to $2\delta_{v}$ profiles for
different $Ra$ and for the same tilt angle collapse on to a single curve quite
well (except perhaps those correspond to the largest $Ra$ for $\theta=2.0$ and
$3.4^{o}$). Note that $z\simeq 2\delta_{v}$ is around where $U$ reaches its
maximum value and beyond this position it decays toward cell centre. So this
position may be taken as the separation between the boundary layer region and
the bulk. The above results suggest that for the same tilt angle the profiles
in the boundary layer region have an invariant shape with respect to different
values of $Ra$. This result is consistent with the finding by Sun et al.
(2008). In figure 5 we also plot the theoretical Prandtl-Blasius profile. It
is seen that within the BL ($z\leq\delta_{v}$) the profiles match the
theoretical solution very well, while in the region just outside the boundary
layer where plume emissions occur, all measured profiles are generally less
steep than the Prandtl-Blasius profile. This feature is also similar to that
observed by (Zhou & Xia, 2010) and will be further discussed in Sec. 3.6. On
the other hand, it is seen from figure 5 that profiles obtained at different
$\theta$ seem to have different degrees of deviation from the Prandtl-Blasius
profile. This can be seen more clearly in figure 6 where we show two examples
in which profiles for different $\theta$ but with values of $Ra$ close to each
other are plotted together along with the theoretical PB profile. This result
suggests that the shape of the velocity profile near the plume-emission region
is modified by the tilting angle. It is also noted that the profiles measured
with $\theta=1.0^{o}$ show strong deviations from the linear dependence with
zero interception. We shall come back to this when discussing boundary layer
scalings in the next section.

Taking $U_{max}$ as the characteristic velocity of LSC, we define the Reynolds
number $Re=U_{max}H/\nu$ and plot $Re$ as a function of $Ra$ and for different
$\theta$ in figure 7(a). When fitting a power-law to the data for different
$\theta$ separately, they all produce an exponent close to 0.43. To better
compare the amplitude of $Re$ for different $\theta$, we fix the scaling
exponent at $0.43$ and fit power laws to the different data sets again. This
gives $Re=(0.185\pm 0.002,0.182\pm 0.003,0.206\pm 0.001,0.203\pm 0.001)\times
Ra^{0.43}$, where the amplitudes in the brackets are for $\theta=0.5^{o}$,
$1.0^{o}$, $2.0^{o}$, and $3.4^{o}$, respectively. These results show that in
general the values of $Re$ with larger $\theta$ are larger than those with
smaller $\theta$. In an earlier study of the effect of cell titling, Ahlers et
al. (2006) have found that $Re$ obtained indirectly from temperature
measurement increases with the tilted angle, which is consistent with the
trend observed here. We note also that the value of the scaling exponent of
$Re$ obtained from many previous studies, and sometimes under nominally
similar conditions, varies over a rather wide range from $0.43$ to $0.55$ (see
for example, Xin et al. (1996); Xin & Xia (1997); Qiu & Xia (1998a, b);
Ashkenazi & Steinberg (1999); Lam et al. (2002); Brown et al. (2007); Sun et
al. (2008); Xie et al. (2012)). The reason for such variations is not
completely clear at present. A detailed study on this issue is beyond the
scope of this paper. For interested readers, we refer to Sun & Xia (2005) who
offered an explanation that can account some of these dispersions in the
exponent.

From the measured profile of the RMS velocity (see figure 8), we can define
another Reynolds number $Re_{\sigma}=\sigma_{max}H/\nu$, which is shown in
figure 7(b) as a function of Ra in a log-log scale for the four tilt angles.
Here it is seen that $Re_{\sigma}$ does not seem to have an obvious dependence
on $\theta$. We therefore fitted a single power law to all four data sets on
the plot, which gave $Re_{\sigma}=0.007Ra^{0.55\pm 0.01}$. The value of the
exponent is somewhat larger than $0.5$ that was obtained from several previous
studies (Xin et al., 1996; Xin & Xia, 1997; Qiu & Xia, 1998a, b; Sun et al.,
2008). But given the uncertainties in the experimental measurements, it is
hard for one to attach too much significance to this difference.

###  3.3 The viscous boundary layer and its scaling with $Ra$ and $Re$

Figure 8:  Determination of the viscous boundary layer thickness $\delta_{v}$
through the slope-method from the mean horizontal velocity profile $U(z)$
(circles) and the thickness $\delta_{\sigma}$ from the standard deviation
profile $\sigma_{U}$ (crosses). The measurement was made near the bottom plate
with tilt angle $\theta=3.4^{o}$ and at $Ra=4.2\times 10^{8}$.

We define the thickness $\delta_{v}$ of the viscous boundary layer through the
“slope-method” as shown in figure 8 where a mean velocity $U(z)$ (circles)
profile and the corresponding standard deviation profile $\sigma_{u}(z)$
(crosses) are shown, which are measured at $Ra=4.2\times 10^{8}$ with
$\theta=3.4^{o}$. It is seen that $\delta_{v}$ is defined as the distance at
which the extrapolation of the linear part of $U(z)$ equals its maximum value
$U_{max}$, i.e. $\delta_{v}=U_{max}[dU/dz|_{z=0}]^{-1}$. A length scale
$\delta_{\sigma}$ can also be defined from the profile of $\sigma_{u}(z)$
where $\sigma_{u}$ reaches its maximum value. For the present example, the
values for the two boundary layer length scales $\delta_{v}$ and
$\delta_{\sigma}$ are found to be $4.20$ and $6.05$ mm, respectively. For ease
of reference, the values of $\delta_{v}$ are listed in Table 1.

Figure 9:  Measured viscous boundary layer thickness $\delta_{v}$ (a)versus
$Ra$ and (b) versus $Re$ for the four tilt angles: $\theta=0.5^{o}$ (inverted
triangles); $1.0^{o}$ (squares); $2.0^{o}$ (triangles); and $3.4^{o}$
(circles). The dashed lines are power-law fits
$\delta_{v}/H=A_{1}Ra^{\beta_{1}}$ and $\delta_{v}/H=A_{2}Re^{\beta_{2}}$ to
the respective data sets, with the fitting results listed in Table 2.

Figure 10: The scaling exponent $\beta_{2}$ versus cell tilt angle $\theta$,
where $\beta_{2}$ is obtained from the power law fit $\delta_{v}/H\sim
Re^{\beta_{2}}$. The dashed line indicates $\beta_{2}=-0.5$ for a Pradtl-
Blasius laminar boundary layer.

We now examine the scalings of the boundary layer thickness with both the
Rayleigh number $Ra$ and the Reynolds number $Re$. In figures 9(a) and (b) we
plot the measured viscous boundary layer thickness $\delta_{v}$ vs,
respectively, $Ra$ and $Re$ for the four tilt angles. The lines in the figures
represent the best power-law fits $\delta_{v}/H=A_{1}Ra^{\beta_{1}}$ and
$\delta_{v}/H=A_{2}Re^{\beta_{2}}$ to the respective data sets and the
obtained fitting parameters are listed in Table 2. Also shown in the Table for
comparison are results obtained in cells with different geometries and using
different methods. It is seen from the table that for small tilt angles
($\theta=0.5$ and $1.0^{o}$), the exponents are essentially the same and
within the experimental uncertainties the $Re$-scaling exponent may be taken
as the same as that predicted for a Prandtl-Blasius boundary layer, i.e.
$\delta_{v}\sim Re^{-1/2}$. For larger titling angles, there appears to be a
trend for both $\beta_{1}$ and $\beta_{2}$ to decrease (absolute value
increases) with increasing $\theta$. It thus appears that titling the cell by
over $1^{o}$ is a rather strong perturbation to the BL, at least for its
scaling. The situation for the amplitude of viscous boundary layer thickness
$\delta_{v}$ is a bit more complicated. From both figures 9(a) and (b) it
seems that at lower values of $Ra$ ($Re$) the BL thickness increases with
increasing tilting angle, except for $\theta=1^{o}$. For this latter titling
angle, $\delta_{v}$ appears to have an overall upward shift from the rest data
sets. While we do not know the exact reason(s) for this, we note from figure 6
that the profiles for this tilt angle seem to have a nonzero intercept on the
horizontal axis. This appears to suggest that the origin of the $z$-axis for
this was somehow shifted. But even if this is the case, the relatively small
“shift” cannot account for the large “deviation” of this $\delta_{v}$ from the
rest data sets (assuming there is indeed something “wrong” with this data
set). Aside from the amplitude, the behavior of the $Ra$\- and $Re$-scaling
exponents may be summarized as follows. For small tilting angle ($\theta\leq
1^{o}$), the effect of tilting is to lock the azimuthal plane of the LSC (or
restrict its azimuthal meandering range) but the BL is otherwise not strongly
perturbed and scaling wise the BL is approximately Prandtl-Blasius type. For
relatively large titling angle ($\theta>1^{o}$), the BL appears to be strongly
perturbed as far as scaling is concerned and the magnitude of the scaling
exponent increases with titling angle, i.e. the BL thickness $\delta_{v}$
decays with increasing $Ra$ ($Re$) with a steeper slope. The situation is
illustrated in figure 10 where $\beta_{2}$ is plotted as a function of the
tilt angle $\theta$.

Quantity | $Ra$ | Pr | Geometry | $A_{1}$ | $-\beta_{1}$ | $A_{2}$ | $-\beta_{2}$ | $\theta(^{0})$ | Source
---|---|---|---|---|---|---|---|---|---
$\delta_{v}/H$ | $10^{8}\sim 10^{10}$ | $\sim 7$ | cylin. | 0.51 | $0.16\pm 0.02$ |  | $0.32$ | 0 | a
| $10^{8}\sim 10^{10}$ | $6\sim 1027$ | cylin. | $0.65Pr^{0.24}$ | $0.16\pm 0.02$ |  | $0.32$ | 0 | b
| $10^{8}\sim 10^{10}$ | $\sim 7$ | cubic | 0.69 | $0.18\pm 0.04$ |  | $0.36$ | 0 | c
|  |  |  | 3.6 | $0.26\pm 0.03$ |  | $0.52$ | 0 | d
| $10^{9}\sim 10^{10}$ | $4.3$ | rectan. | 4.95 | $0.27\pm 0.01$ | $0.64$ | $0.50$ | 0 | e
| $10^{8}\sim 10^{9}$ | $5.4$ | cylin. | 0.745 | $0.19\pm 0.01$ | 0.369 | $0.45\pm 0.04$ | $0.5$ | f
|  |  |  | 1.41 | $0.20\pm 0.01$ | 0.564 | $0.46\pm 0.03$ | $1.0$ | f
|  |  |  | 5.86 | $0.29\pm 0.01$ | 1.41 | $0.61\pm 0.04$ | $2.0$ | f
|  |  |  | 13.3 | $0.32\pm 0.01$ | 6.26 | $0.81\pm 0.01$ | $3.4$ | f
|  |  |  | $A_{3}$ | $-\beta_{3}$ | $A_{4}$ | $-\beta_{4}$ |  |
$\delta_{\sigma}/H$ | $10^{7}\sim 10^{11}$ | $\sim 7$ | cylin. | 1.02 | $0.25\pm 0.02$ |  | $0.5$ | 0 | a
| $10^{8}\sim 10^{10}$ | $\sim 7$ | cubic | 0.95 | $0.25\pm 0.04$ |  | $0.5$ | 0 | c
|  |  |  | 43 | $0.38\pm 0.03$ |  | $1.0$ | 0 | d
| $10^{9}\sim 10^{10}$ | $4.3$ | rectan. | 16.5 | $0.37\pm 0.10$ | $0.69$ | $0.72\pm 0.14$ | 0 | e
| $10^{8}\sim 10^{9}$ | $5.4$ | cylin. | 0.58 | $0.15\pm 0.02$ | $0.14$ | $0.26\pm 0.03$ | $0.5$ | f
|  |  |  | 1.77 | $0.20\pm 0.02$ | $0.27$ | $0.37\pm 0.04$ | $1.0$ | f
|  |  |  | 2.68 | $0.23\pm 0.02$ | $0.32$ | $0.41\pm 0.04$ | $2.0$ | f
|  |  |  | 9.9 | $0.29\pm 0.02$ | $0.75$ | $0.54\pm 0.04$ | $3.4$ | f
Table 2:  Fitting results for the normalized viscous boundary layer thickness
$\delta_{v}/H$ determined from the mean horizontal velocity profile and
$\delta_{\sigma}/H$ determined from the rms horizontal velocity profile. The
fitting parameters $A_{i}$ and $\beta_{i}$ ($i=1,2,3,4$) are defined through
the power laws: $\delta_{v}/H=A_{1}Ra^{\beta_{1}}$,
$\delta_{v}/H=A_{2}Re^{\beta_{2}}$, $\delta_{\sigma}/H=A_{3}Ra^{\beta_{3}}$,
and $\delta_{\sigma}/H=A_{4}Re_{\sigma}^{\beta_{4}}$. The control parameters
$Ra$ and $Pr$ and cell geometry of measurements are also listed. Also shown in
the table are results from some previous experiments. The sources are: a. Xin
et al. (1996); b. Lam et al. (2002); c. Qiu & Xia (1998a) (bottom); d. Qiu &
Xia (1998b) (side wall); e. Sun et al. (2008); and f. present work. (Note: the
cell tilt angle $\theta$ is indicated as $0$ when it was not mentioned in the
respective papers and we assume the cell was nominally leveled in those
cases.)

Figure 11:  Scalings of the boundary layer scale $\delta_{\sigma}$ determined
from the measured rms velocity profiles: (a) versus $Ra$ and (b) versus
$Re_{\sigma}$. The symbols represent: $\theta=0.5^{o}$ (inverted triangles),
$1.0^{o}$ (squares), $2.0^{o}$ (triangles), and $3.4^{o}$ (circles). The lines
are power law fits $\delta_{\sigma}/H=A_{3}Ra^{\beta_{3}}$ and
$\delta_{\sigma}/H=A_{4}Re_{\sigma}^{\beta_{4}}$ to the respective data sets,
with the fitting results listed in Table 2.

In addition to the boundary layer thickness $\delta_{v}$ determined from the
mean horizontal velocity profile, another length scale can also be defined
based on the profile of the horizontal r.m.s. velocity $\sigma_{u}$, which may
be called the r.m.s. velocity boundary layer thickness, as defined in figure
8. In figures 11(a) and (b) we plot $\delta_{\sigma}$ versus $Ra$ and
$Re_{\sigma}$ respectively. The $Ra$-scaling exponent varies from $-0.15$ to
$-0.29$, which appears to follow similar trend as that of $\delta_{v}$, i.e.
its absolute value increases with increasing $\theta$. But it and that of
$Re_{\sigma}$-scaling exponent show significant difference with those obtained
in previous studies. Table 2 shows the fitting results of
$\delta_{\sigma}=A_{3}Ra^{\beta_{3}}$ and
$\delta_{\sigma}=A_{4}Re_{\sigma}^{\beta_{4}}$.

Now we compare our result with previous experimental results obtained in the
cells with different geometries. As shown in Table 2, the value of $\beta_{1}$
obtained in both cylindrical and cubic geometries and measured near the bottom
plate of the cell is $-0.16$. In all these previous measurements, the Reynolds
number based on the maximum horizontal velocity near the plate was also
obtained and they gave a scaling exponent $\gamma=0.5$ via $Re\sim
Ra^{\gamma}$. From this we obtain $\beta_{2}=-0.32$. In these studies, the
convection cell was nominally leveled, i.e. not intensionally tilted. In the
present study, for the small tilting angle cases, where we assume the BL is
not strongly perturbed, the measured $\beta_{1}\simeq-0.19$ when combined with
combined with $\gamma=0.43$ give a $\beta_{2}\simeq-0.45\pm 0.04$ (note that
the actual value of $\beta_{2}$ are obtained from fitting the $\delta_{v}$ vs
$Re$ data, not from the relationship between the exponents). If we take these
values to be close to the Prandtl-Blasius result, then scaling wise the
viscous BL in a cylindrical geometry is also of a Prandtl-Blasius type, as was
already found in a rectangular cell (Sun et al., 2008). For the relatively
large deviations found in the untilted case, it may be attributed to the
random azimuthal motion of the LSC.

Finally we remark that as far as the scaling of the viscous BL is concerned,
there is no theoretical prediction for the dependence of $\delta_{v}$ on $Ra$,
only that on $Re$ (for example, $\delta_{v}\sim Re^{-1/2}$ for the Prandtl-
Blasius BL). In the literature, it is sometimes stated that $\delta_{v}$
should scale as $Ra^{-1/4}$ for the Prandtl-Blasius BL. This is based on the
assumption that $Re\sim Ra^{1/2}$. From above we have seen that the scaling
exponent of $Re$ with $Ra$ varies over a rather wide range. It is therefore
more meaningful to talk about the scaling of $\delta_{v}$ with $Re$, rather
than with $Ra$. We further note that in Sun et al. (2008) it was found that
$\delta_{v}\sim Ra^{-0.27}$ and $Re\sim Ra^{0.55}$, which together give
$\delta_{v}\sim Re^{-0.50}$. In the present case, we have $\delta_{v}\sim
Ra^{-0.2}$ and $Re\sim Ra^{0.43}$, which together give $\delta_{v}\sim
Re^{-0.46\pm 0.03}$. Whether this is fortuitous or there is something deep
here remains remains to be explored.

###  3.4 Fluctuations and statistical properties of the velocity field in the
boundary layer

Figure 12: Time traces of horizontal $u(t)$ (left panels) and vertical $w(t)$
(right panels) velocity components measured at $Ra=2.4\times 10^{8}$ and
$\theta=1^{o}$, at $x=0$ and different distances $z$ from the bottom plate.

Figure 13: Histograms of (a) the horizontal velocity $u(t)$ and (b) vertical
velocity $w(t)$ measured at various distances from the bottom plate with
$Ra=2.4\times 10^{8}$ and $\theta=1^{o}$.

Figure 14: Profiles of (a) the normalized rms velocity $\sigma_{u}$
($\sigma_{w}$) and of (b) the skewness $S_{u}$ ($S_{w}$) measured at
$Ra=2.4\times 10^{8}$ and $\theta=1^{o}$. The vertical distance $z$ is
normalized by the velocity boundary layer thickness $\delta_{v}$. In both
plots the circles represent those for the horizontal velocity component $u$
and the crosses represent those for the vertical velocity component $w$.

In previous BL measurements in the cylindrical cell, owing to the nature of
the dual-beam incoherent cross-correlation technique employed (Xin et al.,
1996; Lam et al., 2002), only time-averaged velocity profiles are measured and
no time-dependent quantities are obtained. It is therefore interesting to
examine these quantities and compare them with similar quantities obtained in
other type of turbulent flows. Figure 12 shows the time series of both the
horizontal component $u(t)$ (left panel) and the vertical component $w(t)$
(right panel) of the velocity, measured at various positions from the plate.
The corresponding velocity histograms are shown in figure 13. The measurements
were made at $Ra=2.4\times 10^{8}$ and $\theta=1^{o}$. We show the velocity
trace at several typical positions: (i) inside the thermal boundary layer,
(ii) around the thermal boundary layer, (iii) around the viscous boundary
layer; (iv) at the position of the maximum velocity; and (v) far away from the
boundary layers. The figures show that the absolute horizontal velocity is
much higher than vertical velocity at each position. One general feature we
observed is that velocity time series and histograms look similar for
different tilting angles. For this reason, we show here results for only one
tilting angle.

At $Ra=2.4\times 10^{8}$, the viscous BL thickness is $\delta_{v}=5.07$ mm. It
is seen from figures 12(a) and 12(b) that at positions inside the BL, the
horizontal velocity $u(t)$ skews toward the positive side, i.e. the velocity
is skewed toward the mean flow direction. This may be understood by the fact
that close to the viscous sublayer the flow speed is very close to zero and a
fluctuation smaller than the mean would mean a flow reversal, which is a
rather rare event. Once outside of the BL, one observes more symmetric
fluctuations around the mean velocity. For the vertical velocity $w(t)$, its
mean velocity is very small at most positions. But the fluctuation increases
significantly when the position is outside of the BL, which are signatures of
plume emissions at these positions. These properties can also be seen from the
velocity histograms shown in figure 13. A notable difference of the present
results from those observed in Sun et al. (2008) is that for positions outside
of the BL the horizontal velocity fluctuates more or less symmetrically around
the mean, rather than skewed toward the negative as seen in the rectangular
cell.

The statistical properties of the velocity may be characterized more
quantitatively by its root-mean-square (r.m.s.) value and its skewness, which
are shown in figure 14. Figure 14(a) plots the velocity r.m.s $\sigma_{u}$ and
$\sigma_{w}$ normalized by maximum horizontal velocity $U_{max}$ versus the
normalized distance $z/\delta_{v}$. Figure 14(b) shows the skewness profiles
$S_{u}=\langle(u-\langle u\rangle)^{3}\rangle/(\langle(u-\langle
u\rangle)^{2}\rangle)^{3/2}$ and $S_{w}=\langle(w-\langle
w\rangle)^{3}\rangle/(\langle(w-\langle w\rangle)^{2}\rangle)^{3/2}$ for the
horizontal and vertical velocities, respectively. Similar to Sun et al.
(2008), our result could not tell whether $\sigma_{w}$ favors a power law or a
logarithmic scaling with the distance $z$, even though our measurement had a
much higher spatial resolution. This is partly due to the limited size of the
measurement area.

###  3.5 Properties of shear stresses and near-wall quantities

Figure 15:  Viscous stress $\tau_{v}$ (solid squares), the Reynolds stress
$\tau_{R}$ (open circles) and the total stress $\tau$ (open squares) as
functions of the normalized distance from the plate for (a) $Ra=4.2\times
10^{8}$ and (b) $Ra=9.9\times 10^{8}$, both with $\theta=3.4^{o}$.

Figure 16:  The Ra-dependence of (a) the wall shear stress $\tau_{w}$, (b) the
friction velocity $u_{\tau}$, (c) the wall thickness (viscous sublayer)
$\delta_{w}$, and (d) the friction coefficient $c_{f}$, with power law fits
shown as solid lines. The symbols represent data for different tilting angles:
$\theta=0.5^{o}$ (inverted triangles), $1^{o}$ (squares), $2^{o}$ (triangles),
and $3.4^{o}$ (circles).

Figure 17:  The Re-dependence of (a) the wall shear stress $\tau_{w}$, (b) the
friction velocity $u_{\tau}$, (c) the wall thickness (viscous sublayer)
$\delta_{w}$, and (d) the friction coefficient $c_{f}$. The symbols represent
data for different tilting angles: $\theta=0.5^{o}$ (inverted triangles),
$1^{o}$ (squares), $2^{o}$ (triangles), and $3.4^{o}$ (circles). Power law
fits are indicated in the figure.

Figure 18:  Measured horizontal velocity profiles normalized by wall units for
four different Ra all taken at tilting angle $\theta=3.4^{o}$.

One of the advantages of PIV measurement is that it enables one to measure the
horizontal and vertical velocities at the same time, so that one can calculate
the Reynolds shear stress $\tau_{R}=-\rho(z)\langle
u^{\prime}w^{\prime}\rangle$. Here $u^{\prime}$ and $w^{\prime}$ are the
fluctuations of the horizontal and vertical velocity components respectively,
$\rho(z)$ is the z-dependent fluid density. Viscous shear stress is defined as
$\tau_{v}=\mu(z)du/dz$, where $\mu(z)$ is the dynamic viscosity dependent on
position $z$. The Reynolds stress represents the transport of momentum by
turbulent fluctuations, whereas the viscous stress describes the momentum
transfer by viscosity. The total shear stress is then
$\tau=\tau_{v}+\tau_{R}$.

Figure 15 plots the profiles of the viscous shear stress, Reynolds stress and
total stress for (a) $Ra=4.2\times 10^{8}$, and (b) $9.9\times 10^{8}$. It is
seen that both Ra have the same qualitative features. Here the examples are
for $\beta=3.4^{o}$, and results for other tilting angles are similar. Near
the plate, it is seen that the Reynolds stress $\tau_{R}$ is close to zero,
while the viscous shear stress $\tau_{v}$ is maximum because of the large
velocity gradient $du/dz$ at the wall. So the total stress at the wall
$\tau_{w}(=\tau(0))$ comes almost entirely from the contribution of the
viscous shear stress. Moving away from the plate, the velocity gradient
becomes smaller and the viscous shear stress decreases to zero. The Reynolds
stress $\tau_{R}$ increases and attains its maximum at $z\approx
1.5\delta_{v}$. It then decreases to around zero $z\approx 2\delta_{v}$ and
becomes negative in the bulk flow. It is also seen clearly from the figure
that $\tau_{R}$ and $\tau_{v}$ cross at $z\approx 1.5\delta_{v}$, where
$\tau_{R}$ is close to its maximum value. This suggests that the momentum
transfer in the outer region is dominated by turbulent fluctuations. But in
the viscous boundary layer, the momentum transfer is still dominated by the
viscous diffusion, which implies that the viscous boundary layer is still
laminar in this range of $Ra$.

With the measured near-wall high-resolution velocity field, we are now in a
position to check the dynamic wall properties in turbulent thermal convection.
We first consider the scaling of four basic wall quantities with both $Ra$ and
$Re$. These are the wall shear stress $\tau_{w}$, the skin-friction velocity
$u_{\tau}=(\tau/\rho_{0})^{1/2}$, the viscous sublayer length scale
$\delta_{w}=\nu_{0}/u_{\tau}$, and the skin-friction coefficient
$c_{f}=\tau/\rho_{0}U_{max}^{2}$. Here $\rho_{0}\equiv\rho(z=0)$ and
$\nu_{0}\equiv\nu(z=0)$. Figure 16 shows the scaling of these quantities with
$Ra$. It is seen that within experimental uncertainties there is no difference
between data with different $\theta$. This suggests that tilting the cell does
not have any appreciable effect on BL properties near the wall. Without
differentiating the different data sets, power law fits to all data yield
$\tau_{w}\sim Ra^{0.63}$, $u_{\tau}\sim Ra^{0.32}$, $\delta_{w}\sim
Ra^{-0.37}$ and $c_{f}\sim Ra^{-0.19}$. In a rectangular cell, Sun et al
(2008) found for the same quantities the fitted power law exponents $0.86$,
$0.44$, $-0.50$, and $-0.28$ respectively. It is seen that the absolute values
of these exponents are all larger than those obtained in the present
experiment. There is no theoretical prediction for the Ra-scaling of these
quantities in turbulent thermal convection, so we do not know what the
difference means.

It will be more useful perhaps to examine the scaling of these quantities with
the Reynolds number $Re$, since theoretical predictions exist for such
scalings for wall-bounded shear flows (Schlichting & Gersten, 2000). Figure 17
plots these quantities as a function of Re, the symbols are the same as in
figure 16. For the quantities $\tau_{w}$, $u_{\tau}$, $\delta_{w}$, and
$c_{f}$ our results give the exponents $1.46$, $0.75$, $-0.86$, and $-0.46$.
For a laminar boundary layer over a flat plate, the theoretically predicted
‘classical’ exponents for these quantities are $3/2$, $3/4$, $-1$, and $-1/2$
respectively. One sees that within the experimental uncertainties there is an
excellent agreement between the present experiment and the theoretical
predictions for all the wall quantities except for $\delta_{w}$, which is a
bit smaller. For reference, the previous measurement in rectangular cell gives
$1.55$, $0.8$, $-0.91$, and $-0.34$ for the corresponding quantities (Sun et
al., 2008).

To further compare the present system with classical boundary layers, we
examine velocity profiles in terms of the wall units. Figure 18 shows the
normalized mean horizontal velocity profiles for four different values of $Ra$
taken at $\theta=3.4^{o}$ in a semi-log plot, here $u^{+}=u(z)/u_{\tau}$ and
$z^{+}=z/\delta_{w}$. The linear scaling of $u^{+}$ over $z^{+}$ in the
viscous sublayer below $z^{+}<5$ is reflected quite well by the measured
profiles confirming that the boundary layer is not turbulent in the present
range of $Ra$ and $Pr$. The velocity normalized by wall unit decrease after
reaching the maximum value in $z^{+}\sim 10$. Comparing to the same quantity
measured in the rectangular cell (Sun et al., 2008), however, our result shows
some deviation from the theoretical profile. This is a reflection of the fact
that in the cylindrical cell it is more difficult to measure the profile
accurately very close to the wall.

###  3.6 Dynamical scaling and the shape of velocity profiles in the boundary
layer

Figure 19:  Comparison between profiles obtained in the dynamical frame
($u^{*}(z^{*})$, red circles) and the laboratory frame ($u(z)$, blue squares),
measured at different tilt angles $\theta$ but with comparable values of $Ra$.
Also shown for comparison is the theoretical Prandtl-Blasius laminar velocity
profile (solid line). (a) $\theta=0.5^{o}$, $Ra=5.77\times 10^{8}$; (b)
$\theta=1.0^{o}$, $Ra=6.00\times 10^{8}$; (c) $\theta=2.0^{o}$, $Ra=5.85\times
10^{8}$; and (d) $\theta=3.4^{o}$, $Ra=6.69\times 10^{8}$.

Figure 20:  The shape factor $H=\delta_{d}/\delta_{m}$ of profiles
$u^{*}(z^{*})$ (red circles) obtained in the dynamical frame and of $u(z)$
(blue squares) obtained in the laboratory frame as a function of $Ra$ and for
different titling angles. (a) $\theta=0.5^{o}$, (b) $1.0^{o}$, (c) $2.0^{o}$,
and (d) $3.4^{o}$. The dashed line represents the value of $2.59$ for the
theoretical Prandtl-Blasius laminar BL.

Figure 21:  Examples of instantaneous horizontal velocity profiles scaled by
the instantaneous kinematic BL thickness and the instantaneous maximum
velocity (measured at $Ra=6.00\times 10^{8}$ and $\theta=1^{o}$). The
corresponding instantaneous shape factor is also indicated on the plot. The
solid curves are the Prandtl-Blasius velocity profiles.

Figure 22:  PDFs of the shape-factor difference $\delta H$ between those of
the rescaled instantaneous profiles and that of the Prandtl-Blasius profile,
measured at (a) $\theta=0.5^{o}$, (b) $1.0^{o}$, (c) $2.0^{o}$, and (d)
$3.4^{o}$.

The dynamic scaling method of Zhou & Xia (2010) has been found to work well
when tested in quasi-2D experiment and 2D numerical simulations (Zhou et al.,
2010, 2011). But it has not been examined in 3D experiments. Here we
investigate how it works in our cylindrical geometry. As the method has been
well documented elsewhere (Zhou & Xia, 2010; Zhou et al., 2010, 2011), we will
only give a brief description of it here. From the measured instantaneous
velocity profile $u(z,t)$ one can obtain an instantaneous viscous boundary
layer thickness $\delta_{v}(t)$ using the same ‘slope’ method as used for the
mean velocity profiles. A local dynamical BL frame can then be constructed by
defining the time-dependent rescaled distance $z^{*}(t)$ from the plate as

$z^{*}(t)\equiv z/\delta_{v}(t).$ (3)

The dynamically time averaged mean velocity profile $u^{*}(z^{*})$ in the
dynamical BL frame is then obtained by averaging over all values of $u(z,t)$
that were measured at different discrete times $t$ but at the same relative
position $z^{*}$, i.e.

$u^{*}(z^{*})\equiv\langle u(z,t)|z=z^{*}\delta_{v}(t)\rangle.$ (4)

Figure 19 shows the mean velocity profiles measured in the laboratory and the
dynamical frames respectively at the four tilting angles and for comparable
values of $Ra$ (as indicated in the figure caption). These results show that
the dynamical scaling method appears to be more effective for larger values of
$\theta$. This may be understood based on the fact that a larger tilt angle
will place stronger restriction on the azimuthal meandering of the LSC so that
it has less fluctuations in the horizontal direction perpendicular to the mean
flow. We note, however, regardless of the tilt angle, the method works less
effectively than it is in quasi-2D experiment and 2D simulations.

A more quantitative approach to characterize the shape of the mean velocity
profiles is to investigate their shape factor $H=\delta_{v}/\delta_{m}$
defined as the ratio between the displacement thickness $\delta_{v}$ and the
momentum thickness $\delta_{m}$, where

$\delta_{d}=\int_{0}^{\infty}[1-\frac{u(z)}{u_{max}}]dz,~{}{and}~{}\delta_{m}=\int_{0}^{\infty}[1-\frac{u(z)}{u_{max}}]\frac{u(z)}{u_{max}}dz.$
(5)

Since $u(z)$ decays after reaching its maximum value, the above integrations
are evaluated only over the range from $z=0$ to where $u(z)=u_{max}$. For our
profiles the obtained shape factors range between $1.9$ to $2.3$, which are
smaller than $H=2.59$, the value for a laminar Prandtl-Blasius boundary layer.
A shape factor smaller than the theoretical value means the corresponding
profile will approach its asymptotic value (the maximum velocity) slower than
the theoretical profile does.

In figure 20 we show the shape factor $H$ for mean velocity profiles obtained
in the laboratory and dynamical frames respectively for the four tilting
angles and for all $Ra$ measured. The dashed lines in the figure indicate the
Prandtl-Blasius value of 2.59. It is seen that, despite the data scatter,
there is a general trend that for both lab- and dynamical-frame profiles the
deviation from the Prandtl-Blasius profile increases with $Ra$. This is no
surprise, since, as the convective flow above the BL becomes more turbulent
with increasing $Ra$, the BL itself will experience stronger fluctuations and
hence larger deviations from the laminar case. This finding that the dynamical
rescaling method works better for smaller Ra than larger ones is consistent
with those found in DNS studies in the same geometry by Stevens et al. (2012)
for the temperature profile and by Shi et al. (2012) and by Scheel et al.
(2012) for the velocity profile. The second feature is that for all $\theta$
and $Ra$ the profiles obtained in the dynamical frame in general show some
degree of improvement towards that of Prandtl-Blasius value as compared to
those obtained in the laboratory frame. We also note that the “degree of
improvement does not seem to have an obvious dependence on $Ra$, which is also
consistent with the findings of Zhou & Xia (2010); Zhou et al. (2010).

Some insight can be obtained by examining the rescaled instantaneous velocity
profiles. Figure 21 show examples of rescaled instantaneous velocity profiles,
where the distance from the plate has been normalized by the instantaneous BL
thickness corresponding to that moment and the velocity has been normalized by
the instantaneous maximum horizontal velocity. It is seen that there are quite
few cases where the rescaled instantaneous velocity profile is rather close to
the theoretical Prandtl-Blasius profile (up to the point of the maximum
velocity) and deviations of the instantaneous shape are likely caused by
distubances such as plume emissions. Also shown in the figure are the shape
factor $H(t)$ of these instantaneous profiles. To quantify how the
instantaneous profiles are distributed with respect to the Prandtl-Blasius
profile, we examine the PDF of the shape factor difference $\delta
H(t)=H(t)-H^{PB}$ where $H^{PB}=2.59$. Figure 22 plots the PDFs of $\delta
H(t)$ for the 4 tilting angles and for all measured $Ra$ respectively. Despite
the seemingly large variations among them, these PDFs show the general trend
that the rescaled instantaneous profiles measured at lower values of $Ra$
($\lesssim 1\times 10^{9}$) are more of the time having a shape closer to that
of the Prandtl-Blasius profile and that for higher values of $Ra$ the peak of
the PDFs shift to smaller values of $H$. This indicates that with increasing
$Ra$ the profiles around the BL thickness becomes more rounded, i.e. the
approach to the maximum velocity becomes slower and slower. We further note
that these general trends are true across all tilt angles. Another feature
observed in the present 3D case is that we did not find any strong correlation
between the instantaneous BL thickness $\delta_{v}$ and the velocity $u(t)$
just above the BL. This is in contrast to the finding in the quasi-2D
experiment where $\delta_{v}$ and $u(t)$ are found to have a strong negative
correlation, i.e. a large velocity above would exert a stronger shear and
therefore thins the BL thickness (Zhou & Xia, 2010). This result suggest that
in certain aspect the BLs in the 3D and in the 2D/quais-2D cases are
dynamically different.

##  4 Summary and conclusions

We have conducted an experimental study of velocity boundary layer properties
in turbulent thermal convection. High-resolution two-dimensional velocity
field was measured using the particle image velocimetry (PIV) technique in a
cylindrical cell of height $H=18.6$ cm and aspect ratio close to unity, with
the Rayleigh number $Ra$ varying from $10^{8}$ to $6\times 10^{9}$ and the
Prandtl number $Pr$ fixed at $\sim 5.4$, with the convection cell tilted with
respect to gravity at angles $\theta=0.5^{o}$, $1^{o}$, $2^{o}$, and
$3.4^{o}$, respectively. Measurements made with small $\theta$ are aimed at
studying BL properties under more steady shear, but the BL itself is assumed
to be unperturbed otherwise. For large values of $\theta$ we wish to examine
how the BL responds to relatively large perturbations. We also examined
effectiveness of the dynamical BL scaling method in a three-dimensional
system.

It is found that the Reynolds number $Re$ ($=U_{max}H/\nu$) based on the
maximum mean horizontal velocity scales with $Ra$ as $Re\sim Ra^{0.43}$ and
the Reynolds number $Re_{\sigma}$ ($=\sigma_{max}H/\nu$) based on the maximum
rms velocity scales with $Ra$ as $Re_{\sigma}\sim Ra^{0.55}$. Both exponents
do not seem to have an apparent dependence on the tilt angle. On the other
hand, the amplitude of $Re$ seem to show a weak increasing trend with
$\theta$.

With the measured horizontal velocity, we obtain two length scales, i.e. the
viscous BL thickness $\delta_{v}$ based on the mean horizontal velocity
profile and the length scale $\delta_{\sigma}$ based on the rms horizontal
velocity profile. It is found that as far as scaling with the Reynolds number
$Re$ is concerned, the behavior of $\delta_{v}$ can be divided into two
regimes according to the tilting angle of the cell. For $\theta\leq 1^{o}$, it
is found that $\delta_{v}\sim Re^{-0.46\pm 0.03}$, which within experimental
uncertainty may be considered to be consistent with that of the Prandtl-
Blasius BL. It thus appears that the main effect of tilting the cell is to
restrict the azimuthal meandering of the large-scale circulation but the BL is
otherwise not strongly perturbed. For $\theta\geq 1^{o}$, the absolute value
of the exponent is found to increase with $\theta$ and in this case the BL may
be considered to be strongly perturbed. It is found that the scaling exponent
of $\delta_{\sigma}$ with respect to $Ra$ ($Re$) does not have a strong
dependence on $\theta$ as $\delta_{v}$ does. But similar to $\delta_{v}$, the
absolute values of these exponents increase with increasing $\theta$.

It is also found that tilting the cell modifies the velocity profile in the BL
region, i.e. for different tilt angles the shape of profiles is different. But
for the same tilting angle the velocity profiles measured at different $Ra$
can be brought to collapse on a single curve when the mean velocity is
normalized by the maximum velocity $U_{max}$ and the distance from the plate
by the viscous BL thickness $\delta_{v}$.

With simultaneously measured horizontal and vertical velocity components, we
also obtain the Reynolds stress $\tau_{R}$ in the velocity boundary layer. It
is found that $\tau_{R}$ is stronger in the mixing zone comparing with the
rectangular cell. The wall quantities such as the wall shear stress$\tau_{w}$,
the viscous sublayer$\delta_{w}$, the friction velocity $u_{\tau}$ are also
measured. Their scaling exponents with the Reynolds number are very close to
those predicted for classical laminar boundary layers, which is also
consistent with the measurement in rectangular cell.

Regarding the dynamical scaling method, we found that the method in general
works better when the cell is tilted at larger angle $\theta$ than it does at
smaller angles, but the effect is somewhat marginal. With respect to the
influence of $Ra$, it is found that in general the mean velocity profile
sampled in both the laboratory and dynamical frames are more closer to the
Prandtl-Blasius profile at smaller values of $Ra$ than they are at larger
$Ra$, which is consistent with findings from previous DNS studies. Moreover,
it is found that for smaller values of $Ra$ ($\lesssim 1\times 10^{9}$) the
PDF’s of the shape factor $H$ for the rescaled instantaneous profiles exhibit
a peak close to that for the Prandtl-Blasius profile, whereas for larger
values of $Ra$ the peaks shift to smaller values of $H$, indicating the
profile’s approach to the maximum velocity becomes slower and slower with
increasing $Ra$. Another finding is that the effectiveness of the dynamical
scaling method, in terms of its ability of bringing the mean velocity profile
closer to that of Prandtl-Blasius profile, does not have any apparent
dependence on $Ra$. Our general conclusion is that as far as the effectiveness
of the dynamical scaling method is concerned the influence of titling angle is
much smaller than that of the Rayleigh number $Ra$. We note that the Prandtl-
Blasius boundary layer theory is a 2D model, so it is perhaps no surprise that
the dynamic method works less well in 3D than in 2D.

We would like to thank X.-D. Shang and S.-Q. Zhou for kindly making their PIV
facility available to us and L. Qu, Y.-C. Xie and S.-D. Huang for helping with
the experiment. This work was supported in part by the Hong Kong Research
Grants Council under Project Nos. CUHK404409 and CUHK403811.

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# No Signalling and Unknowable Bohmian Particle Positions

Robert C. Helling
Ludwig-Maximilians-Universität München
Theresienstraße 39
80333 München
Germany
E-mail<EMAIL_ADDRESS>
###### Abstract

We exhibit in a model with simple dynamics, specifically a particle in a
square box or two particles in one dimensional boxes, that if an experimenter
can prepare the initial wave function of a system, the maximal information
about the positions of Bohmian particles that is compatible with “no
signalling” is that they are distributed according to $|\psi(x)|^{2}$. In
particular, the positions cannot be prepared independently from the wave
function. Any sharper “actual” position of the particle must be inaccessible
since it could be used to send signals instantaneously. This is a consequence
of the non-local character of the Bohmian dynamical law.

##  1 Introduction

Bohmian Mechanics (for an introduction, see Duerr ) is often presented as an
approach to quantum theory that has an “ontology”, meaning that it is about
particles that are characterized by their positions and is in that sense
realistic. The price to pay is that the theory has non-local equations of
motions but is claimed to be observationally indistinguishable from the
textbook version of quantum mechanics (which one could loosely call Copenhagen
style lacking a better name irrespective if one believes in collapse, many
worlds, decoherence or the like).

The Bohmian theory is based on the observation that the quantum mechanical
current

$j=\bar{\psi}\nabla\psi-(\nabla\bar{\psi})\psi$ (1)

is conserved for Schrödinger type Hamilton operators

$H=-\Delta+V(x).$ (2)

From that one defines a velocity field

$v(x)=j/|\psi|^{2}=2\Im(\nabla\psi/\psi)$ (3)

and postulates “particles” whose position $q(t)$ follow the velocity field,
i.e.

$\frac{d}{dt}q=v(q).$ (4)

If one starts with a statistical ensemble of such particles with probability
density $|\psi(x)|^{2}$ at the initial time then at all times the evolved
probability density will be given by $|\psi(x,t)|^{2}$. Note well that the
wave function $\psi$ evolves according to the time dependent Schrödinger
equation and that thus there is no feedback from the particle postions $q$ to
the wave function.

In the Bohmian framework, it is emphasised that all measurements in the end
can be traced back to position measurements (which could be the position of a
pointer on a scale or the position of a black spot on a photo plate in a
double slit experiment) so it is understood that the $q$ are the positions of
the “real” or actual particles. They have objective (realistic, deterministic
and in that sense classical) trajectories but they are quantum in the sense
that their dynamical law (4) is very different from Newton’s second law.
Still, they produce the predictions of standard quantum mechanics as their
probability density traces at all times the probability density implied by the
absolute value square of the wave function assuming it did at an initial time.

The existence of trajectories appears to be in direct conflict with standard
lore of quantum mechanics as well as the existence of classical probability
densities seems to clash with Bell-type inequalities. But this tension is
usually relieved upon the realization that as long as one only considers
observables that are functions of the positions only (and not of momenta)
these all commute. And since one deals only with a commutative algebra of
observables, all states can indeed be realized as probability densities on
some classical (configuration) space.

This argument is only true as long as one considers only positions at one
instant of time as in general the positions at different time fail to commute.
This fact, in the context of the Bohm theory, was emphasised by
CorreggiMorchio ; Werner . In those papers, positional observables that lead
to (violations) of Bell type inequalities were constructed and here, we
strongly build on these works.

In this note, we will analyse a particular simple example of the construction
in CorreggiMorchio  and spell out the consequences.

Our model, consisting of a particle in a box, has the advantage of being
elementary solvable so one has full analytic control at all stages as it comes
without devices like double slits, Stern-Gerlach devices or beam splitters
that do not have explicitly know Hamiltonians and in which technicalities
could be hidden or suspected to be hidden. All observables considered are
diagonal in position representation and thus directly expressible in terms of
Bohmian particle positions. The model’s understanding only requires the most
elementary quantum mechanics.

##  2 The Model

Often, inequalities of Bell type that demonstrate that Quantum Mechanics
cannot be a local, realistic theory (in the very general meaning of having a
space of states that is a simplex so all states can uniquely decomposed into
extremal states) are expressed in terms of entangled qubits which are then
thought of as realized by spin or helicity degrees of freedom. For a
discussion in the Bohmian context this can cause problems or confusion since
their Hamiltonians are generally not of Schrödinger type (2) and the Bohmian
trajectories do not directly apply.

Using Stern-Gerlach type experiments where an inhomogeneous magnetic field
that couples to a spin degree of freedom one can translate spin states to
positions. But those have the problem that they cannot be easily accessible to
analytic study since they lead to complicated dynamics.

Of course, the only structural property of a qubit is that it lives in a two
dimensional Hilbert space and has a non-commutative algebra of operators is
acting on it. This can also be realized as a two dimensional subspace of a
positional Hilbert space and this is what we will do in this note
followingCorreggiMorchio ; Werner .

Our system is simply a free particle in a two dimensional square box, i.e. the
Hilbert space is

${\cal H}=L^{2}([-\pi/2,\pi/2]^{2})\qquad\hbox{and}\qquad H=-\Delta$ (5)

with Dirichlet boundary conditions.

At times, it will be convenient to emphasise the bipartite nature of the two
coordinates by the use of an equivalent tensor product language:

${\cal H}=L^{2}([-\pi/2,\pi/2])\otimes L^{2}([-\pi/2,\pi/2])$ (6)

and (very explicitly) a non-interacting time evolution given in terms of

$H=H_{1}\otimes\mathbb{I}+\mathbb{I}\otimes H_{2}\qquad\hbox{with}\qquad
H_{i}=-\frac{\partial^{2}}{\partial x_{i}^{2}},$ (7)

where $\mathbb{I}$ denotes the identity operator. Thus, instead of one
particle in a two dimensional box, this set-up equivalently describes two
independent particles in one dimensional boxes (i.e. intervals). In the
following, we will employ both these one and two particle interpretations
interchangeably. For the two particle interpretation, it can be beneficial to
think of the two intervals as widely separated to visualize that anything
performed on particle one must not have measurable consequences on particle
two as otherwise no-signalling would be violated. But for the moment, we stick
with the “one particle in a square box” point of view.

The particle is prepared to be in the state given by the wave function

$\psi(x_{1},x_{2})=\frac{\sqrt{2}}{\pi}\left(\cos(x_{1})\sin(2x_{2})-\sin(2x_{2})\cos(x_{1})\right).$
(8)

With the ground state $\psi_{1}(x_{i})=\sqrt{2/\pi}\cos(x_{i})$ of energy 1
and first excited state $\psi_{2}(x_{i})=\sqrt{2/\pi}\sin(2x_{i})$ of $H_{i}$
of energy 4, our state can be written as the entangled state

$\psi=\frac{1}{\sqrt{2}}(\psi_{1}\otimes\psi_{2}-\psi_{2}\otimes\psi_{1}).$
(9)

From this form, it is obvious that $\psi$ is an eigenstate of $H$ with energy
5 and thus stationary. This is also reflected by in the Bohmian theory where
the velocity field vanishes

$v=0$ (10)

as the wave function is real and thus according to (3), the Bohmian particles
do not move at all.

Figure 1: The probability distribution $|\psi(x,y)|^{2}$

We will consider two statements about the particle:

$A\colon\hbox{The particle is in the right half of the box}$ (11)

and

$B\colon\hbox{The particle is in the upper half of the box}.$ (12)

As observables, they can be expressed by the multiplication operators by

$A=\hbox{sgn}(x_{1})=\Sigma\otimes\mathbb{I}\qquad
B=\hbox{sgn}(x_{2})=\mathbb{I}\otimes\Sigma,$ (13)

where $\Sigma$ is the operator that multiplies by the sign function in
position space. As functions of different coordinates, they commute and are
thus simultaneously observable. As can be observed from figure 1, by symmetry,
both have vanishing expectation value but their outcome is anti-correlated.
One could imagine to do this experiment by inserting a horizontal or vertical
wall in the box and then detecting the particle in one of the two now
separated halves.

We can apply a (Heisenberg picture) time evolution and obtain

$A_{t}=e^{itH}Ae^{-itH}=(e^{itH_{1}}\,\Sigma\,e^{-itH_{1}})\otimes\mathbb{I}$
(14)

and similarly for $B_{t}$. As $\psi$ is stationary and $\psi_{1}$ and
$\psi_{2}$ are symmetric and anti-symmetric, respectively, one finds
$\langle\psi_{i},\Sigma\,\psi_{i}\rangle=0$ and thus

$\langle\psi,A_{t}\psi\rangle=\langle\psi,B_{t}\psi\rangle=0.$ (15)

As the time evolution does not mix the two tensor factors, the two observables
even commute for different times

$[A_{s},B_{t}]=0$ (16)

and thus can be observed without one disturbing the measurement of the other.

For the direct application of the Bohmian framework, it is essential that all
these measurements are measurements of positions so that one can relate the
actual observation directly to $q$. To make use of the quantum nature of
entanglement, however, we need a non-commutative set of observables which we
have in observing $A$ or $B$ at different times as the Hamiltonian does not
commute with $A$ or $B$ and thus

$[A_{s},A_{t}]\neq 0\neq[B_{s},B_{t}]\quad\hbox{for }t-s\not\in\pi\mathbb{Z}.$
(17)

A short calculation yields for their correlation

$\langle\psi,A_{s}B_{t}\psi\rangle=-\cos\big{[}(E_{2}-E_{1})(s-t)\big{]}|\langle\psi_{1},\Sigma\psi_{2}\rangle|^{2},$
(18)

where $\langle\psi_{1},\Sigma\psi_{2}\rangle=8/3\pi$. We find that $A$ and $B$
are totally anti-correlated when measured at the same time but then oscillate
to correlation and back.

If one is not happy with the Heisenberg picture of time dependent observables,
one can re-express this time dependence in terms of the Copenhagen
interpretation invoking collapse of the wave function after the first
measurement. One would say that the state is stationary until the first
measurement upon which (depending on the outcome), the wave function is
projected to zero in one half of the box which results in a state that is no
longer an eigenstate of the total Hamiltonian and thus oscillates which
explains the outcome of the second measurement being an oscillating function
of time.

This is the case even though the measurements of $A_{s}$ and $B_{t}$ act in
different factors of the Hilbert space and due to their commutativity do not
influence each other.

The spectral projectors for the two eigen-spaces of the operator $A$ are

$P_{\pm}=\frac{1}{2}(\mathbb{I}\pm\Sigma)\otimes\mathbb{I}=\theta(\pm
x)\otimes\mathbb{I},$ (19)

with the Heavyside step function denoted $\theta$.

According to the collapse prescription, after a measurement of this
observables yields the $i$th eigenvalue, the state collapses to

$\psi_{c}=\frac{P_{i}\psi}{\|P_{i}\psi\|}.$ (20)

As from there on, we are only interested only in particle (or coordinate) two,
we trace over the Hilbert space factor of particle one and obtain a reduced
density matrix

$\rho_{r}=\hbox{tr}_{1}|\psi_{c}\rangle\langle\psi_{c}|.$ (21)

Without the measurement, this reduced density matrix was simply
$\frac{1}{2}\mathbb{I}_{2}$ in the $\psi_{1},\psi_{2}$ basis, but as

$\langle\psi_{1}|\theta|\psi_{2}\rangle=\frac{4}{3\pi}\neq 0,$ (22)

after the measurement contains off-diagonal entries

$\rho_{r}=\begin{pmatrix}\frac{1}{2}&\frac{4}{3\pi}\\\
\frac{4}{3\pi}&\frac{1}{2}\end{pmatrix}.$ (23)

This density matrix is diagonalized in the basis of eigenvectors
$(\psi_{1}\pm\psi_{2})/\sqrt{2}$ with eigenvalues
$\frac{1}{2}\pm\frac{4}{3\pi}=0.924413$ and $0.0755868$.

Clearly, these eigenvectors are no longer eigenfunctions of the Hamiltonian of
particle 2 and thus this state is no longer stationary. The probability
density for the second particle to be found in a specific position is plotted
in Figure 2 and one can see the oscillations that lead to the time dependent
oscillation (18) of the particle two between left an right.

Figure 2: The probability distribution for particle two as a function of time
(pointing up) after the observation of particle one if particle one has been
found in left/right half (top) and sum of the two lacking information about
particle one (bottom)

This is conditioned on the first particle being found in the left half of the
interval. If it would have been found in the right half, the probability
density would oscillation in the opposite direction. If the outcome of the
measurement on particle one is not known, both distributions would have to be
superimposed (mixed) yielding the stationary distribution just as without any
measurement on particle one at all. So the distribution of particle two
oscillates only conditioned on the outcome of the measurement of particle one.

##  3 The Bohmian perspective

After this analysis in the Copenhagen language we will perform the same in the
Bohmian framework. In order for it to cover the oscillating correlation as a
function of the different times of observation it cannot keep the particle
velocity $v$ zero at all times. So, also here does the measurement of the
$x_{1}$ coordinate (“left” or “right”) influence the velocity of the Bohmian
particles in the $x_{2}$-direction: They have to oscillate up and down. Even
thought there is no explicit collapse in the Bohmian framework, the
measurement is again measuring the position of particle one (resulting in a
moving particle two). We don’t have to know about the detailed workings of the
measurement process to conclude that the particle has to start moving as
otherwise one would not measure the two-time correlation function eq. (18)
that depends on the difference of the two times. If it were different there
would be an immediate observational difference between the orthodox and the
Bohmian approach.

The Bohmian ontology is only about the momentary positions of the particles so
there is no immediate problem. It can be seen that this is in fact everything
one can know about these particles. This is clearest in the realization of the
model as two particles moving in two one-dimensional intervals. If one could
observe their velocity, this would violate No Signalling: By observing the
velocity of particle two one would know if a measurement has been performed on
particle one. This is independent of knowing the outcome of the experiment. So
it must be in principle impossible to measure the velocity, it is not just
that so far nobody cared to measure it. The particles only have a position
unlike for example the particles in classical Hamiltonian mechanics which have
both a position and a momentum.

One could try to measure the velocity by measuring the position twice,
separated by a short time interval. But then one could argue that the first
measurement would necessarily disrupt the particle so much that the measured
velocity is no longer the velocity of the particle before trying to measure
it.

But there is more: It is usually assumed that we don’t have any more specific
information about the initial position of the particles except that it is
distributed according to $|\psi|^{2}$ (this is known as the “quantum
equilibrium hypothesis” in the literature). Again, this is not about our
voluntary ignorance. If we had any more specific knowledge about the
particles’ initial position, in particular if there were a way to prepare it
while still preparing the initial wave-function to be $\psi$, we could measure
the position of particle two at time $t$ and thus (possibly probabilistically)
determine if it moved since it had been prepared which would imply that a
measurement at particle one had taken place, possibly at a large space-like
distance.

The oscillating trajectories of the Bohmian particle two after particle one
has been measured (assuming a similar collapse at least as an effective
description of the measurement process) are shown in Fig. 3. Those are key to
our claim that if any information about those particles beyond what is already
contained in $|\psi(x)|^{2}$ would be accessible to observation, this
information could be used to transmit information instantaneously from
particle one to an observer of particle two (the impossibilitiy of knowing the
distribution of the particles has also been discussed in Duerr2 ; valentini1 ;
valentini2  based on general arguments).

To this end, let us assume the system of the two particles has been set up in
the quantum state according to (8). Let us further assume that Alice’s
information about the position of the Bohmian particle two is described by a
probability density $\varrho(x)$.

In particular, if Alice knew the position of that particle, $\varrho$ would be
a $\delta$-function. But her knowledge could be more coarse grained and be
described by a more general probability density. Just knowing that the quantum
state is given by (8) is described by $\varrho(x)$ being equal to
$|\psi(x)|^{2}$, so let us parametrize it as

$\varrho(x)=\lambda(x)|\psi(x)|^{2}$ (24)

Having any more specific information about the location of the Bohmian
particle would correspond to a non-constant $\lambda$.

Now, Bob wants to transmit one bit of information to Alice. If that bit is “0”
he does nothing but if it is “1” he observes if particle one is in the left or
the right half (of course without telling Alice the outcome of his
observation).

For Alice waits a short moment of time. If the bit was “0” then the Bohmian
velocity was vanishing throughout the experiment. No Bohmian particle moves
and she finds her particle two at a position described by the original
probability density $\varrho$.

Figure 3: Bohmian trajectories of particle two after particle one has been
measured in one half of its interval

If, however, Bob had transmitted a “1” and had thus done an observation on
particle one, particle two starts moving with a velocity field (3) and the
probability density changes due to a non-vanishing divergence of the current

$\nabla\left(\lambda|\psi|^{2}2\Im\left(\nabla\psi/\psi\right)\right)=(\nabla\lambda)\cdot|\psi|^{2}v$
(25)

which is non-vanishing as long as $\lambda$ is not constant. Thus this change
in position can be detected by observing the position of particle two which
follows a different distribution than $\varrho$.

Only in the case where $\lambda$ is spatially constant, Alice cannot detect
(not even probabilistically) the difference between Bob sending a 0 and a 1.
So we conclude that no-signalling implies that Alice’s knowledge about the
Bohmian particles cannot be better than what is already given by the
probability density indicated by the wave function. The positions of the
particles, the additional ingredient of the Bohmian interpretation, thus must
not be knowable (beyond what is already known in terms of $\psi$) if no-
signalling holds.

##  4 Signalling and Semi-Classics

Non-local signalling has of course never been observed in the real world and
if it were it would create immediate problems with causality at least as long
as the world is believed to be realtivistic.

Furthermore, there is no experiment that can distinguish between the Bohmian
interpretation of quantum theory and the “orthodox” version. This is because
it is possible to take the Bohmian perspective and then simply ignore the
particle positions $q$ to get back to the orthodox view.

From a structural stand point, this is possible because besides the equation
of motion $(\ref{eq:4})$ for $q$, the particle positions do not appear in any
other equation of motion, in particular they don’t appear in the Schrödinger
equation that governs the time evolution of the wave function. There is no
feed-back. This is in contrast to the situation for example in electrodynamics
with charged particles: There, the motion of the particles feels the electro-
magnetic forces due to the field-strength but the motion of the sources also
influences the electro-magnetic field.

If the Bohmian particle postions would source any other field, one could use
an observation of that field as a proxy for the particle positions and their
motions and build a signalling device based on the set-up described in this
note.

There is an attempt to use Bohmian notions as a semi-classical approximation
Struyve1 ; Struyve2 : The idea is to treat particles quantum mechanically but
couple those to a classical field (for example electro-magnetic or
gravitational) via $\partial_{\mu}F^{\mu\nu}=j_{\nu}$ with (in our notation)

$j^{0}(x,t)=\sum_{k}e\delta(x-q_{k}(t)),\quad
j^{i}=\sum_{k}ev_{k}^{i}(t)\delta(x-q_{k}(t)).$

In the situation of the previous section, particle two is at rest before the
measurement of particle one is performed. It would only create an
electrostatic Coulomb field. But as soon as the measurement of particle one
takes place, particle two starts oscillating and not only creates a magnetic
field but also an electro-magnetic wave. One could use a radio receiver close
to particle two to detect that a measurement of particle one has been
performed possibly very far away in the universe.

This shows that for the Bohmian theory to be non-signalling (and not to get in
tension with causality), it is essential that no observable degree of freedom
couples directly to the particle positions $q$, they have to remain invisible.

##  5 Conclusion

In order not to violate no-signalling, one must not be able to know more about
the particle positions than their $|\psi|^{2}$-distribution, in particular, it
must be impossible to prepare another (possibly purer) initial state (of
knowledge about the position) of the particles while keeping the preparation
of the wave function or just have more information about the initial state
than this particular probability distribution. In classical physics, the fact
that a state is given by a probability density that is not a single
$\delta$-function peak expresses ignorance about the microscopic details.

In a Bohmian world, any further information would immediately lead to the
possibility to send signals faster than the speed of light.

As a consequence, compared to “orthodox quantum mechanics”, Bohmian mechanics
has further elements of reality (the particle positions) with a deterministic
equation of motion but without the experimenter’s influence on obtaining
knowledge of or controlling the initial conditions further than what is given
by the probability distribution already encoded in the wave function. In this
sense, these particles are like the proverbial angels on the tip of a pin:
They exist and but interaction with them is very limited: If we observe their
position we can have no information where they came from.

This should be contrasted to the Copenhagen interpretation: As we have
explained, the collapse of the wave function is also global and instantaneous
(and thus leads to oscillations of the wave function conditioned on the
outcome of the first measurement). But the wave function is not directly
observable. In this approach it is clear that causality is only to be imposed
at the level of observables. And here it is clear that locality (and thus
causality) hold simply by the observation that any observable of the form
$X\otimes\mathbb{I}$ commutes with any observable of the form
$\mathbb{I}\otimes Y$ and thus actions of Bob on his system cannot influence
anything that Alice experiences in her system.

###### Acknowledgements.

I would like to thank various unnamed members of the “Workgroup Mathematical
Foundations of Physics” as well as Tim Maudlin and Ward Struyve for helpful
discussions that helped to sharpen the points made in this paper.

## References

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