1 Introduction
Rayleigh–Bénard convection (RBC), a fluid layer confined between two plates heated from below and cooled from above, continues to attract attention not only because of its relevance to geophysical and astrophysical flows, but also owing to the fact that it is an idealised model for the study of thermally driven turbulence (for reviews, see, for example, Ahlers, Grossmann & Lohse (Reference Ahlers, Grossmann and Lohse2009), Lohse & Xia (Reference Lohse and Xia2010), Chillà & Schumacher (Reference Chillà and Schumacher2012), Xia (Reference Xia2013)). In RBC, buoyancy injects energy into the turbulence and creates vigorous velocity and temperature fluctuations. Understanding the dynamics of these fluctuations has been one of the central issues (Grossmann & Lohse Reference Grossmann and Lohse2004). The RBC system is controlled by two dimensionless parameters – the Rayleigh number
$Ra=\unicode[STIX]{x1D6FC}g\unicode[STIX]{x0394}TH^{3}/(\unicode[STIX]{x1D705}\unicode[STIX]{x1D708})$
and the Prandtl number
$Pr=\unicode[STIX]{x1D708}/\unicode[STIX]{x1D705}$
, where
$g$
is the gravitational acceleration constant,
$H$
the height of the convection cell,
$\unicode[STIX]{x0394}T$
the temperature difference across the top and bottom plates, and
$\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D705}$
and
$\unicode[STIX]{x1D708}$
are the thermal expansion coefficient, the thermal diffusivity and the kinematic viscosity of the working fluid, respectively.
Two turbulence states with different probability density functions (PDFs) of the temperature at the cell centre were observed in turbulent RBC – a ‘soft turbulence’ state with a Gaussian PDF and a ‘hard turbulence’ state with an exponential PDF (Heslot, Castaing & Libchaber Reference Heslot, Castaing and Libchaber1987). In the latter, the r.m.s. temperature
$\unicode[STIX]{x1D70E}_{T}$
normalised by
$\unicode[STIX]{x0394}T$
obeys a
$-0.145$
power law with
$Ra$
. A mixing-length model was proposed to explain this dependence (Castaing et al.
Reference Castaing, Gunaratne, Heslot, Kadanoff, Libchaber, Thomae, Wu, Zaleski and Zanetti1989). However, later studies show that
$\unicode[STIX]{x1D70E}_{T}/\unicode[STIX]{x0394}T$
not only depends on
$Pr$
but also on the plate topography, even for the most widely studied aspect-ratio-unity cylindrical cells (see table 1 for a summary). It is thus natural to ask if there is any universal behaviour of the temperature fluctuations in the bulk flow.
In addition to the value of
$\unicode[STIX]{x1D70E}_{T}$
at the cell centre, the functional form of the root-mean-square (r.m.s.) temperature profile
$\unicode[STIX]{x1D70E}_{T}(z)$
is also of great interest as its shape determines the transport properties across the boundary. Based on different assumptions of the local force balance, theory predicts different profiles of
$\unicode[STIX]{x1D70E}_{T}(z)$
(Adrian Reference Adrian1996). Only very recently, a clear understanding of
$\unicode[STIX]{x1D70E}_{T}(z)$
has been obtained (He & Xia Reference He and Xia2019): while in a region with mean horizontal shear (the viscous force balances the inertia force),
$\unicode[STIX]{x1D70E}_{T}(z)$
obeys a power-law dependence on
$z$
; in regions with abundant plume emissions (the buoyancy balances the inertia force),
$\unicode[STIX]{x1D70E}_{T}(z)$
is in a logarithmic form, which is true for the idealised case – that is, turbulent RBC in cells with smooth surfaces (‘smooth cell’). In the presence of roughness on the top and bottom boundaries (‘rough cell’),
$\unicode[STIX]{x1D70E}_{T}$
is enhanced considerably (Du & Tong Reference Du and Tong2001). An interesting question is whether
$\unicode[STIX]{x1D70E}_{T}$
shows universal behaviour in cells with different plate topographies. In addition to the shape of
$\unicode[STIX]{x1D70E}_{T}(z)$
, there is no generally accepted characteristic temperature scale in the bulk. For instance,
$\unicode[STIX]{x0394}T$
has been widely used as a typical temperature scale, but the maximum value
$\unicode[STIX]{x1D70E}_{T}(z)_{max}$
is also used sometimes (see, for example, Wang et al. (Reference Wang, Xu, He, Yik, Wang, Schumacher and Tong2018)). The
$Ra$
-dependence of
$\unicode[STIX]{x1D70E}_{T}(z)$
scaled by different scales exhibits different features: while
$\unicode[STIX]{x1D70E}_{T}(z)/\unicode[STIX]{x0394}T$
for different
$Ra$
differs (Sun, Cheung & Xia Reference Sun, Cheung and Xia2008; Ahlers et al.
Reference Ahlers, Bodenschatz, Funfschilling, Grossmann, He, Lohse, Stevens and Verzicco2012),
$\unicode[STIX]{x1D70E}_{T}(z)/\unicode[STIX]{x1D70E}_{T}(z)_{max}$
for different
$Ra$
collapses better in the mixing zone. Therefore, the typical temperature scale in the bulk is not clear at present.
Compared to numerous studies on temperature fluctuations, the investigation on velocity fluctuations in turbulent RBC is scarce. The velocity fluctuation is usually studied in terms of a Reynolds number based on the vertical r.m.s. velocity
$\unicode[STIX]{x1D70E}_{w}$
, given by
$Re_{\unicode[STIX]{x1D70E}_{w}}=\unicode[STIX]{x1D70E}_{w}H/\unicode[STIX]{x1D708}$
. It is found that
$Re_{\unicode[STIX]{x1D70E}_{w}}$
scales with
$Ra$
to a 0.5 power law, consistent with the free-fall-like argument (Shen, Xia & Tong Reference Shen, Xia and Tong1995; Daya & Ecke Reference Daya and Ecke2001; Qiu et al.
Reference Qiu, Shang, Tong and Xia2004; Shang, Tong & Xia Reference Shang, Tong and Xia2008). To the best of our knowledge, there have been almost no direct measurements of velocity fluctuations in rough cells. The typical velocity scale in the bulk flow also remains to be explored.
In this paper, we present an investigation of the temperature and the velocity fluctuations in the bulk of turbulent RBC. We demonstrate that the typical temperature and velocity scales in the bulk are, respectively, the convective temperature
$\unicode[STIX]{x1D703}_{\ast }$
and the convective velocity
$w_{\ast }$
. Once scaled by these quantities, the fluctuations in the bulk exhibit universal behaviour. These findings shed new light on the bulk dynamics in convective turbulence.
2 The experimental set-up and relevant parameters
The experiment was carried out in an upright cylindrical cell with an aspect ratio
$\unicode[STIX]{x1D6E4}=D/H\approx 1$
, where
$D=192~\text{mm}$
is the cell diameter and
$H=202~\text{mm}$
is its height. The design and construction of the cell can be found in Xie & Xia (Reference Xie and Xia2017). We mention only its essential features here. The cell consists of three parts: a copper top plate, a copper bottom plate and a Plexiglas sidewall. The bottom plate was heated by rubber electrical heaters and the top plate was cooled by passing temperature-controlled water through a chamber fitted onto its top surface. The temperature of the top (bottom) plate was measured using four (five) thermistors, from which we calculated
$Ra$
,
$Pr$
and the Nusselt number
$Nu=QH/(\unicode[STIX]{x1D712}\unicode[STIX]{x0394}T)$
, where
$Q$
is the heat flux supplied at the bottom plate and
$\unicode[STIX]{x1D712}$
the thermal conductivity of the fluid. Deionised and degassed water was used as the working fluid, with a mean fluid temperature kept at a constant of
$40\,^{\circ }\text{C}$
. Thus the Prandtl number was
$Pr=4.34$
. By changing
$\unicode[STIX]{x0394}T$
, we achieved a
$Ra$
range of
$1.59\times 10^{8}\leqslant Ra\leqslant 8.82\times 10^{9}$
. To study the effects of wall roughness on the fluctuations of the temperature and the velocity, another set of measurements in a rough cell were made. The roughness elements were in the form of square-latticed pyramids with a height
$h$
of 8 mm. The heat transport in rough cells shows three regimes depending on
$Ra$
(Xie & Xia Reference Xie and Xia2017). The
$Ra$
range in the present study in the rough cell is in the heat-transport-enhanced regimes.
Scaling of the r.m.s. temperature (
$\unicode[STIX]{x1D70E}_{T,c}/\unicode[STIX]{x0394}T=ARa^{\unicode[STIX]{x1D6FE}}$
) at the cell centre. In the surface type column, ‘
$S$
’ stands for ‘smooth surface cell’ and ‘
$R$
’ stands for ‘rough surface cell’. The references are:
$(a)$
Castaing et al. (Reference Castaing, Gunaratne, Heslot, Kadanoff, Libchaber, Thomae, Wu, Zaleski and Zanetti1989),
$(b)$
Niemela et al. (Reference Niemela, Skrbek, Sreenivasan and Donnelly2000),
$(c)$
Du & Tong (Reference Du and Tong2001),
$(d)$
Daya & Ecke (Reference Daya and Ecke2001),
$(e)$
Wang & Xia (Reference Wang and Xia2003),
$(f)$
Lakkaraju et al. (Reference Lakkaraju, Stevens, Verzicco, Grossmann, Prosperetti, Sun and Lohse2012),
$(g)$
Wei et al. (Reference Wei, Chan, Ni, Zhao and Xia2014),
$(h)$
Kaczorowski, Chong & Xia (Reference Kaczorowski, Chong and Xia2014),
$(i)$
Wei & Ahlers (Reference Wei and Ahlers2016);
$(j)$
Xie & Xia (Reference Xie and Xia2017).
The r.m.s. temperature,
$\unicode[STIX]{x1D70E}_{T}=\sqrt{\langle (T-\langle T\rangle )^{2}\rangle }$
, inside the cell was measured using a waterproof thermistor with a head diameter of 0.3 mm and a response time of 30 ms, where
$\langle \cdots \rangle$
stands for time averaging. Two sets of temperature fluctuation measurements were made. In the first set, the r.m.s. temperature at the centre
$\unicode[STIX]{x1D70E}_{T,c}$
of the smooth cell and the rough cell were measured as a function of
$Ra$
. In the second set, the vertical r.m.s. temperature profiles
$\unicode[STIX]{x1D70E}_{T}(z)$
were measured in the rough cell. The location of
$z=0~\text{mm}$
was set at the valley of the roughness elements on the bottom plate. At each vertical position
$z$
, a temperature time trace was measured for one hour with a sampling rate of 15 Hz. Good care was taken to ensure that the system was isolated from the surrounding temperature variations. For the detailed thermal isolation method, we refer to Xie, Ding & Xia (Reference Xie, Ding and Xia2018).
The vertical velocity fluctuation, characterised by a Reynolds number
$Re_{\unicode[STIX]{x1D70E}_{w}}=\unicode[STIX]{x1D70E}_{w}H/\unicode[STIX]{x1D708}$
, was measured at the centre of the rough cell, using a laser Doppler velocimeter (LDV). Here
$\unicode[STIX]{x1D70E}_{w}=\sqrt{\langle (w-\langle w\rangle )^{2}\rangle }$
is the vertical r.m.s. velocity. The flow was seeded with tracer particles with a diameter of
$2.893~\unicode[STIX]{x03BC}\text{m}$
. The LDV sampling rate was
${\sim}30~\text{Hz}$
. Typical measurements lasted for 12 h to obtain sufficient statistics of the second-order quantity like the r.m.s. velocity.
Considering the region outside the boundary layers, the relevant physical parameters governing the flow dynamics are
$\unicode[STIX]{x1D6FC}g$
,
$Q_{0}$
and
$H$
, where
$Q_{0}=Q/(\unicode[STIX]{x1D70C}c_{p})$
is the specific heat flux, and
$\unicode[STIX]{x1D70C}$
and
$c_{p}$
are, respectively, the density and the specific heat capacity of the working fluid. A simple dimensional analysis yields the convective temperature
$\unicode[STIX]{x1D703}_{\ast }$
and the convective velocity
$w_{\ast }$
:
$\unicode[STIX]{x1D703}_{\ast }\equiv Q_{0}^{2/3}/(\unicode[STIX]{x1D6FC}gH)^{1/3}$
and
$w_{\ast }\equiv (\unicode[STIX]{x1D6FC}gHQ_{0})^{1/3}$
(Deardorff Reference Deardorff1970). The two scales can be expressed in dimensionless forms in terms of
$Ra$
,
$Pr$
and
$Nu$
as
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D703}_{\ast }/\unicode[STIX]{x0394}T=Nu^{2/3}/(RaPr)^{1/3},\quad Re_{w_{\ast }}=w_{\ast }H/\unicode[STIX]{x1D708}=(RaNuPr^{-2})^{1/3}. & & \displaystyle\end{eqnarray}$$
3 Results and discussions
3.1 The temperature fluctuation in the bulk
We first study the r.m.s. temperature profiles. Figures 1(a) and 1(c) show
$\unicode[STIX]{x1D70E}_{T}(z)$
normalised, respectively, by the temperature difference across the top and bottom plates (
$\unicode[STIX]{x0394}T$
) and the convective temperature (
$\unicode[STIX]{x1D703}_{\ast }$
) measured in the rough cell for
$Ra=3.2\times 10^{8}$
,
$2.2\times 10^{9}$
and
$5.2\times 10^{9}$
. The r.m.s. temperatures (
$\unicode[STIX]{x1D70E}_{T,c}$
) measured at the centre of the smooth cell for
$8.9\times 10^{8}\leqslant Ra\leqslant 9.3\times 10^{9}$
are plotted as hexagons. For comparison, we also plot in these figures
$\unicode[STIX]{x1D70E}_{T}(z)$
measured in smooth cells from Du & Tong (Reference Du and Tong2000) at
$Ra=1.5\times 10^{9}$
and from Wei et al. (Reference Wei, Chan, Ni, Zhao and Xia2014) at
$Ra=6.8\times 10^{8}$
. The horizontal axes are normalised by
$H$
.
It is seen that the profiles measured in either a smooth cell or a rough cell for different
$Ra$
collapse onto each other outside the thermal boundary layer (TBL). With increasing
$z$
,
$\unicode[STIX]{x1D70E}_{T}(z)$
increases to a maximum and then decreases gradually when moving towards the cell centre. These observations are in general consistent with earlier studies (see, for example, Lui & Xia (Reference Lui and Xia1998)). However, considerable differences between
$\unicode[STIX]{x1D70E}_{T}(z)/\unicode[STIX]{x0394}T$
measured in smooth cells and in rough cells are observed (figure 1
a). First, the outer edge of the TBL (that is, the peak position of
$\unicode[STIX]{x1D70E}_{T}(z)/\unicode[STIX]{x0394}T$
) is shifted towards the cell centre, which is because the motionless fluids trapped in the bottom of valleys between roughness elements have very small temperature fluctuations. Second,
$\unicode[STIX]{x1D70E}_{T}$
is enhanced considerably near the edge of the TBL in rough cells owing to more thermal plumes being emitted from the tip of roughness elements. This enhancement is also true at the cell centre since the data in the smooth cell are systematically smaller than those in the rough cell. However, we note, even for a smooth cell, Ahlers et al. (Reference Ahlers, Bodenschatz, Funfschilling, Grossmann, He, Lohse, Stevens and Verzicco2012) showed that
$\unicode[STIX]{x1D70E}_{T}(z)/\unicode[STIX]{x0394}T$
for different
$Ra$
does not collapse in the classical regime of turbulent RBC. The above results suggest that
$\unicode[STIX]{x0394}T$
is not a suitable characteristic scale for
$\unicode[STIX]{x1D70E}_{T}$
. When scaled with
$\unicode[STIX]{x1D703}_{\ast }$
, the r.m.s. temperature at the cell centre, for both the smooth and the rough cells, collapse onto each other (but not for the profile in the mixing zone), and the magnitude of the fluctuation at the edge of the TBL also becomes comparable in two cases (figure 1
c). To compare directly the r.m.s. temperature profiles in the bulk region, in figure 1(b,d), the distance
$z$
of the data measured in the rough cells is offset by the roughness height
$h$
. Remarkably, compared with
$\unicode[STIX]{x1D70E}_{T}(z)/\unicode[STIX]{x0394}T$
(figure 1
b),
$\unicode[STIX]{x1D70E}_{T}(z)/\unicode[STIX]{x1D703}_{\ast }$
in smooth and rough cells collapse onto each other outside the TBL (figure 1
d), suggesting that
$\unicode[STIX]{x1D703}_{\ast }$
is a characteristic temperature scale in the bulk of turbulent convection.
Measured r.m.s. temperature profiles
$\unicode[STIX]{x1D70E}_{T}(z)$
along the cell centreline. The vertical axis is scaled by
$\unicode[STIX]{x0394}T$
(a,b) and by
$\unicode[STIX]{x1D703}_{\ast }$
(c,d). The horizontal axes are scaled by the cell height
$H$
. The legends with ‘smooth’ and ‘rough’ stand for smooth cells and rough cells, respectively. In (b,d), the distance
$z$
for the data measured in the rough cell is offset by the roughness height
$h$
. The solid line in (d) is a power-law fit to the data in the range
$10^{-2}\leqslant z/H\leqslant 2\times 10^{-1}$
given by
$\unicode[STIX]{x1D70E}_{T}/\unicode[STIX]{x1D703}_{\ast }=0.53\times (z/H)^{-0.57\pm 0.03}$
. The data sources are (a) present, (b) Du & Tong (Reference Du and Tong2000) and (c) Wei et al. (Reference Wei, Chan, Ni, Zhao and Xia2014).
Measured r.m.s. temperature profiles along the centreline in a cubic cell (a,b) and in a rectangular cell (c,d). The vertical axes are normalised by
$\unicode[STIX]{x0394}T$
(a,c) and by
$\unicode[STIX]{x1D703}_{\ast }$
(b,d). The solid lines in (b,d) are power-law fits given by (b)
$\unicode[STIX]{x1D70E}_{T}(z)/\unicode[STIX]{x1D703}_{\ast }=0.25(z/H)^{-0.74\pm 0.02}$
and (d)
$\unicode[STIX]{x1D70E}_{T}(z)/\unicode[STIX]{x1D703}_{\ast }=0.20(z/H)^{-0.74\pm 0.02}$
. Insets of (b,d):
$\unicode[STIX]{x1D70E}_{T}(z)/\unicode[STIX]{x1D703}_{\ast }$
in linear–log plots. The data in the cubic cell are taken from Wang & Xia (Reference Wang and Xia2003) and those in the rectangular cell from Sun et al. (Reference Sun, Cheung and Xia2008).
We now examine the universal profiles shown in figure 1(d) in detail. They can be divided into three regions: a TBL region, a mixing zone and a core region, which are marked with different background colours. In the mixing zone spanning
$3\times 10^{-2}\leqslant z/H\leqslant 2.5\times 10^{-1}$
, a power law
$\unicode[STIX]{x1D70E}_{T}/\unicode[STIX]{x1D703}_{\ast }=0.53\times (z/H)^{-0.57\pm 0.03}$
fits the data nicely. This scaling exponent is consistent with results in the literature (Wei & Ahlers Reference Wei and Ahlers2016; Wang et al.
Reference Wang, Xu, He, Yik, Wang, Schumacher and Tong2018; He & Xia Reference He and Xia2019), and also close to the theoretical prediction of
$-1/2$
by Adrian (Reference Adrian1996). This observation further supports that
$\unicode[STIX]{x1D70E}_{T}(z)$
obeys a power-law decay in a region where a mean horizontal shear exists, and it is true for both smooth cells and rough cells. In the core region (
$0.25\leqslant z/H\leqslant 0.65$
),
$\unicode[STIX]{x1D70E}_{T}(z)/\unicode[STIX]{x1D703}_{\ast }$
is nearly a constant independent of
$Ra$
and the plate topography, suggesting a universal temperature fluctuation here.
We next test the convective temperature in other cell geometries. Figures 2(a) and 2(b) show
$\unicode[STIX]{x1D70E}_{T}(z)$
normalised respectively by
$\unicode[STIX]{x0394}T$
and
$\unicode[STIX]{x1D703}_{\ast }$
measured along the centreline in a cubic cell. The r.m.s. temperature profiles obtained along the cell centreline in a rectangular cell are shown in figure 2(c,d). The data in the cubic cell are taken from Wang & Xia (Reference Wang and Xia2003) and those in the rectangular cell are from Sun et al. (Reference Sun, Cheung and Xia2008). Let us focus on the cubic case first. It is seen that the originally
$Ra$
-dependent profiles (figure 2
a) collapse onto each other once they are scaled by
$\unicode[STIX]{x1D703}_{\ast }$
(figure 2
b). The solid line in figure 2(b) is a power-law fit to the data in the range
$2\times 10^{-2}\leqslant z/H\leqslant 3\times 10^{-1}$
, yielding
$\unicode[STIX]{x1D70E}_{T}(z)/\unicode[STIX]{x1D703}_{\ast }=0.25(z/H)^{-0.74\pm 0.02}$
. Note that
$\unicode[STIX]{x1D70E}_{T}(z)/\unicode[STIX]{x1D703}_{\ast }$
for different
$Ra$
do not collapse inside the TBL. This is because
$\unicode[STIX]{x1D703}_{\ast }$
is a temperature scale for the bulk, so is not applicable in the near-wall region where the viscosity and the thermal diffusivity dominate the dynamics. Similar behaviour is observed in the rectangular cell – that is, when compared with
$\unicode[STIX]{x1D70E}_{T}(z)/\unicode[STIX]{x0394}T$
, the
$\unicode[STIX]{x1D70E}_{T}(z)/\unicode[STIX]{x1D703}_{\ast }$
data collapse better for different
$Ra$
, and can be fitted by a power law with a scaling exponent of
$-0.74\pm 0.02$
in the mixing zone. The insets of figures 2(b) and 2(d) plot the same data as the main figures but in log–linear plots. They clearly show that these profiles cannot be fitted by logarithmic functions. Note that the scaling exponent is different from the cylindrical cells, suggesting that turbulent fluctuations in the bulk are cell-geometry-dependent, which may be partially attributed to the geometry dependence of the large-scale flow dynamics.
(a) Normalised r.m.s. temperature at the cell centre
$\unicode[STIX]{x1D70E}_{T,c}/\unicode[STIX]{x0394}T$
versus
$Ra$
. The legends with ‘rough’ mean rough cells and those with ‘cube’ means cubic cells. Details of the various data and the associated power laws are given in table 1; (b) the same data as in (a) but with the vertical axis scaled by
$\unicode[STIX]{x1D703}_{\ast }$
. The upper and lower horizontal lines mark the averaged value of 0.85 in cylindrical cells and 0.34 in cubic cells, respectively. (c) A zoom-in of the region with
$10^{8}\leqslant Ra\leqslant 10^{12}$
.
As
$\unicode[STIX]{x1D70E}_{T}/\unicode[STIX]{x1D703}_{\ast }$
in the core region is a constant (figure 1
b), we next focus on its value at the cell centre. Figure 3(a) plots
$\unicode[STIX]{x1D70E}_{T,c}/\unicode[STIX]{x0394}T$
versus
$Ra$
measured in smooth cells and rough cells from the present study. For comparison, data for
$\unicode[STIX]{x1D70E}_{T,c}/\unicode[STIX]{x0394}T$
adopted from a number of sources are also shown. These data span
$10^{7}\leqslant Ra\leqslant 10^{15}$
and
$0.7\leqslant Pr\leqslant 23.34$
, and were measured in smooth cells and rough cells. Details about their
$Ra$
,
$Pr$
, scaling property and cell type can be found in table 1.
When scaled with
$\unicode[STIX]{x0394}T$
, both the magnitude and the scaling property of
$\unicode[STIX]{x1D70E}_{T,c}$
are seen to vary dramatically among different experiments (figure 3
a). The data reveal that
$\unicode[STIX]{x1D70E}_{T,c}/\unicode[STIX]{x0394}T$
generally decreases with
$Pr$
. The present measurements in rough cells even show a positive
$\unicode[STIX]{x1D6FE}$
(that is,
$\unicode[STIX]{x1D70E}_{T,c}/\unicode[STIX]{x0394}T\sim Ra^{0.1}$
). Figure 3(b) plots the same data as those in figure 3(a), but scaled with
$\unicode[STIX]{x1D703}_{\ast }$
, and figure 3(c) shows a zoomed region with
$10^{8}\leqslant Ra\leqslant 10^{12}$
. To obtain
$\unicode[STIX]{x1D70E}_{T,c}/\unicode[STIX]{x1D703}_{\ast }$
, we first took published data of
$\unicode[STIX]{x1D70E}_{T,c}/\unicode[STIX]{x0394}T$
as a function of
$Ra$
and
$Pr$
from the cited references, then calculated
$\unicode[STIX]{x1D703}_{\ast }/\unicode[STIX]{x0394}T$
using (2.1a
) with
$Ra$
,
$Pr$
and
$Nu$
that were measured together with
$\unicode[STIX]{x1D70E}_{T,c}$
. The ratio
$(\unicode[STIX]{x1D70E}_{T,c}/\unicode[STIX]{x0394}T)/(\unicode[STIX]{x1D703}_{\ast }/\unicode[STIX]{x0394}T)$
is therefore
$\unicode[STIX]{x1D70E}_{T,c}/\unicode[STIX]{x1D703}_{\ast }$
.
We first look at data measured in cylindrical cells. It is seen that
$\unicode[STIX]{x1D70E}_{T,c}/\unicode[STIX]{x1D703}_{\ast }$
from different experiments collapses around a straight line for
$Ra>1\times 10^{8}$
(in the so-called ‘hard turbulence’ regime), suggesting the existence of a universal constant. The upper solid line in figure 3(b) marks the mean value (
$\langle \unicode[STIX]{x1D70E}_{T,c}/\unicode[STIX]{x1D703}_{\ast }\rangle _{Ra,Pr}=0.85$
). A mean value close to unity suggests that
$\unicode[STIX]{x1D703}_{\ast }$
is indeed a representative temperature in the core region. This universal constant indicates that a common mechanism governs temperature fluctuation dynamics in the core region.
For
$\unicode[STIX]{x1D70E}_{T,c}/\unicode[STIX]{x1D703}_{\ast }$
measured in cubic cells, one sees that they decrease with
$Ra$
for
$Ra\leqslant 10^{9}$
and appear to reach another plateau around
$\langle \unicode[STIX]{x1D70E}_{T,c}/\unicode[STIX]{x1D703}_{\ast }\rangle _{Ra,Pr}=0.34$
afterwards (the lower solid line in figure 3
b). If this feature can be verified by data for higher
$Ra$
, then this means the above constant is cell-geometry-dependent. This dependence may be a result of the different azimuthal dynamics of the large-scale flow in the two geometries. A previous study suggests that the scaling of the r.m.s. temperature depends on cell geometry (Daya & Ecke Reference Daya and Ecke2001). Now one sees that
$\unicode[STIX]{x1D70E}_{T,c}/\unicode[STIX]{x1D703}_{\ast }$
reaches different plateaus in cells with different geometries. It can thus be used to characterise and quantify the level of turbulent fluctuations in different cell geometries. The apparent different behaviours in the bulk fluctuation remain to be explained. We note a potential application of the above result is in cases where it is difficult to measure
$\unicode[STIX]{x1D70E}_{T}$
directly. In such situations, one can use global quantities such as
$Ra$
,
$Nu$
and
$Pr$
to obtain an estimate of the level of turbulent fluctuations in the core region.
(a) Measured Reynolds number
$Re_{\unicode[STIX]{x1D70E}_{w,c}}$
in a rough cell.
$Re_{\unicode[STIX]{x1D70E}_{w,c}}$
values obtained in smooth cells from the literature are also shown. The solid lines are power-law fits to the data, yielding
$Re_{\unicode[STIX]{x1D70E}_{w,c}}=0.021Ra^{0.50\pm 0.01}$
(circles, rough cell) and
$Re_{\unicode[STIX]{x1D70E}_{w,c}}=0.014Ra^{0.50\pm 0.01}$
(smooth cell). (b) Values of
$\unicode[STIX]{x1D70E}_{w,c}$
normalised by
$w_{\ast }$
versus
$Ra$
. The two solid lines are power-law fits to the data:
$\unicode[STIX]{x1D70E}_{w,c}/w_{\ast }=0.08Ra^{0.07\pm 0.02}$
(upper line) and
$\unicode[STIX]{x1D70E}_{w,c}/w_{\ast }=0.07Ra^{0.07\pm 0.02}$
(lower line).
3.2 The vertical velocity fluctuation in the bulk
We study the velocity fluctuations in this section. Figure 4(a) shows the Reynolds number
$Re_{\unicode[STIX]{x1D70E}_{w,c}}$
based on the vertical velocity fluctuation
$\unicode[STIX]{x1D70E}_{w,c}$
measured at the centre of the rough cell. For comparison,
$Re_{\unicode[STIX]{x1D70E}_{w,c}}$
obtained in smooth cells from Shen et al. (Reference Shen, Xia and Tong1995), Daya & Ecke (Reference Daya and Ecke2001), Qiu et al. (Reference Qiu, Shang, Tong and Xia2004) and Shang et al. (Reference Shang, Tong and Xia2008) in cylinders and from Daya & Ecke (Reference Daya and Ecke2001) and Kaczorowski et al. (Reference Kaczorowski, Chong and Xia2014) in cubes are also shown. The
$Re_{w,c}$
values in both smooth and rough cells with
$\unicode[STIX]{x1D6E4}=1$
can be described by power laws:
$Re_{\unicode[STIX]{x1D70E}_{w,c}}=0.021Ra^{0.50\pm 0.01}$
(rough cell) and
$Re_{\unicode[STIX]{x1D70E}_{w,c}}=0.014Ra^{0.50\pm 0.01}$
(smooth cell). Note a transition is seen for
$Ra\geqslant 10^{9}$
in the data obtained in a cube by Daya & Ecke (Reference Daya and Ecke2001), implying a geometry dependence of the velocity fluctuation. As these data are taken in a cell with
$\unicode[STIX]{x1D6E4}=0.7$
, they are not used in the fitting. The velocity fluctuations in the rough cell are enhanced by 50 % when compared with those in smooth cells. This directly confirms that not only are the temperature fluctuations enhanced in a rough cell, but also the velocity fluctuations.
To test if the convective velocity
$w_{\ast }$
is a characteristic scale for velocity in the core region, we study
$\unicode[STIX]{x1D70E}_{w,c}/w_{\ast }$
. The
$\unicode[STIX]{x1D70E}_{w,c}/w_{\ast }$
data are obtained by first calculating
$Re_{\unicode[STIX]{x1D70E}_{w,c}}$
, and then
$Re_{w_{\ast }}$
using (2.1b
) with
$Ra$
,
$Pr$
and
$Nu$
values that were measured simultaneously with
$\unicode[STIX]{x1D70E}_{w,c}$
. The ratio between
$Re_{\unicode[STIX]{x1D70E}_{w,c}}$
and
$Re_{w_{\ast }}$
is therefore
$\unicode[STIX]{x1D70E}_{w,c}/w_{\ast }$
. Figure 4(b) plots
$\unicode[STIX]{x1D70E}_{w,c}/w_{\ast }$
versus
$Ra$
. For the data from Daya & Ecke (Reference Daya and Ecke2001), Qiu et al. (Reference Qiu, Shang, Tong and Xia2004) and Shang et al. (Reference Shang, Tong and Xia2008), there are no
$Nu$
data available. To obtain
$Re_{w\ast }$
, we used the heat transport scaling relation
$Nu=0.14Ra^{0.297}Pr^{-0.03}$
, which was obtained in the
$Ra$
range
$2\times 10^{7}\leqslant Ra\leqslant 3\times 10^{10}$
and the
$Pr$
range
$4\leqslant Pr\leqslant 1350$
(Xia, Lam & Zhou Reference Xia, Lam and Zhou2002). For the rest of the data,
$Re_{\unicode[STIX]{x1D70E}_{w,c}}$
and the corresponding
$Nu$
were measured simultaneously. Interestingly, the data in rough cells and smooth cells collapse onto each other, suggesting that
$\unicode[STIX]{x1D70E}_{w,c}/w_{\ast }$
exhibits universal behaviour that is independent of the plate topography. The scaled
$\unicode[STIX]{x1D70E}_{w,c}$
shows a rather weak
$Ra$
-dependence, given by
$\unicode[STIX]{x1D70E}_{w,c}/w_{\ast }\sim Ra^{0.07\pm 0.02}$
, as indicated by the solid lines in the figure. Note that the data from Shang et al. (Reference Shang, Tong and Xia2008) are
${\sim}14\,\%$
lower than the others. We currently do not understand this small difference. It may be due to the systematic error introduced when calculating
$w_{\ast }$
, which involves values of
$Nu$
that were not measured simultaneously with
$\unicode[STIX]{x1D70E}_{w,c}$
.
Profiles of vertical r.m.s. velocity
$\unicode[STIX]{x1D70E}_{w}$
. The vertical axes are scaled by (a) the free-fall velocity
$U_{f}$
and (b) the convective velocity
$w_{\ast }$
. The solid line in (b) is a logarithmic fit to the data in the range
$4\times 10^{-3}\leqslant z/H\leqslant 7\times 10^{-2}$
, yielding
$\unicode[STIX]{x1D70E}_{w}/w_{\ast }=0.22\ln (z/H)+1.22$
. Inset of (b):
$\unicode[STIX]{x1D70E}_{w}(z)/w_{\ast }$
in a log–log plot. The data are taken from Scheel & Schumacher (Reference Scheel and Schumacher2016).
Next, we examine the profiles of the vertical r.m.s. velocity
$\unicode[STIX]{x1D70E}_{w}(z)$
in the mixing zone. Experimentally, obtaining velocity r.m.s. profiles requires long-time and multi-point measurement. Because of this, experimentally measured profiles of
$\unicode[STIX]{x1D70E}_{w}$
are scarce. For this reason, we use profiles from numerical studies. Figure 5(a) shows the horizontally averaged vertical r.m.s. velocity profiles
$\unicode[STIX]{x1D70E}_{w}(z)$
, which are adopted from Scheel & Schumacher (Reference Scheel and Schumacher2016). Both
$Ra$
and
$Pr$
for these profiles vary over a wide range. It is seen that, when normalised by the free-fall velocity
$U_{f}$
, the profiles do not collapse onto a single curve, suggesting that
$U_{f}$
is not a characteristic velocity scale in the bulk. Figure 5(b) plots the same data set as those in figure 5(a), but with the vertical axis scaled by
$w_{\ast }$
. The data for different
$Ra$
and
$Pr$
now collapse onto each other in the mixing zone, suggesting that
$w_{\ast }$
is a proper velocity scale. The
$\unicode[STIX]{x1D70E}_{w,c}/w_{\ast }$
data in the range
$4\times 10^{-3}\leqslant z/H\leqslant 7\times 10^{-2}$
can be fitted by a logarithmic function, given by
$\unicode[STIX]{x1D70E}_{w}/w_{\ast }=0.22\ln (z/H)+1.22$
. This logarithmic dependence is in agreement with the theoretical prediction by Adrian (Reference Adrian1996). It is also observed that the logarithmic region increases with
$Ra$
and
$Pr$
. The inset of figure 5(b) plots the same data as the main figure on a log–log scale, showing that the data cannot be fitted by a power law. Note
$\unicode[STIX]{x1D70E}_{w,c}/w_{\ast }$
values in the core region are larger than those shown in figure 4(b) and they do not collapse onto each other. A possible reason may be that the data presented in figure 4 were taken at a single point (the cell centre), but those in figure 5 were averaged along a horizontal cross-section which includes large velocity fluctuations produced by thermal plumes carried with the large-scale circulation.
3.3 Implications for ultrahigh-Rayleigh-number convection
We now study whether the convective temperature is applicable to the ultrahigh-
$Ra$
(
${\geqslant}10^{13}$
) data and may possibly shed some light on the turbulence in this regime. The data are taken from Ahlers et al. (Reference Ahlers, Bodenschatz, Funfschilling, Grossmann, He, Lohse, Stevens and Verzicco2012) with an adapted plot shown in figure 6(a). Those data were measured in pressurised SF6 gas with
$Ra$
reaching
$1\times 10^{15}$
and
$Pr\approx 0.8$
. It is seen that the logarithmic functions fit
$\unicode[STIX]{x1D70E}_{T}(z)/\unicode[STIX]{x0394}T$
nicely. But data for different
$Ra$
do not collapse, and they seem to have similar decay rates (the pre-factor of the logarithmic term). The observed logarithmic dependence in the plume abundant region, near the sidewall, was also obtained by He & Xia (Reference He and Xia2019) recently. In figure 6(b), we show that once
$\unicode[STIX]{x1D70E}_{T}(z)$
is scaled by
$\unicode[STIX]{x1D703}_{\ast }$
, they fall into two groups. The solid lines in the figure are logarithmic fits to the data in the range
$1.8\times 10^{-2}\leqslant z/H\leqslant 1.46\times 10^{-1}$
, yielding
$\unicode[STIX]{x1D70E}_{T}(z)/\unicode[STIX]{x1D703}_{\ast }=-0.49\ln (z/H)+1.15$
for
$Ra\geqslant 7.90\times 10^{14}$
and
$\unicode[STIX]{x1D70E}_{T}(z)/\unicode[STIX]{x1D703}_{\ast }=-0.30\ln (z/H)+1.20$
for
$Ra\leqslant 1.18\times 10^{13}$
. The difference in the decay rates suggests that the bulk fluctuations undergo a transition. We note that He et al. (Reference He, Funfschilling, Nobach, Bodenschatz and Ahlers2012) have stated that the system has reached the ultimate state of thermal convection for
$Ra\geqslant 5\times 10^{14}$
. A systematic investigation on the
$Ra$
-dependence of
$\unicode[STIX]{x1D703}_{T}(z)/\unicode[STIX]{x1D703}_{\ast }$
could provide more evidence on the existence of an internal flow state transition.
The r.m.s. temperature profiles measured in ultrahigh-
$Ra$
convection. (a) Figure adapted from Ahlers et al. (Reference Ahlers, Bodenschatz, Funfschilling, Grossmann, He, Lohse, Stevens and Verzicco2012). (b) The same set of data as in (a) with the vertical axis scaled by
$\unicode[STIX]{x1D703}_{\ast }$
. The solid lines in (b) are logarithmic fits to the data with solid symbols:
$\unicode[STIX]{x1D70E}_{T}(z)/\unicode[STIX]{x1D703}_{\ast }=-0.49\ln (z/H)+1.15$
(upper line) and
$\unicode[STIX]{x1D70E}_{T}(z)/\unicode[STIX]{x1D703}_{\ast }=-0.30\ln (z/H)+1.20$
(lower line).
4 Conclusions
We have studied the temperature and the velocity fluctuations in the bulk of Rayleigh–Bénard turbulence using new experimental data from the present study and experimental and numerical data from previous studies. We show that, when scaled by the convective temperature
$\unicode[STIX]{x1D703}_{\ast }$
, the r.m.s. temperature at the cell centre is a constant (
$\unicode[STIX]{x1D70E}_{T,c}/\unicode[STIX]{x1D703}_{\ast }\approx 0.85$
) over the Rayleigh number range of
$10^{8}\leqslant Ra\leqslant 10^{15}$
and the Prandtl number range of
$0.7\leqslant Pr\leqslant 23.34$
, and is independent of the surface topographies of the top and bottom plates of the convection cell. A constant close to unity suggests that the convective temperature is a proper measure of the temperature fluctuations in the core region. The vertical r.m.s. velocity, on the other hand, shows a rather weak
$Ra$
-dependence, given by
$\unicode[STIX]{x1D70E}_{w,c}/w_{\ast }\sim Ra^{0.07\pm 0.02}$
, which is also independent of the plate topography. In the mixing zone with a mean horizontal shear, the r.m.s. temperature profile
$\unicode[STIX]{x1D70E}_{T}(z)/\unicode[STIX]{x1D703}_{\ast }$
obeys a power-law dependence on the vertical distance
$z$
from the plate, and the vertical r.m.s. velocity profile
$\unicode[STIX]{x1D70E}_{w}(z)/w_{\ast }$
obeys a logarithmic dependence on
$z$
. The study thus demonstrates that the typical scales for the temperature and the velocity are the convective temperature
$\unicode[STIX]{x1D703}_{\ast }$
and the convective velocity
$w_{\ast }$
, respectively. The discovery of these universal aspects of fluctuations sheds new light on the bulk dynamics in convective turbulence. We further show that
$\unicode[STIX]{x1D703}_{\ast }$
could be used to study internal flow state transitions in the ultrahigh-
$Ra$
turbulent convection. We note that, despite universal properties holding over a wide range of
$Ra$
,
$Pr$
and surface topographies,
$\unicode[STIX]{x1D70E}_{T,c}/\unicode[STIX]{x1D703}_{\ast }$
is found to depend on the cell geometry – it is a constant of 0.34 in cubes and 0.85 in cylinders. The scaling exponent of the
$\unicode[STIX]{x1D70E}_{T}/\unicode[STIX]{x1D703}_{\ast }$
profile also depends on the cell geometry. Finally, the present study focuses on turbulent fluctuations in convection cells with an aspect ratio close to unity. It will be interesting to find out if the universal behaviour observed here will also exist in cells of different aspect ratios.
Acknowledgement
We thank S.-D. Huang and Y.-H. He for helpful discussions. This work was supported by a SUSTech Startup Fund and by the Hong Kong Research Grant Council under grant nos CUHK 14301115 and 14302317.
