1. Introduction
Turbulent thermal convection is an important phenomenon both in nature and in industry. One common model system is the Rayleigh–Bénard cell, which consists in a fluid layer confined inside adiabatic walls, heated from below and cooled from above by horizontal smooth plates. In the limit of Boussinesq conditions, this system is controlled by only three non-dimensional parameters: the Rayleigh number,
$\mathit{Ra}$
, the Prandtl number,
$\mathit{Pr}$
, and the cell aspect ratio
${\it\Gamma}$
.
The Rayleigh number,
$\mathit{Ra}$
, accounts for the thermal forcing,
$$\begin{eqnarray}\mathit{Ra}=\frac{g{\it\alpha}(T_{h}-T_{c})H^{3}}{{\it\nu}{\it\kappa}},\end{eqnarray}$$
with
$g$
the acceleration due to gravity,
${\it\alpha}$
the fluid thermal expansion coefficient,
$T_{h}$
the temperature of the hot plate,
$T_{c}$
the temperature of the cold plate,
$H$
the cell height,
${\it\nu}$
the fluid kinematic viscosity and
${\it\kappa}$
the fluid thermal diffusivity.
The Prandtl number,
$\mathit{Pr}$
, is a property of the fluid and compares the two diffusion mechanisms that impede convection,
$$\begin{eqnarray}\mathit{Pr}=\frac{{\it\nu}}{{\it\kappa}}.\end{eqnarray}$$
The aspect ratio,
${\it\Gamma}$
, is a geometric parameter of the cell,
$$\begin{eqnarray}{\it\Gamma}=\frac{W}{H},\end{eqnarray}$$
where
$W$
is the width of the cell and
$H$
its height.
The system output can be assessed by the Nusselt number,
$Nu$
, which compares the thermal heat flux,
$\dot{q}$
, to the case of a purely diffusive situation where the heat flux,
$\dot{q}_{diff}$
, is given by
$$\begin{eqnarray}\dot{q}_{diff}=\frac{{\it\lambda}(T_{h}-T_{c})}{H},\end{eqnarray}$$
where
${\it\lambda}$
is the fluid thermal conductivity. The Nusselt number can thus be written
$$\begin{eqnarray}Nu=\frac{\dot{q}H}{{\it\lambda}(T_{h}-T_{c})}.\end{eqnarray}$$
An important problem is to be able to predict the system heat flux for a given thermal forcing, i.e. to relate the Nusselt number to
$\mathit{Ra}$
and
$\mathit{Pr}$
, in particular in the case of asymptotically large forcings. Many scaling theories have been proposed, i.e. expressing the Nusselt numbers as a power law,
$Nu={\it\beta}\mathit{Ra}^{{\it\gamma}}$
, see reviews by Ahlers, Grossmann & Lohse (Reference Ahlers, Grossmann and Lohse2009), Lohse & Xia (Reference Lohse and Xia2010) and Chillà & Schumacher (Reference Chillà and Schumacher2012).
In the turbulent regime, the average temperature within the Rayleigh–Bénard cell is mostly homogeneous, except within thin fluid layers near the plates. Those layers, usually referred to as thermal boundary layers, play an important role in the dynamics of the system. One classical argument consists in considering that the heating and cooling plates are independent, thus the heat flux
$\dot{q}$
should not depend on
$H$
. This yields
${\it\gamma}=1/3$
. This prediction catches the experimental behaviour relatively well, though the situation is more complicated. Many efforts have been made to better model the experimental observations and catch the details of the
$Nu$
versus
$\mathit{Ra}$
relation (Castaing et al.
Reference Castaing, Gunaratne, Heslot, Kadanoff, Libchaber, Thomae, Wu, Zaleski and Zanetti1989; Shraiman & Siggia Reference Shraiman and Siggia1990; Grossmann & Lohse Reference Grossmann and Lohse2000; Stevens et al.
Reference Stevens, van der Poel, Grossmann and Lohse2013).
In particular, an open problem is the possible destabilization of these boundary layers and its effect on the asymptotic scaling laws. Half a century ago, Kraichnan predicted that the transition to a turbulent boundary layer would increase the scaling law to
${\it\gamma}=1/2$
with logarithmic corrections (Kraichnan Reference Kraichnan1962). Because this scaling can be derived as a rigorous upper bound (Doering & Constantin Reference Doering and Constantin1996), it means that it will not undergo further qualitative changes as
$\mathit{Ra}$
tends to infinity, and thus this regime is sometimes referred to as the ultimate regime of convection.
There are only a few laboratory set-ups capable of achieving high enough Rayleigh numbers to reach the transition towards this regime. Such an experimental observation was first reported in a cryogenic Rayleigh–Bénard cell using gaseous helium as the working fluid (Chavanne et al.
Reference Chavanne, Chillà, Castaing, Hébral, Chabaud and Chaussy1997). The data are compatible with the prediction from Kraichnan, with logarithmic corrections. The critical Rayleigh number then reported was
$10^{11}$
, though the precise value is somewhat uncertain because it depends on the exponent of the scaling one uses as a reference. A deviation from
$Nu\sim \mathit{Ra}^{2/7}$
was considered in the case of the original paper of Chavanne et al. If one chooses to consider a deviation from
${\it\gamma}=1/3$
instead, then the same dataset yields a critical Rayleigh number closer to
$10^{12}$
, as was subsequently reported by the Grenoble group (Gauthier et al.
Reference Gauthier, Salort, Bourgeois, Garden, du Puits, Thess and Roche2009; Roche et al.
Reference Roche, Gauthier, Kaiser and Salort2010).
New experimental heat-flux measurements have been reported recently by the MPIDS group in Göttingen for Rayleigh numbers beyond
$10^{12}$
, using pressurized SF
$_{6}$
as the working fluid (Ahlers et al.
Reference Ahlers, Grossmann and Lohse2009, Reference Ahlers, He, Funfschilling and Bodenschatz2012b
; Funfschilling, Bodenschatz & Ahlers Reference Funfschilling, Bodenschatz and Ahlers2009; He et al.
Reference He, Funfschilling, Nobach, Bodenschatz and Ahlers2012). They have obtained several possible effective scaling exponents, possibly less steep than
${\it\gamma}=1/3$
, but eventually close to
${\it\gamma}=0.36$
for
$\mathit{Ra}>10^{14}$
. A theoretical interpretation has been proposed by Grossmann & Lohse (Reference Grossmann and Lohse2011), based on an extension of their original unifying theory (Grossmann & Lohse Reference Grossmann and Lohse2001) to the case of turbulent boundary layers and logarithmic velocity profiles.
Our alternative approach in the experimental work presented in this paper aims at triggering the transition to a turbulent boundary layer at a lower Rayleigh number by means of a controlled roughness. Indeed, the critical shear Reynolds number can be lowered in the presence of roughness (Schlichting & Gersten Reference Schlichting and Gersten2000).
Several groups have reported roughness-induced heat-transfer enhancements. Groove-shaped roughness has been added to the already-transiting Rayleigh–Bénard cell in Grenoble, which allowed the scaling exponent to be further increased from an effective 0.38 due to the logarithmic corrections to the purely asymptotic
${\it\gamma}=1/2$
(Roche et al.
Reference Roche, Castaing, Chabaud and Hébral2001). A numerical simulation was performed by Stringano, Pascazio & Verzicco (Reference Stringano, Pascazio and Verzicco2006) in this particular geometry and backed the increase of
${\it\gamma}$
.
Pyramid-shaped roughness was used in Hong Kong. Though it does not always produce a change in
${\it\gamma}$
, it still yields enhancement of plume emissions and a 20–76 % increase for
$Nu$
. This is larger than the increase due to the increase in heating area caused by the roughness (Shen, Tong & Xia Reference Shen, Tong and Xia1996; Du & Tong Reference Du and Tong1998, Reference Du and Tong2000; Qiu, Xia & Tong Reference Qiu, Xia and Tong2005). The result has been extended recently to the case where roughness is added on one plate only (Wei et al.
Reference Wei, Chan, Ni, Zhao and Xia2014).
Spherical roughness was used also by Ciliberto & Laroche (Reference Ciliberto and Laroche1999), distributed either randomly or periodically on the bottom plate. The roughness was made of glass and therefore changed the plate conductivity locally as well. They reported an increase from
${\it\gamma}=2/7$
to a higher exponent, only in the case of randomly distributed spheres.
Our own previous experiment, carried out in Lyon, involved a
$40~\text{cm}\times 40~\text{cm}$
rectangular cell with a rough bottom plate and water as the working fluid (Salort et al.
Reference Salort, Liot, Rusaouen, Seychelles, Tisserand, Creyssels, Castaing and Chillà2014). The controlled roughness consisted in an array of obstacles
$2~\text{mm}$
high and
$5~\text{mm}\times 5~\text{mm}$
square, evenly spaced every 1 cm. Heat-transfer enhancement was observed and agreed fairly well with measurements with a similar kind of roughness inside a larger cylindrical cell (Tisserand et al.
Reference Tisserand, Creyssels, Gasteuil, Pabiou, Gibert, Castaing and Chillà2011).
The careful analysis of the local temperature fluctuations, as well as temperature visualization with background-oriented synthetic schlieren imaging, strongly suggested a destabilization of the laminar boundary layer on the top of the obstacles and a confinement of the flow between the obstacles (referred to as notches). These observations led us to propose a simple model, which accounted for the observed global heat-transfer enhancements (Salort et al. Reference Salort, Liot, Rusaouen, Seychelles, Tisserand, Creyssels, Castaing and Chillà2014).
The roughness-induced heat-transfer enhancement may depend on the details of the roughness geometry, and the influence of the roughness shape may be of great interest for engineers (García et al.
Reference García, Solano, Vicente and Viedma2012). However, in this work, we focus on general effects only, obtained once a transition to turbulence inside the boundary layer has been triggered. We choose one particular set of obstacles for its simplicity, and study how the changes in the boundary layer structure affect the dynamics of the heat transport. The aim is to provide experimental evidence of the effects of boundary layer destabilization on the global heat flux. The understanding of the mechanisms at play is of the utmost importance in predicting more generally the heat transfer in the high-
$\mathit{Ra}$
limit.
The case of periodic box-shaped roughness elements with spanwise dimension equal to the full depth of the cell has also been studied analytically (Shishkina & Wagner Reference Shishkina and Wagner2011) and numerically (Wagner & Shishkina Reference Wagner and Shishkina2015). The effect of the height, width and distance between the obstacles is studied. They find an increase for
${\it\gamma}$
, which then saturates at larger
$\mathit{Ra}$
when the cavities (that we refer to as notches in the present work) are fully washed out. The main differences from the other studies mentioned above are that the height of these roughness elements is much larger than the typical boundary layer thickness, and that the roughness consists only of a set of four obstacles on the plate. Although this might be a different situation than the case considered in the present paper of a rough surface where the roughness has dimensions comparable to those of the boundary layers, and consists of a much larger set of obstacles, it nevertheless triggers changes in the boundary layer structure that can be compared to the situation presented in this work.
In this paper, we present new measurements carried out inside the Barrel of Ilmenau. The Barrel of Ilmenau is the world’s largest experiment (up to
$7.0~\text{m}\times 6.3~\text{m}$
) to study highly turbulent convection in air with unrivalled spatial and temporal resolution. A rectangular cell, with proportions strictly identical to the water cell in Lyon, but six times larger, has been inserted inside the Barrel. As in the Lyon experiment, the top plate is smooth and the bottom plate is rough. This configuration has allowed us to study velocity fields using particle image velocimetry (PIV) near the obstacles, as well as the local heat flux on the bottom plate, and thus to go beyond our previous analysis. The observation of logarithmic profiles of velocity reported at high Rayleigh numbers in this work is direct evidence of the destabilization of the boundary layers. This is not to be confused with the logarithmic profiles of temperature that have been observed above smooth plates, in both the classical and ultimate states (Ahlers et al.
Reference Ahlers, Bodenschatz, Funfschilling, Grossmann, He, Lohse, Stevens and Verzicco2012a
; Ahlers, Bodenschatz & He Reference Ahlers, Bodenschatz and He2014; Wei & Ahlers Reference Wei and Ahlers2014), and which do not necessarily imply a transition to turbulent boundary layers.
2. Experimental set-up
Sketch of the convection cell. The green surfaces represent the laser sheets, either vertical for PIV in the groove and on the obstacle, or horizontal for PIV inside the notch.
The convection cell is a 0.62 m thick,
$2.50~\text{m}\times 2.50~\text{m}$
rectangular cell with 0.50 cm thick walls (see sketch in figure 1). The walls are inserted inside the Barrel of Ilmenau. The bottom and top plates are directly those of the Barrel itself and are described in more detail in previous work by du Puits, Resagk & Thess (Reference du Puits, Resagk and Thess2013). The controlled roughness consists in an array of 1.2 cm high,
$3~\text{cm}\times 3~\text{cm}$
square aluminium obstacles, evenly glued on the bottom plate. As shown in figure 2, because these obstacles are aligned with the walls, it is possible to distinguish between ‘grooves’ washed by the mean wind and ‘notches’ between obstacles where the fluid is confined.
Picture of the roughness on the bottom plate. The blue coloured area is an example of a ‘groove’ portion, washed by the mean wind. The green coloured area is a example of a ‘notch’, where the fluid is confined between obstacles.
In this configuration, the rectangular cell is fully surrounded by the larger cylindrical Rayleigh–Bénard cell, 7.15 m in diameter and 2.50 m in height. The vertical temperature distribution is almost the same inside and outside the enclosure, therefore the lateral heat exchange throughout the walls is negligible. The working fluid is air. The Prandtl number is 0.71.
Experimental conditions. The first column indicates the kind of measurement that has been performed: particle image velocimetry near the bottom rough plate (PIV) and/or thermal heat-flux measurements (HFS). The Nusselt number is the estimation of the local Nusselt number at
$(x,y)=(60,31~\text{cm})$
, see text for details.
Although this work is mainly about PIV and characterization of the velocity boundary layers, the Nusselt numbers have also been estimated, to determine when the critical Nusselt number, where roughness-induced heat-transfer enhancement is expected, has been reached. Because there are only three Rayleigh numbers (given in table 1), it is not possible to infer power-law scalings. The heat flux is computed from three commercial sensors, identical to those used previously in the facility (du Puits, Resagk & Thess Reference du Puits, Resagk and Thess2010; Kaiser & du Puits Reference Kaiser and du Puits2014): two Omega Newport HFS-4 sensors that measure the heat flux on the top of an obstacle,
$\dot{q}_{plot}$
, and the heat flux inside a notch,
$\dot{q}_{notch}$
, and one PhyMeas sensor that measures the heat flux in a groove,
$\dot{q}_{groove}$
.
The radiative heat flux cannot be neglected; it accounts for 10–30 % of the total heat flux in our set-up. To accurately estimate the convective Nusselt number, we subtracted an estimated radiative flux using the same model as Kaiser & du Puits (Reference Kaiser and du Puits2014), with emissivities
${\it\epsilon}_{c}=0.05$
for the cooling plate,
${\it\epsilon}_{h}=0.07$
for the heating plate and
${\it\epsilon}_{w}=0.24$
for the walls because they were coated with a thin foil during the heat-flux measurements.
The total convective heat flux,
$\dot{q}$
, is then estimated by summing the contributions:
$$\begin{eqnarray}\dot{q}={\textstyle \frac{1}{2}}\dot{q}_{groove}+{\textstyle \frac{1}{4}}\dot{q}_{notch}+{\textstyle \frac{1}{4}}\dot{q}_{plot}.\end{eqnarray}$$
The three heat-flux sensors are located near one another, close to
$x=60~\text{cm}$
from the sidewall (horizontal direction in figure 1) and
$y=31~\text{cm}$
(depth direction in figure 1).
Non-dimensional local heat-flux measurements at
$x=60~\text{cm}$
inside the rough cell (red open squares). The error bars represents the 5.5 % uncertainty of the commercial heat-flux sensors on the dimensionless heat flux. The results from Kaiser (Reference Kaiser2015) inside a smooth cell are shown for reference: whole-plate heat-flux measurements from an infrared camera (black circles) with an uncertainty of 7 %, with heat flux averaged over a
$3~\text{cm}\times 3~\text{cm}$
area at
$x=60~\text{cm}$
(down-pointing green triangles) and at
$x=190~\text{cm}$
(up-pointing cyan triangles). The horizontal dashed line indicates the transition Nusselt number,
$Nu_{c}=H/(2h_{0})$
, above which a heat-transfer enhancement is expected.
To assess whether there is a heat-transfer enhancement, the Nusselt number obtained from that heat flux is compared in figure 3 with reference data obtained in the same cell but with smooth boundaries. We use previous measurements by Kaiser (Reference Kaiser2015) as the reference data for the smooth case. Kaiser & du Puits (Reference Kaiser and du Puits2014) have shown, however, that the heat flux is not homogeneous on the plate, so comparison between the present local measurement and whole-plate heat fluxes should be carried out with care. Both local and global reference data are available in the smooth case (Kaiser & du Puits Reference Kaiser and du Puits2014). The global values are averages computed from whole-plate heat-flux maps obtained with an infrared camera.
In the present work, we measured local heat fluxes only, and these can be compared to local smooth estimates, computed from the heat-flux map by averaging over a
$3~\text{cm}\times 3~\text{cm}$
area, at the same position as the sensors in the present work on both sides of the cell, because the local heat flux depends on the wind direction.
The transition towards a roughness-induced enhanced heat-transfer regime is expected to occur when the thermal boundary layer thickness,
${\it\delta}_{th}$
, matches the height of the roughness,
$h_{0}$
(Tisserand et al.
Reference Tisserand, Creyssels, Gasteuil, Pabiou, Gibert, Castaing and Chillà2011). Therefore, the expected critical Nusselt number,
$Nu_{c}$
, is
$$\begin{eqnarray}Nu_{c}=\frac{H}{2h_{0}}.\end{eqnarray}$$
This critical Nusselt number is shown as a dashed horizontal line in figure 3.
The Nusselt number in the rough case, computed with the local heat flux (2.1), is in quantitative agreement with the smooth case at the intermediate Rayleigh number,
$\mathit{Ra}=1.59\times 10^{10}$
, close to the expected transition threshold. At the lowest Rayleigh number,
$\mathit{Ra}=4.66\times 10^{9}$
, the Nusselt number in the rough case might be slightly lower than expected in the smooth case. This can be caused by the additional heat resistance due to the fluid inside the notch, as was suggested previously by Tisserand et al. (Reference Tisserand, Creyssels, Gasteuil, Pabiou, Gibert, Castaing and Chillà2011). The Nusselt number in the rough case at the highest Rayleigh number,
$\mathit{Ra}=3.64\times 10^{10}$
, is higher than the Nusselt number in the smooth case. This can be interpreted as roughness-triggered heat-transfer enhancement.
2.1. Velocity measurements
The flow was seeded with cold-atomized droplets of di-ethyl-hexyl-sebacate with a typical size of
$1~{\rm\mu}\text{m}$
, identical to those used previously by du Puits et al. (Reference du Puits, Li, Resagk and Thess2014). Their size is sufficiently small for them to behave as tracers. As shown in figure 1, these particles were illuminated with either a horizontal (for visualization in the groove and on the top of the obstacles) or vertical (for visualization inside the notch) laser light sheet of about
$70~\text{mm}$
height and
$2~\text{mm}$
thickness, generated by a 2 W cw laser in combination with a beam expander. The fast acquisition of the particle motion was captured using an IOI Flare-2M360-CL
$2048\times 1088$
camera with a frame rate between 340 Hz at the lowest
$\mathit{Ra}$
at full resolution and 902.5 Hz at the highest
$\mathit{Ra}$
at
$2040\times 400$
resolution.
We have recorded three sequences of 20 s every five minutes at the lowest
$\mathit{Ra}$
, or every minute at the highest
$\mathit{Ra}$
, and sequences of 2 s every minute for one hour at all
$\mathit{Ra}$
. The velocity fields are then computed using a cross-correlation PIV algorithm implemented in the CIVx software suite (Fincham & Delerce Reference Fincham and Delerce2000), above and below the critical Nusselt number
$Nu_{c}$
, and near several positions on the rough plate: on top of an obstacle, in a groove and inside a notch (see figure 4).
Summary of the notations and main dimensions of the system.
The fluid inside the notch is almost at rest, as indirectly assumed by Salort et al. (Reference Salort, Liot, Rusaouen, Seychelles, Tisserand, Creyssels, Castaing and Chillà2014). Due to the limitations of our acquisition system, it is not possible to resolve the details of both the slow recirculation inside the notch and the much faster flow away from the plate. That is why we used separate acquisitions: acquisitions above the obstacle (figure 4 a,b) and measurements inside the notch only (figure 4 c,d).
2.2. Notations for the similarity parameters
Our aim is to compare the flow features, in particular the velocity profiles, velocity boundary layer thickness and the turbulent fluctuations, above and below
$Nu_{c}$
, and compare them with smooth experimental data from the literature, and with simple classical theoretical profiles, such as the Prandtl–Blasius viscous velocity profile or the logarithmic velocity profile of isothermal turbulent shear flows (Schlichting & Gersten Reference Schlichting and Gersten2000).
In order to perform such kinds of comparison, it is necessary to specify the non-dimensional parameters and the definition of the boundary layer thickness. The notations that we use in our analysis are summarized in table 2.
As pointed out by previous experimental investigations of the velocity profiles, such as du Puits, Resagk & Thess (Reference du Puits, Resagk and Thess2007), the comparison with Blasius would normally require computation of the similarity parameter
${\it\eta}=z\sqrt{u_{max}/{\it\nu}x}$
, and one possibility would be to follow the implicit assumption of Grossmann & Lohse (Reference Grossmann and Lohse2000) that the development of the boundary layer starts at the outer edge of the plate. But then the experimental profiles drastically differ from the Blasius profile, and this observation holds also for the data reported in this paper. The comparison with Blasius-type profiles seems possible, however, if
$x$
is specified in such a way that the velocity gradients
$\text{d}u/\text{d}z$
of the Blasius prediction and the experimental data are made to match, or equivalently if the profiles are plotted in terms of
$z/{\it\delta}_{\times }$
where
${\it\delta}_{\times }$
is obtained as the distance from the plate at which the extrapolation of the tangent at
$z=0$
crosses
$u_{max}$
(Sun, Cheung & Xia Reference Sun, Cheung and Xia2008; Zhou & Xia Reference Zhou and Xia2010).
However,
$\text{d}u/\text{d}z|_{z=0}$
is ill-defined for some of the profiles above
$Nu_{c}$
. This will be discussed in more detail in the following sections. For this reason, and also to allow comparison of the shear Reynolds number measured in the smooth case by Li et al. (Reference Li, Shi, du Puits, Resagk, Schumacher and Thess2012) and Willert, du Puits & Resagk (Reference Willert, du Puits and Resagk2014), it is useful to use the displacement thickness
${\it\delta}_{displacement}$
as an alternative definition of the boundary layer thickness,
$$\begin{eqnarray}{\it\delta}_{displacement}(x)=\int _{0}^{\infty }\left(1-\frac{\langle u(x,z,t)\rangle _{t}}{u_{max}}\right)\text{d}z,\end{eqnarray}$$
where
$\langle \cdot \rangle _{t}$
is the temporal average. In practice, the upper bound is chosen at the distance
${\it\delta}_{max}$
where
$u(z_{max})=u_{max}$
.
The comparison with logarithmic profiles of classical turbulent shear flows requires one to define
$z^{+}=zU^{\ast }/{\it\nu}$
and
$u^{+}=u/U^{\ast }$
, where
$U^{\ast }$
is a characteristic velocity of the turbulent flow considered, defined such that
$$\begin{eqnarray}{\it\tau}={\it\rho}{U^{\ast }}^{2},\end{eqnarray}$$
where
${\it\tau}$
is the shear stress (Landau & Lifshitz Reference Landau and Lifshitz1987). This shear stress is linked to the Reynolds tensor and the velocity gradient:
$$\begin{eqnarray}{\it\tau}={\it\rho}\langle u^{\prime }v^{\prime }\rangle _{t}+{\it\mu}\frac{\partial u}{\partial z},\end{eqnarray}$$
where
${\it\mu}={\it\nu}{\it\rho}$
is the dynamic viscosity of the fluid.
In experimental works where the velocity gradient is well defined and well resolved at
$z=0$
, such as du Puits et al. (Reference du Puits, Resagk and Thess2007) and Willert et al. (Reference Willert, du Puits and Resagk2014), the shear stress can be computed at the wall, e.g.
${\it\tau}_{w}={\it\mu}\partial u/\partial z$
. In the following, we will rather compute
${\it\tau}$
away from the plate, where the velocity gradient is negligible, i.e.
${\it\tau}_{turb}={\it\rho}\langle u^{\prime }v^{\prime }\rangle$
, and therefore
$$\begin{eqnarray}U^{\ast }=\sqrt{\langle u^{\prime }v^{\prime }\rangle _{t}}.\end{eqnarray}$$
3. The flow field adjacent to the roughness elements
3.1. The flow field below and beyond the transition limit
There are three main simple flow structures that can be considered inside the notch, between roughness elements, sketched in figure 5: (a) stratified fluid inside the notch, below the critical limit of linear instability, with no velocity. This would lead to reduced heat transfer compared to the case of the smooth plate, as was observed by Tisserand et al. (Reference Tisserand, Creyssels, Gasteuil, Pabiou, Gibert, Castaing and Chillà2011) in a cylindrical cell; (b) slow circulation inside the notch with no fluid exchange, caused by internal convection and shear from the wind. In that case, there is only weak additional thermal resistance and the boundary condition at
$z=h_{0}$
is almost unchanged, compared to the top of the obstacle; (c) convection and mixing with the bulk flow. In that case, the notch may contribute to the heat-transfer enhancement or to the plume emission.
Mean velocity fields at
$\mathit{Ra}=4.66\times 10^{9}$
(a,c,e) and
$\mathit{Ra}=4.04\times 10^{10}$
(b,d,f). (a,b) On top of an obstacle, (c,d) inside a notch, (e,f) in a groove. The colour code is given for one Rayleigh number and is identical for the three locations. The scale of the arrows is arbitrary and differs from one plot to another to allow better visualization of the flow. The solid red line is the velocity displacement thickness,
${\it\delta}_{displacement}$
(see (2.3)).
Sketch of possible flow structure inside a notch. (a) Thermally stratified, no convection, (b) internal convection, no fluid exchange, (c) external convection with fluid exchange.
The experimental velocity fields are shown in figure 4 and provide evidence of a change of flow structure inside the notch from configuration (b) at
$\mathit{Ra}=4.66\times 10^{9}$
to configuration (c) at
$\mathit{Ra}=4.04\times 10^{10}$
. This change occurs concomitantly with heat-transfer enhancement and changes in the velocity statistics that we detail in the following subsections.
3.2. Flow structure below the transition
The velocity profiles below the transition Nusselt number
$Nu_{c}$
are shown in figure 6. They compare fairly well with typical profiles previously obtained in smooth cells (du Puits et al.
Reference du Puits, Resagk and Thess2007; Sun et al.
Reference Sun, Cheung and Xia2008; Li et al.
Reference Li, Shi, du Puits, Resagk, Schumacher and Thess2012). The negative mean velocities for
$z<0.5~\text{cm}$
inside the notch are a signature of the slow recirculation. The measurements collapse quickly above the obstacles.
Velocity profiles near the rough plate at
$\mathit{Ra}=4.66\times 10^{9}$
, in a groove (full orange triangles), on the top of a roughness element (full magenta circles) and inside a notch (cyan squares, full symbols obtained with the acquisition inside the notch, open symbols with the acquisition above the notch, away from the plate). The black dashed lines are the theoretical Prandtl–Blasius profiles. (a) Raw profiles where the origin
$z=0$
is the bottom of the roughness elements. (b) Non-dimensional profiles compared with experimental data obtained in smooth cells at
$\mathit{Ra}=7.48\times 10^{11}$
from du Puits et al. (Reference du Puits, Resagk and Thess2007) (open blue circles), at
$\mathit{Ra}=3\times 10^{9}$
from Li et al. (Reference Li, Shi, du Puits, Resagk, Schumacher and Thess2012) (open red squares) and at
$\mathit{Ra}=5.3\times 10^{9}$
from Sun et al. (Reference Sun, Cheung and Xia2008) (open green triangles).
The profiles above an obstacle or above a notch are very similar for
$z>h_{0}$
. The reason is that the boundary condition is close: zero velocity at
$z=h_{0}$
above obstacles, or almost zero velocity at
$z=h_{0}$
above notches. In other words, above obstacles and notches the velocity goes from 0 at
$z=h_{0}$
to
$u_{max}$
at
$z={\it\delta}_{max}$
, and in the groove it goes from 0 at
$z=0$
to
$u_{max}$
at a nearly similar
$z={\it\delta}_{max}$
. For this reason, the profile is much steeper above notches and obstacles than in the groove, and thus
${\it\delta}_{\times }$
, defined from the slope at origin, is much smaller in the former case.
Once shifted in
$z$
(by choosing
$z=0$
on the top of the obstacle, rather than at the bottom of the obstacle for the profiles above the notch and above the roughness element) and rescaled by
${\it\delta}_{\times }$
, the tangents at the origin are indeed collapsed, and the shape of the profile can be compared to other profiles found in the literature, and to theoretical velocity profiles (figure 6
b). The profile above a roughness element differs slightly from the Prandtl–Blasius profile, and is consistent with the results from du Puits et al. (Reference du Puits, Resagk and Thess2007) and Li et al. (Reference Li, Shi, du Puits, Resagk, Schumacher and Thess2012) obtained in the Barrel with smooth plates. On the other hand, the profiles in a groove are much closer to the profiles obtained by Sun et al. (Reference Sun, Cheung and Xia2008) and to the laminar Prandtl–Blasius profile. This may be a consequence of the confinement and the additional drag caused by the rough walls, which yields locally a smaller Reynolds number.
The displacement thickness, plotted in figure 4, is larger than
$h_{0}$
. At the lowest Rayleigh number,
$\mathit{Ra}=4.66\times 10^{9}$
, we find
${\it\delta}_{displacement}=1.4~\text{cm}$
in the groove, or 0.7 cm and 0.56 cm beyond
$z=h_{0}$
, respectively above the notch and above the roughness element. Hence, the thickness of the boundary layer,
${\it\delta}_{v}$
, defined from
$z=0$
at the bottom of roughness elements, lies between 1.4 and 1.9 cm. To compare these observations to previous results, one has to infer an estimate for the thermal boundary layer thickness. Since the Prandtl number is less than 1, the thermal boundary layer is thicker than the kinetic boundary layer. For Prandtl–Blasius boundary layers, one can show that
${\it\delta}_{th}/{\it\delta}_{v}$
scales like
${\sim}\mathit{Pr}^{-1/3}$
for large Prandtl numbers and like
${\sim}\mathit{Pr}^{-1/2}$
for small Prandtl numbers (Shishkina, Horn & Wagner Reference Shishkina, Horn and Wagner2013). In the range of intermediate Prandtl numbers where the present experiment lies, close to the centre of the cell, the direct numerical simulation study by Shishkina et al. (Reference Shishkina, Horn and Wagner2013) yields
$$\begin{eqnarray}{\it\delta}_{th}\approx 2.0~{\it\delta}_{v}.\end{eqnarray}$$
Therefore, the thermal boundary layer thickness can be estimated to lie in the range between 2.8 and 3.8 cm, which is indeed larger than
$h_{0}$
, as expected.
The outer velocity is
$u_{max}=11~\text{cm}~\text{s}^{-1}$
. Thus, the shear Reynolds number,
$$\begin{eqnarray}\mathit{Re}_{s}=\frac{{\it\delta}_{v}u_{max}}{{\it\nu}},\end{eqnarray}$$
is of order 100. This is consistent with the measurements in the same conditions but over smooth surfaces (Willert et al. Reference Willert, du Puits and Resagk2014).
3.3. Flow structure beyond the transition
At high Rayleigh number, the flow structure changes substantially: (i) the notches are fully washed by the mean flow and exchange fluid with the turbulent bulk, (ii) the velocity profile features are very different both quantitatively and qualitatively and (iii) the velocity fluctuations are relatively higher. The profiles substantially differ as well from those obtained at lower Rayleigh number: there are inflection points and changes of slope, and there is no horizontal asymptote (see figure 7). The inflection points and changes of slope for
$z<h_{0}$
may be a consequence of the drag on the rough walls. For these reasons, these data do not allow us to compute an accurate estimate for the velocity boundary layer.
${\it\delta}_{\times }$
is ill-defined because it is not clear how to define
$\partial u/\partial z$
at
$z=0$
with these profiles;
${\it\delta}_{displacement}$
can be estimated by integrating as far away from the plate as possible, yielding possibly biased values (e.g. possibly undervalued). Yet, we find
${\it\delta}_{displacement}=0.90~\text{cm}$
smaller indeed than
$h_{0}$
in the groove, and
${\it\delta}_{displacement}=0.47~\text{cm}$
on the top of the obstacle.
Velocity profiles near the rough plate at
$\mathit{Ra}=4.04\times 10^{10}$
, in a groove (full orange triangles), on the top of a roughness element (full magenta circles) and inside a notch (full cyan squares: acquisition of the low velocities, open cyan squares: acquisition of the fast velocities).
Reynolds tensor at
$\mathit{Ra}=4.66\times 10^{9}$
(a,c,e) and
$\mathit{Ra}=4.04\times 10^{10}$
(b,d,f). (a,b) On top of an obstacle, (c,d) inside a notch, (e,f) in a groove. The colour code is given for one Rayleigh number and is identical for the three locations. (a,c,e) Yields
$U^{\star }=0.77~\text{cm}~\text{s}^{-1}$
at
$\mathit{Ra}=4.66\times 10^{9}$
, (b,d,f)
$U^{\star }=4.96~\text{cm}~\text{s}^{-1}$
at
$\mathit{Ra}=4.04\times 10^{10}$
.
Non-dimensional velocity profile in a groove (orange triangles) and above an obstacle (magenta circles) at
$\mathit{Ra}=4.04\times 10^{10}$
. Green dashed line:
$2.40\log z^{+}+B$
.
It is not possible to find the maximum velocity
$u_{max}$
. The typical value of the wind, however, appears to be only 3–4 times larger than before the transition, while the Reynolds tensor
$\langle u^{\prime }v^{\prime }\rangle$
, on the other hand, is typically 50 times larger after the transition. The fields of
$\langle u^{\prime }v^{\prime }\rangle$
are given in figure 8. The maximum value of the Reynolds tensor can be used as a definition for
$U^{\star }$
. We find
$U^{\star }=4.96~\text{cm}~\text{s}^{-1}$
at
$\mathit{Ra}=4.04\times 10^{10}$
, which allows us to compute the typical scale
$z^{\star }$
, defined as
$$\begin{eqnarray}z^{\star }=\frac{{\it\nu}}{U^{\star }},\end{eqnarray}$$
and an estimate of the thickness of the viscous sublayer,
${\it\delta}$
, classically defined as (Tennekes & Lumley Reference Tennekes and Lumley1987)
$$\begin{eqnarray}{\it\delta}\approx 5z^{\star }.\end{eqnarray}$$
This yields
${\it\delta}\approx 1.7~\text{mm}$
.
The figure shows, after the transition, regions with high values of
$\langle u^{\prime }v^{\prime }\rangle$
, particularly downstream obstacles, yielding regions of high strain near the top of the obstacles. This suggests a possible transition to a turbulent boundary layer, specifically on the top of the obstacles, in agreement with our previous indirect observations in water (Salort et al.
Reference Salort, Liot, Rusaouen, Seychelles, Tisserand, Creyssels, Castaing and Chillà2014). These regions can be seen also in the groove, further downstream than those inside the notch. These regions may be interpreted as wakes produced by the obstacles, advected streamwise, as well as spanwise, downstream of the roughness elements. One may note that the typical turbulent velocity,
$U^{\ast }$
, associated with those structures is found to be nearly identical in the three fields.
To figure out whether these turbulent wakes can trigger a transition towards a turbulent boundary layer on the top of obstacles, the velocity profiles are compared in figure 9 to the classical prediction for
$z^{+}>30$
, where one expects logarithmic velocity profiles (Tennekes & Lumley Reference Tennekes and Lumley1987),
$$\begin{eqnarray}u=U^{\star }(2.40\log z^{+}+B),\end{eqnarray}$$
where
$z^{+}=z/z^{\star }$
and
$B=5.84$
over a smooth surface. The surface can be considered rough when the viscous sublayer is thinner than the typical roughness size. This is indeed the case here since the estimate for the viscous sublayer is
$1.7~\text{mm}$
, much smaller than
$h_{0}=1.2~\text{cm}$
. Thus,
$B$
is expected to be a function of
$k^{+}$
,
$$\begin{eqnarray}k^{+}=\frac{h_{0}}{{\it\delta}}=\frac{h_{0}U^{\ast }}{{\it\nu}}.\end{eqnarray}$$
In the present measurements,
$k^{+}$
is of order 40 and lies in the range of the ‘transition’ regime. The fully rough regime is usually expected for
$k^{+}\gg 100$
(Tennekes & Lumley Reference Tennekes and Lumley1987; Schlichting & Gersten Reference Schlichting and Gersten2000). In this transition regime, however,
$B$
is known to be between
$-5$
and 5, at least in the classical case of sand roughness, but the exact value may differ in the present case of square roughness. The green dashed line in figure 9 is plotted with
$B=-3$
. In the range of scales that we could measure, the experimental data thus appear to be compatible with such a logarithmic profile.
In addition to this destabilization of the boundary layer on top of obstacles, the mean velocity fields in figure 4 show that, while the flow above the notch is mostly unaffected and is nearly horizontal at the lowest Rayleigh number, it gets a vertical component after the transition which allows for matter transfers between the bulk and the notch. Evidence for effects of this kind has also been found in other experimental systems (Du & Tong Reference Du and Tong2000), and they are associated with an increase in the thermal transport without change of the scaling law.
3.4. Comparison with thermal boundary layer measurements
Our results should be compared to those obtained from temperature measurements in water, and to what could be expected if the interpretation of the transition features observed in similar conditions in the water cell does hold. In the previous water experiment in Lyon (Salort et al. Reference Salort, Liot, Rusaouen, Seychelles, Tisserand, Creyssels, Castaing and Chillà2014), which was six times smaller, we measured the thermal boundary layer thickness directly from the mean temperature profiles. The result was that
$$\begin{eqnarray}{\it\delta}_{th,water}\lesssim 0.4~\text{mm},\end{eqnarray}$$
much thinner than the classical laminar predictions or similar experimental measurements over smooth surfaces. This was one of the arguments for proposing a model based on the roughness-triggered destabilization of the boundary layers.
The mechanisms at play in the present situation in air are expected to be similar. Of course, it is possible that the details of the boundary layer structure differ in these two situations because of the Prandtl number (
$\mathit{Pr}=0.71$
in the present experiment;
$\mathit{Pr}$
between 4 and 8 in the water experiment). Yet, both Prandtl numbers are of order 1, and the typical orders of magnitude should agree if the relevant mechanism is similar.
One way to carry out such a comparison is to infer an estimate for the thermal boundary layer thickness from the present velocity boundary layer measurements. The mean velocity boundary layer observed near the rough plate in the present PIV fields can be interpreted in terms of a turbulent boundary layer and viscous sublayer. Because of the efficient mixing in the turbulent boundary layer, one may assume that most of the temperature drop occurs inside the viscous sublayer. Since the Prandtl number is less than 1, the thermal boundary layer is thicker than the kinetic viscous sublayer. The analytical and numerical study by Shishkina et al. (Reference Shishkina, Horn and Wagner2013) shows that the ratio of the thermal and kinetic boundary layer thicknesses depends greatly on the angle
${\it\beta}$
at which the wind attacks the plate. It is not clear how to extrapolate results obtained in the laminar case to the present situation with turbulent boundary layers. Therefore, the following discussion should be understood in terms of orders of magnitude only. For
$\mathit{Pr}=0.786$
, in the case of a laminar boundary layer, Shishkina et al. (Reference Shishkina, Horn and Wagner2013) find
${\it\delta}_{th}/{\it\delta}_{v}$
in the range between 1.08 (for
${\it\beta}={\rm\pi}$
) and 2.37 (for
${\it\beta}={\rm\pi}/2$
). The flow in the logarithmic layer above the viscous sublayer is turbulent and does not yield a constant and homogeneous value for
${\it\beta}$
, thus we may only assume that
$$\begin{eqnarray}1.08{\it\delta}<{\it\delta}_{th}<2.37{\it\delta},\end{eqnarray}$$
i.e.,
$$\begin{eqnarray}1.8~\text{mm}<{\it\delta}_{th}<4.0~\text{mm}.\end{eqnarray}$$
Considering that (1) the thermal boundary layer can be written in terms of the Nusselt number,
$$\begin{eqnarray}{\it\delta}_{th}=\frac{H}{2Nu},\end{eqnarray}$$
and (2) the Nusselt number is a function of
$\mathit{Ra}$
and
$\mathit{Pr}$
only, then
${\it\delta}_{th}$
is expected to be proportional to the cell height
$H$
if the Rayleigh numbers and all the other control parameters are equal, hence allowing a comparison of the results: the estimates of
${\it\delta}_{th}$
are expected to lie within the same ratios with respect to the cell heights.
We previously showed
${\it\delta}_{th,water}\lesssim 0.4~\text{mm}$
in the water experiment at similar Nusselt number (Salort et al.
Reference Salort, Liot, Rusaouen, Seychelles, Tisserand, Creyssels, Castaing and Chillà2014); we thus expect
${\it\delta}_{th}\lesssim 2.4~\text{mm}$
in the present set-up in air with dimensions six times larger. The present estimated range for
${\it\delta}_{th}$
, inferred from the velocity measurements (3.9), is indeed compatible with that prediction.
4. Conclusions
Highly resolved PIV measurements have been undertaken in turbulent Rayleigh–Bénard convection in air with a rough surface of the heating plate. A particular feature of convection at a rough surface is the transition in the scaling of the heat transfer beyond a critical Rayleigh number. Our measurements demonstrate that this transition can be associated clearly with a transition of the flow field around and in between the obstacles.
The interpretation of our previous experimental measurements in water and the phenomenological model derived from them made several implicit assumptions that have been verified in a much more direct fashion in the present work in air. An important assumption was that the flow structure below the roughness-induced transition was identical to the smooth case. The present analysis of the velocity features at
$\mathit{Ra}=4.66\times 10^{9}$
backs up this hypothesis: the velocity profiles are fairly similar to those obtained in smooth cells, and they are identical above obstacles, notches or grooves, a fraction of
$h_{0}$
away from the plate. For Rayleigh numbers lower than the transition value, this work shows indeed that the statistical quantities of the flow, e.g. Reynolds number, mean velocity profile and the mean velocity field, are impacted by the roughness elements only very close to the plate. In all respects, the flow is fairly similar to the smooth case.
There was also indirect evidence in the water experiment, above the critical roughness-induced Rayleigh number, that the boundary layer on the top of the obstacle was no longer laminar. In the present work, we have sought direct evidence of a possible transition to a turbulent boundary layer by means of high-resolution PIV measurements near the roughness elements. We find a high value for the Reynolds tensor
$\langle u^{\prime }v^{\prime }\rangle$
, in quantitative agreement with the expectations in terms of inferred viscous sublayer thickness.
The observations allowed us to go into more detail than the temperature-based analysis in water: we found evidence for both turbulent structures in the wakes of the roughness elements and incursions of the bulk between the obstacles. This leads to a change in the velocity statistics near the plate. In particular, the velocity profiles differ dramatically from the typical profiles obtained in the smooth case.
The incursions in the bulk are still fairly moderate. An important perspective would be to go further, in terms of
$Nu/Nu_{c}$
, and find out how the flow structure is modified when the notches are much more fully washed. Recent simulations by Wagner & Shishkina (Reference Wagner and Shishkina2015) suggest that the roughness-induced heat-flux enhancement would then saturate, with a power-law exponent back to
$1/3$
. This could be done in the future with the use of elements of higher roughness.
Acknowledgements
The authors would like to thank E. Rusaouën for useful discussion. The data analysis has been achieved thanks to the resources of PSMN (Pôle Scientifique de Modélisation Numérique) of ENS Lyon. The access to the Barrel of Ilmenau, and its operating costs, was funded by the ‘European High-performance Infrastructures in Turbulence’ (EuHIT) Infrastructure Transnational Access Program (European Grant Agreement no. 312778).
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