Boundary layer structure in turbulent Rayleigh–Bénard convection in a slim box

Hong-Yue Zou·Wen-Feng Zhou·Xi Chen·Yun Bao·Jun Chen·Zhen-Su She

Abstract Logarithmic boundary layers have been observed in different regions in turbulent Rayleigh–Bénard convection(RBC).However,how thermal plumes correlate with the logarithmic law of temperature and how the velocity profile changes with pressure gradient are not fully understood.Here,we perform three-dimensional(3D)simulations of turbulent Rayleigh–Bénard convection in a slim box without the front and back walls,with aspect ratio width:depth:height=L:D:H=1:1/6:1(corresponding to the x,y, and z coordinates,respectively),in the Rayleigh number Ra=[1×108,1×1010]for Prandtl number Pr=0.7.To investigate the structures of the viscous and thermal boundary layers,we examine the velocity profiles in the streamwise and vertical directions(i.e.U and W)along with the mean temperature profile throughout the plumeimpacting,plume-ejecting, and wind-shearing regions.The velocity profile is successfully quantified by a two-layer function of a stress length?as proposed by She et al.(J Fluid Mech,2017),though it is neither Prandtl–Blasius–Pohlhausen(PBP)type nor logarithmic.In contrast,the temperature profile in the plume-ejecting region is logarithmic for all simulated cases,attributed to the emission of thermal plumes.The coefficient of the temperature log law,A,can be described by the composition of the thermal stress length and the thicknesses of thermal boundary layer and The adverse pressure gradient responsible for turning the wind direction contributes to intensively emitting plumes and the logarithmic temperature profile at the plume-ejecting region.The Nusselt number scaling and the local heat flux in the slim box are consistent with previous results for confined cells.Therefore,the slim-box RBC is a preferred system for investigating in-box kinetic and thermal structures of turbulent convection with the large-scale circulation in a fixed plane.

Keywords Rayleigh–Bénard convection ·Wall-bounded turbulence·Heat transport·Direct numerical simulation

1 Introduction

Rayleigh–Bénard convection(RBC)is commonly used to study natural convection,due to the simplicity of its configuration and the richness of its flow regimes.In this system,fluid is filled in a closed cell,heated on the bottom and cooled on the top,with adiabatic side no-slip wall[1–3].The control parameters are the Rayleigh number Ra=gβ?TH3/(νκ),the Prandtl number,Pr= ν/κ, and the aspect ratio Γ =L/H,where ν is the kinematic viscosity,κ the thermal diffusivity,H is the height of the sample,L is its width,g is the gravitational acceleration,β is the thermal expansion coefficient, and ?T is the temperature difference between the top and bottom plate.Enhancement of the heat transport of a natural convection system such as the RBC is particularly useful in many industrial processes and is of fundamental interest[4,5].

The boundary layer(BL)in RBC exhibits a transition from a laminar to a turbulent regime when Ra exceeds a critical value,Rac.In the laminar regime,the mean velocity profile(MVP)takes a Prandtl–Blasius–Pohlhausen(PBP)type,whereas in the turbulent regime,a logarithmic(log)profile is usually expected,when an analogy is made with a BL passing a flat plate.A BL equation for RBC of Pr>1 was recently developed[6]considering both laminar and turbulent contributions, and a model describing the mean profiles of temperature and its variance in the near-wall region was experimentally tested[7].For Ra

The logarithmic mean temperature profile(MTP)is another issue which has recently garnered much attention in numerical and experimental studies,for Ra ranging from 1010to 1015[12–14].Logarithmic temperature profiles were observed near the sidewall sin experiments and direct numerical simulations(DNS),which showed markedly decreased thickness near the middle of the conducting plate.The mechanism behind the log law of the MTP is still debated—whether it is induced by the momentum transport near the no-slip sidewall,by the heat transport due to emitted plumes,or by both,is still unknown.Some results indicate a correlation between the intensive plume emission and the logarithmic temperature profile.In a cylindrical container at Γ?1,the plumes are found to be abundantly emitted from the top/bottom plate near the sidewalls,leading to an intensive local heat flux[15].Ina recent two-dimensional(2D)DNS study with a horizontal periodic boundary,a vertical log MTP only appears in the regions where plumes accumulate[16].However,the physics in these regions is still not fully understood.

Most previous experimental and numerical studies of the RBC system set the horizontal section in a circular or square geometry,where the large-scale circulation(LSC)exhibits frequent reversal,cessation or azimuthal motion[17,18].In such systems,the plume-emitting regions appear to‘wander’along the conducting plate, and it is difficult to extract the property of a certain flow region free of the influence from other regions.Similar phenomena have been observed in the cubic box(L:D:H=1:1:1),where the large-scale convection is found to exhibit random reorientation of the LSC and low-frequency oscillation perpendicular to the LSC[19].Complex convection flow also appears in a rectangular container for Ra=8×108～1×1010,due to strong secondary flow in the form of horizontal rolls surrounding the core of the cell and orthogonal to the cross-stream rolls[20].Thus,neither a cubic cell nor a rectangular cavity is appropriate to establish a turbulent RBC with a statistically steady LSC over a large range of Ra.

We perform a three-dimensional(3D)simulation with an LSC in a fixed plane so that a statistical mean field can be studied in great detail.This is achieved by reducing the scale in the depth direction to make a slim box,e.g.L:D:H=1:1/6:1,for which the mean flow become side ally confined in the vertical plane.It has been reported that the aspect ratio in the plane sperpendicular to the LSC,D/L,has strong effects on the global heat transport—the increased wall friction and suppressed LSC lead to more coherent and energetic plumes emitted from the conducting plates and thus enhancement of the global heat transport[21].The focus of the present study is to investigate a steady LSC in a plane free from the wall/confinement effect.This helps to settle down a better defined statistical mean field with well-defined flow regions.The periodic condition in the depth(y)direction allows for velocity and temperature fluctuating over a range of scales smaller than 1/6 of the depth of the box.Meanwhile,it is wide enough to develop relevant turbulent thermal convection for the simulated Ra.This configuration will be referred to as the slim-box RBC simulation,in which a turbulent LSC is confined in a fixed plane.Compared to othe r RBC systems of D/L1,the slim-box simulation establishes a stronger and more stable LSC with an unchanged wind direction.It will be seen that the periodic boundary condition in the y direction somewhat mimics the rectangular cell;indeed,the measured velocity and temperature profiles averaged in the depth direction present relevant features of experimental and numerical result spreviously observed.

We focus on the characteristics of the convection flow in different regions.An outcome of the current simulation,in addition to the LSC and the corner roll[22],is the identification of three flow regions,namely plume-impacting,wind shearing and plume-ejecting regions,in a time-averaged velocity field,similar to the results for 3D DNS of the RBC in a circular cylinder[23].Note that the RBC in the slim box has a larger LSC,a result of the absence of friction due to sidewalls.The coherent motion of the stronger LSC yields a thinner viscous BL and hence a larger Reynolds number.The Nusselt number Nu in the present study is also slightly larger than that in a confined cell,as measured at the same Pr[24].

The mean horizontal and vertical velocity(U and W) and temperature profiles in the slim box(averaged in the depth direction)are measured and studied in great detail in three regions,i.e.plume-ejecting,wind-shearing, and plume impacting,for the medium Rayleigh number.Because of the strong adverse pressure gradient in the wind-shearing and plume-ejecting regions,the BL in RBC is very different from a turbulent BL on a flat plate,so that neither a PBP-typenor the log-law BL is indeed observed in the MVP.On the other hand ,the multilayer theory proposed by Sheetal.[25]allows us to analytically quantify the kinetic BL by means of a two layer stress-length function.

For the MTP,the plume-ejecting region is found to hold a logarithmic region in all simulated cases,in agreement with previous studies.This logarithmic layer,however,is one of the multilayer structures for the temperature length, and the log-law coefficient,A,can be related to the thick-ness of the sublayer and buffer layer,which are measured at all stream wise locations and shown to exhibit a two-layer structure away from the ejection corner of the RBC cell.This model,for the first time,yields a two-dimensional(2D)temperature me an field,in agreement with DNS data,which is superior to the result obtained by Grossmann et al.[26]The relation between the log law of MTP and the emission of thermal plumes,previously suggested by a two-dimensional simulation of RBC[16],is confirmed in our 3D simulations, and the role of the adverse pressure gradient is emphasized.

Fig.1 Sketch of the convection box showing the definitions of coordinates and dimensions.The plane marked by a dashed line is the location of the side view of the convection

The paper is organized as follows.In Sect.2,we discuss the numerical simulation method.Section 3 contains the results and discussion,including temperature and pressure distribution,the mean velocity and temperature profiles, and the heat flux distribution in the slim box.Concluding remarks are given in Sect.4.

2 Numerical setup

The choice of the geometrical configuration is based on the following considerations.When the scale in the depth direction is reduced to form a slim box(with e.g.L:D:H=1:1/6:1,where L,D,H denote horizontal x,depth y and vertical z directions,respectively, and with a vertical aspect ratio Γ=L/H=1),the mean flow is confined in the vertical plane(at least approximately).Figure 1 schematically displays the geometry of the slim box and the LSC.The periodic boundary condition set in the y direction allows for velocity and temperature fluctuation over a range of scales smaller than 1/6 of the length of the box.

We numerically integrate the incompressible Navier-Stoke sequation with the Boussinesq approximation and the continuity equation,following Ref.[27]:

where ezis the unity vector pointing in the opposite direction to gravity,u is the velocity vector,p is the reference pressure and θ is the nondimensional temperature(with±1/2 at the bottom and top walls).The integrated equations are normalized using the free-fall velocity U=the reference pressure P0=RaPr(ρκ2)/H2, and the time scale

The fluid in the slim-box RBC is bounded in the x?z plane by the upper and lower isothermal plates and adiabatic sidewalls,so the boundary conditions are?θ/?x|x=0=?θ/?x|x=L=0.No-slip and impenetrability conditions are used for all solid boundaries.Periodicity is assumed in the y direction,i.e.θ|y=0= θ|y=D,?θ/?y|y=0= ?θ/?y|y=D,u|y=0=u|y=D, and ?u/?y|y=0= ?u/?y|y=D).

All the simulations were performed with a second-order finite-difference code,see Ref.[28]for details.Due to the absence of singularity at the origin,the central second order finite difference was applied in the y direction.A time-splitting method extensively discussed in Refs.[29,30]was used for the time advancement.The third-order low-storage Runge–Kutta method,along with the Crank–Nicolson scheme,was implemented to evaluate the nonsolenoidal velocity[30].The finite-difference scheme for the temperature equation is the same as that for velocity,except for pressure-related terms.

Solving the Poisson equation for pressure requires that the solution be sufficiently smooth up to the boundary.The iterative method is inefficient at high Ra when small-scale fluctuations are abundantly developed.We thus applied the fast fourier transformation(FFT)method to alter the parallel diagonal dominant(PDD)solver from 3D to 2D,see Ref.[31]for details.

Keeping the grid spacing smaller than the Kolmogorov scale ηKand the Batchelor scale ηBover the whole domain is important to ensure proper spatial resolution[32],where ηK =(ν3/?u)1/4= Pr1/2Ra?1/4(Nu ? 1)?1/4H, and[33].The time step was chosen to satisfy the Courant–Friedrichs–Lewy(CFL)condition,i.e.CFL?0.2.In our simulations,Nu was calculated by integrating over the whole volume and over time.Table 1 reports the minimum and maximum grid spacings,Δminand Δmax,which are indeed smaller than ηBand ηK;thus,the finest scales in the bulk flow are well resolved.The thermal BL thickness was the n calculated using the relation of λθ=H/(2Nu).The Bolgiano length scale was evaluated using LB=where?θis the thermal dissipation rate[32].

Table1 Numbers of the grids Nx×Ny×Nz,values of the Nusselt number estimated by three different methods Nu1,Nu2,Nu3,the minimum and maximum slim-box sizes Δm in and Δmax,the Bolgiano length scale L B,the Kolmogorov viscous scale ηK, and the Batchelor scale ηB

An additional control for DNS quality is to compare different Nu using different integration procedures.One way is to compute the heat flux directly by integrating along the two conducting walls,i.e.Nu1= ??Θ/?z at the upper(z=H)or the lower(z=0)plates.The second is to compute the volume-averaged temperature dissipation ingθor energy dissipation ingu[34],yielding Nu2=1+Pr?ingu?,respectively,where?·?denotes the ensemble average over both time and space.As reported in Table 1,these Nusselt numbers are in good agreement with each other,ensuring the statistical convergence of the simulated RB system.

A well-resolved simulation at high Ra requires a large computation resource.The simulation semployed up to1024 TH-1A CPUs(central processing units),using 768×128×512 grids for Ra=1×108and 1×109, and 1600×512×1600 grids for Ra=1×1010.In order to examine the turbulent state of the RBC flow,we placed 8 probe sat different heights,from z/H=9.8× 10?3to z/H=0.25 at the centerline,i.e.x/L=0.5 and y/D=0.5.The se probes record point-wise fluctuating temperature θ(t) and velocity( of three components u(t),v(t), and w(t)).Figure 2 shows that the flow is in a developing stage before t=30, and the n reaches a statistically steady state after t=40.The positive correlation between fluctuating velocity,u(t), and fluctuating temperature,θ(t),is seen for the fluid near the bottom plate,as seen in Fig.2c,d,respectively.

Fig.2 Signals of a numerical probe located halfway down the slim box at z/H=9.8×10?3,x/L=0.5, and y/D=0.5,for Ra=1×108

3 Results and discussion

3.1 Temperature and pressure distributions

Figure 3 presents snapshots of the temperature and pressure isosurface for Ra=1×108,showing that the LSC is confined in the x?z plane.It can also be seen that three dimensionality is remarkable in the near-wall region,where thermal plumes are intensively emitted.However,temperature and pressure within the LSC remain almost unchanged in the y direction,indicating that the LSC is characterized by quasi-two-dimensional flow.

The data are collected for comparison with experimental and other DNS studies including the viscous BL thickness and heat flux.This consists of two sets of experimental measurements[35,36],two sets of DNS from Ref.[37]with different aspect ratios and from Stevens et al.[38]for a cylindrical box with aspect ratio Γ =1/2 at higher Ra of 2×1010,2×1011and 2×1012.Mean streamwise velocity measurements from du Puits et al.[39]at several Ras around 1×1011and mean temperature measurements from Ref.[13]are also compared with the present simulations.For convenience,we summarize these DNS and experimental results in Table 2.

Ensemble average is carried out by integrating in the depth(y)direction and in time.The mean velocity,temperature and pressure distribution at Ra=1×108and 1×109are shown in Fig.4.The mean flow is represented by a counterclockwise rotation at the center of the box.As Ra increases,the high-speed fluid shifts towards the border of the box,along with a slower-rotating LSC.A pair of corner rolls,as the secondary flow induced by the LSC,exist in the respective corners on the diagonal line of the box.The y have been reported in previous experimental[40–42] and numerical studies[11,43,44].An analytic model for the corner rolls in turbulent RBC was recently reported by Zhou and Chen[22].

Fig.3 Visualization of the instantaneous a,b,c temperature and d,e,f pressure.The color in plots a–c denotes temperature, and the color in plots d–f denotes pressure.The slices in plots b and e are the corresponding fields in the y? z planes at various stream wise(x)locations.The fields shown in plots c and f are the fields in the x?z plane on the half way point of the depth of the box.The sesnapshots are obtained from the simulation at Ra=1×108 and Pr=0.7

Table2 Parameters of DNS and experiments

Figure 4 shows that the positive high pressure coincides with the regions of intensively ejecting plumes,corresponding to the lower right and upper left corners of the slim box.The center of the box,on the other hand ,corresponds to a low-pressure zone,as shown in Fig.4b,d.This indicates that the pressure gradient maintains a balance with the centrifugal force.Similarly,the cores of two corner rolls are closely as so-ciated with two lower-pressure zones,which were observed in 2D simulations[45].Note that the 3D corner roll is a smaller than the 2D one at Ra=1×108.Furthermore,the 3D simulation presents a more round-shaped LSC and a perceptibly wider wind-shearing region than the 2D simulation,indicating that the three-dimensionality of the near-wall corner flow is remarkable.This result is consistent with the previous numerical study[22].

Fig.4 Time-averaged 2D distribution of pressure and temperature displayed by pseudo-colors at Ra = 1×108 in plots a,b and Ra=1×109 in plots c,d,respectively.The arrows indicate the velocity vectors.The rotatory LSC,the corner roll, and the plume-impacting,w ind-shearing and plume-ejecting regions are marked by dashed lines in plot a.The lines with arrows in plots b and d are the time-averaged streamlines

Based on the averaged flow in the x?z plane,we characterize the RBC flow by five regions:(1)the LSC motion in bulk,(2)the corner roll,(3)the plume-impacting regions above the corner rolls with(0?x?0.25),(4)the wind shearing regions towards the middle of the conducting plates(0.25?x?0.75), and (5)the plume-ejecting region(0.75?x?1).The last three regions clearly possess a turbulent BL at a large aspect ratio,cf.Ref.[16].Note that the change in flow direction at the corners signifies the presence of a high-pressure gradient[46,47],as consistently presented in Fig.4b,d.

According to the correlation between the pressure and velocity fields,one can see that in the w ind-shearing region,the BL flow is driven by the longitudinal pressure gradient.The fluid near the bottom plate is advected from a favorable pressure gradient region to an adverse pressure gradient(APG)region.The APG in relation to the flow direction is present in the plume-ejecting region, and is responsible for the change in the flow direction from horizontal to vertical.This makes RBC-BL distinct from canonical BL under ZPG conditions.Consequently,the viscous BL of turbulent RBC is not always of the PBP-type.In Sect.3.2,we will discuss the structure of the viscous BL by analyzing the MVP in detail.

Figure5a,c shows the distribution of momentum dissipation ingufor Ra=1×108and Ra=1×109,respectively, and the thermal dissipation ingθis shown in Fig.5b,d.Note that the maximum of both inguand ingθare extremely close to the bottom/top plates.On the other hand ,the variation in ingθnear the sidewalls is moderate,due to the adiabatic wall condition,while inguexhibits a significant enhancement in the plume-impacting region(i.e.left-down and right-up corner of Fig.5a,c).In contrast,both ingθand inguare low in the bulk zone,corresponding to the core of the LSC.

3.2 Multilayer structure of velocity profiles

The vertical MVPs at the center of the w ind-shearing region(x/L=0.5)against the PBP-type profiles for different Ra s are shown in Fig.6.It is clearly seen that the MVPs differ markedly from the PBP-type profile,especially near the conducting plates.The deviation from the PBP-type profile has also been reported in the literature(cf.Refs.[36,39]).

Fig.6 MVPs at the box center(Ra=1×108～1×1010),with the PBP pr ofile(solid line).Note that z is normalized by the thickness of the viscous BL,λu,defined by the near-wall maximum velocity

The mean horizontal velocity U(z)at x/L=0.5, and the mean vertical velocity W(x)at z/H=0.5 are shown in Fig.7.A linear profile is found at the core of the LSC for both U(z) and W(x),indicating as olid rotation of fluid in this region(for 0.25?z/H?0.75 in Fig.7a and 0.25?x/L?0.75 in Fig.7b).Linearity of MVPs has been experimentally observed in both rectangular[48] and cylindrical cells[49].The slim-box simulations show that the linear core of the LSC extends to the scale of 0.4L for all Ras,quite similar to that observed in a narrow 3D RBC cell with D/H?1/4[48] and that in a cylindrical cell[49].We also note that the humps of velocities appear at the border of the LSC for the U and W profiles,which determine the characteristic LSC velocities,Ulcand Wlc,respectively.The y are distinct from peak velocities near the wall,as denoted by Unwand Wnw,respectively.

Compared with the confined cell[48],the slim box with periodic boundary condition has a higher Ulcand a relatively large linear core.On the other hand ,the Wlcis on the order of Wnw,see Fig.7.In particular,for the horizontal velocity at x/L=0.5,Ulcis greater than Unwfor all Ra s,indicating that near-wall buoyancy is overwhelmed by the wind shearing.With increasing Ra,both Ulc( and Wlc) and Unw( and Wnw)tend to decrease,but the linear core velocity is more sharply decreased than that near the wall.

The thickness of the viscous BL as a critical quantity for turbulent BL is calculated.Several definitions of BL thickness have been suggested[35].Here we use the thickness of BL,λu,obtained by extrapolation of the linear profile U(z)to reach Unw.Note that λuis a dimensionless parameter normalized by the height of the box,H.Figure 8a shows that λuis smaller than the values me a sured in two experiments in confined cells[35,36].A confined LSC tends to establish a steadier circulation in the cell.The experimental study with a tilted cylindrical cell showed that a larger tilt angle imposes a stronger restriction on the azimuthal motion of the LSC so that it has less fluctuation perpendicular to the wind[36].The experiments with a slender rectangular cell also indicated that the confinement of the LSC tends to stabilize the large-scale flow structure and to establish a thinner viscous BL[35],as also shown in Fig.8a.In the present study,the slim box almost eliminates the azimuthal meandering of the LSC.The relatively small value of λufor the slim-box flow is attributed to the absence of the depth confinement, and thus the wall friction,which allows us to develop a stronger LSC(with larger Unw)at a larger scale,leading to a thinner viscous BL in comparison with the confined cell.The present simulations give a scaling of λu～ Ra?0.22,similar to the scaling of?0.20 in Ref.[36],but different from ?0.27 in Ref.[35].

Fig.7 a Horizontal velocity profile U(z)cut along the z axis(at x=0.5).b Vertical velocity profile W(x)cut along the x axis(at z=0.5)

Fig.8 Effects of the Rayleigh number on the thickness of the viscous BL and the Reynolds number.a Ra-dependence of the viscous boundary thickness normalized by H.b Ra-dependence of Re

The mean velocity of the LSC reaches a maximum Unw,relevant for determining the thickness of the viscous BL.Here,we present the bulk Reynolds numbers defined by Re=UnwH/ν against Ra obtained from the simulations and two other experiments in Fig.8b.Because of the large Unwin the present simulations,the Re s of the simulations are always larger than the experimental measurements.Again,the present results show a scaling(Re∝Ra0.54),close to that of[35](Re∝Ra0.55),in contrast to a smaller slope Re∝Ra0.43[36].

The BL thickness over the entire wind-shearing region is calculated(i.e.0.3

Fig.9 Thickness of the viscous BL as a function λu= λu,0(1?x/L)?0.5,represented by solid lines

To quantify MVP in the wind-shearing region,we follow the SED theory[25]to employ the stress length as the similarity variable,which takes a multilayer formula describing the structure in the direction normal to the wall.Neglecting the relatively small variation along the direction parallel to the wall in the mean momentum equation(i.e.the incompressible BL approximation),one obtains the following balance equations:

Eq.(4)along z and Eq.(5)along x yields

Note that u+=u/uτ,z+=z/inguand

where uτ=and ingu= ν/uτ.Similarly,we have=w/wτ,=x/ingwand where wτ=and ingw= ν/wτ.Introducinggives a normalized mean velocity equation as

We emphasize that Eqs.(8) and (9)hold when the relatively small pressure gradient effect in the wind-shearing region is neglected.Introducing the stress length( andwe have the solutions of Eqs.(8) and (9):

The theory developed by Sheet al.[25]allows us to construct the two-layer similarity solutions forand,which are assumed to possess a dilation group invariance and to take the following analytic form

Figure 10a,b shows that MVPs can be accurately described by the two-layer stress length formula (12) and (13).The two-layer structure region extends up to z+≈30 for Ra=1010.At lower Ra,the flow at z+～20 is bulkdominated,having a relatively thin BL.The Ra-dependence ofand ?0are shown in Fig.10c,d,respectively.The fact is thatdecreases monotonically as power functions—=2.15Ra?0.125for Eq.(12) and=for Eq.(13).The sublayer thickness=holds for two decades of Ra(1×108≤Ra≤1×1010),

whereas the convection in the cylindrical cell has a lower scaling,0.13,as seen in Fig.10c.This implies that the convection near the wall in a rectangular cell is stronger than that in a cylindrical cell, and inspires us to verify these scalings for different configurations over a wide range of Ra.For the sublayer on the sidewalls,the thickness is insensitive to Ra,i.e.6.50.

3.3 Log-law of the thermal boundary layer

The existence of logarithmic temperature profiles in turbulent RBC has attracted much attention in recent years.The log thermal layer is usually attributed to plume emission near the conducting plates[16].Here,we discuss the relation between the statistical behaviors of thermal structures and the logarithmic distribution of temperature.

According to the DNS,thermal dissipation in the horizontal direction can almost be ignored in the plume-ejecting region.Thus the temperature equation in the 2D form can be simplified as:

with normalized vertical coordinate,z?=(z/H)Nu.The thermal BL can be divided into three layers according to the mean momentum equation:(a)the near-wall region(the conduction layer),described by1;(b)the region far from the wall,dominated by1;(c)the layer between these two layers.Note thatrepresents the convective heat flux by normal velocity fluctuations from the wall.The thermal balance in the form of the stress length similar to the momentum transport is obtained, and thus the temperature distribution is extracted.

Fig.11 MTP of three data sets:the present DNS,the DNS from Ref.[38], and the experiment from Ahlers et al.[13].The solid lines indicate Eq.(17).The triangles are the MTP averaged along the horizontal(x)direction.The dashed line is the model of Eq.(31)in Ref.[6], and the gray curve is PBP profile

Fig.12(Color online)The log-law region of thermal BL described by Eq.(18)at Ra=1×108,marked by solid line.The pseudo-color denotes temperature.Cold color represents low temperature, and hot color high temperature.The arrows are the velocity vectors

We follow the SED theory[25]to quantify the MTP by postulating a similar(thermal)sublayer,buffer layer and log layer,with a temperature lengthas the similarity variable,which plays a similar role as the stress length.Specifically,∝ z?5/2for,displaysa three-layer structure,i.e∝ z?3/2for z?<;where the superscript?denotes the variable nor malization by Nu/H.The first exponent 3/2 readily follows from a nearwall expansion,ignoring the fluctuation magnitude,whereand.The third layer has a linear scaling in z,corresponding to the log layer.

The buffer layer,similar to the role of the momentum buffer layer,is considered the transition from a laminar flow near the wall to a fully turbulent flow in the bulk.Inspecting the DNS,we obtain the scaling exponent 5/2 by invoking the integral scale of temperature fluctuation,,proportional to z?9/4near the wall, and a dissipation-to-production ratio for temperature fluctuation,proportional to z in the buffer layer.Sinceit followsin the buffer layer as a semi-empirical result.Finally,the SED theory postulates a generalized dilation symmetry thatdisplays a generalized Lie-group invariance characterizing the transition between the local scaling behaviors,yielding a composite solution of

The stress length of temperatureis a three-layer function expressed as Eq.(16).Transition of the scaling offrom 3/2 to 5/2 occurs at.The next transition of scaling from 5/2 to 1 is present at.

Jointly solving Eqs.(15) and (16)yields an analytic function of the MTP as

The first consequence of the solution is the logarithmic law of MTP.For1,the na logarithmic MTP follows

Figure 11 shows the comparison between the analytic solutions at a fixed location from the sidewall(r=0.0045)for Ra covering six decades,from moderate(Ra=1×109)in DNS to the extremely high Ra experiments(Ra=1.1×1015)[13].The MTP in the plume-ejecting region clearly presents a log law covering at least one decade in z(0.04?z/H?0.4).Close inspection of Figs.4 and 12 shows that the range of log-layer coincides with the in tensive plume emission,indicating the relation between the two phenomena.

Fig.13 a(r) and b(r)for Ra=1×108,1×109 and 1×1010,respectively.The solid lines are the multilayer function of Eq.(20).The solid lines are the multilayer function of Eq.(21)

We calculated the theoretical model,Eq.(31),for Pr?1 from Ref.[6],with parameter c=1,see the inset of Fig.11.The systematic deviations of the MTP(Pr<1)from the equation and the PBP profile predictions are observed.The MTP in the plume-ejecting region no longer represents a typical turbulent BL.Thus,we calculated the MTP averaged along the horizontal(x)direction,which represents the characteristics of BL in the wind-shearing region[6].The inset of Fig.11 shows that the space-averaged MTP is markedly higher than local MTP in the plume-ejecting region, and much closer to the BL equation.Though MTP in the w ind shearing region with small Pr has a trend similar to the case with large Pr,neither the present MTP nor that from Stevens et al.[38](Pr=0.7)can be described by the equation for Pr?1.However,an improved thermal BL equation is capable of describing the flow with a low Prandtl number down to 0.01[50].

Near the side wall(r?rb),andare constants denoted asand.The coefficient,,may be associated with the size of thermal plumes,which decreases for increasing Ra,to be determined in the following section.In the wind-shearing region(r?rb),we have a scalingwhich decreases with increasing r,see Fig.13a.On the other hand ,the thickness of buffer layer,as shown in Fig.13b,increases in r,indicating that the thermal BL becomes thinner moving downstream with the wind(for increasing x or decreasing r).

After comparing the datasets,we find that rb?0.125 and rc?0.35 for the slim box(for DNS–A),but rb?0.0075 and rc?0.35 for the cylindrical cell(DNS–B)—the y are likely independent of Ra.On the other hand ,the coefficientsanddepend on Ra,see Fig.14.Specifically,0.082Ra0.155for the slim-box simulations(DNS–A),but 0.046Ra0.155for the cylindrical cell(DNS–B).The same scaling of 0.155 for Ra indicates similar behavior of the heat transport near the side walls.Moreover,63.0Ra?0.11for the slim box(DNS – A),but 46.0Ra?0.11for the cylindrical cell,also with the same scaling of?0.11(see the inset of Fig.14).Since0.375 is independent of Ra and geometry,matching condition for A?1yields a scaling for the coefficient of the log-law slope,A～Ra0.1225,near the side wall,in good agreement with the experimental result of Ahlers et al.[12]:A～Ra0.123.

The coefficient in the log-law A=1/κθas the function of was first discussed in Ref.[26],suggesting an analytic form,|A|=|A1|/for the cylindrical cell.Note that√A(x)holds a?1/2-scaling(i.e.A ～More recently,Ahlers et al.[13]claimed that the scaling was?0.65,due to an elliptical path(rather than a circular path) of the LSC.In the present study,forEq.(16)can be rewritten asi.e.Based on the aforementioned analysis onand(i.e.and=const.),we have A ～ r?0.60,in agreement with the recent result of Gross man n and Lohse[26].Figure 15 presents the comparison between our predicted coefficient A(r)from DNS at different Ra s,which is superior to the fitting function of Ref.[26];the latter is only valid in a restricted domain(0.02

Fig.14 Ra-dependence of a and b

Fig.15 Parameter A of the log-law function varies with the longitudinal coordinate x for different Ra sin DNS-A.The in set is the result of DNSB.The solid lines represent A

To investigate the dynamics in the log layer,we calculated the profiles of the horizontal and vertical velocity in the plume-ejecting and wind-shearing regions,as presented in Fig.16.No logarithmic region for U(z)is found in the plume-ejecting region.It is reasonable to believe that the log layer of temperature has an origin free from the log viscous BL;in other words,the Reynolds analogy does not hold here.It is interesting to make a comparison between velocities in the wind-shearing region(0.25?x/L?0.75) and those in the plume-ejecting region(x/L?0.75).The flow is governed by strong APG due to the confinement of the sidewalls, and the momentum is changed from a horizontal to an upward direction in the plume-ejecting region.The overlapping of the plume-ejecting region with the log thermal BL indicates that vertical momentum plays the main role in establishing the log law of temperature.In addition,the vertical velocity is found to be approximately proportional to the wall distance,W/U0=0.7(z/H)at 0?z/H?0.4 in the plume-ejecting region,as shown in Fig.16.

Fig.16 Horizontal U(z) and vertical velocity W(z)profiles in the plume-ejecting region at x/L=0.9 and the wind-shearing region at x/L=0.5 for the case of Ra=1×108.The solid line is the fitting of W/U0=0.7(z/H)

Fig.17 Compensated Nu as a function of Ra

3.4 Rayleigh number effect on heat transport

One of the main issues in the study of RBC is determining the dimensionless heat transfer coefficient,the Nusselt number Nu.The heat flux can be calculated by Nu(z)=?wθ/κ? ?θ/?z?x,y,t,where?·?x,y,trepresents averaging over the horizontal plane and sufficient length of time.A comparison between Nu of the present simulation with previous data is presented in Fig.17.Note that the DNS by Kaczorowski et al.[37]examined various as pectratios(varying from Γ=1/8 to Γ=1) and Prandtl numbers(Pr=0.7 and 4.3).The slim-box simulations agree with other data at the same Pr(i.e.Pr=0.7),but our Nu is slightly larger than that of Ref.[37].This is consistent with the fact that the slim box enhances the LSC and thus a more intensive heat transport.

We now examine the contribution of each flow region to total heat transport by calculating Nuloc(x,z) =?wθ/κ ? ?θ?y,t,where?·?y,tdenotes the averaging over the depth direction and sufficient length of time.Figure 18a shows Nuloc(x,0)/Nu at the bottom plate(i.e.z=0),which is dominated by thermal diffusion Nuloc(x,0)≈??θ(x,0).Note that the maximum heat flux appear sat x=0.25,where the thickness of the thermal BL is the thinnest,as shown in Fig.18a,corresponding to where the cold plumes impinge on the heating plate.The total heat flux in the range of 0?x?0.25 corresponds to that by the corner roll.

Fig.18 a Local heat flux Nu loc/Nu(normalized with global Nusselt number,Nu)at the bottom plate(i.e.z=0)varying with x[10,21].b Nu loc at the half height of the convection box(z/H=0.5)varying with x

In the center region(0.25?x/L?0.75),our computation shows a linearly decreasing heat flux with increasing x,corresponding to an increase in the thermal BL thickness,consistent with other data[10,48].However,this decreasing trend is weakened at higher Ra, and the flow becomes more homogeneous in the horizontal direction.Figure 18b shows the local heat flux Nulocat half the height of the box(z/H=0.5).For Ra=1×108,Nulocis more symmetrical with respect to z/H=0.5 than z=0,which is due to the heat convection by the LSC.The three flow regions,i.e.plume-impacting(0?x/L?0.25),wind-shearing(0.25?x/L?0.75) and plume-ejecting(0.75?x/L?1)regions,are rather distinct,as shown in Fig.18a,which is also clearly observed for Ra=1×109.Convective heat transfer is predominant in the heat flux near the centerline z/H=0.5.We find that the re is a symmetry breaking in the vertical velocity W(x)near the centerline,the maximum magnitude of W is 0.271 on the right side,greater than?0.243 on the left side,leading to the higher local heat flux at x=0 side.

4 Concluding rem arks

We performed the 3D DNS of RBC at Pr=0.7 and Ra=1×108～1×1010for a slim box(the ratio of length,depth and height is1:1/6:1),with periodic boundary condition in the depth(y)direction.Two facts for the slim-box RBC are found:(a)the LSC is steadily confined parallel to the x?z plane, and (b)a higher heat flux and stronger LSC is observed,in contrast to the cases in a confined cell,as a result of the absence of front/back walls.Three flow regions(i.e.plume-impacting,wind-shearing, and plume-ejecting)are clearly revealed in the time-averaged distribution of velocity and temperature.

The non-Blasius velocity profiles under the influence of a strong adverse pressure gradient can be described by the multilayer stress-length function,following the symmetry based theory of canonical wall turbulence[25].The MTPs can be satisfactorily described by a multilayer structure of a thermal dissipation stress-length function,which yields an analytic description of the log-law coefficient A for a range of x and Ra.Certain parameters,namely?u0,?w0,?θ0, and zbuf,in the stress-length functions for viscous and thermal BLs are found to be influenced by Ra,Pr, and even the geometry of the convection cell.Parameterization of the multilayer functions for various configurations can be fulfilled by investigating additional experiments and numerical simulations.Moreover,with know ledge of the symmetry in each layer,these stress-length functions can be applied to interpret and predict the convection flow at extreme conditions,for example,high Ra or high/low Pr.

Local heat transport is analyzed and discussed.The Nusselt number scaling and local heat flux of the present simulations are consistent with previous experiments[21] and numerical simulations[37]in the confined rectangular cell.Thus,we conclude that the present slim-box RBC is an ideal system for studying in-box kinetic and thermal structures, and space-time correlations[51],in confined turbulent convection.

AcknowledgementsThe Project was supported by the National Natural Science Foundation of China(Grants 11452002,11521091, and 11372362) and MOST(China)973 Project(Grant 2009CB724100).