Heat transport in horizontally periodic and confined Rayleigh-Bénard convection with no-slip and free-slip plates

Maojing Huang , Xiaozhou He

School of Mechanical Engineering and Automation, Harbin Institute of Technology (Shenzhen), Shenzhen 518055, China

ABSTRACT We report the results of the direct numerical simulations of two-dimensional Rayleigh-Bénard convec- tion(RBC) in order to study the influence of the periodic (PD) and confined (CF) samples on the heat transportNu.The numerical study is conducted with the Rayleigh number (Ra) varied in the range 10 6 ≤Ra≤10 9 at a fixed Prandtl numberPr= 4.3 and aspect ratioΓ= 2 with the no-slip (NS) and free- slip(FS) plates.There exists a zonal flow forRa≥3 ×10 6 with the free-slip plates in the periodic sample.In all the other cases, the flow is the closed large-scale circulation (closed LSC).The striking features are that the heat transportNuis influenced and the temperature profiles do not be influenced when the flow pattern is zonal flow.

Thermal convection with slippery boundary conditions is ubiq- uitous in nature and many engineering applications.For exam- ple, the fluid-fluid interfaces in Venus’s atmosphere and oceans are commonly considered as free-slip boundary conditions [1-3] .In industrial applications, the super-hydrophobic surfaces [4] are used to reduce resistance at the solid surfaces.The convection flow is commonly modeled by Rayleigh-Bénard convection (RBC) [5-7] .This is a system that a fluid is heated from below and cooled from above.The key control parameters are the Rayleigh numberRa≡αgΔH3/(κν), Prandtl numberPr≡ν/κand aspect ratio of the convection sampleΓ=L/H.Hereαis the thermal expansion coef- ficient,gthe gravitational acceleration,Δ=Tb−Ttthe temperature difference between the hot bottom (Tb) and cold top (Tt) plates,κthe thermal diffusivity andνthe kinematic viscosity of the fluid.HandLare the height and length of the sample, respectively.

Many recent studies in direct numerical simulations (DNS) fo- cus on the no-slip (NS) and free-slip (FS) plate boundary conditions in the periodic (PD) and confined (CF) samples [8-12] .It has been found that the efficiencies of heat transport vary in a wide range of the Prandtl numberPrand aspect ratioΓin RBC with the no-slip and free-slip plates [13-19] .In a confined cell, the heat transport enhances 7% for the case of the no-slip plates and free-slip side- walls compared with the case of all no-slip walls [20] .The heat transport for the case of all free-slip walls is 80% larger than those for the case of all no-slip walls [21] .

This is, however, the opposite in turbulent RBC for the peri- odic sample, whose sidewall boundary conditions are periodic.In the periodic sample, the heat transport for the free-slip plates de- creases 80% compared with that for the no-slip plates [22-24] , due to zonal flow suppressing the heat transport.The zonal flow con- sists of two shearing layers of fluid that move in opposite horizon- tal directions.It is not driven directly by buoyancy and is sustained by Reynolds stress [25] .Wang et al.[26] found that the kinetic en- ergy in the zonal flow is mainly produced by the buoyancy force.The zonal flow is common in RBC at the free-slip plates in the pe- riodic cells [23] and also exists in the periodic cells at the no-slip plates for smallΓ[22] .It remains unclear, however, when does the zonal flow form and what are the structures of the tempera- ture boundary layer (BL) profiles of the zonal flow.The structure of the thermal BL could determine the heat transport in turbulent RBC [27] .A temperature BL equation that takes into account the influence of turbulent fluctuations has been derived along a semi- infinite horizontal heated plate.It has yielded mean temperature [28-30] and variance temperature profiles [31,32] .Recently, The temperature variance equations have been derived in the mixing zone and the log layer [33-35] .It has been derived for the general forms of the mean temperature and temperature variance profiles with slippery conducting surfaces [36] .

Here, we numerically explore the heat transport and mass transport in the periodic (PD) and confined (CF) samples.It is worth noting that the sidewall boundary conditions are no-slip in our confined samples.The plates boundary conditions are no-slip (NS) or free-slip (FS) plate boundary conditions.Our results show there exist two distinct flow states in the periodic samples with the free-slip plates, namely, closed large-scale circulation (closed LSC) and zonal flow.The zonal flow, which exists in the periodic sample, suppresses the heat transport.Then, we compare the heat transport and mass transport in the periodic and confined samples as a function of Rayleigh numberRain turbulent RBC with the no- slip and free-slip plates.Finally, we study the effect of the periodic sidewall boundary conditions on the mean temperature and tem- perature variance profiles near the thermal BL.It is found that the sidewall boundary conditions do not influence the mean tempera- ture and temperature variance profiles with the no-slip and free- slip plates.

The governing equations for turbulent RB convection are the incompressible Navier-Stokes equations and the convective heat equation under the Oberbeck-Boussinesq approximation.The non- dimensional forms of these equations read,

whereeyis the unit vector in theydirection opposite to grav- ity.The length, velocity, time, pressure and temperature are made dimensionless by the cell heightH, the free-fall velocityUf=the free-fall timetf=H/Uf, the free-fall pressuregαΔ/Hand the temperature differenceΔacross the cell, respectively.

At the top and bottom plates, we impose no-slip or free-slip boundary conditions for the velocity field.For the temperature field, we employTb= 0.5 andTt= −0.5 at the bottom and top conducting plates, respectively.In the confined sample, the side- wall boundary condition is no-slip for the velocity field and adia- batic for the temperature field.In the periodic sample, the sidewall boundary condition is periodic.

Equations (1) -(3) are solved numerically using the open-source code Nek50 0 0 [37] , which uses a spectral element method to accu- rately resolve the gradients in the velocity fielduand temperature fieldθ.In Nek50 0 0, the time-derivative terms are employed the third-order backward-differentiation formula (BDF3).The nonlinear terms are employed the explicit third-order extrapolation (EXT3) and the linear terms are treated implicitly.The turbulence fields are expanded within each element usingNth-order Lagrangian in- terpolation polynomials as the basis functions on Gauss-Lobatto- Legendre (GLL) collocation points.More details of the numerical scheme and mesh resolution can be seen in the references by Fis- cher [37] and Scheel et al [38] .

The computational mesh of the flow domain is divided inton=nx×nyspectral elements, wherenxandnydenote the num- bers of spectral elements along the horizontal (x-) and vertical (y-) direction, respectively.In all simulation runs, the polynomial order is set toN= 7 so that we have(N+ 1)2= 64 grid points within each element.The mesh resolution satisfies the dimensionless Kol- mogorov length scaleand the Batchelor length scaleηB= 1/[Ra(Nu−1)] 1/4[39] .We employ adaptive time steps to ensure that the Courant number is below 0.5.We run each simulation for at least 200tfto reach the steady state, followed by a continuous running for at least another 500tfto obtain time av- eraging.The initial condition isu=0andθ= 0 for our all cases.

The time average Nusselt numberNuis calculated usingNu≡[ 〈vθ〉 −κ∂y〈θ〉 ]/(κΔ/H).Note that 〈·〉 denotes average in time and over horizontal area.The Reynolds number is defined asRe=where 〈·〉Vdenotes the time and volume average.denote horizontal and ver- tical Reynolds number, respectively [14] .Here,uandvare the hor- izontal and vertical component of the velocity,respectively, andθis the temperature.Table 1 lists a summary of the parameters used for DNS runs.We present the numerical data with the Rayleigh numberRavaried in the range 106≤Ra≤109, Prandtl numberPrfixed at 4.3 and aspect ratioΓfixed at 2.

Table 1 The parameters used in the DNS runs with different Rayleigh numberRaat a fixedPr= 4.3 andΓ= 2 .The pa- rameters include the plate BC, the sidewall BC and the total number of spectral elements in the flow domainn=n x×n y, wheren xandn ydenote horizontal (x-) and vertical (y-) direction,respectively.Also included are the orderNof the polynomials and the numberN BLof grid points used to calculate the thermal BL.Averaged timet a v gis used for averaging the Nusselt numberNu, Reynolds numberRe, horizontal Reynolds numberRe xand vertical Reynolds numberRe y.

Figure 1 shows typical snapshots of temperature for the con- fined (upper) and periodic (lower) samples with different Rayleigh numbers in the two-dimensional (2D) sample ofΓ= 2 at the free- slip plates.Here and below, it is noted that the confined cell refers to the case of the no-slip sidewall boundary condition and the pe- riodic sample refers to the case of the periodic sidewall bound- ary condition.In the periodic cell, there exists zonal flow with the free-slip plates atRa= 108(see Fig.1 d).In all the other cases, the flow is closed LSC.When the flow is closed LSC, the heat trans- ferNuenhances for free-slip plates than for no-slip plates both in the confined and periodic samples.In comparison, the heat trans- ferNufor the free-slip plates is 32% lower than those for the no- slip plates at Rayleigh numberRa= 108in the periodic cell.The zonal flow could suppress the vertical heat transport.For a large aspect ratioΓ= 10 with free-slip plates in the periodic sample, we found that the zonal flow is stable when the initial flow is zonal flow, while two closed LSC exist when the initial flow isu=0andθ= 0 (we didn’t showΓ= 10 in the paper).In the periodic side- walls, the closed LSC is steady atRa= 106with the free-slip plates, as seen in Fig.1 c.The updraft of the hot fluid (and downdraft of the cold fluid) is similar to the bump in steady closed LSC [40] , which would enhance the heat transportNu.

Next, we discuss the dependence ofNuas a function of Ra in the confined and periodic samples with the no-slip and free- slip plates.The Nusselt number follows the scaling relation ofNu～RaγNuwith the effective exponentγNuvarying from 0.28 nearRa= 108to 0.32 nearRa= 10 11 [41] .Recently, it was found that the scaling relation shows Nu ～Ra0.312Nu～Ra0.312in the classi- cal regime [35] .Figure 2 a shows the reducedNu/Ra0.312 as a func- tion ofRa.As discussed above, zonal flow exists in the periodic cell with the free-slip plates forRa≥3 ×106, while the flow is closed LSC in all the other cases.It is found in the figure that the classical scaling also roughly holds for confined and periodic samples with different plate boundary conditions when the flow is the closed LSC.The scaling ofNu(Ra)is Nu ～Ra0.15Nu～Ra0.15for zonal flow, which is close to the previous results [14] .As expected, the heat transfer Nu for the confined and periodic cells with the no-slip plates are the same.However, it is found that the heat transferNufor the periodic cells is larger than that for the confined cell atRa＜3 ×106with no-slip and free-slip plates.The heat trans- ferNuis 15% higher for the periodic cell than for the confined cell with no-slip plates forRa＜3 ×106.The result is close to that the heat transferNuis 14% higher for the free-slip sidewalls than for the no-slip sidewalls with the no-slip plates [22] .In the case of the free-slip plates forRa＜3 ×106, the heat transferNufor the periodic sample is twice as large as that for the confined sample.ForRa＜3 ×106, the flow topology and heat transfer for the pe- riodic sample are similar to those for the free-slip sidewalls (not present here).As shown in Fig.1 c, The closed LSC is steady and the thermal plumes are almost exclusively straight up and down in the vertical direction without significant horizontal structure forRa= 106[42] .Therefore, the interface between the two closed LSC could be equivalent to a free-slip sidewall.

We plot in Fig.2 b the ratioNuF/Nu0 −1 as a function of Ra for the confined and periodic samples.Here,NuFdenotes Nu for free- slip plates andNu0 denotes Nu for no-slip plates.When the flow is the closed LSC, we find the ratioNuF/Nu0 −1 ≃ 1 for the confinedcell andNuF/Nu0 −1 ≃ 2.3 for the periodic cell nearly independent of Ra .The ratioNuF/Nu0 −1 for the periodic cell is larger than that for the confined cell, due to the increased Reynolds number Re .In the periodic sample with free-slip plates, the ratioNuF/Nu0 −1 decreases asRaincreases when the flow is the zonal flow.The ratioNuF/Nu0 −1＜0 denotes that the heat transportNufor the free- slip plates decreases compared with the no-slip plates.Even more, the heat transferNufor the free-slip plates is smaller than that for the no-slip plates forRa＞107in the periodic cell, agreeing with the previous studies [22,23] , due to the zonal flow suppressing the vertical heat transport.

In Fig.3 a and 3 b we show the dependence of the Reynolds numberReand horizontal Reynolds numberRexas a function of Rayleigh numberRa, respectively.The scaling relation of Re as a function ofRais Re ～Ra0.6Re～Ra0.6 for the closed LSC and zonal flow, agreeing with the previous results [14] .The Reynolds numberReand horizontal Reynolds numberRexfor the periodic cell are larger than that for the confined cell because the no-slip sidewalls slow down the flow.Figure 3 c and d show the ratio ofReF/Re0 andRex,F/Rex,0.HereReFandRex,Fdenote the Reynolds number and horizontal Reynolds number for free-slip plates, respectively, andRe0 andRex,0denote the Reynolds number and horizontal Reynolds number for no-slip plates, respectively.In Fig.3 c it is found thatReF/Re0 for the periodic sample is 40% higher than for the con- fined sample.In the zonal flow, the ratio ofReF/Re0for the pe- riodic sample is slightly larger than that for the confined sample, attributing to the horizontal Reynolds numberRex.The zonal flow slows the growth of the heat transportNuand vertical Reynolds numberRey(see Table 1), and does not slow the growth of the horizontal Reynolds numberRex.The ratioRex/Rex,0for the peri- odic cell is 20% higher than the confined cell, as shown in Fig.3 d.

Fig. 1. Visualizations of instantaneous temperature in the 2D confined cells (upper row) and periodic cells (lower row) with free-slip plates for Rayleigh numberRa= 10 6 (a)(c) and 10 8 (b)(d) .

Fig. 2. (a)Nu /Ra0.312 as a function ofRafrom 2D DNS for the periodic (PD) and confined (CF) cells with no-slip and free-slip plates boundary conditions.Symbols correspond to the confined cells with no-slip (blue triangles) and free-slip plates (black squares).Grey crosses and red circles denote no-slip and free-slip plates, respectively, in the periodic cells.(b)N u F /N u0 −1 as a function ofRain the confined (black squares) and periodic (red circles) cells.

Fig. 3. (a)Reand (b)Re x, as a function ofRafrom 2D DNS for the periodic (PD) and confined (CF) cells with no-slip and free-slip plates boundary conditions.Symbols correspond to the confined cells with no-slip (blue triangles) and free-slip plates (black squares).Grey crosses and red circles denote no-slip and free-slip plates, respectively, in the periodic cells.(c)Re F /Re0 and (d) Re x,F /Re x,0 , as a function ofRain the confined (black squares) and periodic (red circles) cells.

Fig. 4. (a)Θand (b)Ω, as a function ofξfor periodic (PD) and confined (CF) cells with no-slip (NS) and free-slip (FS) plate boundary conditions.The red solid and blue dashed lines in (a) represent the calculatedΘ(ξ)using Eq.(4) , and in (b) are the numerical solutionΩ(ξ:p, c, Δ2/η0, f, γ)of Eq.(6) withξ0 = 0.85 .

Finally, in order to better understand the effect of periodic side- wall boundary conditions on the heat transportNu, we turn to simulation results for the mean temperatureΘ(ξ)and tempera- ture varianceΩ(ξ)profiles near the thermal BL in the periodic and confined cells with the no-slip and free-slip plates.Here,ξ=y/λis the vertical distance from the conducting plate normalized by the thermal BL thicknessλ.One considers the effect of BL fluctuations and turbulent thermal diffusivityκt= −〈v＇θ＇〉/(∂y〈θ〉).Here,v＇ is the vertical velocity fluctuation andθ＇is the temperature fluctua- tion.The turbulent thermal diffusivity is considered asκt≈mξpκ, wheremis a positive constant number.The parameterpis equal to 2 for the free-slip plate, whilepequals to 3 for the no-slip plate.One could obtain the mean temperature profileΘ(ξ)near the thermal BL with the no-slip and free-slip plates [36]

The constant numbermis related to the constant numbercby the requirementΘ(∞)= 1 and one has

Figure 4 a shows the mean temperatureΘ(ξ)obtained for the periodic and confined cells at different Rayleigh numbersRaas a function ofξwith the no-slip and free-slip plates.It is found that all theΘ(ξ)data for periodic and confined samples with differ- entRafall into two distinct groups, respectively.For each group, the overlappedΘ(ξ)profiles follow Eq.(4) with the no-slip and free-slip plates.The mean temperature profile in the periodic cell is similar to that in the confined cell with the no-slip or free-slip plates.It is noted that the data for no-slip plates are in good agree- ment with previous results with parameterp= 3 [28] in periodic and confined samples.

In addition to the mean temperatureΘprofile, the temperature varianceΩ≡η/η0profile is also measured in slippery surfaces [36] .Hereη= 〈θ＇2〉 andη0is the maximal value ofη.One could consider the turbulent diffusivitywithκf/κ≈fξp(f＞0).Here, the constantpis equal to 2 for the free-slip plate and equal to 3 for the no-slip plate.The solution of the tempera- ture variance profileΩ(ξ)near the thermal BL with the no-slip and free-slip plates is given by

with the parameterβ=pm(c−1).Equation (6) can be numeri- cally solved with the initial conditionsΩ(ξ0)= 1 and dΩ(ξ0)/dξ= 0 at the peak positionξ0ofΩ(ξ).The final solutionΩ(ξ:p,c,Δ2/η0,f,γ)contains five parameters.The parameterΔ2/η0is a measurable quantity that can be determined from the DNS data.The parameterscandfare used to fit the data of the turbulent diffusivityκtandκf.It has been found that the parameterγis close to 1 andξ0is close to 0.85 in three-dimensional experiments and DNS [32] .

In Fig.4 b, we show the temperature varianceΩ(ξ)profiles ob- tained in the periodic and confined cells for differentRaas a func- tion ofξwith the no-slip and free-slip plates.When the flow is the closed LSC,Ω(ξ)profiles for the periodic sample are similar to those for the confined sample with no-slip and free-slip plates, which are independent ofRa.For the zonal flow, which exists in the periodic cell with the free-slip plates forRa= 108, the nu- merical solutionΩ(ξ)of Eq.(6) is in excellent agreement with the DNS data in the regionξ1 .Beyondξ≃ 1 , there exist some differences between the DNS data and numerical solution, due to the temperature fluctuations by the increasing thermal plumes.It could also be seen in Fig.1 d that a mass thermal plumes erupt from thermal BL.This does not exist a path for the hot (cold) plumes to transport upward (downward).Therefore, the thermal plumes erupt into bulk randomly and the temperature fluctuation becomes larger outside the thermal BL.The results with no-slip plates agree with previous results with parameterp= 3 by Wang et al.[31,32] both in the periodic and confined samples.

To summarize, we have studied the influence of the horizon- tally periodic sidewall boundary conditions on the heat transport in two-dimensional RBC.The simulation runs are conducted with no-slip or free-slip plate conditions at the Rayleigh numberRavar- ied in the range 106≤Ra≤109, Prandtl numberPr= 4.3 and as- pect ratioΓ= 2 .The conclusions are as follows:

(1) There exist two distinct flow states in the periodic samples with the free-slip plates, namely, closed LSC and zonal flow.WhenRaexceeds a threshold value, the zonal flow forms.In our cases, the zonal flow exists forRa≥3 ×106in the periodic sample with free-slip plates.

(2) As expected, the heat transferNufor the periodic cell is similar to that for the confined cell with no-slip plates.Interest- ingly, the heat transferNuis 15% higher for the periodic sample than for the confined sample with no-slip plates forRa＜3 ×106.In the case of free-slip plates forRa＜3 ×106, the heat transfer Nu for the periodic sample is twice as large as that for the confined sample.ForRa＜3 ×106in the periodic sample, the closed LSC is steady and the thermal plumes are almost exclusively straight up and down in the vertical direction so that the interface between the two closed LSC could be equivalent to the free-slip sidewall boundary condition.

(3) The heat transportNuis influenced and the temperature profiles do not be influenced for the periodic sample compared with the confined sample with no-slip and free-slip plates.For the periodic cell, the heat transportNuis smaller than that for the confined cell with the free-slip plates whenRaexceeds 3 ×106.Especially, the heat transportNufor the free-slip plates is smaller than the no-slip plates in the periodic sample forRa＞107, agree- ing with the previous studies.It has been found that the mean temperatureΘand temperature varianceΩfor the periodic cells agree well with those for the confined cells near the thermal BL with the no-slip or free-slip plates.

(4) The efficiency of the heat transportNuis reduced asRain- creases in the periodic sample, due to the zonal flow.WhenRaexceeds a threshold value, the zonal flow generates in the periodic sample.In the periodic cell, the closed LSC becomes unsteady and tilted, and the zonal flow would develop from the unsteady closed LSC in the case of the free-slip plates forRa≥3 ×106.The zonal flow could suppress the heat transportNu.Furthermore, we have shown that the zonal flow suppresses the vertical Reynolds num- berReyand does not suppress the horizontal Reynolds numberRex.

Our work thus provides a common framework for understand- ing the influence of periodic sidewalls with no-slip and free-slip plates on the heat transport, mean temperature and temperature variance profiles.

Declaration of Competing Interest

The authors declare that there is no conflict of interest regard- ing the publication of this paper

Acknowledgement

This work was supported by the Natural Science Foundation of Guangdong Province (Grant No.2020A1515011094), and the Science, Technology and Innovation Commission of Shenzhen Mu- nicipality (Grant No.KQJSCX20180328165817522).