Boundary Layer Structure in Turbulent Thermal Convection

8/4/2019 Boundary Layer Structure in Turbulent Thermal Convection

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Boundary layer structure in turbulent thermal convection andits consequences for the required numerical resolution

Olga Shishkina 1, Richard J. A. M. Stevens 2, SiegfriedGrossmann 3, Detlef Lohse 2

1 DLR - Institute for Aerodynamics and Flow Technology, Bunsenstrae 10, D-37073Gottingen, GermanyE-mail: [email protected] Department of Science and Technology, Impact Institute, and J.M. Burgers Centerfor Fluid Dynamics, University of Twente, P.O. Box 217, 7500 AE Enschede, TheNetherlandsE-mail: [email protected]; [email protected] Fachbereich Physik der Philipps-Universit at, Renthof 6, D-35032 Marburg, GermanyE-mail: [email protected]

Abstract. Results on the PrandtlBlasius type kinetic and thermal boundary layerthicknesses in turbulent RayleighBenard convection in a broad range of Prandtlnumbers are presented. By solving the laminar PrandtlBlasius boundary layerequations, we calculate the ratio of the thermal and kinetic boundary layer thicknesses,which depends on the Prandtl number P r only. It is approximated as 0 .588P r

1 / 2

for

P r

P r and as 0 .982

P r 1 / 3 for

P r

P r , with

P r

0.046. Comparison

of the PrandtlBlasius velocity boundary layer thickness with that evaluated in thedirect numerical simulations by Stevens, Verzicco, and Lohse ( J. Fluid Mech. 643, 495(2010)) gives very good agreement. Based on the PrandtlBlasius type considerations,we derive a lower-bound estimate for the minimum number of the computationalmesh nodes, required to conduct accurate numerical simulations of moderately high(boundary layer dominated) turbulent RayleighBenard convection, in the thermal andkinetic boundary layers close to bottom and top plates. It is shown that the numberof required nodes within each boundary layer depends on N u and P r and grows withthe Rayleigh number Ra not slower than Ra

0 .15 . This estimate agrees excellentlywith empirical results, which were based on the convergence of the Nusselt number innumerical simulations. a r

X i v : 1 1 0 9 . 6 8 7 0 v 1

[ p h y s i c s . f l u - d y n ]

3 0 S e p 2 0 1 1


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Boundary layer structure in turbulent thermal convection 2

1. Introduction

RayleighBenard (RB) convection is the classical system to study properties of thermalconvection. In this system a layer of uid conned between two horizontal platesis heated from below and cooled from above. Thermally driven ows are of utmostimportance in industrial applications and in natural phenomena. Examples include thethermal convection in the atmosphere, the ocean, in buildings, in process technology,and in metal-production processes. In the geophysical and astrophysical context onemay think of convection in Earths mantle, in Earths outer core, and in the outer layerof the Sun. E.g., the random reversals of Earths or the Suns magnetic eld have beenconnected with thermal convection.

Major progress in the understanding of the RayleighBenard system has beenmade over the last decades, see e.g. the recent reviews [ 1, 2]. Meanwhile it has

been well established that the general heat transfer properties of the system, i. e. N u = N u(Ra, P r ) and Re = Re( N u, P r ), are well described by the GrossmannLohse(GL) theory [3, 4, 5, 6]. In that theory, in order to estimate the thicknesses of the kineticand thermal boundary layers (BL) and the viscous and thermal dissipation rates, theboundary layer ow is considered to be scalingwise laminar PrandtlBlasius ow over aplate. We use the conventional denitions: The Rayleigh number is Ra = gH 3 /with the isobaric thermal expansion coefficient , the gravitational acceleration g, theheight H of the RB system, the temperature difference between the heated lowerplate and the cooled upper plate, and the material constants , kinematic viscosity,and , thermal diffusivity, both considered to be constant in the container (Oberbeck

Boussinesq approximation). The Prandtl number is dened as P r = / and theReynolds number Re = UH/ , with the wind amplitude U which forms in the bulk of the RB container.

The assumption of a laminar boundary layer will break down if the shearReynolds number Res in the BLs becomes larger than approximately 420 [ 7]. Mostexperiments and direct numerical simulations (DNS) currently available are in regimeswhere the boundary layers are expected to be still (scalingwise) laminar, see [ 1].Indeed, experiments have conrmed that the boundary layers scalingwise behave asin laminar ow [8], i.e., follow the scaling predictions of the PrandtlBlasius theory[9, 10, 11, 12, 13, 7]. Recently, Zhou et al. [14, 15] have shown that not only the scalingof the thickness, but also the experimental and numerical boundary layer proles inRayleighBenard convection agree perfectly with the PrandtlBlasius proles, if theyare evaluated in the time dependent reference frames, based on the respective momentary thicknesses. This conrms that the PrandtlBlasius boundary layer theory is indeed therelevant theory to describe the boundary layer dynamics in RayleighBenard convectionfor not too large Res .

The aim of this paper is to explore the consequences of the PrandtlBlasius theoryfor the required numerical grid resolution of the BLs in DNSs. Hitherto, convergencechecks can only be done a posteriori , by checking whether the Nusselt number does


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not considerably change with increasing grid resolution [ 16, 17, 18, 19, 20, 21] or byguaranteeing (e.g. in ref. [ 22, 21]) that the Nusselt numbers calculated from the globalenergy dissipation rate or thermal dissipation rate well agree with that one calculatedfrom the temperature gradient at the plates or the ones obtained from the overallheat ux. The knowledge that the proles are of PrandtlBlasius type offers theopportunity to a priori determine the number of required grid points in the BLs forgiven Rayleigh number and Prandtl number, valid in the boundary layer dominatedranges of moderately high Ra numbers.

In section 2 we will rst revisit the PrandtlBlasius BL theory see refs.[9, 10, 11, 12, 13, 7] or for more recent discussions in the context of RB refs. [6, 23] and derive the ratio between the thermal boundary layer thickness and the velocityboundary layer thickness u as functions of the Prandtl number P r extending previouswork (section 3). We will also discuss the limiting cases for large and small P r ,respectively. The transitional Prandtl number between the two limiting regimes turnsout to be surprisingly small, namely P r= 0 .046. The crossover range is found tobe rather broad, roughly four orders of magnitude in P r . In section 4 we note thatthe PrandtlBlasius velocity BL thickness is different from the velocity BL thicknessbased on the position of the maximum r.m.s. velocity uctuations (widely used in theliterature), but well agrees with a BL thickness based on the position of the maximumof an energy dissipation derivate that was recently introduced in ref. [ 24, 21]. We thenderive the estimate for the minimum number of grid points that should be placed inthe boundary layers close the top and bottom plates, in order to guarantee proper gridresolution. Remarkably, the number of grid points that must have a distance smallerthan u from the wall increases with increasing Ra, roughly asRa0.15 . This estimateis compared with a posteriori results for the required grid resolution obtained in variousDNSs of the last three decades, nding good agreement. Section 5 is left to conclusions.

2. Prandtl boundary layer equations

The PrandtlBlasius boundary layer equations for the velocity eld u (x, z ) (assumed tobe two-dimensional and stationary) over a semi-innite horizontal plate [9, 10, 11, 12,13, 7] read

ux x ux + uz zux = z z ux , (1)with the boundary conditions ux(x, 0) = 0, uz(x, 0) = 0, and ux(x, ) = U . Hereux (x, z ) is the horizontal component of the velocity (in the direction x of the large-scale circulation), uz(x, z ) the vertical component of the velocity (in the direction zperpendicular to the plate), and U the horizontal velocity outside the kinetic boundarylayer (wind of turbulence). Correspondingly, the equation determining the (stationary)temperature eld T (x, z ) reads

ux x T + uz zT = z zT, (2)


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with the boundary conditions T (x, 0) = T plate and T (x, ) = T bulk , which underOberbeckBoussinesq conditions is the arithmetic mean of the upper and lower platetemperature. Applying these equations to RB ow implies that we assume thetemperature eld to be passive.

The dimensionless similarity variable for the vertical distance z from the platemeasured at the distance x from the plates edge is

= z U x . (3)Since the ow in Prandtl theory is two-dimensional, a streamfunction canbe introduced, which represents the velocity eld. The streamfunction is non-dimensionalized as = / xU , and the temperature is measured in terms of / 2,giving the non-dimensional temperature eld . Rewriting eqs. ( 1) and (2) in terms of and one obtains

d3/d 3 + 0 .5 d2/d 2 = 0 , (4)d2/d 2 + 0 .5P r d/d = 0 . (5)

Here the boundary conditions are(0) = 0 , d/d (0) = 0 , d/d () = 1 , (6)(0) = 0 , () = 1 . (7)

The temperature and velocity proles obtained from numerically solving equations(4)(7) (for particular Prandtl numbers) are already shown in textbooks [12, 7, 13] andin the context of RB convection in refs. [ 23, 25]: From the momentum equation ( 6) with

above boundary conditions one immediately obtains the horizontal velocity d/d . Thedimensionless kinetic boundary layer thickness u can be dened as that distance fromthe plate at which the tangent to the function d/d at the plate ( = 0) intersects thestraight line d/d = 1 (see gure 1 a). As equation ( 4) and the boundary conditions ( 6)contain no parameter whatsoever, the dimensionless thickness u of the kinetic boundarylayer with respect to the similarity variable is universal, i.e., independent of P r andU or Re,

u = A1 3.012 or A 0.332. (8)Solving numerically equation ( 5) with the boundary conditions ( 7) for any xed

Prandtl number, one obtains the temperature prole with respect to the similarityvariable (see gure 1 b). Note that in contrast to the longitudinal velocity d/d , thetemperature prole depends not only on but also on the Prandtl number, since P rappears in equation ( 5) as the (only) parameter. The distance from the plate at whichthe tangent to the prole intersects the straight line = 1 denes the dimensionlessthickness of the thermal boundary layer,

= C (P r ), (9)where C (P r ) is a certain function of Prandtl number. E.g., one numerically ndsC 3.417, 1.814, and 1.596 for P r = 0 .7, 4.38, and 6.4, respectively (see gure 1b).


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0 u 6

0

1

d / d

(a)

0 6

0

1

(b)

Figure 1. Solution of the PrandtlBlasius equations ( 4)(7): (a) Longitudinalvelocity prole dd () (solid curve) with respect to the similarity variable . Thetangent to the longitudinal velocity prole at the plate ( = 0) and the straight lined/d = 1 (both dashed lines) intersect at = u A 1 3.012, for all P r . We denethis value u as the thickness of the kinetic boundary layer. ( b) Temperature prole() as function of the similarity variable for P r = 0 .7 (black solid curve), P r = 4 .38(red solid curve) and P r = 6 .4 (green solid curve). The tangents to the prole curvesat the plate ( = 0) and the straight line = 1 (dashed lines) dene the edges(thicknesses) of the corresponding thermal boundary layers, i. e., = C (P r ). Forthe presented cases P r = 0 .7, 4.38, and 6.4 one has C (0.7) 3.417, C (4.38) 1.814,and C (6.4) 1.596, respectively.

From ( 8) and (9) one obtains the ratio between the (dimensional) thermal boundarylayer thickness and the (dimensional) kinetic boundary layer thickness u :

u

=u

= AC (P r ). (10)As discussed above, the constant A and the function C = C (P r ) are found from thesolutions of equations ( 4)(7) for different P r . A and C (P r ) reect the slopes of therespective proles,

A =d2d2 (0), C (P r ) =

dd (0)

1. (11)

With ( 3) the physical thicknesses are u = u / U x and = / U x , generallydepending on U and the position x along the plate. The physical thermal BL thicknessthen is

=C (P r )

U/ (x )= U x (0)

1

= z

(0)1

. (12)

Thus, explicitly it depends neither on U nor on the position x along the plate. Remindingthe denition of the thermal current J = uzT

z T , we get z (0) =

1 / 2

T z (0) =


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2 J = 2H 1 N u, i. e., on x-average we have

=H

2 N u. (13)

is the so-called slope thickness, see Sect. 2.4 of reference [23]. In contrast to thethermal BL thickness the physical velocity BL thickness u = A1/ U x dependsexplicitly both on the position x and on the wind amplitude U . In a RayleighBenardcell we choose for x a representative value x = aL = aH . Then the famous Prandtlformula [9] results

u =aH Re

. (14)

Here a = aA2 = A1a. The constant a has been obtained empirically [ 5], based onthe experimental measurements by [26] performed in a cylindrical cell of aspect ratioone, lled with water. The result was [ 5]

a 0.482. (15)We note that this value probably depends on the aspect ratio, on the shape of theRB container, and can also be different for numerical 2D RayleighBenard convection[27, 28, 29]. It will also be different for the slope thickness as considered here or otherdenitions as e. g. the 99% -thickness.

It seems worthwhile to note that similarly to the case of also u can be expressedby a prole slope at the plate. Analogously to the temperature case one calculatesfor the kinetic thickness u = U/ u xz (0). Here U appears explicitly and the derivativemay depend on x . The denominator is the local stress tensor component, which after averaging describes the momentum transport, just as the temperature prolederivative at the plate characterises the heat transport. In combination with eq. ( 14) itsays that the kinetic stress behaves as u xz (0) U

Re/ (aH ).From eqs. ( 10) and (14) we also nd the useful (and known) relation for Prandtl-

Blasius boundary layers

= aC (P r )H

Rewith a = A a 0.160. (16)

From solving equations ( 4)(7) together with relations ( 11) one obtains that the BL

thickness ratio ( 10) has two limiting cases, namely / uP r1/ 2

for very small P r 1and / u P r1/ 3 for very large P r 1. We thus present the ratio of the thermaland kinetic boundary layer thicknesses normalised by P r1/ 3 in gure 2 for different P rfrom P r = 106 to 106. The gure conrms that the scaling of the ratio between thethermal and kinetic boundary layer thicknesses in the low and high Prandtl numberregimes is P r 1/ 2 and P r1/ 3, respectively. Between these two limiting regimes thereis a transition region, whose width is about 4 orders of magnitude in P r . In the nextsection we will derive analytic expressions for the ratio / u in the respective regimes,which will be used in the remainder of the paper to analyse the resolution propertiesfor DNS in the BLs of the RayleighBenard system.


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-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7logP r

-0.10.00.10.20.30.40.50.60.70.8

l o g

(

/ u ) P r 1

/ 3

Figure 2. Double-logarithmic plot of the ratio of the thermal and kinetic boundarylayer thicknesses, normalised by P r

1 / 3 , as obtained from numerical solution of equations ( 4)(7) as function of P r (solid black line). For large P r the curve throughthe data is constant, for small P r the (plotted, reduced) curve behaves P r

1 / 6 .Approximation ( 22) (green dotted line) is indistinguishable from / u in the region

P r < 310 4 . Approximation (24) (blue dashed-dotted line) well represents / u forP r > 0.3; for P r > 3 it practically coincides with approximation ( 25). Approximation(26) (red solid curve) connects the analytical approximations in the transition range3 10 4 P r 3 between the lower and upper Prandtl number regimes.

In the PrandtlBlasius theory the asymptotic velocity amplitude U is a givenparameter; the resulting heat current N u is a performance of the boundary layers only.In contrast, in RayleighBenard convection the heat transport is determined by theBLs together with the bulk ow. Therefore in RB convection the wind amplitude U nolonger is a passive parameter, but U and N u are actively coupled properties of the fullthermal convection process.

The Reynolds number Re is dened as the dimensionless wind amplitude,Re =

UH

. (17)

From the law for the kinetic BL thickness ( 14), the thermal BL thickness (13), andthe BL thickness ratio ( 10) one obtains

Re =aH u

2

=u

2 aH

2

= 4a2 N u2u

2

. (18)

This Re N u2 law is in perfect agreement with the GL theory [ 3, 4, 5, 6]. In thattheory several sub-regimes in the ( Ra, P r ) parameter space are introduced, dependingon the dominance of the BL or bulk contributions. In regimes I and II the BL of thetemperature eld dominates, while in III and VI it is the thermal bulk. Regimes Iand II differ in the velocity eld contributions: It either is the u-BL (I) or the u-bulk(II) which dominates; analogously the pair III and IV is characterized. The labels


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(for lower Pr) and u (for u pper Pr) distinguish the cases in which the thermal BL isthicker or smaller than the kinetic one. All ranges in the GL theory, which are thermalboundary layer dominated, show the Re N u2 behaviour, namely I l, I u , II l, II u . Inthe thermal bulk dominated ranges of RB convection the relation between

Re and

N u

is different. In III u we have Re N u4/ 3, in IV l it is Re N u, and in IV u alsoReN u4/ 3 holds; but here the PrandtlBlasius result ( 18) is not applicable, since theheat transport mainly depends on the heat transport properties of the bulk. In therange I , although boundary layer dominated, also a different relation ( Re N u3)holds; here the upper and the lower kinetic BLs ll the whole volume and thereforethere is no free ow outside the BLs, in contrast to the PrandtlBlasius assumption of an asymptotic velocity with the LSC amplitude U .

3. Approximations for the ratio / u of the temperature and velocity

boundary layer thicknesses

In this section we will derive analytical approximations for the ratio / u for the threeregimes identied in the previous section, cf. gure 2. We start by discussing the low(P r < 3 104) and the high (3 < P r ) Prandtl number regimes, before we discuss thetransition region 3 104 P r 3.3.1. Approximation of / u for P r < 3 104In the case of very small Prandtl number, P r 1, the thickness of the velocity boundarylayer is negligible compared with the thickness of the temperature boundary layer, i.e., u . Hence, in most of the thermal boundary layer it is ux U . Introducing thesimilarity variable as in ref. [13]

=z2 U x , (19)

one obtains the following equation for the temperature as a function of :

d2/d 2 + 2 d/d = 0 , with (0) = 0 , () = 1 .The solution of this boundary value problem is the Gaussian error function

() = erf( ) 2

0 et 2

dt. (20)

According to ( 3) and ( 19), the similarity variable used in the Prandtl equations andthe similarity variable used in the approximation for P r 1 are related as follows

=12P r

1/ 2. (21)

Applying now the formulae ( 20), (21) and (11) we obtain the following equalities:2

=dd

(0) =dd

(0) dd

=1

C (P r ) 2P r1/ 2.


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This leads to the approximation for the function C (P r ) = P r1/ 2 for very small P r .u

= AP r1/ 2 0.588P r1/ 2, P r 1. (22)

In gure 2 one can see that for very small Prandtl numbers, P r < 3 104

, theapproximation ( 22) is as expected indistinguishable from the numerically obtained / u .

3.2. Approximation of / u for P r > 3Meksyn [12], based on the work by Pohlhausen [ 11], derived that the solution of thetemperature equation ( 5), together with relation ( 7) equals

2 = D

/ 2

0eF (t)P r dt, F (t) = 12

t

0(q)dq. (23)

The constant D can be found as usual from the boundary condition at innity and was

approximated in [ 11, 12] for P r > 1 as followsD =

0.478P r 1/ 3c(P r )

, c(P r ) 1 +1

45P r 1

405P r 2+

161601425P r 3

...

From this and ( 23) one derives

0.478P r 1/ 3c(P r )

= D =d

d(/ 2) (0) = 2dd

(0) =2

C (P r ).

This connects c(P r ) and C (P r ) as followsC (P r )

20.478

c(P r )P r1/ 3 2.959c(P r )P r 1/ 3,resulting in the approximation

u

= AC (P r ) = E P r1/ 3c(P r ), E A2

0.478 0.982. (24)For Pr 1, the function c(P r ) approaches 1, hence C (P r ) 2.959P r1/ 3, implying

u

= E P r1/ 3, P r 1. (25)In gure 2 the approximation ( 24) is presented as a blue dash-dotted curve. For P r > 3the function ( / u )P r 1/ 3 almost coincides with the constant E .3.3. Approximation of / u in the crossover range 3 104 P r 3As one can see in gure 2, the approximation ( 22) well represents / u in the region

P r < 3104, while (25) is a good approximation of / u for P r > 3. An approximationof the ratio of the thermal and kinetic boundary layer thicknesses in the transition region3 104 P r 3 is obtained by applying a least square t to the numerical solutionsof the PrandtlBlasius equations ( 4)-(7). One nds:

u P r

0.357+0 .022log P r , 3 104 P r 3. (26)As seen in gure 2, this relation is a good t of the full solution in the transition regime.


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3.4. Summary

For the ratio / u of the thicknesses of the thermal and kinetic boundary layers, whichdepends strongly (and only) on P r , we nd according to (22), (25), and ( 26)

u

=AP r1/ 2, A 0.332, P r < 3 104,P r0.357+0 .022log P r , 3 104 P r 3,E P r1/ 3, E 0.982, P r > 3.

(27)

The crossover Prandtl number P rbetween the asymptotic behaviours, cf. rstand last line of (27), is dened as the intersection point P r = 0 .046 of theasymptotic approximations. Note that this crossover between the small- P r behaviour/ u P r1/ 2 and the large- P r behaviour / u P r1/ 3 does not happen at a Prandtlnumber of order 1, but at the more than 20 times smaller value P r= 0 .046. In thissense most experiments are conducted in the large P r regime. However, also note thatother denitions of the BL thicknesses lead to other crossover Prandtl numbers.

Finally, we also give the thickness of the kinetic BL in the three regimes, as obtainedfrom (27) and (13), namely

u =0.5 N u1P r 1/ 2A11/ 2H , P r < 3 104,0.5 N u1P r 0.3570.022log P r H , 3104 P r 3,0.5 N u1P r 1/ 3E 1H , P r > 3.

(28)

We compare this PrandtlBlasius result ( 28) for the kinetic boundary layerthickness in terms of N u and P r (thus valid if the heat transport is BL dominated)with the estimate given in reference [ 21], where the kinetic boundary layer thickness in

a cylindrical cell is identied as two times that height at which the averaged quantityu := u 2u t,,r (29)

has a maximum, because it was empirically found that the maximum of u isapproximately in the middle of the velocity boundary layer. Here u is the velocityeld and the averaging is over time t, the azimuthal direction , and over the radialdirection 0 .1R < r < 0.9R, with R the radius of the cylindrical convective cell. Therestricted range for the radial direction has been used in order to exclude the singularityregion close to the cylinder axis and the region close to the sidewall, where the denitionmisrepresents the kinetic boundary layer thickness. Figure 3 shows that there is a very

good agreement between the theoretical PrandtlBlasius slope boundary layer thicknessand that obtained using ( 29). The gure also shows that the position of the maximumr.m.s. velocity uctuations is not a good indicator for the velocity boundary layer edge;it rather seems to identify the position where the LSC is the strongest.

4. Resolution requirements within the boundary layers in DNS

We now come to the main point of the paper: What can we learn from the PrandtlBlasius theory for the required mesh resolution in the BLs of DNS of turbulent RBconvection? Obviously, a proper mesh resolution should be used in order to obtain


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0 0.05 0.1

z/H

0.0

0.2

0.4

0.6

0.8

1.0

u / m a x

( u ) , u

r m s

/ m a x u r

m s

(a)

0 0.01 0.02

z/H

0.0

0.2

0.4

0.6

0.8

1.0

u / m a x

( u ) , u

r m s

/ m a x u r

m s

(b)

Figure 3. Proles of u (29) (black), and the r.m.s. velocity uctuations for theazimuthal velocity component u (green) for ( a) Ra = 10 8 and P r = 6 .4 and ( b)Ra = 2 109 and P r = 0 .7. The proles have been normalised with the respectivemaxima for clarity. The vertical black lines indicate the velocity boundary layerthickness based on ( 28). The red dashed and solid lines indicate the heights at whichthe quantity u (29) has a maximum and two times this height, respectively. Thevertical green line indicates the position of the maximum r.m.s. velocity uctuations.

accurate results. In a perfect DNS the local mesh size should be smaller than the local

Kolmogorov K (x , t ) and Batchelor B (x , t ) scales (see e.g. ref. [30]), and the resolutionin the boundary layers should be also sufficient, see e.g. [ 31, 16, 25, 32, 21]. It indeedhas been well established that the Nusselt number is very sensitive to the grid resolutionused in the boundary layers; when DNS is underresolved, the measured Nusselt numberis too high [31, 33, 16, 34, 35, 36, 21]. Hitherto, the standard way to empirically checkwhether the mesh resolution is sufficient is to try a ner mesh and to make sure that theNusselt number is not too different. In this way the minimal number of grid points thatis needed in the boundary layer is obtained by trial and error: Gr otzbach [31] variedthe number of grid points in the boundary layer between 1 and 5 in simulations up to

Ra = 3

105 with

P r = 0 .71 and found that 3 grid points in the boundary layers should

be sufficient. Verzicco and Camussi [33] tested this at Ra = 2 107 and P r = 0 .7 andstated that at least 5 points should be placed in the boundary layers. Stevens et al.[21] tested the grid resolution for Ra = 2 106 to 2 1011 and P r = 0 .7. They foundthat for Ra = 2 109 the minimum number of nodes in the boundary layers should bearound 10 and that this number increases for increasing Ra. Together with the earlierseries of papers the data clearly suggest that indeed there is an increase of required gridpoints in the BL with increasing Rayleigh number.

However, one must be careful. The empirical determination of the required numberof grid points in the BL is not only intensive in computational cost, but also difficult.


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The Nusselt number obtained in the simulations not only depends on the grid resolutionin the BLs at the top and bottom plates, but also on the grid resolution in the bulk andat the side walls where the thermal plumes pass along [ 21]. So obviously a generaltheory-based criterion for the required grid resolution in the thermal and kinematicboundary layers will be helpful for performing future simulations. In this section wewill derive such a universal criterion, harvesting above results from the PrandtlBlasiusboundary layer theory.

We rst dene the (local) kinetic energy dissipation rates per mass,

u (x , t ) 2

i j

u i(x , t )x j

+u j (x , t )

x i

2

. (30)

Its time and space average for incompressible ow with zero velocity b.c. is u t,V =

i ju i ( x ,t )

x j

2t,V . It is connected with the Nusselt number through the exact

relation

u t,V = 3

H 4( N u 1)RaP r2. (31)

This follows directly from the momentum equation for RayleighBenard convection inBoussinesq approximation [ 37]. Here, t,V denotes averaging over the whole volume of the convective cell and over time and (later) t,A denotes averaging over any horizontalplane and time.

We start with the well established criterion that in a perfect DNS simulation the

(local) mesh size must not be larger than the (local) Kolmogorov scale [38] K (x , t ),which is locally dened with the energy dissipation rate of the velocity,

K (x , t ) = 3/ u (x , t )1/ 4 . (32)

K is the length scale at which the inertial term u2r /r and the viscous term u r /r

2

of the Navier-Stokes equation balance, where ur ( u r )1/ 3 has been assumed for the

velocity difference at scale r . A corresponding length scale T follows from the balanceof the advection term ur T r /r and the thermal diffusion term T r /r

2 in the advectionequation; it is

T (x , t ) = 3/ u (x , t )1/ 4 = K (x , t )

P r3/ 4. (33)

However, for large Pr the velocity eld is smooth at those scales at which the temperatureeld is still uctuating. Then the velocity difference ur u /r and advection termand thermal diffusion term balance at the so-called Batchelor scale [39] B , which isdened as

B (x , t ) = 2/ u (x , t )1/ 4 = K (x , t )P r 1/ 2. (34)

For small P r < 1 obviously T > B > K and for comparison with the grid resolution,the Kolmogorov scale K seems to be the most restrictive (i.e., smallest) length scale.In contrast, for large P r > 1 it T < B < K and one may argue that T is the most


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restrictive length scale. This indeed may be the case in the Prandtl number regime inwhich the velocity eld can still be described through Kolmogorov scaling ur ( u r )

1/ 3,but for even larger P r the velocity eld becomes smooth ur

u /r and then the

grid resolution should be compared to the Batchelor scale B as smallest relevant lengthscale. In below analysis, for P r > 1 we will restrict ourselves to this limiting case.

We now dene global Kolmogorov and Batchelor length scales globalK 3 / 4

u1 / 4t,V

and

globalB 1 / 4 1 / 2

u1 / 4t,V

, respectively, (and also the global length scale globalT 3 / 4

u1 / 4t,V

). Using

the exact relation ( 31), one can nd how the global Kolmogorov length globalK dependson Ra, P r , and N u, namely

globalK 3/ 4

u1/ 4t,V

= P r 1/ 2Ra1/ 4( N u 1)1/ 4

H. (35)

The admissible global mesh size hglobal should clearly be smaller than both globalK

andglobalB , which implies that one is on the safe side provided that

hglobal P r 1/ 2

Ra1/ 4( N u 1)1/ 4H for P r 1 (36)

or with the relation ( 34) between the Kolmogorov and Batchelor length

hglobal 1

Ra1/ 4( N u 1)1/ 4H for P r > 1. (37)

A similar way to estimate mesh requirements in the bulk was suggested for the rsttime by Gr otzbach [31]. Note that with these estimates for the required bulk resolutionfor most times and locations one is on the safe side, as equation ( 31) is an estimate forthe volume averaged energy dissipation rate, which is localized in the boundary layers.However, not only the background eld but also plumes detaching from the boundarylayers do require an adequate resolution.

To estimate the number of nodes that should be placed in the boundary layers, wewill rst estimate the area averaged energy dissipation rate in a horizontal plane in thevelocity BL, u t,ABL . Employing eqs. (17), (14) and ( 30), one can nd a lower boundfor this quantity, namely

u t,ABL u xz

2

t,A

u xz t,A

2

U u

2

=

= ReH

Re1/ 2aH

2

= 3Re3a2H 4

. (38)

From eqs. ( 31), (38), (18) and ( 27) it follows a lower bound for the ratio

u t,ABL

u t,V P r 2Re3

a2Ra N u= 64a4 N u5 P

r 2

Rau

6

=643a4A6 N u5P r1Ra1, P r < 3 104,64a4 N u5P r 0.15+0 .132log P r Ra1, 3 104 P r 3,64a4E 6

N u5

Ra1,

P r > 3.

(39)


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For the Kolmogorov length BLK in the velocity BL one can therefore write

BLK 3

u

1/ 4

t,ABL

u t,V

u t,ABL

1/ 4

globalK . (40)

The mesh size hBL in the BL must be smaller than BLK and BLB , i.e., one is on the safeside if

hBL

23/ 2a1 N u3/ 2P r 3/ 4A3/ 23/ 4H , P r < 3 104,23/ 2a1 N u3/ 2P r 0.53550.033log P r H , 3 104 P r 1,23/ 2a1 N u3/ 2P r 0.03550.033log P r H , 1 < P r 3,23/ 2a1E 3/ 2 N u3/ 2H , P r > 3,

(41)

according to ( 39), (40), (36) and (37).From the relations ( 41), (27) and (13) one can estimate the minimum number of

nodes of the computational mesh, which must be placed in each thermal and kineticboundary layer close the plates. We nd that this minimum number of nodes in thethermal boundary layers is

N th.BL

hBL2a N u1/ 2P r3/ 4A3/ 23/ 4, P r < 3 104,2a N u1/ 2P r0.5355+0 .033log P r , 3 104 P r 1,2a N u1/ 2P r0.0355+0 .033log P r , 1 < P r 3,2a N u1/ 2E 3/ 2, P r > 3,(42)

while the minimum number of nodes in the kinetic boundary layers is

N v.BL u

hBL= u

hBL2a N u1/ 2P r1/ 4A1/ 21/ 4, P r < 3 104,2a N u1/ 2P r0.1785+0 .011log P r , 3104 P r 1,2a N u1/ 2P r 0.3215+0 .011log P r , 1 < P r 3,2a N u1/ 2P r 1/ 3E 1/ 2, P r > 3.

(43)

The number of nodes in the thermal boundary layer looks very restrictive for very low

P r ; however, one should realise that for very low P r the thermal boundary layer alsobecomes much thicker than the velocity boundary layer. Hence, the criterion for the

number of nodes in the thermal boundary layers determines the ideal distribution of nodes above the viscous boundary layer. For very high P r the kinetic boundary layerbecomes much thicker than the thermal boundary layer, and hence the restriction forthe velocity boundary layer determines the ideal distribution of nodes above the thermalboundary boundary layer. Note that for large P r equation ( 42) suggests that the numberof grid points in the thermal boundary layer becomes independent of P r (for xed N u):Indeed, as the velocity eld is smooth anyhow, with increasing P r no extra grid pointsare necessary in the thermal BL.

In gure 4 we show the minimum number of nodes N th.BL and N v.BL , respectively,necessary to simulate the cases which have been investigated experimentally so far, for


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0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

l o g

N t h

. B L

(a)

5 6 7 8 9 10 11

logRa0

0.2

0.4

0.6

0.8

1.01.2

1.4

l o g

N v

. B L

(b)

Figure 4. Minimum number of BL nodes necessary in DNS of boundary layerdominated, moderately high RB convection. ( a) N th.BL (42) in the thermal boundarylayers and ( b) N v.BL (43) in the kinetic boundary layers, required to simulate theexperimentally investigated cases, references [ 40] (lilac squares, P r = 0 .67), [41] (blacktriangles, 0 .60 P r 7.00), [42] (blue circles, 0 .68 P r 5.92), [43] (green triangles,0.73 P r 6.00), [44] (red pentagons, 3 .76 P r 5.54), [45] (black crosses,P r = 4 .2) and [46] (black pluses, P r = 7 .0). Dashed lines are ts to the quasi-data(measured values introduced into eqs. ( 42), (43)), with precision O(10 4 ); roundingthe respective numbers to their upper bounds gives ( a) N th.BL 0.35Ra

0 .15 (44) and(b) N v.BL 0.31Ra

0 .15 (45) for the quasi-data in the ranges 10 6 Ra 1010 and0.67 P r 0.73.

different Ra and P r . The data points are generated by introducing the experimentalvalues of Ra, P r , and the (measured) corresponding N u into the formulas ( 42), (43).Based on these quasi-data points, one can give e. g. the following ts for the minimumnumber of nodes within the boundary layers for the case of P r 0.7:

N th.BL 0.35Ra0.15 , 106 Ra 1010 , (44)N v.BL 0.31Ra0.15 , 106 Ra 1010 . (45)

Note that the numerical pre-factors in these estimates signicantly depend on the


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Prandtl number and on the empirically determined (ref. [5]) value of a, cf. eq. (15).The minimum node numbers for other values of P r can be calculated directly using therelations ( 42)(43). Apparently the scaling exponent depends much less on P r . Allthese estimates only give lower bounds on the required number of nodes in the boundarylayers.

As discussed at the beginning of this section, previous studies by Gr otzbach [31],Verzicco and Camussi [ 33], and Stevens et al. [21] found an increasing number of nodesthat should be placed in the thermal and kinetic boundary layers. The theoretical resultsthus conrm all above studies, because the increasing number of nodes was due to theincreasing Ra number at which the tests were performed. To be more specic: accordingto the estimates ( 44) and (45) for P r = 0 .7 the minimum number of nodes that shouldbe placed in the thermal and kinetic boundary layers is N 2.3 for Ra = 3 105,N 4.4 for Ra = 2 107, and N 8.7 for Ra = 2 109. The empirically found valuesat the respective Ra with P r 0.7 are 3 for Ra = 3 10

5, 5 for Ra = 2 10

7, and

10 for Ra = 2 109. Thus there is very good agreement between the theoretical resultsand the empirically obtained values, especially if one considers the difficulties involvedin determining these values empirically, and the empirical value for the constant a (15)that is used in the theoretical estimates. We want to emphasize that not only theboundary layers close to the plates, but also the kinetic boundary layers close to thevertical walls must be well resolved.

To sum up, the mesh resolution should be analysed a priori using the resolutionrequirements in the bulk ( 36), (37) and in the boundary layers ( 42), (43). Havingconducted the DNS, the Kolmogorov and Batchelor scale should be checked a posteriori ,to make sure that the mesh size was indeed small enough (as it has been done, forexample, in refs. [19, 20]).

5. Conclusion

In summary, we used laminar PrandtlBlasius boundary layer theory to determine therelative thicknesses of the thermal and kinetic boundary layers as functions of P r (27).

We found that neither the position of the maximum r.m.s. velocity uctuationsnor the position of the horizontal velocity maximum reect the slope velocity boundarylayer thickness, although many studies use these as criteria to determine the boundarylayer thickness. In contrast to them, the algorithm by Stevens et al. [21] agrees verywell with the theoretical estimate of the kinetic slope boundary layer thickness.

We used the results obtained from the PrandtlBlasius boundary layer theory toderive a lower bound on the minimum number of nodes that should be placed in thethermal and kinetic boundary layers close to the plates. We found that this minimumnumber of nodes increases not slower than Ra0.15 with increasing Ra. This result is inexcellent agreement with results from several numerical studies over the last decades, inwhich this minimum number of nodes was determined empirically. Hence, the derivedestimates can be used as guideline for future direct numerical simulations.


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REFERENCES 17

Acknowledgments

OS expresses her thanks to Prof. Dr.-Ing. Claus Wagner and to the DeutscheForschungsgemeinschaft (DFG) for supporting the work under the grant WA 1510/9.The Twente part of the work is supported by the Foundation for Fundamental Researchon Matter (FOM), sponsored by NWO.

References

[1] Ahlers G, Grossmann S and Lohse D 2009 Rev. Mod. Phys. 81 503[2] Lohse D and Xia K Q 2010Annu. Rev. Fluid Mech. 42 335364[3] Grossmann S and Lohse D 2000 J. Fluid. Mech. 407 2756[4] Grossmann S and Lohse D 2001 Phys. Rev. Lett. 86 33163319

[5] Grossmann S and Lohse D 2002 Phys. Rev. E 66 016305[6] Grossmann S and Lohse D 2004 Phys. Fluids 16 44624472[7] Landau L D and Lifshitz E M 1987 Fluid Mechanics (Oxford: Pergamon Press)[8] Sun C, Cheung Y H and Xia K Q 2008 J. Fluid Mech. 605 79 113[9] Prandtl L 1905 uber ussigkeitsbewegung bei sehr kleiner reibung Verhandlungen

des III. Int. Math. Kongr., Heidelberg, 1904 (Leipzig: Teubner) pp 484491[10] Blasius H 1908Z. Math. Phys. 56 137[11] Pohlhausen K 1921 Z. Angew. Math. Mech. 1 252268

[12] Meksyn D 1961 New Methods in Laminar Boundary Layer Theory (Oxford:Pergamon Press)

[13] Schlichting H 1979 Boundary layer theory 7th ed (New York: McGraw Hill bookcompany)

[14] Zhou Q and Xia K Q 2010Phys. Rev. Lett. 104 104301[15] Zhou Q, Stevens R J A M, Sugiyama K, Grossmann S, Lohse D and Xia K Q 2010

J. Fluid. Mech.

[16] Kerr R 1996 J. Fluid Mech. 310 139179[17] Verzicco R and Camussi R 1997 Phys. Fluids 9 12871295[18] Kerr R and Herring J R 2000 J. Fluid Mech. 419 325344[19] Shishkina O and Wagner C 2007 Phys Fluids. 19 085107[20] Shishkina O and Wagner C 2008 J. Fluid Mech. 599 383404[21] Stevens R J A M, Verzicco R and Lohse D 2010 J. Fluid. Mech. 643 495507[22] Calzavarini E, Lohse D, Toschi F and Tripiccione R 2005 Phys. Fluids 17 055107[23] Ahlers G, Brown E, Fontenele Araujo F, Funfschilling D, Grossmann S and Lohse

D 2006 J. Fluid Mech. 569 409445[24] Stevens R J A M, Clercx H J H and Lohse D 2010 New J. Phys. x y


18/18

REFERENCES 18

[25] Shishkina O and Thess A 2009 J. Fluid Mech. 633 449460[26] Qiu X L and Tong P 2001 Phys. Rev. Lett 87 094501[27] DeLuca E E, Werne J, Rosner R and Cattaneo F 1990 Phys. Rev. Lett. 64 2370

2373[28] Schmalzl J, Breuer M, Wessling S and Hansen U 2004 Europhys. Lett. 67 390396[29] Sugiyama K, Calzavarini E, Grossmann S and Lohse D 2009 J. Fluid Mech. x y[30] Monin A S and Yaglom A M 1975 Statistical Fluid Mechanics (Cambridge,

Massachusetts: The MIT Press)[31] Grotzbach G 1983 J. Comp. Phys. 49 241264[32] Shishkina O, Shishkin A and Thess A 2009 J. Comput. and Appl. Math. 226 336

344[33] Verzicco R and Camussi R 2003 J. Fluid Mech. 477 1949[34] Amati G, Koal K, Massaioli F, Sreenivasan K R and Verzicco R 2005 Phys. Fluids

17 121701[35] Verzicco R and Sreenivasan K R 2008 J. Fluid Mech. 595 203219[36] Verdoold J, van Reeuwijk M, Tummers M J, Jonker H J J and Hanjalic K 2008

Phys. Rev. E 77 016303[37] Siggia E D 1994Annu. Rev. Fluid Mech. 26 137168[38] Kolmogorov A N 1941CR. Acad. Sci. USSR. 30 299[39] Batchelor G K 1959 J. Fluid Mech. 5 113

[40] Ahlers G 2009Physics 2 74[41] Chavanne X, Chilla F, Chabaud B, Castaing B and Hebral B 2001 Phys. Fluids 13

13001320[42] Niemela J and Sreenivasan K R 2003 J. Fluid Mech. 481 355384[43] Roche P E, Castaing B, Chabaud B and Hebral B 2004 J. Low. Temp. Phys. 134

10111042[44] Sun C, Ren L Y, Song H and Xia K Q 2005 J. Fluid Mech. 542 165174[45] Xia K Q, Lam S and Zhou S Q 2002Phys. Rev. Lett. 88 064501

[46] Qiu X L and Xia K Q 1998Phys. Rev. E 58 486491

Boundary Layer Structure in Turbulent Thermal Convection

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