Physica A 392 (2013) 6273–6283

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Physica A

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An investigation into the maximum entropy productionprinciple in chaotic Rayleigh–Bénard convectionR.A.W. Bradford ∗

EDF Energy, Barnett Way, Barnwood, Gloucester, Glos.GL4 3RS, UK

h i g h l i g h t s

• The maximum entropy production principle is applied to Rayleigh–Bénard convection.• The Navier–Stokes equation is not used as a constraint in the optimisation.• The resulting maximal Nusselt number is close to experimental determinations.• The Nusselt number is roughly equal to the number of distinct vertical modes.• MEPP can predict the dynamic steady states of non-equilibrium systems.

a r t i c l e i n f o

Article history:Received 15 April 2013Received in revised form 13 August 2013Available online 24 August 2013

Keywords:Rayleigh–Bénard convectionMaximum entropy productionChaotic

a b s t r a c t

The hypothesis is made that the temperature and velocity fields in Rayleigh–Bénardconvection can be expressed as a superposition of the active modes with time-dependentamplitudes, even in the chaotic regime. The maximum entropy production principle isinterpreted as a variational principle in which the amplitudes of the modes are thevariational degrees of freedom. For a given Rayleigh number, the maximum heat flow forany set of amplitudes is sought, subject only to the constraints that the energy equationbe obeyed and the fluid be incompressible. The additional hypothesis is made that alltemporal correlations between modes are zero, so that only the mean-squared amplitudesare optimising variables. The resulting maximal Nusselt number is close to experimentaldeterminations. The Nusselt number would appear to be simply related to the numberof active modes, in particular the number of distinct vertical modes. It is significant thatreasonable results are obtained for the optimised Nusselt number in that the dynamics (theNavier–Stokes equation) is not used as a constraint. This suggests grounds for optimismthat the maximum entropy production principle, interpreted in this variational manner,can provide a reasonable guide to the dynamic steady states of non-equilibrium systemswhose detailed dynamics are unknown.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

The existence of general extremal principles in non-equilibrium thermodynamics is contentious. A suggestion whichhas gained popularity in some quarters is that a non-equilibrium system in a dynamic steady state will maximise its rateof entropy production. This is referred to as the Maximum Entropy Production Principle (MEPP). There is no proof of thishypothesis in the general case, though attempts in this direction have been made, notably by Dewar, [1,2]. However, whilsta rigorous mathematical proof from more basic physics may not be forthcoming, this does not necessarily prevent the

∗ Tel.: +44 01452653237.E-mail addresses: RickatMerlinHaven@hotmail.com, rick.bradford@edf-energy.com.

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6274 R.A.W. Bradford / Physica A 392 (2013) 6273–6283

principle being of considerable utility in understanding highly complex systems, see for example Bruers [3], Grinstein andLinsker [4], Dewar [5] and Virgo [6]. The generic situation when modelling very complex systems is that some constraintsmay be secure, such as conservation of energy, but formulating the dynamical equations in sufficient detail to make themodel deterministic is often problematical. Deterministic models of complex systems face a choice between including greatdetail, in which case the model becomes forbiddingly complicated with many free parameters, or to simplify, in whichcase there is a danger that the essential behaviour is lost. The role of MEPP is to take the place of the detailed dynamics insuch models. The argument is that, provided the essential constraints are imposed, MEPP will provide a good account ofthe key behaviour of the system without the need for a deterministic dynamical model, see for example Martyushev andSeleznev [7], Martyushev [8], Meysman and Bruers [9], and Niven [10].

In this spirit, the lack of a rigorous physical proof has not prevented application of the principle to specific systems,especially in the fields of ecology, biology and planetary science. Indeed it was the success of the model as applied to theEarth’s atmosphere by Paltridge [11]which didmuch to stimulate interest inMEPP. Since then, proponents of the hypothesishave pointed to its apparent success in modelling the atmospheres not only of the Earth [12] but also of Titan andMars [13].Other apparently successful applications of MEPP have been in rationalising transitions among multiple steady states ofoceanic thermohaline circulation [14], in the behaviour of the Earth’s mantle [15], and in biological systems. The latterincludes the claim that steady-state bacterial photosynthesis operates close to maximum entropy production [16] andalso the claim that the molecular motor ATP synthase has evolved in accordance with MEPP [17]. MEPP has also provedsuccessful in fluid flow problems, Niven [18]. Nevertheless, MEPP has remained suspect in the physics community due tolack of theoretical justification. Worse still, it is unclear what exactly the principle asserts in the general case due to anelement of choice in which constraints are imposed and which variables are used to perform the optimisation.

In view of the absence of a general proof, the credibility of MEPP may be investigated using specific systems whose truebehaviour can be either computed or experimentally determined. A system which has been used for this purpose by someauthors is Rayleigh–Bénard convection (RBC). This involves a horizontal enclosure of fluid of uniform depth which is heatedfrombelow resulting in heat transfer by combined conduction and convection to its upper surface. For temperature gradientsin a certain range, this system produces stable convection patterns such as rolls or hexagonal cells. For larger temperaturegradients the fluid motion becomes increasingly chaotic and ultimately turbulent. A number of authors have examinedMEPP, or maximal heat flow, in the regime of stable convection cells, notably Malkus and Veronis [19], Koschmieder [20],Kita [21,22], Cafaro and Saluzzi [23],Weaver, Dyke andOliver [24], Attard [25,26], Busse [27], and Busse andWhitehead [28].However a consensus has not emerged, some authors claiming that the selected convection mode is that which maximisesentropy production and some concluding the opposite.

A proponent of MEPP might claim that the principle has not yet been tested in the correct regime of RBC. Indeed,proponents ofMEPPdo consistently emphasise that the principlewill hold only for systemswhich are ‘‘sufficiently complex’’.But all the investigations of RBC referenced above involve the sub-chaotic regime in which the system is in a stableconvection-cell state, or transitions between two such states. This may not be the most appropriate regime in which toinvestigate extremal principles in RBC. Rather extremal principles may be more appropriate in the chaotic regime. By thesame token, the failure of MEPP in the chaotic regimewould constitute a more robust refutation. Consequently in this paperwe investigate the principle in the chaotic regime (as well as the sub-chaotic regime). To do so it is first necessary to identifythe degrees of freedomwith respect to which the extremal behaviour is postulated. These are identified here with the time-dependent amplitudes of the marginally stable modes out of which we hypothesise the fluid motion may be regarded ascomposed.

Thepaper is organised as follows. Section 2presents themaximisationproblem inoutline. Section 3presents the algebraicdetails of the formulation. Section 4 presents the numerical method employed for the optimisation and gives the resultingNusselt number. Section 5 is a reminder of the solution for the heat flow for a single stable convection mode. Section 6presents a comparison between our maximal Nusselt number and experimental evaluations. Finally Section 7 discusses thesignificance of the results.

2. The basis of the extremal problem considered

In the case of RBC with prescribed temperature boundary conditions, MEPP becomes synonymous with maximum heatflow. Because of the difficulty of solving the equations of fluid flow and heat transport in RBC, many workers have beenmotivated to derive upper bounds to the heat flow instead, e.g., Malkus and Veronis [19], Howard [29], Worthing [30], Ottoand Seis [31], and Whitehead and Doering [32]. Since these analyses also seek upper bounds to the heat transfer there isa danger of confusing them with the objective of the present paper. The key difference is whether or not the dynamical(Navier–Stokes) equation is used. Analyses such as Refs. [19,29–32] use the full equation set, including the Navier–Stokesequation, but derive upper bounds on the heat transfer as an alternative to solving the equations directly.

In contrast, the present paper seeks to treat the RBC system in a manner analogous to those cases where MEPP would bedeployed ‘‘in earnest’’: that is, to systems forwhich the detailed dynamics are prohibitively difficult to represent explicitly asequations. Consequently the novel aspect of the present work is that the dynamical (Navier–Stokes) equation is deliberatelynot used. Instead it is replaced by MEPP. To be more precise, appeal to Navier–Stokes is confined to the form of the fieldsconsidered, namely as an expansion in terms of the active marginally stable modes (Section 3). But the amplitudes of eachof the modes in the expansion are free variables. The paper seeks to determine the maximum heat transfer with respect

R.A.W. Bradford / Physica A 392 (2013) 6273–6283 6275

to variations in these amplitudes, subject only to the constraint of energy conservation and incompressibility. We wish todiscover whether this maximum compares well with true (experimental) rates of heat transfer. It is important to appreciatethat, a priori, the maximum heat transfer determined without imposing the dynamical Navier–Stokes constraint could bewildly wrong. It is emphasised that this paper does not seek to improve knowledge of themagnitude of heat transfer in RBC.The purpose is to examine the performance of MEPP when the detailed dynamics are not employed as a constraint.

Denoting the prescribed temperatures at the top and bottom of the fluid by TT and TB, and the fluid vertical depth by b,the average temperature gradient is β = (TB − TT ) /b. The energy equation together with incompressibility can be used toderive an integral relation first discussed by Chandrasekhar [33] for the case of time-independent states. A similar relationcan be derived for a chaotic steady state in which suitably time-averaged quantities are constant. In this case the integralrelation becomes,

β ⟨uzT1⟩txyz + κT1∇2T1

txyz =

1κ

⟨uzT1⟩xy

2tz

−1κ

⟨uzT1⟩xyz

2t

(1)

where κ is the fluid thermal diffusivity and the notation ⟨· · ·⟩ denotes averaging over the variables shown as subscripts: tbeing time, z being the upwards vertical coordinate (origin at themid-height of the fluid), and x, y the horizontal coordinates.The fluid velocity vector is uwith uz its vertical component. The term T1 in (1) is defined as the difference between the fluidtemperature and its average over the x, y plane,

T1 ≡ T − ⟨T ⟩xy . (2)

The heat flux averaged over any horizontal plane and over time, ⟨J⟩xyt , can be normalised by the conductive flux, Kβ , whereK = κCρ is the thermal conductivity of the fluid, to form theNusselt number. This can be found in terms of the field variablesuz and T1 thus,

Nu ≡⟨J⟩xytKβ

= 1 +1

κβ⟨uzT1⟩txyz . (3)

Hence the extremal problem to be considered here is to maximise (3) whilst respecting the constraint imposed by (1). Amost important fact is that (1) is derived from the energy equation and incompressibility alone and is independent of theNavier–Stokes equation, i.e., it applies independently of the dynamics.

3. Detailed formulation of the extremal problem considered

3.1. Expansion in terms of active modes

It is proposed that the velocity and temperature fields in a steady chaotic state be represented in the form,

uz =

npmχ

Ωχnpmωχ

npm (z) cos anx · cos dpy (4)

T1 =

npmχ

Ωχnpmθχ

npm (z) cos anx · cos dpy. (5)

Each term in the sums (4) and (5) is a marginally stable convection mode, labelled by the four mode numbers n, p,m, χ ,with Ω

χnpm being time dependent amplitudes for each mode. The amplitudes Ω

χnpm will provide the degrees of freedomwith

respect to which the heat flux, (3), will be maximised. The sums in (4) and (5) extend over only the ‘active’ modes, i.e., onlythose modes whose critical Rayleigh number (Rχ

npm) is less than the applied Rayleigh number (Ra). These terms are nowdefined together with the form of the marginally stable modes.

Following the early observations by James Thomson [34] and the first quality experimental investigations by Bénard[35,36], the theoretical explanation for what later became known as Rayleigh–Bénard convection was published by LordRayleigh [37]. The form of the marginally stable modes is derived in standard texts, e.g., Chandrasekhar [33]. We shallassume the fluid is confined to a rectangular region with sides x = 0, x = Lx, y = 0 and y = Ly and that the fluid slipsfreely over these boundaries. We shall confine attention to the case Lx ≫ b and Ly ≫ b so that the nature of these boundaryconditions is unimportant. In contrast, the fluid is assumed to be confined between rigid plates at its top and bottom surfaces,z = b/2 and z = −b/2, and it is essential to model the effects of viscosity at these boundaries if contact is to be made withexperimental results obtained from such apparatus. The rigid boundaries will cause all the velocity components to be zeroon these z surfaces. The severity of the applied vertical temperature gradient, β , is measured by the dimensionless Rayleighnumber,

Ra =αβgb4

νκ(6)

where α is the volumetric coefficient of thermal expansion of the fluid, ρ its average density, ν its kinematic viscosity, andg the acceleration due to gravity.

6276 R.A.W. Bradford / Physica A 392 (2013) 6273–6283

There is a discrete set of solutions to the energy equation and the Navier–Stokes equation in the Boussinesq approxima-tion assuming incompressibility and in the linear perturbation approximation. These are the modes, labelled by n, p,m, χ ,which are stable when the Rayleigh number marginally exceeds a critical value, Rχ

npm, dependent on mode. The boundaryconditions lead to the following discrete set of possible horizontal wavenumbers,

an =nπLx

, dp =pπLy

and k2 = a2n + d2p (7)

where n, p = 0, 1, 2, 3 . . . can be any positive integers, or zero, except that n and pmay not both be zero. The mode label χtakes the values ‘‘even’’ or ‘‘odd’’ in accord with the z symmetry of the functions ω

χnpm (z) and θ

χnpm (z). The critical Rayleigh

numbers follow from the discrete set of solutions to,

χ = even:

q1 +

√3q2

sinh q1 +

√3q1 − q2

sin q2

cosh q1 + cos q2= −q0 tan

q02

(8)

χ = odd:

q1 +

√3q2

sinh q1 −

√3q1 − q2

sin q2

cosh q1 − cos q2= q0 cot

q02

(9)

where the three real parameters q0, q1, q2 are determined by just one real parameter, τ , thus,

q0 = k√

τ − 1; q1 = k

(1 + τ/2) +

√1 + τ + τ 2

2; q2 = k

− (1 + τ/2) +

√1 + τ + τ 2

2. (10)

The critical Rayleigh number is given in terms of the solution to (8) or (9) for τ by,

Rχnpm = (kb)4 τ 3. (11)

The transcendental equations (8) and (9) have an infinite set of solutionswhichmay be labelled by an integerm = 1, 2, 3 . . .in order of increasing Rχ

npm, so the smallest Rayleigh number for a given n, p, χ is form = 1. The minimum Rχnpm is 1707.762

and occurs for even modem = 1 when kb = 3.117. No convection occurs for Rayleigh numbers less than this value (for thestated boundary conditions). The minimum Rχ

npm for any odd mode is 17,610.55 and occurs for m = 1 when kb = 5.3672.The number of active modes (i.e., modes with Rχ

npm < Ra) increases rapidly with increasing Ra. It also increases with increas-ing aspect ratio Lx/b and Ly/b. For Ra = 109 and a square enclosure of aspect ratio 9 there are 5360,202 active modes. Thefunctions of z appearing in (4) and (5) can now be written explicitly,

χ = even: ωχnpm = cos q0z + A cosh qz + A∗ cosh q∗z (12)

χ = odd: ωχnpm = sin q0z + A sinh qz + A∗ sinh q∗z (13)

χ = even: θχnpm (z) = cos q0z − Aξ ∗ cosh qz − A∗ξ cosh q∗z (14)

χ = odd: θχnpm (z) = sin q0z − Aξ ∗ sinh qz − A∗ξ sinh q∗z (15)

where,

θχnpm =

ντ 2k2

αgθχnpm and q = q1 + iq2. (16)

3.2. Nusselt number and constraint equation in terms of the mode amplitudes

TheNusselt number and the constraint equationmaybe expressed in termsof themode amplitudes,Ωχnpm, by substituting

(4) and (5) into (1) and (3) and carrying out the integrals. The functions ωχnpm and θ

χ ′

npm′ are orthogonal in the sense that,ωχ

npmθχ ′

npm′

z= µχ

npmδχχ ′

δmm′ (17)

which defines the quantities µχnpm. It is convenient to absorb a factor of kτ into the amplitudes, defining Ω

χnpm = kτΩ

χnpm.

The extremal problem is simplified by assuming no correlation between modes. For example, this means that,Ωχ

npm

2 Ω

χ

npm

2t=

Ωχ

npm

2t

Ω

χ

npm

2t. (18)

R.A.W. Bradford / Physica A 392 (2013) 6273–6283 6277

The extremal problem can now be formulated entirely in terms of the mean-squared amplitudes,

Ωχnpm

2t. This extremal

problem can be more succinctly expressed in matrix notation. Define a vector of mean-squared amplitudes by,

f =1Ra

b2

2κ

2 Ωχ

npm

2t

(19)

where the vector index corresponds to the mode number in some specified sequence. Similarly defining the vectors,

H = µχnpm and h = µχ

npm

1 −

Rχnpm

Ra

(20)

the Nusselt number, (3), becomes,

Nu = 1 + H · f (21)

whilst the constraint equation, (1), becomes,

h · f = f T (M) f (22)

where the matrix (M) has components,

(M) =

ωχ

npmθχnpmω

χ

npmθχ

npm

z− µχ

npmµχ

npm

♦npmχnpmχ +

τ

τ

2 ωχ

npmθχ

npm

2z♦mχmχδnnδpp

+

ωχ

npmθχnpmω

χ

npmθχ

npm

z♦mχmχδnnδpp +

ωχ

npmθχnpm

2z−

µχ

npm

2δnnδppδmmδχχ (23)

noting that the mode label npmχ corresponds to one matrix index, and npmχ corresponds to the other, and τ relates tonpmχ whilst τ relates to npmχ . In (23) we have introduced the symbol,

♦xyz...abc... ≡

1 − δx

aδybδ

zc . . .

(24)

which is 1 unless all of abc . . . are equal to the corresponding xyz . . . , in which case it is zero. It is now clear why (1) in theform of (22) constrains the mode amplitudes, since the LHS of (22) is linear in f whilst the RHS is quadratic.

The variational problem is to maximise Nu, given by (21), with respect to variations in f subject to the constraint givenby (22), where H, h and (M) are known in terms of the marginally stable modes as given by (20) and (23).

4. Numerical optimisation

The obvious procedure is to set the derivatives of 1+ H · f +λh · f − f T (M) f

to zero, where λ is a Lagrangemultiplier.

This results in a simple linear equation for f in terms of λ, and the latter is then found by imposing (22). But this approachfails because many of the resulting components of f are negative, an unphysical outcome because the components of f areintended to be mean-squared amplitudes. The required optimum must enforce all components of f to be positive or zero.The outcome of the naive approach suggests that some of the optimised amplitudes will in fact be zero.

The method of steepest ascents was adopted for the optimisation. A maximum of Nu was sought along a line in thedirection of the gradient, ∇fNu, ofNu in f space. If points explored along this line had negative values for some components off thesewere replaced by zero.When the line-maximumhad been found, the gradientwas then re-calculated and the processrepeated until further increases inNu became negligible. Both the gradient and the value ofNuwere calculated subject to theconstraint of remaining on the surface in f space defined by the constraint Eq. (22). Such non-linear optimisation techniquesare not guaranteed to identify the global maximum. However the fact that the present problem is only quadratic gives someconfidence that the global maximum will be found in this case. In fact the process proved robust; widely differing startingamplitudes (i.e., positions in f space) converging to the same optimised Nu. Starting with all components of f equal wassufficient.

An example of the optimised squared-amplitudes obtained for Ra = 105, Lx = Ly = 6b, is shown in Fig. 1. The uniquemode number used as the abscissa in Fig. 1 is defined by listing the active npmχ sequentially, increasing the left-most first.Hence each of the spikes in Fig. 1 gives the squared-amplitudes for the range of n values for a given fixed pmχ . The threeseparate groups of spikes correspond to: even modes with m = 1, even modes with m = 2, and odd modes with m = 1respectively (there are nomodes with higherm in this case). The individual spikes within any group correspond to differentp values.

Fig. 1 is typical of the results at other Rayleigh numbers and aspect ratios. Most amplitudes are optimised to zero. For agivenmχ , the optimisation process tends to leave non-zero amplitudes only for modes with critical Rayleigh numbers closeto the minimum for that mχ . This is easy to understand from (22) which will lead to the greatest amplitudes for large h.Hence in (20) the factor of 1−Rχ

npm/Ra will favourmodeswith Rχnpm ≪ Ra if the objective is tomaximiseNu. This results in np

values giving large Rχnpm being optimised to zero amplitudes, causing the spikes in Fig. 1. Within any group of spikes, i.e., for

6278 R.A.W. Bradford / Physica A 392 (2013) 6273–6283

Fig. 1. Illustration of the optimised mean-squared amplitudes when all modes are retained in the optimisation (Ra = 105 , Lx = Ly = 6b). The amplitudes

plotted are

Ωχnpm

2t=

Ω

χnpm

2t/ (kτ)2 . The x-axis is the mode number (defined in the text), there being 1950 contributing modes.

a given mχ , the height of the spikes tends to be equal. This is partly because the spikes correspond to different p valuesand the same, or almost the same, k, and hence Rχ

npm, can be made from a variety of np pairs. However it is also due in partto the fact that the z integral factors in (17) and (23) are found (by numerical integration) to vary relatively little betweenmodes. Consequently, modes within a sub-set of modes with the same mχ and which have close to the minimum criticalRayleigh number are virtually degenerate in the sense that they contribute equally to Nu. It would matter little if some ofthese modes were omitted from the optimisation, so long as a few such modes are retained for each mχ . The amplitudesof the retained modes merely increase as other modes are dropped, leaving Nu virtually unchanged. It was demonstratedexplicitly in the numerical investigations that the optimised Nu is insensitive to the number of modes which are retainedfor each mχ , providing there are a few modes for every active mχ and these modes have close to the minimum Rχ

npm forthatmχ .

However, the factor of (τ /τ )2 in the second term of (23) varies substantially when themχ and mχ modes are different.

Consequently, eachmχ makes a different contribution to the Nusselt number, and hence there is a group of different, non-zero amplitudes for each mχ . Each mχ group favours those np with Rχ

npm values close to the minimum for that mχ . Whilstthe favouredmodes with the samemχ are degenerate as regards their contribution to Nu, modes with differentmχ provideindependent contributions to Nu with differing optimised magnitudes, as illustrated by the differing height of the peaks inthe three groups of peaks in Fig. 1.

The insensitivity of the optimised Nu to the number of modes retained for each mχ is rather fortunate since otherwisethe optimisation problem would have been prohibitively computationally heavy for large Ra. The complete set of modeswas retained for Ra up to 105. But it was found that for Ra = 105 and Lx = Ly = 6b the optimised Nusselt number obtainedby retaining all 1950 modes differed negligibly from that obtained using just three modes, i.e., one for each different mχ .For a Rayleigh number of 109 and Lx = Ly = 9b there are 5360,202 active modes. Had all these modes been retained theoptimisation would have been most problematical. However, using just 204 modes, six per mχ , gives a very reasonableresult, and Nu is essentially converged using just 884 modes (26 per mχ ). With this number of degrees of freedom theoptimisation is computationally light.

Note that all the retained modes had Rχnpm close to the minimum for the given pmχ . Retaining (say) 11 modes per mχ

meant choosing arbitrarily 11 p values, the n value of the retainedmode being that which minimised Rχnpm for that pmχ . The

crucial feature is that all active mχ be represented. Denoting by Nχm the number of active mχ values, retaining ζ different

p values means using ζNχm modes in all to carry out the optimisation. Values of ζ from 1 to 41 were used to demonstrate

convergence. Most results were obtained for Lx = Ly = 6b, because this was anticipated to be sufficiently large an aspectratio to differ negligibly from the infinite case. To demonstrate this, Lx = Ly = 9b was also used and shown to result in nosignificant difference as regards the optimised Nu.

Fig. 2 plots the total number of active modes, N , versus Ra; Fig. 3 plots our optimised Nusselt number, Nu, against Ra;Fig. 4 plots Nu against the number of activemχ , Nχ

m . The following consistent set of power-law dependences were found inthe chaotic regime (roughly Ra ≥ 105) for the maximised Nusselt number,

N ∝ R0.75a Nu ∝ R0.25

a Nu ∝ N0.33 Nu ≈ Nχm (25)

Nχm ∝ R0.25

a Nn ∝ R0.25a Nu ∝ Nn

R.A.W. Bradford / Physica A 392 (2013) 6273–6283 6279

Fig. 2. Total number of active modes, N , versus Ra (Lx = Ly = 6b).

Fig. 3. Optimised Nusselt number, Nu, versus Ra .

where Nn is the average number of n values defined by Nn =

N/Nχ

m . In particular, our maximised Nusselt number isroughly equal to the number of differentmχ which are active, Nχ

m , Fig. 4.

5. The Nusselt number for a single mode

For a single mode, i.e., in the pre-chaotic phase, there is no optimisation to be done and a deterministic answer for Nu

results. The first three terms in (23) are zero and, introducing the shorthand ηχnpm =

ω

χnpmθ

χnpm

2z, (21) and (22) give,

Nu = 1 +

1 −RχnpmRa

ηχnpm

(µχnpm)

2 − 1

. (26)

In practice, in the single-mode regime, the modes with critical Rayleigh number close to the minimum will be favoured,i.e., them = 1 even modes with Rχ

npm close to 1708. Numerical evaluation of the integrals then shows that (26) reduces to,

Nu ≈ 1 + 1.4451 −

1708Ra

. (27)

The greatest single-mode, stable convection cell Nusselt number is thus 2.305. The optimised multi-mode Nu turns out tobe identical to the single-mode Nu given by (27) whilst there is only one mχ active (i.e., the m = 1 even modes). However,the multi-mode optimised Nu exceeds (27) once the threshold of the odd modes is passed, i.e., for Ra > 17,611.

6280 R.A.W. Bradford / Physica A 392 (2013) 6273–6283

Fig. 4. Optimised Nu versus the number of activemχ values, Nχm ; dashed line is line of equality: Nu = Nχ

m .

Fig. 5. Predicted and experimental Nusselt number versus Rayleigh number for small Ra when convection is in a single pure mode (non-chaotic).

6. Comparison with experimental and numerical Nusselt number evaluations

Fig. 5 plots Nu against Ra in the small Ra regime when convection is in a pure mode with minimum criticalRayleigh number. The prediction, (27), is compared with the classic experimental results of Silveston [38], as quoted byChandrasekhar [33], for a range of liquids (water, silicone oil, ethylene glycol and heptane, with Pr between 4 and 200). It iswell known, of course, that the theory for a single pure mode provides a good representation of the experimental data forsufficiently small Ra.What is newhere is that if all activemodes are permitted andNu ismaximised (subject to no correlationbetween modes) then no increase in Nu results. Our optimised Nu values are identical to the pure-mode value given by (27)until Ra exceeds 17,611when the first oddmode becomes active. This is a non-trivial observation: the hypothesis ofmaximalNu does not undermine the agreement with experimental results in the very small Ra regime.

Fig. 6 plots ourmaximisedNu against Ra in the chaotic regime up to Ra = 109 in comparisonwith a range of experimentalevaluations for Prandtl numbers Pr > 0.5. These involve a range of room-temperature liquids as well as data for air, heliumand liquid helium obtained from Silveston [38], Mull & Reiher [39], Turner [40], Julien et al. [41], Rossby [42], Chavanneet al. [43] and Funfschilling et al. [44] and a curve fitted to a range of experimental data in Kreith et al. [45]. Other sources ofexperimental data could be added but these are sufficiently representative. In Fig. 6 the experimental data has been replacedby an approximate trend line in most cases, to aid the comparison. The data of Silveston and of Mull and Reiher was takenfrom Chandrasekhar [33] and has been represented by approximate upper and lower bounds (UB and LB). A limitation of thecomparison is that the experimental data of Fig. 6 was obtained using a range of different aspect ratios, some quite small andhence not strictly comparable with the large aspect ratio of our calculations. However, aspect ratio dependence is believedto be relatively modest [46].

Fig. 6 shows how the experimental Nu departs from the single-mode curve at around Ra ∼ 104 or just before. Ourmaximised Nu departs from the pure-mode curve when Ra > 17,611, when the first oddmode becomes active. A step in the

R.A.W. Bradford / Physica A 392 (2013) 6273–6283 6281

Fig. 6. Comparison of the maximised Nusselt number with experimental values and extending into the chaotic and turbulent regimes (experimental datafor Pr > 0.5).

Fig. 7. Maximised Nu compared with experimental (liquid metal) and numerical results obtained for Pr ≪ 1.

curve of maximised Nu is evident at this point, Fig. 6. There is a subsequent step at Ra = 75,730 when the firstm = 2 (even)mode becomes active. Remarkably, the maximised Nu provides a good representation of the lower lying experimental dataup to Ra ∼ 106. Thereafter the maximised Nu starts to diverge from the experimental data. This is because the trend of themaximised Nusselt number is Nu ∝ R0.25

a , whereas the experimental data for Pr > 0.5 commonly shows Nu ∝ Rγa with

an exponent of at least γ ≈ 2/7, and possibly γ ≈ 1/3, even perhaps increasing to γ ≈ 1/2 in the ‘‘ultimate’’ regime ofextremely large Ra, well beyond that plotted in Fig. 6 (see the reviews of Ahlers et al. [46] and Manneville [47]).

However, the trendNu ∝ R0.25a has theoretical precedent. For example, Ahlers et al. [46] identify four regimes of behaviour

according towhether boundary or bulk contributions to dissipation dominate. In the case that the boundary, including plumeeffects, dominates over the bulk as regards both kinetic energy and thermal dissipation rates, the arguments of Grossmannand Lohse [48] imply Nu ∝ R0.25

a .Data for Pr ≪ 1 is not as extensive as for Pr > 0.5. However, Fig. 7 plots Nu data obtained using mercury by Cioni

et al. [49] and Rossby [42], and using liquid sodium by Horanyi et al. [50], together with the results of numerical solutionsby Verzicco and Camussi [51] and Kerr and Herring [52]. The results shown in Fig. 7 relate to either Ra = 6 × 105 orRa = 107 and our corresponding maximised Nu are plotted for comparison. Our maximised results lie within the range ofthe experimental and computational data, but cannot reproduce the Pr dependence (since a Pr dependence can only comefrom the Navier–Stokes equation, which we have deliberately ignored).

7. Discussion and conclusions

The agreement between the maximised Nusselt numbers derived here and the experimental and numerical results inthe literature, as shown in Figs. 6 and 7, is reasonably good, given that this is inevitably limited by the absence of a Prandtl

6282 R.A.W. Bradford / Physica A 392 (2013) 6273–6283

number dependence in the former. That the maximised Nu should be even crudely of the right order of magnitude is notimmediately obvious. Indeed the wildly excessive maximised Nusselt number which can result by contriving correlationsbetween modes emphasises the non-trivial nature of the result when correlations are assumed to be zero. Recall that theseresults have been obtainedwithout any appeal to the dynamics (theNavier–Stokes equation), except through themarginallystable modes and their critical Rayleigh numbers.

An unlooked-for outcome of the present work is that, if MEPP is indicative, the Nusselt number is simply related to themode statistics, e.g.,Nu is roughly the number of distinct active verticalmodemχ values (Fig. 4),Nu ≈ Nχ

m . But this is limitedto modest Rayleigh numbers for which Nu ∝ R0.25

a does not depart too much from reality. The fact that our ‘‘maximised’’Nu under-estimates the experimental values at larger Ra suggests that correlations between modes are actually non-zero.However, the degree of correlation needed to reproduce the experimental values is probably very slight.

Two reactions to these results are possible. The first is that the reasonable results obtained by maximising Nu are just afluke and have no fundamental significance. The second is that there is indeed a fundamental basis for variational principlesof this kind. Discrimination between these two possibilities will require the performance of MEPP to be assessed across anumber of different problems. The present results suggest that there are reasons to be optimistic that MEPP does provide areasonable, if approximate, guide to the dynamic steady states of non-equilibrium systems.

References

[1] R.C. Dewar, Information theory explanation of the fluctuation theorem, maximum entropy production and self-organized criticality in non-equilibrium stationary states, J. Phys. A: Math. Gen. 36 (2003) 631.

[2] R.C. Dewar, Maximum entropy production and the fluctuation theorem, J. Phys. A: Math. Gen. 38 (2005) L371.[3] S.A. Bruers, Discussion on maximum entropy production and information theory, J. Phys. A: Math. Theor. 40 (2007) 7441.[4] G. Grinstein, R. Linsker, Comments on a derivation and application of the maximum entropy production principle, J. Phys. A: Math. Theor. 40 (2007)

9717.[5] R.C. Dewar, Maximum entropy production as an inference algorithm that translates physical assumptions into macroscopic predictions: don’t shoot

the messenger, Entropy 11 (2009) 931.[6] N. Virgo, From maximum entropy to maximum entropy production: a new approach, Entropy 12 (2010) 107.[7] L.M. Martyushev, V.D. Seleznev, Maximum entropy production principle in physics, chemistry and biology, Phys. Report 426 (2006) 1–45.[8] L.M. Martyushev, The maximum entropy production principle: two basic questions, Philos. Trans. R. Soc. B 365 (2010) 1333–1334.[9] F.J.R. Meysman, S. Bruers, Ecosystem functioning and maximum entropy production: a quantitative test of hypotheses, Philos. Trans. R. Soc. B 365

(2010) 1405–1416.[10] R.K. Niven, Steady state of a dissipative flow-controlled system and the maximum entropy production principle, Phys. Rev. E 80 (2009) 021113.[11] G.W. Paltridge, The steady-state format of global climate systems, Q.J.R. Meteorolog. Soc. 104 (1975) 927.[12] J. Dyke, A. Kleidon, The maximum entropy production principle: its theoretical foundations and applications to the earth system, Entropy 12 (2010)

613.[13] R.D. Lorenz, J.I. Lunine, P.G. Withers, Titan, mars and earth: entropy production by latitudinal heat transport, Geophys. Res. Lett. 28 (2001) 415.[14] S. Shimokawa, H. Ozawa, On the thermodynamics of the oceanic general circulation: irreversible transition to a state with higher rate of entropy

production, Q.J.R. Meteorolog. Soc. 128 (2002) 2115.[15] R.D. Lorenz, Planets, life and the production of entropy, Int. J. Astrobiol. 1 (2002) 3.[16] D. Juretic, P. Zupanovic, Photosyntheticmodelswithmaximumentropy production in irreversible charge transfer steps, Comput. Biol. Chem. 27 (2003)

541.[17] R.C. Dewar, D. Juretic, P. Zupanovic, The functional design of the rotary enzyme ATP synthase is consistent with maximum entropy production, Chem.

Phys. Lett. 430 (2006) 177.[18] R.K. Niven, Simultaneous extrema in the entropy production for steady-state fluid flow in parallel pipes. arXiv:0911.5014.[19] W.V.R. Malkus, G. Veronis, Finite amplitude cellular convection, J. Fluid. Mech. 4 (1958) 225–260.[20] E.L. Koschmieder, Bénard Cells and Taylor Vortices, Cambridge University Press, 1993.[21] T. Kita, Principle of maximum entropy applied to Rayleigh–Bénard convection, J. Phys. Soc. Japan 75 (2006) 124005. arXiv:cond-mat/0611271.[22] T. Kita, Entropy change through Rayleigh–Bénard convective transition with rigid boundaries, J. Phys. Soc. Japan 76 (2007) 064006. arXiv:0705.3926.[23] E. Cafaro, A. Saluzzi, Behaviour of Bénard convection from non-equilibrium thermodynamics point of view, Associazione Termotecnica Italiana, 50th

Congresso, St.Vicent, 1995.[24] I.Weaver, J.G. Dyke, K. Oliver, Can the principle ofmaximum entropy production be used to predict the steady states of a Rayleigh–Bernard convective

system? in: Beyond The Second Law: Entropy Production and Non-Equilibrium Systems, Springer, New York, US, 31 October 2013.[25] P. Attard, Non-Equilibrium Thermodynamics and Statistical Mechanics: Foundations and Applications, Oxford University Press, 2012, (Chapter 6).[26] P. Attard, Optimising principle for non-equilibrium phase transitions and pattern formation with results for heat convection, 2012. arXiv:1208.5105.[27] F.H. Busse, J. Math. Phys. 46 (1967) 140.[28] F.H. Busse, J.A. Whitehead, Instabilities of convection rolls in a high Prandtl number fluid, J. Fluid Mech. 47 (1971) 305.[29] L.N. Howard, Heat transport in turbulent convection, J. Fluid Mech. 17 (1963) 405–432.[30] R.A. Worthing, Contributions to the Variational Theory of Convection, Ph.D Thesis, Michigan Technological University, 1995.[31] F. Otto, C. Seis, Rayleigh–Bénard convection: improved bounds on the Nusselt number, J. Math. Phys. 52 (2011) 083702.[32] J.P. Whitehead, C.R. Doering, The ultimate regime of two-dimensional Rayleigh–Bénard convection with stress-free boundaries, Phys. Rev. Lett. 106

(2011) 244501.[33] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Oxford University Press, 1961.[34] J. Thomson, On a changing tessellated structure in certain liquids, Proc. Phil. Soc. Glasgow 13 (1882) 464.[35] H. Bénard, Les Tourbillons cellulaires dans une nappe liquide, Revue generale des Sciences pures et appliquees 11 (1900) 1261–1309.[36] H. Bénard, Les Tourbillons cellulaires dans une nappe liquide transportant de la chaleur par convection en regime permanent, Annales de Chimie et

de Physique 23 (1901) 62.[37] Lord Rayleigh, On convective currents in a horizontal layer of fluid when the higher temperature is on the under side, Philos. Mag. 32 (1916) 529.[38] P.L. Silveston, Warmedurchgang in waagerechten Flussigkeitsschichten, part 1, Forsch. Ing. Wes. 24 (1958) 29–59.[39] W. Mull, H. Reiher, Der warmeschutz von luftschichten, Gesundh-Ing. Beihefte, Reihe, 1 28 (1930) 1.[40] J.S. Turner, Buoyancy Effects in Fluids, Cambridge University Press, 1973.[41] K. Julien, S. Legg, J.C. McWilliams, J. Werne, Rapidly rotating turbulent Rayleigh–Bénard convection, J. Fluid Mech. 322 (1996) 243.[42] H.T. Rossby, A study of Bénard convection with and without rotation, J. Fluid Mech. 36 (1969) 309.[43] X. Chavanne, F. Chill, B. Castaing, B. Hebral, B. Chabaud, J. Chaussy, Observation of the ultimate regime in Rayleigh–Bénard convection, Phys. Rev. Lett.

79 (1997) 3648.

R.A.W. Bradford / Physica A 392 (2013) 6273–6283 6283

[44] D. Funfschilling, E. Brown, A. Nikolaenko, G. Ahlers, Heat transport by turbulent Rayleigh–Bénard convection in cylindrical cells with aspect ratio oneand larger, J. Fluid Mech. 536 (2005) 145.

[45] F. Kreith, R.M. Manglik, M.S. Bohn, Principles of Heat Transfer, seventh ed., Cengage Learning Custom Publishing, 2010.[46] G. Ahlers, S. Grossmann, D. Lohse, Heat transfer and large scale dynamics in turbulent Rayleigh–Bénard convection, Rev. Modern Phys. 81 (2009) 503.[47] P. Manneville, Rayleigh–Bénard convection: thirty years of experimental, theoretical, and modeling work, in: Dynamics of Spatio-Temporal Cellular

Structures, in: Springer Tracts in Modern Physics, vol. 207, 2006, p. 41.[48] S. Grossmann, D. Lohse, Thermal convection for large Prandtl number, Phys. Rev. Lett. 86 (2001) 3316.[49] S. Cioni, S. Ciliberto, J. Sommeria, Strongly turbulent Rayleigh–Bénard convection in mercury: comparison with results at moderate Prandtl number,

J. Fluid Mech. 335 (1997) 111.[50] S. Horanyi, L. Krebs, U. Müller, Turbulent Rayleigh–Bénard convection in low Prandtl number fluids, Int. J. Heat Mass Transfer 42 (1999) 3983.[51] R. Verzicco, R. Camussi, Prandtl number effects in convective turbulence, J. Fluid Mech. 383 (1999) 55.[52] R. Kerr, J.R. Herring, Prandtl number dependence of Nusselt number in direct numerical simulations, J. Fluid Mech. 419 (2000) 325.