Laminar boundary layers in convective heat transport
We study Rayleigh–Bénard convection in the high-Rayleigh-number and high-Prandtl-number regime, i.e., we consider a fluid in a container that is exposed to strong heating of the bottom and cooling of the top plate in the absence of inertia effects. While the dynamics in the bulk are characterized by a chaotic convective heat flow, the boundary layers at the horizontal container plates are essentially conducting and thus the fluid is motionless. Consequently, the average temperature exhibits a linear profile in the boundary layers.
In this article, we rigorously investigate the average temperature and oscillations in the boundary layer via local bounds on the temperature field. Moreover, we deduce that the temperature profile is indeed essentially linear close to the horizontal container plates. Our results are uniform in the system parameters (e.g. the Rayleigh number) up to logarithmic correction terms. An important tool in our analysis is a new Hardy-type estimate for the convecting velocity field, which can be used to control the fluid motion in the layer. The bounds on the temperature field are derived with the help of local maximal regularity estimates for convection-diffusion equations.
Rayleigh–Bénard convection is the flow of an incompressible Newtonian fluid in a container with an imposed temperature gradient due to heating of the bottom and cooling of the top plate. It is one of the classical models of fluid dynamics. With its numerous applications in geophysics, oceanography, meteorology, astrophysics and engineering, Rayleigh–Bénard convection played a central role in experimental and theoretical physics since the turn of the last century. Depending on the heating rate, experimentalists and numerical analysts observe a wide range of flow pattern: from purely conducting (i.e., motionless) states, over steady and oscillatory fluid motions, to chaotic pattern and fully developed turbulence. In view of the richness of the observed phenomena, it is not surprising that over many years, Rayleigh–Bénard convection has become a paradigm for nonlinear dynamics, including instabilities, bifurcations and pattern formation, e.g. [4, 9, 11]. We refer to [18, 13, 3, 1] for reviews and further references.
We consider Rayleigh–Bénard convection in the so-called high-Rayleigh-number and high-Prandtl-number regimes. This means, the applied temperature forcing is so strong that the fluid motion is unorganized. Moreover, inertia effects are negligible, so that, in particular, the observed flow pattern is rather chaotic than turbulent. Thinking of a container that is infinitely extended in horizontal directions, boundary effects on the vertical sidewalls of the container become irrelevant. In this situation, the flow pattern shows a clear spatial separation of the relevant heat transfer mechanisms: thin laminar boundary layers in the vicinity of the horizontal plates, in which heat is essentially conducted and a large bulk that is characterized by a convective heat flow. Since near the rigid walls of the container, the fluid is almost at rest, the relevant heat transport mechanism in the boundary layers is conduction, and thus, the temperature profile is essentially linear. Fluctuations around this profile take place on large length scales, with a small amplitude. It is in these boundary layers where the majority of the temperature drop between the hot bottom and the cold top boundaries happens. In the bulk, far from the rigid container walls, convection is only limited by viscous friction. The temperature is essentially equilibrated around the mean temperature of the system. However, due to the strong temperature forcing, overload heat is produced on the bottom, which generates instabilities of the boundary layer: The boundary layer bursts and hot fluid parcels with warm tails, so-called plumes, detach and flow into the much colder bulk due to buoyancy forces. While rising, the plumes push aside the above fluid and are itself in turn deflected. In this way, the plumes take the forms of stalks with caps on their tops — the plumes obtain their characteristic mushroom-like shapes, cf. [20, 13]. (Of course, the corresponding effects can also be observed on the cold top plate. For convenience, we restrict our considerations to the bottom boundary.)
The rigorous understanding of the dynamic flow pattern in the chaotic regime is far from being satisfactory. Until today, most of the work is devoted to the scaling of the Nusselt number, that is, the scaling of the average upward heat flux as a function of the imposed temperature forcing [6, 5, 7, 8, 19, 16]. In this paper, we focus on the boundary layers at the horizontal bottom and top plates of the container. So far, boundary layer theories in Rayleigh–Bénard convection were primarily derived as intermediate steps towards the understanding of the Nusselt number scaling. Since due to the absence of fluid motion, the boundary layers offer the main resistance for the heat flow through the container, the Nusselt number is dominantly determined by the boundary layer, and thus, for scaling theories, the width of these boundary layer is of particular interest. (In fact, the Nusselt number is inversely proportional to the width of the layer.) However, because of their extreme thinness, experimental and numerical investigations of the boundary layers are very difficult, and still, no generally accepted boundary layer theory is available for Rayleigh–Bénard convection. An overview on the current state of experimental and theoretical research for more general (than the one considered in the present paper) Rayleigh–Bénard experiments can be found in [1, Sect. VI].
The present work is a first attempt to partially characterize the observed pattern in chaotic high-Rayleigh-number convection — at least in the phenomenologically most regular region. Our aim is a rigorous justification of the laminar profile and thus the dominant role of conduction in the boundary layer. We rigorously establish local bounds on the temperature field and its gradients in the boundary layers that are (up to logarithmic corrections) uniform in the system parameters (Theorem 1). These bounds indicate that the temperature field close to the horizontal plates is indeed essentially laminar and fluctuations only happen on relatively large length scales, with a weak dependence on the heating rate. In other words, our analysis proves that heat is transported essentially via conduction. Moreover, we can deduce that the average temperature decays linearly and with slope equal to the Nusselt number (Theorem 2), as it is expected in conducting boundary layers.
The remainder of the paper is organized as follows: In Subsection 1.2, we introduce the mathematical model and the Nusselt number; in Subsection 1.3, we present our main results and discuss the method of this paper. Section 2 is devoted to the analysis of the velocity field. Finally, Section 3 contains the proofs of Theorem 1 and 2.
1.2 Model and Nusselt number
Despite the complexity of the observed phenomena, the mathematical model for Rayleigh–Bénard convection is relatively simple. If density variations of the fluid are sufficiently small and the thermal diffusivity is negligible compared to kinematic viscosity, the problem can be modelled by the infinite-Prandtl-number limit of the Boussinesq equations:
Here, is temperature, the fluid velocity, the hydrodynamic pressure, and the upward unit vector. We suppose that the container has the simple form , where we refer to the first coordinates as the horizontal ones, and to the last coordinate as the vertical one. We complete the system with periodic boundary conditions in the horizontal directions; at the rigid top and bottom plates, we set
Thus, we suppose uniform heating/cooling of the bottom/top plate, and no-slip boundary conditions for the fluid velocity. We follow the convention to write the space coordinate and the velocity field as and , respectively.
The system is nondimensionalized and admits a single control parameter: the container height . The side length of the period cell is chosen arbitrary and has no significance in the subsequent investigation. We distinguish two regimes: 1) the linear regime for small container heights, . Here, heat transfer between bottom and top plate is exclusively due to conduction and the temperature field stabilizes in a linear profile. 2) the chaotic regime for large container heights, . In this regime, the fluid experiences a strong temperature forcing, which leads to the formation of chaotic flow pattern, as described in the previous subsection. Notice that the present nondimensionalization differs from the common one in the physics literature, where the control parameter that measures the applied temperature forcing is the Rayleigh number Ra. Both, and Ra are related via , which explains the term “High-Rayleigh-number convection” for the chaotic regime. For the sake of completeness, we like to mention that between these two regimes, there is a critical container height for which the linear profile becomes unstable and a steady circular fluid (convection rolls) flow sets in. These convection rolls become itself unstable for larger heights and a cascade of bifurcations can be observed, until the chaotic behavior sets in for .
The efficiency of the heat transport is measured by the Nusselt number Nu, which is defined as the average upward heat flux,
Here, the brackets denote the horizontal space and time average, i.e.,
for any function . Some equivalent expressions for the Nusselt number can be derived:
Indeed, for (5), we average (1) in horizontal space and time and apply the boundary conditions (4). Identity (6) can be derived by testing the heat equation (1) with , integrating by parts and using the divergence-free condition (2) and the boundary conditions (4), while (7) follows from testing (3) with , integrating by parts and using (2) and (4) again.
Over many years, the dependence of the Nusselt number on the control parameter has been studied extensively, cf. [12, 1] and references therein. In the absence of inertia, as in the present model, the expected scaling of Nu is the following: In the linear regime , when heat transport is essentially due to conduction, the Nusselt number scales like the imposed temperature gradient: . In the chaotic regime , it is conjectured that conductive and convective heat transport are balanced in the system, in the sense that . While the first scaling law can be easily established, cf. [17, Theorem 2], a rigorous derivation of the second one remains reluctant until today. The presently best upper bound on the Nusselt number in the chaotic regime was derived by Otto and the author in  and is optimal only up to a double-logarithmic factor:
Observe that proving rigorous lower bounds on Nu is substantially different from proving upper bounds: Lower bounds depend sensitively on the particular choice of the initial data. Indeed, there are ungeneric solutions to the system (1)–(4) for which the heat flux is less efficient, for instance the purely conducting state , . In this case, it is . Therefore, one can only expect to prove (physically relevant) a-priori upper bounds on the Nusselt number in the chaotic regime.
1.3 Main results of this paper
From now on, we restrict our attention to the chaotic regime, that means, we assume that
Moreover, in order to rule out “ungeneric” configurations like purely conducting solutions (as described in the last subsection) or steady fluid motions, e.g. convection rolls, we focus on solutions for which
We are interested in statistical properties of the temperature field in the boundary layer in the chaotic regime. Our first result gives upper bounds on derivatives of the temperature field up to fourth order. We show that in a boundary layer of order one, the temperature can be controlled in appropriate Sobolev norms uniformly in terms of — modulo logarithmic prefactors.
Assume that and . Let , . Then there exists a such that
In the above statement, the size of the boundary layer is not explicitly fixed to one. Instead, we can rather think of these estimates to hold in any boundary layer whose width is of order one or logarithmic in . However, for simplicity we restrict ourselves in the following to the order-one case. Likewise, for symmetry reasons, the same bounds hold in the upper boundary layer. Moreover, for the benefit of a concise statement and to trim the estimates in the proofs, we do not compute explicit exponents on the logarithms. Upper bounds on (at least for the second and third order derivatives), can be found in the author’s PhD thesis .
The bounds stated in Theorem 1 can be considered in the context of rigorous upper bounds on the Nusselt number initiated by Constantin and Doering. Indeed, in view of (6), the bounds in [5, 8, 16], e.g. (8), can be read as
for some , and thus, estimate (9) appears to be a new contribution in the study of universal bounds on the temperature field in infinite-Prandtl-number Rayleigh–Bénard convection.
The estimates in Theorem 1 can be used to show that, in the vertical boundary layers, the temperature profile is essentially linear:
Assume that and . There exists a such that for any
It follows from the above Theorem that
Again, by symmetry, an analogue statement can be derived in the upper boundary layer. As the majority of the temperature drop between the bottom and the top plate in the Rayleigh–Bénard experiment just happens in the boundary layers, the heat flow must be inversely proportional to the width of the layer. The precise slope can be easily deduced from (5) by choosing and exploiting the boundary conditions (4) for , namely . Now, in Theorem 2, we recover this slope all over the boundary layer. The main benefit of the above estimate is the cubic control of the deviation in vertical direction around the linear profile, which only depends logarithmically and thus weakly on .
The method we use in this paper in order to derive the bounds on the temperature field stated in Theorem 1 are motivated by bounds derived by Otto for the viscous Burgers equation in  and Otto and Ramos for the Navier–Stokes equation in . In both papers, the authors establish maximal-regularity-type estimates with tools borrowed from Harmonic Analysis. Using and refining the techniques from these two papers, we derive new local maximal regularity estimates for convection-diffusion equations, see Propositions 2 and 3 on p. 2f. After differentiating the heat equation (1) in spatial directions and and localizing in the boundary layer, these estimates can be applied to gain control on gradients of the temperature field. We have to comment on the choice of the norms in the assertions of Theorem 1. These are determined by the leading order error terms, which come from differentiating and localizing the transport nonlinearity. Only few tools are known that are applicable in order to bound these terms uniformly in (modulo logarithms): localized bounds on the velocity field in the spirit of those derived in , cf. Proposition 1 on page 1, the maximum principle for , and interpolation inequalities of the form
The value of in the maximal-regularity estimates is chosen optimally with respect to the estimates at hand.
2 Bounds on the velocity field
In this section, we collect a series of bounds on the velocity fields: some known results from  and  and some new results together with their proofs. The main new result, stated in Proposition 1 below, is an estimate on the shear velocity in the boundary layer. As this estimate might be of independent interest, in this section, we state all results for boundary layers of arbitrary width .
We consider the stationary Stokes equation for the velocity field and the hydrodynamic pressure in the Rayleigh–Bénard problem, i.e., equations (2) & (3) equipped with periodic boundary conditions in all horizontal directions and with no-slip conditions on the vertical boundaries, i.e., for . We observe that, because of (2), we have the additional information that for . It is convenient to eliminate the pressure term in (3) with the help of the incompressibility condition (2). This leads to a fourth-order equation for :
Here, denotes the Laplace operator in the horizontal variables. Indeed, applying the divergence operator to (3) yields , and therefore, applying the Laplace operator to the vertical component of (3), we obtain . In a similar way, making additionally use of the boundary conditions, we see that satisfies the equation , and thus by (2), is determined by
Furthermore, since there are no external forces acting on the fluid velocity, we have
At this point, we like to recall some bounds on the vertical velocity component , derived in  and . The key estimate of  (here, slightly extended by including the horizontal gradient ) is the following: Let and be periodic in and satisfy (10). Then
In Lemma 3 below, we will extend this result to the horizontal velocity component . Notice that, as a byproduct of the result in , the r.h.s. of (12) is controlled by the Nusselt number (modulo a logarithmic correction):
In [16, Lemma 3, see also eq. (18)], the authors derive an maximal regularity estimate for the third-order derivatives of from (10). By the Nusselt number representations (6)&(7), the result can be stated as follows:
or, invoking the formula (11):
Lemma 2 ().
The norm in (15) can best be understood on the Fourier level: For any periodic function we define
Here is the Fourier transform of at wave number , i.e.,
Let and be periodic in and satisfy (10). Then for any and it holds
for some . This inequality fails for — but only logarithmically. We exploit this logarithmic failure in two different ways. In the “bulk” , the logarithmic failure of (20) with produces the prefactor in (19). Notice that the estimates leading to (19) are sharp. In particular, the logarithmic prefactor can be recovered in the critical Hardy inequality when choosing (with the appropriate boundary conditions at ). In the “boundary layer” , we prove a subcritical Hardy inequality, i.e., (20) with some , which leads to (18). Notice that in view of the prescribed boundary conditions for , it holds for , which just beats the integrability. Consequently, we can expect (20) to be true only for , i.e., . In light of this observation, our bound for the boundary layer term seems to be suboptimal, since it requires . However, it is optimal in terms of scaling — a fact that will be exploited in Proposition 1 below. The logarithmic prefactor in (18) stems — roughly speaking — from the fact that the boundary terms at have to be estimated via the bulk estimate (19).
Proof of Lemma 3.
We start by recalling the following inequality that has been derived in [8, p. 238&239] and that can be easily reproduced by testing the equation for with and integrating by parts: With we have
Notice that we may assume — applying an approximation argument — that has the same boundary values as , so that boundary terms vanish when integrating by parts. Now the statements in (18) and (19) follow from the three Hardy(-type) estimates
Indeed, for instance estimate (18) can be derived as follows:
It remains to prove statements (22)–(24). Observe that (22) is a standard Hardy estimate. We omit its proof which is very similar to the one of the critical Hardy inequality (23). The latter follows from
via the transformation , , and . This estimate is easily established. We integrate by parts and estimate with the help of the Cauchy–Schwarz inequality:
which immediately yields (23).
The proof of (24) is slightly more involved. We introduce the weight function
and have the obvious estimate
because of . Varying the weight is helpful, since it allows to treat both boundary layer and bulk estimates simultaneously, and thus avoids the appearance of boundary terms at (for which we do not know an appropriate estimate). Using the definition of , we easily compute
where denotes the real part of a complex number . We see via integration by parts that
so that we may rewrite the above formula as
We invoke the Cauchy-Schwarz and Young inequalities to deduce
For any it holds
Moreover, if then
As an immediate consequence of the previous proposition together with the Nusselt number bound (8), we observe for later citations that, in a boundary layer of order one, we have the estimates
for some . Here the first estimate is due to Poincaré’s inequality.
Proof of Proposition 1.
The bounds on and , follow directly from Lemma 1 via
It remains to estimate the -term. Because of (11), we have to show
First, we apply the elementary estimate
to and obtain
For the first term on the r. h. s. of the above equation, we invoke the boundary layer estimate from Lemma 3 with ,
The second term is estimated with the help of Lemma 3 in :
We combine the above estimates and deduce (28) since . ∎
We conclude this section with a global bound on the velocity field, which combines classical Calderon–Zygmund theory with the maximum principle for the temperature.
For any , it holds
Proof of Lemma 4.
and the maximum principle for the temperature in the sense that
cf. (41) below and discussion on page 41. Although (30) is a very classical estimate, we have to argue that the estimate is uniform in the aspect ratio . By rescaling , , and , and redefining , we may w.l.o.g. assume that . Furthermore, since the pressure is unique up to a constant, we suppose that
Our starting point is the analogue result for the Stokes equation on an infinite strip. Consider
in with non-slip boundary conditions at the vertical boundary: for . Then we have
by classical theory, cf. [10, Chapter IV], as long as the r.h.s. is finite. In order to apply this estimate, we introduce a smooth cut-off function with the properties that , for , for , and , . Here, is an arbitrary, but fixed, number that has to be chosen explicitly at the end of the proof. It is readily checked that
In view of the periodicity of , , and in , we may rewrite the above equation as
Notice that, by the homogeneous boundary conditions of at , we can apply the Poincaré inequality in the vertical variable both for and and obtain
Moreover, thanks to (31), the Poincaré inequality in yields
We conclude that