Variational bounds on energy dissipation in incompressible flows. III. Convection

Bayesian emulation and history matching of JUNE

We analyze JUNE: a detailed model of COVID-19 transmission with high spatial and demographic resolution, developed as part of the RAMP initiative. JUNE requires substantial computational resources to evaluate, making model calibration and general uncertainty analysis extremely challenging. We describe and employ the uncertainty quantification approaches of Bayes linear emulation and history matching to mimic JUNE and to perform a global parameter search, hence identifying regions of parameter space that produce acceptable matches to observed data, and demonstrating the capability of such methods. This article is part of the theme issue 'Technical challenges of modelling real-life epidemics and examples of overcoming these'.

Optimal cooling of an internally heated disc

Motivated by the search for sharp bounds on turbulent heat transfer as well as the design of optimal heat exchangers, we consider incompressible flows that most efficiently cool an internally heated disc. Heat enters via a distributed source, is passively advected and diffused, and exits through the boundary at a fixed temperature. We seek an advecting flow to optimize this exchange. Previous work on energy-constrained cooling with a constant source has conjectured that global optimizers should resemble convection rolls; we prove one-sided bounds on energy-constrained cooling corresponding to, but not resolving, this conjecture. In the case of an enstrophy constraint, our results are more complete: we construct a family of self-similar, tree-like 'branching flows' whose cooling we prove is within a logarithm of globally optimal. These results hold for general space- and time-dependent source-sink distributions that add more heat than they remove. Our main technical tool is a non-local Dirichlet-like variational principle for bounding solutions of the inhomogeneous advection-diffusion equation with a divergence-free velocity. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 1)'.

Bounds on heat flux for Rayleigh-Bénard convection between Navier-slip fixed-temperature boundaries

We study two-dimensional Rayleigh-Bénard convection with Navier-slip, fixed temperature boundary conditions and establish bounds on the Nusselt number. As the slip-length varies with Rayleigh number [Formula: see text], this estimate interpolates between the Whitehead-Doering bound by [Formula: see text] for free-slip conditions (Whitehead & Doering. 2011 Ultimate state of two-dimensional Rayleigh-Bénard convection between free-slip fixed-temperature boundaries. Phys. Rev. Lett. 106, 244501) and the classical Doering-Constantin [Formula: see text] bound (Doering & Constantin. 1996 Variational bounds on energy dissipation in incompressible flows. III. Convection. Phys. Rev. E 53, 5957-5981). This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 1)'.

The background method: theory and computations

The background method is a widely used technique to bound mean properties of turbulent flows rigorously. This work reviews recent advances in the theoretical formulation and numerical implementation of the method. First, we describe how the background method can be formulated systematically within a broader 'auxiliary function' framework for bounding mean quantities, and explain how symmetries of the flow and constraints such as maximum principles can be exploited. All ideas are presented in a general setting and are illustrated on Rayleigh-Bénard convection between stress-free isothermal plates. Second, we review a semidefinite programming approach and a timestepping approach to optimizing bounds computationally, revealing that they are related to each other through convex duality and low-rank matrix factorization. Open questions and promising directions for further numerical analysis of the background method are also outlined. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 1)'.

Velocity-informed upper bounds on the convective heat transport induced by internal heat sources and sinks

Three-dimensional convection driven by internal heat sources and sinks (CISS) leads to experimental and numerical scaling laws compatible with a mixing-length-or 'ultimate'-scaling regime [Formula: see text]. However, asymptotic analytic solutions and idealized two-dimensional simulations have shown that laminar flow solutions can transport heat even more efficiently, with [Formula: see text]. The turbulent nature of the flow thus has a profound impact on its transport properties. In the present contribution, we give this statement a precise mathematical sense. We show that the Nusselt number maximized over all solutions is bounded from above by [Formula: see text], before restricting attention to 'fully turbulent branches of solutions', defined as families of solutions characterized by a finite non-zero limit of the dissipation coefficient at large driving amplitude. Maximization of [Formula: see text] over such branches of solutions yields the better upper-bound [Formula: see text]. We then provide three-dimensional numerical and experimental data of CISS compatible with a finite limiting value of the dissipation coefficient at large driving amplitude. It thus seems that CISS achieves the maximal heat transport scaling over fully turbulent solutions. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 1)'.

Steady thermal convection representing the ultimate scaling

Nonlinear simple invariant solutions representing the ultimate scaling have been discovered to the Navier-Stokes equations for thermal convection between horizontal no-slip permeable walls with a distance [Formula: see text] and a constant temperature difference [Formula: see text]. On the permeable walls, the vertical transpiration velocity is assumed to be proportional to the local pressure fluctuations, i.e. [Formula: see text] (Jiménez et al. 2001 J. Fluid Mech., 442, 89-117. (doi:10.1017/S0022112001004888)). Two-dimensional steady solutions bifurcating from a conduction state have been obtained using a Newton-Krylov iteration up to the Rayleigh number [Formula: see text] for the Prandtl number [Formula: see text], the horizontal period [Formula: see text] and the permeability parameter [Formula: see text]-[Formula: see text], [Formula: see text] being the buoyancy-induced terminal velocity. The wall permeability has a significant impact on the onset and the scaling properties of the found steady 'wall-bounded' thermal convection. The ultimate scaling [Formula: see text] has been observed for [Formula: see text] at high [Formula: see text], where [Formula: see text] is the Nusselt number. In the steady ultimate state, large-scale thermal plumes fully extend from one wall to the other, inducing the strong vertical velocity comparable with the terminal velocity [Formula: see text] as well as intense temperature variation of [Formula: see text] even in the bulk region. As a consequence, the wall-to-wall heat flux scales with [Formula: see text] independent of thermal diffusivity, although the heat transfer on the walls is dominated by thermal conduction. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 1)'.

Vigorous convection in porous media

The problem of convection in a fluid-saturated porous medium is reviewed with a focus on 'vigorous' convective flow, when the driving buoyancy forces are large relative to any dissipative forces in the system. This limit of strong convection is applicable in numerous settings in geophysics and beyond, including geothermal circulation, thermohaline mixing in the subsurface and heat transport through the lithosphere. Its manifestations range from 'black smoker' chimneys at mid-ocean ridges to salt-desert patterns to astrological plumes, and it has received a great deal of recent attention because of its important role in the long-term stability of geologically sequestered CO₂. In this review, the basic mathematical framework for convection in porous media governed by Darcy's Law is outlined, and its validity and limitations discussed. The main focus of the review is split between 'two-sided' and 'one-sided' systems: the former mimics the classical Rayleigh-Bénard set-up of a cell heated from below and cooled from above, allowing for detailed examination of convective dynamics and fluxes; the latter involves convection from one boundary only, which evolves in time through a series of regimes. Both set-ups are reviewed, accounting for theoretical, numerical and experimental studies in each case, and studies that incorporate additional physical effects are discussed. Future research in this area and various associated modelling challenges are also discussed.

Upper bound on angular momentum transport in Taylor-Couette flow

We investigate the upper bound on angular momentum transport in Taylor-Couette flow theoretically and numerically by a one-dimensional background field method. The flow is bounded between a rotating inner cylinder of radius R_{i} and a fixed outer cylinder of radius R_{o}. A variational problem is formulated and solved by a pseudo-time-stepping method up to a Taylor number Ta=10^{9}. The angular momentum transport, characterized by a Nusselt number Nu, is bounded by Nu≤cTa^{1/2}, where the prefactor c depends on the radius ratio η=R_{i}/R_{o}. Three typical radius ratios are investigatedi.e., η=0.5,0.714,and0.909, and the corresponding prefactors c=0.0049,0.0075,and0.0086 are found to improve (lower) the rigorous upper bounds by Doering and Constantin [C. Doering and P. Constantin, Phys. Rev. Lett. 69, 1648 (1992)PRLTAO0031-900710.1103/PhysRevLett.69.1648] and Constantin [P. Constantin, SIAM Rev. 36, 73 (1994)SIREAD0036-144510.1137/1036004] by at least one order of magnitude. Furthermore, we show, via an inductive bifurcation analysis, that considering a three-dimensional background velocity field is unable to lower the bound.

Rough-wall turbulent Taylor-Couette flow: The effect of the rib height

In this study, we combine experiments and direct numerical simulations to investigate the effects of the height of transverse ribs at the walls on both global and local flow properties in turbulent Taylor-Couette flow. We create rib roughness by attaching up to 6 axial obstacles to the surfaces of the cylinders over an extensive range of rib heights, up to blockages of 25% of the gap width. In the asymptotic ultimate regime, where the transport is independent of viscosity, we emperically find that the prefactor of the [Formula: see text] scaling (corresponding to the drag coefficient [Formula: see text] being constant) scales with the number of ribs [Formula: see text] and by the rib height [Formula: see text]. The physical mechanism behind this is that the dominant contribution to the torque originates from the pressure forces acting on the rib which scale with the rib height. The measured scaling relation of [Formula: see text] is slightly smaller than the expected [Formula: see text] scaling, presumably because the ribs cannot be regarded as completely isolated but interact. In the counter-rotating regime with smooth walls, the momentum transport is increased by turbulent Taylor vortices. We find that also in the presence of transverse ribs these vortices persist. In the counter-rotating regime, even for large roughness heights, the momentum transport is enhanced by these vortices.

Radiative heating achieves the ultimate regime of thermal convection

The absorption of light or radiation drives turbulent convection inside stars, supernovae, frozen lakes, and Earth's mantle. In these contexts, the goal of laboratory and numerical studies is to determine the relation between the internal temperature gradients and the heat flux transported by the turbulent flow. This is the constitutive law of turbulent convection, to be input into large-scale models of such natural flows. However, in contrast with the radiative heating of natural flows, laboratory experiments have focused on convection driven by heating and cooling plates; the heat transport is then severely restricted by boundary layers near the plates, which prevents the realization of the mixing length scaling law used in evolution models of geophysical and astrophysical flows. There is therefore an important discrepancy between the scaling laws measured in laboratory experiments and those used, e.g., in stellar evolution models. Here we provide experimental and numerical evidence that radiatively driven convection spontaneously achieves the mixing length scaling regime, also known as the "ultimate" regime of thermal convection. This constitutes a clear observation of this regime of turbulent convection. Our study therefore bridges the gap between models of natural flows and laboratory experiments. It opens an experimental avenue for a priori determinations of the constitutive laws to be implemented into models of geophysical and astrophysical flows, as opposed to empirical fits of these constitutive laws to the scarce observational data.

Roughness-Facilitated Local 1/2 Scaling Does Not Imply the Onset of the Ultimate Regime of Thermal Convection

In thermal convection, roughness is often used as a means to enhance heat transport, expressed in Nusselt number. Yet there is no consensus on whether the Nusselt vs Rayleigh number scaling exponent (Nu∼Ra^{β}) increases or remains unchanged. Here we numerically investigate turbulent Rayleigh-Bénard convection over rough plates in two dimensions, up to Ra≈10^{12}. Varying the height and wavelength of the roughness elements with over 200 combinations, we reveal the existence of two universal regimes. In the first regime, the local effective scaling exponent can reach up to 1/2. However, this cannot be explained as the attainment of the so-called ultimate regime as suggested in previous studies, because a further increase in Ra leads to the second regime, in which the scaling saturates back to a value close to the smooth wall case. Counterintuitively, the transition from the first to the second regime corresponds to the competition between bulk and boundary layer flow: from the bulk-dominated regime back to the classical boundary-layer-controlled regime. Our study demonstrates that the local 1/2 scaling does not necessarily signal the onset of ultimate turbulence.

Optimal Wall-to-Wall Transport by Incompressible Flows

We consider wall-to-wall transport of a passive tracer by divergence-free velocity vector fields u. Given an enstrophy budget ⟨|∇u|^{2}⟩≤Pe^{2} we construct steady two-dimensional flows that transport at rates Nu(u)≳Pe^{2/3}/(logPe)^{4/3} in the large enstrophy limit. Combined with the known upper bound Nu(u)≲Pe^{2/3} for any such enstrophy-constrained flow, we conclude that maximally transporting flows satisfy Nu∼Pe^{2/3} up to possible logarithmic corrections. Combined with known transport bounds in the context of Rayleigh-Bénard convection, this establishes that while suitable flows approaching the "ultimate" heat transport scaling Nu∼Ra^{1/2} exist, they are not always realizable as buoyancy-driven flows. The result is obtained by exploiting a connection between the wall-to-wall optimal transport problem and a closely related class of singularly perturbed variational problems arising in the study of energy-driven pattern formation in materials science.

Optimal bounds with semidefinite programming: An application to stress-driven shear flows

We introduce an innovative numerical technique based on convex optimization to solve a range of infinite-dimensional variational problems arising from the application of the background method to fluid flows. In contrast to most existing schemes, we do not consider the Euler-Lagrange equations for the minimizer. Instead, we use series expansions to formulate a finite-dimensional semidefinite program (SDP) whose solution converges to that of the original variational problem. Our formulation accounts for the influence of all modes in the expansion, and the feasible set of the SDP corresponds to a subset of the feasible set of the original problem. Moreover, SDPs can be easily formulated when the fluid is subject to imposed boundary fluxes, which pose a challenge for the traditional methods. We apply this technique to compute rigorous and near-optimal upper bounds on the dissipation coefficient for flows driven by a surface stress. We improve previous analytical bounds by more than 10 times and show that the bounds become independent of the domain aspect ratio in the limit of vanishing viscosity. We also confirm that the dissipation properties of stress-driven flows are similar to those of flows subject to a body force localized in a narrow layer near the surface. Finally, we show that SDP relaxations are an efficient method to investigate the energy stability of laminar flows driven by a surface stress.

Time-stepping approach for solving upper-bound problems: Application to two-dimensional Rayleigh-Bénard convection

An alternative computational procedure for numerically solving a class of variational problems arising from rigorous upper-bound analysis of forced-dissipative infinite-dimensional nonlinear dynamical systems, including the Navier-Stokes and Oberbeck-Boussinesq equations, is analyzed and applied to Rayleigh-Bénard convection. A proof that the only steady state to which this numerical algorithm can converge is the required global optimal of the relevant variational problem is given for three canonical flow configurations. In contrast with most other numerical schemes for computing the optimal bounds on transported quantities (e.g., heat or momentum) within the "background field" variational framework, which employ variants of Newton's method and hence require very accurate initial iterates, the new computational method is easy to implement and, crucially, does not require numerical continuation. The algorithm is used to determine the optimal background-method bound on the heat transport enhancement factor, i.e., the Nusselt number (Nu), as a function of the Rayleigh number (Ra), Prandtl number (Pr), and domain aspect ratio L in two-dimensional Rayleigh-Bénard convection between stress-free isothermal boundaries (Rayleigh's original 1916 model of convection). The result of the computation is significant because analyses, laboratory experiments, and numerical simulations have suggested a range of exponents α and β in the presumed Nu∼Pr(α)Ra(β) scaling relation. The computations clearly show that for Ra≤10(10) at fixed L=2√[2],Nu≤0.106Pr(0)Ra(5/12), which indicates that molecular transport cannot generally be neglected in the "ultimate" high-Ra regime.

New perspectives in turbulent Rayleigh-Bénard convection

Recent experimental, numerical and theoretical advances in turbulent Rayleigh-Bénard convection are presented. Particular emphasis is given to the physics and structure of the thermal and velocity boundary layers which play a key role for the better understanding of the turbulent transport of heat and momentum in convection at high and very high Rayleigh numbers. We also discuss important extensions of Rayleigh-Bénard convection such as non-Oberbeck-Boussinesq effects and convection with phase changes.

Ultimate state of two-dimensional Rayleigh-Bénard convection between free-slip fixed-temperature boundaries

Rigorous upper limits on the vertical heat transport in two-dimensional Rayleigh-Bénard convection between stress-free isothermal boundaries are derived from the Boussinesq approximation of the Navier-Stokes equations. The Nusselt number Nu is bounded in terms of the Rayleigh number Ra according to Nu≤0.2891Ra(5/12) uniformly in the Prandtl number Pr. This scaling challenges some theoretical arguments regarding asymptotic high Rayleigh number heat transport by turbulent convection.

Bounds on heat transport in Bénard-Marangoni convection

For Pearson's model of Bénard-Marangoni convection, the Nusselt number Nu is proven to be bounded as a function Marangoni number Ma according to Nu<or=0.838 x Ma(2/7) for infinite Prandtl number and according to Nu <or=Ma(1/2) uniformly for finite Prandtl number. The analysis is also used to raise the lower bound for the critical Marangoni number for energy stability of the conduction solution from 56.77 to 58.36 when the Prandtl number is infinite.

Comparison of turbulent thermal convection between conditions of constant temperature and constant flux

We report the results of high-resolution direct numerical simulations of two-dimensional Rayleigh-Bénard convection for Rayleigh numbers up to Ra=10;{10} in order to study the influence of temperature boundary conditions on turbulent heat transport. Specifically, we considered the extreme cases of fixed heat flux (where the top and bottom boundaries are poor thermal conductors) and fixed temperature (perfectly conducting boundaries). Both cases display identical heat transport at high Rayleigh numbers fitting a power law Nu approximately 0.138xRa;{0.285} with a scaling exponent indistinguishable from 2/7=0.2857... above Ra=10;{7}. The overall flow dynamics for both scenarios, in particular, the time averaged temperature profiles, are also indistinguishable at the highest Rayleigh numbers.

Ultimate-state scaling in a shell model for homogeneous turbulent convection

An interesting question in turbulent convection is how the heat transport depends on the strength of thermal forcing in the limit of very large thermal forcing. Kraichnan predicted [Phys. Fluids 5, 1374 (1962)] that for fluids with low Prandtl number (Pr), the heat transport measured by the Nusselt number (Nu) would depend on the strength of thermal forcing measured by the Rayleigh number (Ra) as Nu approximately Ra(1/2) with logarithmic corrections at very high Ra. According to Kraichnan, the shear boundary layers play a crucial role in giving rise to this so-called ultimate-state scaling. A similar scaling result is predicted by the Grossmann-Lohse theory [J. Fluid Mech. 407, 27 (2000)], but with the assumption that the ultimate state is a bulk-dominated state in which both the average kinetic and thermal dissipation rates are dominated by contributions from the bulk of the flow with the boundary layers either broken down or playing no role in the heat transport. In this paper, we study the dependence of Nu and the Reynolds number (Re) measuring the root-mean-squared velocity fluctuations on Ra and Pr, for low Pr, using a shell model for homogeneous turbulent convection where buoyancy is acting directly on most of the scales. We find that Nu approximately Ra(1/2)Pr(1/2) and Re approximately Ra(1/2)Pr(-1/2) , which resemble the ultimate-state scaling behavior for fluids with low Pr, and show that the presence of a drag acting on the large scales is crucial in giving rise to such scaling. As a large-scale drag cannot exist by itself in the bulk of turbulent thermal convection, our results indicate that if buoyancy acts on most of the scales in the bulk of turbulent convection at very high Ra, then the ultimate state cannot be bulk dominated.

Scaling in magnetohydrodynamic convection at high Rayleigh number

The theory of Grossmann and Lohse [J. Fluid Mech. 407, 27 (2000)] is extended to include the effect of a magnetic field on convection of an electrically conducting fluid. Different scaling laws are obtained depending on whether the bulk or the boundary layers make the major contribution to the dissipation. Scalings are obtained for both weak and strong magnetic fields. The predictions are shown to be in better agreement with experimental data than earlier theoretical models.

Exponentially growing solutions in homogeneous Rayleigh-Bénard convection

It is shown that homogeneous Rayleigh-Bénard flow, i.e., Rayleigh-Bénard turbulence with periodic boundary conditions in all directions and a volume forcing of the temperature field by a mean gradient, has a family of exact, exponentially growing, separable solutions of the full nonlinear system of equations. These solutions are clearly manifest in numerical simulations above a computable critical value of the Rayleigh number. In our numerical simulations they are subject to secondary numerical noise and resolution dependent instabilities that limit their growth to produce statistically steady turbulent transport.

Smooth and rough boundaries in turbulent Taylor-Couette flow

We examine the torque required to drive the smooth or rough cylinders in turbulent Taylor-Couette flow. With rough inner and outer walls the scaling of the dimensionless torque G is found to be consistent with pure Kolmogorov scaling G approximately Re2. The results are interpreted within the Grossmann-Lohse theory for the relative role of the energy dissipation rates in the boundary layers and in the bulk; as the boundary layers are destroyed through the wall roughness, the torque scaling is due only to the bulk contribution. For the case of one rough and one smooth wall, we find that the smooth cylinder dominates the dissipation rate scaling, i.e., there are corrections to Kolmogorov scaling. A simple model based on an analogy to electrical circuits is advanced as a phenomenological organization of the observed relative drag functional forms. This model leads to a qualitative prediction for the mean velocity profile within the bulk of the flow.

Convective heat transport in a rotating fluid layer of infinite Prandtl number: optimum fields and upper bounds on Nusselt number

By means of the Howard-Busse method of the optimum theory of turbulence we investigate numerically upper bounds on convective heat transport for the case of infinite fluid layer with stress-free vertical boundaries rotating about a vertical axis. We discuss the case of infinite Prandtl number, 1-alpha solution of the obtained variational problem and optimum fields possessing internal, intermediate, and boundary layers. We investigate regions of Rayleigh and Taylor numbers R and Ta, where no analytical bounds can be derived, and compare the analytical and numerical bounds for these regions of R and Ta where such comparison is possible. The increasing rotation has a different influence on the rescaled optimum fields of velocity w(1), temperature theta(1) and the vertical component of the vorticity f(1). The increasing Ta for fixed R leads to vanishing of the boundary layers of w(1) and theta(1). Opposite to this, the increasing Ta leads first to a formation of boundary layers of the field f(1) but further increasing the rotation causes vanishing of these boundary layers. We obtain optimum profiles of the horizontal averaged total temperature field which could be used as hints for construction of the background fields when applying Doering-Constantin method to the problems of rotating convection. The wave number alpha(1) corresponding to the optimum fields follows the asymptotic relationship alpha(1)=(R/5)(1/4) for intermediate Rayleigh numbers. However, when R becomes large with respect to Ta, after a transition region, the power law for alpha(1) becomes close to the power law for the case without rotation. The Nusselt number Nu is close to the nonrotational bound 0.32R(1/3) for the case of large R and small Ta. Nu decreases with increasing Taylor number. Thus, the upper bounds reflect the tendency of inhibiting thermal convection by increasing rotation for a fixed Rayleigh number. For the regions of Rayleigh and Taylor numbers where the numerical and asymptotic bounds on Nu can be compared, the numerical bounds are about 70% lower than the asymptotic bounds.

Ultimate state of thermal convection

The ultimate regime of thermal convection, the so-called Kraichnan regime [R. H. Kraichnan, Phys. Fluids 5, 1374 (1962)]], hitherto has been elusive. Here numerical evidence for that regime is presented by performing simulations of the bulk of turbulence only, eliminating the thermal and kinetic boundary layers and replacing them with periodic boundary conditions.

Thermodynamics of fluid turbulence: a unified approach to the maximum transport properties

Dissipative properties of various kinds of turbulent phenomena are investigated. Two expressions are derived for the rate of entropy increase due to thermal and viscous dissipation by turbulence, and for the rate of entropy increase in the surrounding system; both rates must be equal when the fluid system is in a steady state. Possibility is shown with these expressions that the steady-state properties of several different types of turbulent phenomena (Bénard-type thermal convection, turbulent shear flow, and the general circulation of the atmosphere and ocean) exhibit a unique state in which the rate of entropy increase in the surrounding system by the turbulent dissipation is at a maximum. The result suggests that the turbulent fluid system tends to be in a steady state with a distribution of eddies that produce the maximum rate of entropy increase in the nonequilibrium surroundings.

Upper bounds on convective heat transport in a rotating fluid layer of infinite prandtl number: case of intermediate taylor numbers

By means of the Howard-Busse method of the optimum theory of turbulence we obtain upper bounds on the convective heat transport in a heated from below layer of fluid of infinite Prandtl number rotating with a constant angular velocity about the vertical axis. We consider the region of intermediate Taylor numbers: alpha41<<Ta<<alpha61 where alpha(1) is the wave number connected to the 1-alpha-solution of the variational problem. The studied optimum fields possess a three-layer or four-layer structure: in addition to the internal, intermediate, and boundary layers, Ekman layers could arise between the intermediate and boundary ones. For the discussed interval of Taylor numbers the intermediate layers do not expand in the direction of the internal layers. We present an asymptotic theory for the case of the fluid layer with rigid lower boundary and stress-free upper boundary. We use an improved solution of the Euler-Lagrange equations of the variational problem for the intermediate sublayer of the optimum field. This solution leads also to correction of the thicknesses of the boundary layers and to lowering of the upper bounds on the convective heat transport for the cases of fluid layer with stress-free or with rigid boundaries. Thus the known upper bounds for these cases can be treated as upper bounds on the upper bounds on the convective heat transport. For the case of the fluid layer with stress-free boundaries the four-layer optimum fields leads to bounds on the convective heat transport which change from R(1/3) at the lower boundary of their interval of validity to values slightly large than R(2/7) near the upper boundary of the interval of validity. Finally we discuss the area of application of the obtained bounds with respect to the Taylor number Ta and Rayleigh number R.

Upper bound on the heat transport in a layer of fluid of infinite prandtl number, rigid lower boundary, and stress-free upper boundary

We obtain an upper bound on the convective heat transport in a heated from below horizontal fluid layer of infinite Prandtl number with rigid lower boundary and stress-free upper boundary. Because of the asymmetric boundary conditions the solutions of the Euler-Lagrange equations of the corresponding variational problem are also asymmetric with different thicknesses of the boundary layers on the upper and lower boundary of the fluid. The obtained bound on the convective heat transport and the corresponding wave number are between the values for a fluid layer with two rigid boundaries and a fluid layer with two stress-free boundaries.

Variational principle for the Navier-Stokes equations

A variational principle is presented for the Navier-Stokes equations in the case of a contained boundary-driven, homogeneous, incompressible, viscous fluid. Based upon making the fluid's total viscous dissipation over a given time interval stationary subject to the constraint of the Navier-Stokes equations, the variational problem looks overconstrained and intractable. However, introducing a nonunique velocity decomposition, u(x,t)=phi(x,t) + nu(x,t), "opens up" the variational problem so that what is presumed a single allowable point over the velocity domain u corresponding to the unique solution of the Navier-Stokes equations becomes a surface with a saddle point over the extended domain (phi,nu). Complementary or dual variational problems can then be constructed to estimate this saddle point value strictly from above as part of a minimization process or below via a maximization procedure. One of these reduced variational principles is the natural and ultimate generalization of the upper bounding problem developed by Doering and Constantin. The other corresponds to the ultimate Busse problem which now acts to lower bound the true dissipation. Crucially, these reduced variational problems require only the solution of a series of linear problems to produce bounds even though their unique intersection is conjectured to correspond to a solution of the nonlinear Navier-Stokes equations.