Energy stability bounds on convective heat transport:mNumerical study

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The concept of nonlinear energy stability has recently been extended to deduce bounds on energy dissipation and transport in incompressible flows, even for turbulent flows. In this approach an effective stability condition on ‘‘background’’ flow or temperature profiles is derived, which when satisfied ensures that the profile produces a rigorous upper estimate to the bulk dissipation. Optimization of the test background profiles in search of the lowest upper bounds leads to nonlinear Euler-Lagrange equations for the extremal profile. In this paper, in the context of convective heat transport in the Boussinesq equations, we describe numerical solutions of the Euler-Lagrange equations for the optimal background temperature and present the numerical computation of the implied bounds. @S1063-651X~97!05706-1# PACS number~s!: 47.27.Te, 03.40.Gc, 47.27.Cn, 47.27.Ak The idea that notions of stability—usually reserved for the characterization of static stationary states—may be applied to dynamic, and even turbulent, phenomena has been proposed at various times and in various contexts. The modern concept of spontaneously self-imposed marginal stability was proposed in the context of thermal convection in the 1950s @1#, later resulting in quantitative predictions for the bulk heat transport @2#. More recently, these sentiments have been given a rigorous formulation in the context of the dynamics of incompressible fluids. Building on a mathematical device invented by Hopf @3#, and utilizing a decomposition referred to as the background field method@4,5#, it has now become possible to formulate variational principles for upper bounds on the time averaged rate of heat transport where the key constraint on the background profiles over which the variation takes place is technically identical to a nonlinear energy stability @6# condition. The optimization problem produces nonlinear Euler-Lagrange equations for the ‘‘marginally stable’’ profile yielding the lowest upper bound. Interestingly, in some cases these Euler-Lagrange equations are of the same functional form as those found in Howard’s theory of bounds on flow quantities in statistically stationary states @7#, and with regard to the functional geometry of the constraints further connection has been established with Busse’s multiple boundary layer theory @8# of Howard’s bounds. In this Brief Report we present the results of a numerical study of the solutions of the Euler-Lagrange equations for the optimal background profile. This study represents a practical implementation of the optimal methods developed in Ref. @5#, and it illustrates some of the features that are expected in this kind of analysis. In particular, we observe that the optimal marginally stable profile may exist on different branches of solutions of the Euler-Lagrange equations, with the signal for the qualitative structural change being a change in ‘‘stability’’ of the solution. Consider an incompressible Newtonian fluid confined to the rectangular volume between rigid noslip isothermal plates at z50 and 1. A vertical temperature gradient is imposed, so in the usual nondimensional units the fluid’s velocity vector field u(x,t)5(u 1 ,u 2 ,u 3 ) and temperature field T(x,t) satisfy the Boussinesq equations ]u ]t 1uiiu1ip5sDu1sRakT, ~1!