^{1,a)}

### Abstract

By means of the Howard-Busse method of the optimum theory of turbulence we investigate numerically the upper bounds on the Nusselt number in a heated-from-below horizontal layer of fluid of finite Prandtl number for the case of rigid boundaries. The bounds are obtained by the solutions of the Euler-Lagrange equations of a variational problem possessing up to three wave numbers. The obtained results are compared to the numerical results for the case of fluid layer with stress-free boundaries [N. K. Vitanov and F. H. Busse, “Upper bounds on heat transport in a horizontal fluid layer with stress-free boundaries,” ZAMP48, 310 (1997)] as well as to the numerical and analytical asymptotic results obtained by Howard [“Heat transport by turbulent convection,” J. Fluid Mech.17, 405 (1963)], Busse [“On Howard’s upper bound for heat transport by turbulent convection,” J. Fluid Mech.37, 457 (1969)], and Strauss [“On the upper bounding approach to thermal convection at moderate Rayleigh numbers, II. Rigid boundaries,” Dyn. Atm. Oceans1, 77 (1976)]. We show that for low and intermediate Rayleigh numbers the numerical bounds are positioned below the analytical asymptotic bounds obtained by Howard and Busse. For large Rayleigh numbers the numerical bounds tend to approach the analytical asymptotic bounds. We confirm numerically the bound obtained by Howard for the case of one-wave-number solution of the Euler-Lagrange equations. As the region of validity of the results of the analytical asymptotic theory for solutions of the Euler-Lagrange equations with two and three wave numbers lies in the area of very high Rayleigh numbers the values of the second and third wave numbers are different from their analytical asymptotic values for the values of the Rayleigh number reached by the numerical computation.

This work was supported in part by the Alexander von Humboldt Foundation, and in part by NATO through its Cooperative Science and Technology Subprogram Grant No. PST.CLG.979126.

I. INTRODUCTION

II. MATHEMATICAL FORMULATION OF THE PROBLEM

III. BEHAVIOR OF THE OPTIMUM FIELDS AND COMPARISON TO THE CASE OF STRESS-FREE BOUNDARY CONDITIONS

IV. OPTIMUM WAVE NUMBERS AND UPPER BOUNDS ON THE CONVECTIVE HEAT TRANSPORT

V. CONCLUDING REMARKS

### Key Topics

- Numerical solutions
- 62.0
- Heat transport
- 25.0
- Number theory
- 24.0
- Turbulent flows
- 20.0
- Fluid equations
- 18.0

## Figures

Dependence of the Nusselt number on the wave number for the solution of the Euler-Lagrange equations of variational problem. . The optimum value of Nu for fixed Rayleigh number has to be obtained as the maximum value of the function . For solutions of the Euler-Lagrange equations we have to search for the maximum of the function . For three-wave-number solutions of the Euler-Lagrange equations we need the maximum of , etc.

Dependence of the Nusselt number on the wave number for the solution of the Euler-Lagrange equations of variational problem. . The optimum value of Nu for fixed Rayleigh number has to be obtained as the maximum value of the function . For solutions of the Euler-Lagrange equations we have to search for the maximum of the function . For three-wave-number solutions of the Euler-Lagrange equations we need the maximum of , etc.

Optimum field for the solution of the Euler-Lagrange equations. (a) Case of fluid layer with rigid boundaries. From bottom to the top the profiles are plotted for the following values of Rayleigh number: (marked with 1), (marked with 2), (marked with 3), (marked with 4), (marked with 5), (marked with 6), (marked with 7). (b) Comparison between optimum fields for the cases of rigid and stress-free boundaries. . Solid line: optimum field for fluid layer with rigid boundaries and dot-dashed line: optimum field for fluid layer with stress-free boundaries.

Optimum field for the solution of the Euler-Lagrange equations. (a) Case of fluid layer with rigid boundaries. From bottom to the top the profiles are plotted for the following values of Rayleigh number: (marked with 1), (marked with 2), (marked with 3), (marked with 4), (marked with 5), (marked with 6), (marked with 7). (b) Comparison between optimum fields for the cases of rigid and stress-free boundaries. . Solid line: optimum field for fluid layer with rigid boundaries and dot-dashed line: optimum field for fluid layer with stress-free boundaries.

Optimum fields of the solution of the variational problem for the cases of fluid layer with rigid and stress-free boundaries. (a) Case of fluid layer with rigid boundaries. (solid lines) and (dashed lines) for the solution of the Euler-Lagrange equations. From bottom to the top with respect to the height of the peaks the Rayleigh number has the values: (marked with 1), (marked with 2), (marked with 3), (marked with 4), (marked with 5). (b) Comparison between for the cases of rigid and stress-free boundaries. . Solid lines: for the case of rigid boundaries and dot-dashed lines: for the case of stress-free boundaries.

Optimum fields of the solution of the variational problem for the cases of fluid layer with rigid and stress-free boundaries. (a) Case of fluid layer with rigid boundaries. (solid lines) and (dashed lines) for the solution of the Euler-Lagrange equations. From bottom to the top with respect to the height of the peaks the Rayleigh number has the values: (marked with 1), (marked with 2), (marked with 3), (marked with 4), (marked with 5). (b) Comparison between for the cases of rigid and stress-free boundaries. . Solid lines: for the case of rigid boundaries and dot-dashed lines: for the case of stress-free boundaries.

Optimum fields of the solution of the variational problem for the cases of fluid layer with rigid and stress-free boundaries. Case of fluid layer with rigid boundaries. (solid line); (dashed line); and (dotted line) for the solution of the Euler-Lagrange equations. From bottom to the top with respect to the height of the peaks the Rayleigh number has the values: (marked with 1), (marked with 2).

Optimum fields of the solution of the variational problem for the cases of fluid layer with rigid and stress-free boundaries. Case of fluid layer with rigid boundaries. (solid line); (dashed line); and (dotted line) for the solution of the Euler-Lagrange equations. From bottom to the top with respect to the height of the peaks the Rayleigh number has the values: (marked with 1), (marked with 2).

Optimum fields for the solution of the Euler-Lagrange equations for the case of rigid and stress-free boundary conditions. (a) Optimum fields for the case of rigid boundaries. The profiles from bottom to the top are obtained for the following values of the Rayleigh number: (marked with 1), (marked with 2), (marked with 3), (marked with 4), (marked with 5), (marked with 6), (marked with 7). (b) Comparison between the profiles of optimum fields for the cases of rigid and stress-free boundary conditions. . Solid line: case of rigid boundaries and dot-dashed line: case of stress-free boundaries.

Optimum fields for the solution of the Euler-Lagrange equations for the case of rigid and stress-free boundary conditions. (a) Optimum fields for the case of rigid boundaries. The profiles from bottom to the top are obtained for the following values of the Rayleigh number: (marked with 1), (marked with 2), (marked with 3), (marked with 4), (marked with 5), (marked with 6), (marked with 7). (b) Comparison between the profiles of optimum fields for the cases of rigid and stress-free boundary conditions. . Solid line: case of rigid boundaries and dot-dashed line: case of stress-free boundaries.

Optimum fields (solid lines) and (dashed lines) for the solution of the Euler-Lagrange equations for fluid layers with rigid and stress-free boundaries. (a) Optimum fields for the case of rigid boundaries. Solid lines ; dashed lines: . From bottom to the top with respect to the heights of the maxima of the optimum fields the Rayleigh number has the values (marked with 1), (marked with 2), (marked with 3), (marked with 4), (marked with 5). (b) Comparison among the optimum fields for the case of rigid and stress free boundaries. . Solid lines: case of rigid boundaries and dot-dashed lines: case of stress-free boundaries.

Optimum fields (solid lines) and (dashed lines) for the solution of the Euler-Lagrange equations for fluid layers with rigid and stress-free boundaries. (a) Optimum fields for the case of rigid boundaries. Solid lines ; dashed lines: . From bottom to the top with respect to the heights of the maxima of the optimum fields the Rayleigh number has the values (marked with 1), (marked with 2), (marked with 3), (marked with 4), (marked with 5). (b) Comparison among the optimum fields for the case of rigid and stress free boundaries. . Solid lines: case of rigid boundaries and dot-dashed lines: case of stress-free boundaries.

Optimum fields (solid line); (dashed lines); and for the solution of the Euler-Lagrange equations for fluid layers with rigid and stress-free boundaries. Optimum fields for the case of rigid boundaries. Solid lines ; dashed lines: . From bottom to the top with respect to the heights of the maxima of the optimum fields the Rayleigh number has the values (marked with 1), (marked with 2).

Optimum fields (solid line); (dashed lines); and for the solution of the Euler-Lagrange equations for fluid layers with rigid and stress-free boundaries. Optimum fields for the case of rigid boundaries. Solid lines ; dashed lines: . From bottom to the top with respect to the heights of the maxima of the optimum fields the Rayleigh number has the values (marked with 1), (marked with 2).

Optimum fields for the solution of the Euler-Lagrange equations of the variational problem for the cases of rigid and stress-free boundaries. (a) Profiles of the optimum field for the case of fluid layer with rigid boundaries. The values of the Rayleigh numbers from top to the bottom are (marked with 10), (marked with 9), (marked with 8), (marked with 7), (marked with 6), (marked with 5), (marked with 4), (marked with 3), (marked with 2), (marked with 1). (b) Comparison between the optimum fields for the cases of rigid and stress-free boundaries. . Solid line: Case of rigid boundaries and dot-dashed line: case of stress-free boundaries.

Optimum fields for the solution of the Euler-Lagrange equations of the variational problem for the cases of rigid and stress-free boundaries. (a) Profiles of the optimum field for the case of fluid layer with rigid boundaries. The values of the Rayleigh numbers from top to the bottom are (marked with 10), (marked with 9), (marked with 8), (marked with 7), (marked with 6), (marked with 5), (marked with 4), (marked with 3), (marked with 2), (marked with 1). (b) Comparison between the optimum fields for the cases of rigid and stress-free boundaries. . Solid line: Case of rigid boundaries and dot-dashed line: case of stress-free boundaries.

Optimum fields (solid lines), (dashed lines), and (dotted lines) for the solution of the Euler-Lagrange equations. With respect to the value of the Rayleigh number the fields are marked as follows: (marked with 5), (marked with 4), (marked with 3), (marked with 2), (marked with 1).

Optimum fields (solid lines), (dashed lines), and (dotted lines) for the solution of the Euler-Lagrange equations. With respect to the value of the Rayleigh number the fields are marked as follows: (marked with 5), (marked with 4), (marked with 3), (marked with 2), (marked with 1).

Optimum fields (solid lines), (dashed lines), (dotted lines), and (dotted lines) for the solution of the Euler-Lagrange equations. .

Optimum fields (solid lines), (dashed lines), (dotted lines), and (dotted lines) for the solution of the Euler-Lagrange equations. .

Behavior of the wave numbers . Filled circles: for the solution of the Euler-Lagrange equations (rigid boundaries). Circles: for the solution of the Euler-Lagrange equations (stress-free boundaries). Filled squares: for solution of the Euler-Lagrange equations (stress-free boundaries). Squares: for solution of the Euler-Lagrange equations (rigid boundaries). Filled diamonds: for the solution of the Euler-Lagrange equations (rigid boundaries). Diamonds: for the solution of the Euler-Lagrange equations (stress-free boundaries). Triangles up: for the solution of the Euler-Lagrange equations (stress-free boundaries) Triangles down: for the solution of the Euler-Lagrange equations (stress-free boundaries) Stars: for the solution of the Euler-Lagrange equations (stress-free boundaries). Solid line for the solution of the Euler-Lagrange equations (analytical asymptotic theory of Howard).

Behavior of the wave numbers . Filled circles: for the solution of the Euler-Lagrange equations (rigid boundaries). Circles: for the solution of the Euler-Lagrange equations (stress-free boundaries). Filled squares: for solution of the Euler-Lagrange equations (stress-free boundaries). Squares: for solution of the Euler-Lagrange equations (rigid boundaries). Filled diamonds: for the solution of the Euler-Lagrange equations (rigid boundaries). Diamonds: for the solution of the Euler-Lagrange equations (stress-free boundaries). Triangles up: for the solution of the Euler-Lagrange equations (stress-free boundaries) Triangles down: for the solution of the Euler-Lagrange equations (stress-free boundaries) Stars: for the solution of the Euler-Lagrange equations (stress-free boundaries). Solid line for the solution of the Euler-Lagrange equations (analytical asymptotic theory of Howard).

Numerical and analytical upper bounds on convective heat transport. Filled circles: upper bound on Nu for the solution of the Euler-Lagrange equations (stress-free boundaries). Circles: upper bound on Nu for the solution of the Euler-Lagrange equations (rigid boundaries). Triangles: upper bound on Nu for the solution of the Euler-Lagrange equations (rigid boundaries). Dotted line: upper bound on Nu for the solution of the Euler-Lagrange equations (stress-free boundaries) Stars: upper bound on Nu for the solution of the Euler-Lagrange equations (stress-free boundaries). Dashed line: upper bound on Nu for the solution (analytical asymptotic theory of Howard for rigid boundaries).

Numerical and analytical upper bounds on convective heat transport. Filled circles: upper bound on Nu for the solution of the Euler-Lagrange equations (stress-free boundaries). Circles: upper bound on Nu for the solution of the Euler-Lagrange equations (rigid boundaries). Triangles: upper bound on Nu for the solution of the Euler-Lagrange equations (rigid boundaries). Dotted line: upper bound on Nu for the solution of the Euler-Lagrange equations (stress-free boundaries) Stars: upper bound on Nu for the solution of the Euler-Lagrange equations (stress-free boundaries). Dashed line: upper bound on Nu for the solution (analytical asymptotic theory of Howard for rigid boundaries).

Optimum wave numbers. Solid line: for solution of the Euler-Lagrange equations (numerical calculation). Dashed line: for solution of the Euler-Lagrange equations (numerical calculation). Dot-dashed line: for solution of the Euler-Lagrange equations (asymptotic theory). Double-dot-dashed line: for solution of the Euler-Lagrange equations (asymptotic theory).

Optimum wave numbers. Solid line: for solution of the Euler-Lagrange equations (numerical calculation). Dashed line: for solution of the Euler-Lagrange equations (numerical calculation). Dot-dashed line: for solution of the Euler-Lagrange equations (asymptotic theory). Double-dot-dashed line: for solution of the Euler-Lagrange equations (asymptotic theory).

Numerical and analytical upper bounds on convective heat transport. Solid line: upper bound on Nu for the solution of the Euler-Lagrange equations (numerical calculations by means of symmetric optimum fields). Dashed line: upper bound on Nu for the solution of the Euler-Lagrange equations (analytical asymptotic theory of Busse). Circles: upper bound obtained by Strauss (Ref. 28) by means of antisymmetric optimum fields. This bound lies below the bound obtained by means of symmetric optimum fields.

Numerical and analytical upper bounds on convective heat transport. Solid line: upper bound on Nu for the solution of the Euler-Lagrange equations (numerical calculations by means of symmetric optimum fields). Dashed line: upper bound on Nu for the solution of the Euler-Lagrange equations (analytical asymptotic theory of Busse). Circles: upper bound obtained by Strauss (Ref. 28) by means of antisymmetric optimum fields. This bound lies below the bound obtained by means of symmetric optimum fields.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content