Non-horizontal Boundaries Research Articles

Locally reacting boundary condition over sloping boundaries for the parabolic equation

A boundary condition which models locally reacting boundaries is derived for the generalized parabolic equation. This formulation is valid over nonhorizontal boundary surfaces. The derived condition is discretized using Lagrange polynomials. Particular implementation schemes are discussed for nonhorizontal boundaries. Numerical results compare the accuracy of this approach with other methods that have been used with the PE. Comparison with experimental results may also be presented. [Work supported by ONR and USACECOM.]

The thermally forced circulation in a small, ice‐covered lake1

A simple, one‐dimensional model of the stratification in a small, ice‐covered lake is developed. It is shown that the heat flux from the sediments gives rise to a stable stratification and an organized circulation pattern. Data from a field study are compared with the predictions of the model. The good agreement found suggests that the buoyancy layer at the nonhorizontal boundaries is essential in modeling the thermal circulation in this type of lake.

Flow of a stratified fluid bounded by walls of finite thermal conductance

Abstract A study is made of the behavior of a thermally stratified fluid in a container when the non-horizontal boundaries have finite thermal conductance. The theory of Rahm and Walin is briefly recounted. Numerical solutions to the Navier-Stokes equations for a Boussinesq fluid in a cylinder, adopting a Newtonian heat flux condition at the vertical sidewall, are presented. Results on the details of flow and temperature fields are given over ranges of the Rayleigh number Ra, the container aspect ratio H, and the sidewall conductance S. As S increases, the isotherms in the meridional plane are horizontal at small radii but they diverge at large radii. This creates temperature nonuniformilies in the horizontal direction, and convective motions result. The salient features of the interior temperature profiles are captured by the theoretical model. The velocity field is characterized by two oppositely-directed circulations. As Ra or S varies, the qualitative circulation patterns remain substantially unchanged, but the magnitudes of the convective flows differ by large amounts. The effects of the externally-imposed parameters on the flow and temperature structures are examined.

Consistent coupled mode theory of sound propagation for a class of nonseparable problems

This article examines the effects of boundary condition approximations that arise whenever the coupled mode theories of A. D. Pierce [J. Acoust. Soc. Am. 37, 19–27 (1965)] and D. M. Milder [J. Acoust. Soc. Am. 46, 1259–1263 (1969)], are applied to propagation problems involving range dependent boundaries. This boundary condition approximation requires that the depth derivative, rather than the normal derivative of the field, be continuous across a sloping interface. The approximation is necessary in order to carry out the partial separation of depth and range variables effected in the mathematical formulation of the theory. This article will show that a consequence of this approximation is that conventional coupled mode theory applied to dissipationless media with nonhorizontal boundaries does not conserve energy. It is shown that a correction to coupled mode theory can be derived such that the proper boundary conditions are satisfied and energy is conserved to first order in the local bottom slope. Moreover, the corrections are not prohibitive in terms of added computational complexity. Numerical examples are presented which illustrate the nonconservation of energy effect and the corrections to the theory.

On thermally forced stratified rotating fluids

Axisymmetric steady motion of an inhomogeneous rotating fluid is considered. A system of equations, with appropriate boundary conditions, controlling the smooth interior fields is derived under the assumption of small dissipation and small side-boundary conductance. It is argued that this system, being derived without linearization of the equations, might form the basis of valuable numerical analysis. Assuming sufficiently weak forcing, i.e. high insulation of the non-horizontal boundaries, a linear system is derived. An explicit solution is presented and discussed for a particularly simple and important case.

The effects of boundary condition approximations on coupled mode theory

This paper examines the effect of boundary condition approximations that are inherent to the coupled mode theories proposed by Pierce [J. Acoust. Soc. Am. 37, 19–27 (1965)] and Milder [J. Acoust. Soc. Am. 46, 1259–1263 (1969)]. The boundary condition approximation that is investigated involves the replacing of the continuity of the normal component of partial velocity boundary conditions with the continuity of the z component of partial velocity boundary conditions at sloping interfaces. This approximation is necessary in order to carry out the partial separation of depth and range variables effected in the mathematical formulation of the theory. This paper will show that a consequence of this approximation is that coupled mode theory applied to propagation media with nonhorizontal boundaries does not conserve energy. It will also be shown that a correction to coupled mode theory may be derived such that the proper boundary conditions are satisfied and energy is conserved, both to first order in the local bottom slope. Numerical examples will be presented which illustrate the nonconservation of energy effect and the proposed corrections to the theory. [Work supported by the Naval Ocean Research and Development Activity and the Naval Electronic System Command.]