Natural-convection boundary layers

Abstract

This thesis considers the natural-convection flow for two geometries positioned in the gravitational field: a square cavity differentially heated over the vertical walls and a semi-infinite hot vertical plate in an isothermal environment. The temperature difference results in the natural-convection flow of the fluid inside the cavity and along the plate. When the temperature difference is large (or dimensionless: when the Rayleigh number is large), the flow mainly occurs in a thin layer along the vertical walls, which is the so-called natural-convection boundary layer. The flow for both air and water is calculated by numerically solving the two-dimensional, incompressible Navier-Stokes equations, including the energy equation. The Boussinesq approximation is applied. The spatial derivatives in the equations are discretized with the finite-volume method, whereas the time derivatives are discretized with an implicit scheme using three time levels. Besides the Navier-Stokes equations, the boundary-layer equations are also numerically solved. The accuracy of the numerical results is checked by grid and time step refinement, and the results are compared with existing experimental data. Different flow phenomena related to the natural-convection boundary-layer flow are investigated, giving a complete picture of the flow.

All possible laminar similarity solutions of the boundary-layer equations, with a constant wall temperature and a variable thermal stratification of the environment, are numerically determined. A similarity solution depends on only one transformed coordinate. Some of the similarity solutions have no practical meaning because they cannot be matched with the rest of the flow. If the stratification is stable (i.e., the temperature increases with the height), there is flow reversal in part of the boundary layer.

The proper scalings of the steady laminar flow in the cavity are derived from the steady Navier-Stokes solutions. Proper scalings give a scaled solution that is independent of the Rayleigh number if the Rayleigh number is increased to infinity. The Navier-Stokes equations for large Rayleigh numbers are shown to converge to the boundary-layer equations. Therefore, the boundary-layer equations actually describe the scalings of the cavity flow.

When the Rayleigh number exceeds a critical value (Racr), the steady laminar solution in the cavity becomes unstable and a bifurcation to an unsteady laminar solution is found. The instability is determined by solving the unsteady Navier-Stokes equations. In the case of adiabatic horizontal walls, Racr = 1.7 x 10^6 is found for air and Racr ~ 10^10 for water. The flow for conducting horizontal walls is less stable. Directly beyond Racr, the unsteady flow shows a single frequency; only air in the case of adiabatic horizontal walls shows two frequencies. Arguments are given that support the hypothesis that the instability for conducting horizontal walls is related to the Rayleigh-Bénard instability, whereas the instability for adiabatic horizontal walls is related to a Tollmien-Schlichting instability in the vertical boundary layer. The second frequency for air in the case of adiabatic horizontal walls seems to be related to an instability after the hydraulic jump, which occurs in the upper corner of the cavity where the hot vertical boundary layer bends to a horizontal layer.

The instabilities initiate the laminar-turbulent transition. When the Rayleigh number is further increased, the flow becomes fully turbulent. The turbulent boundary-layer equations, with a k-ε model for the turbulence, are solved for the plate in an isothermal environment. A much better prediction of the wall-heat transfer for the plate is found with the low-Reynolds-number k-ε models of Lam & Bremhorst, Chien, and Jones & Launder than with the standard k-ε model. The low-Reynolds-number models delay the transition and their solution can be nonunique. The differential Reynolds-stress model for the turbulent flow along the plate shows that the use of the eddy-viscosity concept in the k-ε models is not fully justified for natural-convection flows. The proper scalings and wall functions are derived from the numerical results for the turbulent vertical natural-convection boundary layer along the plate. These wall functions are shown to be consistent with the theoretical proposal of George & Capp for wall functions in the outer part of the boundary layer.

The turbulent flow in the cavity is calculated with the Reynolds equations (i.e., the time-averaged Navier-Stokes equations) using the standard k-ε model and the low-Reynolds-number models of Chien and Jones & Launder. A definite conclusion about the accuracy of the turbulence models in the cavity requires the availability of more experimental data. As was also found for the plate, the low-Reynolds-number models delay the transition and their solution can be nonunique. The stratification in the core of the cavity is much smaller for the turbulent flow than for the laminar flow. Due to the small stratification, the averaged wall-heat transfer for the turbulent flow in the cavity is only slightly larger than the averaged wall-heat transfer for the plate in the isothermal environment. The scalings of the turbulent flow in the cavity are close to the scalings for the plate with respect to the Rayleigh-number dependence, but not with respect to the height dependence.