Ventilation, stratification and flow instability driven by vertically distributed buoyancy sources
This thesis consists of three different projects on flows driven by vertically distributed buoyancy sources. Following a storyline, the three serial problems have inherent connections.
In the first project, a theoretical model is developed for the steady multi-layered flow induced by a plane vertically distributed buoyancy source producing a turbulent wall plume in a ventilated box. The boundary condition at the wall for each layer is established by deducing the turbulent entrainment rate. Using conformal mapping techniques and Poisson's integral theorem, closed-form solutions for the streamfunction of the induced flow in each layer are established. While the flow near the ceiling was overlooked in the classic model for the multi-layered stratification, after considering the possible flow scenarios, the stratification is reevaluated herein by incorporating an entraining ceiling current. With a markedly thinner top layer, the refined stratification matches well with available experimental observations -- agreement that was absent for the classic model -- the model overcoming the previous over-prediction in the number of interfaces. The magnitude of dimensionless flow velocity, independent of the wall buoyancy flux and physical scale of the box, decreases significantly with the number of layers. Three types of layer, each with a distinct induced flow pattern, are distinguished and their implications for indoor airflow considered. Notably, the flow in the base layer represents a continual and smooth flushing of air between the inlet opening and wall plume, whereas any intermediate layer is almost entirely comprised of near-stagnant air.
Second, the linear temporal stability characteristics of a thermal plume generated along a heated vertical cylinder is investigated theoretically. Special focus is given to the uniform-wall-buoyancy-flux case whereby the cylinder surface sustains the same linear temperature gradient as the environment. A competition between the axisymmetric and helical wave modes is a remarkable feature of the instability, distinguishing these `annular wall plumes’ from free plumes/jets for which the helical mode is usually dominant. It is found that higher surface curvature stabilises the temporal axisymmetric mode significantly, but only has moderate effects on the helical mode. The most temporally unstable perturbation mode switches from a helical into an axisymmetric mode when the Prandtl number increases beyond a critical value. Both the roles of shear and buoyancy during the destabilisation are identified through an energy analysis which indicates that while (mean) shear work is usually a major source of perturbation energy, buoyancy work manifests for long-wave axisymmetric perturbation modes, and for thin cylinders and high Prandtl numbers. In heat transfer applications, to promote transition to turbulence and enhance convective heat transfer, working fluids of either sufficiently low or high Prandtl numbers are preferred; cylinders of too-small dimensionless radii should be avoided.
The third project presents a numerical approach innovated to overcome the numerical difficulty of tracking the eigenmodes when evaluating a fully numerical dispersion relationship. Based on this numerical approach, the absolute/convective instability of the same annular plume as that in the second project is analysed. Following a discussion about the numerical issue of eigenmode tracking and some previous methods to alleviate it, a bespoke `initial value approach' is proposed, which primarily consists of making an estimation of the target eigenmode via solving a set of successive initial value problems, and identifying the target eigenmode from the exact solutions. Furthermore, this numerical approach is adapted into an iterative scheme, which facilitates the computation of the absolute/convective instability over the parameter space in an automatic and efficient way. For the specific temperature configuration considered herein, an annular wall plume is always absolutely stable, whereas decreasing the cylinder radius from the planar limiting case first decreases and then increases the tendency of the flow towards being absolutely unstable. The helical mode is especially susceptible to being absolutely unstable on very thin cylinders. A high Prandtl number promotes the absolute instability for both azimuthal perturbation modes. Finally, several conditions for the occurrence of plume detrainment are proposed based on the results and a hypothesis which connects the absolute instability to the detrainment phenomenon. The numerical approach also enables the absolute/convective instabilities of multiple branches of a general dispersion relationship to be computed and assessed simultaneously.