2 results
Two-layer model for shallow horizontal convective circulation
- Dominique N. Brocard, Donald R. F. Harleman
-
- Journal:
- Journal of Fluid Mechanics / Volume 100 / Issue 1 / 11 September 1980
- Published online by Cambridge University Press:
- 19 April 2006, pp. 129-146
-
- Article
- Export citation
-
This paper discusses experiments and a theoretical model for the convective circulation driven by a surface buoyancy flux in a horizontal layer of fluid. The layer is closed at one end and, at the other end, the buoyancy has a fixed value over a given depth. Such circulation occurs in side arms of cooling lakes used for waste-heat disposal from power generation. Some geophysical circulations, such as in the Red Sea, are also of the above type.
The experiments were done in a 35 ft long flume using heat transfer between the heated water and the atmosphere to generate the surface buoyancy flux. The observed circulation was characterized by two distinct layers flowing in opposite directions and separated by a density interface. For upper-layer depths less than about half the total depth at the open end, the downflow was observed to be concentrated near the closed end. Circulation flowrates and vertical temperature profiles were measured.
The theoretical model uses a ‘two-layer’ approach. The mass, momentum, and buoyancy conservation equations are integrated vertically on each side of the interface. Mass and buoyancy transfer across the interface are neglected. The interfacial shear stress is proportional to the square of the difference of the average layer velocities. For small layer densimetric Froude numbers, the free surface is shown to be approximately horizontal and the problem reduces to one ordinary differential equation for the thickness of the upper layer. General solutions of this interface equation are presented for horizontal and sloping bottoms. Different configurations are possible depending on the nature of the singular points which occur in the phase plan.
For the convective circulation, the interface is shown to go through a singular point. This condition leads to a simple analytical solution for the circulation flowrate in the case of constant surface buoyancy flux and horizontal bottom. This solution compares well with the experimental data and with measurements on the Red Sea circulation.
Stability and mixing of a vertical plane buoyant jet in confined depth
- Gerhard H. Jirka, Donald R. F. Harleman
-
- Journal:
- Journal of Fluid Mechanics / Volume 94 / Issue 2 / 25 September 1979
- Published online by Cambridge University Press:
- 19 April 2006, pp. 275-304
-
- Article
- Export citation
-
A plane turbulent buoyant jet discharging vertically into a two-dimensional channel of confined depth is considered. The channel opens at both ends into a large outside reservoir, thus defining a steady symmetrical flow field within the channel. The analysis is aimed at two aspects, the stability and the bulk mixing characteristics of the discharge. A stable discharge configuration is defined as one in which a buoyant surface layer is formed which spreads horizontally and does not communicate with the initial buoyant jet region. On the other hand, the discharge configuration is unstable when a recirculating cell exists on both sides of the jet efflux.
It is shown that discharge stability is only dependent on the dynamic interaction of three near-field regions, a buoyant jet region, a surface impingement region and an internal hydraulic jump region. The buoyant jet region is analysed with the assumption of a variable entrainment coefficient in a form corresponding to an approximately constant jet-spreading angle as confirmed by different experimental sources. The properties of surface impingement and internal jump regions are determined on the basis of control volume analyses. Under the Boussinesq approximation, only two dimensionless parameters govern the near-field interaction; these are a discharge densimetric Froude number and a relative depth. For certain parameter combinations, namely those implying low buoyancy and shallow depth, there is no solution to the conjugate downstream condition in the hydraulic jump which would satisfy both momentum and energy conservation principles. Arguments are given which interpret this condition as one which leads to the establishment of a near-field recirculation cell and, thus, discharge instability.
The far-field boundary conditions, while having no influence on discharge stability, determine the bulk mixing characteristics of the jet discharge. The governing equations for the two-layered counterflow system in the far field are solved. The strength of the convective transport, and hence the related dilution ratio, is governed by another non-dimensional parameter, the product of the relative channel length and the boundary friction coefficient.
Experiments in a laboratory flume, covering a range of the governing parameters, are in excellent agreement with the theoretical predictions, both the stability criterion and the bulk mixing characteristics.