Abstract
We derive a new expression for the entrainment coefficient in a turbulent plume using an equation for the squared mean buoyancy. Consistency of the resulting expression with previous relations for the entrainment coefficient implies that the turbulent Prandtl number in a pure plume is equal to 3/5 when the mean profiles of velocity and buoyancy have a Gaussian form of equal width. Entrainment can be understood in terms of the volume flux, the production of turbulence kinetic energy or the production of scalar variance for either active or passive variables. The equivalence of these points of view indicates how the entrainment coefficient and the turbulent Prandtl and Schmidt numbers depend on the Richardson number of the flow, the ambient stratification and the relative widths of the velocity and scalar profiles. The general framework is valid for self-similar plumes, which are characterised by a power-law scaling. For jets and pure plumes it is shown that the derived relations are in reasonably good agreement with results from direct numerical simulations and experiments.
Highlights
An exact expression (2.9b) for the entrainment coefficient has been obtained in terms of budgets associated with the mean buoyancy in a self-similar turbulent plume
For pure plumes with Gaussian velocity and buoyancy profiles of equal width, the relations predict that the turbulent Prandtl is equal to 3/5
Arbitrary plume Richardson numbers can be considered by including a power-law source term in the buoyancy equation
Summary
Following the classical works of Priestley & Ball (1955) and Morton, Taylor & Turner (1956), integral models of turbulent plumes have provided physical insights and a robust means of predicting bulk flow properties in applications ranging from natural ventilation (Linden 1999) to geophysics (Woods 2010). How does entrainment relate to the small-scale behaviour of turbulence? How do the buoyancy of an environment and of a plume influence entrainment? What determines the relative rate of spread of the velocity and buoyancy profile in a turbulent plume?. Turbulent plumes are amenable to theoretical study and mathematical modelling because, under certain circumstances, they evolve in a self-similar fashion, i.e. the dependences of their dynamics and transport properties on their cross-stream (radial) coordinate are independent of height (see e.g. Zel’dovich 1937). An example of one such coefficient is the classical entrainment coefficient (Taylor 1945; Batchelor 1954)
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