Simple Plume Modelling
Quite simple modelling will enable many properties of fully developed, turbulent plumes to be investigated (see e.g.).
- It is usually sufficient to assume that the pressure gradient is set by the gradient far from the plume (this approximation is similar to the usual Boussinesq approximation)
- The distribution of density and velocity across the plume are modelled either with simple Gaussian distributions or else are taken as uniform across the plume (the so-called 'top hat' model).
- Mass entrainment velocity into the plume is given by a simple constant times the local velocity - this constant typically has a value of about 0.08 for vertical jets and 0.12 for vertical, buoyant plumes. For bent-over plumes, the entrainment coefficient is about 0.6.
- Conservation equations for mass flux (including entrainment) and momentum flux (allowing for buoyancy) then give sufficient information for many purposes.
For a simple rising plume these equations predict that the plume will widen at a constant half-angle of about 6 to 15 degrees.
A top-hat model of a circular plume entraining in a fluid of the same density is as follows:
The Momentum M of the flow is conserved so that
- is constant
The mass flux J varies, due to entrainment at the edge of the plume, as
where k is an entrainment constant, r is the radius of the plume at distance x, and A is its cross-sectional area.
This shows that the mean velocity v falls inversely as the radius rises, and the plume grows at a constant angle dr/dx= k'
Read more about this topic: Plume (hydrodynamics)
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