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# hydrodynamics

[hahy-droh-dahy-nam-iks, -di-]
hydrodynamics: see mechanics.
In hydrodynamics, a plume is a column of one fluid moving through another. Several effects control the motion of the fluid, including momentum, buoyancy and density difference. When momentum effects are more important than density differences and buoyancy effects, the plume is usually described as a jet.

Usually, as a plume moves away from its source, it widens because of entrainment of the surrounding fluid at its edges. This usually causes a plume which has initially been 'momentum-dominated' to become 'buoyancy-dominated' (this transition is usually predicted by a dimensionless number called the Richardson number).

A further phenomenon of importance is whether a plume is in laminar flow or turbulent flow. Usually there is a transition from laminar to turbulent as the plume moves away from its source. This phenomenon can be clearly seen in the rising column of smoke from a cigarette.

Another phenomenon which can also be seen clearly in the flow of smoke from a cigarette is that the leading-edge of the flow, or the starting-plume, is quite often approximately in the shape of a ring-vortex (smoke ring).

Plumes are of considerable importance in the dispersion of air pollution. A classic work on the subject of air pollution plumes is that by Gary Briggs.

A thermal plume is one which is generated by gas rising from above heat source. The gas rises because thermal expansion makes warm gas less dense than the surrounding cooler gas.

## Simple Plume Modelling

Quite simple modelling will enable many properties of fully-developed, turbulent plumes to be investigated (see eg ).

1. 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)
2. 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).
3. 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.
4. 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 $rho$ is as follows:

The Momentum M of the flow is conserved so that

$A rho v ^2 = M$ is constant

The mass flux J varies, due to entrainment at the edge of the plume, as

$d J/d x = d A rho v/dx = k r rho v$
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'

## References

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