Plume (fluid dynamics)

Types

A thermal plume occurs when a column of heated gas propagates upward above a thermal source. The plume rises because thermal expansion causes the heated gas to become less dense than the surrounding cooler gas.

Pollutants released to the ground can propagate into groundwater sources, leading to groundwater pollution. The resulting body of polluted water within an aquifer is characterized as a plume, with its migrating edges called plume fronts. These plumes are used to locate, map, and measure water pollution within the aquifer's total body of water. Additionally, plume fronts are used to determine the direction and speed of a given contaminate's spread.

Plumes are of considerable importance in the atmospheric dispersion modelling of air pollution. Gary Briggs has authored a classic work on the subject of air pollution plumes.

Simple plume modeling

Simple modeling can enable many properties of fully developed, turbulent plumes to be investigated. Many of the classic scaling arguments were developed in a combined analytic and laboratory study described in an influential paper by Bruce Morton, G. I. Taylor, and Stewart Turner. This and subsequent work is described in the popular monograph of Stewart Turner:

1. It is usually sufficient to assume that the pressure gradient is set by that which is far from the plume (this approximation is similar to the usual Boussinesq approximation).
2. The distribution of density and velocity across the plume are modeled either with simple Gaussian distributions or else are taken as uniform across the plume (the so-called top hat model).
3. The rate of entrainment into the plume is proportional to the local velocity. Though initially thought to be a constant, recent work has shown that the entrainment coefficient varies with the local Richardson number. Typical values for the entrainment coefficient are about 0.08 for vertical jets and 0.12 for vertical, buoyant plumes, while for bent-over plumes the entrainment coefficient is about 0.6.
4. Conservation equations for mass (including entrainment), and momentum and buoyancy fluxes are sufficient for a complete description of the flow in many cases. For a simple rising plume these equations predict that the plume will widen at a constant half-angle of about 6 to 15 degrees.

The value of the entrainment coefficient is the key parameter in simple plume models. Research continues into assessing how the entrainment coefficient is affected by, for example, the geometry of a plume, suspended particles within a plume, and background rotation.

Gaussian plume modeling

Gaussian plume models can be used in several fluid dynamics scenarios to calculate the concentration distribution of solutes, such as the release by a smoke stack or a contaminant released into a river. Gaussian distributions are established by Fickian diffusion, and follow a Gaussian, or bell-shaped, distribution. For calculating the expected concentration ${\displaystyle C}$ of a point source, we consider a mass ${\displaystyle M}$ released at an instantaneous point in time ${\displaystyle t}$ in a one-dimensional domain along ${\displaystyle x}$. This results in the distribution $${\displaystyle C(x,t)=\frac{M}{\sqrt{4\pi Dt}}\exp \left(\frac{(x-x_0{)}^2}{4D(t-t_0)}\right),}$$ where ${\displaystyle M}$ is the mass released at time ${\displaystyle t=t_0}$ and location ${\displaystyle x=x_0}$, and ${\displaystyle D}$ is the diffusivity in units of meters²/second. This model makes the following four assumptions:

1. The mass ${\displaystyle M}$ is released instantaneously at time ${\displaystyle t}$.
2. The one-dimensional domain ${\displaystyle x}$ is unbounded, or infinite.
3. Mass is spread only through the mechanism of diffusion.
4. Diffusion, quantified with diffusivity ${\displaystyle D}$, is constant, or does not vary as a function of ${\displaystyle x}$.