# UFR 1-06 Best Practice Advice

# Axisymmetric buoyant far-field plume

Underlying Flow Regime 1-06

# Best Practice Advice

## Best Practice Advice for the UFR

### Key Physics

The key physics to be captured in this UFR is the self-similar behaviour of a spreading axisymmetric buoyant plume.

### Numerical Modelling Issues

- The flow can be treated as axisymmetric.
- For a grid-independent resolution of the George
*et al.*[3] plume, at least (40 × 100) grid nodes should be used in the (radial × axial) directions. At least 10 nodes should be used radially to resolve the plume source. - Discretization schemes should be at least second-order accurate.
- For further advice on boundary conditions, see Test Case

### Physical Modelling

- Use of the standard
*k – ε*model with or without the common Standard Gradient Diffusion Hypothesis (SGDH) for the production term due to buoyancy,*G*, will probably result in overprediction of the centreline mean parameters, and underprediction of the spreading rate of the plume. - To obtain more accurate plume predictions with a
*k – ε*model, use the Generalized Gradient Diffusion Hypothesis (GGDH) instead of the Standard Gradient Diffusion Hypothesis (SGDH). More refined models have been suggested which could further improve model predictions, see for example Hossain & Rodi [8]. - Do not neglect the buoyancy source term in the
*ε*-equation as this can lead to problems with numerical stability. - If you are considering only the far-field region of a buoyant plume, where density differences are small, the Boussinesq approximation can be used. If, however, your flow domain includes the region nearer the source of buoyancy where density differences are appreciable, avoid using the Boussinesq approximation.
- For cases where buoyancy is not as strong as in a plume, in the
limit of a non-buoyant axisymmetric jet, be aware of the limitations of
the standard
*k – ε*model. The spreading rate of a non-buoyant round jet is 15% lower than for a two-dimensional, plane jet. However, the standard*k – ε*model predicts the spreading rate for round jets to be 15%*higher*than for the plane jets [67].

### Application Uncertainties

In Van Maele & Merci’s calculations, they substituted
the streamwise normal stress in the GGDH expression for the production
due to buoyancy, *G,* with the turbulent kinetic energy. They
justified this ad-hoc modification on the basis that that the
streamwise normal stress was underpredicted using a linear *k – ε*
model and therefore using *k* was more
appropriate. However, this modification may not be appropriate in more
complex flows where the flow direction is not aligned to one of the
coordinate axes. The effect on Van Maele & Merci’s
results would have been to increase the buoyancy production term
*G*. One could therefore expect to see a slightly less
significant difference between SGDH and GGDH approaches without this
ad-hoc modification.

Van Maele & Merci also did not report any tests that had been performed
to assess the influence of the entrainment boundaries on the flow
predictions. Ideally, simulations should have been performed using a
smaller or larger domain to demonstrate that the presence of the
entrainment boundaries had no effect on the solution.

### Recommendations for further work

It is recommended that further experiments and a systematic
re-evaluation of available data be undertaken to establish with
confidence the self-similar behaviour of axisymmetric buoyant plumes,
similar to the exercise undertaken 30 years ago by Chen & Rodi [1]
Both List [6]
and Dai *et al.* [10]
have called into question whether the measurements by
George *et al.* [3] were carried out sufficiently far
from the source, in the region where self-similar behaviour exists.
This issue was, however, dismissed by
Shabbir & George [11][34].

Contributed by: **Simon Gant** — *UK Health & Safety Laboratory*

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