UFR 1-06 Evaluation

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Axisymmetric buoyant far-field plume in a quiescent unstratified environment

Underlying Flow Regime 1-06

Comparison of CFD Calculations with Experiments

Van Maele & Merci [2] presented the results from a number of simulations that examined the effects of different combinations of models and approximations. The Boussinesq approximation $({\overline {\rho }}\approx \rho _{\infty })$ was shown to have no affect on the model predictions when the SGDH model was used. Indeed, the SGDH source term itself had a negligible influence on the results. When using the GGDH source term, however, the Boussinesq approximation had an effect on the results nearest to the plume exit at z/D = 12, where the assumption of $({\overline {\rho }}\approx \rho _{\infty })$ caused an increase in the peak velocity of around 5%. By assuming that the mean density was the same as the reference density, the buoyancy source term became smaller and so the turbulent kinetic energy and hence the eddy viscosity were also smaller. As a consequence, there was less mixing, the centreline velocity increased and the spreading rate decreased. The effect was significant where the mean density differed most from the reference density, nearest to the plume source, but was negligible in the far field. These results suggest that the Boussinesq approximation can be used in the far field of buoyant plumes where density differences are small. However, if the CFD domain extends from the far field to the source of buoyancy, such as a fire or strongly heated surface where density differences are appreciable, then the Boussinesq approximation should not be used.

It should also be remembered that Van Maele & Merci’s interpretation of the ‘Boussineq approximation’ only involved setting $({\overline {\rho }}\approx \rho _{\infty })$ in the production term, G. The density and other flow properties (molecular viscosity, specific heat etc.) still varied as a function of temperature elsewhere in the transport equations.

Van Maele & Merci examined the effects of SGDH versus GGDH and the effect of switching on and off both the production due to buoyancy term, G, and the source term in the ε–equation, S_εB, on the standard and realizable k – ε models. Table 4 summarizes the cases tested. In the relevant cases, they used the full buoyancy source term G rather than any truncated form of the equation. The ε–equations were different for standard and realizable models, but in both cases, where used, the buoyancy-related source term was given by:

$\ \ \ \ \ \ \ \ S_{\varepsilon B}=c_{\varepsilon 1}\left(1-c_{\varepsilon 3}\right){\frac {\varepsilon }{k}}G\ \qquad {\text{with}}\ \ c_{\varepsilon 3}=0.8\ \ \ \ \ \ (35)$

The results were compared to the experimental data of George&'et al. [3] and the correlations of Shabbir & George [11] which were given by:

$\ \ {\text{Axial velocity:}}\ \ \ \ \ \ WF_{0}^{-1/3}z^{1/3}=3.4\exp \left(-58\eta ^{2}\right)\ \ \ \ \ \ \ \ (36)$

$\ \ {\text{Buoyancy:}}\ \ \ \ \ \ g\beta \Delta TF_{0}^{-2/3}z^{5/3}=9.4\exp \left(-68\eta ^{2}\right)\ \ \ \ \ \ (37)$

where W is the mean axial momentum, ΔT is the difference between the local mean and ambient temperatures, z is the vertical distance from the source, η = r/z is the similarity variable (r is the radial distance from the plume centreline), and β the thermal expansion coefficient. The buoyancy added at the source, F₀ , is found from :

$F_{0}=2\pi \int _{0}^{r_{0}}Wg{\frac {\delta \rho }{\rho _{\infty }}}r{\text{d}}r\ \ \ \ \ \ \ \ \ \ (38)$

Front Page

Description

Test Case Studies

Evaluation

Best Practice Advice

References

Contributed by: Simon Gant — Lea Associates

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