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 :

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

and was 1.0 × 10^-6 cm⁴/s³ in the George et al. plume [3]. To compare to these empirical correlations, Van Maele & Merci used their results taken at a position, z = 1.75 m, equivalent to approximately 28 inlet diameters from the source. To demonstrate that this was sufficiently far from the source to produce self-similar profiles, the dimensionless velocity and temperature profiles were shown to be practically identical at a distance of 2.75 m.

Profiles of the mean axial velocity and buoyancy are compared to the empirical correlations in Figures 7 and 8. A summary of the centreline values and spreading rates for velocity and temperature are given in Table 5. This shows the measured values from the experiments of George et al. [3], the recommended values given by Chen & Rodi [1] from their analysis of plume experiments up to 1980, the subsequent measured values from Shabbir & George [11], the RANS results from Van Maele & Merci [2] and Hossain & Rodi, and the LES results from Zhou et al. [28]. Three results are taken from Hossain & Rodi: Case A which is from a k – ε model without any buoyancy modifications, Case B which is from a k – ε model with buoyancy corrections in both k and ε equations, and Case C which is from Hossain & Rodi’s algebraic stress/flux model.

For the Van Maele & Merci [2] results, both the standard and realizable k – ε models without buoyancy modifications (cases SKE and RKE) predicted overly large centreline velocity and buoyancy values and under-predicted the spreading rates of the plume. When the SGDH source term was used, with or without S_εB , it had practically no effect on the results. This may have been partly due to the particular choice of the constant c_ε3 which made the contribution from the buoyancy production small relatively to shear production in the ε-equation. The SKE, SKE_A and SKE_A* cases all returned very similar predictions to each other and RKE, RKE_A and RKE_A* cases behaved similarly. This was shown by Van Maele & Merci to be a consequence of the SGDH source term being negligible in comparison to other terms in the k and ε equations.

Table 5 shows that for the basic k – ε model without any buoyancy corrections there are significant differences in terms of the predicted spreading rates between the results of Van Maele & Merci [2] and Hossain & Rodi [8] (cases SKE and A, respectively). Van Maele & Merci’s predictions of the spreading rates are nearly 20% higher. This may have been due to the choice of non-standard model constants by Hossain & Rodi [8] who used c_μ = 0.109 and σ_t = 0.614, whilst Van Maele & Merci [2] used c_μ = 0.09 and σ_t = 0.85. The results for the k – ε model with buoyancy corrections are more similar (Cases SKE_A and B, respectively). Here, in addition to the non-standard c_μ and &sigma_t constants, Hossain & Rodi [8] used a model with constant c_ε3 scaled by the Richardson number, which was zero in the vertical buoyant plume, such that the buoyancy production had the same weight as the shear production.

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Test Case Studies

Evaluation

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References

Contributed by: Simon Gant — Lea Associates

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