Axisymmetric buoyant far-field plume in a quiescent unstratified environment

Underlying Flow Regime 1-06

Test Case

Brief Description of the Study Test Case

The experiments used in this UFR are those of George et al. [3] which were conducted in 1974 at the Factory Mutual Research Corporation and were subsequently repeated by Shabbir & George [34] at the University of Buffalo.

Heated air is discharged through a circular orifice into ambient air that is at rest.
The plume source temperature is 300°C and the ambient air is 29°C.
The source has diameter, D = 6.35 cm.
The hot air is discharged at a velocity of U0 = 67 cm/s with a approximately a top-hat profile.
Temperature and velocity fluctuations at the inlet are less than 0.1%.
George et al. [3] present experimentally measured profiles

of both mean and fluctuating components of the temperature and axial velocity in the self-similar region at x/D = 8, 12 and 16 above the source.

Test Case Experiments

The experiments used in this UFR are those of George et al.�[gat77] which were conducted in 1974 at the Factory Mutual Research Corporation and were subsequently repeated by Shabbir & George [sg92] at the University of Buffalo.

The general arrangement is shown in Figure 4. Compressed air is passed through a set of heaters and porous mesh screens before exiting through a nozzle into the enclosure. The nozzle is stated as a 15:1 contraction in [gat77], a 12:1 contraction in [sg92] and appears to be different again in a drawing of the arrangement in [gat77] (see Figure 5). It resulted in a velocity profile through the exit which was uniform to within 2% outside the wall boundary layer. The velocity and temperature fluctuations at the exit were measured to be very low, less than 0.1% in [gat77] and 0.5% in [sg92]. The temperature of the source was 300�C and the ambient environment 29�C. Both were controlled to an accuracy of within 1�C. The discharge velocity was 67 cm/s, as calculated from the measured heat flux. These source conditions corresponded to Reynolds number, Re₀ = 870, and densimetric Froude number, Fr₀ = 1.23[#sdfootnote1sym ¹]. There was no evidence of laminar flow behaviour at a position two inlet diameters downstream from the source. The effective origin of the plume, x₀, was found to be at the same location as the exit (see [gat77] for details of how this was determined).

The screen enclosure around the plume exit was 2.44 � 2.44 metres in cross-section and 2.44 metres high (there is, presumably, an error in [gat77] which suggests that the enclosure is 2.44 � 2.44 � 2.44 mm). In the later Shabbir & George experiments, a 2 � 2 � 5 metre enclosure was used. The purpose of the screens was to minimize the effect of cross-draughts and other disturbances affecting the flow. Two-wire probes were used by George et al.�[gat77] to record velocities and temperature.

[#sdfootnote1anc 1]The densimetric Froude number is calculated here from the source and ambient temperatures, the exit velocity and source diameter given by George et al. [gat77], using Equation (1). However, George et al. [gat77] stated that the densimetric Froude number was 1.4. It is unclear how they determined this value. Using the approach taken by Chen & Rodi [cr80] in which the source density instead of the ambient density is used to make the density difference dimensionless, and Froude number is defined using the square of the expression given in Equation (1), this gives a Froude number of 0.80.

CFD Methods

${\frac {\partial }{\partial x_{j}}}\left({\overline {\rho }}U_{j}\varepsilon \right)={\frac {\partial }{\partial x_{j}}}\left[\left(\mu +{\frac {\mu }{\sigma _{\varepsilon }}}\right){\frac {\partial \varepsilon }{\partial x_{j}}}\right]+c_{\varepsilon 1}{\overline {\rho }}S{\frac {\varepsilon }{k}}-c_{\varepsilon 2}{\overline {\rho }}{\frac {\varepsilon ^{2}}{k+{\sqrt {\nu \varepsilon }}}}+S_{\varepsilon B}$