Unsteady Near-Field Plumes

Underlying Flow Regime 1-07

Best Practice Advice

Best Practice Advice for the UFR

Key Physics

The key physics of this UFR is the transient, unsteady behaviour in the near-field of a turbulent buoyant helium-air plume. The flow features two key instabilities. Firstly, the Rayleigh-Taylor instability related to the presence of dense fluid above less-dense fluid, which gives rise to fingers or spikes of dense fluid separated by rising bubbles of lighter fluid. Secondly, the Kelvin-Helmholtz instability related to the shear-layer interface between the rising plume and the ambient fluid, which produces roll-up vortex sheets on the boundary between the two layers of fluid travelling at different velocities. The flow is very challenging to predict using CFD, due to the sharp density gradients at the plume exit which produce flow conditions where small scales of turbulent motion feed into the larger scales.

Numerical Modelling

For LES, the flow cannot be treated as two-dimensional or axisymmetric. Full three-dimensional time-dependent simulations must be performed.

For simulation of the selected UFR test case, open boundaries should be used on all sides of the flow domain except for the floor. Constant pressure boundaries may be used, although if a fully-compressible code is used, care will need to be taken to ensure that the boundaries are non-reflective.

For simulation of the selected UFR test case, the domain should extend at least 4 metres radially and vertically to minimize any effects of the open boundaries on the development of the plume. Ideally, tests should be performed to ensure that the location of the open boundaries has no significant effect on the results.

The finest mesh should be used given the available computing resources. The results discussed above suggest that a mesh of around 4 million nodes should give good agreement with the experiments in terms of mean flow quantities, but may still be insufficient for good predictions of fluctuations or RMS values. Tieszen^[1] noted that at least 75 cells across the base diameter of the plume are necessary to avoid significant differences in the vertical centreline velocity compared to the measured values. Ideally, a grid-dependence study should be undertaken to investigate the magnitude of these effects.

Physical Modelling

Either the fully-compressible or the low-Mach-number form of the Navier-Stokes equations can be used. The fully-compressible N-S equations require more careful treatment to avoid acoustic waves reflecting back into the domain from open boundaries. Furthermore, they will require a very short time-step, based on the speed of sound instead of the local flow speed, unless special treatments are used. For details of a fully-compressible N-S treatment, see DesJardin et al. [1].

The baroclinic torque is non-zero and therefore should not be neglected.

The Boussinesq approximation, where flow properties are assumed not to vary as a function of temperature or composition, and where buoyancy is only included as an additional body-force term in the momentum equations, should not be used. The Boussinesq approximation is only appropriate for modelling small density differences, equivalent to a temperature difference less than around 15°C in air [80].

If an LES approach is used, the effect of the unresolved small-scale turbulence on the resolved motion can either be accounted for by an explicit sub-grid-scale model, such as the dynamic Smagorinsky model, or by numerical damping in an implicit LES (a “no-model” approach). If an explicit approach is taken, central differencing should ideally be used for convection in the momentum equation but bounded upwind-biased schemes will probably be needed for the scalar equations to prevent unbounded under/overshoots. DesJardin et al. [1] obtained slightly better results with the implicit approach but this is likely to depend on grid resolution. If a coarse grid is used, an explicit LES model should probably be avoided. Both implicit and explicit approaches should ideally be tested to examine the sensitivity of results to the turbulence treatment. Recent work by Blanquart & Pitsch [40] has shown very good predictions for both mean momentum and concentration using the Lagrangian dynamic SGS model of Meneveau et al. [51] for turbulent diffusion terms in both the momentum and helium mass-fraction transport equations.

It is difficult to provide definitive guidance on use of RANS models, since to date it appears that there have only been two relevant studies for this flow, and they produced somewhat contradictory results. Chung & Devaud [39] found in the helium plume experiments of O‘Hern et al. [4] that steady flow behaviour was obtained using k – ε models with SGDH or GGDH and values of the model constant C_ε3 varying between 0 and 1. They also found that assuming the flow to be axisymmetric or using a fully three-dimensional approach gave practically identical results and an axisymmetric approach with 22,882 cells gave a grid-independent solution. In contrast, Nicolette et al. [38] found a “standard” k – ε model produced unsteady flow behaviour using all but the very coarsest of meshes, which had only 56,000 cells for the three-dimensional geometry. For meshes containing 500,000 to 2 million cells, the predicted flow behaviour was unsteady with the finer meshes resolving an increasing proportion of the unsteady flow structures. These differences in resolving steady or unsteady flow behaviour could in part be due to the former study using a steady solution method whilst the latter used a transient time-stepping approach. Nevertheless, Chung & Devaud [39] reported that the residuals in their steady simulations could be reduced to low levels (maximum residuals of 10^-5), whilst usually in flows where there is a tendency towards transient behaviour it is difficult to obtain such good convergence.

\textmd{\textup{Putting these differences to one side, the }}\textmd{\textup{Chung \& Devaud }}\textmd{\textup{[39]}}\textmd{\textup{ study showed that good predictions of the steady flow behaviour in the near{}-field of buoyant plumes could be achieved provided that special care was taken over the choice of the model constant, }}\textmd{\textit{C}}\textmd{\textit{\textsubscript{${\varepsilon}$}}}\textmd{\textup{\textsubscript{3}}}\textmd{\textup{. Different optimum values of }}\textmd{\textit{C}}\textmd{\textit{\textsubscript{${\varepsilon}$}}}\textmd{\textup{\textsubscript{3}}}\textmd{\textup{ were found when using either SGDH or GGDH, and varying the value of }}\textmd{\textit{C}}\textmd{\textit{\textsubscript{${\varepsilon}$}}}\textmd{\textup{\textsubscript{3}}}\textmd{\textup{ produced very marked changes in the flow predictions. It is recommended therefore if studying the near{}-field flow behaviour of plumes using similar models to examine the sensitivity of the results to this parameter. \ }}

\textmd{\textup{The Nicolette }}\textmd{\textit{et al}}\textmd{\textup{. }}\textmd{\textup{[38]}}\textmd{\textup{ study mainly focused on testing their newly{}-developed Buoyant Vorticity Generation (BVG) extension to the }}\textmd{\textit{k}}\textmd{\textup{ {--} }}\textmd{\textit{${\varepsilon}$}}\textmd{\textup{ model. The model showed promising results in comparison to the helium plume experiments from NIST }}\textmd{\textup{[31]}}\textmd{\textup{[32]}}\textmd{\textup{ and the Sandia FLAME facility with a low helium inlet velocity of 0.13 m/s. However, the model gave less }}\textmd{\textup{encouraging predictions when compared to the O{\textquotesingle}Hern }}\textmd{\textit{et al}}\textmd{\textup{. }}\textmd{\textup{[4]}}\textmd{\textup{ experiments where the inlet velocity was higher, due to the delayed predicted onset of laminar to turbulent transition. There were also additional complications with mesh{}-dependent transient behaviour, as mentioned above.}}

{\mdseries\upshape Overall, the two studies indicate that further work is needed before definitive best{}-practice advise can be provided on the use of RANS models in the near{}-field of buoyant plumes. }