# Confined buoyant plume

## Discretisation

The influence of discretisation was not examined in the source study underlying this report.

## Grid and grid resolution

Grid resolution in the region occupied by the plume was found to be an crucial factor affecting model accuracy. It was found possible to obtain reasonable, grid-independent results, with mesh resolution in this region of around 0.01 the smallest horizontal dimension of the room.

## Boundary conditions

It is important that the walls of the enclosure are modeled as adiabatic, no-slip boundaries, to reproduce the flow considered here. The conditions at the openings to the enclosure can be specified by including an exterior space with pressure boundaries on all sides. The pressure boundaries should be at least one enclosure height away from the floor and ceiling in the vertical direction, and at or beyond the walls of the room in the vertical. An alternative approach is not to include an exterior domain but in the mesh to reduce the are of the openings by a factor equal to the expected discharge coefficient. This approach has been used successfully by Cook et al (2003).

## Computational domain

As discussed above, the computational domain must extend for some distance beyond the boundaries of the room if the discharge characteristics of the openings are to be modeled explicitly. Otherwise, if the discharge coefficients are known, these can be incorporated without need to model the external region by reducing the area of the openings by a factor equal to the expected discharge coefficient.

Within the context of the parameter ranges considered here, the orientation, location and number of the openings should not affect the character of the results (providing they are contained wholly within each layer and the upper opening height is included as the datum in the theoretical model). However, in some circumstances discharge coefficients may be affected by buoyancy effects and buoyancy-driven exchange at the upper openings. Mixing may also results across the interface heights as a consequence of these effects, or if the area of the lower opening is relatively small and the momentum of the incoming flow is correspondingly high.

## Turbulence model general domain

The RNG turbulence model was found to given better results than the "standard" k – ε model, although the results of the latter were acceptable. It should be ensured that the appropriate buoyancy production terms are included in the k equation for both models, as given in the appendix.

## Near wall model

Standard wall functions should be included at the solid boundaries.

## Near wall model

Obtaining satisfactory convergence was found to be a crucial aspect for the successful modeling of the flow. The flow was successfully modeled as a steady-state, but a pseudo-transient solution approach was found to be necessary for the later solution iterations in order to obtain satisfactory convergence.

## Application uncertainties

The comparisons with the salt bath models discussed previously indicate that validation can be carried out through comparison of "full size" model results directly with the results of the salt bath experiments, or by comparison with the theoretical model, providing the appropriate nondimensionalisations have been made. The principle variables compared are the interface height and two layer stratification strength. However, it should be noted that within these comparisons there exist a number of uncertainties:

1) The interface will not be perfectly sharp, and the measurement of its location can be subjective.

2) The measurement of upper layer density should be made some distance from the plume and interface, where the fluid is relatively well mixed.

3) The discharge coefficients for the openings are a crucial aspect of the flow and must be acquired in order to compare nondimensional results. Principally, the discharge coefficient values are required to evaluate the governing non-dimensional parameter value of A*, given by equation (1). If discharge coefficients from standard tables are to be used, it is recommended that, as a precautionary measure, it be checked that paramater values are not such that the discharge coefficient for the upper openings are affected by buoyancy effects (Hunt & Holford 2000, Holford & Hunt 2001).

4) A virtual origin correction needs to be made before comparing results. This can be done following the guidance given by Hunt & Kaye (2001).

5) The value of the entrainment constant, α, can also vary from situation to situation. The value of the entrainment constant realized in the CFD simulations may not be the same as that in the salt-bath models, and this does not necessarily point to inaccuracies in the CFD model, if not exactly the same situation is being considered. At present there is no method available for determining the value of α a priori. It can either be obtained through post-analysis of data on a case-by-case basis or a representative value chosen from the literature

6) In cases where the vents are in the walls of the space, the effective height of the space is reduced and a corresponding correction needs to be made if using the theoretical formulae (see Hunt & Linden 2001).

7) It is not guaranteed that the two-layer displacement flow will be realized if parameter values are such that significant mixing across the interface occurs or two-way flow develops at the upper opening. These effects will lead to breakdown of the two layer stratification and quantitative or qualitative changes in the dynamics.

## Suggestions for further work

The studies of Cook & Lomas reported here have demonstrated how the salt-bath two-layer displacement flow can be reproduced in CFD simulations, and have established a methodology for benchmarking such simulations against the salt-bath experiment data and theoretical model.

The studies of Cook & Lomas were not designed to given detailed validation or assessment of particular turbulence models. Two commonly used k – ε were implemented and were found to given satisfactory results. Further work is required to see if additional benefits can be obtained by employing more computationally intensive turbulence models, such as second-order closure Reynolds stress or LES models. The scope of the data obtained from the salt-bath models does not lend itself well to detailed examination of turbulence model accuracies. It is recommended that further experimental work and direct simulation of the salt-bath experiments and original scale would be required to investigate these issues.

Beyond the context of the work considered here, there remain additional effects that are important in real building displacement ventilation flows, in particular the effects of radiative heat transfer and conduction of heat through non-adiabatic surface. Further work to examine the nature of these effects and associated modelling challenges would also be of benefit.

Acknowledgements. We are grateful to Dr. Malcolm Cook (Institute for Energy and Sustainable Development, De Montfort University, Leicester, UK), Dr. Gary Hunt (Department of Civil and Environmental Engineering, Imperial College London, UK) and Professor P.F. Linden (Department of Applied Mechanics and Engineering Science, University California San Diego) for providing the source material for this report and for many useful discussions regarding the content.

## Appendix: details of turbulence models

NOTATION:

gi    gravity vector, component i
${\displaystyle {g_{i}^{'}=g(\rho -\rho _{0})/\rho _{0}}}$   reduced gravity
h   enthalpy
k    turbulence kinetic energy
p    fluid pressure
${\displaystyle {p'=p+{\tfrac {2}{3}}\rho k-\rho _{0}g_{i}x_{i}}}$
ui    fluid velocity, component i
xi    Cartesian spatial coordinate , component i
C1, C2, Cμ    model coefficients
Cp    specific heat
P    shear generated production of turbulent kinetic energy
G    buoyancy generated production of turbulent kinetic energy
β    thermal expansion coefficient
ε    turbulent energy dissipation rate
λ    thermal conductivity
μ    viscosity
μT    turbulent eddy viscosity
μeff = μ + μT    effective viscosity
ρ    fluid density
ρ0    reference density (for Boussinesq approximation)
σh    turbulent Prandtl number for h
σk    turbulent Prandtl number for k
σT    general turbulent Prandtl number
σε    turbulent Prandtl number for ε

The two turbulence models used in the simulations reported here — the ‘standard’ and RNG turbulence models — are widely used for modelling of buoyancy driven flows in buildings and have essentially similar implementations in the commercial CFD codes. For completeness, the implementation used in the CFX package used in the reported simulations is here.

The conservation equations for momentum, mass and enthalpy are:

 ${\displaystyle \displaystyle {{\frac {\partial }{\partial x_{j}}}u_{j}u_{i}=-{\frac {1}{\rho _{0}}}{\frac {\partial }{\partial x_{j}}}p^{\prime }+g_{i}^{\prime }+{\frac {1}{\rho _{0}}}{\frac {\partial }{\partial x_{j}}}\mu _{\text{eff}}{\frac {\partial u_{i}}{\partial x_{j}}}}}$ ${\displaystyle \left(A1\right)}$

 ${\displaystyle \displaystyle {u_{j}{\frac {\partial }{\partial x_{j}}}\rho =0}}$ ${\displaystyle \left(A2\right)}$

 ${\displaystyle \displaystyle {{\frac {\partial }{\partial x_{j}}}u_{j}h={\frac {\partial }{\partial x_{j}}}\left({\frac {\lambda }{C_{p}}}+{\frac {\mu _{T}}{\sigma _{m}}}\right){\frac {\partial h}{\partial x_{j}}}}}$ ${\displaystyle \left(A3\right)}$

The turbulent eddy viscosity is given by:

 ${\displaystyle \displaystyle {\mu _{T}=\rho C_{\mu }{\frac {k^{2}}{\varepsilon }}}}$ ${\displaystyle \left(A4\right)}$

The values of k and ε are given by the solution of the respective transport equations, based on the model of Launder & Spalding (1974):

 ${\displaystyle \displaystyle {{\frac {\partial }{\partial x_{j}}}\rho \mu _{j}k-{\frac {\partial }{\partial x_{j}}}\left(\mu +{\frac {\mu _{T}}{\sigma _{k}}}\right){\frac {\partial k}{\partial x_{j}}}=P+G-\rho \varepsilon }}$ ${\displaystyle \left(A5\right)}$

 ${\displaystyle \displaystyle {{\frac {\partial }{\partial x_{j}}}\rho \mu _{j}\varepsilon -{\frac {\partial }{\partial x_{j}}}\left(\mu +{\frac {\mu _{T}}{\sigma _{k}}}\right){\frac {\partial \varepsilon }{\partial x_{j}}}=C_{1}{\frac {\varepsilon }{k}}P-C_{2}\rho {\frac {\varepsilon ^{2}}{k}}}}$ ${\displaystyle \left(A6\right)}$

where the shear and buoyancy induced turbulent energy production terms, P and G, are given by:

 ${\displaystyle \displaystyle {P=\mu _{\text{eff}}{\frac {\partial u_{i}}{\partial x_{j}}}\left({\frac {\partial u_{i}}{\partial x_{j}}}+{\frac {\partial u_{j}}{\partial x_{i}}}\right)}}$ ${\displaystyle \left(A7\right)}$

 ${\displaystyle \displaystyle {G={\frac {\mu _{\text{eff}}}{\sigma _{T}}}\beta g_{j}{\frac {\partial T}{\partial x_{j}}}}}$ ${\displaystyle \left(A8\right)}$

‘Standard’ k – ε model:

The standard model makes use of (A1) ­– (A8) with values for the empirical coefficients of:

C1 = 1.44
C2 = 1.92
Cμ ­= 0.09
σk = 1.0
σε = 1.217
σh = 0.9

These values are those given by Launder & Spalding (1974) with the exception of σε which in the original model was equal to 1.3. In the CFX implementation, the value of this parameter is obtained from:

${\displaystyle {\sigma _{\varepsilon }={\frac {\kappa ^{2}}{(C_{2}-C_{1}){\sqrt {C_{\mu }}}}}}}$

where κ = 0.4187 is the von Karman constant.

RNG k – ε model:

The RNG model also makes use of (A1) ­– (A8) with the same values for the empirical coefficients as in the standard model, with the exception ofCμ ­ and C1 which are given by:

Cμ ­= 0.085 ,

and

${\displaystyle {\left.C_{1}=1.44-\eta (1-\eta /\eta _{0})(1+\zeta _{0})\right.}}$

in which:

${\displaystyle {\eta =(P/\mu _{T})^{\tfrac {1}{2}}\ k/\varepsilon }}$
η0= 4.38 ,
ζ0= 0.015 .

The essential difference between the standard and RNG models is that in the latter the coefficient C1 becomes a dynamic variable computed at each timestep and note that is only the turbulent dissipation, ε , equation that is affected by this.