Advanced Building Simulation

150 Addington
Gr/Re 2 �� 1, then buoyancy dominates. The last condition is the most common for
buoyant flows in a quiescent ambient, and as such, boundary layer effects become
paramount. Furthermore, diffusion (conduction) from the boundary emerges as an
important determinant of the regime of the boundary layer. The Rayleigh number is
an indication of the balance between diffusive and buoyant forces: the higher the
Rayleigh number, the more likely the boundary layer is turbulent.
Rayleigh number (Ra) � ��g (Ts � Tf) L3 (ratio of buoyancy force to diffusion)
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Generally accepted ranges of the Rayleigh number for buoyant flow in confined
spaces are (Pitts and Sissom 1977):
● conduction regime Ra � 10 3
● asymptotic flow 10 3 � Ra � 3 � 10 4
● laminar boundary layer flow 3 � 10 4 � Ra � 10 6
● transition 10 6 � Ra � 10 7
● turbulent boundary layer flow 10 7 � Ra
Given the formulation of the Rayleigh number, it is evident that the typical buoyant
flow encountered in buildings will have multiple regimes within its boundary layer,
and each regime will have a significantly different heat transfer rate—represented by
the Nusselt number.
Nusselt number (Nu) � hL (ratio of convective transport to diffusive transport)
k
The most interesting characteristic of buoyant flow is that the characteristic length
is contingent on both the regime and the flow type, whereas in forced convection, the
length is a fixed geometric measure. Depending on the flow description, the characteristic
length may be determined by the height of a vertical isothermal surface or the
square root of the area of a horizontal surface. If isothermal vertical surfaces are
closely spaced, the characteristic length reverts to the horizontal spacing, and particularly
interesting relationships emerge if surfaces are tilted (Incropera 1988). As a
result, a large opportunity exists to manipulate the heat transfer from any buoyant
flow. For example, microelectronics cooling strategies depend heavily on the management
of characteristic length to maximize heat transfer to the ambient environment.
Clearly, an approach other than semiempirical turbulence models must be
found to accurately simulate the behavior of the boundary (Table 6.2).
Accurate modeling of buoyant flows thus requires discrete modeling of the boundary
layer. For transition regimes, normally occurring near and above Ra � 10 7 , Direct
Numerical Simulation (DNS) is the only recognized simulation method for determining
turbulence (Dubois et al. 1999). No empirical approximations or numerical simplifications
are used in DNS, rather the Navier–Stokes equations are solved at the
length-scale of the smallest turbulent behavior. Although DNS is commonly used in the
science community, it has found almost no application in the modeling of room air.
The number of nodes needed to discretize all scales of turbulence increases roughly as