Advanced Building Simulation

122 Chen and Zhai
5.2.1 Large-eddy simulation
By filtering the Navier–Stokes and continuity equations in the LES approach, one
would obtain the governing equations for the large-eddy motions as
�u i
�(t)
�u i
�x i
�
(5.1)
(5.2)
where the bar represents grid filtering. The subgrid-scale Reynolds stresses, � ij, in
Equation (5.1),
(5.3)
are unknown and must be modeled with a subgrid-scale model. Numerous subgridscale
models have been developed in the past thirty years. The simplest and probably
the most widely used is the Smagorinsky subgrid-scale model (Smagorinsky 1963)
since the pioneering work by Deardorff (1970). The model assumes that the subgridscale
Reynolds stress, � ij, is proportional to the strain rate tensor,
S ij �
� 0
�
(u
�x iuj) � �
j
� ij � u iu j � u iu j
1
2� �ui �xj �
� ij ��2� SGSS ij
�uj �xi� where the subgrid-scale eddy viscosity, � SGS, is defined as
� SGS � (C SGS�) 2 (2S ij·S ij) 1/2
1 �
�p
�x i
�
� �
2 ui �
�xj �xj � C� 2 (2S ij·S ij) 1/2
(5.4)
(5.5)
(5.6)
The Smagorinsky constant, C SGS, ranges from 0.1 to 0.2 determined by flow types,
and the model coefficient, C, is the square of C SGS. The model is an adaptation of the
mixing length model of RANS modeling to the subgrid-scale model of LES.
5.2.2 RANS modeling
Reynolds (1895) introduced the Reynolds-averaged approach in 1895. He decomposed
the instantaneous velocity and pressure and other variables into a statistically
averaged value (denoted with capital letters) and a turbulent fluctuation superimposed
thereon (denoted with � superscript). Taking velocity, pressure, and a scale
variable as examples:
u i � U i � u i�, p � P � p�, � � � � ��
(5.7)
The statistical average operation on the instantaneous, averaged, and fluctuant
variables have followed the Reynolds average rules. Taking velocity as an example,
��ij �xj