Advanced Building Simulation

the Reynolds average rules can be summarized as:
ui � Ui � Ui, ui� � 0, ui�U j � 0,
ui � uj � Ui � Uj, uiuj � UiUj � ui�u j�
(5.8)
Note that the bars in Equation 5.8 stand for “statistical average” and are different
from those used for LES. In LES, those bars represent grid filtering.
By applying the Reynolds averaging method to the Navier–Stokes and continuity
equation, they become:
�U i
�t
�U i
�x i
(5.9)
(5.10)
where ui�u j� is the Reynolds stress that is unknown and must be modeled. In the last
century, numerous turbulence models have been developed to represent ui�u j�.
Depending on how the Reynolds stress is modeled, RANS turbulence modeling can
be further divided into Reynolds stress models and eddy-viscosity models. For simplicity,
this chapter discusses only eddy-viscosity turbulence models that adopt the
Boussinesq approximation (1877) to relate Reynolds stress to the rate of mean stream
through an “eddy” viscosity �t. (5.11)
where �ij is the Kronecker delta (when i≠j, �ij�0; and when i�j, �ij�1), and k is the
turbulence kinetic energy (k � ui�u i�/2) . Among hundreds of eddy-viscosity models,
the standard k–� model (Launder and Spalding 1974) is most popular. The standard
k–� model solves eddy viscosity through
(5.12)
where C ��0.09 is an empirical constant. The k and � can be determined by solving
two additional transport equations:
where
u i�u j� �
�k
Uj �xj ��
Uj �xj �Ui � Uj �xj �
�u i�
�x i
k
�t � C� 2
�
�
�
P � �
1
t
2
� 0
2
�
3
ijk � �t� �Ui �xj �
�x j�� � �
�
�x j�� � �
� �U i
�x j
� � 1 �
�
�P
�x i
� t
� k� �k
�x j� � P � �
� t
� �� ��
�x j� � [C �1P � C �2�] �
k
�U j
�x i� 2
�
�
�Uj �xi� CFD tools for indoor environmental design 123
�
�xj�� �Ui �xj � u i�u j��
(5.13)
(5.14)
(5.15)