56
Air Pollution and Turbulence: Modeling and Applications
Considering a nonisotropic decaying convective turbulence, the aim of this
section is to derive a formulation for the eddy diffusivities in a RL.
3.5.1.1 Energy Density Spectrum Dynamical Equation
It is possible to derive a spectral form of the turbulent energy equation from the
momentum conservation law, expressed through the NS equations. In the case of a
homogeneous turbulent fl ow, the spectral form of the turbulent energy equation is
(Hinze, 1975; Stull, 1988):
2
0
( , ; )
( , ; )
( , ; )
( , ; ) 2
( , ; )
E k t z
g
M k t z W k t z
H k t z
k E k t z
t
T
∂
=
+
+
− ν
∂
(3.80)
where
t is the time
g/T 0 is the buoyancy parameter
k is the wave number
z is the height above the ground
E(k,t;z) is the 3-D energy density spectrum
W(k,t;z) is the transport term composite of the energy-transfer-spectrum function
that represents the contribution due to the inertial transfer of energy among
different wave numbers or the time-rate-of-change per unit wave number of
the energy spectrum and pressure–velocity correlation
M(k,t;z) is the energy production by mechanical (shear) effects
H(k,t;z) is the production or loss due to buoyancy contribution, and the last term
on the right-hand side of Equation 3.80 is the energy loss due to viscous
dissipation
In this chapter, we take into account the case in which the inertial energy-transfer
terms are important. Under this condition, Equation 3.80 becomes
2
( , )
( , ) 2
( , )
E k t W k t
k E k t
t
∂
=
− ν
∂
(3.81)
A turbulent fl ow contains eddies of different size or different wavelengths. The small
eddies are subjected to the stress generated by large eddies. This fi eld increases the vorticity of small eddies and, consequently, their kinetic energy. Thus, TKE is transferred
from large eddies toward smaller and smaller eddies until the Kolmogorov microscale
is reached, where the energy is dissipated as heat. This process is represented by the
term W(k,t) of Equation 3.81. This term was parameterized according to Pao (1965) for
a turbulent isotropic fl ow on the basis of dimensional analysis, as follows:
(
)
−
∂
= −
α ε
∂
1 1/3 5/3
( , )
( , )
W k t
k E k t
k
(3.82)
where
α is the Kolmogorov constant
ε is the rate of molecular dissipation of TKE
© 2010 by Taylor and Francis Group, LLC
Air Pollution and Turbulence: Modeling and Applications
Considering a nonisotropic decaying convective turbulence, the aim of this
section is to derive a formulation for the eddy diffusivities in a RL.
3.5.1.1 Energy Density Spectrum Dynamical Equation
It is possible to derive a spectral form of the turbulent energy equation from the
momentum conservation law, expressed through the NS equations. In the case of a
homogeneous turbulent fl ow, the spectral form of the turbulent energy equation is
(Hinze, 1975; Stull, 1988):
2
0
( , ; )
( , ; )
( , ; )
( , ; ) 2
( , ; )
E k t z
g
M k t z W k t z
H k t z
k E k t z
t
T
∂
=
+
+
− ν
∂
(3.80)
where
t is the time
g/T 0 is the buoyancy parameter
k is the wave number
z is the height above the ground
E(k,t;z) is the 3-D energy density spectrum
W(k,t;z) is the transport term composite of the energy-transfer-spectrum function
that represents the contribution due to the inertial transfer of energy among
different wave numbers or the time-rate-of-change per unit wave number of
the energy spectrum and pressure–velocity correlation
M(k,t;z) is the energy production by mechanical (shear) effects
H(k,t;z) is the production or loss due to buoyancy contribution, and the last term
on the right-hand side of Equation 3.80 is the energy loss due to viscous
dissipation
In this chapter, we take into account the case in which the inertial energy-transfer
terms are important. Under this condition, Equation 3.80 becomes
2
( , )
( , ) 2
( , )
E k t W k t
k E k t
t
∂
=
− ν
∂
(3.81)
A turbulent fl ow contains eddies of different size or different wavelengths. The small
eddies are subjected to the stress generated by large eddies. This fi eld increases the vorticity of small eddies and, consequently, their kinetic energy. Thus, TKE is transferred
from large eddies toward smaller and smaller eddies until the Kolmogorov microscale
is reached, where the energy is dissipated as heat. This process is represented by the
term W(k,t) of Equation 3.81. This term was parameterized according to Pao (1965) for
a turbulent isotropic fl ow on the basis of dimensional analysis, as follows:
(
)
−
∂
= −
α ε
∂
1 1/3 5/3
( , )
( , )
W k t
k E k t
k
(3.82)
where
α is the Kolmogorov constant
ε is the rate of molecular dissipation of TKE
© 2010 by Taylor and Francis Group, LLC