or for the surface boundary layer (Kaimal and Finnigan 1994):
@ u 0
3 h
0
@t
% Àu 0
3
2 @h
@x 3
þ
gh
02
h m
À
@ h
0 u 02
3
@x 3
À
1
q
h
0 @p 0
@x 3
!
I
II
III
IV
V
ð3:87Þ
Equation (3.87) is like Eq. (3.77) where term II represents the production of
shear stresses; term III is the production by buoyancy; term IV is the turbulent
transport by tangential tensions and term V refers to pressure destruction.
As with Eq. (3.77), the elimination of the correlations is mainly due to pressure
forces, so the viscous dissipation component can be neglected in this equation.
Likewise, term II for turbulent transport in the surface boundary layer is low. Thus,
the final budget relates to term V for pressure destruction and terms of production of
turbulence II, III mechanical and buoyancy, respectively. For the vertical flow of
another scalar, such as carbon dioxide or moisture, the algebraic developments are
analogous, and equations like Eqs. (3.86) and (3.87) can be obtained.
3.5.6 Kinetic Energy Budget Equations
The TKE budget, expressed per unit mass, makes it possible to characterize the
context of the production processes, transport, and loss of turbulent fluctuations. As
previously mentioned, TKE is a measure of turbulence intensity. In the inertial
sublayer, TKE is mechanically derived from shear stresses in the mean flow and
thermally produced by buoyancy forces. This kinetic energy is transferred via the
inertial cascade from larger to smaller eddies and converted into heat by viscous
dissipation.
From Eq. (3.71) on the budget for the variance for velocity fluctuations, the
kinetic energy equations can be deduced (Stull 1994; Tennekes and Lumley 1980):
@e
@t
þ u j
@e
@t
¼
I
II
¼ d i3
g
h V
u 0
i h
0
V
À
@ u 0
j e
@x j
À
1
q
@ u 0
i p 0
ð Þ
@x i
À 2u 0
i u 0
j
@u i
@x j
À e
III
IV
V
VI
VII
ð3:88Þ
where e is the turbulent kinetic energy per unit mass given by 0:5 u
2
1 þ u
2
2 þ u
2
3
À
Á
. In
this equation, term I is the rate of storage of kinetic energy; term II is the advection
by the average kinetic energy field speed; term III is the term production or consumption by buoyancy. This term is either production or loss, depending on
54
3 Characterization of Turbulent Flow in the Surface Boundary Layer
@ u 0
3 h
0
@t
% Àu 0
3
2 @h
@x 3
þ
gh
02
h m
À
@ h
0 u 02
3
@x 3
À
1
q
h
0 @p 0
@x 3
!
I
II
III
IV
V
ð3:87Þ
Equation (3.87) is like Eq. (3.77) where term II represents the production of
shear stresses; term III is the production by buoyancy; term IV is the turbulent
transport by tangential tensions and term V refers to pressure destruction.
As with Eq. (3.77), the elimination of the correlations is mainly due to pressure
forces, so the viscous dissipation component can be neglected in this equation.
Likewise, term II for turbulent transport in the surface boundary layer is low. Thus,
the final budget relates to term V for pressure destruction and terms of production of
turbulence II, III mechanical and buoyancy, respectively. For the vertical flow of
another scalar, such as carbon dioxide or moisture, the algebraic developments are
analogous, and equations like Eqs. (3.86) and (3.87) can be obtained.
3.5.6 Kinetic Energy Budget Equations
The TKE budget, expressed per unit mass, makes it possible to characterize the
context of the production processes, transport, and loss of turbulent fluctuations. As
previously mentioned, TKE is a measure of turbulence intensity. In the inertial
sublayer, TKE is mechanically derived from shear stresses in the mean flow and
thermally produced by buoyancy forces. This kinetic energy is transferred via the
inertial cascade from larger to smaller eddies and converted into heat by viscous
dissipation.
From Eq. (3.71) on the budget for the variance for velocity fluctuations, the
kinetic energy equations can be deduced (Stull 1994; Tennekes and Lumley 1980):
@e
@t
þ u j
@e
@t
¼
I
II
¼ d i3
g
h V
u 0
i h
0
V
À
@ u 0
j e
@x j
À
1
q
@ u 0
i p 0
ð Þ
@x i
À 2u 0
i u 0
j
@u i
@x j
À e
III
IV
V
VI
VII
ð3:88Þ
where e is the turbulent kinetic energy per unit mass given by 0:5 u
2
1 þ u
2
2 þ u
2
3
À
Á
. In
this equation, term I is the rate of storage of kinetic energy; term II is the advection
by the average kinetic energy field speed; term III is the term production or consumption by buoyancy. This term is either production or loss, depending on
54
3 Characterization of Turbulent Flow in the Surface Boundary Layer