38
2 Water at Rest and in Motion
and
1 loT
U = -
udt =I- O.
T 0
(2.41 )
Now an important question arises. How do the mass conservation and momentum principles, which are the foundation of fluid mechanics, change for
turbulent motion? Are they still valid? The answer to this question is yes.
The substitution of representations (2.38) into the mass conservation equation
(2.18) yields similar relationships for the mean motion, i.e.:
au aVow _ 0
ox + oy + OZ - ,
and for the fluctuating part:
au' ov' ow'
-+-+-=0. ox oy oz
(2.42)
(2.43)
The situation becomes more complicated in the case of the momentum principle. Full analysis of the momentum principle for turbulent motion is out
of the scope of this book, and detailed derivations can be found in many hydromechanics books (for example, Schlichting, 1960; Monin and Yaglom, 1971;
Le Mehaute, 1976). Here, we note only that mean motion which is steady
and irrotational, and for which the viscous forces are neglected, still obeys the
Bernoulli equation, despite the fact that the actual turbulent motion is always
unsteady, rotational and dissipative.
The fluctuations of pressure, p', are usually very small when compared with
the mean pressure ]i, so the approximation p ~ p is used. Also, the term
representing the viscous forces in Eq. (2.22) is generally small in comparison
to the inertia terms on the left-hand side of Eq. (2.22) and is often neglected,
except when investigating the boundary layer.
For example, the final form of the momentum equation for turbulent motion
can be written as (only the momentum equation for the x direction is given):
(
Local acceleration
Convective ~celeration )
r " - - . ,
,
aU
aU
aU
oz
p
-
+U-+V-+WW
at
ox
oy
OZ
Pressure forces
,-A-..
Op
ox
Viscous forces
Turbulent fluctuation forces
.
+
in which bars over the expressions denote time averaging.
(2.44)
2 Water at Rest and in Motion
and
1 loT
U = -
udt =I- O.
T 0
(2.41 )
Now an important question arises. How do the mass conservation and momentum principles, which are the foundation of fluid mechanics, change for
turbulent motion? Are they still valid? The answer to this question is yes.
The substitution of representations (2.38) into the mass conservation equation
(2.18) yields similar relationships for the mean motion, i.e.:
au aVow _ 0
ox + oy + OZ - ,
and for the fluctuating part:
au' ov' ow'
-+-+-=0. ox oy oz
(2.42)
(2.43)
The situation becomes more complicated in the case of the momentum principle. Full analysis of the momentum principle for turbulent motion is out
of the scope of this book, and detailed derivations can be found in many hydromechanics books (for example, Schlichting, 1960; Monin and Yaglom, 1971;
Le Mehaute, 1976). Here, we note only that mean motion which is steady
and irrotational, and for which the viscous forces are neglected, still obeys the
Bernoulli equation, despite the fact that the actual turbulent motion is always
unsteady, rotational and dissipative.
The fluctuations of pressure, p', are usually very small when compared with
the mean pressure ]i, so the approximation p ~ p is used. Also, the term
representing the viscous forces in Eq. (2.22) is generally small in comparison
to the inertia terms on the left-hand side of Eq. (2.22) and is often neglected,
except when investigating the boundary layer.
For example, the final form of the momentum equation for turbulent motion
can be written as (only the momentum equation for the x direction is given):
(
Local acceleration
Convective ~celeration )
r " - - . ,
,
aU
aU
aU
oz
p
-
+U-+V-+WW
at
ox
oy
OZ
Pressure forces
,-A-..
Op
ox
Viscous forces
Turbulent fluctuation forces
.
+
in which bars over the expressions denote time averaging.
(2.44)