2. Mean velocity profiles

2.4 Outer Layer (Von Kármán, 1930)
Dimensional parameters U, y, w, , – but not .
Dimensional analysis (5 parameters, 3 independent dimensions) ⇒ 2 independent
dimensionless groups, conveniently taken as
U e −U
y
,
=
u
Then one has the velocity-defect law:
U e −U
= f o ( )
(11)
u
Unlike fw which is expected to be universal, fo( ) will vary with the particular flow.
¢
2.5 Overlap Layer – the Log Law
As noted by C.B. Millikan (1937) the inner and outer layers can only overlap smoothly if the
overlap-region velocity profile is logarithmic.
Outer layer: U − U = f ( )
+ +
e o
Inner layer:
+ +
U = f w ( y )
Introducing + = u / , so that y + =
+
U (
+
) f ( ) + f (
+
)
e
= o w
+ , and adding:
+ to be the sum of separate functions of and + , fw must
For a function fw of the product
be logarithmic. This can be proved formally by differentiating successively with respect to
each variable, as follows.
Differentiate wrt + :
U
′
(
+
) = 0 +
+ ′
e
f w
Differentiate wrt η:
0 = f ′ (
w
+
) +
(
+
)
f ′
(
+
w
+
w
+
)
+ +
= f ′ ( ) ′′
w y + y f ( y
d + df
w
= ( y )
+
+
dy
dy
)
Hence,
+ df
w
y +
dy
= constant
This constant is conventionally written as 1/ , where (≈ 0.41), is von Kármán’s constant.
+ + = df
w
dy
1
y
which integrates to give
1 +
= ln y + B , B another constant.
f w
Turbulent Boundary Layers 2 - 3 David Apsley