058:268 Turbulent Flows
Handout: Channel Flow - Mean Velocity Profiles
Similarity
U = f ( ρ ,ν , δ , y,
dp w
) = f ( ρ ,ν , δ , y, uτ )
dx
⇒ 6 − 3 = 3 non-dimensional groups
U
uτ
y δ
⎛y
⎞
= f ( , ) = f ⎜ , Reτ ⎟
δ δν
⎝δ
⎠
One can do the same analysis for the Velocity Gradient:
dU
uτ
y y
Φ( , )
δ δν
dy
y
δv is the appropriate length scale in the viscous wall region, while δ is the appropriate
length scale in the outer layer
=
The law of the wall
Prandtl (1925): In the inner layer ( y
δ <<1) and at high Reynolds numbers <U> is
determined only by the viscous scales
⇒
dU
dy
=
uτ
y
ΦI ( )
δv
y
for y
δ <<1
(1)
with
u+ =
⇒
du +
dy
+
=
1
y
+
U
uτ
ΦI (y+ )
and y + =
y
δν
(2)
The integral of the previous equation is the Law of the Wall valid in viscous sublayer
and the inner layer (y/d<0.1)
u+ = fw(y+ )
(3)
where
+
y+
fw(y ) = ∫
0
1
y
+'
Φ I ( y + ' )dy + '
Viscous Sublayer
From equation (3)
fw = u+ =
U
uτ
⇒ f w (0) = 0
du +
f w' (0) =
dy +
=
y =0
δν ∂ U
uτ
∂y
| y =0
=
ν τw
uτ2 ρν
=
ρ τw
=1
τw ρ
Hence, the Taylor series expansion for small y+
u + = f w ( y + ) = f w (0) + f w' (0) y + + O ( y +2 )
⇒ u+ = y+
The Log Law
At large Reynolds numbers, the outer part of the inner layer corresponds to large y+
y
where viscosity has little effect ( Φ independent of
)
δν
Φ = Φ(
y + >> 1 ⇒
⇒
Also Φ independent of
y
δ
y
δ
>>
y
δν
y
y
, )
δν δ
=
y δ
δ δv
>> 1
δv
v
=
= Reτ−1
δ uτ δ
for
y
δ
⇒
<< 1 let ' s say
y δν
δν δ
y
δ
< 0.1
< 0.1
⇒ y + < 0.1
δu
δ
= 0.1 τ = 0.1 Reτ
δν
ν
where the edge of the inner layer was defined at 0.1δ.
So if Reτ is such that
y
δ
>> Reτ−1 and y + = y / δ v < 0.1 Reτ
Φ I ( y + ) independent of both y
y
δν (viscosity) and of δ , thus from equation (2):
⇒ Φ I = const =
⇒
du +
dy
+
=
⇒ u+ =
1
κ
1
κy +
1
κ
ln y + + B
with
κ = 0.41 (Von Karman constant)
B=5.2
Velocity Defect Law
In the outer layer ( y + > 50 ) the assumption that Φ is independent of ν implies that, for
large y , Φ tends asymptotically to a function of y/δ only:
δν
dU
=
dy
uτ
y
Φo ( )
y
δ
(4)
By integrating in y from y to δ (channel centerline)
δ
∫
y
Uo − U
uτ
dU
dy
'
y
= FD ( ) with
δ
δ
dy ' = ∫
y
uτ
y
'
Φo(
y'
δ
)dy '
1 1
y
FD ( ) = ∫ ' Φo( y ' )dy '
δ
y y
(5)
δ
Unlike the law of the wall function f w ( y + ) which is universal, the function FD ( ) is
δ
different in different flows.
y
At sufficiently high Reynolds numbers (approximately Re>20,000) there is an overlap
region between the inner layer and the outer layer (see Figs. 7.8 and 7.13). In this region
both equations (1) and (4) are valid
⎛ y
y dU
= Φ I ⎜⎜
uτ dy
⎝ δv
⎞
y
⎟⎟ = Φ 0 ⎛⎜ ⎞⎟ for δ v << y << δ
⎝δ ⎠
⎠
This can be satisfied only if the two functions on the right hand side are equal to the same
constant (1/κ), which leads to
dU
u
⇒
= τ for δ v << y << δ
dy
yκ
For small y/δ, the velocity defect law (equation 5) can be written as:
U0 − U
uτ
1 y
⎛ y⎞
= FD ⎜ ⎟ = − ln + B1
κ δ
⎝δ ⎠
(6)
The value of the flow-dependent constant B1 is ~0.2 based on DNS and ~0.7 based on
experimental data for channel flows.
Statistics in turbulent channel flows: