2.20 - Marine Hydrodynamics, Spring 2005
Lecture 18
2.20 - Marine Hydrodynamics
Lecture 18
4.9 Turbulent Flow – Reynolds Stress
Assume a flow v with a time scale T . Let τ denote a time scale τ << T . We can then
write for each component of the velocity
ui = ūi + ui
(1)
where by definition
1
ūi =
τ
0
τ
ui dt
It immediately follows that
∂
∂ui
u¯i = ui − ūi = ūi − ūi = 0, also
ūi =
etc.
∂x
∂x
Substitute Eq. (1) into continuity and average over τ , i.e., take ( ) ∂ūi
∂ui
∂ui
=
+
= 0,
∂xi
∂xi
∂xi
=⇒
∂ūi
=0
∂xi
↓
0
but
∂ui
= 0 =
∂xi
∂u
+
i ,
∂xi
∂ūi
∂x
i
0
↓
, just shown
1
=⇒
∂ui
=0
∂xi
Substitute Eq. (1) into the momentum equations and take ( )
∂ui
∂ui
1 ∂τij
1 ∂p
+ uj
=
=−
+ ν∇2 ui
ρ ∂xi
∂t
∂xj
ρ ∂xj
∂ui
⎧
2
⎨
ν∇2 ui = ν∇ ui
∂ui
∂ūi
=
+
; similarly
⎩
∂t
∂t
∂t
0
uj
∂p
∂xi
=
∂
(p̄
∂xi
+ p ) =
∂p̄
∂xi
etc.
∂
∂ui ∂ ūi
∂ūi
∂u
∂ = u
¯j + uj
(¯
ui + ui ) = ūj
+ uj
+ ūj i +uj
u
∂xj
∂xj
∂xj
∂xj
∂xj
∂xj i
0
0
but from continuity we have
uj
∂
∂
u
i =
u
u − ui
∂xj
∂xj j i
∂uj
∂xj
0→by continuity
and thus we finally obtain
∂ūi
1 ∂p
∂ ∂ūi
+ ūj
=−
+ ν∇2 ūi −
uu
∂t
∂xj
ρ ∂xi
∂xj i j
1 ∂
τ
ρ ∂xj ij
Reynolds averaged N-S equation:
∂ūi
∂ūi
1 ∂ + ūj
=
τij − ρui uj
∂t
∂xj
ρ ∂xj
Reynolds stress:
τRij ≡ −ρui uj
2
4.10 Turbulent Boundary Layer Over a Smooth Flat Plate
We have already seen that the function of the friction coefficient Cf (ReL ) differs for laminar
and turbulent flows. In this paragraph we will discuss the case of a turbulent boundary
layer.
Following a procedure similar to that for flow past a body of general geometry, we will
use an approximate velocity profile, obtain the P-Flow solution and eventually substitute
everything into von Karman’s momentum integral equation. The velocity profiles used in
practice are either empirical ((1/7)th power) or semi-empirical (logarithmic) laws.
log
y
u
Uo
δ
U
Uo
o
1/7
u
log
y
δ
4.10.1 (1/7)th Power Velocity Profile Law
Let the velocity profile be determined by the following empirical law
y 1/7
ū
=
Uo
δ
(2)
where δ = δ(x) is to be determined.
From equation (2) we can obtain directly δ ∗ and θ
δ
8
7 ∼
θ =
δ = 0.0972 δ
72
δ∗ =
However, we need to use an additional empirical law to determine the skin friction.
From Blasius’ law of friction for pipes we obtain an expression for τo
−1/4
τo
Uo δ
= 0.0227
ν
ρUo2
3
From P-Flow for flow past a flat plate we have U (x) = U0 = const, and dp/dx = 0
Substituting δ ∗ , θ, τo , Uo into von Karman’s moment equation
τo
d
=
(θ) =⇒ 0.0227
2
dx
ρUo
Uo δ
ν
−1/4
=
7 dδ
72 dx
This is a 1s t order ODE for δ. One BC is required. We assume that the the flow is
tripped at x = 0, i.e., at x = 0 the flow is already turbulent. Further on, we assume
that the turbulent boundary layer starts at x = 0, i.e., δ(0) = 0. It follows that
δ (x) ∼
= 0.373x
Uo x
ν
−1/5
=⇒
δ ∼
= 0.373Re−1/5
x
x
Compare:
Laminar Boundary Layer Turbulent Boundary Layer (1/7th power law)
√
δ (x) ∝ x4/5
δ (x) ∝ x
4 1/5
νx
νx
∗ ∼
0.047
1.72
δ∗ ∼
δ
=
=
Uo
Uo
Once the profile has been determined we can evaluate the friction drag
D = 0.036 ρUo2 BL Re−1/5
L
Thus, the friction coefficient for turbulent (tripped and/or ReL > 5 × 105 ) flow over
a flat plate is
D
= 0.073Re−1/5
Cf =
L
1 ρU 2 BL
2 o
4.10.2 Logarithmic Velocity Profile Law
If the velocity profile is determined by the semi-empirical logarithmic velocity pro­
file law, following an approach similar to that for the 1/7th power law, we obtain
Schoenherr’s formula for the friction coefficient
0.242
= log10 (ReL Cf )
Cf
4
4.10.3 Summary of Boundary Layer Over a Flat Plate
Turbulent BL (1/7th power law)
Laminar BL (Blasius)
δ
∝ Re−1/2
x
x
δ ∗ = 1.72xRe−1/2
∝
x
δ
∝ Re−1/5
x
x
√
δ ∗ = 0.047xRe−1/5
∝ x4/5
x
x
τo = 0.0227ρUo2 Re−1/4
δ
τo = 0.332ρUo2 Re−1/2
x
τo = 0.02297ρUo2 Re−1/5
x
D = 0.664ρU02 (BL)Re−1/2
L
D = 0.03625ρU02 (BL)Re−1/5
L
Cf ≡
D
ρUo2 (BL)
= 1.328Re−1/2
L
Cf ≡
D
ρUo2 (BL)
= 0.0725Re−1/5
L
For τo , the cross-over is at Rex ∼ 3.4 x 103 , i.e.,
Cf
(τo )laminar > (τo )turbulent for Rex < 3.4 × 103
(τo )laminar ∼ (τo )turbulent for Rex ∼ 3.4 × 103
(τo )laminar < (τo )turbulent for Rex > 3.4 × 103
C f L ~ RL−
1
2
C fT ~ RL−
1
5
~ 0.01
Therefore, for most prototype scales:
ln (RL)
(Cf )turbulent > (Cf )laminar
(τo )turbulent > (τo )laminar
RL ~ 1.6 x 104
5