General Points - Transport and Coherent Structures in Wall Turbulence - page 31

The displacement thickness can thus be written as

∞ ∞

UU dy − u

( )

∫ ∫

U τ

+

δ =

∞

= δ g η δ , d η

d

U

0 ∞ ∞ 0

which leads to

δ u

() ()

+ +

d

= A δ

U τ

δ [1.46]

1

δ

∞

∞

( )

The quantity

A , which replaces the integral

,

∫

+

g η δη , d

0

depends normally on the Reynolds number

+ . In parallel,

δ

the momentum thickness assumes the form:

2

∞ ∞ ∞

⎛ ⎞ u ⎛ ⎞ u

θ UU d

( ) ( )

∫ ∫ ∫

τ

g

+

d

τ

g

2 +

d

= 1 − η = ηδ , η + ηδ , η

⎜ ⎟ ⎜ ⎟

δ UU U U

⎝ ⎠ ⎝ ⎠

∞ ∞ ∞ ∞

0 0 0

which results in

⎛ ⎞

θ
δ

u u

[1.47]

= A

τ

1 − A

τ

⎜ ⎟

1 2

U U

⎝ ⎠

∞ ∞

with

∞

() ( )

∫

A δ

+

= g

2

η δ ,

+

d η

2

0

The shape factor H is directly obtained from equations
[1.46] and [1.47],

− 1

δ
θ

⎛ u ⎞

H ==−

d

1 A U τ

⎜ ⎟

2

⎝ ⎠

∞