Near-Wall Coherent Structures: History, Identification and Detection - Transport and Coherent Structures in Wall Turbulence - page 150

In an entirely similar way, we can estimate the velocity
vector

in terms of

ux by

( ) ()

ux r +

( )

( )

ˆ

( ) ( ) () () () () ()

ux r ˆ += Eux rux + = a ru x b ru xu x + +

i i ij j ijk j k

[3.16]

() () () ()

+ cruxuxux + ......

ijkl j k l

We can see that relation [3.10] is now generalized to the
nonlinear case [ADR 79, TUN 80, ADR 88, GUE 89]. The
coefficients

ij ar , ijkl cr are determined

equivalently from equation [3.11] by minimizing the error

() ijk br and () ()

{ }

2

( ) ( )

ˆ [3.17]

eEuxr uxr = + − +

i i i

in the sense of the least squares. Thus, we begin by
determining

∂ e ∂ e ∂ e

i i i

= = = .... = 0

∂ a ∂ b ∂ c

ij ijk ijkl

which gives us the system

() () () () () () ()

⎡ ⎤ ⎡ ⎤

Eu xu x a r + Eu xu xu x b r +

⎣ ⎦ ⎣ ⎦

j k ik j k l ikl

() () () () () () ( )

⎡ ⎤ ⎡ ⎤

+ Eu xu xu xu x c r = Eu xu x r +

⎣ ⎦ ⎣ ⎦

j k l m iklm j i

() () () () () () () () ()

Eu xu xu x a r ⎡ ⎤ + Eu xu xu xu x b ⎡ ⎤ r +

⎣ ⎦ ⎣ ⎦

j k l i j k l m lm

() () () () () () () () ( )

+ Eu xu xu xu xu x c ⎡ ⎤ r = Eu xu x ⎡

i ux r

+ ⎤

⎣ ⎦ ⎣ ⎦

j k l m n ilmm j k

In the simple case of the linear estimation
(

, the problem is reduced to the Yule-
Walker system [3.13]. The matrix

) ( ) ( )

ux r ˆ i += a ru x

ij j

given in relation [3.14]

R ij

,

( )

() ()

becomes and the vector

REuxux =

ij i j