Burgers (1950) proposed il model equation similar to the Nwicr Stokes
equation
( 1 )
where thc Kcynolds number Re is defined in tcrins of Ito. the rnis \due of the
velocity ;it tlic initinl time. and of an initial characteristic length scale ,xl).
This cqi1;ttion C;III be trcilted through t h c usc of \he W H expansion. Dt.li11~
:I white noise process Rhicli is statisticdly independent point to point. Gaussian at eccry point. with co-variance givcn by
(2 )
( / ~ l l l ( . ~ l ) ~ f ‘ l ’ ( . ~ z ) )
= b(.v, - .x.)
I’rowcdiiig
in Mcechatn and Sicgel (IYh4), we h a w for thc lirst t w o terms
with
/ / i 2 1 ( ~ l , x 2 ) - / P i ) ( . ~ l ) / P i ’ ( . ~ 2 )
- ci(.V, - . Y ? )
I ICIY time is iniplicit. I t can be s h o w n that the difference argunicnts guarrtnICC IIiilI the pmccss will he st;itistically honioge~icou~ (which wc supp~se 10
hc ;ippropriiitc t o o t i r prahlcni). Take the Fourier transforins, substitute
Ii.cp. ( 3 ) i n ( I ) . and apply siiitahlc avclaging l o find
wticrc wc neglect H term proportional to ( K ‘ 2 ’ ) 2 assuniing that this is ncgligiblc wlicn the expansion converges properly. Oncc t h c kernels have hccn
equation
( 1 )
where thc Kcynolds number Re is defined in tcrins of Ito. the rnis \due of the
velocity ;it tlic initinl time. and of an initial characteristic length scale ,xl).
This cqi1;ttion C;III be trcilted through t h c usc of \he W H expansion. Dt.li11~
:I white noise process Rhicli is statisticdly independent point to point. Gaussian at eccry point. with co-variance givcn by
(2 )
( / ~ l l l ( . ~ l ) ~ f ‘ l ’ ( . ~ z ) )
= b(.v, - .x.)
I’rowcdiiig
in Mcechatn and Sicgel (IYh4), we h a w for thc lirst t w o terms
with
/ / i 2 1 ( ~ l , x 2 ) - / P i ) ( . ~ l ) / P i ’ ( . ~ 2 )
- ci(.V, - . Y ? )
I ICIY time is iniplicit. I t can be s h o w n that the difference argunicnts guarrtnICC IIiilI the pmccss will he st;itistically honioge~icou~ (which wc supp~se 10
hc ;ippropriiitc t o o t i r prahlcni). Take the Fourier transforins, substitute
Ii.cp. ( 3 ) i n ( I ) . and apply siiitahlc avclaging l o find
wticrc wc neglect H term proportional to ( K ‘ 2 ’ ) 2 assuniing that this is ncgligiblc wlicn the expansion converges properly. Oncc t h c kernels have hccn