Outline Expansion of a function Orthogonal set Wiener-Hermite orthogonal set Expansion of a random function Burger Equation W-H expansion for the Burger Eq. ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi Let n (x) be an orthogonal set Let a(x) be a white noise process 2 2 dx dx a a x x 0 0 a a x x a a x x x x x x dx dx n n n n 1 1 2 2 1 1 2 2 n n m m nm nm n n Wiener -Hermite base Let u(x) be an arbitrary function u H 0 H 0 H 1 H 1 x x 1 1 x x a a x x u u dx dx u x x c c x x n n n n n n c c n n n n 2 2 H H x x , , x x a a x x a a x x x x x x n n 1 1 2 2 1 1 2 2 1 1 2 2 ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 1
Wiener -Hermite base Wiener -Hermite base 3 3 H H x x , , x x , , x x a a x x a a x x a a x x a a x x x x x x 0 0 0 0 1 1 2 2 3 3 1 1 2 2 3 3 1 1 2 2 3 3 H H x x H H x x 1 1 a a x x x x x x a a x x x x x x 2 2 3 3 1 1 3 3 1 1 2 2 1 1 1 1 H H x x H H x x x x x x i j i i i j j j 1 1 2 2 1 1 2 2 H H H H 0 0 2 2 2 2 H H x x , , x x H H x x , , x x x x x x x x x x 1 1 2 2 3 3 3 3 1 1 3 3 2 2 4 4 Wiener -Hermite set is complete x x x x x x x x 1 1 4 4 2 2 3 3 ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi Wiener -Hermite Series Burger Equation 2 2 1 1 1 1 u u x x K K x x x x H H x x dx dx u u x x , , t t u u u u 1 1 1 1 1 1 u u Gaussian Gaussian 2 2 2 2 2 2 K K x x x x , , x x x x H H x x , , x x H H x x , , x x dx dx dx dx 2 2 t t x x x x 1 1 2 2 1 1 2 2 1 1 2 2 1 1 2 2 Non Non Gaussian Gaussian 3 3 3 3 K K x x x x , , x x x x , , x x x x H H x x , , x x , , x x dx dx dx dx dx dx ... ... 1 1 2 2 3 3 1 1 2 2 3 3 1 1 2 2 3 3 Non Non Gaussian Gaussian ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 2
Concluding Remarks 2 2 1 1 K K x x x x Expansion of a function 2 2 t t x x Orthogonal set 1 1 2 2 2 2 dx dx K K x x x x K K x x x x , , x x x x 0 0 Wiener-Hermite orthogonal set 1 1 1 1 1 1 x x 2 2 Expansion of a random function 2 2 K K x x x x , , x x x x 2 2 t t x x Burger Equation W-H expansion for the Burger Eq. 1 1 1 1 1 1 K K x x x x K K x x x x 0 0 2 2 x x ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi ME 639-Turbulence G. Ahmadi 3
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