H 1 H 1 x x 1 1 x x a a x x u u - PowerPoint PPT Presentation

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