The Closure Problem of Turbulence Theory
The Closure Problem of Turbulence Theory
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8) ^(a) = (exp(iay)) = / exp(iay)P(y)dy - 00 is the characteristic function. Upon evaluating the moments by use of the relation (5. 9) (y'') = (-i)^d^$(a)/d..^_ a^O we find (5. 10) -y = 3)^ = 10 > + C), ^ C5 + c^.
15 (y^)^ + loc^^ + ^5 ) ■=1* * %• 21 The significance of the cumulants is that they measure the degree and com- plexity of the deviation of P(y), and its moments, from a normal distribu- tion.
If an expansion analogous to (5"7) is made for the characteristic functional [lO, llj of the... one-time velocity distribution, a sequence of approximations to the various moments can be formed by truncating the expansion so as to admit successively higher cumulants. In the first non- trivial approximation, we admit third-order cumulants only. Then [in correspondence to the second of relations (5. I0), with c, = 6] we find (5. 11) U^j^(x, t;x', t;x", t;x"', t) = U^^(x, t;x', t)U^(x", t;x '", t) + U^Jx, t;x", t)U^^(x', t;x"', t) + U.^(x, t;x '", t)U^^(x', t;x", t), which is the quasi -normality approximation employed by Proudman and Reid [21] and Tatsumi [27, 28].
What to read after The Closure Problem of Turbulence Theory?
You can find similar books in the "Read Also" column, or choose other free books by R H Kraichnan to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
8) ^(a) = (exp(iay)) = / exp(iay)P(y)dy - 00 is the characteristic function. Upon evaluating the moments by use of the relation (5. 9) (y'') = (-i)^d^$(a)/d..^_ a^O we find (5. 10) -y = 3)^ = 10 > + C), ^ C5 + c^.
15 (y^)^ + loc^^ + ^5 ) ■=1* * %• 21 The significance of the cumulants is that they measure the degree and com- plexity of the deviation of P(y), and its moments, from a normal distribu- tion.
If an expansion analogous to (5"7) is made for the characteristic functional [lO, llj of the... one-time velocity distribution, a sequence of approximations to the various moments can be formed by truncating the expansion so as to admit successively higher cumulants. In the first non- trivial approximation, we admit third-order cumulants only. Then [in correspondence to the second of relations (5. I0), with c, = 6] we find (5. 11) U^j^(x, t;x', t;x", t;x"', t) = U^^(x, t;x', t)U^(x", t;x '", t) + U^Jx, t;x", t)U^^(x', t;x"', t) + U.^(x, t;x '", t)U^^(x', t;x", t), which is the quasi -normality approximation employed by Proudman and Reid [21] and Tatsumi [27, 28].
What to read after The Closure Problem of Turbulence Theory?
You can find similar books in the "Read Also" column, or choose other free books by R H Kraichnan to read onlineMoreLess
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