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By F. Krause

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Example text

5. b y Gjp(x, x\ t, t 0 ) = lim G$(x, x\ t, t0). 67) W i t h this representation t h e existence of t h e tensor Gjp is guaranteed for a n y bounded velocity field. Let us now consider t h e equation 3Bj W" £jkieimn 1 dumBn ~d^T ~ ^ j " 8umBn jkllmn ( ~xV ' and let GjP(x, x\ t> t') be the Green's tensor of the left-hand side of this equation. t') - £pkl£lmnJJ QX. 69) . 70) x\ t, t0) Bp(x\ t0) dx' e pkl h mn If d*. ~Um{x''n *» ~1)(*'>*'> d*'dl' I t is not necessary t o prove convergence again. 61).

T)d§ = 1. 56) Alternatively, /LtaXlOT < r c o r , we can carry out the r-integration by assuming the scalar product u' curl u' to be constant. 47). £*#. 48), and generalizes them. We can arrive at an alternative expression by introducing the longitudinal correlation function / defined by &e«(i,T) = /(f,T)ft. 62) [2]. 1. Introductory remarks We start our more detailed discussion with the treatment of homogeneous and steady turbulence. For this kind of turbulence all mean quantities are constant with respect to variations in space and time, and thus, in particular, the pseudotensors gij^h introduced in the foregoing chapter are constant.

Ao. 7. On the convergence of the correlation approximation We define a sequence of fields B(v\x, J3(0)(*, t) a r b i t r a r y , t x - e^te» j J — Q4 I t is obvious t h a t B(f(x, convergence, the field Bioo\x, t) = lim B{v\x, v->oo t) = [%G{x - x',t Bf\x, r r o(jr(x t), v = 0,1,2, t 57 ... 61) t0) dx' t') Ax'dt', v = 1, 2, ... 59). Consequently, the field J3(oo)(#, t) is a solution of the initial value problem of the induction equation. G2) v l b \x, t) = B^ \x, t) - B^ \x, (oo) and represent Z?

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