By J. C. R. Hunt, J. C. Vassilicos

The articles during this quantity, derived from a symposium held on the Newton Institute in Cambridge, study a few key questions that experience engaged turbulence researchers for a few years. so much contain mathematical research, yet a few describe numerical simulations and experimental effects that target those questions. besides the fact that, all are addressed to a large cross-section of the turbulence group, particularly mathematicians, engineers and scientists.

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Additional info for Turbulence Structure and Vortex Dynamics

Sample text

Q = n-2. (27) Then the matrix B ' proves to be unit, ^ dC1 = 00^-6*, (28) and the eigenvalues Pk are reciprocates of the eigenvalues of the matrix A'ik. The simplest case is that of a separating system, when fc Q +1 Z9^. _ 0 ; ( i ^ f c ) ; Aik = [ofc(7fc) - l]Sik; = Qfc(7fc); (29) Pk = [ak(jk)-1}-1. (30) Then the system breaks down into (n - 2) )ndependent equattons for the individual components concentrations. Geometrically, this condition means that the system of hyperplanes C\ = const, i = 2 , .

Let us fix one of the compositions and let the other one tend to it. In the limit the direc­ tion of the vector difference between the two states tends to the eigenvector efc, corresponding to an eigenvalue Afc+1 = limV, (0-> 0)0 Obviously, the ek vector lies in the II plane. This implies that for an initial and end states in the plane II there exists a solution of the Riemann problem that lies completely in the plane. ,jk-i = 0,Jk+i = ■■■ = lq = 0, with natu­ ral in-plane variables being C and j k .

Let, for example, C+ = 0; CJ = 0. Then the intersection point should belong to an invariant subspace of lower dimension that corresponds to vanishing of both concentrations, d = In particular, for four-component systems the composition is represented by a point within a three-dimensional tetrahedron of unit height, the faces are invariant subspaces corresponding to vanishing of a concentration, and two concentrations vanish along an edge. e. devoid of different components)] can be connected by a shock pro­ vided that the respective tie-lines intersect at a common point on an edge.