problem 10.1

Spectrum function ?( k ) of dissipation

Rate of dissipation per unit mass in turbulence is given by ? ? ? ² ? = ? ?( k)d k. By using the definition of vorticity ? _i = ? _ijk ? _j u _k and the Fourier representation (10.6) for u _k( x), derive the following relations:

The function ?( k) = 2 ?k ² E( k) thus obtained is the dissipation spectrum.

problem 10.2

Homogeneous isotropic turbulence

In homogeneous turbulence, statistical correlation of velocity component u _i at a point x and u _j at x ? = x + s is expressed by B _ij := ? u _i( x) u _j( x + s) ? = B _ij( s) which is independent of x by the assumption of homogeneity. The velocity field u _i( x) is assumed to satisfy the condition of incompressibility: ? _i u _i( x) = 0.

1. Suppose that the turbulence is isotropic in addition to homogeneity. Show that the second-order tensor B _ij has the following form:

   
   

   where s = s, e _i = s _i /s (unit vector in the direction of s), and F( s) and G