[lucamirtanini]

Hi,
I have two questions
1) on the Tennekes book "A first course in turbulence", there is written that the diagonal of the spectrum tensor represents the kinetic energy. But it seems to me that it is twice the kinetic energy at a given wavenumber vector? (see p-1 and p-2 in attachment)

2) I cannot understand why the correlation tensor is a function of the vector speration onlu, providing the turbulence is homogeneous. If the turbulence is not homogeneous, the correlation is dependent on other parameters? Which parameters? (see p-3)

Thank you

[LuckyTran]

1) Yes it is twice the kinetic energy


2) The spatial two-point correlation function has such a given definition. R is a function of the starting location and separation to all other parts in space. If the turbulence is homogeneous, then the turbulence at any particular location x1 and the same as at any other location x2 so that R is no longer dependent on a particular starting location but only the separation distance. In general,e.g. non-homogeneous turbulence, R depends on the starting location.

That is, look at (8.1.1). You see that Rij is a function of ui(x,t)*uj(x+r,t). That is, Rij is a function of x and r and t. If the turbulence is homogeneous then R depends only on r and t and not x,r, and t. It is a bit misleading to label Rij(r) because that gives the impression that R is not dependent on x. But because the discussion is going towards the direction of talking about homogeneous turbulence, the authors have decided to drop the x early. Really they are talking about the correlation tensor at a specific location, whereas the generalized Rij(x,r) is a tensor field (a field of tensors and not a single tensor).

This property isn't specific to turbulence but the two-point correlation in general w.r.t. to a general field versus a homogeneous field.