Internal Report
Studies in classical turbulence theory
Author 
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Erik Lindborg 
Doctoral thesis 
1996 
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Id 
ISSN 0348467X 
ISRN KTH/MEK/TR97/ 
Abstract
The thesis deals with various problems in classical turbulence theory. The kinematical theory of homogeneous axisymmetric turbulence is given a simple and concise formulation. A representation of twopoint correlation tensors of homogeneous axisymmetric turbulence is developed, such that each measurable correlation corresponds to a single scalar function, and moreover such that the equations of continuity take the most simple form. The Poisson equation for the pressurestrain tensor is solved in terms of measurable velocity correlations. Kolmogorov's third order structure function law is derived from an equation where the pressure terms and mean flow gradient terms are retained. A new inertial range law, relating the twopoint pressurevelocity correlation to the singlepoint pressurestrain tensor, is derived. This law shows that the twopoint pressurevelocity correlation, just as the third order structure function, grows linearly with the separation distance in the inertial range. An inertial range law is also derived for the third order velocityenstrophy structure function of twodimensional turbulence. The turbulence closure problem is analysed, starting from the hierarchy of single time Fourier moment equations. The geometric properties of the equation hierarchy are investigated. The number of independent components of the n:th order isotropic correlation tensor of Fourier velocity components is proved to be n+1 for n>=4 . It is shown that a closure on the fourth order level would lead to inconsistencies. It is argued that the energy spectrum decreases slower than E(k)=exp(bk) in the dissipation range. Here, b is any positive constant. A comparison between EDQNM and DNS calculations of axisymmetric low Reynolds number turbulence is presented. The agreement between the results is found to be rather good. However, it is argued that this does not permit us to draw any conclusion for the high Reynolds number case. The prediction capability of pressurestrain models, within the context of RSTclosures, is investigated. Severe difficulties are found when the antisymmetric, rotational part of the mean flow gradient tensor, has a stronger influence than the symmetric part.
