J. Fluid Mech. (1997), vol. 350, pp. 351–374. Printed in the United Kingdom c  1997 Cambridge University Press 351 Non-isotropic dissipation in non-homogeneous turbulence By MARTIN OBERLACK Institut f¨ ur Technische Mechanik, RWTH Aachen, 52056 Aachen, Germany (Received 20 July 1995 and in revised form 9 July 1997) On the basis of the two-point velocity correlation equation a new tensor length- scale equation and in turn a dissipation rate tensor equation and the pressure– strain correlation are derived by means of asymptotic analysis and frame-invariance considerations. The new dissipation rate tensor equation can account for non-isotropy effects of the dissipation rate and streamline curvature. The entire analysis is valid for incompressible as well as for compressible turbulence in the limit of small Mach numbers. The pressure–strain correlation is expressed as a functional of the two-point correlation, leading to an extended compressible version of the linear formulation of the pressure–strain correlation. In this turbulence modelling approach the only terms which still need ad hoc closure assumptions are the triple correlation of the fluctuating velocities and a tensor relation between the length scale and the dissipation rate tensor. Hence, a consistent formulation of the return term in the pressure–strain correlation and the dissipation tensor equation is achieved. The model has been integrated numerically for several different homogeneous and inhomogeneous test cases and results are compared with DNS, LES and experimental data. 1. Introduction Current second-moment closure models still suffer from three unresolved problems: the closure of the velocity–pressure-gradient correlation Ψ ij , the triple correlation t D ij and the dissipation rate tensor ε ij . The velocity–pressure-gradient correlation is split into the pressure–strain correlation Φ ij and the pressure–diffusion p D ij which is modelled together with the triple correlation as a gradient-type diffusion term. A significant amount of information that is needed for the closure of these terms is contained in the two-point correlation tensor R ij . Given R ij , one can express the Reynolds stress tensor, the dissipation rate tensor, and the rapid part of the pressure–strain tensor (without the wall term) as a functional of R ij : u i u j = R ij (x, r = 0), ε ij = lim r→0 ν  ∂ 2 R ij ∂x k ∂r k − ∂ 2 R ij ∂r k ∂r k  , Φ rapid ij = 1 2π  V ∂¯ u k ∂x l (x + r)  ∂ 2 R il ∂x j ∂r k − ∂ 2 R il ∂r j ∂r k  d 3 r |r| +(i ↔ j ), where (i ↔ j ) indicates the addition of the previous term with interchanged indices i and j . R ij also provides turbulence length-scale information, including the integral length scale and the Taylor microscale.