Based on a direct numerical simulation (DNS) of a temporally evolving mixing layer, we present a detailed study of the turbulent/non-turbulent (T/NT) interface that is defined using the two most common procedures in the literature, namely either a vorticity or a scalar criterion. The different detection approaches are examined qualitatively and quantitatively in terms of the interface position, conditional statistics and orientation of streamlines and vortex lines at the interface. Computing the probability density function (p.d.f.) of the mean location of the T/NT interface from vorticity and scalar allows a detailed comparison of the two methods, where we observe a very good agreement. Furthermore, conditional mean profiles of various quantities are evaluated. In particular, the position p.d.f.s for both criteria coincide and are found to follow a Gaussian distribution. The terms of the governing equations for vorticity and passive scalar are conditioned on the distance to the interface and analysed. At the interface, vortex stretching is negligible and the displacement of the vorticity interface is found to be determined by diffusion, analogous to the scalar interface. In addition, the orientation of vortex lines at the vorticity and the scalar based T/NT interface are analyzed. For both interfaces, vorticity lines are perpendicular to the normal vector of the interface, i.e. parallel to the interface isosurface.

The two-point theory of homogeneous isotropic turbulence is extended to source terms appearing in the equations for higher-order structure functions. For this, transport equations for these source terms are derived. We focus on the trace of the resulting equations, which is of particular interest because it is invariant and therefore independent of the coordinate system. In the trace of the even-order source term equation, we discover the higher-order moments of the dissipation distribution, and the individual even-order source term equations contain the higher-order moments of the longitudinal, transverse and mixed dissipation distribution functions. This shows for the first time that dissipation fluctuations, on which most of the phenomenological intermittency models are based, are contained in the Navier-Stokes equations. Noticeably, we also find the volume-averaged dissipation ε r used by Kolmogorov (J. Fluid Mech., vol. 13, 1962, pp. 82-85) in the resulting system of equations, because it is related to dissipation correlations.

Tangent lines to a given vector field, such as streamlines or vortex lines, define a local unit vector t that points everywhere in the line's direction. The local behavior of the lines is characterized by the eigenvalues of the tensor \documentclass[12pt]{minimal}\begin{document}$\bf T = {\bf \nabla } \cdot \bf t$\end{document}T=∇·t. In case of real eigenvalues, t can be interpreted as a normal vector to a surface element, whose shape is defined by the eigenvalues of T. These eigenvalues can be used to define the mean curvature −H and the Gaussian curvature K of the surface. The mean curvature −H describes the relative change of the area of the surface element along the field line and is a measure for the local relative convergence or divergence of the lines. Different values of (H, K) determine whether field lines converge or diverge (elliptic concave or elliptic convex surface element, stable/unstable nodes), converge in one principal direction and diverge in another (saddle) or spiral inwards or outwards (stable/unstable focus). In turbulent flows, a plethora of local field line topologies are expected to co-exist and it is of interest to find out whether certain topologies are more likely to occur than others. With this question in mind, the joint probability density function (JPDF) of H and K are evaluated for streamlines and vortex lines from several datasets obtained from direct numerical simulations of forced isotropic turbulence at four different Reynolds numbers. The JPDF for streamlines is asymmetric with a long tail towards negative H, implying that stream tubes tend to expand rapidly while shrinking more gently – a manifestation of negative skewness in turbulence. On the other hand, the JPDF for vortex lines is symmetrical with respect to H, indicating that the convergence and divergence of vortex lines is similar, different to streamlines.

We examine finite Reynolds number contributions to the inertial range solution of the third order structure functions D 3,0 and D 1,2 stemming from the unsteady and viscous terms. Under the assumption that the second order correlations f and g are self-similar under a coordinate change, we are able to rewrite the exact second order equations as a function of a normalized scale r only. We close the resulting system of equations using a power law and an eddy-viscosity ansatz. If we further assume K41 scaling, we find the same Reynolds number dependence as previously in the literature. We proceed to extrapolate towards higher Reynolds numbers to examine the unsteady and viscous terms in more detail. We find that the intersection between the two terms, where their contribution to the solution of the structure function equations is relatively small, scales with the Taylor scale λ.

We examine balances of structure function equations up to the seventh order N = 7 for longitudinal, mixed and transverse components. Similarly, we examine the traces of the structure function equations, which are of interest because they contain invariant scaling parameters. The trace equations are found to be qualitatively similar to the individual component's equations. In the even-order equations, the source terms proportional to the correlation between velocity increments and the pseudo-dissipation tensor are significant, while for odd N, source terms proportional to the correlation of velocity increments and pressure gradients are dominant. Regarding the component equations, one finds under the inertial range assumptions as many equations as unknown structure functions for even N, i.e. can solve for them as function of the source terms. On the other hand, there are more structure functions than equations for odd N under the inertial range assumptions. Similarly, there are not enough linearly independent equations in the viscous range r → 0 for orders N > 3.